ASYMPTOTIC GEOMETRY OF THE HITCHIN METRIC · ASYMPTOTIC GEOMETRY OF THE HITCHIN METRIC RAFE MAZZEO,...

ASYMPTOTIC GEOMETRY OF THE HITCHIN METRIC

RAFE MAZZEO JAN SWOBODA HARTMUT WEISS AND FREDERIK WITT

Abstract We study the asymptotics of the natural L2 metric on theHitchin moduli space with group G = SU(2) Our main result whichaddresses a detailed conjectural picture made by Gaiotto Neitzke andMoore [GMN] is that on the regular part of the Hitchin system thismetric is well-approximated by the semiflat metric from [GMN] Weprove that the asymptotic rate of convergence for gauged tangent vectorsto the moduli space has a precise polynomial expansion and hence thatthe the difference between the two sets of metric coefficients in a certainnatural coordinate system also has polynomial decay New work byDumas and Neitzke shows that the convergence is actually exponentialin directions tangent to the Hitchin section

1 Introduction

In this paper we study the asymptotic geometry of the L2 (lsquoWeil-Peterssontypersquo) metric gL2 on the moduli space M2d of irreducible solutions to theHitchin self-duality equations on a SU(2)-bundle E of degree d over a com-pact Riemann surface X modulo unitary gauge transformations We oftenrefer to gL2 as the Hitchin metric onM2d The spaceM2d can also be iden-tified as the moduli space of stable Higgs bundles (AΦ) modulo complexgauge transformations as well as the twisted character variety of irreduciblerepresentations of π1(X) into SL(2C) modulo conjugation The fact thatgL2 is hyperkahler reflects these various realizations All of this can be gen-eralized to the situation where E has higher rank and carries a G-structurewhere G is any compact semisimple group We treat here only the caseG = SU(2) and for simplicity also set d = 0 (the differences needed to handled ne 0 are minor) We denote the moduli space simplyM

Many topological and geometric properties of M are now understoodand in the past few years a detailed picture has started to emerge about itsasymptotic geometric structure at infinity A key role in this story is playedby the spaceMinfin of lsquolimiting configurationsrsquo consisting of the solutions ofa set of decoupled equations which arise as a limiting form of the Hitchin

Date November 7 2018RM supported by NSF Grant DMS-1105050 and DMS-1608223JS amp HW supported by DFG SPP 2026 lsquoGeometry at infinityrsquoThe author(s) acknowledge(s) support from US National Science Foundation grants

DMS 1107452 1107263 1107367 rdquoRNMS Geometric Structures and Representation Va-rietiesrdquo (the GEAR Network)

1

2 RAFE MAZZEO JAN SWOBODA HARTMUT WEISS AND FREDERIK WITT

equations again modulo unitary gauge transformations There are propersurjective mappings π ∶M rarr B and πinfin ∶Minfin rarr B onto the space of holo-morphic quadratic differentials each carries a Higgs bundle (AΦ) to detΦThe subsetM984094

infin of limiting configurations over the lsquofree regionrsquo B984094 sub B of qua-dratic differentials with only simple zeroes was introduced in [MSWW14]later Mochizuki [Mo] extended the definition of limiting configurations toinclude those also lying over the lsquodiscriminant locusrsquo Λ = B ∖B984094 We denotethe preimage πminus1(B984094) byM984094 There is a canonical diffeomorphism

(1) F ∶M984094infin 995275rarrM984094

which we explain later This diffeomorphism allows us to transfer functionsvector fields and tensors from M984094

infin to M984094 and back The maps π and πinfinare quadratic in the Higgs field so the natural Ctimes action on Higgs bundles(AΦ) satisfies π(A tΦ) = t2 detΦ and similarly for πinfin We consider hereonly the restriction of this Ctimes action to an R+ action The space B984094 is a conewith respect to this (quadratic) action whileM984094

infin is lsquosemi-conicrsquo ie it is abundle of tori over the cone B984094 where the fibers along each R+ orbit are allthe same Limiting configurations are one of the two building blocks for theconstruction of diverging families of solutions inM984094 [MSWW14] (the otheris the family of fiducial solutions cf sect4)

Entirely distinct from those developments motivated by supersymmet-ric quantum field theory a beautiful conjectural picture of the asymptoticgeometry of M has been established in the monumental work by GaiottoMoore and Neitzke [GMN] These authors develop the formalism of spectralnetworks on Riemann surfaces out of which they construct a hyperkahlermetric gGMN onM which they conjecture to be precisely equal to the metricgL2 The short survey paper by Neitzke [Ne] contains an overview of thisconstruction Part of their story involves a simpler hyperkahler metric gsfonM984094 called the semiflat metric which is canonically associated to the un-derlying algebraic completely integrable system structure They show thatit is a good approximation to gGMN in the sense that

gGMN sim gsf +O(eminusβt)The error term is a symmetric two-tensor whose norm with respect to gsfdecays at the stated rate where we are identifying the dilation parameter tas a radial variable onM984094 and the exponential decay rate β depends on theparticular R+ orbit and degenerates as this ray converges to B ∖B984094 (Thereis a precise conjectured formula for β which we do not state here)

These two points of view lead to the challenge of understanding theGaiotto-Moore-Neitzke metric and its relationship to gL2 This is the goalof the present paper In more detail we have two main results

Theorem 11 The pullback Flowastgsf of the semiflat metric toM984094infin is a renor-

malized L2 metric onM984094infin

The diffeomorphism F can be defined via the Kobayashi-Hitchin cor-respondence since points on M984094 and M984094

infin are each associated to unique

ASYMPTOTIC GEOMETRY OF THE HITCHIN METRIC 3

points of the complex gauge group orbit (modulo the real gauge group)Alternately at least outside of a large ball it can also be defined via theconstruction in [MSWW14] of lsquolargersquo solutions to the Hitchin equationsFurthermore there are natural maps from T lowastB984094995723Γ to both M984094 and M984094

infinHere Γ is a certain local system of lattices over B984094 which can be describedeither cohomologically or using the algebraic completely integrable systemstructure onM Thus all three spaces are naturally identified and it is moreor less a matter of taste which one of these one considers the most funda-mental Both T lowastB984094995723Γ and M984094

infin have more obvious coordinates and theseinduce coordinates onM984094 It is in terms of these that we write the metriccoefficients for gL2 and gsf later Our second result quantifies the sense inwhich these are close

Theorem 12 There is a convergent series expansion

gL2 = gsf +infin990118j=0

t(4minusj)9957233Gj +O(eminusβt)

as trarrinfin where each Gj is a dilation-invariant symmetric two-tensor Therate β gt 0 of exponential decrease of the remainder is uniform in any closeddilation-invariant sector W subM984094

infin disjoint from πminus1infin (B ∖B984094)The terms in this series are all lower order including those with positive

powers of t Indeed the semi-conic nature of gsf means that its horizontalmetric coefficients (relative to πinfin) grow like t2 and the Gj with j le 4 areonly nonvanishing in those directions

Throughout this article we say that a tensor G onM984094infin is polynomial in

t if it has the form G = tαG984094 for some real number α where G984094 is dilationinvariant or slightly more generally if it has a convergent expansion in termsof such monomial terms

Remark The polynomial correction terms in Theorem 12 arise in a naturalway The calculations which produce gauged tangent vectors to the modulispace and the corresponding metric coefficients lead to expressions of theform

990124Df(t29957233z) q

qwhere q and q are holomorphic quadratic differentials z is a local holomor-phic coordinate in a disk D centered at a zero of q and f is a Cinfin functionwhich decays exponentially in its argument or more generally a convergentsum of such functions The quotient q995723q is meromorphic in z with a simplepole at z = 0 (provided q has simple zeroes and q does not vanish at thesezeroes) A simple calculation shows that these integrals lead to asymptoticexpressions in t as above The precise calculations appear in Sections 5 andlater

In light of the prediction that gL2 minus gsf decays exponentially in t it is ofconsiderable interest to determine whether any of these polynomial correc-tion terms Gj are nonzero Although the basic strategy and many of the


technical aspects of this paper were understood by us two or three yearsago it was written slowly and its final release was delayed for some monthsas we investigated the sharpness of our results Around the time this pa-per was posted David Dumas and Andy Neitzke announced some furtherprogress which has just now appeared [DN] In this they explain a remark-able cancellation that takes place in the difference of metric coefficients inlsquohorizontalrsquo directions tangent to the Hitchin section This is then trans-fered to show the exponential convergence of the horizontal components ofgL2 to g on the Hitchin section over a general compact Riemann surface XThis is accomplished with careful attention to the rate of exponential decaybut unfortunately they miss the conjectured sharp numerical value of thisrate by a factor of 2 Their result has successfully been extended to theentire space M984094 including non-horizontal directions and the region off ofthe Hitchin section in the very recent preprint [Fr18] by Laura Fredrickson

The techniques of the present paper lead to a number of other interestingresults and we hope the approach developed here will be useful in a numberof related problems

We note in particular that even though the relative decay rate of themetric asymptotics has now been proven to be exponential everywhere onM984094 one sees using Proposition 61 below that gauged tangent vectorsthemselves converge to their limits only at a polynomial rate

The terminology and basic definitions needed to fill out the brief discus-sion above will be presented in the next two sections Following that westudy the deformations of the space of limiting configurations and proveTheorem 11 On the actual moduli space one of the main technical issuesis to put infinitesimal deformations of a given solution into gauge The spe-cial types of fields encountered here which arise in this gauge-fixing requiresome novel mapping properties of the inverse of the lsquogauge-fixing operatorrsquoLt These are proved in sect5 The remaining sections use this to systemati-cally compute the metric coefficients in various directions which establishesTheorem 12

The authors wish to extend their thanks to a number of people with whomwe had very helpful conversations The two who should be singled out areNigel Hitchin and Andy Neitzke both of whom contributed substantiallyboth in terms of encouragement and their very thoughtful advice at vari-ous stages We also thank Laura Fredrickson and Sergei Gukov for manyinsightful remarks and Steven Rayan for a very thorough reading of a firstdraft of the paper Finally we are also extremely grateful to the referee foran extraordinarily detailed report which led to many clarifications of thetext and also for pointing out the reference [DH]

2 Preliminaries on the Hitchin system

We begin by recalling some parts of the theory of SL(2C) Higgs bundlesdeveloped initially in Hitchin in [Hi87a] and subsequently extended by very


many authors The moduli space of stable Higgs bundles carries a rich geo-metric structure including a natural hyperkahler structure arising from itsgauge theoretic interpretation as a hyperkahler quotient [HKLR] It is alsoan algebraic completely integrable system [Hi87a Hi87b] and hence a denseopen set (the so-called regular set) is endowed with a semiflat hyperkahlermetric [Fr] We explain all of this now

21 The moduli space of Higgs bundles Let X be a compact Riemannsurface of genus γ ge 2 KX its canonical bundle and p ∶ E rarr X a complexrank 2 vector bundle over X A holomorphic structure on E is equiva-lent to a Cauchy-Riemann operator part ∶ Ω0(E) rarr Ω01(E) so we think of aholomorphic vector bundle as a pair (E part) A Higgs field Φ is an elementΦ isin H0(XEnd(E) otimesKX) ie a holomorphic section of End(E) twistedby the canonical bundle An SL(2C) Higgs bundle is a triple (E partΦ) forwhich the determinant line bundle detE ∶= Λ2E is holomorphically trivial inparticular degE = 0 and the Higgs field Φ is traceless Thus with End0(E)the bundle of tracefree endomorphisms of E Φ isinH0(XEnd0(E)otimesKX) Inthe sequel a Higgs bundle will always refer to this special situation Thusa Higgs bundle is completely specified by a pair (partΦ) Throughout Higgsbundles are considered exclusively on the fixed complex vector bundle E ofdegree 0 which will therefore be suppressed from our notation

The special complex gauge group Gc consisting of automorphisms of E ofunit determinant acts on Higgs bundles by (partΦ)↦ (gminus1 part g gminus1Φg) Thequotient by this action is not well-behaved unless restricted to the subset ofstable Higgs bundles When degE vanishes a Higgs bundle (partΦ) is calledstable if any Φ-invariant subbundle L ie one for which Φ(L) sub L otimesKX has degL lt 0 Note that if part is stable in the usual sense then (partΦ) is astable Higgs bundle for any choice of Φ We call

M= stable Higgs bundles995723Gc

the moduli space of Higgs bundles This is a smooth complex manifold ofdimension 6(γminus1) Furthermore if N denotes the (smooth quasi-projectivemanifold) of stable holomorphic structures on E then T lowastN embeds as anopen dense subset of M The tangent space to M at an equivalence class[(partΦ)] fits into the exact sequence [Ni]

H0(End0(E))995275rarrH0(End0(E)otimesKX)995275rarr T[(partΦ)]M

995275rarrH1(End0(E))995275rarrH1(End0(E)otimesKX)

We use here the abbreviated notation Hj(F ) for Hj(XF ) The holomor-phic structure on End0(E) is inherited from the one on E and the mapsHj(End0(E)) rarr Hj(End0(E) otimes KX) are induced by [Φ sdot] acting on thesheaf of holomorphic sections of End0(E) The restriction of the natu-ral nondegenerate pairing H0(End0(E)otimesKX)timesH1(End0(E))rarr C comingfrom Serre duality gives rise to a holomorphic symplectic form η on M


which extends the natural complex symplectic form of T lowastN Note also thatH0(End0(E)) 984148H1(End0(E)otimesKX) = 0 if part is stable

22 Algebraic integrable systems We next exhibit on the complexsymplectic manifold (M η) the structure of an algebraic integrable sys-tem [Hi87a Hi87b] Let B = H0(K2

X) denote the space of holomorphicquadratic differentials and Λ sub B the discriminant locus consisting of holo-morphic quadratic differentials for which at least one zero is not simpleThis is a closed subvariety which is invariant under the multiplicative actionof Ctimes and hence B984094 ∶= B ∖Λ is an open dense subset of B

The determinant is invariant under conjugation hence descends to a holo-morphic map

det ∶Mrarr B [(partΦ)]↦ detΦ

called the Hitchin fibration [Hi87a] This map is proper and surjective It canbe shown that there exist 3(γ minus 3) linearly independent functions onM984094 ∶=detminus1(B984094) which commute with respect to the Poisson bracket correspondingto the holomorphic symplectic form η HenceM984094 is a completely integrablesystem over this set of regular values cf [GS Section 44] and [Fr] Inparticular generic fibers of det are affine tori Identifying T lowastq B984094 with the

invariant vector fields onM984094q yields a transitive action on the fibers by taking

the time-1 map of the flow generated by these vector fields The kernel Γq is afull rank lattice in T lowastq B984094 (ie its R-linear span equals T lowastq B984094) and Γ = ⋃qisinB984094 Γq

is a local system over B984094 This gives an analytic family of complex toriA = T lowastB984094995723Γ Since Γ is complex Lagrangian for the holomorphic symplecticform ωT lowastB984094 this form descends to a holomorphic symplectic form η on A

We now and henceforth fix a holomorphic square root

Θ =K19957232X

of the canonical bundle We then define the Hitchin section ofM by

H ∶ B rarrM H(q) = 995697(partΘoplusΘlowast Φq)995834 where Φq = 9957380 minusq1 0

995742

Then H(B984094) is complex Lagrangian Hlowastη = 0 since only Φ varies Thisgives a local symplectomorphism between (T lowastB984094ωT lowastB984094) and (M984094 η) Oneach fiber this is the Albanese mapping determined by the pointH(q) isinM984094

q

We must also identify the affine complex torusM984094q algebraically this turns

out to be a subvariety of the Jacobian of the related Riemann surface

Sq = α isinKX 995852 α2 = q(p(α)) subKX

called the spectral curve associated to q Since the zeroes of q are simplepq ∶= p995852Sq ∶ Sq rarrX is a twofold covering between smooth curves with simplebranch points at the zeroes of q hence by the Riemann-Hurwitz formulaSq has genus 4γ minus 3 We think of points of Sq as the eigenvalues of Φ (thisexplains the name spectral curve)

We summarize this discussion in the


Proposition 21 There is a symplectomorphism between (M984094 η) and (A η)which intertwines the Ctimes action on the two spaces

Remark Note that the implicit Ctimes action on T lowastB984094 here is not the standardpullback action The one here dilates the base but acts trivially on the fibersAnother important fact is that the Ctimes action identifies the fibers M984094

q and

M984094t2q for every t isin Ctimes

There is a more intrinsic description of this picture using the holomorphicLiouville form λ isin Ω1(KX) λα(v) = α(plowastv) for any α isin KX v isin TαKX Itspullback by the inclusion map ιq ∶ Sq rarrKX is the Seiberg-Witten differentialon Sq

λSW(q) ∶= ιlowastqλ isinH0(KSq) 984148H10(Sq)which in particular is a closed form If q is clear from the context wesimply write λSW Now denote by σq the involution of Sq obtained byrestricting the map σ which is multiplication by minus1 on the fibers of KX Then σlowastq (plusmnλSW(q)) = ∓λSW(q) are the two ldquoeigenformsrdquo of plowastqΦ ∶ plowastqE rarrplowastqE otimes plowastqKX The two corresponding holomorphic line eigenbundles Lplusmnof plowastqE are interchanged under σq Since L+ otimes Lminus 984148 plowastqK

minus1X we see that

σlowastqL+ 984148 Lminus1+ otimes plowastqKminus1X Twisting by Θq = plowastqΘ we see that σq(L+ otimes Θq) =

(L+ otimes Θq)minus1 ie L+ otimes Θq lies in what we call the Prym-Picard varietyPPrym(Sq) = L isin Pic(Sq) 995852 σlowastL = Llowast

Summarizing any Higgs bundle (partΦ) with detΦ isin B984094 induces a pair(Sq L+) with L+ otimesΘq isin PPrym(Sq) Conversely (partΦ) with q = detΦ isin B984094can be recovered from a line bundle in PPrym(Sq) Consequently the choiceof square root Θq =K19957232

X identifiesM984094q biholomorphically with PPrym(Sq)

This in turn gets identified via the Hitchin section with its Albanese va-riety H0(KPPrym(Sq))lowast995723H1(PPrym(Sq)Z) This shows thatM984094 rarr B984094 is analgebraic integrable system

23 The special Kahler metric A Kahler manifold (M2mω I) is calledspecial Kahler if there exists a flat symplectic torsionfree connection 984162 suchthat regarding I as a TM -valued 1-form d984162I = 0 The basic reference forspecial Kahler metrics is [Fr] and see [HHP] for the case of Hitchin systems

The analytic family of spectral curves S = ⋃qisinB984094 Sq rarr B984094 induces a specialKahler metric on B984094 To see this first identify the Albanese varieties of theprevious section with

Prym(Sq) ∶=H0(KSq)lowastodd995723H1(SqZ)oddwhereH0(KSq)odd andH1(SqZ)odd denote the (minus1)-eigenspaces ofH0(KSq)and H1(SqZ) under the involution σ cf [BL Proposition 1242] More-over considering B984094 as the σ-invariant deformation space of a given spectralcurve Sq we have TqB984094 984148 H0(NSq)odd 984148 H0(KSq)odd where the canonicalsymplectic form dλ on KX is used to identify the normal bundle NSq of Sq

with the canonical bundle of KSq (cf also [Ba HHP]) It follows that T lowastq B984094 984148


H0(KSq)lowastodd 984148 C3γminus3 This contains the integer lattice Γq = H1(SqZ)odd 984148Z6γminus6 Since H1(SqZ)odd 984148 H1(Prym(Sq)Z) we can choose a symplec-tic basis for the intersection form α1(q) αm(q)β1(q) βm(q) m =3γ minus3 in Γq This intersection form (the polarization of Prym(Sq)) is twicethe restriction of the intersection form of Sq (the canonical polarization ofthe Jacobian of Sq) cf [BL p 377]

An important feature of any special Kahler metric is the existence ofconjugate coordinate systems (z1 zm) and (w1 wm) ie holomor-phic coordinates such that (x1 xm y1 ym) where Re(zi) = xi andRe(wi) = minusyi are Darboux coordinates for ω The local system Γ = ⋃qisinB984094 Γq

is spanned locally by differentials of Darboux coordinates (dxi dyi) and in-duces a real torsionfree flat symplectic connection 984162 over B984094 by declaring984162dxi = 984162dyi = 0 for i = 1 m Thus we can choose the coordinates (xi yi)in such a way that conjugate holomorphic coordinates are

(2) zi(q) = 990124αi(q)

λSW (q) wi(q) = 990124βi(q)

λSW (q) i = 1 m

[Fr Proof of Theorem 34] In terms of these the Kahler form equals

ωsK =3γminus3990118i=1

dxi and dyi = minus1

4990118i

(dzi and dwi + dzi and dwi)

There is an alternate and quite explicit expression for ωsK To this endobserve that

dzi(q) = 990124αi(q)

984162GMq λSW dwi(q) = 990124

βi(q)984162GM

q λSW i = 1 m

where 984162GM is the Gauszlig-Manin connection and λSW ∶ B984094 rarr ⋃qisinB984094H10(Sq) is

considered as a section Then 984162GMq λSW is the contraction of dλSW by the

normal vector field Nq corresponding to q By Proposition 1 in [DH] (cfalso Proposition 82 in [HHP]) we have

(3) 984162GMq λSW =

1

2τq

where τq is the holomorphic 1-form on Sq corresponding to q under theisomorphism

(4) TqB984094 =H0(K2X)

984148995275rarrH0(KSq)odd q ↦ τq ∶=q

λSW

There is a seemingly anomalous factor of 12 here compared to the cited

formula in [DH] The reason is that their expression αq which appears inthe right hand side of their formula for the Gauszlig-Manin derivative of λSW

is precisely 19957232 of τq as we have defined it here

Remark The special case where q = q is of particular interest since itgenerates the Ctimes action on B984094 (Recall however that we work only with the


R+ action) For this infinitesimal variation we have τq = λSW and hence

984162GMq λSW =

1

2λSW

The associated Kahler metric gsK(q q) equals ωsK(q Iq) for the constantcomplex structure I = i It is therefore given by

gsK(q q) =i

2990118j

(dzj(q)dwj(q) minus dwj(q)dzj(q))

= i

2990118j990124αj

984162GMq λSW 990124

βj

984162GMq λSW minus 990124

βj

984162GMq λSW 990124

αj

984162GMq λSW

= i

8990118j990124αj

τq 990124βj

τq minus 990124βj

τq 990124αj

τq

= i

8990124Sq

τq and τq =1

8990124Sq

995852τq 9958522 dA

where we have used the Riemann bilinear relations Here dA is the area formon Sq induced from the one on X for any metric in the given conformal classon X and we recall that the quantity 995852α9958522dA is conformally invariant whenα is a 1-form Note also that intc λSW vanishes for any even cycle c since λSW

is odd with respect to σ This identifies the special Kahler metric on TqB984094with an eighth of the natural L2-metric

995858α9958582L2 = i990124Sq

α and α = 990124Sq

995852α9958522 dA

on H0(KSq)odd via the isomorphism q ↦ τq Using τq = q995723λSW and λ2SW = q

we obtain that 995852τq 9958522 = 995852q9958522995723995852q995852 and so the last integral may be converted intoan integral over the base Riemann surface

(5) gsK(q q) =1

8990124Sq

995852τq 9958522 dA =1

8990124Sq

995852q9958522

995852q995852dA = 1

4990124X

995852q9958522

995852q995852dA

This representation of the special Kahler metric will be important later Forany holomorphic quadratic differential q the quantity 995852q995852dA is conformallyinvariant so again the choice of metric in the conformal class is irrelevantWe single out one key consequence of the preceding discussion

Corollary 22 The special Kahler metric gsK depends smoothly on thebasepoint q isin B984094

Proof This may be seen from the following local coordinate expression forτq In a local holomorphic coordinate chart q(z) = f(z)dz2 and q(z) =f(z)dz2 and since z = 0 is a simple zero of q f(0) = 0 but f 984094(0) ne 0Let (zw) be canonical local coordinates on KX so λSW = wdz ThenSq = w2 = f(z) and hence

2wdw = f 984094(z)dz


there In particular λSW = 2w2dw995723f 984094(z) and q = 4w2f(z)dw2995723f 984094(z)2 so

τq =q

λSW= 2f(z)

f 984094(z)dw

This computation shows that τq and hence gsK depends smoothly on q Note that the smoothness asserted in the corollary is not immediately

apparent from some of the other expressions eg the final one in (5)We conclude this section by discussing the conic structure of this metric

Recall the Ctimes-action on B984094ϕλ(q) ∶= λ2q q isin B984094λ isin Ctimes

It is immediate from (2) and the defining relation λ2SW = q on Sq that the

coordinates zi and wi are homogeneous of degree 1 ie

zi(ϕλ(q)) = 990124αi

τλq = λzi(q) wi(ϕλ(q)) = 990124βi

τλq = λwi(q)

for λ isin W where W is a neighborhood of 1 isin Ctimes Eulerrsquos formula for thederivative of homogeneous functions now gives thatsumi zipartwj995723partzi = wj hence

F(q) = 1

2990118j

zjwj

defines a holomorphic prepotential Indeed since partwi995723partzj = partwj995723partzi we get

partF995723partzj = 12(wj +990118

i

zipartwi995723partzj) = 12(wj +990118

i

zipartwj995723partzi) = wj

This holomorphic prepotential is of course homogeneous of degree 2 ieF(ϕλ(q)) = λ2F(q) This establishes B984094 as a conic special Kahler manifoldsee Proposition 6 in [CM]

Computing locally again we find using the Riemann bilinear relationsand the relation τ2q = q that the Kahler potential is given by

K(q) = 1

2Im990118

j

wj zj =i

4990118j

(zjwj minus zjwj)

= i

4990118j990124αj

τq 990124βj

τq minus 990124αj

τq 990124βj

τq

= i

4990124Sq

τq and τq =1

4990124Sq

995852τq 9958522 dA =1

2990124X995852q995852dA

Let S 984094 = q isin B984094 ∶ intX 995852q995852dA = 1 the L1-unit sphere in B984094 By Corollary 4 in[BC] we find that

(6) φ ∶ (R+ times S 984094 dt2 + t2gsK995852S984094)rarr (B984094 gsK) (t q)↦ t2q

is an isometry This establishes that B984094 is a metric cone In particular forq isin B984094 with intX 995852q995852dA = 1 the curve t ↦ t2q is a unit speed geodesic As acheck on this observe that

(7) dφ995852(tq)(partt) = 2tq dφ995852(tq)(q) = t2q


On the other hand

gsK(q q)995852t2q =i

8990124St2q

(q995723λSW) and (q995723λSW)

= i

8t2990124Sq

(q995723λSW) and q995723λSW =1

t2gsK(q q)995852q

so

(8) (9958582tq9958582sK)995852t2q = 4(995858q9958582sK)995852q = 1 (995858t2q9958582sK)995852t2q = t2(995858q9958582sK)995852q

Here we have used that (995858q9958582sK)995852q =14 intX 995852q995852dA =

14 for q isin S 984094 Thus Equations

(7) and (8) indeed reconfirm the conic structure of gsK

24 Hyperkahler metrics A Riemannian manifold (Mg) is called hy-perkahler if it carries three integrable complex structures I J and K whichsatisfy the quaternion algebra relations and such that the associated 2-formsωC(sdot sdot) = g(sdot C sdot) C = I JK are each closed In particular every special-ization (MCωC) is Kahler (this is also true when C = aI + bJ + cK wherea b c are constants with a2+b2+c2 = 1) whence the name hyperkahler Thetwo examples of hyperkahler metrics of interest here are the Hitchin metriconM and the semiflat metric onM984094

241 Semiflat metric If (Mω984162) is any manifold with a special Kahlerstructure with Kahler metric gsK then T lowastM carries a natural semiflathyperkahler metric gsf cf [Fr Theorem 21] The name semiflat comesfrom the fact that gsf is flat on each fiber of T lowastM In particular if Γ is alocal system in T lowastM of full rank then gsf pushes down to a semiflat metricon the torus bundle T lowastM995723Γ We consider this in the special case M = B984094where A = T lowastB984094995723Γ 984148M984094 the analytic family A of complex tori introduced insect22 The existence of such a metric is common to any algebraic integrablesystem [Fr Theorem 38]

To construct gsf note that the connection 984162 induces a distribution ofhorizontal and complex subspaces of T lowastM Then relative to the decompo-sition TαT

lowastM 984148 Tπ(α)M oplusT lowastπ(α)M gsf equals gπ(α)oplus gminus1π(α) the integrability

is ensured by the differential geometric conditions on a special Kahler met-ric It is clearly flat in the fiber directions In local coordinates (xi yi pi qi)of T lowastM induced by Darboux coordinates (xi yi) for ω the Kahler form ωI

for the natural complex structure on T lowastM is

ωI =990118i

dxi and dyi + dpi and dqi

As noted earlier if M = B984094 then gsf descends to the quotient A = T lowastB984094995723Λand thus induces a metric onM984094 which we still denote by gsf The invariantvector fields on the fibers ofM984094 are given by the η-Hamiltonian vector fieldsXf of functions f π where f is a locally defined function on B984094 (see for


instance [GS (445)]) Hence if Xf is a vector field on M984094 tangent to thefibers then

gsf(Xf Xf) = gminus1sK(df df)Computing the dual metric gminus1sK on T lowastq B984094 amounts to computing the metric on

H0(KSq)lowastodd dual to the L2-metric on H0(KSq)odd The complex antilinear

isomorphim H0(KSq)lowast rarr H0(KSq) obtained by dualizing with respect to

the L2-metric simply is the composition

H0(KSq)lowast = H10(Sq)lowast 995275rarrH01(Sq)995275rarrH10(Sq) =H0(KSq)where the first arrow is given by Serre duality and the second one by com-plex conjugation macr ∶ H01(Sq) rarr H10(Sq) exchanging the space of anti-holomorphic and holomorphic forms So if df(q) is dual to α isin H0(KSq)oddthen

gminus1sK(df(q) df(q)) = 990124Sq

995852α9958522 dA =∶ gsf(αα)

This shows that the vertical part of the semiflat metric is the natural L2-metric on Prym(Sq) We return to this fact in Section 3

We also wish to describe the Prym variety in terms of unitary data Infact each line bundle L in Prym(Sq) corresponds to an odd flat unitary con-nection on the trivial complex line bundle In other words L is representedby a connection 1-form η isin Ω1(Sq iR) such that dη = 0 and σlowastη = minusη Thisspace is acted on by odd gauge transformations ie maps g ∶ Sq rarr S1 suchthat g σ = gminus1 We obtain

Prym(Sq) =H1(Sq iR)oddH1

Z(Sq iR)odd

If η isinH1(Sq iR)odd is a harmonic representative of a class in H1(Sq iR)oddthen η = αminusα for α = η10 isinH0(KSq)odd Here we have used thatH1(SqC) =H10(Sq)oplusH01(Sq) So finally

(9) gsf(η η) ∶= gsf(αα) =1

2990124Sq

995852η9958522 dA = 990124X995852η9958522 dA

which is the form of the metric we will use from now on In Section 3 we willreinterpret the space of imaginary odd harmonic 1-forms on Sq as a spaceof L2-harmonic forms with values in a twisted line bundle on the puncturedbase Riemann surface Xtimes reducing the L2-integral over Sq to an integralover X

Parallel to Corollary 22 and its proof we have

Corollary 23 The semiflat metric is smooth onM984094

242 Hitchin metric The second hyperkahler metric we consider is definedon all ofM and stems from a gauge-theoretic reinterpretation ofM Moreconcretely fix a hermitian metric H on E Holomorphic structures part arethen in 1 minus 1-correspondence with special unitary connections After thechoice of a base connection these correspond to elements in Ω01(sl(E))


For such an endomorphism valued form A we denote the correspondingCauchy-Riemann operator by partA The condition Φ isin H0(X sl(E)otimesKX) isequivalent to partAΦ = 0 where Φ is regarded as a section of Λ10T lowastX otimes sl(E)In particular we get an induced Gc-action on (AΦ) We denote this actionby (AgΦg) for g isin Gc Hitchin [Hi87a] proves that in the Gc-equivalenceclass [E partΦ] = [AΦ] there exists a representative (AgΦg) unique up tospecial unitary gauge transformations such that the so-called self-dualityequations or Hitchin equations (with respect to H)

(10) micro(AΦ) ∶= (FA + [Φ andΦlowast] partAΦ) = 0hold Here FA denotes the curvature of A and Φlowast is the hermitian conjugatewe refer to micro as the hyperkahler moment map

Remark Alternatively we can fix a Higgs bundle (partΦ) and ask for ahermitian metric H such that FH + [Φ and ΦlowastH ] = 0 where lowastH is the adjointtaken with respect to H and FH is the curvature of the Chern connection AThe pair (AΦ) is then a solution to the self-duality equation with respectto H

Stability of (EΦ) translates into the irreducibility of (AΦ) If G denotesthe special unitary gauge group it follows that

M 984148 (AΦ) isin Ω1(su(E)) timesΩ10(sl(E)) irreducible solves (10)995723GThe map micro can be interpreted as a hyperkahler moment map with respect tothe natural action of the special unitary gauge group G on the quaternionicvector space Ω01(sl(E))timesΩ10(sl(E)) with its natural flat hyperkahler met-ric

995858(αϕ)9958582L2 = 2i990124XTr(αlowastand α +ϕ andϕlowast)

(note that Ω1(su(E)) 984148 Ω01(sl(E))) Consequently this metric descends toa hyperkahler metric on the quotient M [HKLR] We describe this metricnext Let su(E) denote the tracefree endomorphisms of E which are skew-hermitian with respect to the hermitian metric H fixed above We endowsl(E) with the hermitian inner product given by ⟨AB⟩ = Tr(ABlowast) andextend it to sl(E)-valued forms by choosing a conformal background metricon X Fix a configuration (AΦ) and consider the deformation complex

0rarr Ω0(su(E))D1(AΦ)995275995275995275995275rarr Ω1(su(E))oplusΩ10(sl(E))

D2(AΦ)995275995275995275995275rarr Ω2(su(E))oplusΩ2(sl(E))rarr 0

The first differential

D1(AΦ)(γ) = (dAγ [Φ and γ])

is the linearized action of G at (AΦ) while the second is the linearizationof the hyperkahler moment map

D2(AΦ)(A Φ) = (dAA + [Φ andΦ

lowast] + [Φ and Φlowast] partAΦ + [AΦ])


The tangent space toM at [AΦ] is then identified with the quotient

kerD2(AΦ)995723imD1

(AΦ) 984148 kerD2(AΦ) cap (imD1

(AΦ))perp

Then

990124X⟨dAγ A⟩dA = 990124

X⟨γ dlowastAA⟩dA

and

990124X⟨[Φ and γ] Φ⟩dA = minus990124

X⟨γ i lowast πskew[Φlowastand Φ]⟩dA

where πskew ∶ sl(E) rarr su(E) is the orthogonal projection hence (A Φ) perpimD1

(AΦ) with respect to the L2-metric in (12) below if and only if

(11) (D1(AΦ))

lowast(A Φ) = dlowastAA minus 2πskew(i lowast [Φlowast and Φ]) = 0

If this is satisfied we say that (A Φ) is in Coulomb gauge (in gauge for

short) For tangent vectors (Ai Φi) i = 12 in Coulomb gauge the inducedL2-metric is given by

gL2((α1 Φ1) (α2 Φ2)) = 2990124XRe⟨α1α2⟩ +Re⟨Φ1 Φ2⟩ dA

= 990124X⟨A1 A2⟩ + 2Re⟨Φ1 Φ2⟩ dA

(12)

where αi denotes the (01)-part of Ai i = 12 and dA denote the area formof the background metric

Remark There is a similar construction when the determinants of theHiggs bundles are not holomorphically trivial and it can be shown that theL2-metric on the moduli space is complete if the degree of E is odd

The first goal of this paper is to show that in a sense to be specified belowthe semiflat metric is the asymptotic model for the Hitchin metric

3 The semiflat metric as L2-metric on limiting configurations

Our goal in this section is to understand the semiflat metric onM984094 as alsquoformalrsquo L2-metric on the space of limiting configurations

31 Limiting configurations One of the main results in [MSWW14] isthat the degeneration of solutions (AΦ) to the self-duality equations asq = detΦ rarr infin is described in terms of solutions of a decoupled version ofthe self-duality equations

Definition 31 Let H be a hermitian metric on E and suppose that q isinH0(K2

X) has simple zeroes Set Xtimesq = X ∖ qminus1(0) A limiting configurationfor q is a Higgs bundle (AinfinΦinfin) over Xtimesq which satisfies the equations

(13) FAinfin = 0 [Φinfin andΦlowastinfin] = 0 partAinfinΦinfin = 0on Xtimesq We call a Higgs field Φ which satisfies [Φinfin andΦlowastinfin] = 0 normal


The unitary gauge group G acts on the space of solutions (AinfinΦinfin) to(13) and we define the moduli space

Minfin = all solutions to (13)995723G

Strictly speaking we have only considered solutions over differentials q isin B984094which correspond to the open subsetM984094

infin of this moduli space We refer to[Mo] for the definition and description of limiting configurations over pointsq isin B ∖B984094

There is some ambiguity in this definition in that we can either divide outby gauge transformations which are smooth across the zeroes of q or by oneswhich are singular at these points The latter group is more complicatedto define because it depends on q and most elements in its gauge orbitare singular However it is not so unreasonable to consider since as wediscuss later in this section tangent vectors to Minfin are lsquorenormalizedrsquo tobe in L2 by using differentials of such singular gauge transformations Inthe following we use this definition of the quotient space Minfin At theother extreme it would have been possible to take a view consonant withthe original definition of limiting configurations in [MSWW14] where each(AinfinΦinfin) is assumed to take a particular normal form in discs Dp aroundeach zero of q This is no restriction because any limiting configurationwhich is bounded near the zeroes of q can be put into this form with a(bounded) unitary gauge transformation With this restriction we divideout by unitary gauge transformations which equal the identity in each Dp

Let us note a few properties of this space First it still possesses a Hitchinfibration πinfin ∶ Minfin rarr B πinfin((AinfinΦinfin)) = detΦinfin A priori detΦinfin isonly defined on Xtimesq but is bounded near the punctures hence it extendsholomorphically to all of X Second Minfin has a lsquosemi-conicrsquo structure[(AinfinΦinfin)] ↦ [(Ainfin tΦinfin)] which dilates the Hitchin base and leaves in-variant the Prym variety fibers

This space arises as a limit of M in two separate ways On the onehand it is shown in [MSWW14] that for any Higgs bundle (AΦ) there isa complex gauge transformation ginfin which is singular at the zeroes of q andis unique up to unitary transformations such that (AΦ)ginfin is a limitingconfiguration (AinfinΦinfin) with detΦinfin = detΦ Using that ginfin is the limit ofsmooth complex gauge transformations one may approximate elements ofMinfin by representatives of sequences of elements inM On the other handconsider instead the family of moduli spaces Mt consisting of solutions tothe scaled Hitchin equations

microt(AΦ) ∶= (FA + t2[Φ andΦlowast] partAΦ) = 0

modulo unitary gauge transformations It follows from the main result of[MSWW14] that away from the discriminant locus this family of spacesconverges toMinfin ie

limtrarrinfinM984094

t =M984094infin


This is meant in the following sense The diffeomorphism F described in(1) can be recast as a family of natural diffeomorphisms Ft ∶M984094

infin rarrM984094t

Furthermore each M984094t has its own L2 metric gL2t all naturally identified

with one another by the dilation action We then assert that (M984094tFlowastt gL2t)

converges smoothly on compact sets to (M984094infin gsf) We do not belabor this

point by writing this out more carefully since it is not used here in anysubstantial way Nonetheless this picture is conceptually interesting in thatit identifies the space of limiting configurations with a certain lsquoblowdown atinfinityrsquo ofM1 We shall return to a closer examination of this phenomenonin another paper

Let us now proceed with an alternate description ofM984094infin We may recast

Definition 31 into one involving harmonic metrics

Definition 32 Let (E partE Φ) be a Higgs bundle such that q = detΦ hasonly simple zeroes A limiting metric is a flat hermitian metric Hinfin on Eover Xtimesq = X ∖ qminus1(0) such that Φ is normal with respect to Hinfin ie thelimiting equation

FHinfin = 0 [Φ andΦlowastHinfin ] = 0is satisfied over Xtimesq Here FHinfin is the curvature of the Chern connectionAHinfin of Hinfin

Fixing a hermitian metric H a limiting configuration is obtained froma limiting metric as follows Express Hinfin with respect to H with an H-selfadjoint endomorphism field Ξinfin so Hinfin(σ τ) = H(σΞinfinτ) for any twosections σ τ of E Setting Ξminus1infin = ginfinglowastinfin then H = glowastinfinHinfin and thus Ainfin = Aginfin

and Φinfin = gminus1infinΦginfin constitute a limiting configuration in the complex gaugeorbit of the Higgs bundle (AΦ)

The interpretation of the limiting metric for a Higgs bundle goes backto an observation by Hitchin and is described in detail in [MSWW15] seealso [Mo] We review this now Fix q isin H0(K2

X) with simple zeroes As insect22 let pq ∶ Sq rarr X denote the spectral cover and Lplusmn sub plowastqE the eigenlinesof plowastqΦ these are exchanged by the involution σ Then L+ = L otimes plowastqΘ

lowast

for the previously chosen square root Θ of the canonical bundle KX and aholomorphic line bundle L isin Prym(Sq) ie σlowastL = Llowast Then Lminus = σlowastL+ =Llowast otimes plowastqΘ

lowast Since q is holomorphic (qq)19957234 is a flat hermitian metric onΘlowast over Xtimesq hence on plowastqΘ

lowast over Stimesq and is singular at the puncturesFurthermore since L is a holomorphic line bundle of zero degree it admitsa flat hermitian metric h Altogether we form the singular flat metrich+ = h(qq)19957234 on L+ If Ah and Aq denote the Chern connections of the

metrics h and (qq)19957234 respectively then the Chern connection Ah+ of h+ isthe tensor product of Ah and Aq Pulling back gives the metric hminus = σlowasth+ onLminus so that h+oplushminus is σ-invariant on L+oplusLminus and thus descends to a limitingmetric Hinfin on E (We use here that plowastqE decomposes holomorphically as thedirect sum of the line bundles L+ and Lminus on the punctured spectral curveStimesq )


Varying the holomorphic line bundle L isin Prym(Sq) we obtain all lim-iting configurations associated to q which identifies Prym(Sq) with thetorus Minfin(q) of limiting configurations associated to q see Section 44in [MSWW14] We describe this more concretely Fix a Cinfin-trivializationC = Sq timesC of the underlying line bundle with standard hermitian metric h0With respect to this metric any holomorphic structure on this trivial bundleis represented by a flat unitary connection d+η where η isin Ω1(Sq iR) is closedand odd under the involution σlowastη = minusη Clearly d+ η is the Chern connec-tion of h0 for the holomorphic structure part + η01 and h+ = h0(qq)19957234 givesrise to the limiting metric Hinfin The Chern connections satisfy Ah+ = Aq + ηand Ahminus = Aq minus η on L+ and Lminus respectively

There is also a Hitchin section in Minfin corresponding to any choice of

square root Θ =K19957232X Thus consider E = ΘoplusΘlowast with Higgs field

Φ = 9957380 minusq1 0

995742

This has spectral data L = OSq isin Prym(Sq) corresponding to η = 0 In-deed note that from [BNR Remark 37] E = (pq)lowastM for M = L+ otimes plowastqKX

However (pq)lowastOSq = OX oplusKminus1X so by the push-pull formula

(pq)lowast(plowastqΘ) = (pq)lowast(OSq otimes plowastqΘ) = (pq)lowastOSq otimesΘ = ΘoplusΘlowast

and hence by the spectral correspondence M = plowastqΘ This shows that L+ =plowastqΘ

lowast and so L = OSq as claimed Let Hinfin be the limiting metric for thisHiggs bundle

Lemma 31 The limiting metric on the Higgs bundle (EΦ) above is givenup to scale by

Hinfin = (qq)minus19957234 oplus (qq)19957234

with respect to the decomposition E = ΘoplusΘlowast

Proof It suffices to check that Φ is normal with respect to Hinfin on thepunctured surface Xtimes To that end trivialize Θplusmn1 locally by dzplusmn19957232 so ifq = fdz2 then

Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz

The eigenvectors splusmn = plusmnradicf dz19957232 + dzminus19957232 satisfy Hinfin(s+ s+) = Hinfin(sminus sminus) =

2995852f 99585219957232 and Hinfin(s+ sminus) = 0 on Xtimes as desired

As before we consider the complex vector bundle E with backgroundhermitian metric H = k oplus kminus1 and Chern connection AH = Ak oplus Akminus1 andconsider the limiting configuration (Ainfin(q)Φinfin(q)) corresponding to Hinfin

In the following we write 995852q99585219957232k = (qq)19957234k where 995852 sdot 995852k is the norm on K2X

induced by k


Lemma 32 The limiting configuration corresponding to the limiting metricHinfin = (qq)minus19957234 oplus (qq)19957234 is given by

Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠


Remark Note that if z is a local holomorphic coordinate around a zeroof q such that q = minuszdz2 and k is the flat metric induced by the holomor-phic trivialization these formulaelig reduce to the standard expression for thesingular model solution

Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz

considered in [MSWW14] and called there the limiting fiducial solution

Proof Write Hinfin(σ τ) = H(σΞinfinτ) where Ξinfin is the H-selfadjoint endo-morphism field

Ξinfin = 995738(qq)minus19957234kminus1 0

0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742

then Hminus1infin = ginfinglowastinfin This gives

gminus1infin (partginfin) = part log995734(qq)19957238k199572329957399957381 00 minus1995742

and consequently

Ainfin = AH + gminus1infin partginfin minus (gminus1infin partginfin)lowast

= AH + 2 Im part log995734(qq)19957238k19957232995739995738i 00 minusi995742

and

Φinfin = gminus1infinΦginfin = 9957380 (qq)minus19957234kminus1q

(qq)19957234k 0995742

as desired

Pulled back to the spectral curve the limiting configuration attains theform

plowastqAinfin(q) = (Aq oplusAq)ginfin Φinfin(q) = gminus1infinΦginfin


More generally if (Ainfin(q η)Φinfin(q η)) denotes the limiting configurationcorresponding to an element L isin Prym(Sq) determined by an odd 1-formη isin Ω1(Sq iR) then

plowastqAinfin(q η) = plowastqAinfin(q) + η otimes gminus1infin 9957381 00 minus1995742 ginfin Φinfin(q η) = Φinfin(q)

Observe now that the pull-back bundle plowastqLΦinfin is spanned by the section isinfinwhere

sinfin = gminus1infin 9957381 00 minus1995742 ginfin isin Γ(S

timesq p

lowastq End0(E))

This section sinfin is parallel with respect to Ainfin(q) so plowastqLΦinfin is trivial as aflat line bundle ie isomorphic to iR = Stimesq times iR with the trivial connectionPulling back to Stimesq any section of LΦinfin can be written as f sdot sinfin wheref isin Cinfin(Stimesq iR) is odd with respect to the involution σ Similarly a 1-form with values in LΦinfin corresponds via pull-back to Stimesq to an odd 1-form

η isin Ω1(Stimesq iR) ie σlowastη = minusη so that H1(Stimesq iR)odd =H1(XtimesLΦinfin) Underthese identifications

Ainfin(q η) = Ainfin(q) + η Φinfin(q η) = Φinfin(q)Define H1

Z(Sq iR)odd sub H1(Sq iR)odd as the lattice of classes with peri-ods in 2πiZ and similarly the lattices H1

Z(Stimesq iR)odd sub H1(Stimesq iR)odd and

H1Z(XtimesLΦinfin) subH1(XtimesLΦinfin) cf [MSWW14 sect44]

Proposition 33 The map d + η ↦ Ainfin(q) + η induces a diffeomorphism


Z(Sq iR)odd984148995275rarr H1(XtimesLΦinfin)

H1Z(XtimesLΦinfin)

=Minfin(q)

In order to prove this proposition we need the following

Lemma 34 The restriction map

H1(Sq iR)odd rarrH1(Stimesq iR)odd =H1(XtimesLΦinfin)is an isomorphism

Proof In the following imaginary coefficients are understood Since Stimesq isa σ-invariant subset of Sq there is a long exact cohomology sequence

rarrHp(Sq Stimesq )odd rarrHp(Sq)odd rarrHp(Stimesq )odd rarrHp+1(Sq S

timesq )odd rarr

By excision Hp(Sq Stimesq ) 984148 995947k

i=1Hp(DiD

timesi ) where (DiD

timesi ) 984148 (DDtimes) are

disks around the punctures p1 pk where k = 4γ minus 4 Using the longexact sequence for the pair (DDtimes) together with the observation thatH0(Dtimes)odd = 0 (constants are even) and H1(Dtimes)odd 984148 H1(S1)odd = 0 (theangular form dθ is even) we obtain that H1(DDtimes)odd =H2(DDtimes)odd = 0It follows that the map H1(Sq)odd rarrH1(Stimesq )odd is an isomorphism

For later use we record


Corollary 35 The restriction of the unique harmonic representative of aclass in H1(Sq iR)odd yields a distinguished closed and coclosed representa-tive of the corresponding class in H1(XtimesLΦinfin) This representative lies inL2 ie is an L2-harmonic 1-form

Proof Since the restriction of the canonical projection π ∶ Sq rarr Xtimes toπminus1(Xtimes) is a conformal map and the space of L2-harmonic 1-forms is con-formally invariant in 2 dimensions it follows that L2-harmonic 1-forms arepreserved under pull-back along π Definition 33 Let

H1(XtimesLΦinfin) = 995743η isin Ω1(Xtimes LΦinfin) ∶ plowastqη isinH1(Sq iR)odd995747

be the corresponding space of L2-harmonic forms on Xtimes

Proof of Proposition 33 It remains to check that the isomorphism fromLemma 34 is compatible with the integer lattices This is clearly the casefor the map H1(Sq iR)odd rarr H1(Stimesq iR)odd Now η isin Ω1(Stimesq iR)odd rep-

resents a class in H1Z(Stimesq iR)odd if and only if it is of the form g = d log g

for g isin Cinfin(Stimesq S1)odd Since g corresponds to a unitary gauge transfor-

mation commuting with Φinfin on Xtimes this is equivalent to η isin Ω1(XtimesLΦinfin)representing a class in H1

Z(XtimesLΦinfin) As a final remark here we include the

Proposition 36 The family of lattices H1Z(Sq iR)odd 984148H1

Z(XtimesLΦinfin) overB984094 are naturally identified with the local system Γ which is defined using thealgebraic completely integrable system structure cf Proposition 21 There-fore as noted in the introduction there is a natural diffeomorphism betweenthe quotients

A = T lowastB984094995723Γ 984148M 984094infin

which intertwines the Ctimes action on both sides

32 Horizontal directions Recall that that the Gauszlig-Manin connectionon the Hitchin fibration gives rise to a splitting of each tangent space ofM984094 into a direct sum of vertical and horizontal subspaces This is the sensein which the terms horizontal and vertical are used in the following Theremainder of this section is devoted to deriving useful expressions for themetric applied to horizontal vertical and mixed pairs of tangent vectors

The Hitchin section is a horizontal Lagrangian submanifold inM984094 as fol-lows from the local symplectomorphism between (T lowastB984094ωT lowastB984094) and (M984094 η)cf sect22 Any smooth family of holomorphic quadratic differentials q(s) isin B984094can thus be lifted to a family of Higgs bundles H(s) = (EΦ(s)) in theHitchin section Fixing a hermitian metric H on E we denote the familyof limiting configurations corresponding to (AH Φ(s)) by (Ainfin(s)Φinfin(s))Setting q ∶= q(0) and q ∶= part

parts995853s=0 q(s) then a brief calculation shows that

Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝

0 995852q995852minus19957232k 995734minus12 Re(q995723q)q + q995739

12 995852q995852

19957232k Re(q995723q) 0

⎞⎠

Assuming the zeroes of q do not coincide with those of q or equivalentlythe deformation is not radial then Ainfin has double poles at the zeroes of qso Ainfin 995723isin L2 However Ainfin is pure gauge and (Ainfin Φinfin) can be transformedto lie in L2 albeit with a singular gauge transformation In addition thisgauged variation even satisfies the Coulomb gauge condition (11) and itsL2 norm turns out to be simply the semiflat metric

To be more precise set

(14) γinfin ∶= minus1

4Im(q995723q)995738i 0

0 minusi995742

Thenαinfin ∶= Ainfin minus dAinfinγinfin = 0

and

ϕinfin ∶= Φinfin minus [Φinfin and γinfin] =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)

so clearly (αinfinϕinfin) = (0ϕinfin) is in L2We next show that (0ϕinfin) satisfies the Coulomb gauge condition again

with the caveat that this is accomplished only by a singular gauge transfor-mation

Lemma 37 The pair (0ϕinfin) satisfies dlowastAinfinαinfinminus2πskew(ilowast [Φlowastinfinandϕinfin]) = 0

Proof Since αinfin = 0 it suffices to show that [Φlowastinfin andϕinfin] = 0 Using the local

holomorphic frame dzplusmn19957232 for E = ΘoplusΘlowast

H = 995738κ 00 κminus1

995742

and hence

Φinfin = 9957380 995852f 995852minus19957232κminus1f

995852f 99585219957232κ 0995742dz

Now one easily calculates

Φlowastinfin = 9957380 995852f 995852minus19957232κminus1

995852f 995852minus19957232κf 0995742dz ϕinfin = 995738

0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally

[Φlowastinfin andϕinfin] =1

2(995852f 995852f995723f minus 995852f 995852minus1f f)9957381 0

0 minus1995742dz and dz = 0

as claimed Finally the following result follows directly from the definitions and for-

mulaelig above


Proposition 38 One has the identity

gsK(q q) = 990124X995852ϕinfin9958522 dA

where ϕinfin is defined by (15)

We have now shown that the restriction of gsf and this renormalized L2

metric (ie the L2 metric obtained on M984094infin by admitting singular gauge

transformations to put tangent vectors into Coulomb gauge) are the same ontangent vectors to the Hitchin section on the space of limiting configurations

To make the analogous computations at limiting configurations which arenot on the Hitchin section we construct more general horizontal lifts offamilies q(s) in B984094 Recall that if q isinH0(K2

X) is fixed and (AinfinΦinfin) is anybase point in πminus1(q) then any element in this fiber takes the form

(16) (Ainfin + ηΦinfin) where [η andΦinfin] = 0 and dAinfinη = 0Write Ainfin(s) Φinfin(s) and η(s) for the horizontal lifts and assume that((Ainfin(0)Φinfin(0)) lies in the Hitchin section over q then differentiating thedefining conditions [η(s) andΦinfin(s)] = 0 and dAinfin(s)η(s) = 0 gives

(17) [η andΦinfin] + [η and Φinfin] = 0and

(18) dAinfin η + [Ainfin and η] = 0

at s = 0 These two equations characterize the tangent vectors (Ainfin+ η Φinfin)to the space of limiting configurationsMinfin in πminus1(q)

We shall use γinfin the infinitesimal gauge transformation which regularizesAinfin to generate all horizontal lifts of q Note that since dAinfinγinfin = Ainfin wehave

dAinfin+ηγinfin = dAinfinγinfin + [η and γinfin] = Ainfin + [η and γinfin]

Lemma 39 Setting η = [ηandγinfin] then equations (17) and (18) are satisfied

hence (Ainfin + η Φinfin) is the horizontal lift of q at (Ainfin + ηΦinfin)

Proof By the Jacobi identity

[η andΦinfin] + [η and Φinfin] = [[η and γinfin]Φinfin] + [η and Φinfin]= [γinfinand[Φinfinandη]]minus[ηand[Φinfinandγinfin]]+[ηandΦinfin] = [γinfinand[Φinfinandη]]+[ηandϕinfin] = 0

since ϕinfin = 12qqΦinfin and [η andΦinfin] = 0 Furthermore

dAinfin η + [Ainfin and η] = dAinfin[η and γinfin] + [Ainfin and η]= [dAinfinη and γinfin] minus [η and dAinfinγinfin] + [Ainfin and η] = 0

using dAinfinη = 0 and dAinfinγinfin = Ainfin By definition Ainfin + η = dAinfin+ηγinfin is

pure gauge which means that (Ainfin + η Φinfin) is horizontal with respect tothe Gauszlig-Manin connection


As before applying γinfin to Φinfin gives the gauge equivalent infinitesimaldeformation (0ϕinfin) of (Ainfin + ηΦinfin) The following is then an immediateconsequence of the fact that the Hitchin fibration is a Riemannian submer-sion

Corollary 310 One has

gsf(qhor qhor) = 990124X995852ϕinfin9958522 dA

where qhor denotes the horizontal lift of q isinH0(K2X)

33 Vertical directions Now fix q isin H0(K2X) and (AinfinΦinfin) isin πminus1(q)

As we have remarked up to gauge any element in πminus1(q) takes the form(Ainfin+ηΦinfin) where η isin Ω1(LΦinfin) satisfies dAinfinη = 0 The infinitesimal gaugeaction shifts η by dAinfinγ γ isin Ω0(LΦinfin) Hence the vertical tangent space isidentified with the cohomology space

H1(LΦinfin) =ker(dAinfin ∶Ω1(LΦinfin)rarr Ω2(LΦinfin))im (dAinfin ∶Ω0(LΦinfin)rarr Ω1(LΦinfin))

Each class in H1(XtimesLΦinfin) possesses a distinguished closed and coclosedL2 representative αinfin By Lemma 34 and Corollary 35 αinfin is the restric-tion of the unique harmonic representative of the corresponding class inH1(Sq iR)odd

Lemma 311 If (Ainfin Φinfin) = (αinfin0) where αinfin isin Ω1(LΦinfin) is the harmonicrepresentative then

dlowastAinfinAinfin minus 2πskew(i lowast [Φlowastinfin and Φinfin]) = 0

Proof This is a trivial consequence of αinfin being coclosed and Φinfin = 0 Proposition 312 If αinfin is as above then

gsf(αinfinαinfin) = 990124X995852αinfin9958522dA

Proof This follows from the above discussion along with Equation (9) 34 Mixed terms

Lemma 313 If vhor = (Ainfin Φinfin) is the horizontal lift of q isin H0(K2X) and

wvert = (αinfin0) is a vertical tangent vector with η harmonic then

⟨vhor wvert⟩ equiv 0pointwise Therefore the L2 inner product of these two vectors vanishesHence the off-diagonal parts of the L2 inner product and the semiflat innerproduct agree

Proof The gauged tangent vector corresponding to a horizontal deforma-tion (Ainfin Φinfin) is of the form (0ϕinfin) while the gauged tangent vector corre-sponding to a vertical deformation is of the form (αinfin0) These are clearlyorthogonal pointwise On the other hand the orthogonality of vertical andhorizontal tangent vectors in the semiflat metric is part of the definition


4 The approximate moduli space

Our goal is to understand the asymptotics of the L2 metric on the opensubsetM984094 of the Hitchin moduli space In this section we recall and slightlyrecast the construction of approximate solutions from [MSWW14] in termsof parametrized families of data and solutions and then use these familiesto define and study the L2 metric onM984094

In more detail consider a smooth slice Sinfin in the lsquopremoduli spacersquo PM984094infin

which consists of the solutions to the uncoupled Hitchin equations beforepassing to the quotient by unitary gauge transformations The slice Sinfin givesa coordinate chart onM984094

infin The construction in [MSWW14] produces fromthe elements in Sinfin a smooth family of approximate solutions Sapp of theself-duality equations and then perturbs each element of Sapp to an exactsolution We add to this cf the discussion in sect10 the observation that thisfinal perturbation map is smooth in these parameters so we obtain a slice Sin the space of solutions to the Hitchin equations which in turn correspondsto a coordinate chart inM984094

In the previous section we studied the L2 inner products of renormalizedgauged tangent vectors on PM984094

infin and showed that these correspond preciselyto the inner products for the semiflat metric The construction above yieldstangent vectors initially to the slice Sapp and then to the slice S To analyzethe L2 metric we first put these tangent vectors into Coulomb gauge andthen compute the appropriate integrals defining the metric Each of thesesteps introduces correction terms to gsf The next four sections containdetails of this for pairs of tangent vectors to the approximate moduli spacewhich are respectively horizontal radial vertical and lsquomixedrsquo The maincorrection terms arise here The final sect10 shows that only an exponentiallysmall further correction is introduced when passing from the approximateto the true moduli space

The construction of an approximate solution is based on a gluing con-struction In the initial step a limiting configuration Sinfin = (AinfinΦinfin) ismodified in a neighborhood of each zero of q = detΦinfin by replacing itthere with a desingularizing lsquofiducialrsquo solution (Afid

t Φfidt ) This yields a

pair Sappt = (Aapp

t Φappt ) which is an approximate solution for the Hitchin

equations in the sense that micro(Sappt ) = O(eminusβt) for some β gt 0 It is straight-

forward to check that this construction may be done smoothly in all pa-rameters Thus from a smooth finite dimensional family Sinfin of limitingconfigurations transverse to the gauge orbits we obtain a smooth finite di-mensional family of fields Sapp We think of this family as a submanifold ofa premoduli space (PMapp)984094 of approximate solutions which hence deter-mines a coordinate chart in the approximate moduli space (Mapp)984094 Sincethis discussion is local in the moduli spaces we may work entirely with theseslices and so do not need to define this approximate moduli space carefullyFor convenience however we shall frequently refer to tangent vectors to


(Mapp)984094 which are tangent vectors to Sapp which have been further mod-ified to satisfy the gauge condition All of this is done of course only insome fixed neighborhood of infinity in the Hitchin base B984094capq ∶ 995858q995858L1 ge t20

To be more specific fix q isin B984094 and let (AinfinΦinfin) denote the unique limitingconfiguration for the Hitchin section with detΦinfin = q By (16) a generallimiting configuration takes the form (Ainfin + ηΦinfin) where η is a suitabledAinfin-closed 1-form commuting with Φinfin The connection Ainfin is flat and hasnontrivial monodromy around each zero of q hence H1(Dtimes dAinfin) = 0 cf[MSWW14 Eq (32)] Thus η = dAinfinγ on each such punctured disk As

follows from [MSWW14 Prop 47] 995852γ995852 = O(r19957232) Therefore we may modifyAinfin+η by an exact LΦinfin-valued 1-form so as to assume that η equiv 0 on 995927pisinpDp

Following [MSWW14 sect32] we define the family of desingularizationsSappt ∶= (Aapp

t + η tΦappt ) by

Aappt = AH + 99573412 + χ(995852q995852k)(4ft(995852q995852k) minus

12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝

0 995852q995852minus19957232k eminusχ(995852q995852k)ht(995852q995852k)q

995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)

Here ht(r) is the unique solution to (rpartr)2ht = 8t2r3 sinh2ht on R+ withspecific asymptotic properties at 0 and infin and ft ∶= 1

8 +14rpartrht Further

χ ∶ R+ rarr [01] is a suitable cutoff-function The parameter t can be removed

from the equation for ht by substituting ρ = 83 tr

39957232 thus if we set ht(r) =ψ(ρ) and note that rpartr = 3

2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ

This is a Painleve III equation there exists a unique solution which decaysexponentially as ρ rarr infin and with asymptotics as ρ rarr 0 ensuring that Aapp

tand Φapp

t are regular at r = 0 More specifically

995176 ψ(ρ) sim minus log(ρ19957233 995734suminfinj=0 ajρ4j9957233995739 ρ984100 0

995176 ψ(ρ) simK0(ρ) sim ρminus19957232eminusρsuminfinj=0 bjρminusj ρ984098infin

995176 ψ(ρ) is monotonically decreasing (and strictly positive) for ρ gt 0

These are asymptotic expansions in the classical sense ie the differencebetween the function and the first N terms decays like the next term inthe series and there are corresponding expansions for each derivative Thefunction K0(ρ) is the Bessel function of imaginary argument of order 0

In the following result and for the rest of the paper any constant C whichappears in an estimate is assumed to be independent of t

Lemma 41 [MSWW14 Lemma 34] The functions ft(r) and ht(r) havethe following properties


(i) As a function of r ft has a double zero at r = 0 and increases monoton-ically from ft(0) = 0 to the limiting value 19957238 as r 984098infin In particular0 le ft le 1

8 (ii) As a function of t ft is also monotone increasing Further limt984098infin ft =

finfin equiv 18 uniformly in Cinfin on any half-line [r0infin) for r0 gt 0

(iii) There are estimates

suprgt0

rminus1ft(r) le Ct29957233 and suprgt0

rminus2ft(r) le Ct49957233

(iv) When t is fixed and r 984100 0 then ht(r) sim minus12 log r+b0+ where b0 is an

explicit constant On the other hand 995852ht(r)995852 le C exp(minus83 tr

39957232)995723(tr39957232)19957232for t ge t0 gt 0 r ge r0 gt 0

(v) Finally

suprisin(01)

r19957232eplusmnht(r) le C t ge 1

It follows from the results in [MSWW14] that the approximate solutionSappt satisfies the self-duality equations up to an exponentially decaying error

as trarrinfin and there is an exact solution (AtΦt) in its complex gauge orbit(unique up to real gauge transformations) which is no further than Ceminusβt

pointwise away for some β gt 0

5 Gauge correction

The L2 metric is defined in terms of infinitesimal deformations which areorthogonal to the gauge group action An arbitrary tangent vector can bebrought into this form by solving the gauge-fixing equation on all of X Wefirst describe gauge-fixing in general and then estimate the gauge correctionterm in this particular instance

At the end of sect242 we introduced the deformation complex and its dif-ferentialsD1

(AΦ) andD2(AΦ) as well as the condition (11) for an infinitesimal

deformation (A Φ) to be in gauge

Lemma 51 (Infinitesimal gauge fixing) If (A Φ) is an infinitesimal de-formation of a solution (AΦ) to the Hitchin equations then there exists a

unique ξ isin Ω0(su(E)) such that (A Φ) minusD1(AΦ)ξ is in gauge The same is

true if (AΦ) is sufficiently close to a solution to the Hitchin equations

Proof First suppose that micro(AΦ) = 0 The transformed pair (A minus dAξ Φ minus[Φ and ξ]) is in gauge if and only if

(D1(AΦ))

lowast((A Φ) minusD1(AΦ)ξ) = 0

or equivalently

(21) L(AΦ)ξ = dlowastAA minus 2πskew(i lowast [Φlowast and Φ])where

(22) L(AΦ) ∶= (D1(AΦ))

lowastD1(AΦ) =∆A minus 2πskew(i lowast [Φlowast and [Φ and sdot]])


This operator already played a role in [MSWW14] albeit acting on isu(E)rather than su(E) Now

⟨Lξ ξ⟩ = 995858dAξ9958582 + 2995858 [Φ and ξ] 9958582so solutions to Lξ = 0 are parallel and commute with Φ But as alreadyused in [MSWW14] if q = detΦ is simple then the solution (AΦ) must beirreducible This implies that L is bijective and so (21) admits a uniquesolution

If (AΦ) is sufficiently close to an exact solution then L(AΦ) remainsinvertible and hence the conclusion is true then as well

For an approximate solution Sappt = (Aapp

t tΦappt ) define

Mtξ ∶=MΦappt

ξ ∶= minus2πskew(i lowast [(Φappt )

lowast and [Φappt and ξ]])

and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt

ξ + [η and ξ] t[Φappt ξ])

Ltξ ∶= (D1t )lowastD1

t ξ =∆Aappt +ηξ minus 2t2πskew(i lowast [(Φapp

t )lowast and [Φapp

t and ξ]])

Note that for any pair (At tΦt)Lt =∆At + t2Mt

51 Analysis of Lminus1t We now study the inverse Gt = Lminus1t recalling from[MSWW14 Proposition 52] that Lt is uniformly invertible when t is large

(23) 995858Gtf995858L2(X) le C995858f995858L2(X)

where C does not depend on t This estimate controls the size of the gauge-fixing terms below However we require finer information about these termsso we now examine the structure and mapping properties of this inverse moreclosely

By construction the approximate solution (Aappt tΦapp

t ) is precisely equalto a fiducial solution inside each Dp This simplifies the results and argu-ments below though these all have analogues if this is not the case egwhen (A tΦ) is an exact solution

We first examine the scaling properties of the operator Lt in each Dp Set

983172 = t29957233r (note the difference with the previous change of variables ρ = 83 tr

39957232

used earlier) The coefficients of At depend only on 983172 and the dθ in At

does not need to be transformed Write ∆At = rminus2995779∆t where 995779∆t = minus(rpartr)2 +(minusipartθ + a(t29957233r))2 for some hermitian matrix a Now rpartr = 983172part983172 so 995779∆t can

be reexpressed (in Dp) as an operator 995779∆ρ which depends on (983172 θ) but not

on t The prefactor rminus2 equals t49957233983172minus2 so

∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172

The second term t2Mt appearing in Lt behaves similarly Indeed thematrix entries of Φt and Φlowastt equal r19957232 times functions of t29957233r = 983172 so that

t2Mt = t2r995779Mρ ∶= t49957233M983172


where M983172 = ρ995779M983172 is an endomorphism with coefficients depending only on(983172 θ)

Altogether in each Dp

(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172

The operator L983172 is smooth on R2 and converges exponentially quickly asρrarrinfin to

(25) Linfin =∆infin +Minfin

here ∆infin is the Laplacian for Afidinfin and Minfin = minus2πskew(ilowast[(Φfid

infin )lowastand[Φfidinfin andsdot]])

both expressed in terms of 983172It follows from (24) that if we consider the operator Lt evaluated at a

fiducial solution (Afidt Φfid

t ) acting on some space of fields (with specifieddecay) on the entire plane R2 then the Schwartz kernel of its inverse Gfid

t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)

(Note that we might expect an additional factor of tminus49957233 on the right side ofthis equation this actually does appear because of the homogeneity of thestandard Lebesgue measure dσ(z) on C cf also the proof of Proposition 53below) To check this we calculate

LtGfidt (z z) = t49957233(L983172G983172)(t29957233z t29957233z) = t49957233δ(t29957233z minus t29957233z) = δ(z minus z)

since the delta function in two dimensions is homogeneous of degree minus2We next check that Gfid

t is uniformly bounded in L2 for t ge 1 (and indeed

its norm decreases as trarrinfin) To this end define (Utf)(w) = tminus29957233f(tminus29957233w)so that Ut ∶ L2(dσ(z))rarr L2(dσ(w)) is unitary for all t We then write

u(z) = Gfidt f(z) = 990124 G983172(t29957233z t29957233z)f(z)dσ(z)

= tminus29957233990124 G983172(t29957233z w)(Utf)(w)dσ(w)

so that

(Utu)(w) = tminus49957233G983172(Utf)(w)or finally

Gfidt = tminus49957233Uminus1t G983172Ut

which proves the claimWe define X 984094 ∶=X ∖995927pisinp Dp and refer to this set as the exterior region in

the following If (AinfinΦinfin) is the limiting configuration used in the approx-imate solution Sapp

t let Gext denote an inverse (or even just a parametrixup to smoothing error) for the corresponding operator Linfin on the exteriorregion Writing Dp(a) for the disk of radius a around p choose a partition

of unity χ1χ2 subordinate to the open cover 995927Dp and X ∖ 995927Dp(79957238)Choose two further cutoff functions χ1 and χ2 so that χj = 1 on the support


of χj and with supp χ1 sub 995927Dp supp χ2 sub X ∖ 995927Dp(39957234) Then define theparametrix for Lt

Gt = χ1Gfidt χ1 + χ2G

extχ2

As an equation of distributions on X timesX

GtLt = Id minusRt

this remainder term

Rt = χ1Gfidt [Ltχ1] + χ2G

ext[Ltχ2] + χ2Rextχ2

is a smoothing operator indeed the support of χj(z) does not intersect thesupport of 984162χj(z) j = 12 and the Green functions are singular only alongthe diagonal so the first two terms have smooth kernels The remainingterm Rext is the smoothing error GextLt = Id minusRext

Suppose now that ut and ft satisfy Ltut = ft or equivalently ut = GtftApplying Gt to ft instead gives that

(27) ut = Gtft +Rtut

We are interested in two specific mapping properties The first one whenft is supported in the exterior region outside the disks and the second whenft is supported in one of these balls and has the form ft(r θ) = f(t29957233r θ)We consider these in turn

Proposition 52 Suppose that Ltut = f where f is Cinfin and supported inthe exterior region X 984094 Then for any k ge 0 995858u995858Hk+2(X) le Ctm995858f995858Hk(X)where m =m(k) gt 0 and C is independent of t

Proof Since Lminus1t ∶ L2 rarr L2 is bounded uniformly for t ge 1 we have 995858ut995858L2 leC995858f995858L2 (on all of X) where C is independent of t Next the coefficients of∆At = Lt minus t2MΦt and of MΦt are uniformly bounded in Cinfin on X 984094 so em-ploying local elliptic estimates there and using the estimate above for the L2

norm of ut shows that 995858ut995858Hk+2(X984094) le Ct2995858f995858Hk(X) again with C indepen-dent of t We turn this estimate into one over Dp as follows We first extendut from X 984094 to a function vt on X such that 995858vt995858Hk+2(X) le Ct2995858f995858Hk(X)In particular the difference wt ∶= ut minus vt satisfies Dirichlet boundary condi-tions on Dp and vanishes on X 984094 Also the restriction to Dp of wt satisfiesLtwt = minusLtvt Because the coefficients of the operator Lt are polynomiallybounded in t it follows that 995858Ltwt995858Hk(Dp) le Ctm1995858f995858Hk(X) for some m1 =m1(k) ge 2 Arguing now exactly as in the proof of [MSWW14 Proposition52 (ii)] it follows that 995858wt995858Hk+2(Dp) le Ctm995858f995858Hk(X) for some further con-

stant m =m(k) gem1 Therefore 995858ut995858Hk+2(X) le 995858wt995858Hk+2(X) + 995858vt995858Hk+2(X) leCtm995858f995858Hk(X) proving the claim

We now come to a key concept The class of functions (or fields) whicharise in the rest of this paper have the property that they decay exponentiallyas t rarr infin away from the zeroes of q but concentrate with respect to thenatural dilation near each of these zeroes We call the building blocks ofsuch functions exponential packets


Definition 51 A family of functions microt(z) on R2 is called an exponential

packet if it is of the form microt(z) = (t29957233995852z995852)τmicro(t29957233z) where995176 microt(z) = micro(t29957233z) where micro(w) is smooth and decays like eminusβ995852w995852

39957232along

with all of its derivatives for some β gt 0995176 τ gt 0

An exponential packet of weight σ is a function of the form tσmicrot(z) whereσ isin R and microt(z) is an exponential packet Finally we say simply thata function microt on X is a convergent sum of exponential packets if in thestandard holomorphic coordinate in each Dp it is a Cinfin convergent sum of

exponential packets and decays like eminusβt for some β gt 0 along with all itsderivatives outside of the Dp If the exponential packets involve factors of

(t29957233995852z995852)τ as above then the sense in which these sums converge must bemodified In the applications below we shall only encounter the same extrafactor (t29957233995852z995852)19957232 in all terms of the sum so it may be simply pulled out ofthe sum

Proposition 53 Suppose that ft(z) is an exponential packet supported in

some Dp Then ut = Gtft is an exponential packet tminus49957233microt(t29957233z) of weightminus43

Proof We have

990124 Gfidt (z z)f(t29957233z)dσ(z) = tminus49957233990124 Gfid

t (z tminus29957233w)f(w)dσ(w)

Thus if we set w = t29957233z then the right hand side equals

tminus49957233990124 Gfidt (tminus29957233w tminus29957233w)f(w)dσ(w)995852w=t29957233z = t

minus49957233microt(z)

This computation shows thatGfidt ft is exponentially small outside of Dp(19957232)

sayNow fix a cutoff function χ which equals 1 in Dp(39957234) and which vanishes

outside Dp(79957238) and set ut = χGfidt ft (In other words we localize the

function Gfidt f from R2 to the disk) Then

Lt(ut minus ut) = [Ltχ]Gfidt ft + χft minus ft ∶= ht

The calculation above shows that ht decays exponentially Hence writingut = ut minus vt then vt = Gtht decays exponentially first in any Sobolev normthen in Cinfin This proves the result

The preceding results now give the following useful result

Corollary 54 If ft is a convergent sum of exponential packets then ut =Gtft is also a convergent sum of exponential packets More precisely

ft =990118j

tσminus2j9957233fjt +O(eminusβt)995278rArr ut =990118j

tσminus49957233minus2j9957233ujt +O(eminusβt)


52 Smooth dependence on parameters The considerations above willbe applied in the next sections to prove the existence of expansions as trarrinfinfor the various components of the L2 metric An important addendum is thatthese are true polyhomogeneous expansions ie the derivatives with respectto various parameters of these metric coefficients have the correspondingdifferentiated expansions For certain derivatives eg those with respect tot this is not hard to deduce However it is much less obvious for derivativesin other directions particularly those with respect to q We now discuss thereasoning which will lead to this conclusion in all cases

The first key point is the fact that the spectral curve Sq varies smoothlyas q varies in B984094 This follows immediately from the nonsingularity of thedefining relation λ2

SW minus q = 0 when q lies away from the discriminant locusWe have also already described the normal vector field Nq arising from thevariation Sq+sq It is evident from the discussion in sect23 that Nq is tangentto the zero section 0 of KX at the intersection points Sq cap 0 ie at thezeroes of q

The second key point is that the (sums of) exponential packets encoun-tered below are mostly of a very special type in that they lift to restric-tions to Sq of globally defined functions on KX which decay exponentiallyalong the fibers To make this precise we define the class of global ex-ponential packets and their sums By definition a sum of global expo-nential packets is a function micro on the total space of KX which is smoothaway from the zero section has an integrable polyhomogeneous singular-ity at 0 and decays exponentially as 995852w995852 rarr infin in each fiber of KX Thelast two conditions here mean that in standard coordinates (zw) on KX micro(zw) sim summicroj(zargw)995852w995852γj as w rarr 0 where each microj is smooth and the

exponents γj rarr infin and 995852micro(zw)995852 le Ceminusβ995852w995852 as w rarr infin (The examples hereare all of the form γj = j or γj = j + 19957232 j isin N)

Proposition 55 Let micro be a convergent sum of global exponential packetson KX and microq the restriction of micro to the spectral curve Sq Then the familyof integrals

q 995207rarr 990124Sq

microq dA

has a convergent expansion as 995858q995858L2 rarr infin in B984094 which holds along with allits derivatives

Proof Let q vary along a transversal to the R+ action and consider thefunction

(t q)995207rarr 990124Stq

microtq dA = 990124tSq

microtq dA

The restrictions of these integrals to any fixed region 995852w995852 ge c gt 0 in KX decayexponentially in t uniformly as q varies in a small set Thus we may restrictto disks Di in Sq centered at the zeroes of q and write the correspondingintegrals in local coordinates For q fixed the integral of an exponentialpacket on a fixed disk is a monomial ctα for some α so the integral of a


convergent sum of exponential packets becomes a convergent sum of suchmonomials This is clearly polyhomogeneous in t The smoothness in t isalso straightforward from these local coordinate expressions

The smoothness in q is also now clear since the spectral curve variessmoothly with q There is one small point to mention however If micro has apolyhomogeneous singularity along the zero section we must use that thevariation of Sq is tangent to the zero section Indeed we can write thecontribution on the disk around q as an integral on a varying family of diskstransverse to the zero section in KX The derivative of this integral withrespect to q is then the integral of the derivative of micro with respect to thevariation vector field However micro is polyhomogeneous along the zero sectionso differentiating it with respect to vector fields tangent to the zero sectiondoes not change its regularity nor the form of its asymptotic expansion atthe zero section This implies that the derivative in q of the integral alongthis family of disks is smooth in q

6 Horizontal asymptotics of the L2-metric

In this and the next few sections we put into gauge the infinitesimaldeformations of the families of approximate solutions and then evaluate theL2 metric on these We begin now by considering the horizontal tangentvectors on (Mapp)984094

Henceforth fix an approximate solution

Sappt = (Aapp

t + η tΦappt ) isin (M

app)984094Now consider the variations of (19) and (20) with respect to q

Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq

qIm part log 995852q995852k minus 2ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝

0 eminusht(995852q995852k)995852q995852minus12

k (q minus qQ)eht(995852q995852k)995852q99585219957232k Q 0

⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp

t + η tΦappt ) η = [η and γinfin] is

tangent to (Mapp)984094 at Sappt cf Lemma 39

The gauge-correction is a two-step process First we employ an infini-tesimal gauge-transformation adapted to the local structure of Sapp

t nearthe zeroes of q The remaining correction term is found using the globalmethods from sect5

61 Initial gauge correction step The infinitesimal gauge transforma-tion

γt ∶= minus2ft(995852q995852k) Imq

q995738i 00 minusi995742


is the obvious desingularization of the field γinfin used in sect3 to remove themain singularity of the limiting configuration We thus define

(αt tϕt) ∶= (Aappt + η tΦapp

t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly

αt ∶= Aappt + η minus dAapp

t +ηγt

tϕt ∶= tΦappt minus t[Φapp

t and γt](30)

This is a tangent vector to a small perturbation of a point in (Mapp)984094 atradius t so it is natural to rescale this tangent vector by a factor of t andshow that it converges as t rarr infin In other words we consider convergenceof the pair (tminus1αtϕt) Since γt rarr γinfin in Cinfin away from the zeroes of q wesee that

(tminus1αtϕt)rarr (0ϕinfin) = (Ainfin Φinfin) minusD1Sinfinγinfin as trarrinfin

(In fact αt tends to 0 away from each Dp even without the extra factor oftminus1) Direct calculation shows that this pair is closer by a factor tminusm m gt 0to being in gauge than (Aapp

t tΦappt )

We now examine αt and ϕt more closely First

dAappt +ηγt = [η and γt] minus 2995735f 984094t(995852q995852k) Im

q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742

whence recalling that η = [η and γinfin]

αt = Aappt + η minus dAapp

t +ηγt

= [η and (γinfin minus γt)] + 4f 984094t(995852q995852k) Imq

qd995852q995852k 995738

i 00 minusi995742

(31)

As for the other term

[Φappt and γt] = 4ift(995852q995852k) Im

q

q

⎛⎝

0 995852q995852minus12

k eminusht(995852q995852k)q

minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that

ϕt = Φappt minus [Φapp

t and γt]

=⎛⎜⎝

0 99573512 minus 995852q995852kh984094t(995852q995852k)995740eminusht(995852q995852k)995852q995852minus

12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)

We next analyze the asymptotics of the family (tminus1αtϕt) in each disk Dp

Proposition 61 Fix ϕinfin ne 0 as in (15) Then in each disk Dp

tminus1αt =infin990118j=0

Ajtt(1minus2j)9957233


and

ϕt minusϕinfin =infin990118j=0

Bjtt(1minus2j)9957233

as t rarr infin where the coefficients Ajt and Bjt are exponential packets andthe sum is convergent Outside the union of the disks Dp

995852tminus1αt995852 + 995852ϕt minusϕinfin995852 le Ceminusβt

Proof The exponential decay outside the Dp is clear so we focus on thebehavior inside one of the disks With a holomorphic coordinate z for whichq = zdz2 we have q = fdz2 for some holomorphic f We assume further thatH is the standard flat metric on the local holomorphic frame dzplusmn19957232 andthat η vanishes on Dp Then in this region

αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝

0 995734(12 minus rh984094t(r))eminusht(r) minus 1

2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)

We now recall that ft ht and (rpartr)ht are all functions of ρ = tr39957232 and satisfy

ft(ρ) rarr 19957238 and ht(ρ) le Ceminusβρ A brief calculation shows that f 984094t(r) is t29957233times a smooth exponentially decreasing function of ρ The assertions nowfollow once we expand f in a Taylor series and write each rj as (t29957233r)jtminus2j9957233in the expression for αt and rjminus19957232 = (t29957233r)jminus19957232t(1minus2j)9957233 in the expressionfor ϕt minusϕinfin

We briefly describe the regularity of the coefficients in (33) when pulledback to the spectral curve

First up to constant multiples the coefficients in αt have the form

f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522

where we consider the right side as a function of λ isin KX However ft(r)has a double zero hence f 984094t(r) vanishes at r = 0 so f 984094t(995852λ9958522) vanishes to order2 and altogether this expression has a simple zero at the zero section

On the other hand the upper right coefficient in ϕt minusϕinfin has the form

microt(995852q995852k)995852q995852minus19957232k q = microt(995852λ9958522)995852λ995852

q

where microt is an exponential packet This has a simple pole at the zero sectionof KX and as we now check its restriction to the spectral curve is boundedIndeed choose the usual coordinate w2 = z so q = fdz2 = 4fw2dw2 and

λ = wdz = 2w2dw These give that q995723995852λ995852 = 2f w2

995852w9958522995852dw995852dw2 The discussion for

the coefficient in the lower left is analogous


In either case the terms are global exponential packets of precisely thesort considered in Proposition 55

62 Second gauge correction step Following (21) we now solve

(34) Ltξt = Rt ∶= dlowastAappt +ηαt minus 2t2πskew(i lowast [(Φapp

t )lowast andϕt])

Lemma 62 The error term Rt is a convergent sum of exponential packetsof weights 2 minus 2j9957233 in each Dp

Rt =infin990118j=0

t2minus2j9957233kjt(z)995738i 00 minusi995742 kjt(z) = kj(t29957233z)

Proof As before choose a holomorphic coordinate z in Dp so that q = zdz2and assume that hermitian metric is trivial on the frame dzplusmn19957232 Followingthe discussion in sect4 assume also that η and hence η = [ηandγinfin] both vanishon Dp

Using (33) we calculate that

dlowastAappt

αt = 4dlowast995734f 984094t(r) Im(f995723z)dr995739 995738i 00 minusi995742

= 4995734minuspartr(f 984094t(r)rminus1) minus f 984094t(r)rminus2 minus (f 984094t(r)rminus2)rpartr995739 Im(eminusiθf)995738i 00 minusi995742

This can then be simplified using

f 984094t(r)rminus2 = 2t2 sinh(2ht(r)) and

partr(f 984094t(r)rminus1) = partr(2t2r sinh(2ht(r))) = 2t2(1 + rpartr) sinh(2ht(r))In addition

minus 2t2πskew(i lowast [(Φappt )

lowast andϕt]) =

4t2Re(ieminusiθf) (sinh(2ht) + 2(rpartrht) cosh(2ht))995738i 00 minusi995742

The rest of the argument is exactly as in the proof of (61) We now invoke the detailed mapping properties for Lminus1t = Gt from Propo-

sitions 52 and 53 and Corollary 54 to conclude the following

Proposition 63 The gauge correction field ξt is a convergent sum of ex-ponential packets plus an exponentially small remainder term

ξt =infin990118j=0

ξjt(z)t(2minus2j)9957233 +O(eminusβt) ξjt(z) = χj(t29957233z)

and hence the actual gauge correction term D1t ξt is also of this type

(35) D1t ξt =

infin990118j=0

ηjt(z)t(4minus2j)9957233 +O(eminusβt) ηjt(z) = ηj(t29957233z)


Note that we must also include the scaling by tminus1 ie the gauge correctionof (tminus1αtϕt) is tminus1D1

t ξt which is a sum of exponential packets starting with

t19957233η0tThe relationship between the gauged infinitesimal deformations to the

approximate moduli space and to the space of limiting configurations isthen

(36) (tminus1αtϕt) minus tminus1D1t ξt = (0ϕinfin) +

infin990118j=0

Cjt(1minus2j)9957233 +O(eminusβt)

and hence(37)995858(tminus1αtϕt) minus tminus1D1

t ξt9958582L2

= 995858ϕinfin9958582L2 + 2⟨ϕinfininfin990118j=0

Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0

Cjt(1minus2j)99572339958582L2 +O(eminusβt)

= 995858ϕinfin9958582L2 +infin990118j=0

Sjtminus(2+j)9957233 +O(eminusβt)

The shift by the factor tminus49957233 in the final series is due to the Jacobian factorin the integration This same shift appears several times below

This is the equation which expresses the difference between the metriccoefficients for the Hitchin and semiflat metrics in this particular directionBy polarization we can obtain a similar expansion for the mixed horizontal

metric coefficients Thus if (vhor)(j) = (A(j)infin + η(j) Φ(j)infin minusD1t (γ

(j)t + ξ(j)t ))

j = 12 are two different gauged horizontal deformations then

tminus2⟨(vhor)(1) (vhor)(2)⟩L2

= tminus2⟨(vhor)(1) (vhor)(2)⟩sf +infin990118j=0

S984094j((vhor)(1) (vhor)(2))tminus(2+j)9957233

where the S984094j are symmetric 2-tensors on horizontal tangent vectors whichare independent of t

Proposition 55 ensures that all expansions here may be differentiatedso that these are lsquoclassicalrsquo expansions (cf the discussion preceding Lemma41) for the horizontal part of the metric

Observe from Propositions 61 and 63 that the two terms (tminus1αtϕtminusϕinfin)and tminus1D1

t ξt are both sums of exponential packets with the same leading

order exponent t19957233 This leaves open the possibility of some unexpectedcancellations so that S0 and perhaps some or all of the remaining Sj mightvanish

As already mentioned in the introduction it has emerged in very recentwork by David Dumas and Andy Neitzke that this cancellation actuallydoes occur at least along the Hitchin section and in horizontal directionsTheir paper [DN] presents a beautiful formula which proves that the integralexpressing the difference between the semiflat and Hitchin metrics for the


model case of the Hitchin section over C actually vanishes This relies ona very interesting integral identity the full meaning of which is not yetclear These authors go on to prove that the rate of convergence for thehorizontal metric coefficients over the Hitchin section on a general surfaceX is exponential

7 Asymptotics in the radial direction

Amongst the horizontal directions already analyzed in sect6 the radial di-rection is distinguished This is of course the direction where q = q so inparticular the term q995723q appearing in many formulaelig in that section equals 1

Let (Ainfin + ηΦinfin) be a limiting configuration associated with q (normal-ized so that intX 995852q995852 = 1) and (Aapp

t + ηΦappt ) the corresponding family of

approximate solutions Then from (15) and the fact that Im(q995723q) = 0 weobtain αt = 0

ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝

0 (12 minus 995852q995852kh984094t(995852q995852k))eminusht(995852q995852k)q995723995852q99585219957232k

(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠

Subtracting these or more simply using q995723q = 1 in (33) we have

ϕt minusϕinfin =

⎛⎝

0 995734(12 minus 995852q995852kh984094t(995852q995852k))eminusht(995852q995852k) minus 1

2995739q995723995852q99585219957232k

995734(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k) minus 1

2995739995852q995852

12

k 0

⎞⎠dz

Previously the infinite Laurent expansion of q995723q led to an infinite sum ofweighted exponential packets while here each of the two nonzero entries inϕt minusϕinfin is a single weighted exponential packet

We summarize these observations in the following proposition

Proposition 71 This difference has the form

ϕt minusϕinfin = ϕt minus 12Φinfin = t

minus19957233 995738 0 A1t (z)

B1t (z) 0

995742 +O(eminusβt)

where the two off-diagonal terms are exponential packets of weight minus19957233

Next we put (0ϕt) into Coulomb gauge by solving

Ltξt = Rt ∶= minus2tπskew(i lowast [(Φappt )

lowast andϕt])

Notice that the approximate solution (Aappt +η tΦ

appt ) is altered by ϕt rather

than tϕt which explains why there is only the single factor t rather thanthe factor t2 in (34)


Inspecting the terms above and also rewriting ϕt = ϕinfin + (ϕt minus ϕinfin)Φfidt = Φinfin + (Φfid

t minus Φinfin) so as to take advantage of normality at t = infin weobtain

Proposition 72 The error term Rt is a diagonal exponential packet ofweight 19957233

Rt = t19957233Ht +O(eminusβt)Consequently ξt = tminus1Jt + O(eminusβt) and D1

t ξt = tminus19957233Kt + O(eminusβt) where Jtand Kt are exponential packets

Proposition 73 The L2 metric on a radial tangent vector has the follow-ing expansion

995858(0ϕt) minusD1t ξt9958582L2 = 995858ϕinfin9958582L2 + atminus59957233 +O(eminusβt)

Note finally that by (8) 995858ϕinfin9958582L2 = 995858tq9958582sK at the point t2q and this equals19957234 provided intX 995852q995852 = 1

8 Asymptotics in fiber directions

We now consider variations in the fiber directions Just as in the previ-ous section we first compute the infinitesimal deformations of approximatesolutions and then use a similar two-step correction to put these into gauge

Fix a limiting configuration which to simplify notation we write simplyas (AinfinΦinfin) rather than (Ainfin + ηΦinfin) even though it is not necessarilyin the Hitchin section By Proposition 33 and Corollary 35 a fiberwiseinfinitesimal deformation of (AinfinΦinfin) is an element of H1(XtimesLinfin) whichin turn is identified with a unique L2 harmonic representative in

H1(XtimesLinfin) = αinfin isin Ω1(Xtimes Linfin) ∶ plowastqαinfin isinH1(Sq iR)oddwhere pq ∶ Sq rarr Xtimes is the spectral cover We use the notation that the

complex line bundle LCinfin = γ isin sl(E) 995852 [Φinfin and γ] = 0 on Xtimes splits into the

sum of real line bundles Linfin ∶= LCinfin cap su(E) and iLinfin of skew-hermitian and

hermitian elements respectivelyWe first replace this infinitesimal deformation with one supported in the

union of annuli Ap ∶= Dp ∖Dp(19957232)

Lemma 81 For each αinfin isin H1(XtimesLinfin) there exists ξinfin isin Ω0(Xtimes Linfin)with supp ξinfin sub 995927pisinpDp and ξinfin(z) = suminfinj=0 ξinfinjr

j+19957232 near each p so that

(38) βinfin ∶= αinfin minus dAinfinξinfinis supported outside each Dp(19957232) Furthermore dAinfinβinfin = 0 and R ∶=dlowastAinfinβinfin isin Ω

0(XtimesLinfin) is supported in 995927pisinpAp

Proof Choose coordinates on each Dp such that z = pq(w) = w2 Thenplowastqαinfin = fdw minus fdw where f is holomorphic and even with respect to theinvolution σ(w) = minusw (observe that dw dw are odd with respect to σ) Wethen choose a local primitive F (w) for f(w) which by replacing F (w) by


(F (w) minus F (minusw))9957232 we may as well assume to be odd and this then gives alocal primitive ξinfin995852Dp for αinfin by

plowastqξinfin995852Dp = F minus F

Since F is odd 995852F (w)995852 = O(995852w995852) so ξinfin995852Dp = O(r19957232)Now patch these local primitives ξinfin995852Dp together to obtain ξinfin using smooth

cutoff functions with gradients supported in 995927pisinpAp The assertions aboutthe supports of ξinfin and βinfin are now obvious Since dAinfinαinfin = 0 and FAinfin = 0we obtain dAinfinβinfin = 0 Last since dlowastAinfinαinfin = 0 we see that R = minusdlowastAinfindAinfinξinfinhas support in 995927pisinpAp

We can view βinfin as an ungauged tangent vector to the space of approx-

imate solutions at (Aappt Φapp

t ) Indeed (Aappt Φapp

t ) = (AinfinΦinfin)gappt for a

(singular) complex gauge transformation which we can assume equals theidentity outside each Dp(19957232) Hence its differential preserves βinfin Thisyields for each t the gauged tangent vector

(39) (αtϕt) ∶= (βinfin0) minusD1t ξt

where ξt isin Ω0(su(E)) is the unique solution to

(40) Ltξt = (D1t )lowast(βinfin0) = dlowastAinfin βinfin = E

where we have assumed without loss of generality that Aappt = Ainfin on

suppβinfin To estimate ξt we write ξt = (ξinfin + ξt) minus ξinfin and consider theequivalent equation

(41) Lt(ξt + ξinfin) = Rt

whereRt = R + Ltξinfin = Ltξinfin minus∆Ainfinξinfin

However recall that Linfinξinfin = ∆Ainfinξinfin since ξinfin commutes with Φinfin anddlowastAinfinξinfin = 0 Thus

Rt = (Lt minusLinfin)ξinfin = (∆At minus∆Ainfin)ξinfin + t2(MΦt minusMΦinfin)ξinfin

Proposition 82 Both

Rt =infin990118j=1

ρjt(z)t1minus2j9957233+O(eminusβt) and ξt+ξinfin =infin990118j=0

bjt(z)t(minus1minus2j)9957233+O(eminusβt)

are convergent sums of exponential packets of weights 1minus2j9957233 and (minus1minus2j)9957233respectively

Proof As in Lemma 81 near each p

ξinfin =infin990118j=0

ξinfinj(z)t(minus1minus2j)9957233

where the coefficients are independent of t Next following the rescalingcalculation in sect51

Lt minus (∆infin + t2Minfin) = t49957233((L983172 minusLinfin) + (M983172 minusMinfin))


and the coefficients of ∆infin minus ∆983172 and Minfin minusM983172 are weighted exponentialpackets The conclusions then follow immediately

We next analyze the difference between the initial vertical tangent vector(αinfin0) and the gauged one


Proposition 83 The difference (αtϕt) minus (αinfin0) is a convergent sum ofexponential packets

(αtϕt) minus (αinfin0) =infin990118j=0

cjt(z)t(1minus2j)9957233 +O(eminusβt)

Proof By (39)

(αtϕt) = (βinfin0) minusD1t ξt = (αinfin0) minus (dAinfinξinfin0) minus (dAtξt t[Φt ξt])

Now write

dAinfinξinfin + dAtξt = (dAinfin minus dAt)ξinfin + dAt(ξinfin + ξt)

and observe also that

dAinfin minus dAt = minus9957342ft(r) minus 14995739995738i 0

0 minusi995742dθ

Since 2ft(r) minus 14 = η(983172) and 995852dθ995852 = r

minus1 = 983172minus29957233t29957233 we see that

(dAinfin minus dAt)ξinfin =infin990118j=1

bjt(z)t(1minus2k)9957233 +O(eminusβt)

is a sum of exponential packets and by Proposition 82 so is

dAt(ξinfin + ξt) =infin990118j=1


This shows that dAinfinξinfin + dAtξt has the correct formFor the other term note that [Φinfin ξinfin] = 0 so that

t[Φt ξinfin] = t[Φt minusΦinfin ξinfin]

and since this difference of Higgs fields is a weighted exponential packet thesame conclusion holds

Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0

Cjtminus(2+j)9957233 +O(eminusβt)

as trarrinfin in particular 995858(αtϕt)9958582L2 minus 995858(αinfin0)9958582L2 = O(tminus29957233)


If (α(j)t ϕ(j)t ) j = 12 are two gauged vertical tangent vectors then

⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233

We make some comments about why these expansions may be differen-tiated Note first that by construction ξinfin is smooth on Sq The termR = dlowastAinfinβinfin = minusd

lowastAinfin

dAinfinξinfin is smooth away from the zero section and has apolyhomogeneous singularity there The operator Lt varies smoothly withthe spectral curve and the derivatives of its coefficients with respect to tparttdo not change form As we have seen this ensures that the solution ξt toLtξt = R also has a smooth expansion This allows us to conclude that allthe expansions in this section may be differentiated

9 Asymptotics of mixed terms

The horizontal and vertical directions are orthogonal with respect to thesemiflat metric but the L2 metric has some nontrivial mixed terms Wenow study their asymptotics

We have proved above that if vhor and wvert are horizontal and verticaltangent vectors in Coulomb gauge above t2q then

1

tvhor = 1

tvhorinfin +

infin990118j=0

vjt(1minus2j)9957233 +O(eminusβt)

wvert = wvertinfin +

infin990118j=0

wjt(1minus2j)9957233 +O(eminusβt)

where the vj and wj are exponential packets In analyzing the inner productbetween a vertical and a horizontal vector the only terms potentially ofconcern are those of the form

990124X⟨microt(995852q995852k)995852q995852minus19957232k q η⟩

where microt is an exponential packet and η is one of the terms in the expan-sion of D1

t ξt To analyze such an expression write q = fdz2 = 4fw2dw2

995852q99585219957232k = 995852λ995852 = 995852w995852995852dz995852 = 2995852w9958522995852dw995852 and η = hdz2 = 4hw2dw2 Then the integrandbecomes

microt(995852q995852k)995852q995852minus19957232k ⟨q η⟩ = 8microt(995852q995852k)Re(f h)995852w9958522which is smooth on Sq and smooth as q varies

In summary we obtain

Corollary 91

⟨tminus1vhor wvert⟩L2 = ⟨tminus1vhor wvert⟩sf +infin990118j=0


The same types of arguments as before relying on Proposition 55 showthat these expansions are convergent and may be differentiated at will


10 Proof of Theorem 12

We now come to the final steps in the proof of Theorem 12 by showingthat the true moduli space M984094 is an exponentially small perturbation ofthe approximate moduli space (Mapp)984094 More specifically we construct adiffeomorphism F ∶ M984094 rarr (Mapp)984094 such that the difference between thepullback of the L2 metric on (Mapp)984094 and the L2 metric on (Mapp)984094 decaysexponentially as trarrinfin

The subtleties in the discussion below involve gauge choices so we de-scribe the procedure carefully Recall from sect4 that we have actually beenworking at the level of slices in the premoduli spaces Thus the construc-tion of the family of approximate solutions corresponds to a diffeomorphismK1 ∶ Sinfin rarr Sapp while the deformation to a true solution corresponds to afurther map K2 ∶ Sapp rarr S The parametrization of a neighborhood inM984094

by a neighborhood inM984094infin is represented by the composition K2 K1 while

the diffeomorphism F is induced by Kminus12 We must do two things first we show that K2 is indeed smooth and

second we compute the induced map on gauged tangent vectorsThe first of these is a straightforward extension from the original existence

theorem Indeed we obtain the complex gauge transformation γt for whichexp(γt)(Sapp

t ) = St by writing the first part of the hyperkahler moment mapmicro as a nonlinear map acting on γ and expanding this equation in a Taylorseries about γ = 0 This takes the form

Ltγ = micro(Sappt ) +Q(γ)

where Q is a smooth function of γ (but not its derivatives) which vanishesquadratically as γ rarr 0 The map Lt is invertible as a map on hermitianinfinitesimal gauge transformations for each Sapp

t and it is a standard matterto show that its inverse depends smoothly on Sapp

t Furthermore the error

term micro(Sappt ) is C0-bounded by Ceminusβt The inverse function theorem applies

directly to prove that there exists a smooth map

T ∶ B rarr Ω0(X isu(E)) Sappt ↦ γt = T (Sapp

t )

defined on the ball B sub Sapp about 0 of some radius C 984094eminusβt such that

micro(exp(T (Sappt )) equiv 0

This proves the first claimThe next step which is slightly more difficult is to show that if vt =

(αtϕt) is a tangent vector to the premoduli space of approximate solutionswhich satisfies the gauge fixing condition then there is a well-defined tangentvector v984094t = (α984094tϕ984094t) to the space of solutions of the Hitchin equation whichis also in gauge and moreover that

995858vt minus v984094t995858 le Ceminusβt

for some β gt 0


This map is a composition of dK2 and a further map to put the result intogauge Thus supposing that vt satisfies the gauge condition we first notethat the estimates for the field γt imply that 995858(dK2 minus Id)vt995858 le Ceminusβt Thefact that vt is in gauge with respect to Sapp

t means that dK2(vt) is nearly ingauge with respect to St or more specifically the correction field ξt satisfies(D1

t )lowast(dK2(vt) minusDt1ξt) = 0 ie

Ltξt = (Dt1)lowastdK2(vt)

Using that the operator norm of Lminus1t is uniformly bounded in t cf [MSWW14Proposition 52] it then follows that ξt is bounded in norm by Ceminusβt andhence the gauged image vector dK2(vt) minusD1

t ξt is within this exponentiallysmall distance from vt

The conclusion of the above estimates is that the gauged tangent vectorstoM984094 are exponentially close to the gauged tangent vectors to (Mapp)984094

By identifyingM984094 via the diffeomorphism φ in (6) with a torus fibrationover (0infin) times S 984094 we decompose

TM984094 = T rM984094 oplus T hM984094 oplus T vM984094

Here the radial subspace T rM984094 is spanned by horizontal lifts of partt ThM984094 is

spanned by horizontal lifts of tangent vectors to S 984094 and T vM984094 is the verticaltangent bundle Let T rlowastM T hlowastM and T vlowastM denote the respective dualbundles

The exponentially small correction fromMapp toM constructed in thissection combined with the vertical horizontal and radial metric estimatesof sections sectsect7ndash9 imply finally the

Theorem 101 There is a decomposition

gL2 minus gsf = hrr + t2hhh + hvv + hrv + thrh + thhvwhere

hrr isin Γ(9959522T rlowastM984094) hhh isin Γ(9959522

T hlowastM984094) hvv isin Γ(9959522T vlowastM984094)

hrh isin Γ(T rlowastM984094995952T hlowastM984094) hrv isin Γ(T rlowastM984094995952T vlowastM984094)hhv isin Γ(T hlowastM984094995952T vlowastM984094)

with convergent expansions

hrr = tminus53arr +O(eminusβt) hrh =

infin990118j=0

tminus(3+j)9957233ajrh +O(eminusβt)

hhh =infin990118j=0

tminus(2+j)9957233ajhh +O(eminusβt) hrv =

infin990118j=0

tminus(3+j)9957233ajrv +O(eminusβt)

hvv =infin990118j=0

tminus(2+j)9957233ajvv +O(eminusβt) hhv =infin990118j=0

tminus(2+j)9957233ajhv +O(eminusβt)

where arr ajhh a

jvv a

jrh a

jrv and ajhv are t-independent tensors


Inserting the expansions for these various parts one obtains Theorem 12as stated in the introduction

References

[Ba] D Balduzzi Donagi-Markman cubic for Hitchin systems Math Res Lett 13(2006) no 5-6 923ndash933

[BC] O Baues V Cortes Proper affine hyperspheres which fiber over projective specialKahler manifolds Asian J Math 7 (2003) no 1 115ndash132

[BNR] A Beauville M Narasimhan and S Ramanan Spectral curves and the gen-eralised theta divisor J Reine Angew Math 398 (1989) 169ndash179

[BL] C Birkenhake H Lange Complex abelian varieties Second Edition Grundlehrender Mathematischen Wissenschaften 302 Springer-Verlag Berlin 2004

[CM] V Cortes T Mohaupt Special geometry of Euclidean supersymmetry III Thelocal r-map instantons and black holes J High Energy Phys 2009 no 7 066 64pp

[DH] A Douady J Hubbard On the density of Strebel differentials Invent Math 30(1975) no 2 175ndash179

[DN] D Dumas A Neitzke Asymptotics of Hitchinrsquos metric on the Hitchin sectionComm Math Phys to appear

[Fr18] L Fredrickson Exponential decay for the asymptotic geometry of the Hitchinmetric preprint 2018 arXiv181001554

[Fr] D Freed Special Kahler manifolds Comm Math Phys 203 (1999) no 1 31ndash52[GMN] D Gaiotto G Moore and A Neitzke Wall-crossing Hitchin systems and

the WKB approximation Adv Math 234 (2013) 239ndash403[GS] V Guillemin S Sternberg Symplectic techniques in physics Second Edition

Cambridge University Press Cambridge 1990[HHP] C Hertling L Hoevenaars and H Posthuma Frobenius manifolds pro-

jective special geometry and Hitchin systems J Reine Angew Math 649 (2010)117ndash165

[Hi87a] N Hitchin The self-duality equations on a Riemann surface Proc LondonMath Soc (3) 55 (1987) no 1 59ndash126

[Hi87b] N Hitchin Stable bundles and integrable systems Duke Math J 54 (1987) no1 91ndash114

[HKLR] N Hitchin A Karlhede U Lindstrom and M Rocek Hyper-Kahler met-rics and supersymmetry Comm Math Phys 108 (1987) no 4 535ndash589

[MSWW14] R Mazzeo J Swoboda H Weiszlig and F Witt Ends of the moduli spaceof Higgs bundles Duke Math J 165 (2016) no 12 2227ndash2271

[MSWW15] R Mazzeo J Swoboda H Weiszlig and F Witt Limiting configurationsfor solutions of Hitchinrsquos equation Semin Theor Spectr Geom 31 (2012-2014)91ndash116

[Mo] T Mochizuki Asymptotic behaviour of certain families of harmonic bundles onRiemann surfaces J Topol 9 (2016) no 4 1021ndash1073

[Ne] A Neitzke Notes on a new construction of hyperkahler metrics Homological mir-ror symmetry and tropical geometry 351ndash375 Lect Notes Unione Mat Ital 15Springer Cham 2014

[Ni] N Nitsure Moduli space of semistable pairs on a curve Proc London Math Soc(3) 62 (1991) no 2 275ndash300


Department of Mathematics Stanford University Stanford CA 94305 USAEmail address mazzeomathstanfordedu

Mathematisches Institut der Universitat Munchen Theresienstraszlige 39 Dndash80333 Munchen Germany

Email address swobodamathlmude

Mathematisches Seminar der Universitat Kiel Ludewig-Meyn-Straszlige 4 Dndash24098 Kiel Germany

Email address weissmathuni-kielde

Institut fur Geometrie und Topologie der Universitat Stuttgart Pfaf-fenwaldring 57 Dndash70569 Stuttgart Germany

Email address frederikwittmathematikuni-stuttgartde


equations again modulo unitary gauge transformations There are propersurjective mappings π ∶M rarr B and πinfin ∶Minfin rarr B onto the space of holo-morphic quadratic differentials each carries a Higgs bundle (AΦ) to detΦThe subsetM984094

infin of limiting configurations over the lsquofree regionrsquo B984094 sub B of qua-dratic differentials with only simple zeroes was introduced in [MSWW14]later Mochizuki [Mo] extended the definition of limiting configurations toinclude those also lying over the lsquodiscriminant locusrsquo Λ = B ∖B984094 We denotethe preimage πminus1(B984094) byM984094 There is a canonical diffeomorphism

(1) F ∶M984094infin 995275rarrM984094

which we explain later This diffeomorphism allows us to transfer functionsvector fields and tensors from M984094

infin to M984094 and back The maps π and πinfinare quadratic in the Higgs field so the natural Ctimes action on Higgs bundles(AΦ) satisfies π(A tΦ) = t2 detΦ and similarly for πinfin We consider hereonly the restriction of this Ctimes action to an R+ action The space B984094 is a conewith respect to this (quadratic) action whileM984094

infin is lsquosemi-conicrsquo ie it is abundle of tori over the cone B984094 where the fibers along each R+ orbit are allthe same Limiting configurations are one of the two building blocks for theconstruction of diverging families of solutions inM984094 [MSWW14] (the otheris the family of fiducial solutions cf sect4)

Entirely distinct from those developments motivated by supersymmet-ric quantum field theory a beautiful conjectural picture of the asymptoticgeometry of M has been established in the monumental work by GaiottoMoore and Neitzke [GMN] These authors develop the formalism of spectralnetworks on Riemann surfaces out of which they construct a hyperkahlermetric gGMN onM which they conjecture to be precisely equal to the metricgL2 The short survey paper by Neitzke [Ne] contains an overview of thisconstruction Part of their story involves a simpler hyperkahler metric gsfonM984094 called the semiflat metric which is canonically associated to the un-derlying algebraic completely integrable system structure They show thatit is a good approximation to gGMN in the sense that

gGMN sim gsf +O(eminusβt)The error term is a symmetric two-tensor whose norm with respect to gsfdecays at the stated rate where we are identifying the dilation parameter tas a radial variable onM984094 and the exponential decay rate β depends on theparticular R+ orbit and degenerates as this ray converges to B ∖B984094 (Thereis a precise conjectured formula for β which we do not state here)

These two points of view lead to the challenge of understanding theGaiotto-Moore-Neitzke metric and its relationship to gL2 This is the goalof the present paper In more detail we have two main results

Theorem 11 The pullback Flowastgsf of the semiflat metric toM984094infin is a renor-

malized L2 metric onM984094infin

The diffeomorphism F can be defined via the Kobayashi-Hitchin cor-respondence since points on M984094 and M984094

infin are each associated to unique














990124Df(t29957233z) q































Θ =K19957232X



995742


q









q and




minus1X we see that














(2) zi(q) = 990124αi(q)

λSW (q) wi(q) = 990124βi(q)

λSW (q) i = 1 m




4990118i



dzi(q) = 990124αi(q)

984162GMq λSW dwi(q) = 990124

βi(q)984162GM

q λSW i = 1 m




(3) 984162GMq λSW =

1

2τq


(4) TqB984094 =H0(K2X)


λSW







984162GMq λSW =

1

2λSW


gsK(q q) =i

2990118j


= i

2990118j990124αj

984162GMq λSW 990124

βj


βj

984162GMq λSW 990124

αj

984162GMq λSW

= i

8990118j990124αj

τq 990124βj

τq minus 990124βj

τq 990124αj

τq

= i

8990124Sq

τq and τq =1

8990124Sq

995852τq 9958522 dA



995858α9958582L2 = i990124Sq

α and α = 990124Sq

995852α9958522 dA



(5) gsK(q q) =1

8990124Sq

995852τq 9958522 dA =1

8990124Sq

995852q9958522

995852q995852dA = 1

4990124X

995852q9958522

995852q995852dA




2wdw = f 984094(z)dz



τq =q

λSW= 2f(z)

f 984094(z)dw






zi(ϕλ(q)) = 990124αi


τλq = λwi(q)


F(q) = 1

2990118j

zjwj


partF995723partzj = 12(wj +990118

i


i




K(q) = 1

2Im990118

j

wj zj =i

4990118j

(zjwj minus zjwj)

= i

4990118j990124αj

τq 990124βj

τq minus 990124αj

τq 990124βj

τq

= i

4990124Sq

τq and τq =1

4990124Sq

995852τq 9958522 dA =1

2990124X995852q995852dA






On the other hand


8990124St2q

(q995723λSW) and (q995723λSW)

= i

8t2990124Sq

(q995723λSW) and q995723λSW =1

t2gsK(q q)995852q

so











ωI =990118i











995852α9958522 dA =∶ gsf(αα)




Z(Sq iR)odd


(9) gsf(η η) ∶= gsf(αα) =1

2990124Sq

995852η9958522 dA = 990124X995852η9958522 dA
























(AΦ))perp

Then

990124X⟨dAγ A⟩dA = 990124


and

990124X⟨[Φ and γ] Φ⟩dA = minus990124




(11) (D1(AΦ))





= 990124X⟨A1 A2⟩ + 2Re⟨Φ1 Φ2⟩ dA

(12)





















t =M984094infin



infin rarrM984094t













lowast









Φ = 9957380 minusq1 0

995742










Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz





induced by k



Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References










































990124Df(t29957233z) q































Θ =K19957232X



995742


q









q and




minus1X we see that














(2) zi(q) = 990124αi(q)

λSW (q) wi(q) = 990124βi(q)

λSW (q) i = 1 m




4990118i



dzi(q) = 990124αi(q)

984162GMq λSW dwi(q) = 990124

βi(q)984162GM

q λSW i = 1 m




(3) 984162GMq λSW =

1

2τq


(4) TqB984094 =H0(K2X)


λSW







984162GMq λSW =

1

2λSW


gsK(q q) =i

2990118j


= i

2990118j990124αj

984162GMq λSW 990124

βj


βj

984162GMq λSW 990124

αj

984162GMq λSW

= i

8990118j990124αj

τq 990124βj

τq minus 990124βj

τq 990124αj

τq

= i

8990124Sq

τq and τq =1

8990124Sq

995852τq 9958522 dA



995858α9958582L2 = i990124Sq

α and α = 990124Sq

995852α9958522 dA



(5) gsK(q q) =1

8990124Sq

995852τq 9958522 dA =1

8990124Sq

995852q9958522

995852q995852dA = 1

4990124X

995852q9958522

995852q995852dA




2wdw = f 984094(z)dz



τq =q

λSW= 2f(z)

f 984094(z)dw






zi(ϕλ(q)) = 990124αi


τλq = λwi(q)


F(q) = 1

2990118j

zjwj


partF995723partzj = 12(wj +990118

i


i




K(q) = 1

2Im990118

j

wj zj =i

4990118j

(zjwj minus zjwj)

= i

4990118j990124αj

τq 990124βj

τq minus 990124αj

τq 990124βj

τq

= i

4990124Sq

τq and τq =1

4990124Sq

995852τq 9958522 dA =1

2990124X995852q995852dA






On the other hand


8990124St2q

(q995723λSW) and (q995723λSW)

= i

8t2990124Sq

(q995723λSW) and q995723λSW =1

t2gsK(q q)995852q

so











ωI =990118i











995852α9958522 dA =∶ gsf(αα)




Z(Sq iR)odd


(9) gsf(η η) ∶= gsf(αα) =1

2990124Sq

995852η9958522 dA = 990124X995852η9958522 dA
























(AΦ))perp

Then

990124X⟨dAγ A⟩dA = 990124


and

990124X⟨[Φ and γ] Φ⟩dA = minus990124




(11) (D1(AΦ))





= 990124X⟨A1 A2⟩ + 2Re⟨Φ1 Φ2⟩ dA

(12)





















t =M984094infin



infin rarrM984094t













lowast









Φ = 9957380 minusq1 0

995742










Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz





induced by k



Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References

























































Θ =K19957232X



995742


q









q and




minus1X we see that














(2) zi(q) = 990124αi(q)

λSW (q) wi(q) = 990124βi(q)

λSW (q) i = 1 m




4990118i



dzi(q) = 990124αi(q)

984162GMq λSW dwi(q) = 990124

βi(q)984162GM

q λSW i = 1 m




(3) 984162GMq λSW =

1

2τq


(4) TqB984094 =H0(K2X)


λSW







984162GMq λSW =

1

2λSW


gsK(q q) =i

2990118j


= i

2990118j990124αj

984162GMq λSW 990124

βj


βj

984162GMq λSW 990124

αj

984162GMq λSW

= i

8990118j990124αj

τq 990124βj

τq minus 990124βj

τq 990124αj

τq

= i

8990124Sq

τq and τq =1

8990124Sq

995852τq 9958522 dA



995858α9958582L2 = i990124Sq

α and α = 990124Sq

995852α9958522 dA



(5) gsK(q q) =1

8990124Sq

995852τq 9958522 dA =1

8990124Sq

995852q9958522

995852q995852dA = 1

4990124X

995852q9958522

995852q995852dA




2wdw = f 984094(z)dz



τq =q

λSW= 2f(z)

f 984094(z)dw






zi(ϕλ(q)) = 990124αi


τλq = λwi(q)


F(q) = 1

2990118j

zjwj


partF995723partzj = 12(wj +990118

i


i




K(q) = 1

2Im990118

j

wj zj =i

4990118j

(zjwj minus zjwj)

= i

4990118j990124αj

τq 990124βj

τq minus 990124αj

τq 990124βj

τq

= i

4990124Sq

τq and τq =1

4990124Sq

995852τq 9958522 dA =1

2990124X995852q995852dA






On the other hand


8990124St2q

(q995723λSW) and (q995723λSW)

= i

8t2990124Sq

(q995723λSW) and q995723λSW =1

t2gsK(q q)995852q

so











ωI =990118i











995852α9958522 dA =∶ gsf(αα)




Z(Sq iR)odd


(9) gsf(η η) ∶= gsf(αα) =1

2990124Sq

995852η9958522 dA = 990124X995852η9958522 dA
























(AΦ))perp

Then

990124X⟨dAγ A⟩dA = 990124


and

990124X⟨[Φ and γ] Φ⟩dA = minus990124




(11) (D1(AΦ))





= 990124X⟨A1 A2⟩ + 2Re⟨Φ1 Φ2⟩ dA

(12)





















t =M984094infin



infin rarrM984094t













lowast









Φ = 9957380 minusq1 0

995742










Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz





induced by k



Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References
































q and




minus1X we see that














(2) zi(q) = 990124αi(q)

λSW (q) wi(q) = 990124βi(q)

λSW (q) i = 1 m




4990118i



dzi(q) = 990124αi(q)

984162GMq λSW dwi(q) = 990124

βi(q)984162GM

q λSW i = 1 m




(3) 984162GMq λSW =

1

2τq


(4) TqB984094 =H0(K2X)


λSW







984162GMq λSW =

1

2λSW


gsK(q q) =i

2990118j


= i

2990118j990124αj

984162GMq λSW 990124

βj


βj

984162GMq λSW 990124

αj

984162GMq λSW

= i

8990118j990124αj

τq 990124βj

τq minus 990124βj

τq 990124αj

τq

= i

8990124Sq

τq and τq =1

8990124Sq

995852τq 9958522 dA



995858α9958582L2 = i990124Sq

α and α = 990124Sq

995852α9958522 dA



(5) gsK(q q) =1

8990124Sq

995852τq 9958522 dA =1

8990124Sq

995852q9958522

995852q995852dA = 1

4990124X

995852q9958522

995852q995852dA




2wdw = f 984094(z)dz



τq =q

λSW= 2f(z)

f 984094(z)dw






zi(ϕλ(q)) = 990124αi


τλq = λwi(q)


F(q) = 1

2990118j

zjwj


partF995723partzj = 12(wj +990118

i


i




K(q) = 1

2Im990118

j

wj zj =i

4990118j

(zjwj minus zjwj)

= i

4990118j990124αj

τq 990124βj

τq minus 990124αj

τq 990124βj

τq

= i

4990124Sq

τq and τq =1

4990124Sq

995852τq 9958522 dA =1

2990124X995852q995852dA






On the other hand


8990124St2q

(q995723λSW) and (q995723λSW)

= i

8t2990124Sq

(q995723λSW) and q995723λSW =1

t2gsK(q q)995852q

so











ωI =990118i











995852α9958522 dA =∶ gsf(αα)




Z(Sq iR)odd


(9) gsf(η η) ∶= gsf(αα) =1

2990124Sq

995852η9958522 dA = 990124X995852η9958522 dA
























(AΦ))perp

Then

990124X⟨dAγ A⟩dA = 990124


and

990124X⟨[Φ and γ] Φ⟩dA = minus990124




(11) (D1(AΦ))





= 990124X⟨A1 A2⟩ + 2Re⟨Φ1 Φ2⟩ dA

(12)





















t =M984094infin



infin rarrM984094t













lowast









Φ = 9957380 minusq1 0

995742










Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz





induced by k



Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References

































(2) zi(q) = 990124αi(q)

λSW (q) wi(q) = 990124βi(q)

λSW (q) i = 1 m




4990118i



dzi(q) = 990124αi(q)

984162GMq λSW dwi(q) = 990124

βi(q)984162GM

q λSW i = 1 m




(3) 984162GMq λSW =

1

2τq


(4) TqB984094 =H0(K2X)


λSW







984162GMq λSW =

1

2λSW


gsK(q q) =i

2990118j


= i

2990118j990124αj

984162GMq λSW 990124

βj


βj

984162GMq λSW 990124

αj

984162GMq λSW

= i

8990118j990124αj

τq 990124βj

τq minus 990124βj

τq 990124αj

τq

= i

8990124Sq

τq and τq =1

8990124Sq

995852τq 9958522 dA



995858α9958582L2 = i990124Sq

α and α = 990124Sq

995852α9958522 dA



(5) gsK(q q) =1

8990124Sq

995852τq 9958522 dA =1

8990124Sq

995852q9958522

995852q995852dA = 1

4990124X

995852q9958522

995852q995852dA




2wdw = f 984094(z)dz



τq =q

λSW= 2f(z)

f 984094(z)dw






zi(ϕλ(q)) = 990124αi


τλq = λwi(q)


F(q) = 1

2990118j

zjwj


partF995723partzj = 12(wj +990118

i


i




K(q) = 1

2Im990118

j

wj zj =i

4990118j

(zjwj minus zjwj)

= i

4990118j990124αj

τq 990124βj

τq minus 990124αj

τq 990124βj

τq

= i

4990124Sq

τq and τq =1

4990124Sq

995852τq 9958522 dA =1

2990124X995852q995852dA






On the other hand


8990124St2q

(q995723λSW) and (q995723λSW)

= i

8t2990124Sq

(q995723λSW) and q995723λSW =1

t2gsK(q q)995852q

so











ωI =990118i











995852α9958522 dA =∶ gsf(αα)




Z(Sq iR)odd


(9) gsf(η η) ∶= gsf(αα) =1

2990124Sq

995852η9958522 dA = 990124X995852η9958522 dA
























(AΦ))perp

Then

990124X⟨dAγ A⟩dA = 990124


and

990124X⟨[Φ and γ] Φ⟩dA = minus990124




(11) (D1(AΦ))





= 990124X⟨A1 A2⟩ + 2Re⟨Φ1 Φ2⟩ dA

(12)





















t =M984094infin



infin rarrM984094t













lowast









Φ = 9957380 minusq1 0

995742










Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz





induced by k



Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References































984162GMq λSW =

1

2λSW


gsK(q q) =i

2990118j


= i

2990118j990124αj

984162GMq λSW 990124

βj


βj

984162GMq λSW 990124

αj

984162GMq λSW

= i

8990118j990124αj

τq 990124βj

τq minus 990124βj

τq 990124αj

τq

= i

8990124Sq

τq and τq =1

8990124Sq

995852τq 9958522 dA



995858α9958582L2 = i990124Sq

α and α = 990124Sq

995852α9958522 dA



(5) gsK(q q) =1

8990124Sq

995852τq 9958522 dA =1

8990124Sq

995852q9958522

995852q995852dA = 1

4990124X

995852q9958522

995852q995852dA




2wdw = f 984094(z)dz



τq =q

λSW= 2f(z)

f 984094(z)dw






zi(ϕλ(q)) = 990124αi


τλq = λwi(q)


F(q) = 1

2990118j

zjwj


partF995723partzj = 12(wj +990118

i


i




K(q) = 1

2Im990118

j

wj zj =i

4990118j

(zjwj minus zjwj)

= i

4990118j990124αj

τq 990124βj

τq minus 990124αj

τq 990124βj

τq

= i

4990124Sq

τq and τq =1

4990124Sq

995852τq 9958522 dA =1

2990124X995852q995852dA






On the other hand


8990124St2q

(q995723λSW) and (q995723λSW)

= i

8t2990124Sq

(q995723λSW) and q995723λSW =1

t2gsK(q q)995852q

so











ωI =990118i











995852α9958522 dA =∶ gsf(αα)




Z(Sq iR)odd


(9) gsf(η η) ∶= gsf(αα) =1

2990124Sq

995852η9958522 dA = 990124X995852η9958522 dA
























(AΦ))perp

Then

990124X⟨dAγ A⟩dA = 990124


and

990124X⟨[Φ and γ] Φ⟩dA = minus990124




(11) (D1(AΦ))





= 990124X⟨A1 A2⟩ + 2Re⟨Φ1 Φ2⟩ dA

(12)





















t =M984094infin



infin rarrM984094t













lowast









Φ = 9957380 minusq1 0

995742










Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz





induced by k



Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References































τq =q

λSW= 2f(z)

f 984094(z)dw






zi(ϕλ(q)) = 990124αi


τλq = λwi(q)


F(q) = 1

2990118j

zjwj


partF995723partzj = 12(wj +990118

i


i




K(q) = 1

2Im990118

j

wj zj =i

4990118j

(zjwj minus zjwj)

= i

4990118j990124αj

τq 990124βj

τq minus 990124αj

τq 990124βj

τq

= i

4990124Sq

τq and τq =1

4990124Sq

995852τq 9958522 dA =1

2990124X995852q995852dA






On the other hand


8990124St2q

(q995723λSW) and (q995723λSW)

= i

8t2990124Sq

(q995723λSW) and q995723λSW =1

t2gsK(q q)995852q

so











ωI =990118i











995852α9958522 dA =∶ gsf(αα)




Z(Sq iR)odd


(9) gsf(η η) ∶= gsf(αα) =1

2990124Sq

995852η9958522 dA = 990124X995852η9958522 dA
























(AΦ))perp

Then

990124X⟨dAγ A⟩dA = 990124


and

990124X⟨[Φ and γ] Φ⟩dA = minus990124




(11) (D1(AΦ))





= 990124X⟨A1 A2⟩ + 2Re⟨Φ1 Φ2⟩ dA

(12)





















t =M984094infin



infin rarrM984094t













lowast









Φ = 9957380 minusq1 0

995742










Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz





induced by k



Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References






























On the other hand


8990124St2q

(q995723λSW) and (q995723λSW)

= i

8t2990124Sq

(q995723λSW) and q995723λSW =1

t2gsK(q q)995852q

so











ωI =990118i











995852α9958522 dA =∶ gsf(αα)




Z(Sq iR)odd


(9) gsf(η η) ∶= gsf(αα) =1

2990124Sq

995852η9958522 dA = 990124X995852η9958522 dA
























(AΦ))perp

Then

990124X⟨dAγ A⟩dA = 990124


and

990124X⟨[Φ and γ] Φ⟩dA = minus990124




(11) (D1(AΦ))





= 990124X⟨A1 A2⟩ + 2Re⟨Φ1 Φ2⟩ dA

(12)





















t =M984094infin



infin rarrM984094t













lowast









Φ = 9957380 minusq1 0

995742










Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz





induced by k



Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References





































995852α9958522 dA =∶ gsf(αα)




Z(Sq iR)odd


(9) gsf(η η) ∶= gsf(αα) =1

2990124Sq

995852η9958522 dA = 990124X995852η9958522 dA
























(AΦ))perp

Then

990124X⟨dAγ A⟩dA = 990124


and

990124X⟨[Φ and γ] Φ⟩dA = minus990124




(11) (D1(AΦ))





= 990124X⟨A1 A2⟩ + 2Re⟨Φ1 Φ2⟩ dA

(12)





















t =M984094infin



infin rarrM984094t













lowast









Φ = 9957380 minusq1 0

995742










Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz





induced by k



Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References
















































(AΦ))perp

Then

990124X⟨dAγ A⟩dA = 990124


and

990124X⟨[Φ and γ] Φ⟩dA = minus990124




(11) (D1(AΦ))





= 990124X⟨A1 A2⟩ + 2Re⟨Φ1 Φ2⟩ dA

(12)





















t =M984094infin



infin rarrM984094t













lowast









Φ = 9957380 minusq1 0

995742










Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz





induced by k



Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References








































t =M984094infin



infin rarrM984094t













lowast









Φ = 9957380 minusq1 0

995742










Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz





induced by k



Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References































infin rarrM984094t













lowast









Φ = 9957380 minusq1 0

995742










Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz





induced by k



Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References

































Φ = 9957380 minusq1 0

995742










Hinfin = 995738995852f 995852minus19957232 0

0 995852f 99585219957232995742 and Φ = 9957380 f1 0

995742dz





induced by k



Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References































Ainfin(q) = AH +1

2995734Im part log 995852q995852k995739 995738

i 00 minusi995742

and

Φinfin(q) =⎛⎝

0 995852q995852minus19957232k q

995852q99585219957232k 0

⎞⎠



Afidinfin =

1

89957381 00 minus1995742995736

dz

zminus dz

z995741 Φfid

infin =⎛⎝

0995771995852z995852

z995771995852z995852

0⎞⎠dz




0 (qq)19957234k995742

If we then set

ginfin = 995738(qq)19957238k19957232 0

0 (qq)minus19957238kminus19957232995742



and consequently



and


(qq)19957234k 0995742

as desired








timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References


































timesq p

lowastq End0(E))










H1Z(XtimesLΦinfin)

=Minfin(q)






timesq )odd rarr


i=1Hp(DiD

timesi ) where (DiD





















Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References














































Ainfin ∶=part

parts995855s=0

Ainfin(s) = minus1

4d Im(q995723q)995738i 0

0 minusi995742


and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References






























and

Φinfin ∶=part

parts995855s=0

Φinfin(s) =⎛⎝


12 995852q995852

19957232k Re(q995723q) 0

⎞⎠




4Im(q995723q)995738i 0

0 minusi995742


and


0 12 995852q995852minus19957232k q

12 995852q995852

19957232k q995723q 0

⎞⎠(15)






H = 995738κ 00 κminus1

995742

and hence


995852f 99585219957232κ 0995742dz




0 12 995852f 995852

minus19957232κminus1f12 995852f 995852

19957232κf995723f 0995742dz

and finally





mulaelig above



gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References































gsK(q q) = 990124X995852ϕinfin9958522 dA

















































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References

























































pair Sappt = (Aapp









t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References


































t + η tΦappt ) by


12)995739 Im part log 995852q995852k 995738

i 00 minusi995742(19)

Φappt =

⎛⎝


995852q99585219957232k eχ(995852q995852k)ht(995852q995852k) 0

⎞⎠(20)






2ρpartρ then

(ρpartρ)2ψ =1

2ρ2 sinh2ψ


tand Φapp













suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References


































suprgt0






(v) Finally

suprisin(01)





5 Gauge correction









(D1(AΦ))


or equivalently


(22) L(AΦ) ∶= (D1(AΦ))







t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References


































t tΦappt ) define

Mtξ ∶=MΦappt



and also set

D1t ξ ∶=D1

(Aappt +ηtΦapp

t )ξ = (dAappt





t and ξ]])



(23) 995858Gtf995858L2(X) le C995858f995858L2(X)






39957232





∆At = t49957233983172minus2995779∆983172 ∶= t49957233∆983172


t2Mt = t2r995779Mρ ∶= t49957233M983172




(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References
































(24) Lt = t49957233L983172 where L983172 =∆983172 +M983172








t

satisfies

(26) Gfidt (z z) = G983172(t29957233z t29957233z)








so that










extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References
































extχ2


GtLt = Id minusRt

this remainder term















39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References
































39957232along







Proof We have














ft =990118j










q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References




































q 995207rarr 990124Sq

microq dA



(t q)995207rarr 990124Stq


microtq dA








Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References



































Sappt = (Aapp



Aappt ∶= d

dε995855ε=0

Aappt (q + εq)

= 9957354f 984094t(995852q995852k)995852q995852kReq


q

q995740995738i 0

0 minusi995742 (28)

and

(29) Φappt ∶= d

dε995855ε=0

Φappt (q + εq) =

⎛⎝



⎞⎠

where Q = 12 + 995852q995852kh

984094t(995852q995852k)Re

qq Then (Aapp







q995738i 00 minusi995742




t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References
































t ) minusD1Sappt

γt isin TSapptMapp

or more explicitly


t +ηγt


t and γt](30)




t tΦappt )



q

qd995852q995852k + ft(995852q995852k)d Im

q

q995740995738i 0

0 minusi995742



t +ηγt


qd995852q995852k 995738

i 00 minusi995742

(31)



q

q

⎛⎝

0 995852q995852minus12


minus995852q99585212

k eht(995852q995852k) 0

⎞⎠

so that


t and γt]

=⎛⎜⎝


12

k q

99573512 + 995852q995852kh984094t(995852q995852k)995740eht(995852q995852k)995852q995852

12

kqq 0

⎞⎟⎠dz

(32)






and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References






























and






αt = 4f 984094t(r) Imf

zdr 995738i 0

0 minusi995742 and

ϕtminusϕinfin =

⎛⎝


2995739rminus

12 f

995734(12 + rh984094t(r))eht(r) minus 1

2995739r

12fz 0

⎞⎠dz

(33)





f 984094t(995852q995852k) Im995736q

q995741d995852q995852k = f 984094t(995852λ9958522) Im995736

q

λ2995741d995852λ9958522




q









t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References

































t )lowast andϕt])


Rt =infin990118j=0




dlowastAappt







lowast andϕt]) =





ξt =infin990118j=0



(35) D1t ξt =

infin990118j=0








infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References



































infin990118j=0



t ξt9958582L2


Cjt(1minus2j)9957233⟩L2 + 995858

infin990118j=0


= 995858ϕinfin9958582L2 +infin990118j=0





(j)t + ξ(j)t ))


















ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References




































ϕinfin =⎛⎝

0 12 995852q995852minus19957232k q

12 995852q995852

19957232k 0

⎞⎠dz = 1

2Φinfin

and by (32)

ϕt = Φappt =

⎛⎝


(12 + 995852q995852kh984094t(995852q995852k))eht(995852q995852k)995852q99585219957232k 0

⎞⎠


ϕt minusϕinfin =

⎛⎝


2995739q995723995852q99585219957232k


2995739995852q995852

12

k 0

⎞⎠dz





minus19957233 995738 0 A1t (z)

B1t (z) 0





lowast andϕt])













































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References






































































Proposition 82 Both

Rt =infin990118j=1
















Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References




































Proof By (39)


Now write




0 minusi995742dθ











Corollary 84

995858(αtϕt)9958582L2(X) = 995858(αinfin0)9958582L2(X) +

infin990118j=0





⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References































⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩L2

= ⟨(α(1)t ϕ(1)t ) (α

(2)t ϕ

(2)t )⟩sf +

infin990118j=0

S984094984094j ((α(1)t ϕ

(1)t ) (α

(2)t ϕ

(2)t )t

minus(2+j)9957233


lowastAinfin





1

tvhor = 1

tvhorinfin +

infin990118j=0



infin990118j=0









Corollary 91




















t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References













































t )






for some β gt 0





















infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References

















































infin990118j=0


hhh =infin990118j=0


infin990118j=0


hvv =infin990118j=0



where arr ajhh a

jvv a

jrh a




References































References





























ASYMPTOTIC GEOMETRY OF THE HITCHIN METRIC · ASYMPTOTIC GEOMETRY OF THE HITCHIN METRIC RAFE MAZZEO,...

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