Parallel Transport over Path Spaces arXiv:0906.1864v3 ... · tempts to construct a group-valued...

Parallel Transport over Path Spaces

Saikat Chatterjee Amitabha Lahiri and Ambar N Sengupta

Abstract We develop a differential geometric framework for par-allel transport over path spaces and a corresponding discrete the-ory an integrated version of the continuum theory using a category-theoretic framework

1 Introduction

A considerable body of literature has grown up around the notion oflsquosurface holonomyrsquo or parallel transport on surfaces motivated by theneed to have a gauge theory of interaction between charged string-likeobjects Approaches include direct geometric exploration of the spaceof paths of a manifold (Cattaneo et al [5] for instance) and a verydifferent category-theory flavored development (Baez and Schreiber[2] for instance) In the present work we develop both a path-spacegeometric theory as well as a category theoretic approach to surfaceholonomy and describe some of the relationships between the two

As is well known [1] from a group-theoretic argument and also fromthe fact that there is no canonical ordering of points on a surface at-tempts to construct a group-valued parallel transport operator for sur-faces leads to inconsistencies unless the group is abelian (or an abelianrepresentation is used) So in our setting there are two interconnectedgauge groups G and H We work with a fixed principal G-bundleπ P rarrM and connection A then viewing the space of A-horizontalpaths itself as a bundle over the path space of M we study a particu-lar type of connection on this path-space bundle which is specified bymeans of a second connection A and a field B whose values are in the

2010 Mathematics Subject Classification Primary 81T13 Secondary 58Z0516E45

Key words and phrases Gauge Theory Higher Gauge Theories Path SpacesDifferential Forms

1

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906

1864

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] 1

6 Ju

n 20

10

2 SAIKAT CHATTERJEE AMITABHA LAHIRI AND AMBAR N SENGUPTA

Lie algebra LH of H We derive explicit formulas describing parallel-transport with respect to this connection As far as we are awarethis is the first time an explicit description for the parallel transportoperator has been obtained for a surface swept out by a path whoseendpoints are not pinned We obtain in Theorem 23 conditions forthe parallel-transport of a given point in path-space to be independentof the parametrization of that point viewed as a path We also discussH-valued connections on the path space of M constructed from thefield B In section 3 we show how the geometrical data including thefield B lead to two categories We prove several results for these cate-gories and discuss how these categories may be viewed as lsquointegratedrsquoversions of the differential geometric theory developed in section 2

In working with spaces of paths one is confronted with the problemof specifying a differential structure on such spaces It appears bestto proceed within a simpler formalism Essentially one continues touse terms such as lsquotangent spacersquo and lsquodifferential formrsquo except that ineach case the specific notion is defined directly (for example a tangentvector to a space of paths at a particular path γ is a vector field alongγ) rather than by appeal to a general theory Indeed there is a goodvariety of choices for general frameworks in this philosophy (see forinstance Stacey [16] and Viro [17]) For this reason we shall make noattempt to build a manifold structure on any space of paths

Background and MotivationLet us briefly discuss the physical background and motivation for

this study Traditional gauge fields govern interaction between pointparticles Such a gauge field is mathematically a connection A ona bundle over spacetime with the structure group of the bundle be-ing the relevant internal symmetry group of the particle species Theamplitude of the interaction along some path γ connecting the pointparticles is often obtained from the particle wave functions ψ coupledtogether using quantities involving the path-ordered exponential inte-gral P exp(minus

intγA) which is the same as the parallel-transport along

the path γ by the connection A If we now change our point of viewconcerning particles and assume that they are extended string-like en-tities then each particle should be viewed not as a point entity butrather a path (segment) in spacetime Thus instead of the two parti-cles located at two points we now have two paths γ1 and γ2 in place ofa path connecting the two point particles we now have a parametrizedpath of paths in other words a surface Γ connecting γ1 with γ2 Theinteraction amplitudes would one may expect involve both the gauge

PARALLEL TRANSPORT OVER PATH SPACES 3

Figure 1 Point particles interacting via a gauge field

field A as expressed through the parallel transports along γ1 and γ2and an interaction between these two parallel transport fields Thishigher order or higher dimensional interaction could be described bymeans of a gauge field at the higher level it would be a gauge fieldover the space of paths in spacetime

Comparison with other works

The approach to higher gauge theory developed and explored byBaez [1] Baez and Schreiber [2 3] and Lahiri [13] and others citedin these papers involves an abstract category theoretic framework of2-connections and 2-bundles which are higher-dimensional analogs ofbundles and connections There is also the framework of gerbes (Chat-terjee [6] Breen and Messing [4] Murray [14])

We develop both a differential geometric framework and category-theoretic structures We prove in Theorem 23 that a requirement ofparametrization invariance imposes a constraint on a quantity calledthe lsquofake curvaturersquo which has been observed in a related but more ab-stract context by Baez and Schreiber [2 Theorem 23] Our differentialgeometric approach is close to the works of Cattaneo et al [5] Pfeiffer[15] and Girelli and Pfeiffer [11] However we develop in additionto the differential geometric aspects the integrated version in terms ofcategories of diagrams an aspect not addressed in [5] also it shouldbe noted that our connection form is different from the one used in [5]To link up with the integrated theory it is essential to explore the effectof the LH-valued field B To this end we determine a lsquobi-holonomyrsquoassociated to a path of paths (Theorem 24) in terms of the field Bthis aspect of the theory is not studied in [5] or other works

Our approach has the following special features

bull we develop the theory with two connections A and A as well asa 2-form B (with the connection A used for parallel-transportalong any given string-like object and the forms A and B usedto construct parallel-transports between different strings)bull we determine in Theorem 24 the lsquobi-holonomyrsquo associated to

a path of paths using the B-field


Figure 2 Gauge fields along paths c1 and c2 interactingacross a surface

bull we allow lsquoquadrilateralsrsquo rather than simply bigons in the cat-egory theoretic formulation corresponding to having stringswith endpoints free to move rather than fixed-endpoint strings

Our category theoretic considerations are related to notions about dou-ble categories introduced by Ehresmann [9 10] and explored furtherby Kelly and Street [12]

2 Connections on Path-space Bundles

In this section we will construct connections and parallel-transportfor a pair of intertwined structures path-space bundles with structuregroups G and H which are Lie groups intertwined as explained belowin (21) For the physical motivation it should be kept in mind that Gdenotes the gauge group for the gauge field along each path or stringwhile H governs along with G the interaction between the gauge fieldsalong different paths

An important distinction between existing differential geometric ap-proaches (such as Cattaneo et al [5]) and the lsquointegrated theoryrsquo en-coded in the category-theoretic framework is that the latter necessar-ily involves two gauge groups a group G for parallel transport alongpaths and another group H for parallel transport between paths (inpath space) We shall develop the differential geometric frameworkusing a pair of groups (GH) so as to be consistent with the lsquointe-gratedrsquo theory Along with the groups G and H we use a fixed smoothhomomorphism τ H rarr G and a smooth map

GtimesH rarr H (g h) 7rarr α(g)h


such that each α(g) is an automorphism of H such that the identities

τ(α(g)h

)= gτ(h)gminus1

α(τ(h)

)hprime = hhprimehminus1

(21)

hold for all g isin G and h hprime isin H The derivatives τ prime(e) and αprime(e) will bedenoted simply as τ LH rarr LG and α LG rarr LH (This structureis called a Lie 2-group in [1 2])

To summarize very rapidly anticipating some of the notions ex-plained below we work with a principal G-bundle π P rarr M over amanifold M equipped with connections A and A and an α-equivariantvertical 2-form B on P with values in the Lie algebra LH We thenconsider the space PAP of A-horizontal paths in P which forms aprincipal G-bundle over the path-space PM in M Then there is anassociated vector bundle E over PM with fiber LH using the 2-form Band the connection form A we construct for any section σ of the bun-dle P rarrM an LH-valued 1-form θσ on PM This being a connectionover the path-space in M with structure group H parallel-transportby this connection associates elements of H to parametrized surfacesin M Most of our work is devoted to studying a second connectionform ω(AB) which is a connection on the bundle PAP which we con-struct using a second connection A on P Parallel-transport by ω(AB)

is related to parallel-transport by the LH-valued connection form θσ

Principal bundle and the connection A

Consider a principal G-bundle

π P rarrM

with the right-action of the Lie group G on P denoted

P timesGrarr P (p g) 7rarr pg = Rgp

Let A be a connection on this bundle The space PAP of A-horizontalpaths in P may be viewed as a principal G-bundle over PM the spaceof smooth paths in M

We will use the notation pK isin TpP for any point p isin P andLie-algebra element K isin LG defined by

pK =d

dt

∣∣∣t=0p middot exp(tK)

It will be convenient to keep in mind that we always use t to denote theparameter for a path on the base manifold M or in the bundle spaceP we use the letter s to parametrize a path in path-space


The tangent space to PAPThe points of the space PAP are A-horizontal paths in P Although

we call PAP a lsquospacersquo we do not discuss any topology or manifoldstructure on it However it is useful to introduce certain differentialgeometric notions such as tangent spaces on PAP It is intuitively clearthat a tangent vector at a lsquopointrsquo γ isin PAP ought to be a vector fieldon the path γ We formalize this idea here (as has been done elsewhereas well such as in Cattaneo et al [5])

If PX is a space of paths on a manifold X we denote by evt theevaluation map

(22) evt PX rarr X γ 7rarr evt(γ) = γ(t)

Our first step is to understand the tangent spaces to the bundlePAP The following result is preparation for the definition (see also [5Theorem 21])

Proposition 21 Let A be a connection on a principal G-bundle π P rarrM and

Γ [0 1]times [0 1]rarr P (t s) 7rarr Γ(t s) = Γs(t)

a smooth map and

vs(t) = partsΓ(t s)

Then the following are equivalent

(i) Each transverse path

Γs [0 1]rarr P t 7rarr Γ(t s)

is A-horizontal(ii) The initial path Γ0 is A-horizontal and the lsquotangency condi-

tionrsquo

(23)partA(vs(t))

partt= FA

(parttΓ(t s) vs(t)

)holds and thus also

(24) A(vs(T )

)minus A

(vs(0)

)=

int T

0

FA(parttΓ(t s) vs(t)

)dt

for every T s isin [0 1]

Equation (23) and variations on it is sometimes referred to as theDuhamel formula and sometimes a lsquonon-abelian Stokes formularsquo Wecan write it more compactly by using the notion of a Chen integralWith suitable regularity assumptions a 2-form Θ on a space X yieldsa 1-form denoted

intΘ on the space PX of smooth paths in X if c is


such a path a lsquotangent vectorrsquo v isin Tc(PX) is a vector field t 7rarr v(t)along c and the evaluation of the 1-form

intΘ on v is defined to be

(25)

(intΘ

)c

v =

(intc

Θ

)(v) =

int 1

0

Θ(cprime(t) v(t)

)dt

The 1-formint

Θ or its localization to the tangent space Tc(PX) iscalled the Chen integral of Θ Returning to our context we then have

(26) evlowastTAminus evlowast0A =

int T

0

FA

where the integral on the right is a Chen integral here it is by defini-tion the 1-form on PAP whose value on a vector vs isin TΓs

PAP is given

by the right side of (23) The pullback evlowasttA has the obvious meaning

Proof From the definition of the curvature form FA we have

FA(parttΓ partsΓ) = partt

(A(partsΓ)

)minusparts

(A(parttΓ)

)minusA([parttΓ partsΓ]︸︷︷︸

0

)+[A(parttΓ) A(partsΓ)

]

So

partt(A(partsΓ)

)minus FA(parttΓ partsΓ) = parts

(A(parttΓ)

)minus[A(parttΓ) A(partsΓ)

]= 0 if A(parttΓ) = 0

(27)

thus proving (23) if (i) holds The equation (24) then follows byintegration

Next suppose (ii) holds Then from the first line in (27) we have

(28) parts(A(parttΓ)


]= 0

Now let s 7rarr h(s) isin G describe parallel-transport along s 7rarr Γ(s t)then

hprime(s)h(s)minus1 = minusA(partsΓ(s t)

) and h(0) = e

Then

parts

(h(s)minus1A

(parttΓ(t s)

)h(s)

)= Ad

(h(s)minus1

) [parts(A(parttΓ)


](29)

and the right side here is 0 as seen in (28) Therefore

h(s)minus1A(parttΓ(t s)

)h(s)

is independent of s and hence is equal to its value at s = 0 Thus if Avanishes on parttΓ(t 0) then it also vanishes in parttΓ(t s) for all s isin [0 1]In conclusion if the initial path Γ0 is A-horizontal and the tangencycondition (23) holds then each transverse path Γs is A-horizontal


In view of the preceding result it is natural to define the tangentspaces to PAP as follows

Definition 21 The tangent space to PAP at γ is the linear space ofall vector fields t 7rarr v(t) isin Tγ(t)P along γ for which

(210) partA(v(t))parttminusFA (γprime(t) v(t)) = 0

holds for all t isin [0 1]

The vertical subspace in TγPAP consists of all vectors v(middot) for whichv(t) is vertical in Tγ(t)P for every t isin [0 1]

Let us note one consequence

Lemma 22 Suppose γ [0 1] rarr M is a smooth path and γ an A-horizontal lift Let v [0 1] rarr TM be a vector field along γ and v(0)any vector in Tγ(0)P with πlowastv(0) = v(0) Then there is a unique vectorfield v isin TγPAP whose projection down to M is the vector field v andwhose initial value is v(0)

Proof The first-order differential equation (210) determines thevertical part of v(t) from the initial value Thus v(t) is this verticalpart plus the A-horizontal lift of v(t) to Tγ(t)P

Connections induced from B

All through our work B will denote a vertical α-equivariant 2-formon P with values in LH In more detail this means that B is anLH-valued 2-form on P which is vertical in the sense that

B(u v) = 0 if u or v is vertical

and α-equivariant in the sense that

RlowastgB = α(gminus1)B for all g isin Gwherein Rg P rarr P p 7rarr pg is the right action of G on the principalbundle space P and

α(gminus1)B = dα(gminus1)|eBrecalling that α(gminus1) is an automorphism H rarr H

Consider an A-horizontal γ isin PAP and a smooth vector field X

along γ = π γ take any lift Xγ of X along γ and set

(211) θγ(X)def=

(intγ

B

)(Xγ) =

int 1

0

B(γprime(u) Xγ(u)

)du

This is independent of the choice of Xγ (as any two choices differ by avertical vector on which B vanishes) and specifies a linear form θγ on


Tγ(PM) with values in LH If we choose a different horizontal lift ofγ a path γg with g isin G then

(212) θγg(X) = α(gminus1)θγ(X)

Thus one may view θ to be a 1-form on PM with values in the vectorbundle E rarr PM associated to PAP rarr PM by the action α of G onLH

Now fix a section σ M rarr P and for any path γ isin PM letσ(γ) isin PAP be the A-horizontal lift with initial point σ

(γ(0)

) Thus

σ PM rarr PAP is a section of the bundle PAP rarr PM Then wehave the 1-form θσ on PM with values in LH given as follows for anyX isin Tγ(PM)

(213) (θσ)(X) = θσ(γ)(X)

We shall view θσ as a connection form for the trivial H-bundle overPM Of course it depends on the section σ of PAP rarr PM butin a lsquocontrolledrsquo manner ie the behavior of θσ under change of σ isobtained using (212)

Constructing the connection ω(AB)

Our next objective is to construct connection forms on PAP Tothis end fix a connection A on P in addition to the connection A andthe α-equivariant vertical LH-valued 2-form B on P

The evaluation map at any time t isin [0 1] given by

evt PAP rarr P γ 7rarr γ(t)

commutes with the projections PAP rarr PM and P rarr M and theevaluation map PM rarrM We can pull back any connection A on thebundle P to a connection evlowasttA on PAP

Given a 2-form B as discussed above consider the LH-valued 1-form Z on PAP specified as follows Its value on a vector v isin TγPAPis defined to be

(214) Z(v) =

int 1

0

B (γprime(t) v(t)) dt

Thus

(215) Z =

int 1

0

B

where on the right we have the Chen integral (discussed earlier in (25))of the 2-form B on P lifting it to an LH-valued 1-form on the space


of (A-horizontal) smooth paths [0 1]rarr P The Chen integral here isby definition the 1-form on PAP given by

v isin TγPAP 7rarrint 1

0


Note that Z and the form θ are closely related

(216) Z(v) = θγ(πlowastv)

Now define the 1-form ω(AB) by

(217) ω(AB) = evlowast1A+ τ(Z)

Recall that τ H rarr G is a homomorphism and for any X isin LHwe are writing τ(X) to mean τ prime(e)X here τ prime(e) LH rarr LG is thederivative of τ at the identity The utility of bringing in τ becomesclear only when connecting these developments to the category theo-retic formulation of section 3 A similar construction but using onlyone algebra LG is described by Cattaneo et al [5] However as wepointed out earlier a parallel transport operator for a surface cannotbe constructed using a single group unless the group is abelian Toallow non-abelian groups we need to have two groups intertwined inthe structure described in (21) and thus we need τ

Note that ω(AB) is simply the connection evlowast1A on the bundle PAP shifted by the 1-form τ(Z) In the finite-dimensional setting it is astandard fact that such a shift by an equivariant form which vanisheson verticals produces another connection however given that our set-ting is technically not identical to the finite-dimensional one we shallprove this below in Proposition 22

Thus

(218) ω(AB)(v) = A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt

We can rewrite this as

(219) ω(AB) = evlowast0A+[evlowast1(Aminus A)minus evlowast0(Aminus A)

]+

int 1

0

(FA + τB

)

To obtain this we have simply used the relation (24) The advantagein (219) is that it separates off the end point terms and expressesω(AB) as a perturbation of the simple connection evlowast0A by a vector inthe tangent space Tevlowast

0AA where A is the space of connections on the

bundle PAP Here note that the lsquotangent vectorsrsquo to the affine spaceA at a connection ω are the 1-forms ω1minusω with ω1 running over A Adifference such as ω1 minus ω is precisely an equivariant LG-valued 1-formwhich vanishes on vertical vectors


Recall that the group G acts on P on the right

P timesGrarr P (p g) 7rarr Rgp = pg

and this induces a natural right action of G on PAP

PAP timesGrarr PAP (γ g) 7rarr Rgγ = γg

Then for any vector X in the Lie algebra LG we have a vertical vector

X(γ) isin TγPAPgiven by

X(γ)(t) =d

du

∣∣∣u=0

γ(t) exp(uX)

Proposition 22 The form ω(AB) is a connection form on the prin-cipal G-bundle PAP rarr PM More precisely

ω(AB)

((Rg)lowastv

)= Ad(gminus1)ω(AB)(v)

for every g isin G v isin Tγ(PAP

)and

ω(AB)(X) = X

for every X isin LG

Proof It will suffice to show that for every g isin G

Z((Rg)lowastv

)= Ad(gminus1)Z(v)

and every vector v tangent to PAP and

Z(X) = 0

for every X isin LGFrom (215) and the fact that B vanishes on verticals it is clear

that Z(X) is 0 The equivariance under the G-action follows also from(215) on using the G-equivariance of the connection form A and of the2-form B and the fact that the right action of G carries A-horizontalpaths into A-horizontal paths

Parallel transport by ω(AB)

Let us examine how a path is parallel-transported by ω(AB) At theinfinitesimal level all we need is to be able to lift a given vector fieldv [0 1]rarr TM along γ isin PM to a vector field v along γ such that

(i) v is a vector in Tγ(PAP

) which means that it satisfies the

equation (210)

(220)partA(v(t))

partt= FA (γprime(t) v(t))


(ii) v is ω(AB)-horizontal ie satisfies the equation

(221) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0

The following result gives a constructive description of v

Proposition 23 Assume that A A B and ω(AB) are as specifiedbefore Let γ isin PAP and γ = π γ isin PM its projection to a pathon M and consider any v isin TγPM Then the ω(AB)-horizontal liftv isin TγPAP is given by

v(t) = vhA

(t) + vv(t)

where vhA

(t) isin Tγ(t)P is the A-horizontal lift of v(t) isin Tγ(t)M and

(222) vv(t) = γ(t)

[A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du

]wherein

(223) v(1) = vhA(1) + γ(1)X

with vhA(1) being the A-horizontal lift of v(1) in Tγ(1)P and

(224) X = minusint 1

0

τB(γprime(t) vh

A(t))dt

Note that X in (224) is A(v(1)

)

Note also that since v is tangent to PAP the vector vv(t) is alsogiven by

(225) vv(t) = γ(t)

[A(v(0)

)+

int t

0

FA(γprime(u) vh

A(u))du

]Proof The ω(AB) horizontal lift v of v in Tγ

(PAP

)is the vector

field v along γ which projects by πlowast to v and satisfies the condition(221)

(226) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0

Now for each t isin [0 1] we can split the vector v(t) into an A-horizontalpart and a vertical part vv(t) which is essentially the elementA

(vv(t)

)isin

LG viewed as a vector in the vertical subspace in Tγ(t)P

v(t) = vhA

(t) + vv(t)

and the vertical part here is given by

vv(t) = γ(t)A(v(t)

)


Since the vector field v is actually a vector in Tγ(PAP

) we have from

(220) the relation

A(v(t)

)= A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du

We need now only verify the expression (223) for v(1) To this endwe first split this into A-horizontal and a corresponding vertical part

v(1) = vhA(1) + γ(1)A(v(1)

)The vector A

(v(1)

)is obtained from (226) and thus proves (223)

There is an observation to be made from Proposition 23 Theequation (224) has on the right side the integral over the entire curveγ Thus if we were to consider parallel-transport of only say the lsquolefthalfrsquo of γ we would in general end up with a different path of paths

Reparametrization Invariance

If a path is reparametrized then technically it is a different pointin path space Does parallel-transport along a path of paths depend onthe specific parametrization of the paths We shall obtain conditionsto ensure that there is no such dependence Moreover in this case weshall also show that parallel transport by ω(AB) along a path of pathsdepends essentially on the surface swept out by this path of pathsrather than the specific parametrization of this surface

For the following result recall that we are working with Lie groupsG H smooth homomorphism τ H rarr G smooth map α GtimesH rarrH (g h) 7rarr α(g)h where each α(g) is an automorphism of H and themaps τ and α satisfy (21) Let π P rarr M be a principal G-bundlewith connections A and A and B an LH-valued α-equivariant 2-formon P vanishing on vertical vectors As before on the space PAP ofA-horizontal paths viewed as a principal G-bundle over the space PMof smooth paths in M there is the connection form ω(AB) given by

ω(AB) = evlowast1A+

int 1

0

τB

By a lsquosmooth pathrsquo s 7rarr Γs in PM we mean a smooth map

[0 1]2 rarrM (t s) 7rarr Γ(t s) = Γs(t)

viewed as a path of paths Γs isin PM With this notation and framework we have

Theorem 23 Let

Φ [0 1]2 rarr [0 1]2 (t s) 7rarr (Φs(t)Φt(s))


be a smooth diffeomorphism which fixes each vertex of [0 1]2 Assumethat

(i) either

(227) FA + τ(B) = 0

and Φ carries each s-fixed section [0 1] times s into an s-fixedsection [0 1]times Φ0(s)

(ii) or

(228)[evlowast1(Aminus A)minus evlowast0(Aminus A)

]+

int 1

0

(FA + τB) = 0

Φ maps each boundary edge of [0 1]2 into itself and Φ0(s) =Φ1(s) for all s isin [0 1]

Then the ω(AB)-parallel-translate of the point Γ0 Φ0 along the path

s 7rarr (Γ Φ)s is Γ1 Φ1 where Γ1 is the ω(AB)-parallel-translate of Γ0

along s 7rarr ΓsAs a special case if the path s 7rarr Γs is constant and Φ0 the identity

map on [0 1] so that Γ1 is simply a reparametrization of Γ0 thenunder conditions (i) or (ii) above the ω(AB)-parallel-translate of the

point Γ0 along the path s 7rarr (Γ Φ)s is Γ0 Φ1 ie the appropriatereparametrizaton of the original path Γ0

Note that the path (Γ Φ)0 projects down to (Γ Φ)0 which bythe boundary behavior of Φ is actually that path Γ0 Φ0 in otherwords Γ0 reparametrized Similarly (Γ Φ)1 is an A-horizontal lift ofthe path Γ1 reparametrized by Φ1

If A = A then conditions (228) and (227) are the same and so inthis case the weaker condition on Φ in (ii) suffices

Proof Suppose (227) holds Then the connection ω(AB) has theform

evlowast0A+[evlowast1(Aminus A)minus evlowast0(Aminus A)

]

The crucial point is that this depends only on the end points ie ifγ isin PAP and V isin TγPAP then ω(AB)(V ) depends only on V (0) and

V (1) If the conditions on Φ in (i) hold then reparametrization has theeffect of replacing each Γs with ΓΦ0(s) Φs which is in PAP and the

vector field t 7rarr parts(ΓΦ0(s)Φs(t)) is an ω(AB)-horizontal vector because

its end point values are those of t 7rarr parts(ΓΦ0(s)(t)) since Φs(t) equals tif t is 0 or 1

Now suppose (228) holds Then ω(AB) becomes simply evlowast0A In

this case ω(AB)(V ) depends on V only through the initial value V (0)


Thus the ω(AB)-parallel-transport of γ isin PAP along a path s 7rarrΓs isin PM is obtained by A-parallel-transporting the initial point γ(0)along the path s 7rarr Γ0(s) and shooting off A-horizontal paths lyingabove the paths Γs (Since the paths Γs do not necessarily have thesecond component fixed their horizontal lifts need not be of the formΓs Φs except at s = 0 and s = 1 when the composition ΓΦs Φs

is guaranteed to be meaningful) From this it is clear that paralleltranslating Γ0 Φ0 by ω(AB) along the path s 7rarr Γs results at s = 1

in the path Γ1 Φ1

The curvature of ω(AB)

We can compute the curvature of the connection ω(AB) This is bydefinition

Ω(AB) = dω(AB) +1

2[ω(AB) and ω(AB)]

where the exterior differential d is understood in a natural sense thatwill become clearer in the proof below More technically we are usinghere notions of calculus on smooth spaces see for instance Stacey [16]for a survey and Viro [17] for another approach

First we describe some notation about Chen integrals in the presentcontext If B is a 2-form on P with values in a Lie algebra then itsChen integral

int 1

0B restricted to PAP is a 1-form on PAP given on

the vector V isin Tγ(PAP

)by(int 1

0

B

)(V ) =

int 1

0

B(γprime(t) V (t)

)dt

If C is also a 2-form on P with values in the same Lie algebra we havea product 2-form on the path space PAP given on X Y isin Tγ

(PAP

)by

(int 1

0

)2

[BandC](X Y )

=

int0leultvle1

[B(γprime(u) X(u)

) C(γprime(v) Y (v)

)]du dv

minusint

0leultvle1

[C(γprime(u) X(u)

) B(γprime(v) Y (v)

)]du dv

=

int 1

0

int 1

0

[B(γprime(u) X(u)


)]du dv

(229)


Proposition 24 The curvature of ω(AB) is

Ωω(AB) = evlowast1FA + d

(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)

where the integrals are Chen integrals

Proof From


int 1

0

τB

we have

Ωω(AB) = dω(AB) +1


= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]

= [evlowast1A(X) evlowast1A(Y )]

+

[evlowast1A(X)

int 1

0

τB(γprime(t) Y (t)

)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0

[τB(γprime(u) X(u)

) τB

(γprime(v) Y (v)

)]du dv

= [evlowast1A evlowast1A](X Y ) +

[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)

In the case A = A and without τ the expression for the curvaturecan be expressed in terms of the lsquofake curvaturersquo FA +B For a resultof this type for a related connection form see Cattaneo et al [5 The-orem 26] have calculated a similar formula for curvature of a relatedconnection form


Figure 3 The section σ applied to a path c

A more detailed exploration of the fake curvature would be of in-terest

Parallel-transport of horizontal paths

As before A and A are connections on a principal G-bundle π P rarr M and B is an LH-valued α-equivariant 2-form on P vanishingon vertical vectors Also PX is the space of smooth paths [0 1] rarr Xin a space X and PAP is the space of smooth A-horizontal paths inP

Our objective now is to express parallel-transport along paths inPM in terms of a smooth local section of the bundle P rarrM

σ U rarr P

where U is an open set in M We will focus only on paths lying entirelyinside U

The section σ determines a section σ for the bundle PAP rarr PM if γ isin PM then σ(γ) is the unique A-horizontal path in P with initialpoint σ

(γ(0)

) which projects down to γ Thus

(233) σ(γ)(t) = σ(γ(t))a(t)

for all t isin [0 1] where a(t) isin G satisfies the differential equation

(234) a(t)minus1aprime(t) = minusAd(a(t)minus1

)A ((σ γ)prime(t))

for t isin [0 1] and the initial value a(0) is eRecall that a tangent vector V isin Tγ

(PM

)is a smooth vector field

along the path γ Let us denote σ(γ) by γ

γdef= σ(γ)


Note for later use that

(235) γprime(t) = σlowast(γprime(t)

)a(t) + γ(t)a(t)minus1aprime(t)︸︷︷︸

vertical

Now define the vector

(236) V = σlowast(V ) isin Tγ(PAP

)to be the vector V in Tγ

(PAP

)whose initial value V (0) is

V (0) = σlowast(V (0)

)

The existence and uniqueness of V was proved in Lemma 22Note that V (t) isin Tγ(t)P and (σlowastV )(t) isin Tσ(γ(t))P are generally

different vectors However (σlowastV )(t)a(t) and V (t) are both in Tγ(t)Pand differ by a vertical vector because they have the same projectionV (t) under πlowast

(237) V (t) = (σlowastV )(t)a(t) + vertical vector

Our objective now is to determine the LG-valued 1-form

(238) ω(AAB) = σlowastω(AB)

on PM defined on any vector V isin Tγ(PM) by

(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)

We can now work out an explicit expression for this 1-form

Proposition 25 With notation as above and V isin Tγ(PM

)

(240)

ω(AAB)(V ) = Ad(a(1)minus1

)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt

where Cσ denotes the pullback σlowastC on M of a form C on P anda [0 1]rarr G describes parallel-transport along γ ie satisfies

a(t)minus1aprime(t) = minusAd(a(t)minus1

)Aσ(γprime(t)

)with initial condition a(0) = e The formula for ω(AAB)(V ) can alsobe expressed as

ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1

)(Aσ minus Aσ) (V (1))minus (Aσ minus Aσ) (V (0))

]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)


Note that in (241) the terms involving Aσ and FAσ cancel each

other out

Proof From the definition of ω(AB) in (217) and (214) we seethat we need only focus on the B term To this end we have from(235) and (237)

B(γprime(t) V (t)

)= B

(σlowast(γprime(t)

)a(t) + vertical (σlowastV )(t)a(t) + vertical

)= B


)a(t) (σlowastV )(t)a(t)

)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)

Now recall the relation (21)

τ(α(g)h

)= gτ(h)gminus1 for all g isin G and h isin H

which implies

τ(α(g)K

)= Ad(g)τ(K) for all g isin G and K isin LH

As usual we are denoting the derivatives of τ and α by τ and α againApplying this to (242) we have

τB(γprime(t) V (t)

)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)

and this yields the result

Suppose

Γ [0 1]2 rarr P (t s) 7rarr Γ(t s) = Γs(t) = Γt(s)

is smooth with each Γs being A-horizontal and the path s 7rarr Γ(0 s)being A-horizontal Let Γ = π Γ We will need to use the bi-holonomyg(t s) which is specified as follows parallel translate Γ(0 0) alongΓ0|[0 t] by A then up the path Γt|[0 s] by A back along Γs-reversedby A and then down Γ0|[0 s] by A then the resulting point is

(243) Γ(0 0)g(t s)

The path

s 7rarr Γs

describes parallel transport of the initial path Γ0 using the connectionevlowast0A In what follows we will compare this with the path

s 7rarr Γs

which is the parallel transport of Γ0 = Γ0 using the connection evlowast1AThe following result describes the lsquodifferencersquo between these two con-nections


Proposition 26 Suppose


is smooth with each Γs being A-horizontal and the path s 7rarr Γ(0 s)being A-horizontal Then the parallel translate of Γ0 by the connectionevlowast1A along the path [0 s]rarr PM u 7rarr Γu where Γ = π Γ results inΓsg(1 s) with g(1 s) being the lsquobi-holonomyrsquo specified as in (243)

Proof Let Γs be the parallel translate of Γ0 by evlowast1A along the

path [0 s] rarr PM u 7rarr Γu Then the right end point Γs(1) traces

out an A-horizontal path starting at Γ0(1) Thus Γs(1) is the result ofparallel transporting Γ(0 0) by A along Γ0 then up the path Γ1|[0 s] by

A If we then parallel transport Γs(1) back by A along Γs|[0 1]-reversed

then we obtain the initial point Γs(0) This point is of the form Γs(0)bfor some b isin G and so

Γs = Γsb

Then parallel-transporting Γs(0) back down Γ0|[0 s]-reversed by Aproduces the point Γ(0 0)b This shows that b is the bi-holonomyg(1 s)

Now we can turn to determining the parallel-transport process bythe connection ω(AB) With Γ as above let now Γs be the ω(AB)-

parallel-translate of Γ0 along [0 s] rarr PM u 7rarr Γu Since Γs and Γsare both A-horizontal and project by πlowast down to Γs we have

Γs = Γsbs

for some bs isin G Since ω(AB) = evlowast1A+τ(Z) applied to the s-derivative

of Γs is 0 and evlowast1A applied to the s-derivative of Γs is 0 we have

(244) bminus1s partsbs + Ad(bminus1

s )τZ(partsΓs) = 0

Thus s 7rarr bs describes parallel transport by θσ where the section σsatisfies σ Γ = Γ

Since Γs = Γsg(1 s) we then have

dbsdsbminus1s = minusAd

(g(1 s)minus1

)τZ(partsΓs)

= minusAd(g(1 s)minus1

) int 1

0

τB(parttΓ(t s) partsΓ(t s)

)dt

(245)

To summarize

Theorem 24 Suppose



is smooth with each Γs being A-horizontal and the path s 7rarr Γ(0 s)being A-horizontal Then the parallel translate of Γ0 by the connectionω(AB) along the path [0 s] rarr PM u 7rarr Γu where Γ = π Γ resultsin

(246) Γsg(1 s)τ(h0(s)

)

with g(1 s) being the lsquobi-holonomyrsquo specified as in (243) and s 7rarrh0(s) isin H solving the differential equation

(247)dh0(s)

dsh0(s)minus1 = minusα

(g(1 s)minus1

) int 1

0

B(parttΓ(t s) partsΓ(t s)

)dt

with initial condition h0(0) being the identity in H

Let σ be a smooth section of the bundle P rarrM in a neighborhoodof Γ([0 1]2)

Let at(s) isin G specify parallel transport by A up the path [0 s] rarrM v 7rarr Γ(t v) ie the A-parallel-translate of σΓ(t 0) up the path[0 s]rarrM v 7rarr Γ(t v) results in σ(Γ(t s))at(s)

On the other hand as(t) will specify parallel transport by A along[0 t]rarrM u 7rarr Γ(u s) Thus

(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)

The bi-holonomy is given by

g(1 s) = a0(s)minus1as(1)minus1a1(s)a0(1)

Let us look at parallel-transport along the path s 7rarr Γs by theconnection ω(AB) in terms of the trivialization σ Let Γs isin PAP be

obtained by parallel transporting Γ0 = σ(Γ0) isin PAP along the path

[0 s]rarrM u 7rarr Γ0(u) = Γ(0 u)

This transport is described through a map

[0 1]rarr G s 7rarr c(s)

specified through

(249) Γs = σ(Γs)c(s) = Γsa0(s)minus1c(s)

Then c(0) = e and

(250) c(s)minus1cprime(s) = minusAd(c(s)minus1

)ω(AAB)

(V (s)

)

where Vs isin TΓsPM is the vector field along Γs given by

Vs(t) = V (s t) = partsΓ(t s) for all t isin [0 1]


Equation (250) written out in more detail is

c(s)minus1cprime(s) = minusAd(c(s)minus1

)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)

minus1)τBσ(Γprimes(t) Vs(t)

)dt]

(251)

where as(t) isin G describes Aσ-parallel-transport along Γs|[0 t] By(246) c(s) is given by

c(s) = a0(s)g(1 s)τ(h0(s))

where s 7rarr h0(s) solves(252)dh0(s)

dsh0(s)minus1 = minus

int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ

(parttΓ(t s) partsΓ(t s)

)dt

with initial condition h0(0) being the identity in H The geometricmeaning of as(t)a0(s) is that it describes parallel-transport first by Aσup from (0 0) to (0 s) and then to the right by Aσ from (0 s) to (t s)

3 Two categories from plaquettes

In this section we introduce two categories motivated by the differ-ential geometric framework we have discussed in the preceding sectionsWe show that the geometric framework naturally connects with certaincategory theoretic structures introduced by Ehresmann [9 10] and de-veloped further by Kelley and Street [12]

We work with the pair of Lie groups G and H along with maps τand α satisfying (21) and construct two categories These categorieswill have the same set of objects and also the same set of morphisms

The set of objects is simply the group G

Obj = G

The set of morphisms is

Mor = G4 timesHwith a typical element denoted

(a b c dh)

It is convenient to visualize a morphism as a plaquette labeled withelements of G

To connect with the theory of the preceding sections we shouldthink of a and c as giving A-parallel-transports d and b as A-parallel-transports and h should be thought of as corresponding to h0(1) of


Nd

Ia

Ic

Nbh

Figure 4 Plaquette

Theorem 24 However this is only a rough guide we shall return tothis matter later in this section

For the category Vert the source (domain) and target (co-domain)of a morphism are

sVert(a b c dh) = a

tVert(a b c dh) = c

For the category Horz

sHorz(a b c dh) = d

tHorz(a b c dh) = b

We define vertical composition that is composition in Vert usingFigure 5 In this figure the upper morphism is being applied first andthen the lower

Horizontal composition is specified through Figure 6 In this figurewe have used the notation opp to stress that as morphisms it is theone to the left which is applied first and then the one to the right

Our first observation is

Proposition 31 Both Vert and Horz are categories under the spec-ified composition laws In both categories all morphisms are invertible

Proof It is straightforward to verify that the composition lawsare associative The identity map ararr a in Vert is (a e a e e) and inHorz it is (e a e a e) These are displayed in in Figure 7 The inverseof the morphism (a b c dh) in Vert is (c bminus1 a dminus1α(d)hminus1) theinverse in Horz is (aminus1 d cminus1 bα(a)hminus1)

The two categories are isomorphic but it is best not to identifythem

We use H to denote horizontal composition and V to denotevertical composition


Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime

Nbprimebh(α(dminus1)hprime

)Ndprime

Iaprime = c

Icprime

Nbprimehprime

Figure 5 Vertical Composition

Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec

Nbprime(α(aminus1)hprime

)h

Figure 6 Horizontal Composition (for b = dprime)

We have seen earlier that if A A and B are such that ω(AB) reduces

to evlowast0A (for example if A = A and FA+ τ(B) is 0) then all plaquettes


Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz

Figure 7 Identity Maps

(a b c dh) arising from the connections A and ω(AB) satisfy

τ(h) = aminus1bminus1cd

Motivated by this observation we could consider those morphisms(a b c dh) which satisfy

(31) τ(h) = aminus1bminus1cd

However we can look at a broader class of morphisms as well Suppose

h 7rarr z(h) isin Z(G)

is a mapping of the morphisms in the category Horz or in Vert into thecenter Z(G) of G which carries composition of morphisms to productsin Z(G)

z(h hprime) = z(h)z(hprime)

Then we say that a morphism h = (a b c dh) is quasi-flat with respectto z if

(32) τ(h) = (aminus1bminus1cd)z(h)

A larger class of morphisms could also be considered by replacingZ(G) by an abelian normal subgroup but we shall not explore thishere


Proposition 32 Composition of quasi-flat morphisms is quasi-flatThus the quasi-flat morphisms form a subcategory in both Horz andVert

Proof Let h = (a b c dh) and hprime = (aprime bprime cprime dprimehprime) be quasi-flatmorphisms in Horz such that the horizontal composition hprime H h isdefined ie b = dprime Then

hprime H h = (aprimea bprime cprimec d α(aminus1)hprimeh)

Applying τ to the last component in this we have

aminus1τ(hprime)aτ(h) = aminus1(aprimeminus1bprimeminus1cprimedprime)a(aminus1bminus1cd)z(h)z(hprime)

=((aprimea)minus1bprime

minus1(cprimec)d

)z(hprime H h)

(33)

which says that hprime H h is quasi-flatNow suppose h = (a b c dh) and hprime = (aprime bprime cprime dprimehprime) are quasi-

flat morphisms in Vert such that the vertical composition hprime V h isdefined ie c = aprime Then

hprime V h = (a bprimeb cprime dprimedhα(dminus1)hprime)


τ(h)dminus1τ(hprime)d = (aminus1bminus1cd)dminus1(aprimeminus1bprimeminus1cprimedprime)dz(h)z(hprime)

=(aprimeminus1

(bprimeb)minus1cprimedprimed)z(hprime V h)

(34)

which says that hprime V h is quasi-flat

For a morphism h = (a b c dh) we set

τ(h) = τ(h)

If h = (a b c dh) and hprime = (aprime bprime cprime dprimehprime) are morphisms then we saythat they are τ -equivalent

h =τ hprime

if a = aprime b = bprime c = cprime d = dprime and τ(h) = τ(hprime)

Proposition 33 If h hprime hprimeprime hprimeprime are quasi-flat morphisms for which thecompositions on both sides of (35) are meaningful then

(35) (hprimeprimeprime H hprimeprime) V (hprime H h) =τ (hprimeprimeprime V hprime) H (hprimeprime V h)

whenever all the compositions on both sides are meaningful

Thus the structures we are using here correspond to double cate-gories as described by Kelly and Street [12 section 11]


Proof This is a lengthy but straight forward verification Werefer to Figure 8 For a morphism h = (a b c dh) let us write

τpart(h) = aminus1bminus1cd

For the left side of (35 ) we have

(hprime H h) = (aprimea bprime cprimec d α(aminus1)hprimeh)

(hprimeprimeprime H hprimeprime) = (cprimec bprimeprime f primef dprime α(cminus1)hprimeprimeprimehprimeprime)

hlowastdef= (hprimeprimeprime H hprimeprime) V (hprime H h) = (aprimea bprimeprimebprime f primef dprimedhlowast)

(36)

where

(37) hlowast = α(aminus1)hprimehα(dminus1cminus1)hprimeprimeprimeα(dminus1)hprimeprimeApplying τ gives

τ(hlowast) = aminus1τ(hprime)z(hrsquo)a middot τ(h)z(h)dminus1cminus1τ(hprimeprimeprime)cdmiddotz(hprimeprimeprime) middot dminus1τ(hprimeprime)dz(hprimeprime)

= (aprimea)minus1(bprimeprimebprime)minus1(f primef)(dprimed)z(hlowast)

(38)

where we have used the fact from (21) that α is converted to a con-jugation on applying τ and the last line follows after algebraic simpli-fication Thus

(39) τ(hlowast) = τpart(hlowast)z(hlowast)

On the other hand by an entirely similar computation we obtain

(310) hlowastdef= (hprimeprimeprime V hprime) H (hprimeprime V h) = (aprimea bprimeprimebprime f primef dprimedhlowast)

where

(311) hlowast = α(aminus1)hprimeα(aminus1bminus1)hprimeprimeprimehα(dminus1)hprimeprimeApplying τ to this yields after using (21) and computation

τ(hlowast) = τpart(hlowast)z(hlowast)

Since τ(hlowast) is equal to τ(hlowast) the result (35) follows

Ideally a discrete model would be the exact lsquointegratedrsquo version ofthe differential geometric connection ω(AB) However it is not clear ifsuch an ideal transcription is feasible for any such connection ω(AB) onthe path-space bundle To make contact with the differential picture wehave developed in earlier sections we should compare quasi-flat mor-phisms with parallel translation by ω(AB) in the case where B is such

that ω(AB) reduces to evlowast0A (for instance if A = A and the fake curva-

ture FA+τ(B) vanishes) more precisely the h for quasi-flat morphisms(taking all z(h) to be the identity) corresponds to the quantity h0(1)


Ndprime

Ic

If

Ndprimeprimehprimeprime Ndprimeprime

Icprime

If prime

Nbprimeprimehprimeprimeprime

Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime

Figure 8 Consistency of Horizontal and Vertical Compositions

specified through the differential equation (247) It would be desirableto have a more thorough relationship between the discrete structuresand the differential geometric constructions even in the case when z(middot)is not the identity We hope to address this in future work

4 Concluding Remarks

We have constructed in (217) a connection ω(AB) from a connectionA on a principal G-bundle P over M and a 2-form B taking valuesin the Lie algebra of a second structure group H The connectionω(AB) lives on a bundle of A-horizontal paths where A is anotherconnection on P which may be viewed as governing the gauge theoreticinteraction along each curve Associated to each path s 7rarr Γs of pathsbeginning with an initial path Γ0 and ending in a final path Γ1 in M isa parallel transport process by the connection ω(AB) We have studiedconditions (in Theorem 23) under which this transport is lsquosurface-determinedrsquo that is depends more on the surface Γ swept out by thepath of paths than on the specific parametrization given by Γ of thissurface We also described connections over the path space of M withvalues in the Lie algebra LH obtained from the A and B We developedan lsquointegratedrsquo version or a discrete version of this theory which ismost conveniently formulated in terms of categories of quadrilateraldiagrams These diagrams or morphisms arise from parallel transport


by ω(AB) when B has a special form which makes the parallel transportssurface-determined

Our results and constructions extend a body of literature rangingfrom differential geometric investigations to category theoretic onesWe have developed both aspects clarifying their relationship

Acknowledgments We are grateful to the anonymous referee foruseful comments and pointing us to the reference [12] Our thanksto Urs Schreiber for the reference [16] We also thank SwarnamoyeePriyajee Gupta for preparing some of the figures ANS acknowledgesresearch supported from US NSF grant DMS-0601141 AL acknowl-edges research support from Department of Science and TechnologyIndia under Project No SRS2HEP-00062008

References

[1] J Baez Higher Yang-Mills Theory at httparxivorgabshep-th

0206130

[2] J Baez and U Schreiber Higher Gauge Theory at httparXivhep-th

0511710v2

[3] J Baez and U Schreiber Higher Gauge Theory II 2-connections on 2-bundlesat httparxivorgabshep-th0412325

[4] L Breen and W Messing Differential Geometry of Gerbes available online athttparxivorgabsmath0106083

[5] Alberto S Cattaneo P Cotta-Ramusino and M Rinaldi Loop and Path Spacesand Four-Dimensional BF Theories Connections Holonomies and Observ-ables Commun Math Phys 204 (1999) 493-524

[6] D Chatterjee On Gerbs PhD thesis University of Cambridge (1998)[7] Kuo-Tsai Chen Algebras of Iterated Path Integrals and Fundamental Groups

Transactions of the American Mathematical Society Vol 156 May 1971 (May1971) pp 359-379

[8] Kuo-Tsai Chen Iterated Integrals of Differential Forms and Loop Space Ho-mology The Annals of Mathematics 2nd Ser Vol 97 No 2 (Mar 1973) pp217-246

[9] C Ehresmann Categories structurees Ann Sci Ecole Norm Sup 80 (1963)349-425

[10] C Ehresmann Categories et structures Dunod Paris (1965)[11] F Girelli and H Pfeiffer Higher gauge theory - differential versus integral

formulation J Math Phys 45 (2004) 3949-3971 Online at httparxiv

orgabshep-th0309173

[12] G M Kelly and Ross Street Review of the Elements of 2-Categories CategorySeminar (Proc Sem Sydney 19721973) pp 75ndash103 Lecture Notes in MathVol 420 Springer Berlin 1974

[13] A Lahiri Surface Holonomy and Gauge 2-Group Int J Geometric Methodsin Modern Physics 1 (2004) 299-309

[14] M Murray Bundle gerbes J London Math Soc 54 (1996) 403416


[15] H Pfeiffer Higher gauge theory and a non-abelian generalization of 2-formelectrodynamics Ann Phys 308 (2003) 447-477 Online at httparxiv

orgabshep-th0304074

[16] Andrew Stacey Comparative Smootheology Online at httparxivorg

abs08022225

[17] Oleg Viro httpwwwpdmirasru~olegvirotalkshtml

Saikat Chatterjee and Amitabha Lahiri S N Bose National Centre forBasic Sciences Block JD Sector III Salt Lake Kolkata 700098 WestBengal INDIA

E-mail address saikatboseresin amitabhaboseresin

Ambar Sengupta Department of Mathematics Louisiana State Uni-versity Baton Rouge Louisiana 70803 USA

E-mail address senguptagmailcomURL httpwwwmathlsuedusimsengupta

1 Introduction




References


Lie algebra LH of H We derive explicit formulas describing parallel-transport with respect to this connection As far as we are awarethis is the first time an explicit description for the parallel transportoperator has been obtained for a surface swept out by a path whoseendpoints are not pinned We obtain in Theorem 23 conditions forthe parallel-transport of a given point in path-space to be independentof the parametrization of that point viewed as a path We also discussH-valued connections on the path space of M constructed from thefield B In section 3 we show how the geometrical data including thefield B lead to two categories We prove several results for these cate-gories and discuss how these categories may be viewed as lsquointegratedrsquoversions of the differential geometric theory developed in section 2

In working with spaces of paths one is confronted with the problemof specifying a differential structure on such spaces It appears bestto proceed within a simpler formalism Essentially one continues touse terms such as lsquotangent spacersquo and lsquodifferential formrsquo except that ineach case the specific notion is defined directly (for example a tangentvector to a space of paths at a particular path γ is a vector field alongγ) rather than by appeal to a general theory Indeed there is a goodvariety of choices for general frameworks in this philosophy (see forinstance Stacey [16] and Viro [17]) For this reason we shall make noattempt to build a manifold structure on any space of paths

Background and MotivationLet us briefly discuss the physical background and motivation for

this study Traditional gauge fields govern interaction between pointparticles Such a gauge field is mathematically a connection A ona bundle over spacetime with the structure group of the bundle be-ing the relevant internal symmetry group of the particle species Theamplitude of the interaction along some path γ connecting the pointparticles is often obtained from the particle wave functions ψ coupledtogether using quantities involving the path-ordered exponential inte-gral P exp(minus

intγA) which is the same as the parallel-transport along

the path γ by the connection A If we now change our point of viewconcerning particles and assume that they are extended string-like en-tities then each particle should be viewed not as a point entity butrather a path (segment) in spacetime Thus instead of the two parti-cles located at two points we now have two paths γ1 and γ2 in place ofa path connecting the two point particles we now have a parametrizedpath of paths in other words a surface Γ connecting γ1 with γ2 Theinteraction amplitudes would one may expect involve both the gauge




















τ(α(g)h

)= gτ(h)gminus1

α(τ(h)


(21)






π P rarrM





pK =d

dt











a smooth map and






tionrsquo

(23)partA(vs(t))

partt= FA

(parttΓ(t s) vs(t)


(24) A(vs(T )

)minus A

(vs(0)

)=

int T

0


)dt







(25)

(intΘ

)c

v =

(intc

Θ

)(v) =

int 1

0

Θ(cprime(t) v(t)

)dt

The 1-formint



int T

0

FA


PAP is given




(A(partsΓ)

)minusparts

(A(parttΓ)


0


]

So

partt(A(partsΓ)


(A(parttΓ)



(27)





]= 0



) and h(0) = e

Then

parts

(h(s)minus1A

(parttΓ(t s)

)h(s)

)= Ad

(h(s)minus1

) [parts(A(parttΓ)


](29)



)h(s)



















(211) θγ(X)def=

(intγ

B

)(Xγ) =

int 1

0

B(γprime(u) Xγ(u)

)du







(γ(0)

) Thus


(213) (θσ)(X) = θσ(γ)(X)








(214) Z(v) =

int 1

0


Thus

(215) Z =

int 1

0

B





0








Thus

(218) ω(AB)(v) = A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt



]+

int 1

0

(FA + τB

)











X(γ)(t) =d

du

∣∣∣u=0

γ(t) exp(uX)


ω(AB)

((Rg)lowastv



)and

ω(AB)(X) = X

for every X isin LG


Z((Rg)lowastv

)= Ad(gminus1)Z(v)


Z(X) = 0







equation (210)

(220)partA(v(t))




(221) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0



v(t) = vhA

(t) + vv(t)

where vhA


(222) vv(t) = γ(t)

[A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du

]wherein

(223) v(1) = vhA(1) + γ(1)X



0

τB(γprime(t) vh

A(t))dt


)


(225) vv(t) = γ(t)

[A(v(0)

)+

int t

0

FA(γprime(u) vh

A(u))du


(PAP

)is the vector


(226) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0


(vv(t)

)isin


v(t) = vhA

(t) + vv(t)


vv(t) = γ(t)A(v(t)

)



) we have from

(220) the relation

A(v(t)

)= A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du


v(1) = vhA(1) + γ(1)A(v(1)

)The vector A

(v(1)







int 1

0

τB




Theorem 23 Let




(i) either

(227) FA + τ(B) = 0


(ii) or


]+

int 1

0

(FA + τB) = 0











]










in the path Γ1 Φ1



Ω(AB) = dω(AB) +1




int 1



)by(int 1

0

B

)(V ) =

int 1

0

B(γprime(t) V (t)

)dt


(PAP

)by

(int 1

0

)2

[BandC](X Y )

=

int0leultvle1

[B(γprime(u) X(u)


)]du dv

minusint

0leultvle1

[C(γprime(u) X(u)


)]du dv

=

int 1

0

int 1

0

[B(γprime(u) X(u)


)]du dv

(229)




(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)


Proof From


int 1

0

τB

we have



= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]


+

[evlowast1A(X)

int 1

0


)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0


) τB

(γprime(v) Y (v)

)]du dv


[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)








σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References




















τ(α(g)h

)= gτ(h)gminus1

α(τ(h)


(21)






π P rarrM





pK =d

dt











a smooth map and






tionrsquo

(23)partA(vs(t))

partt= FA

(parttΓ(t s) vs(t)


(24) A(vs(T )

)minus A

(vs(0)

)=

int T

0


)dt







(25)

(intΘ

)c

v =

(intc

Θ

)(v) =

int 1

0

Θ(cprime(t) v(t)

)dt

The 1-formint



int T

0

FA


PAP is given




(A(partsΓ)

)minusparts

(A(parttΓ)


0


]

So

partt(A(partsΓ)


(A(parttΓ)



(27)





]= 0



) and h(0) = e

Then

parts

(h(s)minus1A

(parttΓ(t s)

)h(s)

)= Ad

(h(s)minus1

) [parts(A(parttΓ)


](29)



)h(s)



















(211) θγ(X)def=

(intγ

B

)(Xγ) =

int 1

0

B(γprime(u) Xγ(u)

)du







(γ(0)

) Thus


(213) (θσ)(X) = θσ(γ)(X)








(214) Z(v) =

int 1

0


Thus

(215) Z =

int 1

0

B





0








Thus

(218) ω(AB)(v) = A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt



]+

int 1

0

(FA + τB

)











X(γ)(t) =d

du

∣∣∣u=0

γ(t) exp(uX)


ω(AB)

((Rg)lowastv



)and

ω(AB)(X) = X

for every X isin LG


Z((Rg)lowastv

)= Ad(gminus1)Z(v)


Z(X) = 0







equation (210)

(220)partA(v(t))




(221) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0



v(t) = vhA

(t) + vv(t)

where vhA


(222) vv(t) = γ(t)

[A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du

]wherein

(223) v(1) = vhA(1) + γ(1)X



0

τB(γprime(t) vh

A(t))dt


)


(225) vv(t) = γ(t)

[A(v(0)

)+

int t

0

FA(γprime(u) vh

A(u))du


(PAP

)is the vector


(226) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0


(vv(t)

)isin


v(t) = vhA

(t) + vv(t)


vv(t) = γ(t)A(v(t)

)



) we have from

(220) the relation

A(v(t)

)= A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du


v(1) = vhA(1) + γ(1)A(v(1)

)The vector A

(v(1)







int 1

0

τB




Theorem 23 Let




(i) either

(227) FA + τ(B) = 0


(ii) or


]+

int 1

0

(FA + τB) = 0











]










in the path Γ1 Φ1



Ω(AB) = dω(AB) +1




int 1



)by(int 1

0

B

)(V ) =

int 1

0

B(γprime(t) V (t)

)dt


(PAP

)by

(int 1

0

)2

[BandC](X Y )

=

int0leultvle1

[B(γprime(u) X(u)


)]du dv

minusint

0leultvle1

[C(γprime(u) X(u)


)]du dv

=

int 1

0

int 1

0

[B(γprime(u) X(u)


)]du dv

(229)




(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)


Proof From


int 1

0

τB

we have



= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]


+

[evlowast1A(X)

int 1

0


)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0


) τB

(γprime(v) Y (v)

)]du dv


[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)








σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References









a smooth map and






tionrsquo

(23)partA(vs(t))

partt= FA

(parttΓ(t s) vs(t)


(24) A(vs(T )

)minus A

(vs(0)

)=

int T

0


)dt







(25)

(intΘ

)c

v =

(intc

Θ

)(v) =

int 1

0

Θ(cprime(t) v(t)

)dt

The 1-formint



int T

0

FA


PAP is given




(A(partsΓ)

)minusparts

(A(parttΓ)


0


]

So

partt(A(partsΓ)


(A(parttΓ)



(27)





]= 0



) and h(0) = e

Then

parts

(h(s)minus1A

(parttΓ(t s)

)h(s)

)= Ad

(h(s)minus1

) [parts(A(parttΓ)


](29)



)h(s)



















(211) θγ(X)def=

(intγ

B

)(Xγ) =

int 1

0

B(γprime(u) Xγ(u)

)du







(γ(0)

) Thus


(213) (θσ)(X) = θσ(γ)(X)








(214) Z(v) =

int 1

0


Thus

(215) Z =

int 1

0

B





0








Thus

(218) ω(AB)(v) = A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt



]+

int 1

0

(FA + τB

)











X(γ)(t) =d

du

∣∣∣u=0

γ(t) exp(uX)


ω(AB)

((Rg)lowastv



)and

ω(AB)(X) = X

for every X isin LG


Z((Rg)lowastv

)= Ad(gminus1)Z(v)


Z(X) = 0







equation (210)

(220)partA(v(t))




(221) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0



v(t) = vhA

(t) + vv(t)

where vhA


(222) vv(t) = γ(t)

[A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du

]wherein

(223) v(1) = vhA(1) + γ(1)X



0

τB(γprime(t) vh

A(t))dt


)


(225) vv(t) = γ(t)

[A(v(0)

)+

int t

0

FA(γprime(u) vh

A(u))du


(PAP

)is the vector


(226) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0


(vv(t)

)isin


v(t) = vhA

(t) + vv(t)


vv(t) = γ(t)A(v(t)

)



) we have from

(220) the relation

A(v(t)

)= A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du


v(1) = vhA(1) + γ(1)A(v(1)

)The vector A

(v(1)







int 1

0

τB




Theorem 23 Let




(i) either

(227) FA + τ(B) = 0


(ii) or


]+

int 1

0

(FA + τB) = 0











]










in the path Γ1 Φ1



Ω(AB) = dω(AB) +1




int 1



)by(int 1

0

B

)(V ) =

int 1

0

B(γprime(t) V (t)

)dt


(PAP

)by

(int 1

0

)2

[BandC](X Y )

=

int0leultvle1

[B(γprime(u) X(u)


)]du dv

minusint

0leultvle1

[C(γprime(u) X(u)


)]du dv

=

int 1

0

int 1

0

[B(γprime(u) X(u)


)]du dv

(229)




(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)


Proof From


int 1

0

τB

we have



= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]


+

[evlowast1A(X)

int 1

0


)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0


) τB

(γprime(v) Y (v)

)]du dv


[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)








σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References




(25)

(intΘ

)c

v =

(intc

Θ

)(v) =

int 1

0

Θ(cprime(t) v(t)

)dt

The 1-formint



int T

0

FA


PAP is given




(A(partsΓ)

)minusparts

(A(parttΓ)


0


]

So

partt(A(partsΓ)


(A(parttΓ)



(27)





]= 0



) and h(0) = e

Then

parts

(h(s)minus1A

(parttΓ(t s)

)h(s)

)= Ad

(h(s)minus1

) [parts(A(parttΓ)


](29)



)h(s)



















(211) θγ(X)def=

(intγ

B

)(Xγ) =

int 1

0

B(γprime(u) Xγ(u)

)du







(γ(0)

) Thus


(213) (θσ)(X) = θσ(γ)(X)








(214) Z(v) =

int 1

0


Thus

(215) Z =

int 1

0

B





0








Thus

(218) ω(AB)(v) = A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt



]+

int 1

0

(FA + τB

)











X(γ)(t) =d

du

∣∣∣u=0

γ(t) exp(uX)


ω(AB)

((Rg)lowastv



)and

ω(AB)(X) = X

for every X isin LG


Z((Rg)lowastv

)= Ad(gminus1)Z(v)


Z(X) = 0







equation (210)

(220)partA(v(t))




(221) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0



v(t) = vhA

(t) + vv(t)

where vhA


(222) vv(t) = γ(t)

[A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du

]wherein

(223) v(1) = vhA(1) + γ(1)X



0

τB(γprime(t) vh

A(t))dt


)


(225) vv(t) = γ(t)

[A(v(0)

)+

int t

0

FA(γprime(u) vh

A(u))du


(PAP

)is the vector


(226) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0


(vv(t)

)isin


v(t) = vhA

(t) + vv(t)


vv(t) = γ(t)A(v(t)

)



) we have from

(220) the relation

A(v(t)

)= A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du


v(1) = vhA(1) + γ(1)A(v(1)

)The vector A

(v(1)







int 1

0

τB




Theorem 23 Let




(i) either

(227) FA + τ(B) = 0


(ii) or


]+

int 1

0

(FA + τB) = 0











]










in the path Γ1 Φ1



Ω(AB) = dω(AB) +1




int 1



)by(int 1

0

B

)(V ) =

int 1

0

B(γprime(t) V (t)

)dt


(PAP

)by

(int 1

0

)2

[BandC](X Y )

=

int0leultvle1

[B(γprime(u) X(u)


)]du dv

minusint

0leultvle1

[C(γprime(u) X(u)


)]du dv

=

int 1

0

int 1

0

[B(γprime(u) X(u)


)]du dv

(229)




(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)


Proof From


int 1

0

τB

we have



= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]


+

[evlowast1A(X)

int 1

0


)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0


) τB

(γprime(v) Y (v)

)]du dv


[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)








σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References


















(211) θγ(X)def=

(intγ

B

)(Xγ) =

int 1

0

B(γprime(u) Xγ(u)

)du







(γ(0)

) Thus


(213) (θσ)(X) = θσ(γ)(X)








(214) Z(v) =

int 1

0


Thus

(215) Z =

int 1

0

B





0








Thus

(218) ω(AB)(v) = A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt



]+

int 1

0

(FA + τB

)











X(γ)(t) =d

du

∣∣∣u=0

γ(t) exp(uX)


ω(AB)

((Rg)lowastv



)and

ω(AB)(X) = X

for every X isin LG


Z((Rg)lowastv

)= Ad(gminus1)Z(v)


Z(X) = 0







equation (210)

(220)partA(v(t))




(221) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0



v(t) = vhA

(t) + vv(t)

where vhA


(222) vv(t) = γ(t)

[A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du

]wherein

(223) v(1) = vhA(1) + γ(1)X



0

τB(γprime(t) vh

A(t))dt


)


(225) vv(t) = γ(t)

[A(v(0)

)+

int t

0

FA(γprime(u) vh

A(u))du


(PAP

)is the vector


(226) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0


(vv(t)

)isin


v(t) = vhA

(t) + vv(t)


vv(t) = γ(t)A(v(t)

)



) we have from

(220) the relation

A(v(t)

)= A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du


v(1) = vhA(1) + γ(1)A(v(1)

)The vector A

(v(1)







int 1

0

τB




Theorem 23 Let




(i) either

(227) FA + τ(B) = 0


(ii) or


]+

int 1

0

(FA + τB) = 0











]










in the path Γ1 Φ1



Ω(AB) = dω(AB) +1




int 1



)by(int 1

0

B

)(V ) =

int 1

0

B(γprime(t) V (t)

)dt


(PAP

)by

(int 1

0

)2

[BandC](X Y )

=

int0leultvle1

[B(γprime(u) X(u)


)]du dv

minusint

0leultvle1

[C(γprime(u) X(u)


)]du dv

=

int 1

0

int 1

0

[B(γprime(u) X(u)


)]du dv

(229)




(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)


Proof From


int 1

0

τB

we have



= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]


+

[evlowast1A(X)

int 1

0


)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0


) τB

(γprime(v) Y (v)

)]du dv


[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)








σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References






(γ(0)

) Thus


(213) (θσ)(X) = θσ(γ)(X)








(214) Z(v) =

int 1

0


Thus

(215) Z =

int 1

0

B





0








Thus

(218) ω(AB)(v) = A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt



]+

int 1

0

(FA + τB

)











X(γ)(t) =d

du

∣∣∣u=0

γ(t) exp(uX)


ω(AB)

((Rg)lowastv



)and

ω(AB)(X) = X

for every X isin LG


Z((Rg)lowastv

)= Ad(gminus1)Z(v)


Z(X) = 0







equation (210)

(220)partA(v(t))




(221) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0



v(t) = vhA

(t) + vv(t)

where vhA


(222) vv(t) = γ(t)

[A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du

]wherein

(223) v(1) = vhA(1) + γ(1)X



0

τB(γprime(t) vh

A(t))dt


)


(225) vv(t) = γ(t)

[A(v(0)

)+

int t

0

FA(γprime(u) vh

A(u))du


(PAP

)is the vector


(226) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0


(vv(t)

)isin


v(t) = vhA

(t) + vv(t)


vv(t) = γ(t)A(v(t)

)



) we have from

(220) the relation

A(v(t)

)= A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du


v(1) = vhA(1) + γ(1)A(v(1)

)The vector A

(v(1)







int 1

0

τB




Theorem 23 Let




(i) either

(227) FA + τ(B) = 0


(ii) or


]+

int 1

0

(FA + τB) = 0











]










in the path Γ1 Φ1



Ω(AB) = dω(AB) +1




int 1



)by(int 1

0

B

)(V ) =

int 1

0

B(γprime(t) V (t)

)dt


(PAP

)by

(int 1

0

)2

[BandC](X Y )

=

int0leultvle1

[B(γprime(u) X(u)


)]du dv

minusint

0leultvle1

[C(γprime(u) X(u)


)]du dv

=

int 1

0

int 1

0

[B(γprime(u) X(u)


)]du dv

(229)




(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)


Proof From


int 1

0

τB

we have



= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]


+

[evlowast1A(X)

int 1

0


)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0


) τB

(γprime(v) Y (v)

)]du dv


[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)








σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References




0








Thus

(218) ω(AB)(v) = A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt



]+

int 1

0

(FA + τB

)











X(γ)(t) =d

du

∣∣∣u=0

γ(t) exp(uX)


ω(AB)

((Rg)lowastv



)and

ω(AB)(X) = X

for every X isin LG


Z((Rg)lowastv

)= Ad(gminus1)Z(v)


Z(X) = 0







equation (210)

(220)partA(v(t))




(221) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0



v(t) = vhA

(t) + vv(t)

where vhA


(222) vv(t) = γ(t)

[A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du

]wherein

(223) v(1) = vhA(1) + γ(1)X



0

τB(γprime(t) vh

A(t))dt


)


(225) vv(t) = γ(t)

[A(v(0)

)+

int t

0

FA(γprime(u) vh

A(u))du


(PAP

)is the vector


(226) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0


(vv(t)

)isin


v(t) = vhA

(t) + vv(t)


vv(t) = γ(t)A(v(t)

)



) we have from

(220) the relation

A(v(t)

)= A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du


v(1) = vhA(1) + γ(1)A(v(1)

)The vector A

(v(1)







int 1

0

τB




Theorem 23 Let




(i) either

(227) FA + τ(B) = 0


(ii) or


]+

int 1

0

(FA + τB) = 0











]










in the path Γ1 Φ1



Ω(AB) = dω(AB) +1




int 1



)by(int 1

0

B

)(V ) =

int 1

0

B(γprime(t) V (t)

)dt


(PAP

)by

(int 1

0

)2

[BandC](X Y )

=

int0leultvle1

[B(γprime(u) X(u)


)]du dv

minusint

0leultvle1

[C(γprime(u) X(u)


)]du dv

=

int 1

0

int 1

0

[B(γprime(u) X(u)


)]du dv

(229)




(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)


Proof From


int 1

0

τB

we have



= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]


+

[evlowast1A(X)

int 1

0


)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0


) τB

(γprime(v) Y (v)

)]du dv


[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)








σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References








X(γ)(t) =d

du

∣∣∣u=0

γ(t) exp(uX)


ω(AB)

((Rg)lowastv



)and

ω(AB)(X) = X

for every X isin LG


Z((Rg)lowastv

)= Ad(gminus1)Z(v)


Z(X) = 0







equation (210)

(220)partA(v(t))




(221) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0



v(t) = vhA

(t) + vv(t)

where vhA


(222) vv(t) = γ(t)

[A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du

]wherein

(223) v(1) = vhA(1) + γ(1)X



0

τB(γprime(t) vh

A(t))dt


)


(225) vv(t) = γ(t)

[A(v(0)

)+

int t

0

FA(γprime(u) vh

A(u))du


(PAP

)is the vector


(226) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0


(vv(t)

)isin


v(t) = vhA

(t) + vv(t)


vv(t) = γ(t)A(v(t)

)



) we have from

(220) the relation

A(v(t)

)= A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du


v(1) = vhA(1) + γ(1)A(v(1)

)The vector A

(v(1)







int 1

0

τB




Theorem 23 Let




(i) either

(227) FA + τ(B) = 0


(ii) or


]+

int 1

0

(FA + τB) = 0











]










in the path Γ1 Φ1



Ω(AB) = dω(AB) +1




int 1



)by(int 1

0

B

)(V ) =

int 1

0

B(γprime(t) V (t)

)dt


(PAP

)by

(int 1

0

)2

[BandC](X Y )

=

int0leultvle1

[B(γprime(u) X(u)


)]du dv

minusint

0leultvle1

[C(γprime(u) X(u)


)]du dv

=

int 1

0

int 1

0

[B(γprime(u) X(u)


)]du dv

(229)




(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)


Proof From


int 1

0

τB

we have



= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]


+

[evlowast1A(X)

int 1

0


)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0


) τB

(γprime(v) Y (v)

)]du dv


[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)








σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References



(221) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0



v(t) = vhA

(t) + vv(t)

where vhA


(222) vv(t) = γ(t)

[A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du

]wherein

(223) v(1) = vhA(1) + γ(1)X



0

τB(γprime(t) vh

A(t))dt


)


(225) vv(t) = γ(t)

[A(v(0)

)+

int t

0

FA(γprime(u) vh

A(u))du


(PAP

)is the vector


(226) A(v(1)

)+

int 1

0

τB(γprime(t) v(t)

)dt = 0


(vv(t)

)isin


v(t) = vhA

(t) + vv(t)


vv(t) = γ(t)A(v(t)

)



) we have from

(220) the relation

A(v(t)

)= A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du


v(1) = vhA(1) + γ(1)A(v(1)

)The vector A

(v(1)







int 1

0

τB




Theorem 23 Let




(i) either

(227) FA + τ(B) = 0


(ii) or


]+

int 1

0

(FA + τB) = 0











]










in the path Γ1 Φ1



Ω(AB) = dω(AB) +1




int 1



)by(int 1

0

B

)(V ) =

int 1

0

B(γprime(t) V (t)

)dt


(PAP

)by

(int 1

0

)2

[BandC](X Y )

=

int0leultvle1

[B(γprime(u) X(u)


)]du dv

minusint

0leultvle1

[C(γprime(u) X(u)


)]du dv

=

int 1

0

int 1

0

[B(γprime(u) X(u)


)]du dv

(229)




(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)


Proof From


int 1

0

τB

we have



= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]


+

[evlowast1A(X)

int 1

0


)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0


) τB

(γprime(v) Y (v)

)]du dv


[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)








σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References



) we have from

(220) the relation

A(v(t)

)= A(v(1)

)minusint 1

t

FA(γprime(u) vh

A(u))du


v(1) = vhA(1) + γ(1)A(v(1)

)The vector A

(v(1)







int 1

0

τB




Theorem 23 Let




(i) either

(227) FA + τ(B) = 0


(ii) or


]+

int 1

0

(FA + τB) = 0











]










in the path Γ1 Φ1



Ω(AB) = dω(AB) +1




int 1



)by(int 1

0

B

)(V ) =

int 1

0

B(γprime(t) V (t)

)dt


(PAP

)by

(int 1

0

)2

[BandC](X Y )

=

int0leultvle1

[B(γprime(u) X(u)


)]du dv

minusint

0leultvle1

[C(γprime(u) X(u)


)]du dv

=

int 1

0

int 1

0

[B(γprime(u) X(u)


)]du dv

(229)




(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)


Proof From


int 1

0

τB

we have



= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]


+

[evlowast1A(X)

int 1

0


)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0


) τB

(γprime(v) Y (v)

)]du dv


[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)








σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References



(i) either

(227) FA + τ(B) = 0


(ii) or


]+

int 1

0

(FA + τB) = 0











]










in the path Γ1 Φ1



Ω(AB) = dω(AB) +1




int 1



)by(int 1

0

B

)(V ) =

int 1

0

B(γprime(t) V (t)

)dt


(PAP

)by

(int 1

0

)2

[BandC](X Y )

=

int0leultvle1

[B(γprime(u) X(u)


)]du dv

minusint

0leultvle1

[C(γprime(u) X(u)


)]du dv

=

int 1

0

int 1

0

[B(γprime(u) X(u)


)]du dv

(229)




(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)


Proof From


int 1

0

τB

we have



= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]


+

[evlowast1A(X)

int 1

0


)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0


) τB

(γprime(v) Y (v)

)]du dv


[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)








σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References




in the path Γ1 Φ1



Ω(AB) = dω(AB) +1




int 1



)by(int 1

0

B

)(V ) =

int 1

0

B(γprime(t) V (t)

)dt


(PAP

)by

(int 1

0

)2

[BandC](X Y )

=

int0leultvle1

[B(γprime(u) X(u)


)]du dv

minusint

0leultvle1

[C(γprime(u) X(u)


)]du dv

=

int 1

0

int 1

0

[B(γprime(u) X(u)


)]du dv

(229)




(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)


Proof From


int 1

0

τB

we have



= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]


+

[evlowast1A(X)

int 1

0


)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0


) τB

(γprime(v) Y (v)

)]du dv


[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)








σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References




(int 1

0

τB

)+

[evlowast1Aand

int 1

0

τB

]+

(int 1

0

)2

[τBandτB]

(230)


Proof From


int 1

0

τB

we have



= evlowast1dA+ d

int 1

0

τB +W

(231)

where

W (X Y ) = [ω(AB)(X) ω(AB)(Y )]


+

[evlowast1A(X)

int 1

0


)dt

]+

[int 1

0

τB(γprime(t) X(t)

)dt evlowast1A(Y )

]+

int 1

0

int 1

0


) τB

(γprime(v) Y (v)

)]du dv


[evlowast1Aand

int 1

0

τB

](X Y )

+

(int 1

0

)2

[τBandτB](X Y )

(232)








σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References







σ U rarr P



(γ(0)


(233) σ(γ)(t) = σ(γ(t))a(t)





(PM



γdef= σ(γ)





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References





vertical




(PAP



)







(239) ω(AAB)(V ) = ω(AB)

(σlowastV

)



)

(240)


)Aσ (V (1))+

int 1

0

Ad(a(t)minus1

)τBσ

(γprime(t) V (t)

)dt



)Aσ(γprime(t)


ω(AAB)(V )

= Aσ (V (0)) +[Ad(a(1)minus1


]+

int 1

0

Ad(a(t)minus1

)(FAσ + τBσ

)(γprime(t) V (t)

)dt

(241)



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References



other out


B(γprime(t) V (t)

)= B



)= B



)= α

(a(t)minus1

)Bσ

(γprime(t) V (t)

)

(242)


τ(α(g)h


which implies

τ(α(g)K




)= Ad

(a(t)minus1

)τBσ

(γprime(t) V (t)

)


Suppose



(243) Γ(0 0)g(t s)

The path

s 7rarr Γs


s 7rarr Γs











Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References










Γs = Γsb




Γs = Γsbs








(g(1 s)minus1

)τZ(partsΓs)


) int 1

0


)dt

(245)

To summarize

Theorem 24 Suppose




(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References



(246) Γsg(1 s)τ(h0(s)

)


(247)dh0(s)


(g(1 s)minus1

) int 1

0


)dt





(248) Γ(t s) = σ(Γ(t s)

)a0(s)as(t)








specified through


Then c(0) = e and


)ω(AAB)

(V (s)

)






)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References




)[Ad(as(1)minus1

)Aσ(Vs(1)

)+

int 1

0

Ad(as(t)


)dt]

(251)


c(s) = a0(s)g(1 s)τ(h0(s))



int 1

0

α(as(t)a0(s)g(1 s)

)minus1Bσ


)dt






Obj = G



(a b c dh)




Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References


Nd

Ia

Ic

Nbh

Figure 4 Plaquette



sVert(a b c dh) = a

tVert(a b c dh) = c


sHorz(a b c dh) = d

tHorz(a b c dh) = b









Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References


Nd

Ia

Ic = aprime

Nbh

= Ndprimed

Ia

Icprime


)Ndprime

Iaprime = c

Icprime

Nbprimehprime


Nd

Ia

Ic

Nbh opp Ndprime

Iaprime

Icprime

Nbprimehprime = Nd

Iaprimea

Icprimec


)h





Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References


Ne

Ia

Ia

Nee

Identity for Vert

Na

Ie

Ie

Nae

Identity for Horz




















minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References








minus1(cprimec)d

)z(hprime H h)

(33)






=(aprimeminus1


(34)



τ(h) = τ(h)


h =τ hprime













(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References








(36)

where




(38)





where








Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References


Ndprime

Ic

If


Icprime

If prime


Nd

Ia

Ic

Nbh Nb

Iaprime

Icprime

Nbprimehprime









References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References





References


0206130


0511710v2










orgabshep-th0309173






orgabshep-th0304074


abs08022225






1 Introduction




References



orgabshep-th0304074


abs08022225






1 Introduction




References

Parallel Transport over Path Spaces arXiv:0906.1864v3 ... · tempts to construct a group-valued...

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Transcript of Parallel Transport over Path Spaces arXiv:0906.1864v3 ... · tempts to construct a group-valued...