Gauge Theory and Topological Strings Geometry Conference in honour of Nigel Hitchin - RHD, C. Vafa,...
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Transcript of Gauge Theory and Topological Strings Geometry Conference in honour of Nigel Hitchin - RHD, C. Vafa,...
Gauge Theory andTopological Strings
Geometry Conference in honour of Nigel Hitchin
- RHD, C. Vafa, E.Verlinde, hep-th/0602087 - J. de Boer, M. Chang, RHD, J. Manschot, E. Verlinde,
hep-th/0608059
Robbert DijkgraafUniversity of Amsterdam
integrable systemsintegrable systemsspecial holonomiesspecial holonomies
hyper-Kahlerhyper-Kahler
calibrationscalibrations
(generalized) CY(generalized) CYspectral curvesspectral curves
mirror symmetrymirror symmetry
self-dual geometryself-dual geometry
quantization quantization
instantons instantons
Nigel Hitchin’s Circle of
Ideas
monopolesmonopoles
Higgs-bundlesHiggs-bundles
integrable systemsintegrable systemsspecial holonomiesspecial holonomies
hyper-Kahlerhyper-Kahler
calibrationscalibrations
(generalized) CY(generalized) CYspectral curvesspectral curves
mirror symmetrymirror symmetry
self-dual geometryself-dual geometry
quantization quantization
instantons instantonsmonopolesmonopoles
Higgs-bundlesHiggs-bundles
integrable systemsintegrable systemsspecial holonomiesspecial holonomies
hyper-Kahlerhyper-Kahler
calibrationscalibrations
(generalized) CY(generalized) CYspectral curvespectral curve
mirror symmetrymirror symmetry
self-dual geometryself-dual geometry
quantization quantization
instantons instantonsRandom Walkmonopolesmonopoles
Higgs-bundlesHiggs-bundles
X simply-connected Kähler manifold, dimC X=3, c1(X) = 0, no torsion.
X
Calabi-Yau threefolds
Diffeomorphism type of X is completely fixed by b3(X) and b2(X) plus classical invariants
F cl0 (t) =
Z
X
16t3; t 2 H 2(X ;Z)
F cl1 (t) =
112
Z
Xt ^c2
Miles Reid’s Fantasy:“There is only one CY space”
M g
b2 = 0
All CY connected through conifoldtransitions S3 → S2
b2 = 1Kähler CYs
complexstructuremoduli
Gromov-Witten InvariantsExact instanton sum 2 ( , )d H X
genus g X
Moduli stack of stable maps
GWg;d =
Z
[M g (X ;d)]v i r
1 2 Q
Topological String (A model)
F qug (t) =
X
d
GWg;de¡ dt
Quantum corrections, tH2(X,C)
Ztop(t;¸) = expX
g
¸2g¡ 2Fg(t)
Partition function
Fg(t) = F clg (t) +F qu
g (t)
Topological String (B model)Complex moduli, tH2,1(X)
Localizes on (almost) constant maps df=0
f
Kodaira-Spencer field theory
• genus 0: classical Variation of Hodge Structures
• genus 1: analytic Ray-Singer torsion
• genus 2 and higher: quantum corrections
quantization of complex structuremoduli space M X
CY
3S
3T
CY fibered by special Lagrangian T3
[Strominger, Yau, Zaslov]
network ofsingularities
S1 shrinks
D-Branes
X0 ( ) ( )evenY K X H X
coherent sheaves
A
X
13( ) ( )Y K X H X
special Lagrangians+ gauge bundle
B
homological mirror
symmetry)
derived category Fukaya category
Symplectic vector space V = ¤ C »= H 3(X ;C)
Z
X®^¯
Charge Lattice (B-model)¤B = K 1(X ) »= H 3(X ;Z)
Period Map & Quantization
moduli space of CY MX
hol 3-form dz1 dz2 dz3
V
Lagrangian cone L=graph (dF0) semi-classical state ψ ~ exp F0
L
Topological String Partition Function
2 2exp ( ), gtop g
g
Z F t
Transforms as a wave function (metaplectic representation) under Sp(2n,Z)
change of canonical basis (A,B)
A-Model
complexifiedKähler cone
symplectic vector space
complexifiedKähler volume
H ev(X ;C)
h®;¯i = index D® ¯ ¤
1¸
ek+iB
+ GW quantum corrections
F cl0 =
t3
6 2
Charged objects: D-branes
3CY time
charged particles
( , ) ( , )evp q H X
electric-magnetic charges
Large volume:•q electric D0-D2•p magnetic D4-D6
Gauge Theory Invariants
Coherent sheaf E ! X
(conjectured) Donaldson-Thomas invariant
D(¹ ) =R
[M ¹ ]v i r 1
Moduli space of stable sheaves
Charge ¹ = [E] 2 K 0(X ) = ¤
Gauge Theory
D() is conjectured to be the partition function of a 6-dim topologically twisted gauge theory
S =
Z
XjF j2 + jDÁi j2 + [Ái ;Áj ]2 + :::
Localizes to 6-dim version of Hitchin’s equations
Generating function
Choose polarization
K 0(X ) = ¤ = ¤+ ©¤¡
¹ = (p;q) 2 ¤
To make contact with GW-theory
¤+ = H 0 ©H 2; ¤¡ = H 4 ©H 6
Partition function
Zgauge(p;Á) =X
q2¤ ¡
D(p;q)eiq¢Á
GW-DT Equivalence[Maulik, Nekrasov, Okounkov, Pandharipande]
Consider the case of rank one, p = (1,0)(ideal sheaves)
where
Zgauge(p;Á) = Ztop(t;¸)
Á = (t;¸) 2 H 2(X ) ©H 0(X )
Donaldson-Thomas Invariants
U(1) gauge theory + singularities q=(d,n)
n = ch3 » TrF 3
d = ch2 » TrF 2
instanton strings
Zgauge =X
d;n
D(d;n)edt+n¸
Strong-weak coupling
GW ¸ ! 0
Ztop = expX
g;d
GWg;d¸2g¡ 2edt
DT ¸ ! 1
Zgauge =X
n;d
Dn;den¸ edt
Two expansion of single analytic function?
Gopakumar-Vafa invariants
charges q
( , ) log dettop qF t
1CY S
M-theory limit
virtual loopsof M2 branes
¸ ! 1
¸
GV Partition function
Gas of 5d charged & spinning black holes
Z(¸;t) =Y
n1;n2d ;m
³1¡ e (n1+n2+m)+td
´ ¡ N md
GV-invariants (integers)N m
d »p
d3 ¡ m2
Infinite products of Borcherds type.Automorphic properties?
Topological String Triality
top. stringsGromov-Witten
M-theoryGopakumar-
Vafa
gauge theoryDonaldson-
Thomas
1 2
2
2,0
0
1( )
, 0
3d partitions
exp
1
g gtop g
g
n n
n n
Z GW
e
e
Stat-Mech: 3d Partitions
GW
GV
DT
OSV Conjecture[Ooguri, Strominger, Vafa]
Consider the limit
where
p! 1
Zgauge(p;Á) » jZtop(t;¸)j2
p+ iÁ =
µt¸
;1¸
¶2 H 2(X ) ©H 0(X )
Hitchin’s theory of 3-forms
0
1 Im( )
2
Legendre
XS F
i
3 61 2 3( ) , e e e
( )i
0
d d
integrable complex structure
Integral structure
3( , )H X
“attractive”CY’s
3( , )H X
Bohr-Sommerfeld quantization of moduli space MX
N free non-relativistic fermions
p
Zgauge =X
f er mions
e¡ ¸ E +iµP
C2 = E =X
i
12p2
i ; C1 = P =X
i
pi
Black Hole statesYM instantons
/( ) ngauge
n
Z D n e
( )( )n
S nD n e
Free fermion states
Egauge
E
Z e
31
24N
gaugeZ e
Ztop(t;¸) =I
dxx
Y
p2Z ¸ 0+ 12
³1+ xept+ 1
2 ¸ p2´ ³
1+ x¡ 1ept¡ 12 ¸ p2
´
= expX
g
¸2g¡ 2Fg(t)
quasi-mdoular form of wt 6g-6
Two conjectures,related by modular transformation?
¸ ! 1=
OSV
Z(p;Á) » jZtop(t;¸)j2
p! 1
p+ iÁ =
µt¸
;1¸
¶
DT
Z(p;Á) = Ztop(t;¸)
p= 1
Á = (t;¸)
Rank zero, divisor
CY X
P
p= (0;c1) = (0;[P ]) q= (ch2;ch3) = (d;n)
Zgauge elliptic genus of modulus space M P
Elliptic GenusIf M is a CY k-fold
S1-equivariant y-genus of the loop space LM
weak Jacobi-form of wt 0 and index k/2
elll(M ;z;¿)
ell(¿;z + m¿ + n) = e¡ ¼id(m2¿+2mz)=2ell(¿;z)
ellµ
a¿ + bc¿ + d
;z
c¿ + d
¶
= e¼i k cz 22(c¿ + d) ell(¿;z)
Elliptic Genus
elll(M ;z;¿) =X
d;n
D(d;n)e2¼idze2¼in¿
Fourier expansion
Modular properties
Zgauge(t;¸) = ell(M P ;t;¸)
Topological String Theory
• Universal, deep, but mysterious object that captures many interesting connections between physics and geometry.