Localization of VortexPartition Function of N=(2,2)

Super Yang-Mills Theory

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吉田豊Y. YoshidaarXiv:1101.0872[hep-th]

Introduction

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Moore, Nekrasov & Shatashivli (1998), Nekrasov(2002)

Instanton partition function in N=2 4-dim SYM

k-Instanton partition function by Localization formula ex) G=U(N) vector multiplet

Instanton number

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Instanton partition function with surface operator in N=2 SYM 　Alday et al(2009), Alday & Tachikawa, Bruzzo et al(2010)

Instanton number The first Chern number

Dimofte, Gukov & Hollands (2010)

: Vortex partition function in N=(2,2) 2dim SQED ?

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The moduli space of abelian vortex with vortex number k in two dimension is isomorphic to

Jaffe & Taubes(1980)

Equivariant character

k-vortex partition function for N=(2,2) SQED with single chiral multiplet

contour integral representation

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Contribution from a vector multiplet

Contribution from a chiral multiplet

vortex partition function of N=(2,2) SQED with chiral multiplet ？

twisted mass

Vortex counting from topological vertex

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5d Nekrasov partition(K-theoretic instanton counting)

Introduction of Surface operatorIntroduction of A-brane

Closed A-model on toric CY

G=U(1) 4-dim pure N=2 SYM ex)

Theory induced on the surface operator isN=(2,2) U(1) SQED with single chiral mutiplet

string side gauge theory side

Kozcaz, Pasquetti & Wyllard(2010)

content 1. Introduction 2.Vortices in 2d super Yang-Mills

theories 3. Localization of vortex in N=(2,2) SYM 4. Vortex partition and equivariant

character 5. Relation to geometric indices 6. Summary

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Vortices in 2d super Yang-Mills Theories

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Vortex equation (Bogomol’nyi equation) with G=U(N)

1.This equation preserves half of the supersymmetry.2. On-shell action.

Vortex number is defined by the first Chern number

complexified FI-parameter

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Super YM theory with 8 SUSY (2-dim N=(4,4) SYM)

The vector multiplet in N=(4,4) SYM consists of

Hypermultiplets in N=(4,4) theory consists of

matter content of N=(4,4) theory

N=(2,2) vector multiplet

N=(2,2) adjoint chiral multiplet

N=(2,2) fundametnal chiral multiplet

N=(2,2) anti-fundametnal chiral multiplet

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Vacuum (Higgs branch)r:FI-parameter

Symmetry group of Vacuum

Bosonic part of Lagrangian

Global gauge groupFlavor group

twisted mass

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k-vortex moduli space in (p+2)-dim U(N) SYM with 8 SUSYby k Dp- N D(p+2) brane construction(Hanany & Tong 2002)

0 1 2 3 4 5 6 7 8 9NS5 o o o o o oD2 o o o D0 o

vortex partition function(zero mode theory) in N=(4,4) SYMfrom brane system

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D0-D0

D0-D2

I : orientational moduli

B : translational moduli

DRED of vector with gauge group

DRED of adjoint chiral multiplet

DRED of chiral malutiplet

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:k-vortex partition functions Chen and Tong (2006)

Mass deformation

D-term condition

The moduli space of k-vortex Eto et al(2005)Hanany & Tong(2002)

We consider mass deformation N=(4,4) theory. 　 Taking large mass limit, we obtain N=(2,2) SYM with N chiral multiplets.

Edalati & Tong (2007)

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DRED of 2d (0,2) chiral multipet

DRED of 2d (0,2) fermi multipet

In the presence of the mass term, vortex partition function is deformed

multiplets decouple from the vortex theory

heavy mass limit

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k-vortex partition function for N=(2,2) U(N) SYM with N-fundamental matter

with

This action is expressed in Q-exact form

Localization of vortex partition functions in N=(2,2) SYM

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SUSY transformation generates the following vector field on

Nekrasov (2002) Bruzzo et al (2002)

Superdeterminant

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k-vortex parition function in G=U(N) 　 N=(2,2) SYM N-flavor

Vortex partition function in G=U(1) 　 N=(2,2) SQED

This agree with the result from the equivariant character

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Vortex partition and equivariant character

We introduce the following torus action

Vortex moduli space

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At the fixed points, we can decompose the representation space as

Gauge transformation

Restriction map

Fixed point condition

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2d partition(Young diagram)

1d partition

In the case of 4-dim instanton…

In the case of 2-dim vortex

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•character of each spaces

Infinitesimal gauge transformation

Tangent space of k-vortex moduli space

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equivariant character

3d vortex partition function

Replacement

Relation to geometric indices

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-genus of complex manifold M

Equivariant case

The fixed points

The weight at the point

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3d vortex partition function

This corresponds to geometric genus

This corresponds to Euler number

N=(2,2) case

N=(4,4) case

Summary We have obtained N=(2,2) vortex partition function from the mass

deformation of N=(4,4) vortex partition function.

N=(2,2) vortex partition function can be written with Q-exact form⇒ We can apply Localization formula　　・especially we reproduce abelian vortex from open BPS state counting

or equivariant character of

Vortex parition function is expressed by 1d partitionCf) Nekrasov partition is expressed by 2d partition(Young diagram).

3d vortex partition is related to certain geometric indices of the k-vortex moduli space

Future direction Relation to integrable structure( KP hierarchy, spin chain), etc…

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