Localization of Vortex Partition Function of N=(2,2) Super Yang-Mills Theory 2011 年 4 年 27 年 SAL@KEK 1 年年年 Y. Yoshida arXiv:1101.0872[hep-th]
Feb 24, 2016
Localization of VortexPartition Function of N=(2,2)
Super Yang-Mills Theory
2011 年 4 月 27 日 SAL@KEK 1
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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