in supersymmetric gauge theories

(1)

Counting non-Abelian vortices

in supersymmetric gauge theories

Toshiaki Fujimori (INFN, University of Pisa)

Taro Kimura (University of Tokyo, RIKEN) Muneto Nitta (Keio University)

Keisuke Ohashi (Kyoto University) In collaboration with

(2)

1. Introduction

[Abrikosov, 1953]

[Nielsen-Olesen, 1973]

[Hanany-Tong, 2003]

[Auzzi-Bolognesi-Evslin-Konishi-Yung, 2003]

● Higgs phase of Non-Abelian gauge theory

● internal degrees of freedom (orientational moduli)

vortices in the Abelian Higgs model

Non-Abelian vortices

● BPS configurations in supersymmetric gauge theories

Vortices : topological soliton (non-trivial winding of phase)

(3)

vortices : codim = 2

4d : string-like object

● correspondence between 4d gauge theory

and 2d effective theory on vortex worldsheet

3d : particle-like object

● 3d mirror symmetry (duality)

2d : instanton-like object

● non-perturbative effects ( twisted superpotential )

BPS vortices in supersymmetric gauge theories

vortices charged particles

2d counterpart of 4d Yang-Mills instantons

[Intriligator-Seiberg, 1996] , etc

[Dorey, 1998], [Shifman-Yung, 2004], [Hanany-Tong, 2004], etc

(4)

vortices : codim = 2

2d : instanton-like object

● non-perturbative effects ( twisted superpotential )

4d : string-like object

3d : particle-like object

● 3d mirror symmetry (duality)

vortices charged particles

[Intriligator-Seiberg, 1996] , etc

[Dorey, 1998], [Shifman-Yung, 2004], [Hanany-Tong, 2004], etc

2d counterpart of 4d Yang-Mills instantons

● correspondence between 4d gauge theory

and 2d effective theory on vortex worldsheet

(5)

4d instanton counting [Nekrasov, 2002] , etc

● prepotential

2d vortex counting

● twisted superpotential

● instanton partition function

● vortex partition function

[Shadchin, 2006] , etc

N = 2 SU (N )_C+F

CP^N^¡1 SU (N )_C+F

SU(N ¡ 1) £ U(1)

=

N = (2; 2)

vortex conting in 2d supersymmetric gauge theories

(6)

NS5

N D4

x⁶ x^7,8,9

1 2 3 4 5 6 7 8 9

N D4 ● ● ● ●

NS5 ● ● ● ● ●

x^4,5

● Higgs branch of U(N) gauge theory with N fundamental hypermultiples

(7)

D-brane construction of vortex moduli space

NS5

k D2

N D4

1 2 3 4 5 6 7 8 9

N D4 ● ● ● ●

NS5 ● ● ● ● ●

k D2 ● ●

● k D2 branes = k vortices

x⁶ x^7,8,9 x^4,5

[Hanany-Tong, 2003]

● Higgs branch of U(N) gauge theory with N fundamental hypermultiples

(8)

ADHM like construction of the vortex moduli space NS5

NS5

k D2

N D4

● k D2 branes = k vortices

x⁶ x^7,8,9 x^4,5

● U(k) gauge theory with one adjoint , N fundamental

(9)

vortex moduli space =

half dimensional subspace of instanton moduli space Kahler quotient construction from ADHM construction

ADHM construction (hyperKahler quotient construction )

Kahler quotient construction for vortex moduli space

[Hanany-Tong, 2003]

(10)

N-tuple of integers (1d Young tableaux) N-tuple of Young tableaux

N = 2 SU (N )

_C+F

CP

^N^¡1

SU (N )

_C+F

SU(N ¡ 1) £ U(1)

=

N = (2; 2)

2d U(N) gauge theory with N fundamentals

partition functions fixed points of a torus action

from instanton partition function of 4d U(N) SYM vortex moduli space =

half dimensional subspace of instanton moduli space

[Yoshida, 2011], [Bonelli-Tanzini-Zhao, 2011]

[Shadchin, 2006],

(11)

D-brane construction

field theoretical construction moduli matrix formalism

Kahler quotient

vortex conting in 2d supersymmetric gauge theories

(12)

Kahler quotient

Kahler metrics do not agree !!

(13)

Kahler quotient

question: if two descriptions give the same

vortex partition function or not vortex conting in 2d supersymmetric gauge theories

fermionic zero modes?

●

Kahler metrics do not agree !!

(14)

Kahler quotient

fermionic zero modes?

vortex partition function from moduli matrix formalism

●

agreement of the partition functions

question: if two descriptions give the same

vortex partition function or not Kahler metrics do not agree !!

(15)

Plan of talk

1. introduction

2. BPS vortices in the Higgs phase

3. vortex effective action and localization

4. torus action on zero modes

5. summary

● moduli matrix formalism

● fixed points of a torus action

● bosonic and fermionic zero modes

(16)

2. BPS Vortices in the Higgs phase

Higgs phase ・・・ completely broken gauge symmetry

gauge theory with chirals in fundamental rep.

is an matrix

gauge and flavor symmetries

2d supersymmetric gauge theories (4 SUSY)

(17)

: FI parameter Lagrangian (bosonic part which is relevant to vortex configuration)

scalar potential (D-term potential)

completely broken gauge symmetry

(18)

vortex configuration ・・・ non-trivial winding

・・・ number of vortices

[Fujimori et al, 2007]

● U(N) case

● more generaly

( : simple Lie group )

(19)

BPS equations

Bogomol’nyi bound for Euclidean action

moduli space of BPS vortices

(20)

BPS equations in the holomorphic gauge : complexified gauge group

gauge symmery V-transformation (equivalence relation)

(21)

BPS equations in the holomorphic gauge

gauge symmery V-transformation (equivalence relation) moduli matrix formalism

: complexified gauge group

moduli matrix

[Eto et al, 2006]

(22)

gauge invariant quantities

”baryonic’’ invariant

k vortex configurations boundary condition for

boundary condition [Eto et al, 2007]

(23)

: complex Grassmaniann constant : moduli space of vacua

: equivalence relation

: moduli paramters (coordinates of the moduli space) Example 1

(24)

U(2) gauge theory with fundamental scalars

: vortex position

fixed form of the moduli matrix

: inhomogeneous coordinate of CP^1

moduli space of a single U(2) vortex

for

(25)

vortex effective theory ・・・ non-linear sigma model

3. effective theory and localization

: Kahler metric on the moduli space path integral

vortex partition function

vortex effective theory

[Shadchin, 2006]

localization

field theory description

: moduli parameters

(26)

Kahler metric

perpendicular to the gauge zero modes basis of zero modes

linearized BPS equations

(27)

fermionic zero modes around BPS background

U(N) gauge theory with N fundamental chiral multiplets

supermultiplets in the effective theory : positive definite ・・・ no zero mode

(28)

twisted boundary conditions (compact direction)

deformations

potential on the moduli space dimensional reduction

with the twisted boundary condition

torus action deformation

spatial rotation

(29)

vortex effective action

: Killing vector of the torus action

dimensional reduction with twisted boundary condition mass and Ω-deformations in the effective action

(30)

“BRST“ transforamtion

● The action is invariant under the BRST transformation

● The action is BRST exact

: Kahler metric : holomorphic Killing vector

(31)

one parameter deformation

the saddle point approximation becomes exact

: fixed points of the torus action extrema of the effective action is t independent

localization

(32)

Gaussian integral

: generator of the torus action on the zero modes

quadratic part of the effective action

= tangent space

(33)

Moduli matrix corresponding to the fixed points

neighborhood of the fixed point

fixed points are labeled by integers

(34)

fixed form

: coordinates around the fixed point

(35)

torus action on

torus action on the coordinates

vortex partition function for

U(N) gauge theory with N fundamental chiral multiplets

This agrees with the vortex partition function

obtained from the D-brane construction

4d instantons in U(N) super Yang-Mills

(36)

addional anti-fundamental chiral multiplets

no additional bosonic moduli parameter

●

● additional fermionic zero modes

● structure of the fixed points is the same

these fermionic zero modes are not paired

with bosonic zero modes

(37)

this also agrees with the D-brane construction

4d instantons in U(N) theory with N (anti-)fundamentals contribution of the additional fermionic zero modes

: generator of the torus action on

the additional fermionic zero modes

(38)

superpotential

mass for vector multiplet scalar

U(N) N fundamental + N anti-fundamental + 1 adjoint

torus action is modefied

enhancement of SUSY

(39)

additional fermionic zero modes

4d instantons in theory ( with adjoint mass) this agrees with the D-brane construction

(40)

4. summary

● The vortex partition function from the field theoretical point of view

● moduli matrix ・・・ torus action around the fixed points

● agreement with the D-brane construction

● our method can be easily generalized to the theory without D-brane construction

● Application of our method for the non-perturbative physics in 2d