Defect in Gauge Theory and Quantum Hall States Taro Kimuraand Norton Lee Institut de Mathématiques de Bourgogne Université Bourgogne Franche-Comté France

2025-04-26 0 0 874.79KB 46 页 10玖币

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Defect in Gauge Theory and Quantum Hall States

Taro Kimura♠and Norton Lee♦

♠Institut de Mathématiques de Bourgogne, Université Bourgogne Franche-Comté, France

♦Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Republic of

Korea

E-mail: taro.kimura@u-bourgogne.fr,norton.lee@ibs.re.kr

Abstract: We study the surface defect in N= 2∗U(N)gauge theory in four dimensions

and its relation to quantum Hall states in two dimensions. We ﬁrst prove that the defect

partition function becomes the Jack polynomial of the variables describing the brane positions

by imposing the Higgsing condition and taking the bulk decoupling limit. Further tuning the

adjoint mass parameter, we may obtain various fractional quantum Hall states, including

Laughlin, Moore-Read, and Read-Rezayi states, due to the admissible condition of the Jack

polynomial.

arXiv:2210.05949v3 [hep-th] 22 Mar 2023

Contents

1 Introduction 2

2 Four Dimensional N= 2∗Gauge Theory 3

2.1 Introducing Surface defect 5

2.2 qq-character and eigenvalue equation 7

2.3 Bulk decoupling limit 10

2.4 Two particles case 12

2.4.1 Center of mass frame 12

2.4.2 Defect partition function 13

3 Jack Polynomial 15

3.1 Concrete expressions 16

4 Higgsing the Coulomb Moduli Parameters 16

4.1 Young Tableaux representation 22

4.2 Higher dimensions 25

5 Quantum Hall States 26

5.1 Laughlin State 26

5.2 Moore–Read state 28

5.3 Admissible Condition 30

5.3.1 Instanton sum formula 31

5.3.2 Gauge theory perspective: further Higgsing 36

6 Discussion and Future Direction 37

A Special Functions 39

A.1 Random Partition 39

A.2 Elliptic Functions 39

A.3 Higher rank Theta function 40

A.4 Orbifolded Partition 40

– 1 –

1 Introduction

The relation of low-energy physics of supersymmetric gauge theory and integrable system

has been an active research for decades [1–3]. One of the best-known story is the Seiberg-

Witten curve of the N= 2 supersymmetric gauge theories can be identiﬁed as the spectral

curve of the integrable systems. This correspondence was later extended to the quantum

level by Nekrasov and Shatashivilli in [4,5], with the gauge theories subjected to the Ω-

deformation. This deformation introduces two parameters (ε1, ε2)associated to the rotation

on the two orthogonal plane in R4=C2. The partition function Zand BPS observables

can be computed exactly by localization technique for a variety of gauge theories [6]. In the

limit (ε1, ε2)→(0,0), the classical integrable system is recovered. The Nekrasov-Shatashivilli

limit (NS-limit for short) ε1→~and ε2→0results in an N= (2,2) supersymmetry being

preserved in the ﬁxed plane. One expects to get the quantum integrable system.

From gauge theory to integrable model

One is naturally to ask the question of computing the wavefunction of the integrable sys-

tem. The stationary state wave function, in the context of Bethe/gauge correspondence, are

the vacua of the two-dimensional N= (2,2) theory. In order to get the stationary wave-

function, we compute the expectation value of a special observable in the two dimensional

theory - a surface defect in the four dimensional theory [7–12]. It turns out that induction

of co-dimensional two surface defect provides a powerful tool in the study of Bethe/gauge

correspondence. The parameter of the defect becomes the coordinates that the wavefunction

depends on. The four dimensional theory with a co-dimensional two surface defect can be

realized as a theory on an orbifold. The localization computations extend so as to compute

the defect partition function and expectation value of BPS observables.

Our scope is on the class of qq-characters observable in the gauge theory [6]. The main

statement in [13] proves certain vanishing conditions for the expectation values of the qq-

observables, both with or without defects. These vanishing conditions, called non-perturbative

Dyson-Schwinger equations, can be used to construct KZ-type equations [14] satisﬁed by the

partition function [15,16]. In the NS-limit, the KZ-equations becomes a Schrödinger-type

equation satisﬁed by the partition function.

Jack polynomial and quantum Hall state

The Laughlin wavefunction has provied a key to understand the quantum Hall eﬀect (QHE).

It models the simplest abelian FQH and is the building blocks of model wavefuntion of more

general states, both abelian and non-abelian such as Moore-Read and Read-Rezayi state.

The wavefunctions of such models, aside from the Gaussian factor which we will drop, are

conformally-invariant multivariable polynomials. All three of Laughlin, Moore-Read, and

Read-Rezayi state wavefunctions are proven to be special cases of the Jack polynomial J

with the Jack parameter κtaking negative rational value [17–19].

– 2 –

Summary and organization

In this paper we will establish the relations between three objects: the surface operator in

the 4-dimensional N= 2∗theory, the Jack polynomials, and fractional quantum Hall states.

The main end-result is to realize the fractional quantum Hall states as instanton partition

function of 4-dimensional N= 2∗gauge theory with the presence of full-type surface defect

in the following simultaneous limits

(i) Nekrasov-Shatashivili limit ε2→0,

(ii) Bulk-decoupling limit q=e2πiτ →0,

(iii) Higgsing the Coulomb moduli parameters {aα}to sum of adjoint mass mand Ω-

deformation paranmeter ε1,

(iv) Tuning the ratio between the adjoint mass mand ε1to control the ﬁlling factor of the

quantum Hall states.

The paper is organized as follows:

•In section 2we will review the instanton partition function of N= 2∗and prove

that in the Nekrasov-Shatashivili limit ε2→0(i) the defect partition function is the

eigenfunction of the elliptic Calogero-Moser system .

•In section 2.3, we will show that in the trogonometric limit τ→i∞(ii) the Calogero-

Moser Hamiltonian becomes the Laplace-Beltrami operator after a canonical transfor-

mation. The Jack polynomials are the eigenfunction of the Laplace-Beltrami operator.

•In section 3we will review some basic property of Jack polynomials.

•In section 4we will impose Higgsing condition (iii) to the N= 2∗supersymmetric

gauge theory. The Higgsing truncates the inﬁnite summation of the instanton partition

function. By using the Young Tableaux representation of the instanton conﬁguration,

we prove that the defect partition function becomes the Jack polynomial after Higgsing.

•In section 5we recover both the Laughlin and Moore-Read quantum Hall states from

the defect partition function with a ﬁle tuning of the adjoing mass m(iv). We also

discuss about the admissible condition satisﬁled by the Jack polynomial.

•We end this paper with discussion about potential future work in section 6.

2 Four Dimensional N= 2∗Gauge Theory

We consider N= 2∗U(N)gauge theory in four dimensions with adjoint mass m. The

vacuum of the theory is characterized by Coulomb moduli parameters a= (a1, . . . , aN)and

– 3 –

exponentiated complex gauge coupling

q=e2πiτ , τ =4πi

g2+ϑ

2π(2.1)

The instanton partition function can be calculated via supersymmetric localization com-

putation in the presence of an Ω-background, whose deformation parameters are (ε1, ε2).

The instanton conﬁguration is labeled by a set of Young diagrams λ= (λ(1), . . . , λ(N)),

λ(α)= (λ(α)

1, λ(α)

2, . . . )satisfying

λ(α)

i≥λ(α)

i+1 ≥0(2.2)

which denotes the number of boxes on each row in the Young diagrams. We deﬁne the formal

sum of the exponentials

α=1

eaα,K=

α=1 X

(i,j)∈λ(α)

eaα+(i−1)ε1+(j−1)ε2.(2.3)

The pseudo-measure associated to the instanton conﬁguration is deﬁned using the index

functor Ethat converts the additive Chern class character to multiplicative class

E"X

naexa#=











x−na

a(rational)

(1 −e−xa)−na(trigonometric)

θ(e−xa;p)−na(elliptic)

(2.4)

where na∈Zis the multiplicity of the Chern root xa.θ(z;p)is the theta function deﬁned

in (A.8). We remark the hierarchical structure, θ(e−x;p)p→0

−−−→ 1−e−x=x+O(x2). In this

paper, we mostly apply the rational convention, which corresponds to four dimensional gauge

theory. The pseudo measure associated to the instanton conﬁguration λis computed by:

Z[λ] = E[(1 −em)(NK∗+q1q2N∗K−P1P2KK∗)] .(2.5)

qi=eεiare the exponentiated Ω-deformation parameters with Pi= 1 −qi. Given a virtual

character X=Panaexawe denote by X∗=Panae−xaits dual virtual character.

The supersymmetric localization equates the supersymmetric partition function of the

Ω-deformed b

A0U(N)theory of the grand canonical ensemble

Zinst(a, m, ~ε;q) = X

q|λ|Z(a, m, ~ε)[λ](2.6)

The pseudo-measure Z[λ]can be expressed in terms of products of Γ-functions

Z[λ] = Y

(αi)6=(βj)

Γ(ε−1

2(xαi −xβj −ε1))

Γ(ε−1

2(xαi −xβj )) ×Γ(ε−1

2(˚xαi −˚xβj ))

Γ(ε−1

2(˚xαi −˚xβj −ε1))

×Γ(ε−1

2(xαi −xβj −m))

Γ(ε−1

2(xαi −xβj −m−ε1)) ×Γ(ε−1

2(˚xαi −˚xβj −m−ε1))

Γ(ε−1

2(˚xαi −˚xβj −m)) .(2.7)

– 4 –

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摘要：

DefectinGaugeTheoryandQuantumHallStatesTaroKimuraandNortonLee}InstitutdeMathématiquesdeBourgogne,UniversitéBourgogneFranche-Comté,France}CenterforGeometryandPhysics,InstituteforBasicScience(IBS),Pohang37673,RepublicofKoreaE-mail:taro.kimura@u-bourgogne.fr,norton.lee@ibs.re.krAbstract:Westudythesur...

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Defect in Gauge Theory and Quantum Hall States Taro Kimuraand Norton Lee Institut de Mathématiques de Bourgogne Université Bourgogne Franche-Comté France

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