Superconformal index of low-rank gauge theories via the Bethe Ansatz

Benini, Francesco; Rizi, Giovanni

doi:10.1007/jhep05(2021)061

Cited by 19 publications

(8 citation statements)

References 54 publications

(66 reference statements)

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“…we see that at the geometric level the Euler angles are quotiented by 1 m whereas the thermal circle pinches. 29 This quotient is exactly the same as what one expects for a lens space L(m, 1).…”

Section: Jhep11(2021)047supporting

confidence: 68%

“…At large-N , the black hole entropy is exactly reproduced from a Legendre transformation of the index and corresponds to the extremization of an entropy functional à la the attractor mechanism [16]. Further developments on the N = 4 index and its relation to AdS 5 black holes include [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. These methods have also been extended to the analysis of superconformal field theories with N = 1 supersymmetry [32][33][34][35][36][37][38][39][40].…”

Section: Jhep11(2021)047 1 Introduction and Summarymentioning

confidence: 99%

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SL(3, ℤ) Modularity and New Cardy limits of the $$ \mathcal{N} $$ = 4 superconformal index

Jejjala

Lei

Leuven

et al. 2021

J. High Energ. Phys.

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The entropy of 1/16-th BPS AdS5 black holes can be microscopically accounted for by the superconformal index of the $$ \mathcal{N} $$ N = 4 super-Yang-Mills theory. One way to compute this is through a Cardy-like limit of a formula for the index obtained in [1] using the “S-transformation” of the elliptic Γ function. In this paper, we derive more general SL(3, ℤ) modular properties of the elliptic Γ function. We then use these properties to obtain a three integer parameter family of generalized Cardy-like limits of the $$ \mathcal{N} $$ N = 4 superconformal index. From these limits, we obtain entropy formulae that have a similar form as that of the original AdS5 black hole, up to an overall rescaling of the entropy. We interpret this both on the field theory and the gravitational side. Finally, we comment on how our work suggests a generalization of the Farey tail to four dimensions.

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Section: Jhep11(2021)047supporting

confidence: 68%

Section: Jhep11(2021)047 1 Introduction and Summarymentioning

confidence: 99%

SL(3, ℤ) Modularity and New Cardy limits of the $$ \mathcal{N} $$ = 4 superconformal index

Jejjala

Lei

Leuven

et al. 2021

J. High Energ. Phys.

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“…A larger set of saddles have been classified in [19], but the full set of important/contributing saddles is not understood in either approach. In particular, interesting continuum configurations of the Bethe-ansatz equations have been recently discovered in [37][38][39] whose role in the large-N limit is not fully understood.…”

Section: Jhep10(2021)207mentioning

confidence: 99%

The 4d superconformal index near roots of unity and 3d Chern-Simons theory

Ardehali

Murthy

2021

J. High Energ. Phys.

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We consider the S3×S1 superconformal index ℐ(τ) of 4d $$ \mathcal{N} $$ N = 1 gauge theories. The Hamiltonian index is defined in a standard manner as the Witten index with a chemical potential τ coupled to a combination of angular momenta on S3 and the U(1) R-charge. We develop the all-order asymptotic expansion of the index as q = e2πiτ approaches a root of unity, i.e. as $$ \overset{\sim }{\tau } $$ τ ~ ≡ mτ+n → 0, with m, n relatively prime integers. The asymptotic expansion of log ℐ(τ) has terms of the form $$ \overset{\sim }{\tau } $$ τ ~ k, k = −2, −1, 0, 1. We determine the coefficients of the k = −2, −1, 1 terms from the gauge theory data, and provide evidence that the k = 0 term is determined by the Chern-Simons partition function on S3/ℤm. We explain these findings from the point of view of the 3d theory obtained by reducing the 4d gauge theory on the S1. The supersymmetric functional integral of the 3d theory takes the form of a matrix integral over the dynamical 3d fields, with an effective action given by supersymmetrized Chern-Simons couplings of background and dynamical gauge fields. The singular terms in the $$ \overset{\sim }{\tau } $$ τ ~ → 0 expansion (dictating the growth of the 4d index) are governed by the background Chern-Simons couplings. The constant term has a background piece as well as a piece given by the localized functional integral over the dynamical 3d gauge multiplet. The linear term arises from the supersymmetric Casimir energy factor needed to go between the functional integral and the Hamiltonian index.

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“…In this approach, the index is written as a sum over solutions to some Bethe-Ansatz-like equations. See for example [67][68][69][70].…”

Section: Discussionmentioning

confidence: 99%

The exact Schur index in closed form

Pan¹,

Peelaers²

2021

Preprint

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The Schur limit of the superconformal index of a four-dimensional N = 2 superconformal field theory encodes rich physical information about the protected spectrum of the theory. For a Lagrangian model, this limit of the index can be computed by a contour integral of a multivariate elliptic function. However, surprisingly, so far it has eluded exact evaluation in closed, analytical form. In this paper we propose an elementary approach to bring to heel a large class of these integrals by exploiting the ellipticity of their integrand. Our results take the form of a finite sum of (products of) the well-studied flavored Eisenstein series. In particular, we derive a compact formula for the fully flavored Schur index of all theories of class S of type a 1 , we put forward a conjecture for the unflavored Schur indices of all N = 4 super Yang-Mills theories with gauge group SU (N ), and we present closed-form expressions for the index of various other gauge theories of low ranks. We also discuss applications to non-Lagrangian theories, modular properties, and defect indices.

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Superconformal index of low-rank gauge theories via the Bethe Ansatz

Cited by 19 publications

References 54 publications

SL(3, ℤ) Modularity and New Cardy limits of the $$ \mathcal{N} $$ = 4 superconformal index

SL(3, ℤ) Modularity and New Cardy limits of the $$ \mathcal{N} $$ = 4 superconformal index

The 4d superconformal index near roots of unity and 3d Chern-Simons theory

The exact Schur index in closed form

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