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

Jejjala, Vishnu; Lei, Yang; Leuven, Sam van; Li, Wei

doi:10.1007/jhep11(2021)047

Cited by 20 publications

(6 citation statements)

References 68 publications

(256 reference statements)

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“…The paper [85], which appeared on the arXiv the same day as the first version of this paper, has some overlap with our section 3. The paper [86], which appeared on the arXiv soon after, has some overlap with our section 2.…”

Section: Rational Pointsmentioning

confidence: 90%

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

Ardehali¹,

Murthy²

2021

Preprint

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We consider the S 3 × S 1 superconformal index I(τ ) of 4d 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 S 3 and the U (1) R-charge. We develop the all-order asymptotic expansion of the index as q = e 2πiτ approaches a root of unity, i.e. as τ ≡ mτ + n → 0, with m, n relatively prime integers. The asymptotic expansion of log I(τ ) has terms of the form 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 S 3 /Z m . We explain these findings from the point of view of the 3d theory obtained by reducing the 4d gauge theory on the S 1 . 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 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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Section: Rational Pointsmentioning

confidence: 90%

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

Ardehali¹,

Murthy²

2021

Preprint

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“…Computing such indices one can obtain explicit expressions which often exhibit interesting (mock/quasi/etc) modular behavior and its generalizations. For example, in the supersymmetric index in four dimensions [668][669][670] one can see hints of SL(3, Z) modularity [671][672][673][674]. Specializing to N = 2 theories and to Schur indices [74,675] these hints of SL(3, Z) modularity become hints of SL(2, Z) modularity [676].…”

Section: Generalizations Of Automorphymentioning

confidence: 99%

A Panorama Of Physical Mathematics c. 2022

Bah¹,

Freed²,

Moore³

et al. 2022

Preprint

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What follows is a broad-brush overview of the recent synergistic interactions between mathematics and theoretical physics of quantum field theory and string theory. The discussion is forwardlooking, suggesting potentially useful and fruitful directions and problems, some old, some new, for further development of the subject. This paper is a much extended version of the Snowmass whitepaper on physical mathematics [1].

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“…10 Alternatively, one may be able to integrate (using residue calculus in particular) and then asymptotically analyze. See [23,26,34,35] for such analyses in the present context.…”

Section: Jhep01(2022)062mentioning

confidence: 99%

Decomposition of BPS moduli spaces and asymptotics of supersymmetric partition functions

Ardehali

Hong

2022

J. High Energ. Phys.

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We present a prototype for Wilsonian analysis of asymptotics of supersymmetric partition functions of non-abelian gauge theories. Localization allows expressing such partition functions as an integral over a BPS moduli space. When the limit of interest introduces a scale hierarchy in the problem, asymptotics of the partition function is obtained in the Wilsonian approach by i) decomposing (in some suitable scheme) the BPS moduli space into various patches according to the set of light fields (lighter than the scheme dependent cut-off Λ) they support, ii) localizing the partition function of the effective field theory on each patch (with cut-offs set by the scheme), and iii) summing up the contributions of all patches to obtain the final asymptotic result (which is scheme-independent and accurate as Λ → ∞). Our prototype concerns the Cardy-like asymptotics of the 4d superconformal index, which has been of interest recently for its application to black hole microstate counting in AdS5/CFT4. As a byproduct of our analysis we obtain the most general asymptotic expression for the index of gauge theories in the Cardy-like limit, encompassing and extending all previous results.

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

Cited by 20 publications

References 68 publications

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

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

A Panorama Of Physical Mathematics c. 2022

Decomposition of BPS moduli spaces and asymptotics of supersymmetric partition functions

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