Expanding on the Cardy-like limit of the SCI of 4d $$ \mathcal{N} $$ = 1 ABCD SCFTs

Amariti, Antonio; Fazzi, Marco; Segati, Alessia

doi:10.1007/jhep07(2021)141

Cited by 24 publications

(17 citation statements)

References 60 publications

(103 reference statements)

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“…As noted in [41] for N = 4 SYM, and in [42,43] for more general groups, Z dyn is related to the partition function of pure Chern-Simons theory [64] on S 3 as…”

Section: Jhep10(2021)207mentioning

confidence: 95%

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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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“…As noted in [41] for N = 4 SYM, and in [42,43] for more general groups, Z dyn is related to the partition function of pure Chern-Simons theory [64] on S 3 as…”

Section: Jhep10(2021)207mentioning

confidence: 95%

“…More recently, the complete action (1.5) for (m, n) = (1, 0) was obtained in [41] by a direct method, involving a careful analysis of all perturbative terms in the Cardy-like limit. (See [42,43] for more recent related work. )…”

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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“…The entropy function is the Legendre transform of the black hole entropy; it can also be written as the complexified on-shell action of the Euclidean black hole geometry [36,37]. These results for the computation of the superconformal index have also been extended to the case of more general N = 1 gauge theories [25,[38][39][40] [30,33,34,[41][42][43][44].…”

Section: Introductionmentioning

confidence: 92%

The large-$N$ limit of 4d superconformal indices for general BPS charges

Colombo¹

2021

Preprint

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We study the superconformal index of N = 1 quiver theories at large -N for general values of electric charges and angular momenta, using both the Bethe Ansatz formulation and the more recent elliptic extension method. We are particularly interested in the case of unequal angular momenta, J 1 = J 2 , which has only been partially considered in the literature. We revisit the previous computation with the Bethe Ansatz formulation with generic angular momenta and extend it to encompass a large class of competing exponential terms. In the process, we also provide a simplified derivation of the original result. We consider the newly-developed elliptic extension method as well; we apply it to the J 1 = J 2 case, finding a good match with the Bethe Ansatz results. We also investigate the relation between the two different approaches, finding in particular that for every saddle of the elliptic action there are corresponding terms in the Bethe Ansatz formula that match it at large -N. Contents 1 Introduction 1 2 The superconformal index of quiver theories 4 3 Large -N saddle points and the effective action 8 3.1 Contour deformation 3.2 Continuum limit 3.3 String-like saddles 3.4 General saddles 4 The large -N limit with the Bethe Ansatz formula 22 4.1 The Bethe Ansatz formula 4.2 BAE solutions and saddle points of the elliptic action 4.3 Evaluation of the index 4.4 Relation with previous work and competing exponential terms 5 Summary and discussion 35 A The elliptic gamma and related functions 36 B Subleading terms in the Bethe Ansatz formula 40Recently, the case of Kerr-Newman BPS black holes in AdS 5 has attracted a lot of interest. There are known BPS black hole solutions for type IIB supergravity in AdS 5 × S 5 which

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“…A detailed analysis of the 3d supersymmetric Chern-Simons contact terms on the second sheet was carried out in [30]. 2 Complementary studies of the superconformal index have led to similar conclusions about the asymptotic behavior in [31,32].…”

Section: Introductionmentioning

confidence: 85%

Anomaly Matching Across Dimensions and Supersymmetric Cardy Formulae

Ohmori¹,

Tizzano²

2021

Preprint

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t Hooft anomalies are known to induce specific contributions to the effective action at finite temperature. We present a general method to directly calculate such contributions from the anomaly polynomial of a given theory, including a term which involves a U(1) connection for the thermal circle isometry. Based on this observation, we show that the asymptotic behavior of the superconformal index of 4d N = 1 theories on the "second sheet" can be calculated by integrating the anomaly polynomial on a particular background. The integration is then performed by an equivariant method to reproduce known results. Our method only depends on the anomaly polynomial and therefore the result is applicable to theories without known Lagrangian formulation. We also present a new formula that relates the behavior of 6d N = (1, 0) SCFTs on the second sheet to the anomaly polynomial.

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Expanding on the Cardy-like limit of the SCI of 4d $$ \mathcal{N} $$ = 1 ABCD SCFTs

Cited by 24 publications

References 60 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

The large-$N$ limit of 4d superconformal indices for general BPS charges

Anomaly Matching Across Dimensions and Supersymmetric Cardy Formulae

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