$SL(3,\mathbb{Z})$ Modularity and New Cardy Limits of the $\mathcal{N}=4$ Superconformal Index

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

doi:10.48550/arxiv.2104.07030

Cited by 4 publications

(16 citation statements)

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“…At leading order the resulting expression for the superconformal index does indeed match the entropy function of AdS 5 × S 5 black holes [22], both in the large-N limit [21,[23][24][25][26][27] and in the limit of large conserved charges (i.e. the Cardy limit) [20,24,[28][29][30][31][32][33][34][35]. 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].…”

Section: Introductionmentioning

confidence: 77%

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

Colombo¹

2021

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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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Section: Introductionmentioning

confidence: 77%

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

Colombo¹

2021

Preprint

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“…In recent years, superconformal and topologically twisted indices [1-3][4] have been useful tools to understand the statistical meaning of the Bekenstein-Hawking entropy of BPS black holes in AdS 4 [5] and AdS 5 [6][7][8]. As a natural extension of some of these developments, in the context of AdS 5 /CFT 4 , the superconformal index has allowed to uncover zerotemperature phases of SU (N ) N = 4 SYM [9,10]: these phases emerge in certain limits of a chemical potential [9,11,12], and in broad terms, two analytic approaches (to be recalled below), have been useful to study them. The goal of this paper is to revisit and sharpen one of these approaches.…”

Section: Introductionmentioning

confidence: 99%

“…2) integrate and expand the result [9,12,[20][21][22]. 2 Once all the dust settles, we expect the answer from the two approaches to match.…”

Section: Introductionmentioning

confidence: 99%

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On the 4d superconformal index near roots of unity: Bulk and Localized contributions

Cabo-Bizet¹

2021

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This paper studies one particular approach to the expansion near roots of unity of the superconformal index of 4d SU (N ) N = 4 SYM. In such a limit, middledimensional walls of non-analyticity emerge in the complex analytic extension of the integrand. These walls intersect the integration contour at infinitesimal vicinities and come from both, the vector and chiral multiplet contributions. We will call these intersections bits and the complementary region bulk. For N = 2 the index can be conveniently divided in three integrals, two of them over regions covering the vector and chiral bits, the third one over the bulk. At very leading order in an expansion near roots of unity, these three sub-integrals match the contributions of the three Bethe roots that localize the SU (2) superconformal index. The dominating contribution comes from the integral over vector bits, the next subleading one comes from the integral over chiral bits, and the next-tonext subleading one comes from the integral over part of the bulk. Some observations are made suggesting that the expansion near roots of unity could be a useful tool to complete the localization formulas that have been recently argued to compute the SU (N ) index for N ≥ 3 .

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“…Using the index as a tool, a landscape of exotic phases has been recently identified [23] in certain limits [23,24] [ [25][26][27]. These phases accumulate along the real axis located at τ 2 = 0 , more precisely when τ approaches generic rational points, within appropriate angular sections.…”

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confidence: 99%

Quantum Phases of $4d$ $SU(N)$ $\mathcal{N}=4$ SYM

Cabo-Bizet¹

2021

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It is argued that 4d SU (N ) N = 4 SYM has an accumulation line of zerotemperature topologically ordered phases. Each of these phases corresponds to N boundstates forming a representation of a Z(1) N one-form symmetry. Each of the N bound-states is made of two Dyonic components each of them extended over a two dimensional surface. They are localized at the fixed loci of a rotational action, and are argued to correspond to conformal primaries of an SU (N ) 1 WZNW model on a two-torus.

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$SL(3,\mathbb{Z})$ Modularity and New Cardy Limits of the $\mathcal{N}=4$ Superconformal Index

Cited by 4 publications

References 68 publications

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

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

On the 4d superconformal index near roots of unity: Bulk and Localized contributions

Quantum Phases of $4d$ $SU(N)$ $\mathcal{N}=4$ SYM

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