Residues, modularity, and the Cardy limit of the 4d $$ \mathcal{N} $$ = 4 superconformal index

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

doi:10.1007/jhep04(2021)216

Cited by 38 publications

(49 citation statements)

References 45 publications

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“…The analysis of the Cardy-like limit on the second sheet has played an important role in displaying the correct asymptotic growth accounting for the black hole microstates, see [28,[31][32][33][34][35][36][37][38]. This is complementary to the study of the large-N limit of the index, discussed in [29,[39][40][41][42][43][44][45][46][47]. If one takes the large-N limit first, there are many saddles (some of which have a direct holographic interpretation within the classical gravitational theory).…”

Section: Introductionmentioning

confidence: 99%

EFT and the SUSY index on the 2nd sheet

Komargodski

2021

SciPost Phys.

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The counting of BPS states in four-dimensional \mathcal{N}=1𝒩=1 theories has attracted a lot of attention in recent years. For superconformal theories, these states are in one-to-one correspondence with local operators in various short representations. The generating function for this counting problem has branch cuts and hence several Cardy-like limits, which are analogous to high-temperature limits. Particularly interesting is the second sheet, which has been shown to capture the microstates and phases of supersymmetric black holes in AdS_55. Here we present a 3d Effective Field Theory (EFT) approach to the high-temperature limit on the second sheet. We use the EFT to derive the behavior of the index at orders \beta^{-2},\beta^{-1},\beta^0β−2,β−1,β0. We also make a conjecture for O(\beta)O(β), where we argue that the expansion truncates up to exponentially small corrections. An important point is the existence of vector multiplet zero modes, unaccompanied by massless matter fields. The runaway of Affleck-Harvey-Witten is however avoided by a non-perturbative confinement mechanism. This confinement mechanism guarantees that our results are robust.

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Section: Introductionmentioning

confidence: 99%

EFT and the SUSY index on the 2nd sheet

Komargodski

2021

SciPost Phys.

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“…The first one requires an opportune Cardy-like limit [8][9][10][11][12], while the second approach consists of an exact evaluation of the index in terms of (a set of solutions to) the so-called Bethe Ansatz equations (BAE) [13,14]. 1 Many generalizations of these results have since then appeared [15][16][17][18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

The SCI of $$ \mathcal{N} $$ = 4 USp(2Nc) and SO(Nc) SYM as a matrix integral

Amariti

Fazzi

Segati

2021

J. High Energ. Phys.

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We study the superconformal index of 4d $$ \mathcal{N} $$ N = 4 USp(2Nc) and SO(Nc) SYM from a matrix model perspective. We focus on the Cardy-like limit of the index. Both in the symplectic and orthogonal case the index is dominated by a saddle point solution which we identify, reducing the calculation to a matrix integral of a pure Chern-Simons theory on the three-sphere. We further compute the subleading logarithmic corrections, which are of the order of the center of the gauge group. In the USp(2Nc) case we also study other subleading saddles of the matrix integral. Finally we discuss the case of the Leigh-Strassler fixed point with SU(Nc) gauge group, and we compute the entropy of the dual black hole from the Legendre transform of the entropy function.

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“…Two main technical approaches have emerged, one rooted in saddle point approximations [1,2], and one in a Bethe-Ansatz (BA) formula of the index [3]; a systematic discussion comparing both approaches including sub-leading contributions and extending the results to include 4d N = 1 theories was presented in [15]. Other approaches to the evaluation of the index, include, for example, those rooted in doubly-periodic extensions [16,17], direct numerical evaluation [18,19]; a partial list of results includes [20][21][22][23].…”

Section: Jhep06(2021)126mentioning

confidence: 99%

“…where the contour is the unit circle surrounding z = 0 taken in the conventional direction. This picks up the residues inside the unit circle, which can be identified from the product representation of the integrand, with the poles coming from the denominator of (B.3) as worked out in [21]. Alternatively, by truncating the infinite product at some finite order, the index becomes the integral of a rational function of the form…”

Section: Jhep06(2021)126mentioning

confidence: 99%

The Bethe-Ansatz approach to the $$ \mathcal{N} $$ = 4 superconformal index at finite rank

Lezcano

Hong

Liu

et al. 2021

J. High Energ. Phys.

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We investigate the Bethe-Ansatz approach to the superconformal index of $$ \mathcal{N} $$ N = 4 supersymmetric Yang-Mills with SU(N) gauge group in the context of finite rank, N. We explicitly explore the role of the various types of solutions to the Bethe-Ansatz Equations in recovering the exact index for N = 2, 3. We classify the Bethe-Ansatz Equations solutions as standard (corresponding to a freely acting orbifold T2/ℤm× ℤn) and non-standard. For N = 2, we find that the index is fully recovered by standard solutions and displays an interesting pattern of cancellations. However, for N ≥ 3, the standard solutions alone do not suffice to reconstruct the index. We present quantitative arguments in various regimes of fugacities that highlight the challenging role played by the continuous families of non-standard solutions.

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Residues, modularity, and the Cardy limit of the 4d $$ \mathcal{N} $$ = 4 superconformal index

Cited by 38 publications

References 45 publications

EFT and the SUSY index on the 2nd sheet

EFT and the SUSY index on the 2nd sheet

The SCI of $$ \mathcal{N} $$ = 4 USp(2Nc) and SO(Nc) SYM as a matrix integral

The Bethe-Ansatz approach to the $$ \mathcal{N} $$ = 4 superconformal index at finite rank

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