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

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.

Section: Jhep10(2021)207mentioning

confidence: 99%

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

Ardehali

Murthy

2021

“…Note that the BA formula (2.9) is derived by applying the Cauchy's integral formula to (2.6) under the assumption that all the solutions to the BAE (2.8) are isolated. As it has been observed in the literature [38][39][40], however, the BAE (2.8) have continuous family of solutions and therefore the BA formula (2.9) has to be refined to incorporate their contributions.…”

Section: Jhep10(2021)145mentioning

confidence: 99%

“…To circumvent the first issue about continuous family of BAE solutions, in this paper we focus on the contribution from isolated BAE solutions to the twisted index so that the BA formula (2.9) can be used without modifications. In particular, we focus on standard BAE solutions [31,38,39], which we will review in the remaining part of this subsection. Then, to take care of the second issue about 'relevant' BAE solutions that contribute to the twisted index, we assume that any standard BAE solution is enclosed by the contour C in (2.1) and therefore contributes to the twisted index.…”

Section: Jhep10(2021)145mentioning

confidence: 99%

“…Since the BAE is a highly involved system of transcendental equations, however, it is difficult to derive its general solutions. Hence we focus on a particular set of BAE solutions found in [31] and dubbed as standard solutions in [38,39], namely…”

Section: Jhep10(2021)145mentioning

confidence: 99%

“…Each standard BAE solution {u i } {m,n,r} distributes holonomies u i 's evenly over the torus T 2 and thereby define a freely acting orbifold T 2 /Z m × Z n . All the other solutions to the BAE (2.8) are called non-standard [38,39].…”

Section: Jhep10(2021)145mentioning

confidence: 99%

See 2 more Smart Citations

The topologically twisted index of $$ \mathcal{N} $$ = 4 SU(N) Super-Yang-Mills theory and a black hole Farey tail

Hong

2021

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We investigate the large-N asymptotics of the topologically twisted index of $$ \mathcal{N} $$ N = 4 SU(N) Super-Yang-Mills (SYM) theory on T2 × S2 and provide its holographic interpretation based on the black hole Farey tail [1]. In the field theory side, we use the Bethe-Ansatz (BA) formula, which gives the twisted index of $$ \mathcal{N} $$ N = 4 SYM theory as a discrete sum over Bethe vacua, to compute the large-N asymptotics of the twisted index. In a dual $$ \mathcal{N} $$ N = 2 gauged STU model, we construct a family of 5d extremal solutions uplifted from the 3d black hole Farey tail, and compute the regularized on-shell actions. The gravitational partition function given in terms of these regularized on-shell actions is then compared with a canonical partition function derived from the twisted index by the inverse Laplace transform, in the large-N limit. This extends the previous microstate counting of an AdS5 black string by the twisted index and thereby improves holographic understanding of the twisted index.

Decomposition of BPS moduli spaces and asymptotics of supersymmetric partition functions

Ardehali

Hong

2022

Self Cite

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.