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

Abstract

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.

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

“…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].…”

“…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].…”

A gravity interpretation for the Bethe Ansatz expansion of the $\mathcal{N}=4$ SYM index

“…There are several different methods that have been used to compute the index in the large N limit. One is the so-called Cardy limit [6,[54][55][56], a sort of "high-temperature" limit taken on the chemical potentials in which the integral expression of the index considerably simplifies, and that can then be easily followed by the large N limit. Another one is the Bethe Ansatz method [7], described below.…”

Logarithmic Corrections to the Entropy of Rotating Black Holes and Black Strings in AdS$_5$

“…In [39], the logarithmic corrections were extended to other gauge groups beyond S U(N). More recently, the results of [37] were rederived using an effective field theory approach, which clarifies the organization of the index in inverse powers of |τ| and further confirms the logarithmic term as certain degeneracy of vacua [40,41].…”

“…where k i j is the inverse matrix of k i j , and P 2 ≡ p i p j k i j . Note that the saddle-point values τ 0 and µ 0i are parametrically small, which is reminiscent of the 4d Cardy limit originally used in [10,11] and recently clarified in [40,41].…”