Cardy formula for 4d SUSY theories and localization

Abstract

Abstract:We study 4d N = 1 supersymmetric theories on a compact Euclidean manifold of the form S 1 × M 3 . Partition functions of gauge theories on this background can be computed using localization, and explicit formulas have been derived for different choices of the compact manifold M 3 . Taking the limit of shrinking S 1 , we present a general formula for the limit of the localization integrand, derived by simple effective theory considerations, generalizing the result of [1]. The limit is given in terms of an effective potential for the holonomies around the S 1 , whose minima determine the asymptotic behavior of the partition function. If the potential is minimized in the origin, where it vanishes, the partition function has a Cardy-like behavior fixed by Tr(R), while a nontrivial minimum gives a shift in the coefficient. In all the examples that we consider, the origin is a minimum if Tr(R) ≤ 0.

The asymptotic growth of states of the 4d $$ \mathcal{N}=1 $$ superconformal index

“…The remaining two terms determine an effective potential for the gauge holonomies and are similar to those first found in [22] (see also [27]). However we have two important differences: the argument of the functions V 1 , V 2 is ρ · u + r 2 instead of ρ · u (because we are taking n 0 = −1 while the setup of [22,27] corresponds to n 0 = 0), and σ, τ are generically complex (while they were assumed purely imaginary in [22,27], which corresponds to real fugacities p = e 2πiσ , q = e 2πiτ ). Recently, similar asymptotic formulae have been discussed in the specific case of N = 4 SYM in [5,6] by allowing for complex values of the fugacities.…”

“…The first term on the right hand side of (2.14), proportional to the 't Hooft anomaly TrR (see (2.18) below for its definition) and independent of the gauge holonomies, was first obtained in [21]. The remaining two terms determine an effective potential for the gauge holonomies and are similar to those first found in [22] (see also [27]). However we have two important differences: the argument of the functions V 1 , V 2 is ρ · u + r 2 instead of ρ · u (because we are taking n 0 = −1 while the setup of [22,27] corresponds to n 0 = 0), and σ, τ are generically complex (while they were assumed purely imaginary in [22,27], which corresponds to real fugacities p = e 2πiσ , q = e 2πiτ ).…”

“…The Cardy-like ("high-temperature") limit of the superconformal index was studied in [21,27] using effective field theory arguments on S 3 × S 1 backgrounds, and in [22] using asymptotic properties of the elliptic gamma functions. 5 Assuming real values of the fugacities and taking the limit of small chemical potentials σ, τ , these works found a universal exponential contribution to the index weighted by the 't Hooft anomaly TrR ∝ (a − c) (this can receive corrections in special instances [22]).…”

’t Hooft anomalies and the holomorphy of supersymmetric partition functions

“…The 1/β S 1 term was studied in [33]; it is fully gauge invariant and supersymmetric. The finite term W (0) 3d roughly corresponds to the supersymmetric partition function of the 3d N = 2 theory on M 3 obtained from the 4d N = 1 theory by dimensional reduction, up to important subtleties that will not affect our discussion-see [61,62,63,41]. Here, we are interested in the order-β S 1 functional in (3.76).…”

Quantum black hole entropy from 4d supersymmetric Cardy formula

“…We mainly study supersymmetric index of the SU(N) N = 4 SYM on S 1 × M 3 . We compute an asymptotic behavior of the index in the limit of shrinking S 1 for arbitrary N by using a refinement [24,25] of supersymmetric Cardy formula [26]. Therefore our approach for the superconformal index case (M 3 = S 3 ) is basically the same as the one in [20].…”