SL(3, ℤ) Modularity and New Cardy limits of the $$ \mathcal{N} $$ = 4 superconformal index
Vishnu Jejjala, Yang Lei, Sam van Leuven, Wei Li
Abstract
A bstract The entropy of 1 / 16-th BPS AdS 5 black holes can be microscopically accounted for by the superconformal index of the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 super-Yang-Mills theory. One way to compute this is through a Cardy-like limit of a formula for the index obtained in [1] using the “ S -transformation” of the elliptic Γ function. In this paper, we derive more general SL (3 , ℤ) modular properties of the elliptic Γ function. We then use these properties to obtain a three integer parameter family of generalized Cardy-like limits of the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 superconformal index. From these limits, we obtain entropy formulae that have a similar form as that of the original AdS 5 black hole, up to an overall rescaling of the entropy. We interpret this both on the field theory and the gravitational side. Finally, we comment on how our work suggests a generalization of the Farey tail to four dimensions.