Superconformal indices of 3d $$ \mathcal{N} $$ = 2 SCFTs and holography

A bstract We study the superconformal index of 3d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 superconformal field theories on S 1 × ω S 2 in the Cardy-like limit where the radius of the S 1 is much smaller than that of the S 2 . We show that the first two leading terms in this Cardy-like expansion are dictated by the Bethe Ansatz formulation of the topologically twisted index of the same theory. We apply this relation to 3d $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 2 holographic superconformal field theories describing the low-energy dynamics of N M2-branes and derive closed form expressions, valid to all orders in the 1 /N expansion, for the two leading terms in the Cardy-like expansion of the superconformal index. We also discuss the implications of our results for the entropy of supersymmetric Kerr-Newman black holes in AdS 4 and the four-derivative corrections to 4d gauged supergravity.