Superconformal indices of 3d N = 2 SCFTs and holography

We study the superconformal index of 3d N=2 superconformal field theories on S¹_ S² in the Cardy-like limit where the radius of the S¹ is much smaller than that of the S². 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 N=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 AdS₄ and the four-derivative corrections to 4d gauged supergravity.