Constants in Sequences of M2-brane Partition Functions

Abstract

We determine in closed form the $N$-independent constant terms in the all-order $1/N$ expansions of the topologically twisted index and of the associated Bethe potential for the ABJM theory, as well as the Bethe potential constant of the ADHM theory. We reconstruct these constants analytically from high-precision Bethe-Ansatz numerics, verify them down to the level of non-perturbative corrections, and find them to be closely related to the constant map function $A$ governing the round three-sphere partition function. The resulting expressions for the ADHM and ABJM constants pass the non-trivial test dictated by 3d mirror symmetry. Via recently established factorization relations, they also determine in closed form the $N$-independent constant contribution to the squashed three-sphere partition function, through the first two leading orders in its large-squashing expansion. These constants supply precisely the piece left undetermined in the recent exact results for 3d supersymmetric partition functions to all orders in the $1/N$ expansion, and thereby mark an important step toward completing them, both in field theory and in the dual quantum gravity description.