Large N partition functions of the ABJM theory

Abstract

We study the large N limit of some supersymmetric partition functions of the U(N)k × U(N)−k ABJM theory computed by supersymmetric localization. We conjecture an explicit expression, valid to all orders in the large N limit, for the partition function on the U(1) × U(1) invariant squashed sphere in the presence of real masses in terms of an Airy function. Several non-trivial tests of this conjecture are presented. In addition, we derive an explicit compact expression for the topologically twisted index of the ABJM theory valid at fixed k to all orders in the 1/N expansion. We use these results to derive the topologically twisted index and the sphere partition function in the ’t Hooft limit which correspond to genus g type IIA string theory free energies to all orders in the α′ expansion. We discuss the implications of our results for holography and the physics of AdS4 black holes.

Exact strong coupling results in $$ \mathcal{N} $$ = 2 Sp(2N) superconformal gauge theory from localization

“…One may wonder whether to match the full expression for the Apkq term in F in (3.7) one needs to include contributions of other M2-brane BPS states to the M-theory 1-loop partition function. 17 By analogy with a discussion in [24,25] one may conjecture that this may lead to a modification of the measure in the proper-time integral in (3.19) like…”

“…Explicitly, one finds the following expressions for f n [17] (A is Glaisher constant) where all p h and p h,s are rational coefficients.…”

“…(3.29) 17 In particular, one may consider contributions of M2-branes wrapped on 2-cycles of CP 3 part of S 7 {Z k (which, in the perturbative 10d string limit, are related to the type IIA string world-sheet instantons [46,47] but here play a role of massive modes propagating in the loop). These may be the analogs of M2-branes wrapped on 2-cycles of CY space in [25].…”

“…to perturbative expansion of the M-theory partition function on AdS 4 ˆS7 {Z k or weak string coupling expansion of string theory effective action on AdS 4 ˆCP 3 . Related work appeared in [9,10] and [11,12,13,14,15] and also in [16,17,18]. In the M-theory the expansion parameter is the inverse of the effective dimensionless M2-brane tension [3] T 2 " 1 p2πq 2 L 3 11…”

“…In [22] this prefactor was derived as a 1-loop correction in the M2-brane world-volume theory and thus it was concluded that this 1-loop M2 brane correction effectively sums up all large tension terms at all orders in the weak string coupling expansion in the dual type IIA theory. 5 It would be interesting also to try to do a similar matching in the case of the topological indices (or special partition functions on S 2 ˆS1 ) for which localization results were discussed in [17,18].…”