New recursion relations for M2-brane matrix models

Abstract

In this paper we investigate the finite $N$ exact values of the $S^3$ partition function of the $\mathcal{N}=4$ super Yang-Mills theory with one adjoint hypermultiplet and $N_{\text{f}}$ fundamental hypermultiplets, which describes $N$ M2-branes on ${\mathbb{C}}^2\times {\mathbb{C}}^2/{\mathbb{Z}}_{N_{\text{f}}}$, with mass and FI deformations. We claim that the grand canonical sum of the partition function obeys a bilinear difference relation with respect to the shifts of the mass parameters of the fundamental hypermultiplets, which results in a new recursion relation for the partition function with respect to $N$. As an application, we also determine the analytic expression for the leading $1/N$ non-perturbative correction to the free energy of these models, which would correspond holographically to the contribution from an M2-brane wrapped on a 3d volume in the internal space of ${\text{AdS}}_4\times S^7/{\mathbb{Z}}_{N_{\text{f}}}$.

44 pages, 6 figures