New recursion relations for M2-brane matrix models

A bstract In this paper we investigate the finite N exact values of the S 3 partition function of the $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 super Yang-Mills theory with one adjoint hypermultiplet and N f fundamental hypermultiplets, which describes N M2-branes on $$ {\mathbb{C}}^2\times {\mathbb{C}}^2/{\mathbb{Z}}_{N_{\textrm{f}}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℂ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:msup> <mml:mi>ℂ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>ℤ</mml:mi> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:msub> </mml:math> , 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 AdS 4 × S 7 / $$ {\mathbb{Z}}_{N_{\textrm{f}}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ℤ</mml:mi> <mml:msub> <mml:mi>N</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:msub> </mml:math> .