SU(N) q-Toda equations from mass deformed ABJM theory

TL;DR

This paper extends the known bilinear relations of ABJM theory's partition functions to include mass deformations, revealing connections to q-deformed affine SU(N) Toda equations through exact checks and specific parameter choices.

Contribution

It proposes a new bilinear relation for mass-deformed ABJM theory's partition functions, linking it to q-deformed affine SU(N) Toda equations, supported by multiple non-trivial checks.

Findings

Bilinear relations hold for mass-deformed ABJM partition functions.

Special mass parameter choices relate to q-deformed affine SU(N) Toda equations.

Exact partition function values confirm the proposed relations.

Abstract

It is known that the partition functions of the U(N) x U(N+M) ABJM theory satisfy a set of bilinear relations, which, written in the grand partition function, was recently found to be the q-Painleve III_3 equation. In this paper we have suggested a similar bilinear relation holds for the ABJM theory with N=6 preserving mass deformation for an arbitrary complex value of mass parameter, to which we have provided several non-trivial checks by using the exact values of the partition functions for various N,k,M and the mass parameter. For particular choices of the mass parameters labeled by integers $\nu,a$ as $m_1=m_2=-\pi i(\nu-2a)/\nu$, the bilinear relation corresponds to the q-deformation of the affine SU($\nu$) Toda equation in $\tau$-form.