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

It is known that the partition functions of the U(N)(k) x U(N + M)(-k) ABJM theory satisfy a set of bilinear relations, which, written in the grand partition function, was recently found to be the q-Painleve III3 equation. In this paper we have suggested that 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 function 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.

Automated summary

The study reveals that both the U(N)(k) x U(N + M)(-k) ABJM theory and the ABJM theory with N = 6 exhibit bilinear relations in their partition functions with respect to mass parameters, leading to different equations.