Mass-deformed ABJ and ABJM theory, Meixner-Pollaczek polynomials, and $su(1,1)$ oscillators

Abstract

We give explicit analytical expressions for the partition function of $U(N)_{k}\times U(N+M)_{-k}$ ABJ theory at weak coupling ($k\rightarrow \infty )$ for finite and arbitrary values of $N$ and $M$ (including the ABJM case and its mass-deformed generalization). We obtain the expressions by identifying the one-matrix model formulation with a Meixner-Pollaczek ensemble and using the corresponding orthogonal polynomials, which are also eigenfunctions of a $su(1,1)$ quantum oscillator. Wilson loops in mass-deformed ABJM are also studied in the same limit and interpreted in terms of $su(1,1)$ coherent states.