Universal Chern-Simons partition functions as quadruple Barnes’ gamma-functions

Abstract

We show that both perturbative and non-perturbative parts of universal partition functions of Chern-Simons theory on 3d sphere are ratios of four over four Barnes' quadruple gamma functions with arguments given by linear combinations of universal parameters. Since nonperturbative part of partition function is essentially a universal compact simple Lie group's volume, latter appears to be expressed through quadruple Barnes' functions, also. For SU(N) values of parameters recurrent relations on Barnes' functions give the proof of level-rank duality of complete partition function, thus extending that duality on non-integer level and rank. We note that integral representation of universal partition function is defined on few disjoint regions in parameters' space, corresponding to different signs of real parts of parameters, and introduce a framework for discussion of analytic continuation of partition functions(s) from these regions. Although initial integral representation is symmetric under all permutations of parameters (which corresponds particularly to $N \rightarrow -N$ duality of gauge theories with classical groups), analytic continuations are not symmetric under transposition of parameters with different signs of their real parts. For the particular case of SU(N) Chern-Simons this asymmetry appears to be the Kinkelin's functional equation (reflection relation) for Barnes' G-function.