Universal volume of groups and anomaly of Vogel’s symmetry

Abstract

We show that integral representation of universal volume function of compact simple Lie groups gives rise to six analytic functions on $CP^2$, which transform as two triplets under group of permutations of Vogel's projective parameters. This substitutes expected invariance under permutations of universal parameters by more complicated covariance. We provide an analytical continuation of these functions and particularly calculate their change under permutations of parameters. This last relation is universal generalization, for an arbitrary simple Lie group and an arbitrary point in Vogel's plane, of the Kinkelin's reflection relation on Barnes' $G(1+N)$ function. Kinkelin's relation gives asymmetry of the $G(1+N)$ function (which is essentially the volume function for $SU(N)$ groups) under $N\leftrightarrow -N$ transformation (which is equivalent of the permutation of parameters, for $SU(N)$ groups), and coincides with universal relation on permutations at the $SU(N)$ line on Vogel's plane. These results are also applicable to universal partition function of Chern-Simons theory on three-dimensional sphere. This effect is analogous to modular covariance, instead of invariance, of partition functions of appropriate gauge theories under modular transformation of couplings.