Universal volume of groups and anomaly of Vogel's symmetry

We show that integral representation of universal volume function of compact simple Lie groups gives rise to six analytic functions on CP2, 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 calculate their change (anomaly) under permutations of parameters (Vogel's symmetry). This last relation is universal generalization, for an arbitrary simple Lie group and moreover to an arbitrary point in Vogel's plane, of the Kinkelin's reection relation on Barnes' G(1 + N) function. Kinkelin's relation gives asymmetry of the G(1 + N) function (which is essentially reciprocal of the volume function for SU(N) groups) under N $ 􀀀N transformation (which
is equivalent of the permutation of Vogel's parameters for SU(N) groups),
and coincides with abovementioned anomaly of permutations at the SU(N)
line on Vogel's plane. Our results also give an anomaly of Vogel's symmetry of the universal partition function of Chern-Simons theory on three-dimensional sphere. This eect is analogous to modular covariance, instead of invariance, of
partition functions of appropriate gauge theories under modular transformation
of couplings.
Mathematics Subject Classication. 17B20, 17B37, 22E99, 33E20.
Keywords. Simple Lie algebras, simple Lie groups, Barnes' functions,
Vogel plane, universal Lie algebra.