On universal quantum dimensions

Abstract

We represent in the universal form restricted one-instanton partition function of supersymmetric Yang-Mills theory. It is based on the derivation of universal expressions for quantum dimensions (universal characters) of Cartan powers of adjoint and some other series of irreps of simple Lie algebras. These formulae also provide a proof of formulae for universal quantum dimensions for low-dimensional representations, needed in derivation of universal knot polynomials (i.e. colored Wilson averages of Chern-Simons theory on 3d sphere). As a check of the (complicated) formulae for universal quantum dimensions we prove numerically Deligne's hypothesis on universal characters for symmetric cube of adjoint representation.

X 2 series of universal quantum dimensions

“…With this data it is easy to obtain universal formulae for dimensions [4] and quantum dimensions [12] for k-th Cartan power of the adjoint representation. Namely, numerators and denominators of consecutive roots of the given segment of roots cancel (21), so for each segment there remains a number of the first denominators and the same number of the last numerators, which finally lead to the universal formulae.…”

“…Deligne assumes that the standard relations of characters (recall that quantum dimensions are characters on the Weyl line) namely, the product of characters of two representations is equal to the sum of characters of their decomposition, should be satisfied on the entire Vogel's plane (and not on the points of Vogel's table, only). Deligne's hypothesis is checked in some cases [21], particularly for symmetric cube of the adjoint representation. At this time it is not known whether it is possible to satisfy one or both of these requirements, as well as the very existence of universal formulae is not guaranteed.…”

“…the permutation of two parameters. In that way we obtain the final formula (21) below. All this, however, does not combine into formal derivation and altogether should be considered as an educated guess.…”

“…Remark 2, on tables. The entries of the tables 13 and 14 for a given algebra and k are the representation(s), denoted by highest weight, the quantum dimension of which is given by our main formula (21). Remark 3, on the connection with the dimension formulae [7].…”

“…The proof of the main formula is carried out case by case: for each set of parameters α, β, γ from Vogel's table (except C n ) we compare the expression (21) with the Weyl quantum dimension (6) for the corresponding algebra.…”