Partition function of Chern–Simons theory as renormalized q-dimension

Abstract

We calculate q-dimension of k-th Cartan power of fundamental representation Λ0, corresponding to affine root of affine simply laced Kac-Moody algebras, and show that in the limit q → 1, and with natural renormalization, it is equal to universal partition function of Chern-Simons theory on three-dimensional sphere.

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.…”

“…Dimension E 7 =133, number of positive roots |∆ + | = 63, Vogel's parameters (α, β, γ, t) = (−2, 8,12,18). For E 7 λ = ω 2 , in Dynkin's labeling of roots.…”

“…In subsequent works applications to physics were developed, particularly the universality of the partition function of Chern-Simons theory on a sphere [10,11,22], and its connection with q-dimension of kΛ 0 representation of affine Kac-Moody algebras [12] were shown, the universal knot polynomials for 2-and 3-strand torus knots [13,14,15,21] were calculated.…”