Conformal blocks and equivariant cohomology

TL;DR

This paper links conformal blocks to equivariant cohomology integrals, compares special cases with classical formulas, computes related Selberg integrals, and offers a geometric construction for tensor products in Lie algebra representations.

Contribution

It introduces a new geometric perspective on conformal blocks via equivariant cohomology and provides explicit integral formulas and constructions for Lie algebra representations.

Findings

Conformal blocks can be described as equivariant cohomology integrals.

Proportionality coefficients are Selberg type integrals, explicitly computed.

A geometric construction for tensor products of $rak{gl}(m)$) vector representations is proposed.

Abstract

We show that the conformal blocks constructed in the previous article by the first and the third author may be described as certain integrals in equivariant cohomology. When the bundles of conformal blocks have rank one, this construction may be compared with the old integral formulas of the second and the third author. The proportionality coefficients are some Selberg type integrals which are computed. Finally, a geometric construction of the tensor products of vector representations of the Lie algebra $\frak{gl}(m)$ is proposed.