Geometrization of the TUY/WZW/KZ connection

TL;DR

This paper demonstrates that the flat projective connection on nonabelian theta functions aligns with the WZNW conformal blocks connection, providing a geometric construction of the KZ connection on the configuration space of points.

Contribution

It establishes the flatness and equivalence of two key connections in geometric representation theory, and offers a geometric construction of the KZ connection.

Findings

The bundle of nonabelian theta functions and WZNW conformal blocks are connected via a flat projective connection.

The identification between these bundles respects the flat connections from both constructions.

A geometric approach to the KZ connection on the configuration space is developed.

Abstract

Given a simple, simply connected, complex algebraic group G, a flat projective connection on the bundle of nonabelian theta functions on the moduli space of semistable parabolic G-bundles over any family of smooth projective curves with marked points was constructed by the authors in an earlier paper. Here, it is shown that the identification between the bundle of nonabelian theta functions and the bundle of WZNW conformal blocks is flat with respect to this connection and the one constructed by Tsuchiya-Ueno-Yamada. As an application, we give a geometric construction of the Knizhnik-Zamolodchikov connection on the trivial bundle over the configuration space of points in the projective line whose typical fiber is the space of invariants of tensor product of representations.