The KZB equations on Riemann surfaces

TL;DR

This paper provides explicit formulas for the Friedan--Shenker connection on conformal blocks over moduli spaces of Riemann surfaces, linking it to dynamical r-matrices and establishing a universal flatness property.

Contribution

It introduces explicit expressions for the Friedan--Shenker connection using dynamical r-matrices and proves a universal flatness property beyond conformal blocks.

Findings

Explicit formulas for the Friedan--Shenker connection.

Connection expressed via dynamical r-matrices.

Universal flatness of the covariant derivatives.

Abstract

In this paper, based on the author's lectures at the 1995 les Houches Summer school, explicit expressions for the Friedan--Shenker connection on the vector bundle of WZW conformal blocks on the moduli space of curves with tangent vectors at $n$ marked points are given. The covariant derivatives are expressed in terms of ``dynamical $r$-matrices'', a notion borrowed from integrable systems. The case of marked points moving on a fixed Riemann surface is studied more closely. We prove a universal form of the (projective) flatness of the connection: the covariant derivatives commute as differential operators with coefficients in the universal enveloping algebra -- not just when acting on conformal blocks.