Hitchin connection on the Veech curve

TL;DR

This paper derives explicit formulas for the Hitchin connection's pullback on a Veech curve, computes its monodromy, and constructs quantum representations of certain pseudo-Anosov elements in genus two mapping class groups.

Contribution

It provides explicit expressions for the Hitchin connection and its monodromy on a Veech curve, linking geometric structures to quantum representations.

Findings

Explicit formula for the pullback of the Hitchin connection.

Expression for the monodromy representation in terms of iterated integrals.

Quantum representations of infinitely many pseudo-Anosov elements.

Abstract

We give an expression for the pull back of the Hitchin connection from the moduli space of genus two curves to a ten-fold covering of a Teichm\"uller curve discovered by Veech. We then give an expression, in terms of iterated integrals, for the monodromy representation of this connection. As a corollary we obtain quantum representations of infinitely many pseudo-Anosov elements in the genus two mapping class group.