A canonical connection on bundles on Riemann surfaces and Quillen connection on the theta bundle

TL;DR

This paper constructs a natural holomorphic connection on vector bundles over Riemann surfaces, establishing a symplectic isomorphism between moduli spaces of connections and theta line bundle connections, varying holomorphically with the surface.

Contribution

It introduces a canonical holomorphic connection on vector bundles outside the theta divisor and proves a symplectic isomorphism between moduli spaces of connections and theta bundle connections.

Findings

Constructed a natural holomorphic connection on vector bundles outside the theta divisor.

Established a symplectic isomorphism between moduli spaces of pairs and theta bundle connections.

Demonstrated the holomorphic variation of this isomorphism over families of Riemann surfaces.

Abstract

We investigate the symplectic geometric and differential geometric aspects of the moduli space of connections on a compact Riemann surface $X$. Fix a theta characteristic $K^{1/2}_X$ on $X$; it defines a theta divisor on the moduli space ${\mathcal M}$ of stable vector bundles on $X$ of rank $r$ degree zero. Given a vector bundle $E \in {\mathcal M}$ lying outside the theta divisor, we construct a natural holomorphic connection on $E$ that depends holomorphically on $E$. Using this holomorphic connection, we construct a canonical holomorphic isomorphism between the following two: \begin{enumerate} \item the moduli space $\mathcal C$ of pairs $(E, D)$, where $E\in {\mathcal M}$ and $D$ is a holomorphic connection on $E$, and \item the space ${\rm Conn}(\Theta)$ given by the sheaf of holomorphic connections on the line bundle on $\mathcal M$ associated to the theta divisor. \end{enumerate} The above isomorphism between $\mathcal C$ and ${\rm Conn}(\Theta)$ is symplectic structure preserving, and it moves holomorphically as $X$ runs over a holomorphic family of Riemann surfaces.