With I. Biswas. **Opers and Theta Functions**.

We construct natural maps (the Klein and Wirtinger maps) from moduli
spaces of vector bundles on an algebraic curve $X$ to affine spaces,
as quotients of the nonabelian theta linear series. We prove a
finiteness result for these maps over generalized Kummer varieties
(moduli of torus bundles), leading us to conjecture that the maps
are finite in general. The conjecture provides canonical explicit
coordinates on the moduli space. The finiteness results give
low--dimensional parametrizations of Jacobians (in $\Pp^{3g-3}$ for
generic curves), described by $2\Theta$ functions or second
logarithmic derivatives of theta. We interpret the Klein and
Wirtinger maps in terms of opers on $X$. Opers are generalizations
of projective structures, and can be considered as differential
operators, kernel functions or special bundles with connection. The
matrix opers (analogues of opers for matrix differential operators)
combine the structures of flat vector bundle and projective
connection, and map to opers via generalized Hitchin maps. For
vector bundles off the theta divisor, the Szeg\"o kernel gives a
natural construction of matrix oper. The Wirtinger map from bundles
off the theta divisor to the affine space of opers is then defined
as the determinant of the Szeg\"o kernel. This generalizes the
Wirtinger projective connections associated to theta
characteristics, and the assoicated Klein bidifferentials.