  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.
