With I. Biswas. **Theta Functions and Szeg\"o
Kernels**.

We study relations between two fundamental constructions associated
to vector bundles on a smooth complex projective curve: the theta
function (a section of a line bundle on the moduli space of vector
bundles) and the Szeg\"o kernel (a section of a vector bundle on the
square of the curve). Two types of relations are demonstrated.
First, we establish a higher--rank version of the prime form,
describing the pullback of determinant line bundles by difference
maps, and show the theta function pulls back to the determinant of
the Szeg\"o kernel. Next, we prove that the expansion of the Szeg\"o
kernel at the diagonal gives the logarithmic derivative of the theta
function over the moduli space of bundles for a fixed, or moving,
curve. In particular, we recover the identification of the space of
connections on the theta line bundle with moduli space of flat
vector bundles, when the curve is fixed. When the curve varies, we
identify this space of connections with the moduli space of {\em
extended connections}, which we introduce.