Definition, interpretation as sections of line bundles. The associated maps into projective space. The heat equation. The Heisenberg group.

References

Any good book on abelian varieties and theta functions, like:

O. Debarre. Tores et variétés abéliennes complexes. Cours Spécialisés 6. Société Mathématique de France, EDP Sciences (1999).

Moduli spaces and their Picard group. The determinant bundle. The case of vector bundles: the linear system $|\mathcal{L}|$ and its geometry, the $\theta$ divisor of a vector bundle.

References

A. Beauville. Vector Bundles on Curves and Generalized Theta Functions: Recent Results and Open Problems. Current topics in complex algebraic geometry, MSRI Publications 28, 17-33; Cambridge University Press (1995).

The moduli space as a double coset space of an infinite dimensional group. Applications: the Picard group; the space of conformal blocks.

References

Ch. Sorger. La formule de Verlinde. Exp. 794 du Séminaire Bourbaki, Astérisque 237 (1996), 87–114.

Conformal field theories and conformal blocks; the Verlinde formula.

References

A. Beauville. Vector bundles on Riemann surfaces and Conformal Field Theory. Algebraic and Geometric Methods in Mathematical Physics, 145-166; Kluwer (1996).

Topological field theories. The Chern-Simons action. The heat equation for generalized theta functions.

References

E. Witten. Quantum field theory and the Jones polynomial. Comm. Math. Phys. 121 (1989), 351–399.

M. Atiyah. The geometry and physics of knots. Lezioni Lincee, CUP, Cambridge,1990.

N. Hitchin. Flat connections and geometric quantization. Comm. Math. Phys. 131 (1990), 347–380.