Abstract

Elliptic functions and associated modular forms are a central topic in modern mathematics since their inception by Abel and Jacobi in the early 19th century. The modular group SL(2,Z) whose finite dimensional representations serve to classify rational conformal field theories (RCFT) is also a source of duality transformations (starting with the modular inversion used by Kramers and Wannier back in 1941 to relate high and low temperature behaviour in a 2-dimensional ferromagnet). The lectures are planned to provide an introduction to both the mathematics and the physics of the subject. We shall demonstrate, in particular, that finite temperature correlation functions are (doubly periodic) elliptic functions for RCFT in any even number of space-time dimensions. We also give conditions under which the mean value of the energy in an equilibrium state is a modular form of weight D.