From classical theta functions to topological quantum field theory

Abstract

Abelian Chern-Simons theory relates classical theta functions to the topological quantum field theory of the linking number of knots. In this paper we explain how to derive the constructs of abelian Chern-Simons theory directly from the theory of classical theta functions. It turns out that the theory of classical theta functions, from the representation theoretic point of view of A. Weil, is just an instance of Chern-Simons theory. The group algebra of the finite Heisenberg group is described as an algebra of curves on a surface, and its Schrodinger representation is obtained as an action on curves in a handlebody. A careful analysis of the discrete Fourier transform yields the Reshetikhin-Turaev formula for invariants of 3-dimensional manifolds. In this context, we give an explanation of why the composition of discrete Fourier transforms and the non-additivity of the signature of 4-dimensional manifolds under gluings obey the same formula.