  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.
