The Heisenberg Group and Conformal Field Theory

A mathematical construction of the conformal field theory (CFT) associated to a compact torus, also called the “nonlinear σ-model” or “lattice-CFT”, is given. Underlying this approach to CFT is a unitary modular functor, the construction of which follows from a “Quantization commutes with reduction”type of theorem for unitary quantizations of the moduli spaces of holomorphic torusbundles and actions of loop groups. This theorem in turn is a consequence of general constructions in the category of affine symplectic manifolds and their associated generalized Heisenberg groups.