The Heisenberg group and conformal field theory

TL;DR

This paper constructs a mathematical model of conformal field theory associated with a compact torus using Heisenberg groups and modular functors, advancing the understanding of the algebraic and geometric structures in CFT.

Contribution

It introduces a novel approach to CFT via affine symplectic manifolds and generalized Heisenberg groups, establishing a new link between quantization and moduli spaces.

Findings

Construction of a unitary modular functor for lattice CFTs

Proof of a 'Quantization commutes with reduction' theorem

Application of affine symplectic geometry to CFT structures

Abstract

A mathematical construction of the conformal field theory (CFT) associated to a compact torus, also called the "nonlinear Sigma-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 torus-bundles 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.