Classical Chern-Simons theory, Part 1

TL;DR

This paper explores the rich geometric structures of classical Chern-Simons theory, focusing on line bundles over moduli spaces and extending the theory to 2-manifolds with boundary, highlighting its non-trivial topological features.

Contribution

It develops the geometric framework of classical Chern-Simons theory, including line bundles and gluing laws, for simply connected gauge groups, with extensions to manifolds with boundary.

Findings

Detailed construction of line bundles with connection over moduli space

Development of gluing laws for the theory

Extension of the theory to 2-manifolds with boundary

Abstract

There is a large mathematical literature on classical mechanics and field theory, especially on the relationship to symplectic geometry. One might think that the classical Chern-Simons theory, which is topological and so has vanishing hamiltonian, is completely trivial. However, this theory exhibits interesting geometry that is usually absent from ordinary field theories. (The same is true on the quantum level; topological quantum field theories exhibit geometric properties not usually seen in ordinary quantum field theories, and they lack analytic properties which are usually seen.) In this paper we carefully develop this geometry. Of particular interest are the line bundles with connection over the moduli space of flat connections on a 2-manifold. We extend the usual theory to cover 2-manifolds with boundary. We carefully develop ``gluing laws'' in all of our constructions, including the line bundle with connection over moduli space. The corresponding quantum gluing laws are fundamental. Part 1 covers connected and simply connected gauge groups; Part 2 will cover arbitrary compact Lie groups.