Intersection Pairings on Spaces of Connections and Chern-Simons Theory on Seifert Manifolds

TL;DR

This paper explores the computation of Chern-Simons path integrals on Seifert manifolds, revealing connections to characteristic classes, universal bundles, and Riemann-Roch formulas, with extensions to include Wilson lines.

Contribution

It provides a novel integral formula for Chern-Simons path integrals on Seifert manifolds involving characteristic classes and universal bundles, generalizing to include Wilson lines.

Findings

Path integral expressed as an integral over G-connections involving characteristic classes

Special case reduces to a Riemann-Roch type formula in infinite dimensions

Inclusion of Wilson lines along the fiber direction in the analysis

Abstract

Let M be a U(1) bundle over a smooth Riemann surface. I show that for Chern-Simons theory on M, with structure group G, the path integral is an integral over the space of G-connections on the Riemann surface involving characteristic classes as well as a certain 4-dimensional class that comes from a universal bundle. When M is the product of a Riemann surface with a circle the 4-dimensional class does not enter and the path integral takes the form of a Riemann-Roch formula albeit in infinite dimensions. The discussion is generalised to include Wilson lines along the fibre direction in M.