Topological Sectors and Measures on Moduli Space in Quantum Yang-Mills on a Riemann Surface

TL;DR

This paper extends the path integral formulation of quantum Yang-Mills theory on Riemann surfaces to explicitly handle individual topological sectors, enabling exact calculations of Wilson line expectations and proposing new measures on the moduli space.

Contribution

It introduces a formulation that isolates topological sectors in Yang-Mills path integrals and proposes measures on the moduli space, connecting to the small-volume limit of the Yang-Mills measure.

Findings

Exact Wilson line expectations can be computed within each topological sector.

Two new measures on the moduli space are proposed, one matching the small-volume Yang-Mills limit.

The approach clarifies the role of topological types in quantum Yang-Mills on Riemann surfaces.

Abstract

Previous path integral treatments of Yang-Mills on a Riemann surface automatically sum over principal fiber bundles of all possible topological types in computing quantum expectations. This paper extends the path integral formulation to treat separately each topological sector. The formulation is sufficiently explicit to calculate Wilson line expectations exactly. Further, it suggests two new measures on the moduli space of flat connections, one of which proves to agree with the small-volume limit of the Yang-Mills measure. \copyright {\em 1996 American Institute of Physics.}