Energy in Yang-Mills on a Riemann Surface

TL;DR

This paper extends Sengupta's lower bound for Yang-Mills action to a broader space of connections on Riemann surfaces, linking critical sets to Morse theory and potentially illuminating Atiyah and Bott's conjecture.

Contribution

It generalizes Sengupta's lower bound to a larger space of connections with finite action, and relates critical sets to Morse theory on the space of connections.

Findings

Lower bound can be saturated in the extended space

Critical sets correspond to energy action critical points

Potential insights into Atiyah and Bott's Morse theory conjecture

Abstract

Sengupta's lower bound for the Yang-Mills action on smooth connections on a bundle over a Riemann surface generalizes to the space of connections whose action is finite. In this larger space the inequality can always be saturated. The Yang-Mills critical sets correspond to critical sets of the energy action on a space of paths. This may shed light on Atiyah and Bott's conjecture concerning Morse theory for the space of connections modulo gauge transformations.