Regularisable and minimal orbits for group actions in infinite dimensions

TL;DR

This paper develops a framework for regularisable infinite dimensional principal fibre bundles, extending finite dimensional concepts, and introduces notions of minimality for their orbits with applications to gauge theories.

Contribution

It introduces regularisable infinite dimensional principal fibre bundles and extends the concept of minimal orbits using heat-kernel and zeta function regularisation methods.

Findings

Orbits have well-defined regularised volumes.

Two notions of minimality coincide in the infinite dimensional setting.

Infinite dimensional Hsiang's theorem characterizes minimal orbits as extremal volume orbits.

Abstract

We introduce a class of regularisable infinite dimensional principal fibre bundles which includes fibre bundles arising in gauge field theories like Yang-Mills and string theory and which generalise finite dimensional Riemannian principal fibre bundles induced by an isometric action. We show that the orbits of regularisable bundles have well defined, both heat-kernel and zeta function regularised volumes. We introduce two notions of minimality (which extend the finite dimensional one) for these orbits, using both heat-kernel and zeta function regularisation methods and show they coincide. For each of these notions, we give an infinite dimensional version of Hsiang's theorem which extends the finite dimensional case, interpreting minimal orbits as orbits with extremal (regularised) volume.