Regularisable and minimal orbits for group actions in infinite dimensions

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.