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 a notion of μ-minimality ( ) for these orbits which extend the finite dimensional one. Our approach uses heat-kernel methods and yields both “heat-kernel” (obtained via heat-kernel regularisation) and “zeta function” (obtained via zeta function regularisation) minimality for specific values of the parameter μ. For each of these notions, we give an infinite dimensional version of Hsiang's theorem which extends the finite dimensional case, interpreting μ-minimal orbits as orbits with extremal (μ-regularised) volume.