Norm-square localization and the quantization of Hamiltonian loop group spaces

Abstract

In an earlier article we introduced a new definition for the ‘quantization’ of a Hamiltonian loop group space M, involving the equivariant L2-index of a Dirac-type operator D on a non-compact finite dimensional submanifold Y of M. In this article we study a deformation of this operator, similar to the work of Tian-Zhang and Ma-Zhang. We obtain a formula for the index with infinitely many non-trivial contributions, indexed by the components of the critical set of the norm-square of the moment map. This is the main part of a new proof of the [Q,R]=0 theorem for Hamiltonian loop group spaces.