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 $\mathcal{M}$, involving the equivariant $L^2$-index of a Dirac-type operator $\mathcal{D}$ on a non-compact finite dimensional submanifold $\mathcal{Y}$ of $\mathcal{M}$. In this article we study a Witten-type 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.

arXiv comment36 pages, title changed, some corrections and additions in section 6.2 based on referee suggestions

References in corpus (5)

• A local global principle for regular operators in Hilbert C*-modules
• Guide to Boundary Value Problems for Dirac-Type Operators
• Index of transversally elliptic operators
• Witten non abelian localization for equivariant K-theory, and the [Q,R]=0 theorem
• Norm-square localization for Hamiltonian $LG$-spaces

Cited by in corpus (2)

• Geometric quantization of b-symplectic manifolds
• A KK-theoretic perspective on deformed Dirac operators