Spinor modules for Hamiltonian loop group spaces

2017/06/22 by Yiannis Loizides, Eckhard Meinrenken, Loizides, Yiannis +3
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1706.07493

openalex publication_date 2017/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let LG be the loop group of a compact, connected Lie group G. We show that the tangent bundle of any proper Hamiltonian LG-space M has a natural completion TM to a strongly symplectic LG-equivariant vector bundle. This bundle admits an invariant compatible complex structure within a natural polarization class, defining an LG-equivariant spinor bundle S_TM, which one may regard as the Spin_c-structure of M. We describe two procedures for obtaining a finite-dimensional version of this spinor module. In one approach, we construct from S_TM a twisted Spin_c-structure for the quasi-Hamiltonian G-space associated to M. In the second approach, we describe an `abelianization procedure', passing to a finite-dimensional T⊂ LG-invariant submanifold of M, and we show how to construct an equivariant Spin_c-structure on that submanifold.

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