A Universal Property of the Groups Spin^c and Mp^c

TL;DR

This paper establishes a universal property of spin^c and Mp^c structures, showing their fundamental role in defining spinor bundles on Riemannian and symplectic manifolds, respectively.

Contribution

It proves that spin^c and Mp^c structures are universal among structures enabling spinor bundle construction, clarifying their foundational significance.

Findings

Spin^c structures have a universal property for spinor bundle construction.

A similar universal property is established for Mp^c structures on symplectic manifolds.

The results unify the understanding of spinor structures across different geometric contexts.

Abstract

It is well known that spinors on oriented Riemannian manifolds cannot be defined as sections of a vector bundle associated with the frame bundle. For this reason spin and spin^c structures are often introduced. In this paper we prove that spin^c structures have a universal property among all other structures that enable the construction of spinor bundles. We proceed to prove a similar result for metaplectic^c structures on symplectic manifolds.