A gerbe obstruction to quantization of fermions on odd dimensional manifolds with boundary

TL;DR

This paper investigates the topological obstructions in the quantization of fermions on odd-dimensional manifolds with boundary, revealing a gerbe-based obstruction characterized by the Dixmier-Douady class.

Contribution

It introduces a gerbe obstruction to smooth quantization of fermions with boundary conditions, linking topological gerbe theory to quantum field boundary problems.

Findings

Identifies a gerbe obstruction in fermion quantization on manifolds with boundary.

Calculates the Dixmier-Douady class associated with the gerbe.

Shows the obstruction prevents smooth dependence on boundary conditions.

Abstract

We consider the canonical quantization of fermions on an odd dimensional manifold with boundary, with respect to a family of elliptic hermitean boundary conditions for the Dirac hamiltonian. We show that there is a topological obstruction to a smooth quantization as a function of the boundary conditions. The obstruction is given in terms of a gerbe and its Dixmier-Douady class is evaluated.