Geometry and 2-Hilbert Space for Nonassociative Magnetic Translations

TL;DR

This paper develops a geometric framework for quantising nonassociative structures in magnetic and string theory contexts, introducing 2-Hilbert spaces and parallel transport on bundle gerbes.

Contribution

It introduces a novel geometric approach using bundle gerbes and 2-Hilbert spaces for quantising nonassociative magnetic translations and flux backgrounds.

Findings

Constructs a parallel transport on bundle gerbes on .

Shows this transport yields weak projective 2-representations of translation groups.

Provides a new perspective on the fake curvature condition.

Abstract

We suggest a geometric approach to quantisation of the twisted Poisson structure underlying the dynamics of charged particles in fields of generic smooth distributions of magnetic charge, and dually of closed strings in locally non-geometric flux backgrounds, which naturally allows for representations of nonassociative magnetic translation operators. We show how one can use the 2-Hilbert space of sections of a bundle gerbe in a putative framework for canonical quantisation. We define a parallel transport on bundle gerbes on $\mathbb{R}^d$ and show that it naturally furnishes weak projective 2-representations of the translation group on this 2-Hilbert space. We obtain a notion of covariant derivative on a bundle gerbe and a novel perspective on the fake curvature condition.