On the prequantisation map for 2-plectic manifolds

TL;DR

This paper extends the concept of prequantisation to 2-plectic manifolds by constructing a geometric structure involving $PU(H)$-bundles and Lie groupoids, and establishes a Lie 2-algebra quasi-isomorphism linking observables to symmetries.

Contribution

It introduces a novel prequantisation framework for 2-plectic manifolds using gerbes, Lie groupoids, and Lie 2-algebras, generalizing classical prequantisation methods.

Findings

Constructed a $PU(H)$-bundle and Lie groupoid over the manifold.

Established a Lie 2-algebra quasi-isomorphism for non-degenerate forms.

Generalized the prequantisation map of Kostant and Souriau.

Abstract

For a manifold $M$ with an integral closed 3-form $\omega$, we construct a $PU(H)$-bundle and a Lie groupoid over its total space, together with a curving in the sense of gerbes. If the form is non-degenerate, we furthermore give a natural Lie 2-algebra quasi-isomorphism from the observables of $(M,\omega)$ to the weak symmetries of the above geometric structure, generalising the prequantisation map of Kostant and Souriau.