Groupoids, Loop Spaces and Quantization of 2-Plectic Manifolds

TL;DR

This paper extends the groupoid approach to quantize 2-plectic manifolds by analyzing their loop spaces, with applications to quantum geometry in M-branes and string theory.

Contribution

It introduces a method to quantize loop spaces of 2-plectic manifolds using groupoids, bridging geometric quantization and string/M-theory.

Findings

Quantization of loop spaces of R^3, T^3, and S^3.

Matching physical expectations from string theory.

Extension of groupoid quantization to 2-plectic manifolds.

Abstract

We describe the quantization of 2-plectic manifolds as they arise in the context of the quantum geometry of M-branes and non-geometric flux compactifications of closed string theory. We review the groupoid approach to quantizing Poisson manifolds in detail, and then extend it to the loop spaces of 2-plectic manifolds, which are naturally symplectic manifolds. In particular, we discuss the groupoid quantization of the loop spaces of R^3, T^3 and S^3, and derive some interesting implications which match physical expectations from string theory and M-theory.