Quantum Observables of Quantized Fluxes

TL;DR

This paper systematically analyzes the quantization of flux observables in gauge theories, revealing that topological quantum observables form a homology Pontrjagin algebra, with implications for higher gauge fields like in supergravity.

Contribution

It provides a new framework for lifting Poisson-brackets to higher moduli stacks and identifies the algebraic structure of flux observables in both abelian and non-abelian gauge theories.

Findings

Flux observables form the homology Pontrjagin algebra of loop spaces.

Quantization laws extend to higher and non-abelian gauge fields.

Results connect to quantum effects in 11d supergravity and Hypothesis H.

Abstract

While it has become widely appreciated that defining (higher) gauge theories requires, in addition to ordinary phase space data, also "flux quantization" laws in generalized differential cohomology, there has been little discussion of the general rules, if any, for lifting Poisson-brackets of (flux-)observables and their quantization from traditional phase spaces to the resulting higher moduli stacks of flux-quantized gauge fields. In this short note, we present a systematic analysis of (i) the canonical quantization of flux observables in Yang-Mills theory and (ii) of valid flux quantization laws in abelian Yang-Mills, observing (iii) that the resulting topological quantum observables form the homology Pontrjagin algebra of the loop space of the moduli space of flux-quantized gauge fields. This is remarkable because the homology Ponrjagin algebra on loops of moduli makes immediate sense in broad generality for higher and non-abelian (non-linearly coupled) gauge fields, such as for the C-field in 11d supergravity, where it recovers the quantum effects previously discussed in the context of "Hypothesis H".