3-Cocycles, Non-Associative Star-Products and the Magnetic Paradigm of R-Flux String Vacua

TL;DR

This paper explores the non-associative star-product algebra in string theory's R-flux backgrounds, revealing how magnetic charge distributions induce non-associativity and affect symmetry and quantization.

Contribution

It introduces a duality-invariant star-product algebra on phase space and links non-associativity to magnetic charge distributions, providing new insights into string vacua and quantization.

Findings

Star-product algebra accounts for non-associativity via 3-cocycles.

Magnetic charge distributions break angular symmetry and induce non-associativity.

Poincare vector restores symmetry and enables quantization in magnetic monopole fields.

Abstract

We consider the geometric and non-geometric faces of closed string vacua arising by T-duality from principal torus bundles with constant H-flux and pay attention to their double phase space description encompassing all toroidal coordinates, momenta and their dual on equal footing. We construct a star-product algebra on functions in phase space that is manifestly duality invariant and substitutes for canonical quantization. The 3-cocycles of the Abelian group of translations in double phase space are seen to account for non-associativity of the star-product. We also provide alternative cohomological descriptions of non-associativity and draw analogies with the quantization of point-particles in the field of a Dirac monopole or other distributions of magnetic charge. The magnetic field analogue of the R-flux string model is provided by a constant uniform distribution of magnetic charge in space and non-associativity manifests as breaking of angular symmetry. The Poincare vector comes to rescue angular symmetry as well as associativity and also allow for quantization in terms of operators and Hilbert space only in the case of charged particles moving in the field of a single magnetic monopole.