(Non-)commutative closed string on T-dual toroidal backgrounds

TL;DR

This paper explores the relationship between non-geometry and non-commutativity in closed strings by analyzing T-dual backgrounds, deriving string mode commutators, and establishing a link to Q-flux and boundary conditions.

Contribution

It provides the first steps towards quantizing strings on a twisted torus and connects non-commutativity to non-geometric fluxes in T-dual backgrounds.

Findings

Derived non-vanishing coordinate commutator in non-geometric background.

Linked non-commutativity to boundary conditions and Q-flux.

Performed canonical quantization for twisted torus.

Abstract

In this paper we investigate the connection between (non-)geometry and (non-)commutativity of the closed string. To this end, we solve the classical string on three T-dual toroidal backgrounds: a torus with H-flux, a twisted torus and a non-geometric background with Q-flux. In all three situations we work under the assumption of a dilute flux and consider quantities to linear order in the flux density. Furthermore, we perform the first steps of a canonical quantization for the twisted torus, to derive commutators of the string expansion modes. We use them as well as T-duality to determine, in the non-geometric background, a commutator of two string coordinates, which turns out to be non-vanishing. We relate this non-commutativity to the closed string boundary conditions, and the non-geometric Q-flux.