Non-commutativity and non-associativity of the doubled string in non-geometric backgrounds

TL;DR

This paper investigates the non-commutative and non-associative properties of doubled strings in non-geometric backgrounds using a T-duality invariant approach, revealing how monodromy influences string dynamics.

Contribution

It introduces a T-duality invariant action to analyze doubled strings in non-geometric backgrounds, deriving Dirac brackets solely from monodromy, and clarifies conditions for non-commutativity and non-associativity.

Findings

Found non-commutativity in the three-torus with H-flux background.

Did not observe non-associativity in the same background.

Linked the results to the exotic $5^2_2$ brane with similar monodromy.

Abstract

We use a T-duality invariant action to investigate the behaviour of a string in non-geometric backgrounds, where there is a non-trivial global $O(D,D)$ patching or monodromy. This action leads to a set of Dirac brackets describing the dynamics of the doubled string, with these brackets determined only by the monodromy. This allows for a simple derivation of non-commutativity and non-associativity in backgrounds which are (even locally) non-geometric. We focus here on the example of the three-torus with H-flux, finding non-commutativity but not non-associativity, and also comment on the relation to the exotic $5^2_2$ brane, which shares the same monodromy.