Non-topological non-commutativity in string theory

TL;DR

This paper explores non-commutative structures in string theory beyond topological cases, establishing a generalized framework for non-associative effective actions and their measures in curved backgrounds, with implications for D-brane physics.

Contribution

It introduces a generalized non-commutative framework in string theory that extends beyond topological sectors, including non-associative effective actions and measures in curved backgrounds.

Findings

Non-associative effective actions are consistent in string theory.

The measure reduces to the symplectic measure in the topological limit.

Non-singular measures are possible even with degenerate Poisson structures.

Abstract

Quantization of coordinates leads to the non-commutative product of deformation quantization, but is also at the roots of string theory, for which space-time coordinates become the dynamical fields of a two-dimensional conformal quantum field theory. Appositely, open string diagrams provided the inspiration for Kontsevich's solution of the long-standing problem of quantization of Poisson geometry by virtue of his formality theorem. In the context of D-brane physics non-commutativity is not limited, however, to the topolocial sector. We show that non-commutative effective actions still make sense when associativity is lost and establish a generalized Connes-Flato-Sternheimer condition through second order in a derivative expansion. The measure in general curved backgrounds is naturally provided by the Born--Infeld action and reduces to the symplectic measure in the topological limit, but remains non-singular even for degenerate Poisson structures. Analogous superspace deformations by RR--fields are also discussed.