Non-Abelian Geometry

TL;DR

This paper introduces the concept of non-Abelian geometry, a unified framework where spatial noncommutativity and non-Abelian gauge symmetries are intertwined, especially in the context of multiple D-branes in string theory.

Contribution

It proposes a new algebraic structure called non-Abelian geometry, linking noncommutative space and non-Abelian symmetry in a unified manner, with explicit models and computations.

Findings

Derived a new associative algebra for non-Abelian geometry.

Showed how non-Abelian geometry arises in D-brane configurations with background fields.

Discussed potential supergravity duals of the proposed structure.

Abstract

Spatial noncommutativity is similar and can even be related to the non-Abelian nature of multiple D-branes. But they have so far seemed independent of each other. Reflecting this decoupling, the algebra of matrix valued fields on noncommutative space is thought to be the simple tensor product of constant matrix algebra and the Moyal-Weyl deformation. We propose scenarios in which the two become intertwined and inseparable. Therefore the usual separation of ordinary or noncommutative space from the internal discrete space responsible for non-Abelian symmetry is really the exceptional case of an unified structure. We call it non-Abelian geometry. This general structure emerges when multiple D-branes are configured suitably in a flat but varying B field background, or in the presence of non-Abelian gauge field background. It can also occur in connection with Taub-NUT geometry. We compute the deformed product of matrix valued functions using the lattice string quantum mechanical model developed earlier. The result is a new type of associative algebra defining non-Abelian geometry. Possible supergravity dual is also discussed.