Non-commutative World-volume Geometries: Branes on SU(2) and Fuzzy Spheres

TL;DR

This paper investigates how non-zero B-fields in curved backgrounds lead to non-commutative geometries on D-brane world-volumes, specifically analyzing fuzzy spheres in the SU(2) WZW model using boundary conformal field theory.

Contribution

It introduces a method to quantize world-volume geometries in curved backgrounds with H-flux, linking them to fuzzy spheres and non-associative deformations, expanding understanding of D-branes in such settings.

Findings

Establishes a relation between open strings in SU(2) WZW model and fuzzy spheres.

Demonstrates non-commutative and non-associative geometries on D-branes.

Provides a framework applicable to D-branes near NS5-branes.

Abstract

The geometry of D-branes can be probed by open string scattering. If the background carries a non-vanishing B-field, the world-volume becomes non-commutative. Here we explore the quantization of world-volume geometries in a curved background with non-zero Neveu-Schwarz 3-form field strength H = dB. Using exact and generally applicable methods from boundary conformal field theory, we study the example of open strings in the SU(2) Wess-Zumino-Witten model, and establish a relation with fuzzy spheres or certain (non-associative) deformations thereof. These findings could be of direct relevance for D-branes in the presence of Neveu-Schwarz 5-branes; more importantly, they provide insight into a completely new class of world-volume geometries.