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

A. Yu. Alekseev, A. Recknagel, V. Schomerus

TL;DR

This work provides a non-perturbative window into D-brane world-volume geometries in curved backgrounds with NS flux by exploiting boundary conformal field theory in the SU(2) WZW model. It shows that, at infinite level ${\sf k}$, branes wrap fuzzy two-spheres (associative matrix algebras), while finite ${\sf k}$ introduces non-associativity tied to quantum groups and $6J$-symbol data. The analysis reveals a deep link between boundary OPEs in CFT and deformed geometric spaces, connecting coadjoint-orbit quantization, Podleś spheres, and twisted universal enveloping algebras in a unified brane framework. The results extend the understanding of brane geometry beyond flat backgrounds and suggest broad applicability to other groups and supersymmetric cases, with implications for branes near NS5-branes and AdS3×S3 settings.

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.

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

TL;DR

This work provides a non-perturbative window into D-brane world-volume geometries in curved backgrounds with NS flux by exploiting boundary conformal field theory in the SU(2) WZW model. It shows that, at infinite level , branes wrap fuzzy two-spheres (associative matrix algebras), while finite introduces non-associativity tied to quantum groups and -symbol data. The analysis reveals a deep link between boundary OPEs in CFT and deformed geometric spaces, connecting coadjoint-orbit quantization, Podleś spheres, and twisted universal enveloping algebras in a unified brane framework. The results extend the understanding of brane geometry beyond flat backgrounds and suggest broad applicability to other groups and supersymmetric cases, with implications for branes near NS5-branes and AdS3×S3 settings.

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.

Paper Structure

This paper contains 7 sections, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. World-volume geometry -- from the flat case to arbitrary backgrounds
  3. D-branes in the SU(2) WZW model
  4. Semi-classical analysis.
  5. Exact CFT description.
  6. D-brane geometry, fuzzy two-spheres, and quantum groups
  7. Summary and outlook

Figures (1)

  • Figure 1: World-sheet diagrams for closed resp. open string interaction. Having assigned vertex operators to the legs, they can be read as structure constants for the multiplication of two operators, projected on the third channel. In the closed string case, the in-coming operators can be interchanged with the help of world-sheet diffeomorphisms, while the ordering of open string vertices is fixed up to cyclic permutations.