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Branes on Group Manifolds, Gluon Condensates, and twisted K-theory

Stefan Fredenhagen, Volker Schomerus

TL;DR

The work probes D-brane dynamics on group manifolds in the stringy regime, using boundary conformal field theory to classify symmetric branes and track their condensation via RG flows. A central achievement is formulating discrete, RG-invariant brane charges and deriving explicit constraints for the charge group $C(SU(N),K)$, including untwisted and twisted sectors, by exploiting the absorption of boundary spin and fusion data. The results provide nontrivial consistency checks with Bouwknegt–Mathai twisted K-theory and yield concrete predictions for the structure of $K^*_H(SU(N))$, especially in low-rank cases; for higher ranks, the analysis highlights where higher-differential effects may alter the group structure. Overall, the paper connects worldsheet brane dynamics in curved backgrounds to K-theoretic charge classifications, with implications for understanding tachyon-like condensation and noncommutative geometries in string theory.

Abstract

In this work we study the dynamics of branes on group manifolds G deep in the stringy regime. After giving a brief overview of the various branes that can be constructed within the boundary conformal field theory approach, we analyze in detail the condensation processes that occur on stacks of such branes. At large volume our discussion is based on certain effective gauge theories on non-commutative `fuzzy' spaces. Using the `absorption of the boundary spin'-principle which was formulated by Affleck and Ludwig in their work on the Kondo model, we extrapolate the brane dynamics into the stringy regime. For supersymmetric theories, the resulting condensation processes turn out to be consistent with the existence of certain conserved charges taking values in some non-trivial discrete abelian groups. We obtain strong constraints on these charge groups for G = SU(N). The results may be compared with a recent proposal of Bouwknegt and Mathai according to which charge groups on curved spaces X (with a non-vanishing NSNS 3-form field strength H) are given by the twisted K-groups K*_H(X).

Branes on Group Manifolds, Gluon Condensates, and twisted K-theory

TL;DR

The work probes D-brane dynamics on group manifolds in the stringy regime, using boundary conformal field theory to classify symmetric branes and track their condensation via RG flows. A central achievement is formulating discrete, RG-invariant brane charges and deriving explicit constraints for the charge group , including untwisted and twisted sectors, by exploiting the absorption of boundary spin and fusion data. The results provide nontrivial consistency checks with Bouwknegt–Mathai twisted K-theory and yield concrete predictions for the structure of , especially in low-rank cases; for higher ranks, the analysis highlights where higher-differential effects may alter the group structure. Overall, the paper connects worldsheet brane dynamics in curved backgrounds to K-theoretic charge classifications, with implications for understanding tachyon-like condensation and noncommutative geometries in string theory.

Abstract

In this work we study the dynamics of branes on group manifolds G deep in the stringy regime. After giving a brief overview of the various branes that can be constructed within the boundary conformal field theory approach, we analyze in detail the condensation processes that occur on stacks of such branes. At large volume our discussion is based on certain effective gauge theories on non-commutative `fuzzy' spaces. Using the `absorption of the boundary spin'-principle which was formulated by Affleck and Ludwig in their work on the Kondo model, we extrapolate the brane dynamics into the stringy regime. For supersymmetric theories, the resulting condensation processes turn out to be consistent with the existence of certain conserved charges taking values in some non-trivial discrete abelian groups. We obtain strong constraints on these charge groups for G = SU(N). The results may be compared with a recent proposal of Bouwknegt and Mathai according to which charge groups on curved spaces X (with a non-vanishing NSNS 3-form field strength H) are given by the twisted K-groups K*_H(X).

Paper Structure

This paper contains 16 sections, 3 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. D-branes on group manifolds
  3. The geometry of branes on group manifolds
  4. Conformal field theory
  5. Supersymmetric WZW models
  6. Brane dynamics on group manifolds
  7. Brane dynamics at large volume
  8. Condensation in the stringy regime
  9. Conserved charges and twisted K-theory
  10. The charge of untwisted branes on $\boldsymbol{SU (N)}$
  11. Charges of twisted branes on $\boldsymbol{SU (N)}$
  12. Comparison with twisted K-theory
  13. Conclusions and open problems
  14. Some representation theory
  15. Some lemmas used in Section 4
  16. ...and 1 more sections

Key Result

Lemma 1

Suppose

Figures (1)

  • Figure 1: Brane dynamics on $S^3$: A stack of D0-branes at $e$ can decay to a D2-brane. Putting more and more D0-branes at $e$ the resulting brane will be localized further and further away from the group unit and eventually the decay product will be a single D0-brane at $-e$.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof