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On the hierarchy of symmetry breaking D-branes in group manifolds

Thomas Quella

TL;DR

The paper addresses the problem of classifying D-branes in group-manifold backgrounds that preserve a continuous subgroup, by constructing a boundary WZNW functional for symmetry-breaking branes localized on products of twisted conjugacy classes. It connects this geometric construction to an algebraic framework that uses a chain of subgroup embeddings to realize a rational structure, and introduces a target-space reinterpretation G^{new} that clarifies the open/closed string couplings and spectra. The authors apply the framework to the background SL(2,ℝ)×SU(2), revealing a rich hierarchy of symmetry-breaking and non-factorizing D-branes, and demonstrate agreement between the geometric picture and algebraic boundary states, including open-string ground-state spectra. The work further links string theory on group manifolds to asymmetric coset constructions, suggesting broad applicability to more complex backgrounds and potential extensions to stability, charges, and supersymmetric generalizations.

Abstract

We construct the boundary WZNW functional for symmetry breaking D-branes on a group manifold which are localized along a product of a number of twisted conjugacy classes and which preserve an action of an arbitrary continuous subgroup. These branes provide a geometric interpretation for the algebraic formulation of constructing D-branes developed recently in hep-th/0203161. We apply our results to obtain new symmetry breaking and non-factorizing D-branes in the background SL(2,R) x SU(2).

On the hierarchy of symmetry breaking D-branes in group manifolds

TL;DR

The paper addresses the problem of classifying D-branes in group-manifold backgrounds that preserve a continuous subgroup, by constructing a boundary WZNW functional for symmetry-breaking branes localized on products of twisted conjugacy classes. It connects this geometric construction to an algebraic framework that uses a chain of subgroup embeddings to realize a rational structure, and introduces a target-space reinterpretation G^{new} that clarifies the open/closed string couplings and spectra. The authors apply the framework to the background SL(2,ℝ)×SU(2), revealing a rich hierarchy of symmetry-breaking and non-factorizing D-branes, and demonstrate agreement between the geometric picture and algebraic boundary states, including open-string ground-state spectra. The work further links string theory on group manifolds to asymmetric coset constructions, suggesting broad applicability to more complex backgrounds and potential extensions to stability, charges, and supersymmetric generalizations.

Abstract

We construct the boundary WZNW functional for symmetry breaking D-branes on a group manifold which are localized along a product of a number of twisted conjugacy classes and which preserve an action of an arbitrary continuous subgroup. These branes provide a geometric interpretation for the algebraic formulation of constructing D-branes developed recently in hep-th/0203161. We apply our results to obtain new symmetry breaking and non-factorizing D-branes in the background SL(2,R) x SU(2).

Paper Structure

This paper contains 17 sections, 69 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The Lagrangian approach
  3. The algebraic point of view
  4. Target space reinterpretation
  5. The coupling of boundary states to closed strings
  6. The spectrum of open strings
  7. Applications
  8. Preliminaries
  9. Symmetry breaking D-branes in $SL(2,\mathbb{R})$ and $SU(2)$ -- a review
  10. Symmetry breaking D-branes in $SL(2,\mathbb{R})$
  11. Symmetry breaking D-branes in $SU(2)$
  12. On the hierarchy of D-branes in $SL(2,\mathbb{R})\times SU(2)$
  13. Factorizing D-branes
  14. Non-factorizing D-branes from diagonal embedding
  15. Non-factorizing D-branes from group interchanging twists
  16. ...and 2 more sections

Figures (4)

  • Figure 1: Parametrization of $SL(2,\mathbb{R})$.
  • Figure 2: Representatives of maximally symmetric D-branes in $SL(2,\mathbb{R})$. From left to right we have the following types: point-like, $dS_2$, $H_2$ and $AdS_2$ branes.
  • Figure 3: Certain classes of symmetry breaking D-branes on $SL(2,\mathbb{R})$. They are obtained from those in figure \ref{['fig:SLMax']} by a simultaneous $(\theta,\tau)$-rotation.
  • Figure 4: Maximally symmetric and symmetry breaking D-branes on $SU(2)$. The latter arise from the rotation indicated in the central picture. They generically cover a $3$-dimensional subset of $S^3$ but leave open a "window" of a certain size.