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D-branes from conformal field theory

Matthias R Gaberdiel

TL;DR

The paper develops and applies boundary conformal field theory techniques to construct and classify D-branes as boundary states in string theory. Using Ishibashi states, gluing conditions, and the Cardy consistency condition, it demonstrates how open-closed duality constrains brane spectra and provides explicit classifications in tractable models, including a complete analysis for a single free boson and its SU(2) level-1 enhancement at the self-dual radius. It then extends the discussion to radii away from self-duality, incorporates fermions, and surveys ten-dimensional brane configurations, distinguishing stable BPS, unstable non-BPS, and stable non-BPS branes via worldsheet and GSO considerations. The work shows that many D-brane configurations can be exactly characterized in terms of group-theoretic data (e.g., SU(2) representations, SL(2, C) elements) and classifying algebras, with implications for string dualities and stringy geometry. Overall, it provides a powerful, exact framework for understanding D-branes through conformal field theory across bosonic, supersymmetric, and ten-dimensional contexts.

Abstract

An introduction to the construction of D-branes using conformal field theory methods is given. A number of examples are discussed in detail, in particular the construction of all conformal D-branes for the theory of a single free boson on a circle.

D-branes from conformal field theory

TL;DR

The paper develops and applies boundary conformal field theory techniques to construct and classify D-branes as boundary states in string theory. Using Ishibashi states, gluing conditions, and the Cardy consistency condition, it demonstrates how open-closed duality constrains brane spectra and provides explicit classifications in tractable models, including a complete analysis for a single free boson and its SU(2) level-1 enhancement at the self-dual radius. It then extends the discussion to radii away from self-duality, incorporates fermions, and surveys ten-dimensional brane configurations, distinguishing stable BPS, unstable non-BPS, and stable non-BPS branes via worldsheet and GSO considerations. The work shows that many D-brane configurations can be exactly characterized in terms of group-theoretic data (e.g., SU(2) representations, SL(2, C) elements) and classifying algebras, with implications for string dualities and stringy geometry. Overall, it provides a powerful, exact framework for understanding D-branes through conformal field theory across bosonic, supersymmetric, and ten-dimensional contexts.

Abstract

An introduction to the construction of D-branes using conformal field theory methods is given. A number of examples are discussed in detail, in particular the construction of all conformal D-branes for the theory of a single free boson on a circle.

Paper Structure

This paper contains 12 sections, 77 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The boundary conformal field theory approach
  3. Some illustrative examples
  4. A free boson on a circle of radius $R$
  5. The conformal branes for SU(2) level one
  6. The conformal branes for other radii
  7. Introducing fermions
  8. The ten-dimensional theories
  9. Stable BPS branes
  10. Unstable non-BPS branes
  11. Stable non-BPS branes
  12. Conclusions

Figures (4)

  • Figure 1: The brane configurations for $R=2/3$ corresponding to special group elements: on the left, $g$ is of the form (\ref{['Dirgroup']}) and the boundary state describes two Dirichlet branes $D$ and two anti-Dirichlet branes $\overline{D}$ distributed evenly over the target circle; on the right, $g$ is of the form (\ref{['Neugroup']}), and we have three non-BPS Neumann branes $\hat{N}$ with Wilson lines that are evenly distributed over the dual circle.
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