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Lectures on Non-BPS Dirichlet branes

Matthias R Gaberdiel

TL;DR

Gaberdiel develops a boundary-state framework for Dirichlet branes and applies it to non-BPS D-branes, illustrating how world-sheet duality and Cardy consistency fix boundary-state normalizations and spectra. He constructs a stable non-BPS D-particle in the Type IIB orbifold $(-1)^{F_L}{\cal I}_4$ and analyzes the orbifold spectrum, establishing a duality web with Type IIA on K3 and the heterotic string on $T^4$. The discussion extends to non-BPS states such as a D-string and a non-BPS $\widehat{D3}$-brane in dual theories, mapping their charges, stability regions, and decay channels across the dual pair. A key point is the appearance of Bose-Fermi degeneracy at special moduli, which yields vanishing open-string partition functions and brane-brane forces, offering a mechanism for stable non-BPS configurations within string dualities and non-supersymmetric compactifications.

Abstract

A comprehensive introduction to the boundary state approach to Dirichlet branes is given. Various examples of BPS and non-BPS Dirichlet branes are discussed. In particular, the non-BPS states in the duality of Type IIA on K3 and the heterotic string on T4 are analysed in detail.

Lectures on Non-BPS Dirichlet branes

TL;DR

Gaberdiel develops a boundary-state framework for Dirichlet branes and applies it to non-BPS D-branes, illustrating how world-sheet duality and Cardy consistency fix boundary-state normalizations and spectra. He constructs a stable non-BPS D-particle in the Type IIB orbifold and analyzes the orbifold spectrum, establishing a duality web with Type IIA on K3 and the heterotic string on . The discussion extends to non-BPS states such as a D-string and a non-BPS -brane in dual theories, mapping their charges, stability regions, and decay channels across the dual pair. A key point is the appearance of Bose-Fermi degeneracy at special moduli, which yields vanishing open-string partition functions and brane-brane forces, offering a mechanism for stable non-BPS configurations within string dualities and non-supersymmetric compactifications.

Abstract

A comprehensive introduction to the boundary state approach to Dirichlet branes is given. Various examples of BPS and non-BPS Dirichlet branes are discussed. In particular, the non-BPS states in the duality of Type IIA on K3 and the heterotic string on T4 are analysed in detail.

Paper Structure

This paper contains 15 sections, 165 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The boundary state approach
  3. World-sheet duality
  4. A first example: Type IIA and IIB
  5. A second example: Type 0A and 0B
  6. More abstract point of view: Conformal field theory with boundaries
  7. The non-BPS D-particle in IIB$/(-1)^{F_L} {\cal I}_4$
  8. The spectrum of the orbifold
  9. Non-BPS states in Heterotic -- Type II duality
  10. The setup
  11. BPS states
  12. Non-BPS states
  13. Non-BPS D-string
  14. Non-BPS $\widehat{D3}$-brane
  15. Bose-Fermi degeneracy

Figures (5)

  • Figure 1: World-sheet duality
  • Figure 2: D1-brane anti-brane pair with a relative ${\sf Z\!\!Z}_2$ Wilson line and the D0-brane.
  • Figure 3: The non-BPS $\widehat{D1}$-brane and the two fractional D0-branes into which it can decay.
  • Figure 4: A D2-brane anti-brane pair and the non-BPS $\widehat{D1}$-brane into which it can decay. The twisted R-R charge of each D2-brane at each of the four corners is one half of that of the non-BPS $\widehat{D1}$-brane.
  • Figure 5: A non-BPS $\widehat{D3}$-brane and the D2-brane anti-brane pair into which it can decay.