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Constructing D-Branes from K-Theory

Kasper Olsen, Richard J. Szabo

Abstract

A detailed review of recent developments in the topological classification of D-branes in superstring theory is presented. Beginning with a thorough, self-contained introduction to the techniques and applications of topological K-theory, the relationships between the classic constructions of K-theory and the recent realizations of D-branes as tachyonic solitons, coming from bound states of higher dimensional systems of unstable branes, are described. It is shown how the K-theory formalism naturally reproduces the known spectra of BPS and non-BPS D-branes, and how it can be systematically used to predict the existence of new states. The emphasis is placed on the new interpretations of D-branes as conventional topological solitons in other brane worldvolumes, how the mathematical formalism can be used to deduce the gauge field content on both supersymmetric and non-BPS branes, and also how K-theory predicts new relationships between the various superstring theories and their D-brane spectra. The implementations of duality symmetries as natural isomorphisms of K-groups are discussed. The relationship with the standard cohomological classification is presented and used to derive an explicit formula for D-brane charges. Some string theoretical constructions of the K-theory predictions are also briefly described.

Constructing D-Branes from K-Theory

Abstract

A detailed review of recent developments in the topological classification of D-branes in superstring theory is presented. Beginning with a thorough, self-contained introduction to the techniques and applications of topological K-theory, the relationships between the classic constructions of K-theory and the recent realizations of D-branes as tachyonic solitons, coming from bound states of higher dimensional systems of unstable branes, are described. It is shown how the K-theory formalism naturally reproduces the known spectra of BPS and non-BPS D-branes, and how it can be systematically used to predict the existence of new states. The emphasis is placed on the new interpretations of D-branes as conventional topological solitons in other brane worldvolumes, how the mathematical formalism can be used to deduce the gauge field content on both supersymmetric and non-BPS branes, and also how K-theory predicts new relationships between the various superstring theories and their D-brane spectra. The implementations of duality symmetries as natural isomorphisms of K-groups are discussed. The relationship with the standard cohomological classification is presented and used to derive an explicit formula for D-brane charges. Some string theoretical constructions of the K-theory predictions are also briefly described.

Paper Structure

This paper contains 42 sections, 343 equations, 1 figure, 8 tables.

Table of Contents

  1. Introduction and Overview
  2. Outline
  3. Elements of Topological K-Theory
  4. The Grothendieck Group
  5. The Group ${\rm K}(X)$
  6. Reduced K-Theory
  7. Higher K-Theory and Bott Periodicity
  8. Multiplicative Structures
  9. Relative K-Theory
  10. Computing the K-Groups
  11. Clifford Algebras and the Atiyah-Bott-Shapiro Construction
  12. Type IIB D-Branes and ${\rm K}(X)$
  13. The Brane-Antibrane System
  14. The Bound State Construction
  15. Type IIA D-Branes and ${\rm K}^{-1}(X)$
  16. ...and 27 more sections

Figures (1)

  • Figure 1: The relationships between different D-branes in Type II superstring theory. The squares represent stable supersymmetric BPS branes or a combination of such a brane with its antibrane, while the circles depict unstable non-BPS configurations. The horizontal arrows represent the result of quotienting the theory by the operator ${(-1)^{F_{\rm L}}}$, the vertical arrows the effect of constructing a tachyonic kink solution in the brane worldvolume field theory, and the diagonal arrows the usual $T$-duality transformations.