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.

Quiver gauge theory of non-Abelian vortices and noncommutative instantons in higher dimensions

“…with that of the class of the line bundle L which represents the Bott generator of K(CP 1 ) = Z [11]. The topological equivalence (4.1) then implies the equivalence of brane-antibrane systems on M q × CP 1 and M q , with the brane and antibrane systems each carrying a single unit of monopole charge.…”

“…Given that the charges of configurations of D-branes in string theory are classified topologically by K-theory [10,11,13], let us now seek the K-theory representation of the above physical situation. The one-monopole bundle L is a crucial object in establishing the Bott periodicity isomorphism…”

“…It is based on an SU(2)-equivariant generalization of the (noncommutative) Atiyah-Bott-Shapiro (ABS) construction of tachyon field configurations [9]- [11].…”

Quantum gravity from noncommutative spacetime

“…This leads to a remarkable point that there is the so-called stable regime at N > p/2, where π p (GL(N, C)) is independent of N. In this stable regime, the homotopy groups of GL(N, C) or U(N) define a generalized cohomology theory, known as Ktheory [50][51][52][53]. In K-theory, which also involves vector bundles and gauge fields, any smooth manifold X is assigned an Abelian group K(X).…”

“…[49] long ago. In Section IV.C, we define a stable state in a large-N gauge theory and relate it to the K-theory [50][51][52][53]. With the correspondence in Eq.…”