K-theory

In mathematics, K-theory is, roughly speaking, the study of certain kinds of invariants of large matrices. It originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras.

K-theory involves the construction of families of K-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include Bott periodicity, the Atiyah-Singer index theorem and the Adams operations.

K-theory (physics)

In string theory, the K-theory classification refers to a conjectured application of K-theory (in abstract algebra and algebraic topology) to superstrings, to classify the allowed Ramond–Ramond field strengths as well as the charges of stable D-branes.

In condensed matter physics K-theory has also found important applications, specially in the topological classification of topological insulators, superconductors and stable Fermi surfaces (Kitaev (2009), Horava (2005)).

History

This conjecture, applied to D-brane charges, was first proposed by Minasian & Moore (1997). It was popularized by Witten (1998) who demonstrated that in type IIB string theory arises naturally from Ashoke Sen's realization of arbitrary D-brane configurations as stacks of D9 and anti-D9-branes after tachyon condensation.

Such stacks of branes are inconsistent in a non-torsion Neveu–Schwarz (NS) 3-form background, which, as was highlighted by Kapustin (2000), complicates the extension of the K-theory classification to such cases. Bouwknegt & Varghese (2000) suggested a solution to this problem: D-branes are in general classified by a twisted K-theory, that had earlier been defined by Rosenberg (1989).