K-theory (physics)
In string theory, 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 (, ). History This conjecture, applied to D-brane charges, was first proposed by . It was popularized by 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 , complicates the extension of the K-theory classification to such cases. suggested a solution to this problem: D-branes are in general classified by a twisted K-theory, that had earlier been defined by . Applications The K-theory classification of D-branes has had numerous applications. For example, used it to argue that there are eight species of orientifold one-plane. applied the K-theory classification to derive new consistency conditions for flux compactifications. K-theory has also been used to conjecture a formula for the topologies of T-dual manifolds by . Recently K-theory has been conjectured to classify the spinors in compactifications on generalized complex manifolds. Open problems Despite these successes, RR fluxes are not quite classified by K-theory. argued that the K-theory classification is incompatible with S-duality in IIB string theory. In addition, if one attempts to classify fluxes on a compact ten-dimensional spacetime, then a complication arises due to the self-duality of the RR fluxes. The duality uses the Hodge star, which depends on the metric and so is continuously valued and in particular is generically irrational. Thus not all of the RR fluxes, which are interpreted as the Chern characters in K-theory, can be rational. However Chern characters are always rational, and so the K-theory classification must be replaced. One needs to choose a half of the fluxes to quantize, or a polarization in the geometric quantization-inspired language of Diaconescu, Moore, and Witten and later of . Alternately one may use the K-theory of a 9-dimensional time slice as has been done by . K-theory classification of RR fluxes In the classical limit of type II string theory, which is type II supergravity, the Ramond–Ramond field strengths are differential forms. In the quantum theory the well-definedness of the partition functions of D-branes implies that the RR field strengths obey Dirac quantization conditions when spacetime is compact, or when a spatial slice is compact and one considers only the (magnetic) components of the field strength which lie along the spatial directions. This led twentieth century physicists to classify RR field strengths using cohomology with integral coefficients. However some authors have argued that the cohomology of spacetime with integral coefficients is too big. For example, in the presence of Neveu–Schwarz H-flux or non-spin cycles some RR fluxes dictate the presence of D-branes. In