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 (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).
Applications The K-theory classification of D-branes has had numerous applications. For example, Hanany & Kol (2000) used it to argue that there are eight species of orientifold one-plane. Uranga (2001) 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 Bouwknegt, Evslin & Varghese (2004). 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. Diaconescu, Moore & Witten (2003) 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 Varghese & Sati (2004). Alternately one may use the K-theory of a 9-dimensional time slice as has been done by Maldacena, Moore & Seiberg (2001).
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 the former case this is a consequence of the supergravity equation of motion which states that the product of a RR flux with the NS 3-form is a D-brane charge density. Thus the set of topologically distinct RR field strengths that can exist in brane-free configurations is only a subset of the cohomology with integral coefficients. This subset is still too big, because some of these classes are related by large gauge transformations. In QED there are large gauge transformations which add integral multiples of two pi to Wilson loops. The p-form potentials in type II supergravity theories also enjoy these large gauge transformations, but due to the presence of Chern-Simons terms in the supergravity actions these large gauge transformations transform not only the p-form potentials but also simultaneously the (p+3)-form field strengths. Thus to obtain the space of inequivalent field strengths from the forementioned subset of integral cohomology we must quotient by these large gauge transformations. The Atiyah–Hirzebruch spectral sequence constructs twisted K-theory, with a twist given by the NS 3-form field strength, as a quotient of a subset of the cohomology with integral coefficients. In the classical limit, which corresponds to working with rational coefficients, this is precisely the quotient of a subset described above in supergravity. The quantum corrections come from torsion classes and contain mod 2 torsion corrections due to the Freed-Witten anomaly. Thus twisted K-theory classifies the subset of RR field strengths that can exist in the absence of D-branes quotiented by large gauge transformations. Daniel Freed has attempted to extend this classification to include also the RR potentials using differential K-theory.
K-theory classification of D-branes K-theory classifies D-branes in noncompact spacetimes, intuitively in spacetimes in which we are not concerned about the flux sourced by the brane having nowhere to go. While the K-theory of a 10d spacetime classifies D-branes as subsets of that spacetime, if the spacetime is the product of time and a fixed 9-manifold then K-theory also classifies the conserved D-brane charges on each 9-dimensional spatial slice. While we were required to forget about RR potentials to obtain the K-theory classification of RR field strengths, we are required to forget about RR field strengths to obtain the K-theory classification of D-branes.
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