In the classical limit of type II string theory, which is type II supergravity, the Ramond–Ramond field strengths are differential forms. Alternately one may use the K-theory of a 9-dimensional time slice as has been done by Maldacena, Moore & Seiberg (2001). 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). However Chern characters are always rational, and so the K-theory classification must be replaced. Thus not all of the RR fluxes, which are interpreted as the Chern characters in K-theory, can be rational. The duality uses the Hodge star, which depends on the metric and so is continuously valued and in particular is generically irrational.
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. Diaconescu, Moore & Witten (2003) argued that the K-theory classification is incompatible with S-duality in IIB string theory. Recently K-theory has been conjectured to classify the spinors in compactifications on generalized complex manifolds.ĭespite these successes, RR fluxes are not quite classified by K-theory. K-theory has also been used to conjecture a formula for the topologies of T-dual manifolds by Bouwknegt, Evslin & Varghese (2004). Uranga (2001) applied the K-theory classification to derive new consistency conditions for flux compactifications. For example, Hanany & Kol (2000) used it to argue that there are eight species of orientifold one-plane. The K-theory classification of D-branes has had numerous applications. 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). 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.
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. This conjecture, applied to D-brane charges, was first proposed by Minasian & Moore (1997). 10 References (condensed matter physics).5 Reconciling twisted K-theory and S-duality.4.3 Twisted K-theory from MMS instantons.
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