K-theory (physics) - Reconciling Twisted K-theory and S-duality

Reconciling Twisted K-theory and S-duality

Diaconescu, Moore, and Witten have pointed out that the twisted K-theory classification is not compatible with the S-duality covariance of type IIB string theory. For example, consider the constraint on the Ramond-Ramond 3-form field strength G₃ in the Atiyah-Hirzebruch spectral sequence (AHSS):

where d₃=Sq3+H is the first nontrivial differential in the AHSS, Sq3 is the third Steenrod square and the last equality follows from the fact that the nth Steenrod square acting on any n-form x is xx.

The above equation is not invariant under S-duality, which exchanges G₃ and H. Instead Diaconescu, Moore, and Witten have proposed the following S-duality covariant extension

where P is an unknown characteristic class that depends only on the topology, and in particular not on the fluxes. Diaconescu, Freed & Moore (2007) have found a constraint on P using the E₈ gauge theory approach to M-theory pioneered by Diaconescu, Moore, and Witten.

Thus D-branes in IIB are not classified by twisted K-theory after all, but some unknown S-duality-covariant object that inevitably also classifies both fundamental strings and NS5-branes.

However the MMS prescription for calculating twisted K-theory is easily S-covariantized, as the Freed-Witten anomalies respect S-duality. Thus the S-covariantized form of the MMS construction may be applied to construct the S-covariantized twisted K-theory, as a set, without knowing having any geometric description for just what this strange covariant object is. This program has been carried out in a number of papers, such as Evslin & Varadarajan (2003) and Evslin (2003a), and was also applied to the classification of fluxes by Evslin (2003b). Bouwknegt et al. (2006) use this approach to prove Diaconescu, Moore, and Witten's conjectured constraint on the 3-fluxes, and they show that there is an additional term equal to the D3-brane charge. Evslin (2006) shows that the Klebanov-Strassler cascade of Seiberg dualities consists of a series of S-dual MMS instantons, one for each Seiberg duality. The group, of universality classes of the supersymmetric gauge theory is then shown to agree with the S-dual twisted K-theory and not with the original twisted K-theory.

Some authors have proposed radically different solutions to this puzzle. For example, Kriz & Sati (2005) propose that instead of twisted K-theory, II string theory configurations should be classified by elliptic cohomology.