$E_8$ gauge theory, and a derivation of $K$-theory from $M$-theory

Abstract

The partition function of Ramond-Ramond p-form fields in Type IIA supergravity on a tenmanifold X contains subtle phase factors that are associated with T -duality, self-duality, and the relation of the RR fields to K-theory. The analogous partition function of M -theory on X × S 1 contains subtle phases that are similarly associated with E 8 gauge theory. We analyze the detailed phase factors on the two sides and show that they agree, thereby testing M -theory/Type IIA duality as well as the K-theory formalism in an interesting way. We also show that certain D-brane states wrapped on nontrivial homology cycles are actually unstable, that (−1) F L symmetry in Type IIA superstring theory depends in general on a cancellation between a fermion anomaly and an anomaly of RR fields, and thatType IIA superstring theory with no wrapped branes is well-defined only on a spacetime with W 7 = 0.

Twisted K-theory from monodromies

“…When the zero-coupling restriction is relaxed mortal NS5-branes also must be considered, and these lead to further identifications which modify the twisted K-theory classification. Such corrections may be expected, for example, from the lack of S-duality covariance of the twisted K-theory classification of fluxes in type IIB [4]. The final states of brane-decay processes provide examples of fluxes which are nontrivial in compactly supported cohomology but trivial in ordinary cohomology.…”

“…The fact that treating NS fluxes on equal footing with RR fluxes must lead to a breakdown of the twisted K-theory classification was seen already in, for example, Ref. [4] which demonstrated the incompatibility of the K-theory classification with S-duality in IIB.…”

“…To begin to understand why such configurations are missed by the twisted K-theory classification we consider a more powerful classification scheme, which incorporates all of the data of the NS and RR fields of IIA. This is the E 8 gauge bundle formalism [12,4,13,14], which is conjectured to contain within it the twisted K-theory classifications of branes ignoring fluxes and fluxes in the absence of branes as special cases [15,14]. Included here is a preview of how these classifications are realized which leads to a justification for the absence of the D-brane decay residues in twisted K-theory.…”

Effects of brane-flux transition on black holes in string theory

“…Here we briefly review the connection between cohomology and K-theory in terms of the Atiyah-Hirzebruch Spectral Sequence (AHSS) (for further details, see [39,3,27]). Essentially AHSS is an algorithm which relates integral cohomology with twisted K-theory.…”

Global Worldsheet Anomalies from M-Theory

“…For completeness, in this section we show how one may also derive the shift (1.1) in the periods of F by using the K-theory formula [21,22] for the Ramond-Ramond four-form G 4 . For simplicity, we set the NS field to zero for the rest of the paper.…”