Some Relations between Twisted K-theory and E8 Gauge Theory

TL;DR

This work connects M-theory’s E8 gauge-theoretic topological data with Type IIA via twisted K-theory, explicitly incorporating nontrivial NS flux $H_3$ and circle-fibration geometry. By adapting the Diaconescu–Moore–Witten and Moore–Saulina frameworks, it constructs a twisted K-theory torus and theta-function description that encode the RR fields as elements of $K(X,H)$ with $F/2\pi=ch_H(x)\sqrt{\hat A(X)}$ and analyzes the M-theory phase through eta-invariants and eta-forms in the adiabatic limit, relating twelve-dimensional Chern–Simons data to ten-dimensional twisted cohomology. It shows that RR field dynamics in IIA correspond to twisted cohomology classes $H^{\text{even}}(X,H_3)$ and that the partition function naturally arises from a theta function on the twisted K-theory torus, with a polarization determined by a Lagrangian decomposition of the lattice $K(X,H)/K(X,H)_{tors}$. The paper highlights key open problems, including lifting twisted classes, the role of eta-forms in nontrivial circle bundles, and the implementation of T-duality/anomalies at the level of twisted K-theory theta functions, pointing to rich interactions between mathematics and string theory.

Abstract

Recently, Diaconescu, Moore and Witten provided a nontrivial link between K-theory and M-theory, by deriving the partition function of the Ramond-Ramond fields of Type IIA string theory from an E8 gauge theory in eleven dimensions. We give some relations between twisted K-theory and M-theory by adapting the method of Diaconescu-Moore-Witten and Moore-Saulina. In particular, we construct the twisted K-theory torus which defines the partition function, and also discuss the problem from the E8 loop group picture, in which the Dixmier-Douady class is the Neveu-Schwarz field. In the process of doing this, we encounter some mathematics that is new to the physics literature. In particular, the eta differential form, which is the generalization of the eta invariant, arises naturally in this context. We conclude with several open problems in mathematics and string theory.