Some Relations between Twisted K-theory and E8 Gauge Theory

TL;DR

This paper explores the connections between twisted K-theory and M-theory, constructing the twisted K-theory torus for partition functions and introducing new mathematical tools like the eta differential form in the context of E8 gauge theory.

Contribution

It adapts the Diaconescu-Moore-Witten method to relate twisted K-theory with M-theory, including the construction of the twisted K-theory torus and the use of the eta differential form.

Findings

Constructed the twisted K-theory torus for partition functions

Linked the Dixmier-Douady class to the Neveu-Schwarz field via E8 loop group

Introduced the eta differential form into the physics literature

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.