$E_8$ Gauge Theory and Gerbes in String T

Abstract

The reduction of the $E_8$ gauge theory to ten dimensions leads to a loop group, which in relation to twisted $K$-theory has a Dixmier–Douady class identified with the Neveu–Schwarz $H$-field. We give an interpretation of the degree two part of the eta form by comparing the adiabatic limit of the eta invariant with the one loop term in type IIA. More generally, starting with a $G$-bundle, the comparison for manifolds with String structure identifies $G$ with $E_8$ and the representation as the adjoint, due to an interesting appearance of the dual Coxeter number. This makes possible a description in terms of a generalized Wess-Zumino-Witten model at the critical level. We also discuss the relation to the index gerbe, the possibility of obtaining such bundles from loop space, and the symmetry breaking to finite-dimensional bundles. We discuss the implications of this and we give several proposals.