Abstract. We give a precise formulation of the M-theory 3-form potential C in a fashion applicable to topologically nontrivial situations. In our model the 3-form is related to the Chern-Simons form of an E8 gauge field. This leads to a precise version of the Chern-Simons interaction of 11-dimensional supergravity on manifolds with and without boundary. As an application of the formalism we give a formula for the electric C-field charge, as an integral cohomology class, induced by self-interactions of the 3-form and by gravity. As further applications, we identify the M theory Chern-Simons term as a cubic refinement of a trilinear form, we clarify the physical nature of Witten's global anomaly for 5-brane partition functions, we clarify the relation of M-theory ux quantization to K-theoretic quantization of RR charge, and we indicate how the formalism can be applied to heterotic M-theory.

INTRODUCTION

This paper summarizes a talk given at the conference on Elliptic Cohomology at the Isaac Newton Institute, in December, 2002

In this paper we will discuss the relation of M-theory to E8 gauge theory in 10, 11, and 12 dimensions. Our basic philosophy is that formulating M-theory in a mathematically precise way, in the presence of nontrivial topology, challenges our understanding of the fundamental formulation of the theory, and therefore might lead to a deeper understanding of how one should express the unified theory of which 11-dimensional supergravity and the five 10-dimensional string theories are distinct limits. To be more specific, let us formulate three motivating problems for the formalism we will develop.