On the geometry of the supermultiplet in M-theory

The massless supermultiplet of 11-dimensional supergravity can be generated from the decomposition of certain representation of the exceptional Lie group F₄ into those of its maximal compact subgroup Spin(9). In an earlier paper, a dynamical KaluzaKlein origin of this observation is proposed with internal space the Cayley plane, P², and topological aspects are explored. In this paper we consider the geometric aspects and characterize the corresponding forms which contribute to the action as well as cohomology classes, including torsion, which contribute to the partition function. This involves constructions with bilinear forms. The compatibility with various string theories are discussed, including reduction to loop bundles in ten dimensions.