On the Geometry of the Supermultiplet in M-Theory

Abstract

The massless supermultiplet of eleven-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 Kaluza-Klein origin of this observation is proposed with internal space the Cayley plane, OP 2 , 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. * 2. The identity representation of F 4 , i.e. the one corresponding to Dynkin index [0, 0, 0, 0], generates the three representations of Spin(9) [51] Id(F 4 ) −→ (44, 128, 84), the numbers on the right hand side correctly matching the number of degrees of freedoms of the massless bosonic content of eleven-dimensional supergravity with the individual summands corresponding, respectively, to the graviton, the gravitino, and the C-field (see the beginning of section 2).The main Idea of this paper, presented in section 2.1 is interpreting Ramond's triplets as arising from OP 2 bundles with structure group F 4 over our eleven-dimensional manifold Y 11 , on which M-theory is 'defined'. We first discuss in section 2.2 the geometric properties of OP 2 , including Spin(9)-structures and characteristic classes. This leads to one of the main results, theorem 2.9, that the massless fields of M-theory are encoded in the spinor bundle of OP 2 . We then relate Spin(9)-structures on the 9-dimensional vector space V 9 to the geometry of the eight-sphere S 8 which in turn, by [24], is related to Killing spinors on the cone over S 8 . This shows that the unification of the fields, as well as their supersymmetry, in M-theory can be seen from the eight-sphere over the Cayley plane (cf. proposition 2.11). We then show that fields can be given yet another interpretation via the index of the twisted (Kostant) Dirac operator of [44]. The identity representation of F 4 encoding the supergravity multiplet is the space of twisted harmonic spinors on OP 2 , which is propositon 2.12.After studying structures on OP 2 , we use that space itself as the fiber over eleven-dimensional spacetime. In section 2.4.1 we explore the consequences of this idea by relating the geometry and the characteristic classes of the base to that of the total space, using the knowledge of that of the fiber studied in section 2. If the base Y 11 has positive Ricci curvature then so does the total space M 27 . This and related matters are discussed in section 2.4.1. In section 2.5 we relate structures, such as Fivebrane structures [57] [58], as well as genera on the base space to genera on the total space. This includes elliptic genera, Witten genera, Ochanine genera and is the content of proposition 2.15. We use this to r...

Topological actions via gauge variations of higher structures

“…The constructions and results in [16] are established for rational numbers, but they readily extend over the reals, especially for smooth manifolds as we consider here. Note that we are considering only topological terms and our expressions are cohomological, and so we are in a setting akin to that of a topological field theory.…”

“…Such higher structures (beyond Spin) have been recast using Spin structures. We have shown in [16] that (i) For every rational Spin-Fivebrane class F ∈ H 7 (Q; Q), the pullback ρ * F is a rational Fivebrane class.…”

“…As in the case of the 3-dimensional M2-brane worldvolume admitting a String structure [11], the extended M5/NS5-brane worldvolume automatically admits a Fivebrane structure by virtue of it being 7-dimensional. Note that if H 4 (M ; Q) is torsion, then the set of Fivebrane classes and Spin-Fivebrane classes coincide [16]. Compactification manifolds for which this occurs include the following manifolds as in the realistic Kaluza-Klein list [32].…”

“…Indeed, in [16] we defined and characterized these rational Spin-String, Spin-Fivebrane, and Spin-Ninebrane structures. It is our aim here to apply these to describe how topological action functionals in M-theory and string theory can be interpreted as global gauge transformations corresponding to vari-ations of such structures.…”

“…Interpreting the cup products from [16] as topological Lagrangians for brane sigma models, means to regard these branes as propagating not on spacetime M, but on the total space of the Spin bundle Q, i.e., we have sigma models on spacetimes which are "extended". Indeed, in [22], the authors describe the heterotic string not as an algebraically defined CFT, but as a geometric sigma model on the Spin bundle Q of spacetime M .…”