Smooth loops and loop bundles

A loop is a rather general algebraic structure that has an identity element and division, but is not necessarily associative. Smooth loops are a direct generalization of Lie groups. A key example of a non-Lie smooth loop is the loop of unit octonions. In this paper, we study properties of smooth loops and their associated tangent algebras, including a loop analog of the Mauer-Cartan equation. Then, given a manifold, we introduce a loop bundle as an associated bundle to a particular principal bundle. Given a connection on the principal bundle, we define the torsion of a loop bundle structure and show how it relates to the curvature, and also consider the critical points of some related functionals. Throughout, we see how some of the known properties of $G_{2}$-structures can be seen from this more general setting.Comment: 101 pages, 3 figure

Moduli spaces of G2 manifolds

This paper is a review of current developments in the study of moduli spaces of G2 manifolds. G2 manifolds are 7-dimensional manifolds with the exceptional holonomy group G2. Although they are odd-dimensional, in many ways they can be considered as an analogue of Calabi-Yau manifolds in 7 dimensions. They play an important role in physics as natural candidates for supersymmetric vacuum solutions of M-theory compactifications. Despite the physical motivation, many of the results are of purely mathematical interest. Here we cover the basics of G2 manifolds, local deformation theory of G2 structures and the local geometry of the moduli spaces of G2 structures.Comment: 31 pages, 2 figure

Minisuperspace Models in M-theory

We derive the full canonical formulation of the bosonic sector of 11-dimensional supergravity, and explicitly present the constraint algebra. We then compactify M-theory on a warped product of homogeneous spaces of constant curvature, and construct a minisuperspace of scale factors. First classical behaviour of the minisuperspace system is analysed, and then a quantum theory is constructed. It turns out that there similarities with the "pre-Big Bang" scenario in String Theory.Comment: 35 pages, 2 figures, added additional discussion of gauge fixing and self-adjointness of the Hamiltonian, added reference

Isometric Flows of G 2-structures

We survey recent progress in the study of flows of isometric G 2-structures on seven-dimensional manifolds, that is, flows that preserve the metric, while modifying the G 2-structure. In particular, heat flows of isometric G 2-structures have been recently studied from several different perspectives, in particular in terms of 3-forms, octonions, vector fields, and geometric structures. We will give an overview of each approach, the results obtained, and compare the different perspectives

Deformations of G2-structures with torsion

Non UBCUnreviewedAuthor affiliation: Stony Brook UniversityPostdoctora