18 research outputs found
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
-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