The canonical 8-form on manifolds with holonomy group Spin(9)

An explicit expression of the canonical 8-form on a Riemannian manifold with a Spin(9)-structure, in terms of the nine local symmetric involutions involved, is given. The list of explicit expressions of all the canonical forms related to Berger's list of holonomy groups is thus completed. Moreover, some results on Spin(9)-structures as G-structures defined by a tensor and on the curvature tensor of the Cayley planes, are obtained

Composition algebras and the two faces of G2G_{2}

We consider composition and division algebras over the real numbers: We note two r\^oles for the group $G_{2}$: as automorphism group of the octonions and as the isotropy group of a generic 3-form in 7 dimensions. We show why they are equivalent, by means of a regular metric. We express in some diagrams the relation between some pertinent groups, most of them related to the octonions. Some applications to physics are also discussed.Comment: 11 pages, 3 figure

Categorizing perceived causal events

Over the last few decades, Causal Model Theory (CMT) has become a dominant framework for human causal-based reasoning, including categorization and inference. CMT prescribes how people should reason about probabilistic events in terms of causal models. In typical causal-based categorization experiments, subjects are provided with verbal descriptions of causally linked features, generally including probabilistic information. Another line of research focuses on perceived or experienced causal events, rather than on verbal descriptions. In this work we asked whether effects which are consistent with CMT, and that have been obtained with verbal descriptions, generalize to visually perceived events. In two experiments, we presented subjects with videos of a 3D A→B causal event rather than verbal descriptions. In Exp. 1, we found that subjects who saw the causal event did not show the coherence effect in categorization (i.e., subjects tend to rate the null ¬A¬B event as a category member). However, subjects who did see the null event during training did show the effect. In Exp. 2, we ruled out the possibility that Exp. 1’s results were simply an effect of how frequently events were experienced during training. We conclude that a one-shot perceived causal event is not sufficient for people to show causal-based reasoning as CMT predicts

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

The G_2 sphere over a 4-manifold

We present a construction of a canonical G_2 structure on the unit sphere tangent bundle S_M of any given orientable Riemannian 4-manifold M. Such structure is never geometric or 1-flat, but seems full of other possibilities. We start by the study of the most basic properties of our construction. The structure is co-calibrated if, and only if, M is an Einstein manifold. The fibres are always associative. In fact, the associated 3-form results from a linear combination of three other volume 3-forms, one of which is the volume of the fibres. We also give new examples of co-calibrated structures on well known spaces. We hope this contributes both to the knowledge of special geometries and to the study of 4-manifolds.Comment: 13 page