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Clifford algebras, Moufang Loops, G2 structures and deformations

Grant number: 18/10367-2
Support type:Scholarships in Brazil - Master
Effective date (Start): October 01, 2018
Effective date (End): September 30, 2020
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Cooperation agreement: Coordination of Improvement of Higher Education Personnel (CAPES)
Principal Investigator:Roldão da Rocha Junior
Grantee:Aquerman Yanes Martinho
Home Institution: Centro de Matemática, Computação e Cognição (CMCC). Universidade Federal do ABC (UFABC). Ministério da Educação (Brasil). Santo André , SP, Brazil

Abstract

We shall investigate octonionic product deformations which come from the parallelizable torsion on the 7-sphere S7, extending the Moufang identity to these products and obtaining a family of geometries over S7, which arise as new solutions of the moviment equation from the Lagrangian formalism. This is done by considering the spontaneous compactification ADS4xS7, where ADS4 denotes the anti-de Sitter space in four dimensions and its generalizations. Besides the conventional Riemannian geometry and the ones proposed by Cartan and Schouten, we shall obtain solutions in geometries with torsion, as well as in more general seven dimensional spaces. Such formalism will be subsequently derived in the 7-sphere S7 with parallelizable torsion, locally given by the structure constants of a non-associative geodesic loop in the conneceted afine space, which will be afterwards also deformed from the generalization of the so called X-products. G2 structures in 7-manifolds will also be considered, with an introduction of complex octonions and the correspondent G2 structures. We shall rewrite the equation for the Killing spinor in seven dimensions in terms of an complex octonion fiber section. Our goal is to define a complexified G2 structure corresponding to that section and show that its torsion is in the G2 class of complex representations. We will also analize if the Killing equation implies in a G flux 4-form satisfying the Bianchi identity: dG equals 0