Extended geometries - Karlsson, Robin (My University, 2018)
The low-energy effective field theories of string theory/M-theory compactified on torii possess “hidden” symmetries given by the exceptional Lie groups. Moreover a discrete version of these, U-duality, appears to be unbroken in the full theory. Generically such transformations mix gravitational and non-gravitational degrees of freedom and a covari- ant formalism calls for a merger of the underlying fields. The goal of extended geometries is therefore to generalise ordinary geometry in order to include non-gravitational degrees of freedom as a part of an extended geometry. This is done by introducing space-time coordinates in a module of an arbitrary structure group and by the introduction of gen- eralised diffeomorphisms. The physically most interesting cases, being double geometry and exceptional geometry based on even-dimensional orthogonal groups and exceptional groups respectively, are reviewed before the general construction of extended geometries is introduced. Especially we focus on closure of the algebra of generalised diffeomorphisms which for consistency requires a local embedding of ordinary geometry specified by the so-called section constraint. Moreover, an invariant action for the generalised metric is de- rived as well as geometrical objects such as torsion and a generalised Ricci-tensor. Lastly exceptional geometry based on the rank seven exceptional group with a global solution to the section constraint is used to study properties of flux compactifications from eleven- dimensional supergravity to four dimensions.
Keywords: Grundläggande vetenskaper; Geometri; Matematisk fysik; Relativitetsteori, gravitation; Basic Sciences; Geometry; Mathematical physics; Theory of relativity, gravitation