Gauged extended field theory and generalized Cartan geometry

Cartan geometry provides a unifying algebraic construction of curvature and torsion, based on an underlying model Lie algebra; a viewpoint that can be extended naturally to the higher algebraic structures underlying supergravity. We present a Cartan-geometric framework for generalized geometries governed by a differential graded Lie algebra, extending previous results. The extended tangent bundle admits the action of both a global duality group <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi mathvariant="script">G</a:mi> </a:math> and a local gauge group <d:math xmlns:d="http://www.w3.org/1998/Math/MathML" display="inline"> <d:mi>H</d:mi> </d:math> . This algebraic structure is implemented via a brane current algebra; the phase space Poisson structure of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:mi>p</f:mi> </f:math> -branes. Within this Cartan-inspired framework, we define a hierarchy of generalized connections and compute their linearized torsion and curvature tensors, including the higher curvatures required by the tensor hierarchy. This provides a systematic construction of curvature and torsion tensors in generic generalized geometries.