Non-associative differential geometry and L algebras towards a description of non-geometric closed string vacua

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The foundations of a non-associative gravity induced by locally non-geometric R-flux backgrounds of string theory were laid in [1]. Moving on the same direction, we study the relation of invariance under non-associative diffeomorphisms on spacetime of our geometry, the classical diffeomorphism symmetry of closed string theory and the generalized diffeomorphism symmetries of double field theory. Furthermore, we aim to construct an appropriate action functional for the on-associative Einstein equations, and further study the pullbacks of phase space Torsion and Riemann curvatures to the underlying spacetime. Towards this goal we shall express Einstein-Cartan gravity in the language of L algebras, as done recently for usual gauge theories [2]. Following that, we aim to twist the underlying geometrical structure. This line of research should provide evidence for the expected relevance of non associative geometry as a low energy limit of closed string theory