Leibniz-Chern-Simons Theory and Phases of Exceptional Field Theory

We discuss a generalization of Chern-Simons theory in three dimensions based on Leibniz (or Loday) algebras, which are generalizations of Lie algebras. Special cases of such theories appear in gauged supergravity, where the Leibniz algebra is defined in terms of the global (Lie) symmetry algebra of the ungauged limit and an embedding tensor. We show that the Leibniz algebra of generalized diffeomorphisms in exceptional field theory can similarly be obtained from a Lie algebra that describes the enhanced symmetry of an ungauged phase' of the theory. Moreover, we show that a topological phase’ of ${\mathrm{E}}_{8(8)}$ exceptional field theory can be interpreted as a Chern-Simons theory for an algebra unifying the three-dimensional Poincar'e algebra and the Leibniz algebra of ${\mathrm{E}}_{8(8)}$ generalized diffeomorphisms.