The Embedding Tensor, Leibniz-Loday Algebras, and Their Higher Gauge Theories

Abstract

We show that the data needed for the method of the embedding tensor employed in gauging supergravity theories are precisely those of a Leibniz algebra (with one of its induced quotient Lie algebras embedded into a rigid symmetry Lie algebra that provides an additional "represtentation constraint"). Every Leibniz algebra gives rise to a Lie n-algebra in a canonical way (for every $n\in \mathbb{N}\cup \{\infty \}$). It is the gauging of this $L_{\infty }$-algebra that explains the tensor hierarchy of the bosonic sector of gauged supergravity theories. The tower of p-from gauge fields corresponds to Lyndon words of the universal enveloping algebra of the free Lie algebra of an odd vector space in this construction. Truncation to some $n$ yields the reduced field content needed in a concrete spacetime dimension.

26 pages, 1 figure, addition of references and a new subsection entitled "The gauge field sector of gauged maximal supergravity in d = 4". Version accepted for publication in Comm. Math. Phys