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

## TL;DR

This paper reveals that the data for the embedding tensor in gauged supergravity correspond precisely to Leibniz algebras, which induce Lie n-algebras explaining the tensor hierarchy in supergravity theories.

## Contribution

It establishes a canonical construction from Leibniz algebras to Lie n-algebras, elucidating the algebraic structure behind gauged supergravity tensor hierarchies.

## Key findings

- Leibniz algebras correspond to embedding tensor data in supergravity.
- Lie n-algebras are canonically derived from Leibniz algebras.
- The tensor hierarchy relates to Lyndon words in universal enveloping algebras.

## 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.