# Leibniz Gauge Theories and Infinity Structures

## TL;DR

This paper introduces infinity-enhanced Leibniz algebras to systematically construct tensor hierarchies in gauge theories, extending previous low-level frameworks and linking to $L_{
abla}$-algebras for topological field theories.

## Contribution

It defines infinity-enhanced Leibniz algebras that enable consistent tensor hierarchies at arbitrary levels and explores their relation to $L_{
abla}$-algebras.

## Key findings

- Defined infinity-enhanced Leibniz algebras for arbitrary tensor hierarchy levels
- Connected these algebras to $L_{
abla}$-algebras and topological theories
- Extended mathematical structures underlying gauge theories

## Abstract

We formulate gauge theories based on Leibniz(-Loday) algebras and uncover their underlying mathematical structure. Various special cases have been developed in the context of gauged supergravity and exceptional field theory. These are based on `tensor hierarchies', which describe towers of $p$-form gauge fields transforming under non-abelian gauge symmetries and which have been constructed up to low levels. Here we define `infinity-enhanced Leibniz algebras' that guarantee the existence of consistent tensor hierarchies to arbitrary level. We contrast these algebras with strongly homotopy Lie algebras ($L_{\infty}$ algebras), which can be used to define topological field theories for which all curvatures vanish. Any infinity-enhanced Leibniz algebra carries an associated $L_{\infty}$ algebra, which we discuss.