# $L_\infty$ algebras for extended geometry

## TL;DR

This paper explores the use of $L_
abla$ algebras to describe gauge transformations in extended geometry frameworks related to string and M-theory dualities, highlighting their algebraic structure.

## Contribution

It reveals that gauge transformations in extended geometry are governed by an $L_
abla$ algebra derived from Borcherds superalgebras, providing a universal algebraic description.

## Key findings

- Gauge transformations are infinitely reducible.
- They form an $L_
abla$ algebra derived from superalgebras.
- Universal expressions for brackets are established.

## Abstract

Extended geometry provides a unified framework for double geometry, exceptional geometry, etc., i.e., for the geometrisations of the string theory and M-theory dualities. In this talk, we will explain the structure of gauge transformations (generalised diffeomorphisms) in these models. They are generically infinitely reducible, and arise as derived brackets from an underlying Borcherds superalgebra or tensor hierarchy algebra. The infinite reducibility gives rise to an $L_\infty$ structure, the brackets of which have universal expressions in terms of the underlying superalgebra.