# Higher curvature Bianchi identities, generalised geometry and $L_{\infty}$ algebras

## TL;DR

The paper investigates higher derivative corrections to Bianchi identities in supergravity, revealing that these modifications challenge existing geometric frameworks but can be described using $L_{
abla}$-algebras, extending the algebraic understanding of fluxes.

## Contribution

It demonstrates that higher order R^4 corrections lead to an $L_{
abla}$-algebra structure, generalizing the gauge algebra beyond Courant brackets in supergravity.

## Key findings

- Higher derivative corrections modify Bianchi identities.
- The gauge algebra forms an $L_{
abla}$-algebra, not a Courant bracket.
- The algebra terminates at a ten-bracket level for seven-form flux.

## Abstract

The Bianchi identities for bosonic fluxes in supergravity can receive higher derivative quantum and string corrections, the most well known being that of Heterotic theory $d H = \tfrac{1}{4}\alpha'(\text{tr } F^2 - \text{tr } R^2)$. Less studied are the modifications at order $R^4$ that may arise, for example, in the Bianchi identity for the seven-form flux of M theory compactifications. We argue that such corrections appear to be incompatible with the exceptional generalised geometry description of the lower order supergravity, and seem to imply a gauge algebra for the bosonic potentials that cannot be written in terms of an (exceptional) Courant bracket. However, we show that this algebra retains the form of an $L_{\infty}$ gauge field theory, which terminates at a level ten multibracket for the case involving just the seven-form flux.