$L_\infty$ algebras for extended geometry from Borcherds superalgebras

TL;DR

This paper constructs an $L_ abla$ algebra framework for extended geometry gauge transformations using Borcherds superalgebras, explicitly detailing brackets and their properties, with implications for tensor hierarchy algebras.

Contribution

It introduces a novel $L_ abla$ algebra structure for extended geometry gauge transformations based on Borcherds superalgebras, including explicit brackets and Bernoulli number coefficients.

Findings

Explicit $L_ abla$ brackets involving ghosts are derived.

All even brackets above 2-brackets vanish.

Coefficients are given by Bernoulli numbers.

Abstract

We examine the structure of gauge transformations in extended geometry, the framework unifying double geometry, exceptional geometry, etc. This is done by giving the variations of the ghosts in a Batalin-Vilkovisky framework, or equivalently, an $L_\infty$ algebra. The $L_\infty$ brackets are given as derived brackets constructed using an underlying Borcherds superalgebra ${\scr B}({\mathfrak g}_{r+1})$, which is a double extension of the structure algebra ${\mathfrak g}_r$. The construction includes a set of "ancillary" ghosts. All brackets involving the infinite sequence of ghosts are given explicitly. All even brackets above the 2-brackets vanish, and the coefficients appearing in the brackets are given by Bernoulli numbers. The results are valid in the absence of ancillary transformations at ghost number 1. We present evidence that in order to go further, the underlying algebra should be the corresponding tensor hierarchy algebra.