Forms and algebras in (half-)maximal supergravity theories

TL;DR

This paper explores the algebraic structures underlying forms in (half-)maximal supergravity theories across various dimensions, revealing their connection to Borcherds and Kac-Moody superalgebras and discussing gauging via algebraic deformations.

Contribution

It systematically derives Bianchi identities for all forms, showing their compatibility with supersymmetry and linking form spectra to Borcherds and Kac-Moody algebras, including E11.

Findings

Bianchi identities are consistent and solvable for all forms.

Form spectra match those of associated Kac-Moody algebras.

Gauging is achieved by deforming Bianchi identities with a new algebraic element.

Abstract

The forms in D-dimensional (half-)maximal supergravity theories are discussed for 3 $\leq$ D $\leq$ 11. Superspace methods are used to derive consistent sets of Bianchi identities for all the forms for all degrees, and to show that they are soluble and fully compatible with supersymmetry. The Bianchi identities determine Lie superalgebras that can be extended to Borcherds superalgebras of a special type. It is shown that any Borcherds superalgebra of this type gives the same form spectrum, up to an arbitrary degree, as an associated Kac-Moody algebra. For maximal supergravity up to D-form potentials, this is the very extended Kac-Moody algebra E11. It is also shown how gauging can be carried out in a simple fashion by deforming the Bianchi identities by means of a new algebraic element related to the embedding tensor. In this case the appropriate extension of the form algebra is a truncated version of the so-called tensor hierarchy algebra.