Half-maximal supergravity in three dimensions: supergeometry, differential forms and algebraic structure

TL;DR

This paper explores the geometric and algebraic structures of half-maximal supergravity in three dimensions, focusing on superspace formulation, p-form fields, and their relation to Borcherds superalgebras, revealing new higher-degree forms.

Contribution

It provides a detailed superspace analysis of supergravity, classifies p-form field strengths, and links their algebraic structure to Borcherds superalgebras, extending known form sets.

Findings

Explicit construction of p=2,3,4 forms

Identification of five-forms in supergravity

Non-trivial contributions of six-forms at order α'

Abstract

The half-maximal supergravity theories in three dimensions, which have local $SO(8)\xz SO(n)$ and rigid SO(8,n) symmetries, are discussed in a superspace setting starting from the superconformal theory. The on-shell theory is obtained by imposing further constraints; it is essentially a non-linear sigma model that induces a Poincar\'e supergeometry. The deformations of the geometry due to gauging are briefly discussed. The possible $p$-form field strengths are studied using supersymmetry and SO(8,n) symmetry. The set of such forms obeying consistent Bianchi identities constitutes a Lie super co-algebra while the demand that these identities admit solutions places a further constraint on the possible representations of SO(8,n) that the forms transform under which can be easily understood using superspace cohomology. The dual Lie superalgebra can then be identified as the positive sector of a Borcherds superalgebra that extends the Lie algebra of the duality group. In addition to the known $p=2,3,4$ forms, which we construct explicitly, there are five-forms that can be non-zero in supergravity, while all forms with $p>5$ vanish. It is shown that some six-forms can have non-trivial contributions at order $\a'$.