The Action of the (free) $(4,0)$-theory

TL;DR

This paper constructs the first explicit action principle for the free six-dimensional (4,0) supergravity theory, detailing its variables, supersymmetry invariance, and algebraic closure, thus advancing understanding of this exotic, highly symmetric model.

Contribution

It provides the first explicit formulation of the free (4,0) theory's action, including variables, supersymmetry transformations, and algebraic closure, which was previously unknown.

Findings

Explicit free (4,0) action derived

Supersymmetry invariance proven

Algebra closes with (4,0) supersymmetry

Abstract

The $(4,0)$ theory in six dimensions is an exotic theory of supergravity that has been argued to emerge as the strong coupling limit of theories having $N=8$ supergravity as their low energy effective theory in five spacetime dimensions. It has maximal supersymmetry and is superconformal. Very little is known about this intriguing theory. While the spectrum of fields occurring in its description has been given and their equations of motion in the absence of interactions have been written down, no action principle has been formulated, even in the free case. We extend here previous analyses by writing explicitly the action of the free $(4,0)$ theory from which the equations of motion derive. The variables of the variational principle are prepotentials adapted to the self-duality properties of the fields. The "exotic gravitini", described by chiral fermionic two-forms, are given special attention. The supersymmetry transformations are written down and the invariance of the action is explicitly proven. Even though the action is not manifestly covariant, the symmetry transformations are shown to close according to the $(4,0)$-extended Poincar\'e supersymmetry algebra. We also discuss exotic supergravity models with fewer supersymmetries. Remarks on dimensional reduction close the paper.