On Exotic Six-Dimensional Supergravity Theories

The exotic 𝒩 = (3,1) and 𝒩 = (4,0) supergravity theories in six dimensions are theories describing potentials with mixed-symmetry indices satisfying a second order self-duality relation, and giving rise to maximal supergravity in five dimensions upon dimensional reduction. After reviewing how the linearised equations are constructed, we show how introducing additional gauge symmetries one can derive the same equations from first-order self-duality relations. We discuss how this result can be used to obtain a covariant lagrangian.