The Action of the (Free) $\mathcal{N}=(3,1)$ Theory in Six Spacetime Dimensions

The action of the free $\mathcal{N}=(3,1)$ theory in six spacetime dimensions is explicitly constructed. The variables of the variational principle are prepotentials adapted to the self-duality conditions on the fields. The $(3,1)$ supersymmetry variations are given and the invariance of the action is verified. The action is first-order in time derivatives. It is also Poincar'e invariant but not manifestly so, just like the Hamiltonian action of more familiar relativistic field theories.