Off-shell Supersymmetric Continuous Spin Gauge Theory

We construct, for the first time, an off-shell supersymmetric continuous spin gauge theory in 4-dimensional Minkowski spacetime, in both constrained and unconstrained Lagrangian formulations. As an extension to the on-shell description [1], we employ an auxiliary field to close the algebra of supersymmetry transformations off-shell. The 4d $\mathcal{N}=1$ massless continuous spin supermultiplet is then denoted by $(\mathrm{\Phi}, \mathrm{H} \,; \mathrm{\Psi})$, comprised of a dynamical and a non-dynamical complex scalar continuous spin gauge fields $\mathrm{\Phi}$ and $\mathrm{H}$, as well as a Dirac continuous spin gauge field $\mathrm{\Psi}$. In particular, we demonstrate that the off-shell continuous spin supermultiplet, in a limit, reproduces off-shell supersymmetry transformations of the known scalar supermultiplet $(\,{\scriptstyle 0}\,,\, {\scriptstyle 1/2}\,)$, all integer-spin supermultiplets $(\,s\,,\, s\, {\scriptstyle +\, 1/2}\,),\, s\geqslant1$, and all half-integer spin supermultiplets $(\, s\, {\scriptstyle -\, 1/2}\,,\, s \,\,),\, s\geqslant1$.