Exploring the Abelian 4D, $\mathcal{N}$ = 4 Vector-Tensor Supermultiplet and Its Off-Shell Central Charge Structure

An abelian 4D, $\mathcal{N}$ = 4 vector supermultiplet allows for a duality transformation to be applied to one of its spin-0 states. The resulting theory can be described as an abelian 4D, $\mathcal{N}$ = 4 vector-tensor supermultiplet. It is seen to decompose into a direct sum of an off-shell 4D, $\mathcal{N}$ = 2 vector supermultiplet and an off-shell 4D, $\mathcal{N}$ = 2 tensor supermultiplet. The commutator algebra of the other two supersymmetries are still found to be on-shell. However, the central charge structure in the resulting 4D, $\mathcal{N}$ = 4 vector-tensor supermultiplet is considerably simpler that that of the parent abelian 4D, $\mathcal{N}$ = 4 vector supermultiplet. This appears to be due to the replacement of the usual SO(4) symmetry associated with the abelian 4D, $\mathcal{N}$ = 4 vector supermultiplet being replaced by a GL(2,$\mathbb{R}$)$\otimes$GL(2,$\mathbb{R}$) symmetry in the 4D, $\mathcal{N}$ = 4 vector-tensor supermultiplet. The $Mathematica$ code detailing the calculations is available open-source at the HEPTHools Data Repository on GitHub.