Four-dimensional vector multiplets in arbitrary signature

Understand

We derive a necessary and sufficient condition for Poincar\'e Lie superalgebras in any dimension and signature to be isomorphic.

• This reduces the classification problem, up to certain discrete operations, to classifying the orbits of the Schur group on the vector space of superbrackets.
• We then classify four-dimensional ${\cal N}=2$ supersymmetry algebras, which are found to be unique in Euclidean and in neutral signature, while in Lorentz signature there exist two algebras with R-symmetry groups $\mathrm{U}(2)$ and $\mathrm{U(}1,1)$, respectively.
• By dimensional reduction we construct two off shell vector multiplet representations for each possible signature, and find that the corresponding Lagrangians always have a different relative sign between the scalar and the Maxwell term.