Four-dimensional vector multiplets in arbitrary signature (I)

Abstract

We derive a necessary and sufficient condition for Poincaré 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 [Formula: see text] 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 [Formula: see text] and [Formula: see text], respectively.

Supersymmetry algebras in arbitrary signature and their R-symmetry groups

“…Lower-dimensional supergravity theories in nonstandard signatures have been constructed using dimensional reduction in [6][7][8][9][10][11]. A Euclidean version of the special geometry of N = 2 vector and hypermultiplets has been developed in [12][13][14][15], while N = 2 vector multiplets in arbitrary signature were constructed in [16,17]. Four-dimensional supersymmetric solutions in neutral signature have been investigated in [18,19], brane-like solutions in arbitrary dimension and signature have been constructed in [20] and supersymmetric solutions of five-dimensional vector multiplets coupled to supergravity have recently been studied for arbitrary signature in [21].…”

“…Both groups are contained in the Clifford algebra Cl(V ). A precise criterion for two Poincaré Lie superalgebras to be isomorphic is given by Theorem 1 of [17]. As illustrated in [17] by the classification of four-dimensional supersymmetry algebras with eight real supercharges for arbitrary signature, this classification can be done case by case but requires some work.…”

“…A precise criterion for two Poincaré Lie superalgebras to be isomorphic is given by Theorem 1 of [17]. As illustrated in [17] by the classification of four-dimensional supersymmetry algebras with eight real supercharges for arbitrary signature, this classification can be done case by case but requires some work. However there is a sufficient condition for two supersymmetry algebras to be non-isomorphic which is easier to check, namely that their R-symmetry groups are different.…”

“…In appendix C we present JHEP10(2021)203 details on the complexification of spinor modules, which are relevant for understanding the precise relation between the odd parts g 1 and g C 1 of real and complex supersymmetry algebras. In appendix D we review the matrix notation for Weyl spinors introduced in [17], which is used in the main part to disentangle the actions of the spin and R-symmetry group in even dimensions by doubling the auxiliary multiplicity space of our construction. Alternatively one can work without doubling, in which case the R-symmetry group operates on spinor indices, in a way that we also explain in appendix D. In appendix E we provide the details of several isomorphisms which are needed to show that the supersymmetry algebras that we have constructed are classified, essentially, by their R-symmetry groups.…”

“…In summary, real semi-spinors and complex semi-spinors are equivalent to each other and correspond to Weyl spinors of either chirality, which are equivalent, as real spin representations, to Majorana spinors. 14 For more examples we refer to [16,17] where the spinor modules and Schur algebras have been worked for D = 4, 5 for all signatures. This already provides examples of all possible cases.…”