Abstract
We start developing a formalism which allows to construct supersymmetric theories systematically across space-time signatures. Our construction uses a complex form of the supersymmetry algebra, which is obtained by doubling the spinor representation. This allows one to partially disentangle the Lorentz and R-symmetry group and generalizes symplectic Majorana spinors. For the case where the spinor representation is complex-irreducible, the R-symmetry only acts on an internal multiplicity space, and we show that the connected groups which occur are SO(2), SO0(1, 1), SU(2) and SU(1, 1).As an application we construct the off-shell supersymmetry transformations and supersymmetric Lagrangians for five-dimensional vector multiplets in arbitrary signature (t, s). In this case the R-symmetry groups are SU(2) or SU(1, 1), depending on whether the spinor representation carries a quaternionic or para-quaternionic structure. In Euclidean signature the scalar and vector kinetic terms differ by a relative sign, which is consistent with previous results in the literature and shows that this sign flip is an inevitable consequence of the Euclidean supersymmetry algebra.
Highlights
Introduction and summary of resultsNon-Lorentzian space-time signatures are relevant for a variety of reasons
Our construction uses a complex form of the supersymmetry algebra, which is obtained by doubling the spinor representation
For the case where the spinor representation is complexirreducible, the R-symmetry only acts on an internal multiplicity space, and we show that the connected groups which occur are SO(2), SO0(1, 1), SU(2) and SU(1, 1)
Summary
Non-Lorentzian space-time signatures are relevant for a variety of reasons. For Euclidean signature this is obvious, since the study of non-perturbative effects, such as instantons, makes use of the Euclidean functional integral formalism. In [13] analytic continuation of the Killing spinor equations were used to obtain the supersymmetry variations of the fermions, and, by imposing closure of the algebra, the bosonic terms of the on-shell Lagrangian for five-dimensional vector multiplets coupled to supergravity for all signatures. As we will explain in more detail in the paper, the essential part of defining a supersymmetry algebra is to choose a so-called admissible bilinear form on the spinor module S of the Poincare Lie superalgebra sp(V ) = V + so(V ) + S, where V = Rt,s is a space-time with signature (t, s). We compare our results to those of [13], where the bosonic on-shell Lagrangians for five-dimensional vector multiplets coupled to supergravity have been obtained for all signature by analytic continuation of the Killing spinor equations of the Lorentzian theory. Appendices B and C contain the relevant background on para-quaternions and quaternions
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