Five-dimensional vector multiplets in arbitrary signature

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), SO 0 (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.

Supersymmetry algebras in arbitrary signature and their R-symmetry groups

“…, D − 1. 16 We remark that our choice for A is not unique, and for Lorentz signature there are conventions which differ from ours by a factor −1 or ±i.…”

“…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].…”

“…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.…”

“…This is natural, because, as we will see later, τ and another parameter σ are invariants which characterize the properties of a complex bilinear form with Gram matrix C, for which signature does not have an invariant meaning. 16 In general we use µ = 1, . .…”

“…We will use this in section 5 to insure that the vector-valued bilinear form obtained by imposing a reality condition on the complex vector-valued form is real-valued. See also [16,17] for how this freedom is used when constructing supersymmetric theories. To simplify notation, we will omit the superscript (α) whenever the value of the phase is unimportant.…”