Supersymmetric Killing Structures

Abstract

In this text we combine the notions of supergeometry and supersymmetry. We construct a special class of supermanifolds whose reduced manifolds are (pseudo) Riemannian manifolds. These supermanifolds allow us to treat vector fields on the one hand and spinor fields on the other hand as equivalent geometric objects. This is the starting point of our definition of supersymmetric Killing structures. The latter combines subspaces of vector fields and spinor fields, provided they fulfill certain field equations. This naturally leads to a superalgebra that extends the supersymmetry algebra to the case of non-flat reduced space. We examine in detail the additional terms that enter into this structure and we give a lot of examples.

Killing vector fields and harmonic superfield theories

“…3.8 is taken as the definition of the Lie derivative in [Kli05] (up to a global sign). For semi-Riemannian supermetrics, we have the following characterisation, which naturally generalises the analogous formula in the classical case.…”

The Torsion of Spinor Connections and Related Structures

“…. , 1), compare [25]. Furthermore, we draw the attention to [3] for a classification of bilinear forms also for the case of extended supersymmetry algebras.…”

“…Then condition (5.2) holds for the two parallel pure spinors. 6 A denotes the matrix which defines the charge conjugation ϕ C := Aϕ * compare [25].…”

“…In this case Ξ = (ξ, ξ C ) is real and B(R; Ξ, Ξ) does not vanish. This has been used in [25] to show that the real supersymmetric Killing structure is not finite in the case of quaternionic spin representation where a twist of the spinor bundle is necessary to yield a real structure. Nevertheless, it has been shown that in this case there exist two isomorphic finite sub-structures.…”

“…This yields a connection between the graded Jacobi identity on the superalgebra of vector fields and the Bianchi identities derived in Section 3. The motivation for the introduction of supermanifolds and the consideration of the canonical vector field is taken from the constructions in [24] and [25]. The canonical vector field we consider has also been discussed in [30] from another point of view: One of the vector fields is considered as first order opera-tor on the bundle of exterior powers of the spin bundle and it is asked when this operator is a differential.…”