Abstract
We investigate the relationship between conformal and spin structure on lorentzian manifolds and see how their compatibility influences the formulation of rigid supersymmetric field theories. In dimensions three, four, six and ten, we show that if the Dirac current associated with a generic spinor defines a null conformal Killing vector then the spinor must obey a twistor equation with respect to a certain connection with torsion. Of the theories we consider, those with classical superconformal symmetry in Minkowski space can be reformulated as rigid supersymmetric theories on any lorentzian manifold admitting twistor spinors. In dimensions six and ten, we also describe rigid supersymmetric gauge theories on bosonic minimally supersymmetric supergravity backgrounds.
Highlights
Recent interest in this topic stems primarily from the impressive results obtained by Pestun in [10] for Wilson loops in maximally supersymmetric Yang-Mills theory on S4
We investigate the relationship between conformal and spin structure on lorentzian manifolds and see how their compatibility influences the formulation of rigid supersymmetric field theories
If a field theory on a lorentzian spin manifold M has a rigid supersymmetry generated by a spinor ǫ and its associated Dirac current ξ generates a null conformal isometry of M that is compatible with the spin structure, we will see that ǫ must obey a twistor spinor equation with respect to a certain connection with torsion, whose form is dictated by the isotropy group of ǫ
Summary
Given a non-zero element ψ = ψA eA, built from a pair of complex Dirac spinors (ψ1, ψ2) which transform in the defining representation of usp(2), one may impose an alternative reality condition (ψA)∗ = εAB BψB only if B∗B = −1. It defines the symplectic Majorana spinor representation. In this case, B corresponds to a quaternionic structure on C2d and requires σCσd = −1 which occurs only in d = 5, 6, 7, 8 mod 8. We shall make use of these results in our subsequent analysis
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