The Geometry of Supergravity Torsion Constraints

This paper, prepared for the 2001 Park City Research Program in Supergeometry, is an exposition of [4]. References to relevant earlier papers can be found in the bibliography of [4]. As explained in Jim Gates’ lectures, an essential ingredient of the superspace formulation of supergravity is a nonzero torsion tensor. Furthermore, there are torsion constraints in the sense that only certain components of the torsion tensor are allowed to be nonzero. These torsion constraints must be stringent enough to give physically relevant solutions, but flexible enough to allow for nonflat solutions. For example, for N = 1 supergravity on a Lorentzian 3-manifold, Jim Gates motivated the choice that Tαβ c = i(γ)αβ, Tαb γ can be nonzero, Tab γ can be nonzero and the other torsion components must be zero. The existence of a nonzero torsion tensor in supergravity theories, and the constraints on the torsion tensor, are perhaps surprising from the viewpoint of standard differential geometry. In particular, their geometric meaning is not immediately clear. To give a somewhat analogous situation from conventional geometry, suppose that M is an almost complex manifold with a Hermitian metric. Complexifying TM and using notation that will be explained later, suppose that we are told that a desirable set of torsion conditions is