Six-dimensional supermultiplets from bundles on projective spaces

Abstract

The projective variety of square-zero elements in the six-dimensional minimal supersymmetry algebra is isomorphic to P 1 × P 3 . We use this fact, together with the pure spinor superfield formalism, to study supermultiplets in six dimensions, starting from vector bundles on projective spaces. We classify all multiplets whose derived invariants for the supertranslation algebra form a line bundle over the nilpotence variety; one can think of such multiplets as being those whose holomorphic twists have rank one over Dolbeault forms on spacetime. In addition, we explicitly construct multiplets associated to natural higher-rank equivariant vector bundles, including the tangent and normal bundles as well as their duals.Among the multiplets constructed are the vector multiplet and hypermultiplet, the family of O(n)-multiplets, and the supergravity and gravitino multiplets. Along the way, we tackle various theoretical problems within the pure spinor superfield formalism. In particular, we give some general discussion about the relation of the projective nilpotence variety to multiplets and prove general results on short exact sequences and dualities of sheaves in the context of the pure spinor superfield formalism.

The Derived Pure Spinor Formalism as an Equivalence of Categories

“…The equivalence of categories allows for a program to classify families of multiplets starting from the algebraic geometry of the (derived) nilpotence variety. This direction is explored in [34], where (among other things) a description of all six-dimensional N = (1, 0) multiplets whose derived invariants form a single line bundle on the projective nilpotence variety is given.…”