Six-dimensional supermultiplets from bundles on projective spaces

TL;DR

This paper classifies and constructs six-dimensional supermultiplets using bundles on projective spaces and the pure spinor formalism, revealing new multiplets and their geometric properties.

Contribution

It introduces a classification of supermultiplets based on derived invariants and constructs explicit examples from higher-rank vector bundles, advancing the understanding of supermultiplet geometry.

Findings

Classified supermultiplets with line bundle invariants over the nilpotence variety.

Constructed explicit multiplets from tangent, normal, and dual bundles.

Analyzed relations between projective nilpotence variety and supermultiplet structures.

Abstract

The projective variety of square-zero elements in the six-dimensional minimal supersymmetry algebra is isomorphic to $\mathbb{P}^1 \times \mathbb{P}^3$. 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 $\mathcal{O}(n)$-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.