Twisting pure spinor superfields, with applications to supergravity

TL;DR

This paper explores twists of supergravity and supersymmetric theories using pure spinor superfields, revealing geometric structures of component fields and computing minimal twists of supergravity theories, confirming a conjecture in the field.

Contribution

It introduces a novel geometric framework for understanding twisted supermultiplets via equivariant Koszul homology and applies this to compute minimal twists of supergravity theories, confirming a conjecture.

Findings

Component fields correspond to holomorphic vector bundles from equivariant Koszul homology.

BRST/BV differentials are induced by super Lie algebra brackets.

Confirmed a conjecture relating IIB multiplet to BCOV theory.

Abstract

We study twists of supergravity theories and supersymmetric field theories, using a version of the pure spinor superfield formalism. Our results show that, just as the component fields of supersymmetric multiplets are the vector bundles associated to the equivariant Koszul homology of the variety of square-zero elements in the supersymmetry algebra, the component fields of the holomorphic twists of the corresponding multiplets are the holomorphic vector bundles associated to the equivariant Koszul homology of square-zero elements in the twisted supersymmetry algebra. The BRST or BV differentials of the free multiplet are induced by the brackets of the corresponding super Lie algebra in each case. We make this precise in a variety of examples; applications include rigorous computations of the minimal twists of eleven-dimensional and type IIB supergravity, in the free perturbative limit. The latter result proves a conjecture by Costello and Li, relating the IIB multiplet directly to a presymplectic BV version of minimal BCOV theory.