Nilpotence varieties

TL;DR

This paper systematically studies algebraic varieties associated with Lie superalgebras, especially super-Poincaré algebras, revealing their role as moduli spaces for twists of supersymmetric theories and connecting them to pure-spinor formalism and cohomology applications.

Contribution

It classifies nilpotence varieties for super-Poincaré algebras, analyzes their stratification and symmetry actions, and links these structures to physical theories and cohomological frameworks.

Findings

Classification of all possible twists of super-Poincaré theories.

Identification of stratification of nilpotence varieties with twists.

Application of the formalism to construct supersymmetric multiplets and compute cohomology.

Abstract

We consider algebraic varieties canonically associated to any Lie superalgebra, and study them in detail for super-Poincar\'e algebras of physical interest. They are the locus of nilpotent elements in (the projectivized parity reversal of) the odd part of the algebra. Most of these varieties have appeared in various guises in previous literature, but we study them systematically here, from a new perspective: as the natural moduli spaces parameterizing twists of a super-Poincar\'e-invariant physical theory. We obtain a classification of all possible twists, as well as a systematic analysis of unbroken symmetry in twisted theories. The natural stratification of the varieties, the identification of strata with twists, and the action of Lorentz and $R$-symmetry on the varieties are emphasized. We also include a short and unconventional exposition of the pure-spinor superfield formalism, from the perspective of twisting, and demonstrate that it can be applied to construct familiar multiplets in four-dimensional minimally supersymmetric theories; in all dimensions and with any amount of supersymmetry, this technique produces BRST or BV complexes of supersymmetric theories from the Koszul complex of the cone point over the coordinate ring of the nilpotence variety, possibly tensored with a module over that coordinate ring. In addition, we remark on a natural emergence of nilpotence varieties in the Chevalley-Eilenberg cohomology of supertranslations, and give two applications related to these ideas: a calculation of Chevalley-Eilenberg cohomology for the six-dimensional $\mathcal{N}=(2,0)$ supertranslation algebra, and a BV complex matching the field content of type IIB supergravity from the coordinate ring of the corresponding nilpotence variety.