Nilpotence varieties
Abstract
We consider algebraic varieties canonically associated to any Lie superalgebra, and study them in detail for superPoincaré 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 superPoincaréinvariant 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 purespinor superfield formalism, from the perspective of twisting, and demonstrate that it can be applied to construct familiar multiplets in fourdimensional 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 ChevalleyEilenberg cohomology of supertranslations, and give two applications related to these ideas: a calculation of ChevalleyEilenberg cohomology for the sixdimensional $\mathcal{N}=(2,0)$ supertranslation algebra, and a BV complex matching the field content of type IIB supergravity from the coordinate ring of the corresponding nilpotence variety.
 Publication:

arXiv eprints
 Pub Date:
 July 2018
 arXiv:
 arXiv:1807.03766
 Bibcode:
 2018arXiv180703766E
 Keywords:

 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Algebraic Geometry;
 Mathematics  Representation Theory;
 81T60;
 17B08;
 17B81;
 14D21
 EPrint:
 66 pages, two figures, two tables