Perspectives on the pure spinor superfield formalism

TL;DR

This paper offers a new algebraic and geometric perspective on the pure spinor superfield formalism, connecting it to super Lie algebra multiplets and nilpotence varieties, with broad examples across dimensions.

Contribution

It formalizes and generalizes the pure spinor superfield formalism using homological algebra and algebraic geometry, linking supermultiplets to nilpotence varieties.

Findings

Relates supermultiplets to equivariant graded modules over nilpotence varieties

Uses homotopy transfer theorem to connect to component-field formulations

Provides multiple examples across various dimensions

Abstract

In this note, we study, formalize, and generalize the pure spinor superfield formalism from a rather nontraditional perspective. To set the stage, we review the notion of a multiplet for a general super Lie algebra, working in the context of the BV and BRST formalisms. Building on this, we explain how the pure spinor superfield formalism can be viewed as constructing a supermultiplet out of the input datum of an equivariant graded module over the ring of functions on the nilpotence variety. We use the homotopy transfer theorem and other computational techniques from homological algebra to relate these multiplets to more standard component-field formulations. Physical properties of the resulting multiplets can then be understood in terms of algebrogeometric properties of the nilpotence variety. We illustrate our discussion with many examples in various dimensions.