Perspectives on the pure spinor superfield formalism

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