Pure Spinors, Free Differential Algebras, and the Supermembrane

TL;DR

This paper develops a pure spinor formalism for the supermembrane in 11-dimensional supergravity, establishing a BRST-invariant, supersymmetric, and Lorentz-invariant action that connects free differential algebras to M-theory symmetries.

Contribution

It introduces a gauge-fixed supermembrane action in the pure spinor framework, linking FDA structures to BRST symmetry and superalgebra extensions in M-theory.

Findings

Constructed a manifestly BRST-invariant supermembrane action.

Clarified the role of FDA in supermembrane gauge fixing.

Detailed solutions to pure spinor constraints.

Abstract

The lagrangian formalism for the supermembrane in any 11d supergravity background is constructed in the pure spinor framework. Our gauge-fixed action is manifestly BRST, supersymmetric, and 3d Lorentz invariant. The relation between the Free Differential Algebras (FDA) underlying 11d supergravity and the BRST symmetry of the membrane action is exploited. The "gauge-fixing" has a natural interpretation as the variation of the Chevalley cohomology class needed for the extension of 11d super-Poincare' superalgebra to M-theory FDA. We study the solution of the pure spinor constraints in full detail.