Pure spinors and D=11 supergravity
Abstract
In this Thesis we study first- and second-quantized approaches describing D=11 supergravity using pure spinor variables. We introduce the so-called D=11 pure spinor superparticle through BRST cohomology arguments starting from the semi-light-cone gauge D=11 Brink-Schwarz-like superparticle. After performing a light-cone gauge analysis of the pure spinor BRST cohomology at ghost number three, we find the linearized equations of motion of D=11 supergravity in D=9 superspace. In addition, we construct a BRST-closed, ghost number one vertex operator made out of worldline fields and D=11 supergravity superfields, and we run into an inconsistency when constructing a ghost number zero vertex operator satisfying a standard descent equation. We then introduce the non-minimal version of the D=11 pure spinor superparticle, in which a composite b-ghost can be constructed satisfying \Q,b\ = P2. However, its complicated expression makes it difficult to check its nilpotency. We show that introducing an SO(1,10) fermionic vector a simplifies the form of the b-ghost considerably, which allows us to verify that \Q,b\ = P2 and \b,b\= BRST-exact. Using this b-ghost we propose an alternative ghost number zero vertex operator satisfying a standard descent equation. However, its expression will depend on non-minimal pure spinor variables in a very complicated fashion. After discussing this first-quantized approach for D=11 supergravity, we move on to discussing the pure spinor master actions introduced by Cederwall for studying maximally supersymmetric gauge theories. We show that these actions indeed describe D=10 super-Yang-Mills, D=10 super-Born-Infeld and D=11 supergravity by extracting the equations of motion in ordinary superspace for each one of these theories.