Pure Spinor Formalism for Osp(N|4) backgrounds

TL;DR

This work develops a covariant pure spinor formulation for strings on $AdS_4$-based backgrounds by deriving pure spinor constraints from the Maurer–Cartan structure of the $OSp(\mathcal{N}|4)$ supergroups and analyzing a family of cosets $G/H$ to balance bosonic and fermionic degrees of freedom. The authors ghostify the Maurer–Cartan forms, implement BRST symmetry, and identify viable backgrounds, with $OSp(6|4)/U(3)\times SO(1,3)$ yielding the critical $AdS_4\times\mathbb{P}^3$ case in which a consistent pure spinor sigma model can be constructed and quantized. They also study other cosets, showing that only a limited set lead to consistent backgrounds and DOF matching, while a fully gauged bosonic coset removes residual pure spinor degrees of freedom. The results provide a systematic framework for manifestly supersymmetric worldsheet descriptions on $AdS_4$ backgrounds and pave the way for RR-coupling analysis, amplitudes, and possible AdS/CFT interpretations involving Chern–Simons theories.

Abstract

We start from the Maurer-Cartan (MC) equations of the Osp(N|4) superalgebras satisfied by the left-invariant super-forms realized on supercoset manifolds of the corresponding supergroups and we derive some new pure spinor constraints. They are obtained by "ghostifying" the MC forms and extending the differential d to a BRST differential. From the superalgebras G =Osp(N|4) we single out different subalgebras H contained in G associated with the different cosets G/H: each choice of H leads to a different weakening of the pure spinor constraints. In each case, the number of parameter is counted and we show that in the cases of Osp(6|4)/U(3) x SO(1,3), Osp(4|4)/SO(3) x SO(1,3) and finally Osp(4|4) U(2)} x SO(1,3) the bosonic and fermionic degrees of freedom match in order to provide a c=0 superconformal field theory. We construct both the Green-Schwarz and the pure spinor sigma model for the case Osp(6|4)/U(3)x SO(1,3) corresponding to AdS_4 x P^3. The pure spinor sigma model can be consistently quantized.