Geometry of pure spinor formalism in superstring theory

Projective embedding of an isotropic Grassmannian OGr+(5,10) into projective space of spinor representation S can be characterized with a help of Γ-matrices by equations Γαβiλαλβ=0. A polynomial function of degree N with values in S defines a map to OGr+(5,10) if its coefficients satisfy a 2N+1 quadratic equations. Algebra generated by coefficients of such polynomials is a coordinate ring of the quantum isotropic Grassmannian. We show that this ring is based on a lattice; its defining relations satisfy straightened law. This enables us to compute the Poincaré series of the ring.