New BRST Charges in RNS Superstring Theory and deformed Pure Spinors

We show that new BRST charges in RNS superstring theory with nonstandard ghost numbers, constructed in our recent work, can be mapped to deformed pure spinor (PS) superstring theories, with the nilpotent pure spinor BRST charge $Q_{PS}=\oint{\lambda^\alpha}d_\alpha$ still retaining its form but with singular operator products between commuting spinor variables $\lambda^\alpha$. Despite the OPE singularities, the pure spinor condition $\lambda\gamma^m\lambda=0$ is still fulfilled in a weak sense, explained in the paper. The operator product singularities correspond to introducing interactions between the pure spinors. We conjecture that the leading singularity orders of the OPE between two interacting pure spinors is related to the ghost number of the corresponding BRST operator in RNS formalism. Namely, it is conjectured that the BRST operators of minimal superconformal ghost numbers $n=1,2,3...$ can be mapped to nilpotent BRST operators in the deformed pure spinor formalism with the OPE of two commuting spinors having a leading singularity order $\lambda(z)\lambda(w)\sim{O}(z-w)^{-2(n^2+6n+1)}$. The conjecture is checked explicitly for the first non-trivial case $n=1$.