(Non)triviality of Pure Spinors and Exact Pure Spinor - RNS Map

Abstract

All the BRST-invariant operators in pure spinor formalism in $d=10$ can be represented as BRST commutators, such as $V=\lbrace{Q_{brst}},{{\theta_{+}}\over{\lambda_{+}}}V\rbrace$ where $\lambda_{+}$ is the U(5) component of the pure spinor transforming as $1_{5\over2}$. Therefore, in order to secure non-triviality of BRST cohomology in pure spinor string theory, one has to introduce "small Hilbert space" and "small operator algebra" for pure spinors, analogous to those existing in RNS formalism. As any invariant vertex operator in RNS string theory can also represented as a commutator $V=\lbrace{Q_{brst}},LV\rbrace$ where $L=-4c\partial\xi{\xi}e^{-2\phi}$, we show that mapping ${{\theta_{+}}\over{\lambda_{+}}}$ to L leads to identification of the pure spinor variable $\lambda^{\alpha}$ in terms of RNS variables without any additional non-minimal fields. We construct the RNS operator satisfying all the properties of $\lambda^\alpha$ and show that the pure spinor BRST operator $\oint{\lambda^\alpha{d_\alpha}}$ is mapped (up to similarity transformation) to the BRST operator of RNS theory under such a construction. We also observe that BRST cohomology of pure spinor superstring theory contains vertex operators non-trivially coupled to the pure spinor ghost variable. The physical interpretation of these operators remains unclear.