@MISC{Polyakov08(non)trivialityof, author = {Dimitri Polyakov}, title = {(Non)triviality of Pure Spinors and Exact Pure Spinor- RNS Map}, year = {2008} }

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Abstract

All the BRST-invariant operators in pure spinor formalism in d = 10 can be represented as BRST commutators, such as V = {Qbrst, θ+ λ+ V} where λ+ is the U(5) component of the pure spinor transforming as 1 5. Therefore, in order to secure non-triviality of BRST 2 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 = {Qbrst, LV} where L = −4c∂ξξe −2φ, we show that mapping θ+ λ+ to L leads to identification of the pure spinor variable λ α in terms of RNS variables without any additional non-minimal fields. We construct the RNS operator satisfying all the properties of λ α and show that the pure spinor BRST operator ∮ λ α dα is mapped (up to similarity transformation) to the BRST operator of RNS theory under such a construction.