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

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 2 . 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 = {Qbrst, LV } where L = −4c∂ξξe , we show that mapping θ+ λ+ to L leads to identification of the pure spinor variable λ in terms of RNS variables without any additional non-minimal fields. We construct the RNS operator satisfying all the properties of λ and show that the pure spinor BRST operator ∮ λdα is mapped (up to similarity transformation) to the BRST operator of RNS theory under such a construction. . October 2008 † [email protected],[email protected]; the address after January 5,2009: National Institute for Theoretical Physics, Department of Physics and Centre for Theoretical Physics, University of the Witwatersrand, Wits 2050, South Africa Introduction Pure spinor formalism for superstrings has been proposed by Berkovits several years ago [1] as an alternative method of covariant quantization of Green-Schwarz superstring theory [2]. It involves the remarkably simple worldsheet action: S = ∫ dz{ 1 2 ∂Xm∂̄X m + pα∂̄θ α + p̄α∂θ̄ α + λα∂̄w α + λ̄α∂w̄} (1) where pα is conjugate to θα [3] and the commuting spinors λ α and w are the bosonic ghosts which, roughly speaking, are related to the fermionic gauge κ-symmetry in GS superstring theory. The action (1) is related to the standard GS action by substituting the constraint dα = pα − 1 2 (∂X + 1 4 θγ∂θ)(γθ)α = 0 (2) and the corresponding BRST operator Qbrst = ∮ dz 2iπ λdα(z) (3) is nilpotent provided that λ satisfies the pure spinor condition: