Note on N = 0 string as N = 1 string

A similarity transformation, which brings a particular class of the N = 1 string to the N = 0 one, is explicitly constructed. It enables us to give a simple proof for the argument recently proposed by Berkovits and Vafa. The N = 1 BRST operator is turned into the direct sum of the corresponding N = 0 BRST operator and that for an additional topological sector. As a result, the physical spectrum of these N = 1 vacua is shown to be isomorphic to the tensor product of the N = 0 spectrum and the topological sector which consists of only the vacuum. This transformation manifestly keeps the operator algebra. E-mail address: [email protected] E-mail address: [email protected] The aim of this short note is to give a simple proof to the argument initiated by Berkovits and Vafa[1] that the N = 0 (bosonic) string can be regarded as a special case of the N = 1 (fermionic) string. The idea of ref. [1] is the following. For any given c = 26 matter with its energymomentum tensor Tm(z), we have c = 15 (ĉ = 10) system by coupling spin (3/2,−1/2) fermionic ghosts which we denote as (b1, c1). Then this system can be considered as a matter system of critical N = 1 fermionic string because of the existence of N = 1 super conformal algebra in the system: TN=1 = Tm − 3 2 b1∂c1 − 1 2 ∂b1 c1 + 1 2 ∂2(c1∂c1) , (1) GN=1 = b1 + c1(Tm + ∂c1 b1) + 5 2 ∂2c1 , (2) which satisfy TN=1(z)TN=1(w) ∼ 15/2 (z − w) + 2 (z − w) TN=1(w) + 1 z − w ∂TN=1(w) , (3) TN=1(z)GN=1(w) ∼ 3/2 (z − w) GN=1(w) + 1 z − w ∂GN=1(w) , (4) GN=1(z)GN=1(w) ∼ 10 (z − w) + 2 z − w TN=1(w) . (5) Tensoring spin (2,−1) reparametrization ghosts (b, c) and spin (3/2,−1/2) super ghosts (β, γ) to the above, we can construct whole Fock space of N = 1 fermionic string FN=1 = (Tm) ⊗