Manifest Spacetime Supersymmetry and the Superstring

The algebra of spacetime supersymmetry generators in the RNS formalism for the superstring closes only up to a picture-changing operation. After adding non-minimal variables and working in the"large"Hilbert space, the algebra closes without picture-changing and spacetime supersymmetry can be made manifest. The resulting non-minimal version of the RNS formalism is related by a field redefinition to the pure spinor formalism.


Introduction
Although the RNS formalism for the superstring has a beautiful geometric foundation coming from worldsheet super-reparameterization invariance, the lack of manifest spacetime supersymmetry complicates the computation of multiloop scattering amplitudes and the description of Ramond-Ramond backgrounds. As shown in [1], spacetime supersymmetry generators q α can be constructed in this formalism which satisfy the algebra {q α , q β } = γ m αβP m whereP m = dz e −φ ψ m is in the −1 picture. But since the picturechanging operation which relatesP m with the usual translation generator P m = dz ∂x m is only valid on onshell states, this algebra only closes onshell and spacetime supersymmetry is not manifest.
On the other hand, the pure spinor formalism for the superstring [2] has manifest spacetime supersymmetry and can describe Ramond-Ramond backgrounds, but lacks a geometric foundation where the pure spinor BRST operator comes from gauge-fixing a worldsheet symmetry. There is complete agreement between all scattering amplitudes which have been computed using the two formalisms, but equivalence has not yet been proven and multiloop amplitudes in the two formalisms have different types of subtleties.
In this paper, the RNS and pure spinor formalisms will be shown to be equivalent by relating them to a manifestly spacetime supersymmetric generalization of the RNS formalism. In addition to the usual RNS matter and ghost variables, this generalization will involve non-minimal variables that include both pure spinor variables and the fermionic θ α and conjugate momenta d α variables of the Green-Schwarz-Siegel formalism. Since the new formalism includes features of the pure spinor, RNS and Green-Schwarz-Siegel formalisms, it will be called the B-RNS-GSS formalism.
The first step to making spacetime supersymmetry manifest in the RNS formalism is to work in the "large" Hilbert space including the ghost zero modes (ξ 0 , η 0 ) and to define the extended BRST operator Q ′ = Q RNS − η 0 . As discussed in [3] and [4], picturechanging can be interpreted as a BRST-trivial gauge transformation using Q ′ in the large Hilbert space. The next step is to add non-minimal variables to the RNS formalism and construct an operator And the final step is to find a similarity transformation which maps q ′ α into the spacetime supersymmetry generators of the Green-Schwarz-Siegel formalism.
The resulting worldsheet action and BRST operator for this B-RNS-GSS formalism are manifestly spacetime supersymmetric, and BRST-invariant vertex operators are easily constructed in terms of d=10 superfields. Furthermore, a field redefinition is found which maps this worldsheet action, BRST operator and vertex operators into the worldsheet action, BRST operator and vertex operators of the pure spinor formalism. It is hoped that by studying this intermediate B-RNS-GSS formalism, the relation between the multiloop subtleties in the RNS and pure spinor formalisms will be better understood.
The left-moving contribution to the worldsheet action and stress tensor are where the RNS worldsheet action, stress tensor, and BRST operator are defined as 3) 5) and the right-moving variables will be ignored throughout this paper. Although only the open superstring will be discussed in this paper, all results can be easily generalized to the closed superstring by taking the "left-right product" of two open superstrings.
The natural BRST operator for this extended formalism is since the non-minimal term dz[Λ α p α + w α r α ] implies that states in the cohomology of Q nonmin are states in the cohomology of Q RNS which are independent of the spacetime In the RNS formalism [1], the spacetime supersymmetry generator in the − 1 2 picture is q α = dze − 1 2 φ Σ α where Σ α is the spin field of conformal weight 5 8 constructed from ψ m and the (β, γ) ghosts have been fermionized as β = ∂ξe −φ and γ = ηe φ . Since {q α , q β } = γ m αβ dz e −φ ψ m , the spacetime supersymmetry algebra only closes up to picture-changing.
As discussed in [3] and [4], picture-changing can be treated as a BRST-trivial operation if one extends the space of states to the "large" Hilbert space which includes the zero mode of ξ and defines the BRST operator as And it is easy to show that any state in the cohomology of Q ′ can be expressed as a state in the small Hilbert space in the By working in the large Hilbert space with Q ′ = Q nonmin − η 0 and shifting q α by a BRST-trivial quantity, one can now make spacetime supersymmetry manifest by defining the generator where ρ α is chosen such that {q α , q β } = dz ∂x m γ m αβ . Furthermore, after performing a similarity transformation q α → e −R q α e R , this generator can be mapped to the Green-Schwarz-Siegel supersymmetry generator [5] (2.10) The resulting BRST operator Q B−RNS−GSS = e −R (Q nonmin − η 0 )e R is manifestly spacetime supersymmetric and is are the usual GSS spacetime supersymmetric operators [5] for fermionic and bosonic momenta. Note that T of (2.2) can be written in the manifestly spacetime supersymmetric (2.13) To find the shift ρ α and similarity transformation R that map the supersymmetry generator and BRST operator into (2.10) and (2.11), first consider the shift and similarity One finds that up to terms proportional to θ α , e −R Q ′ eR is equal to Q B−RNS−GSS of (2.11) where (A α , A m , W α , F mn ) are the usual d = 10 super-Yang-Mills superfields satisfying and D α = ∂ ∂θ α + 1 2 (γ m θ) α ∂ m is the d = 10 supersymmetric derivative. Note that the similarity transformation of (2.15) mixes different pictures, but one can choose a representative of the BRST cohomology so that the massless super-Yang-Mills vertex operator V of (2.16) is in the zero picture. And by constructing massive superstring vertex operators from the OPE's of massless vertex operators, one can similarly find vertex operators at zero picture for all GSO-projected onshell superstring states. For states which are not GSO-projected, i.e. which have square-root cuts with e − 1 2 φ Σ α , the vertex operator will need to involve spin fields in Λ α and θ α so that the operator does not have square-root cuts with the term dz Λ α (d α − e − 1 2 φ Σ α ) in the BRST operator of (2.11).

Relation with Pure Spinor Formalism
In this section, the BRST operator of (2.11) in the B-RNS-GSS formalism will be mapped to the pure spinor BRST operator by a field redefinition and similarity transformation. The field redefinition will "dynamically twist" the RNS variables ψ m of spin 1 2 and (β, γ) of spin ( 3 2 , − 1 2 ) into the variables (Γ m , Γ m ) of spin (0, 1) and ( β, γ) of spin (2, −1) [7]. The twisting of the ψ m variables to (Γ m , Γ m ) variables shifts their central charge contribution from +5 to −10, and is compensated by the replacement of the (β, γ) with ( β, γ) variables which shifts their central charge contribution from +11 to +26. After performing a similarity transformation, the BRST operator of (2.11) in terms of these twisted variables is mapped to Q pure − η 0 where Q pure = dz(λ α d α + w α r α ) is the pure spinor BRST operator and η 0 is the zero mode coming from fermionizing β = ∂ ξe − φ and γ = ηe φ .
To dynamically twist the ten ψ m spin-half variables into five spin-zero and five spinone variables, it will be useful to construct a pure spinor λ α out of Λ α and λ α as So λ α and λ α satisfy λγ m λ = λγ m λ = 0, and their 11 complex components (in Wickrotated Euclidean space) parameterize the complex space SO(10) U(5) × C. Using the pure spinor variables (λ α , λ α ) to covariantly choose the direction of the twisting, one can now dynamically twist the ten spin-half ψ m variables to spin-zero Γ m variables and spin-one Γ m variables defined by so that After expressing ψ m in terms of Γ m and Γ m , GSO-projected states only depend on even powers of the γ ghost. So it will be useful to define which carries conformal weight −1, and define β of conformal weight +2 to be the conjugate momentum to γ. Fermionizing ( γ, β) as γ = ηe φ and β = ∂ ξe − φ and requiring that ( η, ξ, φ) have no poles with Γ m or Γ m , one finds that One can also express the unhatted ghost variables in terms of the twisted variables as In terms of these twisted variables, the BRST operator of (2.11) is where u m is defined by Λ α = λ α + 1 2(λλ) u m (γ m λ) α , , v m and Γ m are constrained to satisfy v m (γ m λ) α = Γ m (γ m λ) α = 0, and w α and w α are defined to have no poles with each other or with (Γ m , Γ m ). In terms of w α and w α , one finds that (3.10) The next step to relating (3.7) to the pure spinor BRST operator is to perform the One finds that Since w α of (3.13) commutes with the constraints v m (γ m λ) α = Γ m (γ m λ) α = 0 up to the gauge transformation δ w α = f m (λγ m ) α , one can easily verify that (3.12) also commutes with these constraints.