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Covariant Quantization of Superstrings Without Pure Spinor Constraints

We construct a covariant quantum superstring, extending Berkovits' approach by introducing new ghosts to relax the pure spinor constraints. The central charge of the underlying Kac-Moody algebra, which would lead to an anomaly in the BRST charge, is treated as a new generator with a new b-c system. We construct a nilpotent BRST current, an anomalous ghost current and an anomaly-free energy-momentum tensor. For open superstrings, we find the correct massless spectrum. In addition, we construct a Lorentz invariant B-field to be used for the computation of the integrated vertex operators and amplitudes.


Introduction and Summary
Recently, a new formulation of superstrings was developed which is explicitly super-Poincaré invariant in 9 + 1 dimensions [1] [2] [3] [4]. It is based on a free conformal field theory on the world-sheet and a nilpotent BRST charge Q which defines the physical vertices as representatives of its cohomology. In addition to the conventional variables x m and θ α of the Green-Schwarz formalism, a conjugate momentum p α for θ α and a set of commuting ghost fields λ α are introduced. The latter are complex Weyl spinors satisfying the pure spinor conditions λ α γ m αβ λ β = 0 [1] [5]. This equation can be solved by decomposing λ with respect to a non-compact U (5) subgroup of SO(9, 1) into a singlet 1, a vector 5, and a tensor 10. The vector can be expressed in terms of the singlet and tensor, hence there are 11 independent complex variables in λ α .
The pure spinors λ α are needed to cancel the central charge of the conformal algebra (+10 from the bosonic coordinates x m , −32 from the spinor variables p α and θ α , and +22 from the pure spinors λ α ), to obtain the correct double poles in the Lorentz algebra and, last but not least, to render the operator Q nilpotent.
Of course, the problem of the covariant quantization of superstrings is one of the fundamental problems in string theory. The subject has a long history. We give in [6] some of the early papers on the subject which we used, but the list is far from complete.
As shown by Berkovits in [3], the cohomology of the BRST charge contains exactly the physical spectrum of the superstring and, in particular for the massless states, it provides the covariant equations of motion, those of super Yang-Mills theory in 10 dimensions for the open superstring or those of N = 2 supergravity for closed superstrings. For massive states he could not use the formalism based on the U (5)-like subgroup; it was possible to use an SO (8) subgroup, but this introduced an infinite set of ghost-for-ghosts although the cohomology did not depend on these fields. Although Berkovits's approach provides a way to by-pass many of the difficulties of the Green-Schwarz formalism for a super-Poincaré covariant description of superstrings, an explicit parametrization of pure spinors λ α is needed at several points in his construction. Using such an explicit parametrization for pure spinors, he was able to define Lorentz currents which satisfy the covariant Lorentz algebra [7]. However, the solution of the pure spinor constraints breaks the explicit covariance and some expressions cannot be written in a covariant way. For example, the action is not covariant and also intermediate steps in the computation of amplitudes are clearly affected. In addition, a conjugate momentum for the pure spinors can only be constructed in an explicit parametrization, resulting in OPE's which are not manifestly covariant. The tree level amplitudes are not manifestly super-Poincaré invariant and massive vertices can only be constructed in terms of a specific non-covariant parametrization of the pure spinors.
The cohomology of [1] [2] [3] [4] is a constrained cohomology. We therefore decided to try to relax the pure spinor conditions and construct a new BRST operator such that its unconstrained cohomology coincides with the constrained cohomology of Berkovits's approach. The extension of the BRST symmetry to unconstrained λ α led us to enlarge the field space by adding more ghost fields: an anticommuting vector ξ m , a commuting spinor χ α , an anticommuting vector 1-form ω m z , and their corresponding antighosts, and further an anticommuting b − c z system with conformal weights 0 and 1, respectively.
(1.1) Our final BRST current is given by 2) and the energy-momentum tensor is ∂ z x m ∂ z x m − p zα ∂ z θ α we are dealing with a free conformal field theory. The ghost current we use is given by J gh z = − (β mz ξ m + κ α z χ α + β zα λ α + b c z + η m ω zm ) . (1.4) There is also a composite field B zz which will be discussed in the text. Our main results are: (1) we keep manifest SO(9, 1) invariance at every step. No pure spinor constraints or U (5) decompositions appear. As a consequence we can take λ α to be real which solves one of the puzzles of [1] where λ α had to be complex in order that the pure spinor constraint have a solution, but where the real θ α transforms into complex λ α .
(2) We construct a BRST operator with the properties one would like it to have: its current is nilpotent and a primary field. Our BRST charge is based on a Kac-Moody algebra for spacetime symmetries, and it describes the geometry of a D = (9, 1) superspace. In the RNS approach the BRST charge describes the dynamics since it starts with c z T zz + . . ..
We also construct a composite operator B zz which satisfies the fundamental relation T zz = [Q, B zz ] where T zz is given in (1.3). We believe that B zz plays the same role as the zero mode b 0 in the RNS approach. The integrated vertices dzV (0) z are constructed from the unintegrated vertices U (1) (4) We require that the unintegrated vertex operators should satisfy [Q, U (1) ] = 0. This selects all possible deformations of the Kac-Moody algebra, in other words, the BRST charge, with Kac-Moody generators replaced by Kac-Moody generators plus connections, is still nilpotent. To be physical observables these deformations must be primary spin-1 fields, so V (0) z should have conformal spin 1. This second constraint brings in the dynamics. For the massless sector of the open superstring, the first constraint, namely [Q, U (1) ] = 0, implies that the deformations are described by a super gauge-connection A α , A m and field strength A α (usually denoted by W α ) which satisfies the proper Bianchi identities. The second constraint implies the equations of motion for super Yang-Mills theory (at the linearized level). The conditions F αβ = F αm = 0 for the superspace curvature of the gauge connection follow as a consequence of the equations of motion.
The paper is organized in the following way: in section 2, we discuss the construction of the BRST charge based on an extension of Berkovits's approach and promote the central charge of the Kac-Moody algebra to an operator. In section 3, we prove the BRST invariance of the action and derive the boundary conditions for the open superstring needed to maintain BRST invariance and supersymmetry at the boundaries. Section 4 is devoted to the construction of the energy-momentum tensor, the field B zz , the ghost current and to compute their OPE's. In section 5, we discuss briefly the relation with the RNS approach and, finally, in section 6 we study the massless sector of open string. Section 7 contains some comments and speculations.

The BRST charge
On a flat worldsheet, the free left-and right-moving contributions to the classical covariant Green-Schwarz superstring action [8] can be written as . Furthermore, p α (or its anti-holomorphic partnerp α ) is related to x m and θ α by the constraint d α = 0 with [9] Setting d α equal to zero, (2.1) leads to an interacting superstring whose left-movers do not interact with its right-movers, and which therefore is not equal to the Green-Schwarz superstring. The variables x m and θ α are worldsheet scalar fields. On the other hand, the conjugate momenta p α , d α and Π m = ∂x m + 1 2 θ α γ m αβ ∂θ β carry a world-sheet vector index, i.e. p z,α , d z,α , Π z,m . The index z will be omitted in the following when there is no ambiguity. The symbols γ m αβ and γ m αβ are real 16 × 16 symmetric matrices which are the off-diagonal elements of the 32 × 32 Dirac-matrices 5 and which satisfy γ m αβ γ n βγ + γ n αβ γ m βγ = 2η mn δ γ α and γ m (αβ γ m γ)δ = 0. The latter relation makes Fierz rearrangements very easy.
fields are given by (2.5) The susy commutator vanishes on p α in agreement with the vanishing of the anticommutator of the θ-dependent terms in the r.h.s. of (2.4) with p α . We require the BRST charge to be susy invariant and therefore it is constructed on the basis (Π m z , ∂ z θ α , d zα ). All the ghosts and antighosts will be susy inert.
The variables (x m , θ α ) are the coordinates of N = 1 superspace in D = (9, 1) dimensions and (Π m z , ∂ z θ α , d zα ) form a basis for susy-invariant super 1-forms. At the classical level, a general super 1-form can be written as for the holomorphic sector, or for the closed string sector. These expressions will be useful for the cohomology of the string. For the open string the functions A m , A α and A α are arbitrary superfields of the holomorphic supercoordinates (x m L (z), θ α (z)) and for Type II B strings the G mn , . . . , G αα are superfields on the superspace defined by x m (z,z), θ α (z), θα(z). The fields x m and θ α are on-shell, so that we can use conformal field theory techniques. For Type II A strings, we should useθ α instead ofθ α .
Using the standard [12] free OPE's of the worldsheet fields 7 one obtains the following BRST transformation rules From these transformation rules the nilpotency of Q B when (2.8) holds is obvious. For later use, we also need the variation of Π m Since the conjugate momenta p α are independent of the variables θ α , we must also determine their BRST variations Since p α is related to d α (see eq. (2.2)), it is natural to use the latter as the fundamental variable, since its variation has a simpler form (2.14) 6 The cohomology of the operator Q should be computed in the infinite dimensional space of x m , ∂x m , ∂∂x m , . . . , θ α , ∂θ α , ∂∂θ α , . . . , λ α , ∂λ α , ∂∂λ α , . . ., but one can restrict one's attention to particular subsets by using the usual filtration techniques (for example, the linear subspace of zero forms (functions) in (6.4)). 7 As usual, we combine left-and right-moving parts of the fields of the open string into a single field on the double interval. This eliminates cross-terms such as ln(z −w) from the propagators.
It is also natural to require compatibility of the supersymmetry transformations (2.4) with the symmetry generated by Q B . Defining [Q ǫ , λ α ] = 0, we have Q ǫ , Q B = 0. This concludes our review of those results of Berkovits's program which we need. The BRST variation of x m given in (2.11) is not nilpotent if λ does not satisfy any constraints. We therefore introduce a new anticommuting worldsheet-scalar spacetimevector ξ m , whose properties will be determined as we proceed, and define Similarly, since (2.14) is not nilpotent, we introduce a new commuting worldsheet-scalar spacetime-spinor χ α , and define The last term in (2.16) is induced by the extra term in (2.15). Since the BRST variation of the sum of the last two terms is a total derivative we introduced ∂ z χ α instead of a 1-form χ zα . Requiring nilpotency of Q ′ B on x m and d zα yields then the transformation laws for ξ m and χ α The basic principle of our approach is to make the θ variation as simple as possible, hence we still define {Q ′ B , θ α } = λ α , and nilpotency on θ α leads then to [Q ′ B , λ α ] = 0. Nilpotency on ξ m is then obvious, but nilpotency on χ α holds also as one may check by a Fierz transformation. The ghost numbers are assigned such that Q ′ B has ghost number +1. Then x m , θ α and d zα (or p zα ) have ghost number zero, while ξ m , λ α and χ α have ghost number +1. Finally, the transformation rule for Π m becomes These modified BRST transformation rules can be incorporated in a modified BRST current. Introducing the antighosts β m z , κ α z , and β zα for ξ m , χ α , and λ α respectively, the BRST current takes the following form Using the free OPE's, the OPE of two BRST currents can be evaluated The double poles cancel due to the statistics of the ghost fields. The nonvanishing of the OPE for j B′ z (z)j B′ w (w) implies that the BRST charge is not nilpotent. Therefore, it does not make sense to study the cohomology of this Q. Another related problem is that the central charge of the stress tensor does not vanish. Both problems will be solved by a new ingredient.
The nonclosure of Q ′ B is not very surprising. The BRST charge has the form where W zM (z) are local generators of a (super) Kac-Moody algebra, f M NR are the structure constants of the underlying (super) Lie algebra, and C M (z) and B M z (z) are the ghosts and the antighosts, respectively, associated to the generators W zM (z). However, our algebra [9] has a central charge: The Jacobi identities follow from the Fierz identities for the Dirac matrices.
To construct a nilpotent BRST charge, we promote the central charge to an operator and we introduce a new anticommuting b − c z system associated to this operator. The field b is a 0-form with ghost number −1 and conformal spin 0, and c z is a 1-form with ghost number 1 and conformal spin 1. They have the OPE Following [13], but with the full field b instead of only the zero mode b 0 , the correct BRST current j B z (z) is then given by 8 ( 2.26) The last term is added to make this current nilpotent It is easy to check that the contributions to (2.27) from the central charge in (2.22) are cancelled by the contributions to (2.27) due to (2.25).
Because the new BRST charge Q = dzj B z is nilpotent its cohomology is well-defined. We postpone the derivation of the transformation rules of the antighosts until we discuss the BRST invariance of the quantum action.

Action and Boundary Conditions
The dynamics of the model is fixed by postulating free OPE's for all fields. It is natural to ask whether the free action is BRST invariant. This represents a consistency check between the structure of the BRST charge and the free OPE's used to construct it. For the closed string we impose as usual periodic boundary conditions. For the open string we shall in this way derive the boundary conditions on the fields. The proof that the action is invariant is equivalent to the statement that the two-point Ward identities are satisfied at the tree graph level.
Consider the tree level action Another nilpotent charge depending also on b and c z is given by We perform the BRST variation 9 of the left-movers To obtain this result we partially integrated the BRST variation of the action to bring thē ∂ derivatives on ξ m , λ α and χ α , because then the variations can be canceled by suitable BRST variations of β m , β α and κ α .
The boundary terms which we produce in this way are To these terms we should add the corresponding terms from the antiholomorphic sector. Next, we replace the total derivatives ∂ and∂ by 1 2 ∂ σ and find then the following conditions at σ = 0, π for the BRST symmetry of the open string

(3.4)
A solution of all these boundary conditions, assuming that θ α =θ a , is The BRST variation of the classical action in (3.2) can be written in a simpler form if we use the conjugate momenta Π m and d α , Since it is proportional to the equations of motions of the ghost fields, it can be compensated by adding a ghost-antighost action and defining the variations of the antighost fields β zm , β zα , κ α z and b in a suitable way. These transformation rules contain various terms bilinear in the ghosts and antighosts, needed to cancel the variations of the ghosts in (3.7). An easy way to obtain the transformation rules of the antighosts is to use the BRST charge as we now discuss.
From the BRST charge we derive the following transformation rules for the antighost fields For the new ghost c z , we find Using again integration by parts and Fierz identities, it is easy to show that the action is BRST invariant Q, S L + S R + S L,ghost + S R,ghost = 0 .
In the case of the open string, taking (3.5) into account, the last manipulations lead to the following boundary terms 3.10) whose solution is given by b =b at σ = 0, π.
To derive the equations of motion of the antighost fields β µ z , β α and κ α z , one has to integrate the action by parts. This implies new boundary conditions on the antighost fields [14] which take the form β m σ =β m σ , β σα =β σα , κ α σ =κ α σ , c σ =ĉ σ , at σ = 0, π . (3.11) Notice that since all ghost fields are supersymmetrically invariant, the new terms in the action do not spoil the invariance of the theory under supersymmetry transformations. However, for the open string we must make sure that the sum of the boundary terms, which are produced by susy variations, cancels [14]. For susy we find the following boundary terms 12) and further terms with three θ's. Replacing the total derivative ∂ and∂ by 1 2 ∂ σ one finds the combination Since ∂ τ x m does not satisfy any conditions, we find as condition for susy of the open string (ǫγ m θ) = (ǫγ mθ ) at σ = 0, π . (3.14) Similarly, the terms with three θ's yield The solution of these conditions is ǫ =ǫ.

The Energy-Momentum Tensor, the field B zz and the Ghost Current
The antighost b zz for world-sheet diffeomorphisms of the bosonic string and the RNS version of the superstring plays an important role in the construction of higher-loop amplitudes and in the parametrization of moduli of Riemann surfaces. We believe, therefore, that also for the superstring a composite field B zz , constructed from the fundamental fields of the world-sheet theory and playing the same role as the fundamental b-field, must exist. In addition, in order to maintain an explicit covariant formulation, this composite field B zz should be invariant under super-Poincaré transformations of the target space.
Since the theory is a free conformal field theory, it is easy to write down the proper energy-momentum tensor. It follows from the action and is given by the formula This current T zz is BRST invariant, as can be directly verified by using the BRST transformation rules of the fields (there are no double contractions between Q and two fields in (4.1)). The fact that T zz is BRST invariant is encouraging because the composite field B zz should satisfy the relation T zz = {Q, B zz }. The first two terms in T zz can be rewritten as − 1 2 ∂ z x m ∂ z x m − p zα ∂ z θ α , as follows from the definition of Π m and d zα . In the case of open strings, T zz should share boundary conditions with the antiholomorphic partnerTzz. Due to the boundary conditions (3.5), (3.10) and (3.11), T andT can be combined into a single expression which is defined in the whole complex z-plane.
One can compute the central charge in the OPE of T zz with itself, using the free OPE's The value of the central charge is obtained by summing all the contributions: Here (n×m) A,B denotes the contribution from an A, B system, with n the number of components of the fields, and m the central charge for a single field. Relative signs are given by the statistics of the fields. The resulting central charge does not vanish and, therefore, the conformal symmetry is lost. However, we can compensate the non-vanishing central charge by an additional anticommuting spacetime-vector spin 0 -spin 1 system ω m z , η m . The final energy momentum tensor is then given by Due to the contribution from the ω m z , η m system the central charge now vanishes. In order not to spoil the rest of the construction, we assume that these new fields are inert under the BRST symmetry and supersymmetry. In the following, we will argue that these new fields are needed for maintaining complete covariance, and in the next section we show how they appear when one relaxes the pure spinor constraint.
Another important operator in the construction of the superstring amplitudes is the ghost current. In the present formalism, we have the following expression which satisfies Using the free OPE's one can compute the anomaly in the ghost current J z and obtains The anomaly is obtained summing all the contributions: (+10) ξ,β + (−16) λ,β +(−16) χ,κ + (+1) b,c + (+10) ηω = −11.
Following the idea that the theory should be described by a topological field theory in two dimensions, see the appendix, we only need the BRST charge, the energy-momentum tensor, a U (1) charge and, finally, an antighost field B zz to construct amplitudes (for a pedagogical explanation see [15]). The last one is a spin-2 worldsheet-tensor (spacetimescalar) with ghost number −1. It should satisfy the condition that it has no singular OPE with itself [16] and T zz = {Q, B zz } . In addition, to obtain a complete twisted N = 2 superconformal algebra, the OPE with the BRST current should be given by where T is given in (4.1) and in (4.2), the ghost current J gh (w) is given in (4.4), and 2 c/3 = −22.
Notice that in our framework B zz is a composite field, in contrast to the RNS formulation. The requirement that B zz has no singular terms with itself, and the requirement that T zz = {Q, B zz }, put severe restrictions on the possible solutions for B zz .
Since the b field is a Grassmann variable, we can decompose the composite operator where the upper index denotes the ghost charge of B zz . The condition T zz = {Q, B zz } leads to a solution This proposal for B zz is, however, not nilpotent. The requirement that B zz satisfies T zz = {Q, B zz } is not enough to fix B zz completely. In fact, one can always add a trivial part B zz → B zz + Q, X zz where X zz is a local polynomial with ghost number −2. Using this freedom, we construct other expressions that enjoy different algebraic properties. Finally, we show that one particular realization of B zz corresponds to the usual BRST charge for the bosonic string (there are no worldsheet fermions in our theory).
For instance, another expression for B zz is given by where now (4.11) This expression again satisfies T zz = {Q, B zz }. It is easy to check that the OPE with the BRST current j B z (z) produces double poles which are field dependent. The bilinear part of these field-dependent double poles coincides with the ghost current (4.4). However, the other singularities in this OPE shows that the B zz in (4.10) and in (4.11) has neither the OPE in (4.7) with j B z (z) nor is nilpotent. A third proposal for B zz has much better properties than the previous two proposals. It differs again from the previous ones by trivial BRST terms and is given by (4.12) It satisfies the OPE in (4.7) with c = 0. Moreover it is a primary field for α = 35/2 However, it is still not nilpotent since it has a singular OPE with itself. Now observe that the terms proportional to ∂ z b can be written as the BRST variation of b J z (z) using (4.5). Furthermore, ∂ z b is BRST invariant. Therefore, we can eliminate these terms from B zz (z) to arrive to a remarkably simple expression. Consider the following proposal for B zz B zz (z) = bT zz (z) + α∂ 2 z b . (4.13) This new composite operator satisfies T zz (z) = {Q, B zz (z)}, since {Q, b} = 1. Computing the OPE's of B zz with the ghost charge and with the energy-momentum tensor, one can check that it is a ghost number −1, anticommuting primary field with spin 2 if α = − 1 2 . By extracting the term proportional to ∂ z bc z in T zz , we end up with the expression (4.14) whereT zz (z) is given by (4.1) without the last term. The new field B zz (z) satisfies where the r.h.s. can be interpreted as an anomaly (just like the nonclosure term ∂c z ∂ 2 c z /(z − w) in the product of two BRST currents in RNS approach is an anomaly). The operator B zz (z)dz has the same structure (with c → b) as the BRST charge of a bosonic string whose matter part is represented not only by the complete set of superstring matter fields x m , d zα and θ α , but also by the ghost fields ξ m , λ a and χ α , ω m z . To complete the correspondence with the usual bosonic BRST charge, one has to view the fields b, c z as the diffeomorphism ghosts c z , b zz of the usual formulation. This follows from twisting, as we now explain. The b, c z system which we introduced to take care of the central charge in the Kac-Moody algebra is related to the diffeomorphism ghosts b zz , c z of the RNS by twisting twice: once by T bc = ∂ z bc z → T ′ = T − 1 2 ∂ z j z where j z = −bc z which gives c z → ψ 3 2 and b → ψ − 1 2 , and once more by Applying the twisting b → c z and c z → b zz to the field B zz in (4.14), one obtains the usual BRST charge of the bosonic string, except that the central charge ofT zz is +2 instead of +26 and the coefficient of Conversely, our BRST charge is unitarily equivalent to dz c z which is mapped, by twisting, into dz b zz . Hence, our B zz dz and Q are mapped under twisting into Q RNS and dz b zz , respectively. On the other hand, in our approach, the charge B zz dz should contribute to the measure for higher-loop calculations [17] in a similar way as dz b zz is used in the RNS approach.
As is well known, in the RNS formulation (or in the bosonic string), the vertex operators for the open string are defined as the cohomological classes of the BRST operator. They can be separated into unintegrated U (1) (with ghost number +1) and integrated ones V ]. Indeed, there is no nontrivial cohomology in this sector because Q can be mapped into dz c z by a similarity transformation, see the section on comments, and the solution of dz c z |ψ >= 0 is |ψ >= c z |φ >. This means that given an unintegrated vertex U (1) , one is able to construct the integrated one, namely V (0) z , by acting with δ z , up to a BRST trivial part, denoted here by N (−1) z . By simple manipulations, it is easy to verify that δ z = dz B zz . Hence (4.18) and we can directly check that [Q, dzV z ] = 0 10 . This confirms that the field B zz constructed out the fundamental fields of the superstring plays the same role as the RNS b-fields, and therefore it can be used to fix the moduli of the Riemann surfaces.

Relation with the RNS superstring
In the previous section we suggested an indentification of the ghost b, c z with the ghosts c z , b zz of the RNS approach. In the present section, we will provide further relations between the RNS superstring and the present formulation. Some aspects of this problem have been already discussed in [17] and [18]. In the following, we decompose SO(9, 1) spinors and vectors with respect to the U (5)-like subgroup as follows where θ + and p + are opposite-chirality U (5) singlets, θ a and x a are vectors in the 5, and p a and x a are vectors in the 5 * . The components θ [ab] and p [ab] are tensors in the 10 and 10 * , respectively.
Step 0: We start from the RNS fields (x m , ψ m , b, c, β, γ), where (b, c) are the conventional diffeomorphism ghosts and (β, γ) are their superpartners. In total, we have 12 bosonic variables and 12 fermionic ones. As shown in [18], these fields can be mapped into a subset of the Green-Schwarz variables: (x m , θ + , p + , θ a , p a , β + , λ + ). The opposite chirality fields p + , p a , describe the six momenta conjugate to θ + , θ α . The two chiral bosons β + , λ + take into account the contribution of the (β, γ) system, while p + , θ + take into account (b zz , c z ). The worldsheet RNS fermions ψ m are decomposed into ψ a , ψ a and are mapped into the spacetime fermions p a , θ a . We should point out that this step requires intermediate bosonization and fermionization of (b, c) and of (β, γ) (see also [15] for the D=6 case and [19] [20] for Calabi-Yau and K3 compactifications), with b + = e t and λ + = e s .
Step 1: In order to form a complete Green-Schwarz fermion θ α , we have to add the tensor components θ [ab] and their conjugated momenta p [ab] . This can be done in a "topological" way, by adding a BRST quartet consisting of the tensors θ [ab] and p [ab] , and containing a further set of tensor fields (v [ab] , u [ab] ) with opposite statistic. By topological we mean that the BRST charge associated to them is given by Together with the chiral boson s, the spinor λ α = (e s , u a , u [ab] ) forms a pure spinor (see (2.8) [17]. The conjugate momenta β α = (e t , 0, v [ab] ) are identified with the eleven independent components of the pure spinor λ α . Counting the act with Q on b(w) and V z (z). Double contractions from Q to b(w) and V z (z) vanish. One which agrees with our (4.18).
degrees of freedom, the number of bosons minus fermions cancels again as in RNS superstring. The present setting coincides with the pure spinor formulation [1], the BRST charge is nilpotent and the Lorentz algebra can be computed in terms of covariant combinations of pure spinors.
Step 2: In order to arrive at a completely covariant formulation, we let all the components of the spinor λ α become independent. Namely we want to remove the pure spinor constraint by introducing a new independent field u a and its conjugate momentum β a in order to reconstruct a complete spinor. Again, we introduce a BRST trivial quartet consisting of the spinor parts u a and v a , and half of a spacetime vector ξ a and its conjugate momentum β a . Notice that the statistics of the new fields should be opposite to those of β α and λ α , and, therefore, ξ a and β a are anticommuting vectors.
We can also understand the necessity of the new degrees of freedom by observing that, relaxing the pure spinor constraint λγ a λ = 0 is equivalent to set Then, finally substituting u a into the 5 * part of equation (5.3), one gets {Q, ξ a } = e −s {Q, ξ b }u [ba] which implies that the vector ξ m is a constrained vector. The new fields u a , v a and ξ a , β a form again a BRST quartet.
Step 3: To remove the vectorial constraint, we can further enlarge the field space by introducing another quartet formed by the missing components of the vector (ξ a , β a ) (removing the constraint) and introducing a new vector χ a with its conjugate momentum κ a . The latter can be viewed as the 5-part of χ α and of its conjugate momentum κ α . As above, the introduction of these new fields does not affect the physics of the theory.
Step 4: All the constrains are now removed, however the spinor χ α and its conjugate momentum are treated non-covariantly. A completely covariant approach requires 11 more anticommuting fields. One of these new fields, namely the scalar b − c z system, already entered the formalism by requiring the nilpotency of the BRST charge following [13]. The further 10 fields ω m z and η m are necessary for the vanishing of the central charge. Together with the 1-and 10-part of χ α , they saturate the number of degrees of freedom to make the formalism covariant, preserving unitarity, the conformal invariance and the cohomology of the BRST operator.

Massless states for open strings
The cohomology of the operator Q B in [3] -denoted by H n (Q B ), where n is the ghost number -is a constrained cohomology with the condition (2.8) for pure spinors λ α . Berkovits argued that H 1 (Q B ) must be linear in the field λ α , because of the nonlinearity of the action of the Lorentz generators in the fields λ ab . It yields the spectrum of the superstring for SYM [21] (or SUGRA for closed strings), but for massive modes, an explicit parametrization of λ α seems to be needed.
In our approach based on the Kac-Moody algebra (2.22), we must find the appropriate constraints on the kinematics and the dynamics of physical states. We need therefore both a kinematical constraint which relates the curvatures to the connections (supermultiplets of N = 1 superspace D = (9,1) in the case of open strings [21]) and a dynamical constraint which imposes the correct equations of motion on the connections (for example the super Yang-Mills equations of motion in D = (9, 1) [21]).
In the conventional RNS formalism [8] [22], the dynamical properties of the physical states are encoded in the BRST symmetry which implements, at the quantum level, the super-Virasoro algebra of the spinning string, while the supersymmetry multiplets are selected by means of the GSO projection on the BRST cohomology. The BRST cohomology describes the possible deformations of the super-Virasoro algebra in terms of unintegrated vertex operators U (1) with ghost number +1 or, equivalently, in terms of integrated vertex operators dzV (0) z . They are related by the equation where V (0) z has zero ghost number. This equation implies that dzV (0) z is BRST invariant. In our case the BRST algebra encodes the Kac-Moody algebra and the corresponding BRST cohomology should contain all the possible deformations of the underlying algebra. In the following, we will show that indeed the ghost number one BRST cohomology of Q, represented here by the unintegrated vertex U (1) with ghost number +1, describes all the admissible deformations. The constraints on the deformations are obtained as Bianchi identities on the curvatures of the gauge superpotentials A α (x, θ), A m (x, θ) and of the field strenght A α . This analysis does not lead to the field equations which are implemented by requiring that the possible deformations preserve the energy-momentum tensor T in equation (4.1).
First, we discuss the BRST cohomology and then we discuss the deformations which preserve the energy-momentum tensor.
Consider the unintegrated vertex U (1) and the computation of the cohomology for the operator Q. The physical space of vertex operators is contained in the cohomology of Q at ghost number 1, which means that U (1) satisfies where Ω is a ghost number zero superfield. Notice that U (1) could depend on the auxiliary fields η m and ω zm . Nevertheless, since η m and ω zm , as well as the current Q η ≡ dz η m ω zm , are trivial under the BRST transformations, we expect that the physical observables are independent of them. The requirement that the physical vertex U (1) does not contain η m can be written as Since U (1) is a scalar, it can not depend on ω m z . To solve (6.2), we decompose U (1) as 4) where A α , . . . , F αβ are arbitrary superfields of x m , θ α . Terms with derivative ∂ z x m , ∂ z θ α and d zα cannot be present since U (1) is a worldsheet scalar. Perhaps one may construct a composite field C z (after suitable bosonization and fermionization to allow contravariant vectors) but the cohomology derived from (6.4) should not change. Note that, because b has ghost number −1 and b(z)b(w) ∼ 0, we can only include terms which are at most bilinear in ξ m , χ α and λ α . Acting on U (1) with the BRST charge Q (given in (2.26)), and collecting the independent contributions, we find the following equations (6.5) where D α ≡ ∂/∂θ α + 1 2 θ β γ m αβ ∂/∂x m . The normalization is chosen such that From the terms containing b, we get These equations are written in terms of superfields, and therefore, supersymmetry is manifest. Equations (6.6) correspond to Bianchi identities for the curvature defined in (6.5), so they are automatically satisfied when the curvatures are expressed in terms of the potentials as in (6.5). Notice that in D = (9, 1)-superspace, imposing F αβ = F mα = 0 one gets the equations [23] γ αβ These equations imply this other system of equations The latter, together with the Bianchi identities (6.6), imply the equations of motion for linearized super Yang-Mills [23] γ m αβ ∂ m A β = 0 , ∂ m F mn = 0 . (6.9) In ref [24] it is shown (in the Wess-Zumino gauge) that (6.9) implies (6.8) and (6.8) implies (6.7). Hence, all three set of equations of motion are equivalent. The potentials A α , A m of the gauge connection and their corresponding curvatures F αβ , . . . , F α m parametrize all the possible deformation of the Kac-Moody algebra. Indeed, we can observe that, if we define the new BRST Q U = Q + U (1) , the nilpotency of the new BRST charge implies (up to terms quadratic in the vertex U (1) ) equation (6.2), namely Q, U (1) = 0. The new BRST charge is given by (6.10) where the Kac-Moody generators Π m z , ∂ z θ α , d zα are shifted by the gauge potentials A m , A α and A α . In the same way, the energy momentum tensor T zz (z) in equations (4.1) is modified into a new tensor T A zz (z) (6.11) Finally, by requiring that this T A zz (z) satisfies the usual OPE, namely we find the constraints on the gauge potentials and on the field strenghts. The double poles yield (6.13) and the simple poles give further equations (6.14) From these equations, we immediately get the equations of motion and the gauge-fixing Here f is a constant. These correspond to SYM equations of motion. To see this, it is convenient to expand the superfields A m and A α to first power of θ a in terms of gauge field a m (x) and of gaugino u α (x) (in the Wess-Zumino gauge θ α A α = 0 [25]) A α (x, θ) = u α (x) + γ mn,α β θ β ∂ m a n (x) + . . . , (6.16) where the coefficients in front of the θ-terms are fixed by supersymmetry. The equation ∂ 2 A m = 0 implies that ∂ 2 a m (x) = 0, while γ m αβ ∂ m A β = 0 implies that γ m αβ ∂ m u β (x) = 0 at zero order in θ. The latter coincides with the gaugino equation of motion. However, we still have a condition at the same order by means of ∂ n A n = D α A α (where the constant f is set to zero for convenience). By inserting the expansions (6.16), we deduce ∂ m a m = 0, hence the gauge field a m satisfies the usual Maxwell equation of motion in the Landau gauge. As shown in [24] these equations of motion, with the Bianchi identites (6.6) imply the constraints F αβ = F αm = 0.
Notice that F αβ = F αm = 0 parametrize a particular set of all the possible deformations of the Kac-Moody algebra. Namely, those deformations which do not modify the non-abelian Lie algebra of zero modes dz Π m z and dz d zα . Essentially, this means that among all possible deformations of the Kac-Moody algebra, the physical observables are those which do not modify the non-abelian part of the algebra. Clearly, the abelian part of the algebra due to Π Π ∼ z −2 , ∂θ ∂θ ∼ 0 , Π ∂θ ∼ 0 and d ∂θ ∼ z −2 is deformed by the curvatures F mn , F α m and F β α . Finally, we have to point out that the introduction of the b−c z system to promote the central charge of the Kac-Moody algebra (2.22) to a generator renders the BRST cohomology trivial (as pointed out in [13]), however this parametrizes correctly all the possible deformations of the algebra. It is only the requirement on the deformed energy-momentum tensor which implies the correct equations of motion.
Using the method discussed in [24] to eliminate the auxiliary fields from the superfield A α , one arrives at the expressions [25] A α = (γ m θ) α a m − 2 3 (θγ m u) − 1 8 (θγ m γ nr θ)f nr + · · · , A m = a m − (θγ m u) − 1 4 (θγ m γ nr θ)(f nr + 2 3 γ [n ∂ r] u) + · · · , A α = u α + 1 2 (γ mn θ) α (f mn − θγ [m ∂ n] u) + · · · . (6.17) The superfields in (6.17) are written exclusively in terms of the physical gauge field a m , its field strength f mn , and the gaugino u α ; all auxiliary fields have been eliminated. Moreover, the gauge-fixing condition θ α A α = 0 is automatically satisfied. As indicated in (6.1) one can determine the integrated vertex V z ] = ∂ z U (1) . We have found the complete all the other susy transformation rules for the antifields being zero. Moreover, the combination of antifields θ * α + 1 2 x * m γ m αβ θ β is supersymmetric. On the other hand, by differentiating the action S s in(9.1) with respect to fields, we derive the BRST variations of the antifields 3) There are duality relations among fields and antifields, for example the linear shift of x m by means of the vector ξ m corresponds to the linear shift of ξ * m by means of x * m . In the same way, the variation of d α contains the holomorphic derivative of χ α , which is dual to the variation of χ * α into −∂ z d * ,α . This duality between fields and antifields is typical for topological field theories quantized with the BV formalism. To reveal such topological aspects, it is convenient to introduce new variables ξ m = ξ m + 1 2 θ α γ m αβ λ β , d zα = d zα + ∂ z x m γ mαβ θ β + 1 6 γ m αβ θ β θ γ γ mγδ ∂ z θ δ , χ α = χ α + ξ m γ mαβ θ β + 1 6 γ m αβ θ β θ γ γ mγδ λ β .

(9.4)
Inserting the expression for d zα into (9.4), we find that d zα = j ǫ zα where j ǫ zα is the current for the supersymmetric charge given in (2.4). In terms of the new variables the BRST transformations simplify to which clearly show the topological nature of the BRST symmetry generated by Q. Notice that although the BRST symmetry is very simple in terms of the new variables, the variables ξ m , d zα and χ α are no longer susy invariant. In terms of the old variables the BRST symmetry is rather complicated, but Π m , d α and all the ghost fields are supersymmetric (clearly x m and θ α are not supersymmetric being the coordinates of the superspace).