N=4 Superconformal Symmetry for the Covariant Quantum Superstring

We extend our formulation of the covariant quantum superstring as a WZNW model with N=2 superconformal symmetry to N=4. The two anticommuting BRST charges in the N=4 multiplet of charges are the usual BRST charge Q_S and a charge Q_V proposed by Dijkgraaf, Verlinde and Verlinde for topological models. Using our recent work on"gauging cosets", we then construct a further charge Q_C which anticommutes with Q_S + Q_V and which is intended for the definition of the physical spectrum.


Introduction and Conclusions
The past four years a new approach to the covariant quantization of the superstring has been developed. The starting point is a BRST operator Q B = λ α d zα in the leftmoving sector of the superstring [1], depending on free spacetime coordinates x m , θ α and their conjugate momenta p(θ) zα (m = 0, . . . , 9; α = 1, . . . , 16), and commuting ghosts λ α . The constraints d zα ≈ 0 define the conjugate momenta of θ α , and this is the only information of the classical Green-Schwarz string that is kept. The OPE's produce further currents Π zm and ∂ z θ α . Nilpotency of Q B can be achieved by imposing the pure spinor constraint λγ m λ = 0, but in our approach we have relaxed this constraint, and this produced new ghost pairs (ξ m , β zm ) (anticommuting) and (χ α , κ α z ) (commuting) and a conjugate momentum w zα for λ α (we supress from now on the index z most of the time).
We discovered in this approach that the superstring is a "gauged" WZNW model [2], based on a non-semisimple nonreductive superalgebra A which decomposes into coset generators Q α (associated with d α and λ α , w α ) and abelian subgroup generators, namely P m (associated with Π m and ξ m , β m ) and fermionic central charges K α (associated with ∂θ α and χ α , κ α ). The matter currents J M = {d α , Π m , ∂θ α } depend only on x m , θ α and p α , and generate A. The gauging leads to a second set of such matter currents J h M depending on new variables x m h , θ α h and p h α . In terms of these currents a particular superconformal algebra was constructed, with BRST charge j W zB containing the sum of the currents J M + J h M , a stress tensor, a ghost current, and a spin 2 field B zz which contains the difference of the currents J M − J h M . The central charge of this system vanishes. Some preliminary study suggested to us that a second BRST charge was needed to define the physical spectrum. We called this unknown charge Q C (with C for constraints). 3 Two BRST charges suggest the presence of an N=4 algebra, but then one would expect that the superconformal algebra of the WZNW model should be a N=2 subalgebra. We found it not to be an N=2 algebra, but rather a Kazama algebra [4]; however, it is known that one can add a gravitational topological quartet (which we call the Koszul quartet 4 3 A similar analyis has been performed by M. Chesterman [3]. 4 This quartet consists of the ghosts (c ′ z , b ′ zz , γ ′ z , β ′ zz ) with conformal spins (−1, 2, −1, 2) and ghost K ′ below) and modify the Kazama currents such that the sums of the currents of the combined system form a genuine N = 2 algebra [5,6]. In particular the BRST charge of this combined system is the sum of the separate charges, Q W S + Q K ′ S , but the spin 2 current B W of WZNW model is modified intoB W by adding terms depending on the fields of K ′ .
The fact that such a Koszul quartet is a gravitational topological quartet was welcome news, because it enables us to introduce worldsheet diffeomorphisms into our work. It is known from the work of Dijkgraaf-Verlinde-Verlinde [7], that there exist two BRST charges in topological models: a charge Q K S for the Koszul quartet and a charge Q V which is related to diffeomorphisms and which has the form Here T W is the stress tensor of the matter topological system which in our case corresponds to the WZNW system, andB W is the modified spin 2 field mentioned above. The two charges Q V and Q K S anticommute. However, as noticed recently [8], in order that Q V and Q W S + Q K ′ S anticommute, the Koszul quartet needed to turn the Kazama algebra into N=2 algebra cannot be the same as the Koszul quartet needed to construct Q V . Thus there are two Koszul quartets, which we already denoted above by K ′ and K. The quartet K ′ modifies the current B W of the WZNW model, while K enters in the construction of Q V . At this point we have the following BRST charges: . . The first one is a spacetime object, while the latter two are worldsheet objects. They all anticommute.
Although we have now constructed three BRST charges, none of them contains the information that the theory originally contained the pure spinor constraints. So the problem of finding the BRST charge Q C remained. We decided to start a study of general Lie algebras and constraints of the kind encountered in the superstring [9]. In this study we divided generators into the commuting set of Cartan generators, and coset generators.
subalgebra. Since [Q, P ] ∼ K this algebra is nonreductive as well as nonsemisimple. We then "gauged the coset generators". By this provocative statement we meant that we imposed constraints on the ghosts associated to the coset generators (corresponding to the pure spinor constraints [1] ), and then relaxed these constraints in such a way that the cohomology remained unchanged. In the process we found the second BRST charge Q C , but one has to introduce a doubling of the subgroup ghosts as well as an another copy of the subgroup ghosts which vanishing ghost number. In our case these new fields There is a separate BRST charge for the coset fields which we denote by Q co S and a contribution of the coset fields to Q V which we denote by Q co V . Following the procedure of [9] the BRST charge Q W C for the WZNW model with K ′ quartet was recently constructed in [8], but it was found not to anticommute with the total We construct below a charge Q C which does anticommute with Q S + Q V . Our construction is based on the observation that all currents so far have been constructed without bosonization, so that the zero modes η z and η ′ z due to bosonization of the superghosts of the two Koszul quartets K and K ′ trivially anticommute with all the other currents. We propose to take the zero mode η ′ z and to make a similarity transformation with the whole BRST charge Q S as follows C the charge given in [9]. Of course Q S itself remains unchanged under this similarity transformation and Q W C is of the form η ′ z + Q W C + . . . and is independent of K. The extra terms denoted by .... are needed in order that Q W C anticommute with Q S . Having constructed the extra charge Q W C which we expect to be needed to define the correct physical spectrum, we return to the issue of an N = 4 superconformal algebra. A small N = 4 superconformal algebra needs a triplet of SU(2) currents, which for a twisted model (the case we are considering) have spins (0, 1, 2) and ghost numbers (2, 0, −2) [10].
We use the free fields of the K quartet to construct the Wakimoto representation of these SU(2) currents [11]. There are now at least two ways to proceed: use Q S and Q V , or use Q W S + Q K ′ S + Q co S and Q W C to construct another N=4 algebra. In this paper we perform the first construction. It may clarify if we summarize the various charges in a diagram, and indicate the various N = 2 and N = 4 subalgebras which could conceivably be constructed. Those whose existence is only conjectured are indicated by a question mark. From this picture another conjecture emerges: the various N=4 algebras are all subalgebras of an enveloping N = 8 superconformal algebra.

Without Coset Fields
With Coset Fields Mutually Anticommuting BRST Charges of N = 2, 4 Subalgebras In the spacetime sector we begin with the BRST charge Q W S of the WZNW model [2] (see the left upper part of the diagram). The BRST charge Q W S + Q K ′ S belongs to an N=2 algebra [2]. The BRST charge 5 η ′ z anticommutes Q W S + Q K ′ S and these two charges might be part of N = 4 algebra. The coset fields are needed to construct Q W C according to [9] and hence one finds the BRST charge Q co S for the coset fields in the right upper part of the diagram. Comparing the left-and right-hand side of the diagram, we conjecture that the BRST charges Q W C which we discussed above and Q W S + Q K ′ S + Q co S are part of an another N = 4 algebra.
In the worldsheet sector we find the BRST charge Q K S + Q V which is part an N=2 algebra, as discussed in [7], see the lower left part of the diagram. The zero mode η z forms another anticommuting BRST charge, and together these two BRST charges form an N=4 superconformal algebra as shown by Berkovits and Vafa [10]. We can repeat our procedure in the spacetime sector and make a similarity transformation on η z with the BRST charge of the worldsheet sector to obtain Q top C , see the lower right part of the diagram. The formula reads Q top The construction of X top z follows along the lines of [9]: one starts with the Lie derivative along γ [12], which is the analog of the constraints involving d α , and then one relaxes this constraint [13]. In this context γ plays the role of λ α , c plays the role of (ξ m , χ α ), while c ′ corresponds to (ξ ′ m , χ ′ α ), and γ ′ corresponds to (ϕ m , ϕ α ). Note that "gauging" of the coset of the topological quartet K brings in the quartet K ′ of the spacetime sector. This is the more fundamental reason that one needs two quartets.
Perhaps again Q K S + Q V and Q top C are part of an N=4 algebra (see the lower right part of the diagram).
Finally, we come to the contents of this paper. We show that Q S and Q V do indeed belong to an N=4 superconformal algebra. We construct this algebra in steps. In section 2 we construct an N=4 algebra for the quartet K with the Wakimoto triplet, in section 3 we add the coset fields, and in section 4 we add the WZNW model coupled to the quartet K ′ . The final result is given by eq. (45).
It is also easy to construct a charge Q C which anticommute with Q S + Q V , namely Q C = e −R (rη + sη ′ )e R with arbitrary r, s, and R = {Q S + Q V , X}. One choice for In order that physics after the similarity transformation is different from physics before, we expect that a suitable filtration (grading condition [14] 6 ) is needed.
Despite several important results of the pure spinor formalism [15] obtained by N. The two BRST charges Q W S and Q K S are present in any topological model, so they can not be used to eliminate the dependence on x h and θ h . We need another anticommuting operator, like an antighost, to eliminate this dependence. Moreover, if one has two BRST charges, it seems likely that one is dealing with an N = 4 model.
An N = 2 model with two spin-one BRST charges suggest that it is part of a twisted N = 4 model, which should consist of two spin-one BRST currents G + (z) andG + (z), two spin 2 B-fields G − (z) andG − (z), a stress tensor T zz with vanishing anomaly, and further SU(2) currents. In a twisted N = 4 model the SU(2) currents have spin 0, 1, 2, rather than spin 1 [10]. We thus need a spin (0, 1, 2) triplet of currents which separately form a closed algebra.
At this point we may recall that the well-known Wakimoto representation [11] of currents constructed from ghost fields satisfies these properties. One is thus led to study the original N = 2 gravitational Koszul quartet together with the Wakimoto triplet of currents, and try to extend this model to an N = 4 model. This quartet (b zz , c z , β zz , γ z ) has spins (2, −1, 2, −1) and ghost charges (−1, 1, −2, 2).
The ghosts b zz and c z are anticommuting with propagator c(z)b(w) ∼ (z − w) −1 , while γ z and β zz are commuting and satisfy the OPE γ(z)β(z) ∼ (z − w) −1 . The currents of this N = 2 model are given by The stress tensor is simply the sum of the stress tensors of two spin (2, −1) doublets, and the factor 2 in the ghost current yields the ghost charges (2, −2) for γ z and β zz . The B field B zz has spin 2 and ghost number −1, and the parameter µ is a free parameter (to be fixed to µ = 1 later). The spin-1 BRST current j B z and the spin-2 field B zz are the twists of the two spin 3/2 currents of an untwisted N = 2 multiplet. From now on we shall drop the super-and subscripts z when no confusion is possible.

The Wakimoto representation is given by
The superscripts denote the ghost number. The ghost current is identified with J 3 . These currents satisfy the following OPE Closure of the algebra fixes all coefficients in the currents. We could rescale these currents such that the terms with double poles in J ++ J ++ and J 3 J 3 become equal, but the formulas are simpler by keeping the present normalization.
We now present the N = 4 extension of the N = 2 Koszul model. This result has been obtained before in [16] with µ = 0, but we keep µ arbitrary. For completeness we give the derivation with µ = 0. The stress tensor and SU(2) triplet are unchanged, while we have the following anticommuting currents The conformal spin of G + and G − is 1 and 2, respectively [2], while it is straightforward to verify thatG ± (w) have the same conformal spin as G ± The crucial test is whether the OPE's of two fermionic currents close. They do indeed close. We find the following OPE's For our work it is important that the two BRST G + and G + charges are nilpotent and anticommute. This is indeed the case ForG + (z)G + (w) we directly checked that the terms with µ cancel, but the vanishing of this OPE follows already from (13)  3) The OPE's of a twisted N = 4 model are for example given in [10]. We obtain agrement with these OPE's if we rescale our current by factors ±i.

An N = 4 model for one Koszul quartet and coset two BRST currents of the coset and Koszul systems
We can now construct the currentsG + andG − by acting with J ++ and J −− on j co+K zB ≡ G + z and B co+K zz ≡ G − zz . One finds easily The calculation ofG + is more involved. We start from We obtainG Triple and double poles nicely cancel here, confirming the coefficient 9/2 of the term with ∂γ in J ++ . The crucial question is whether the simple structure ofG + in the coset sector also holds in the Koszul sector. We find bc∂c + 2γβ∂c + γc∂β = c 1 2 Hence, the totalG + is indeed of a simple form Also J ++ can be written in this way Thus we obtain the following N = 4 superconformal currents for the WZWN model coupled to coset fields and two Koszul quartets The current J ++,K contains a term x∂γ while the currentG + contains a term y∂ 2 c.
The same analysis as performed for the coset fields shows that also these currents satisfy an N = 4 superconformal algebra. The only parameters to be fixed are the values of x and y. We fix x by requiring that the double poles with γ in the numerator cancel in the following OPE We find This yields the value Confirmation is obtained from which reproduces x = 17.
We find The triple poles cancel for x=17, confirming again the result for x. Then also the double poles cancel, while from the simple poles we find thatG + contains a term −17∂ 2 c. This yields again y = −17.