Space-time supersymmetry in WZW-like open superstring field theory

We investigate space-time supersymmetry in the WZW-like open superstring field theory, whose complete action was recently constructed. Starting from a natural space-time supersymmetry transformation at the linearized level, we construct a nonlinear transformation so as to keep the complete action invariant. Then we show that the transformation satisfies the supersymmetry algebra up to an extra transformation, unphysical on the asymptotic string fields. This guarantees that the constructed transformation in fact acts as space-time supersymmetry on the physical S-matrix.


§1. Introduction
Construction of a complete action including both the Neveu-Schwarz (NS) sector representing space-time bosons and the Ramond sector representing space-time fermions are a long-standing problem in superstring field theory. While the action for the NS sector was constructed based on two different formulations, the WZW-like formulation 1) and the homotopy-algebra-based formulation, 2) it had been difficult to incorporate the Ramond sector in a Lorentz-covariant way. Only recently, however, a complete action has been constructed for the WZW-like formulation, 3) and soon afterwards for the homotopy-algebrabased formulation. 4) Interestingly enough, in these complete actions, the string field in each sector appears quite asymmetrically. In the WZW-like formulation, for example, the string field Φ in the NS sector is in the large Hilbert space, characterizing the WZW-like formulation, but the string field Ψ in the Ramond sector is in the restricted small Hilbert space defined using the picture-changing operators. Then the question is how space-time supersymmetry is realized between these two apparently asymmetric sectors. The purpose of this paper is to answer this question by explicitly constructing the space-time supersymmetry transformation in the WZW-like formulation. * ) In the first quantized formulation, space-time supersymmetry is generated by the supercharge obtained by using the covariant fermion emission vertex, 6) which interchanges each physical state in the NS sector with that in the Ramond sector. Therefore, it is natural to expect first that the space-time supersymmetry transformation in superstring field theory is realized as a linear transformation using this first-quantized supercharge. 7) We will see, however, that this expectation is true only for the free theory, while the action including the interaction terms is not invariant under this linear transformation. We modify it so as to be a symmetry of the complete action, and then verify whether the constructed nonlinear transformation satisfies the supersymmetry algebra. We find that the supersymmetry algebra holds, up to the equations of motion and gauge transformation, only except for a nonlinear transformation. It is shown, however, that this extra transformation can also be absorbed into the gauge transformation up to the equations of motion at the linearized level. Under the assumption that the asymptotic condition holds also for the string field theory, this implies, at least perturbatively, that the constructed transformation acts as space-time supersymmetry on the physical states defined by the asymptotic string fields. This guarantees that supersymmetry is realized on the physical S-matrix. * * ) The rest of the paper is organized as follows. In section 2, we summarize the known results on the complete action for the WZW-like open superstring field theory. In addition, restricting the background to the flat space-time, we introduce the GSO projection operator, which is essential to make the physical spectrum supersymmetric. For later use, some basic ingredients, such as the Maurer-Cartan equations and the covariant derivatives, are extended to those based on general derivations of the string product which can be noncommutative. After this preparation, the space-time supersymmetry transformation is constructed in section 3. Using the first-quantized supercharge, a linear transformation is first defined so as to be consistent with the restriction in the Ramond sector. Since this transformation is only a symmetry of the free theory, we first construct the nonlinear transformation perturbatively by requiring it to keep the complete action invariant. Based on some lower-order results, we suppose the full nonlinear transformation δ S in a closed form, and prove that it is actually a symmetry of the action. In section 4, the commutator of two transformations is calculated explicitly. We show that it provides the space-time translation δ p , up to the equations of motion and gauge transformation, except for a nonlinear transformation δp that can be absorbed into the gauge transformation only at the linearized level. Thus the supersymmetry algebra holds only on the physical states, and hence the physical S-matrix, defined by the asymptotic string fields under appropriate assumptions on asymptotic properties of the string fields. Although this extra symmetry is unphysical in this sense, it is nontrivial in the total Hilbert space including unphysical degrees of freedom. It produces further unphysical symmetries by taking commutators with supersymmetries or themselves successively. We have a sequence of unphysical symmetries corresponding to the first-quantized charges obtained by taking successive commutators of the supercharge and the unconventional translation charge with picture number p = −1. Section 5 is devoted to summary and discussion, and two appendices are added. In Appendix A, we summarize the conventions for the SO (1,9) spinor and the Ramond ground states, which are needed to identify the physical spectrum although they do not appear in this paper explicitly. The triviality of the extra transformation in the Ramond sector, which remains to be shown, is given in Appendix B. Further nonlinear transformations obtained by taking the commutator of two unphysical transformations, [δp 1 , δp 2 ] are also discussed. All the extra symmetries obtained by taking commutators with δ S or δp repeatedly are shown to be unphysical. §2. Complete gauge-invariant action On the basis of the Ramond-Neveu-Schwarz (RNS) formulation of superstring theory, an open superstring field is a state in the conformal field theory (CFT) consisting of the matter sector, the reparametrization ghost sector, and the superconformal ghost sector. We assume in this paper that the background space-time is ten-dimensional Minkowski space, for which the matter sector is described by string coordinates X µ (z) and their partners ψ µ (z) (µ = 0, 1, · · · , 9). The reparametrization ghost sector and superconformal ghost sector are described by a fermion pair (b(z), c(z)) and a boson pair (β(z), γ(z)), respectively. The superconformal ghost sector has another description by a fermion pair (ξ(z), η(z)) and a chiral boson φ(z). 6) The two descriptions are related through the bosonization relation: The Hilbert space for the βγ system is called the small Hilbert space and that for the ξηφ system is called the large Hilbert space. The theory has two sectors depending on the boundary condition on the world-sheet fermions ψ µ , β, and γ. The sector in which the world-sheet fermion obeys an antiperiodic boundary condition is known as the Neveu-Schwarz (NS) sector, and describes the spacetime bosons. The other sector in which the world-sheet fermion obeys a periodic boundary condition is known as the Ramond (R) sector, and describes the space-time fermions. We can obtain the space-time supersymmetric theory by suitably combining two sectors. 9)

String fields and constraints
In the WZW-like open superstring field theory, we use the string field Φ in the large Hilbert space for the NS sector. It is Grassmann even, and has ghost number 0 and picture number 0. Here we further impose the BRST-invariant GSO projection * ) 2) where G N S is defined by . This is necessary to remove the tachyon and makes the spectrum supersymmetric. 9) For the Ramond sector, we use the string field Ψ constrained on the restricted small where X and Y are the picture-changing operator and its inverse acting on the states in the small Hilbert space with picture numbers −3/2 and −1/2, respectively. They are defined by and satisfy The string field Ψ is Grassmann odd, and has ghost number 1 and picture number −1/2. The picture-changing operator X is BRST exact in the large Hilbert space, and can be written using the Heaviside step function as X = {Q, Θ(β 0 )}. Here, instead of Θ(β 0 ), we introduce We impose the BRST-invariant GSO projection as where G R is given by The gamma matrixΓ 11 is defined by using the zero-modes of the world-sheet fermion ψ µ (z) asΓ We summarize the convention on how the zero modes ψ µ 0 act on the Ramond ground states in Appendix A. * ) * ) In the context of string field theory, the GSO projections are also needed to make the Grassmann properties of string fields Φ and Ψ consistent with those of the coefficient space-time fields.

Complete gauge-invariant action
By use of the string fields introduced in the previous subsection, the complete action for the WZW-like open superstring field theory is given by 3) 13) and is invariant under the gauge transformations with O = ∂ t , η, or δ, which are analogs of (components) of the right-invariant one form, satisfying the Maurer-Cartan-like equation is the graded commutator of the two string field A 1 and A 2 : is defined by the operator acting on the string field A as which is nilpotent: (D η (t)) 2 = 0 . Then the linear map F (t) on a general string field Ψ in the Ramond sector is defined by The map F (t) has a property that changes D η (t) into η : Using F (t), we can define a homotopy operator for D η (t) as F (t)Ξ satisfying 3) which trivializes the D η -cohomology as well as the η-cohomology in the large Hilbert space. From the definition (2 . 18), we can show that the homotopy operator F Ξ is BPZ even and satisfies for a string field A . It is useful to note that we can define the projection operators onto the Ramond string field annihilated by D η and its orthogonal complement, respectively.
The BPZ inner product in the small Hilbert space ·, · is related to that in the large Hilbert space ·, · as where A and B are in the small Hilbert space, and also in the Ramond sector for the equations in the first line. Using a general variation of the map F (t) on a string field A , from which we find the equations of motion, Before closing this section, we generalize several ingredients for later use. We can define η, or δ , but also for any other derivations of the string product. Although such general O's are not in general commutative, we assume that they satisfy a closed algebra with respect to the graded commutator of derivations, As an analog of the linear map F (t) in the Ramond sector, we can also define the linear map f (t) on a general string field Φ in the NS sector by (2 . 31) A homotopy operator for D η (t) in the NS sector is given by the BPZ even operator f (t)ξ 0 : We can define the projection operators onto the NS string field annihilated by D η and its orthogonal complement, respectively. §3. Space-time supersymmetry Now let us discuss how space-time supersymmetry is realized in the WZW-like formulation. Starting from a natural linearized transformation exchanging the NS string field Φ and the Ramond string field Ψ , we construct a nonlinear transformation that is a symmetry of the complete action (2 . 13). We show that the transformation satisfies the supersymmetry algebra, up to the equations of motion and gauge transformation, except for an unphysical symmetry.

Space-time supersymmetry transformation
At the linearized level, a natural space-time supersymmetry transformation of string fields in the small Hilbert space, ηΦ and Ψ , is given by is the first-quantized space-time supersymmetry charge with the parameter ǫ α . The spin operator S α (z) in the matter sector can be constructed from ψ µ (z) using the bosonization technique. 6) This S(ǫ) is a (Grassmann-even) derivation of the string product, and is com- This is equivalent to the space-time translation operator p(v) = v µ dz 2πi i∂X µ (z) (center of mass momentum of the string) in the sense that, for example, 7) We frequently omit specifying the parameters explicitly and denote, for example, S(ǫ 1 ) by S 1 . Since ηΦ and Ψ are in the small Hilbert space containing the physical spectrum, (3 . 1) is the transformation law given in Ref. 7) except that the local picture-changing operator at the midpoint is replaced by the X in (2 . 5) so that the transformation is closed in the restricted space. As a transformation of Φ in the large Hilbert space, we adopt here that This is consistent with (3 . 1) but is not unique. A different choice, however, can be obtained by combining (3 . 7) and an Ω-gauge transformation, for example, Using the fact that S is BPZ odd, it is easy to see that the quadratic terms of the action (2 . 13), are invariant under the transformation However, the action at the next order, is not invariant under δ S but is transformed as We have thus to modify the transformation by adding S S (1) and δ (0) S S (2) , which are again nonzero and require to add The procedure is not terminated, so we suppose a full transformation consistent with these results, and then show that it is in fact a symmetry of the complete action.

Complete space-time supersymmetry transformation
Here we suppose that the complete transformation is given by We calculate each of these three terms, which we denote (I), (II), and (III), separately. First, using (2 . 21) and the cyclicity of the inner product, the second term is calculated as For the third term, we find where we have used (2 . 21), (2 . 19), and the fact that X is BPZ even with respect to the inner product in the small Hilbert space, XA , B = A , XB , and QΨ + XηF Ψ is in the restricted small Hilbert space. In order to calculate the first term (I), some consideration is necessary. In addition to the cyclicity, we need the following relation for two graded commutative derivations of the string product, Here the second term vanishes owing to (3 . 9) and (2 . 24): The first term in (3 . 23) can further be calculated as up to the equations of motion (2 . 27) and gauge transformation (2 . 14) generated by some field-dependent parameters, where δ p(v 12 ) is the space-time translation defined by with the parameter v 12 in (3 . 3). In this section, we show that the algebra (4 . 1) is slightly modified, but still the transformation (3 . 18) can be identified with space-time supersymmetry.
Since any transformation of the string field is a special case of the general variation, (4 . 3) holds for any symmetry transformation δ I , This is the case even for the commutator of the two transformations [δ I , δ J ] , with which can be shown by explicit calculation using (2 . 28) and (2 . 31) if we assume (4 . 4) with some field-dependent Ω I . Therefore if the algebra of the transformation is closed on A η , we have or equivalently, the algebra is also closed on e Φ : with some field-dependent Ω IJ . Here in (4 . 7) we used that A η is invariant under the Ω-gauge transformation, A δ Ω = D η Ω, as seen from (4 . 4a).

[δ S 1 , δ S 2 ]
Now let us explicitly calculate the supersymmetry algebra on A η and Ψ , which is easier to calculate than the algebra on the fundamental string fields Φ (or e Φ ) and Ψ due to their Ω-gauge invariance and enough to know that on the fundamental string fields as was shown in the previous subsection. From (3 . 18) we find ) Here we used the relations which hold for a general string field A . The commutator of two transformations on Ψ , (4 . 14) Then, using [D η , D S ] = 0 , The second form can be obtained using (2 . 28), and will be used below.
In order to calculate the algebra on A η , we first calculate the transformation of F Ψ using (2 . 25): where the third equality follows from (2 . 22), and the symbol ∼ = denotes an equation which holds up to the equations of motion. Then the commutator of two transformations on A η can be calculated similarly to that on Ψ . Since the first term can be calculated as we find with the gauge parameters given in (4 . 16) and (4 . 6). The last term absent in (4 . 1) is a new symmetry defined by where the former is determined so as to induce This extra contribution can be absorbed into the gauge transformation, up to the equations of motion, at the linearized level as we will see shortly.
Let us consider the transformation (4 . 22) at the linearized level: We can similarly show that the transformation of Ψ in (4 . 24b) can also be written as a gauge transformation up to the equation of motion at the linearized level as shown in Appendix B. Here we assume that the asymptotic condition 10) holds for string field theory as well as the conventional (particle) field theory. Then, at least perturbatively, we can identify that the transformation (4 . 24), or (4 . 25) and (B . 5) can be interpreted, with appropriate (finite) renormalization, as that of asymptotic string fields. If we further assume asymptotic completeness, this implies that the extra transformation (4 . 24) acts trivially on the on-shell physical states defined by these asymptotic string fields, and thus the physical S-matrix. Thus the supersymmetry algebra is realized on the physical S-matrix, and we can identify the transformation (3 . 18) with space-time supersymmetry.

Extra unphysical symmetries
We have shown that the supersymmetry algebra is realized on the physical S-matrix but this is not the end of the story. The extra transformation δp produces another extra transformation if we consider the nested commutator [δ S 1 , [δ S 2 , δ S 3 ]] . The extra contribution comes from the commutator [δ S , δp] which is non-trivial because the first-quantized charges S andp are not commutative: [S,p] = 0 . In fact, we can show that the algebra holds with the gauge parameters, and Ω Sp in (4 . 6). The new transformation δ [S,p] is defined by and in particular ζ µα = ǫ α v µ on the right-hand side of (4 . 26). This new symmetry is also unphysical in a similar sense to δp. At the linearized level, the transformation (4 . 28) becomes * )   In this paper, we have explicitly constructed a space-time supersymmetry transformation of the WZW-like open superstring field theory in flat ten-dimensional space-time. Under the GSO projections, we have extended a linear transformation expected from space-time supersymmetry in the first-quantized theory to a nonlinear transformation so as to be a symmetry of the complete action (2 . 13). We have also shown that the transformation satisfies the supersymmetry algebra up to gauge transformation, the equations of motion and a transformation δp acting trivially on the asymptotic physical states defined by the asymptotic string fields. This unphysical transformation produces a series of transformations δ [S,p] , δ [pp] , · · · by taking commutators with δ S or δp repeatedly. All of these symmetries also act trivially on the asymptotic physical states, and thus are unphysical, but it is interesting to clarify their complete structure, which is nontrivial in the total Hilbert space including unphysical degrees of freedom.
In any case, except for such an unphysical complexity, we have now understood how space-time supersymmetry is realized in superstring field theory, and therefore are ready to study various consequences of space-time supersymmetry 11)-14) on a firm basis. We have to (re)analyze them precisely using the techniques developed in conventional quantum field theory. * ) We hope to report on them in the near future. stimulating atmosphere.

Appendix A Spinor conventions and Ramond ground states
In this paper, although it is mostly implicit, we adopt the chiral representation for SO (1,9) gamma matrices Γ µ , in which Γ µ is given by where γ µ andγ µ satisfy The charge conjugation matrix C satisfies the relations and is given in the chiral representation by The matrices CΓ µ are symmetric, or equivalently The world-sheet fermion ψ µ (z) in the Ramond sector has zero-modes that satisfy the The degenerate ground states therefore become the space-time spinor, on which ψ µ 0 act as space-time gamma matrices. We summarize here the related convention. We denote the ground state spinor as | α |α , on which ψ µ 0 acts as ThenΓ 11 defined by (2 . 12) acts on the ground states aŝ by which the definition of the GSO projection (2 . 10) is supplemented. Similarly, the BPZ conjugate of the ground state spinor ( α |, α |) satisfies with the normalization The nontrivial matrix elements of ψ µ 0 are then given by We can see that the gauge parameter in (B . 5), is in the restricted small Hilbert space, if we note that {η,M} = 0 .
As was mentioned in section 4.3, the commutator [δp 1 , δp 2 ] produces another unphysical transformation δ [p,p] : where the field-dependent parameters are given by Finally we show that all the extra symmetries obtained from the repeated commutators of δ S 's and δp's act trivially on the physical states defined by the asymptotic string fields.
For this purpose, it is enough to consider the transformations of ηΦ and Ψ at the linearized level for a similar reason to that discussed in Section 4. Using the linearized form of (4 . respectively. Hence all the extra symmetries obtained as repeated commutators of δ S 's and δp's act trivially on the on-shell physical states, and thus the physical S-matrix, defined by the asymptotic string fields.