Supertwistors and Cubic String Field Theory for Open N=2 Strings

The known Lorentz invariant string field theory for open N=2 strings is combined with a generalization of the twistor description of anti-self-dual (super) Yang-Mills theories. We introduce a Chern-Simons-type Lagrangian containing twistor variables and derive the Berkovits-Siegel covariant string field equations of motion via the twistor correspondence. Both the purely bosonic and the maximally space-time supersymmetric cases are considered.


Introduction
It was recently shown by Witten [1] that B-type open topological string theory with the supertwistor space CP 3|4 as a target space is equivalent to holomorphic Chern-Simons (hCS) theory on the same space (for related works see [2]- [7]). This hCS theory in turn is equivalent to supersymmetric N =4 anti-self-dual Yang-Mills (ASDYM) theory in four dimensions. The N =4 super ASDYM model is governed by the Siegel Lagrangian [8]. Its truncation to the bosonic sector describes N =0 ASDYM theory with an auxiliary field of helicity −1 [8,9].
It may be of interest to generalize the twistor correspondence to the level of string field theory (SFT). This could be done using the approach proposed in [5] or in the more general setting of [6]. Alternatively, one could concentrate on (an appropriate extension of) SFT for N =2 string theory. At tree level, open N =2 strings are known to reduce to the ASDYM model in a Lorentz noninvariant gauge [10]; their SFT formulation [11] is based on the N =4 topological string description [12,13]. The latter contains twistors from the outset: The coordinate ζ ∈ CP 1 , the linear system, the integrability and the classical solutions with the help of twistor methods were all incorporated into N =2 open string field theory in [14,15]. Since this theory [11] generalizes the Wess-Zumino-Witten-type model [16] for ASDYM theory and thus describes only anti-self-dual gauge fields (having helicity +1), it is not Lorentz invariant. Its maximally supersymmetric extension, N =4 super ASDYM theory, however, does admit a Lorentz-invariant formulation [8,9]. This theory and its truncation to N =0 features pairs of fields of opposite helicity. In [17] it was proposed to lift the corresponding Lagrangians to SFT.
In the present paper the twistor description of both the purely bosonic and the N =4 supersymmetric ASDYM models [8,9] is raised to the SFT level. In contrast to previous proposals [1]- [7], we allow the string to vibrate only in part of the supertwistor space. The remaining coordinates of this space are not promoted to word-sheet fields but kept as non-dynamical string field parameters. Concretely, we propose a cubic action containing an integration over the supertwistor space CP 3|4 and show that its hCS-type equations of motion are equivalent to the covariant string field equations introduced in [17]. This model may be regarded as a specialization of Witten's supertwistor SFT and may even be equivalent to it. In any case, it is directly related with N =4 super ASDYM theory in four dimensions. We also consider its proper truncation to the bosonic sector, which yields a twistor SFT related to non-supersymmetric ASDYM theory.

Covariant string field theory for open N=strings
Open N=2 strings. From the worldsheet point of view critical open N =2 strings in a flat fourdimensional space-time of signature (− − + +) or (+ + + +) are nothing but N =2 supergravity on a two-dimensional (pseudo) Riemannian surface with boundaries, coupled to two chiral N =2 matter multiplets (X, ψ). The latter's components are complex scalars (the four imbedding coordinates) and Dirac spinors (their four NSR partners) in two dimensions. In the ghost-free formulation of the N =2 string one employs the extension of the c=6, N =2 superconformal algebra to the "small" N =4 superconformal algebra 1 1 We raise and lower indices with ǫ 12 = −ǫ 21 = −1, ǫ12 = −ǫ21 = 1, and similarly for ǫαβ, ǫαβ and ǫαβ, ǫαβ.
Covariant string field theory. Following Berkovits and Siegel [17], we introduce two Lie-algebra valued fermionic string fields A α [X, ψ] and three Lie-algebra valued bosonic string fields G αβ [X, ψ] (symmetric in α and β). Although we suppress it in our notation, string fields are always multiplied using Witten's star product (midpoint gluing prescription) [19]. 2 The index structure reveals that the fields G αβ parametrize the self-dual 3 tensor G αα,ββ = ǫαβG αβ on the target space.
The Lorentz invariant string field theory action [17] reads where . . . means integration over all modes of X and ψ, the trace "tr" is taken over the Lie algebra indices and Note that the action ofD α on any string field B is defined in conformal field theory language as taking the contour integral [11] The covariant string field equations of motion following from the action (4) read For a supersymmetric generalization of the action (4) Berkovits and Siegel [17] introduce a multiplet of string fields imitating the N =4 ASDYM multiplet [8]. Here, A α and χ αi are fermionic whileχ i , φ ij and G αβ are bosonic. Ref. [17] proposes the following action for this super SFT: where the Grassmann parity |B| equals 0 or 1 for bosonic or fermionic fields B, respectively. Due to the large number of string fields this model seems unattractive. However, as we shall see in the coming section, all these fields appear as components of one string field living in a twistor extended target space.

Cubic string field theory for open N=2 strings
Supertwistor space notation. In the Appendix we describe the supertwistor space P 3|4 ǫ of the space (R 4 , g ǫ ) with the metric g ǫ = diag(−ǫ, −ǫ, 1, 1) and ǫ = ±1. It is fibered over the real two-dimensional space Σ ǫ with Σ −1 = CP 1 and Σ +1 = H 2 covered by two patches U ǫ ± . The space P 3|4 ǫ is parametrized by four even complex coordinates (x αα ) ∈ C 4 subject to the reality conditions x 22 =x 11 and x 21 = ǫx 12 , complex coordinates ζ ± ∈ U ǫ ± and odd (Grassmann) coordinates θ i ± , i = 1, . . . , 4. The space P 3|4 ǫ is a Calabi-Yau supermanifold [1]. From now on we shall work on the patch U ǫ + of Σ ǫ , and for notational simplicity we shall omit the subscript "+" in ζ + ∈ U ǫ + , θ i + etc. For further use we introduce BRST operator. Let us introduce the operator taking values in the holomorphic line bundle O(1). We notice that the operators ζ α ∂ z X αβ act as derivatives on string fields. Their zero mode parts, ζ α ∂ ∂x αβ , form two type (0,1) vector fields on the bosonic twistor space P 3 ǫ fibered over Σ ǫ (see the Appendix for more details). Recall that P 3 ǫ being an open subset of CP 3 is the twistor space of (R 4 , g ǫ ). In order to obtain a general type (0,1) vector field along the twistor space one should therefore extend the operator (12) by adding the type (0,1) derivative along Σ ǫ ,∂ Assuming that string fields now depend on the extra variable ζ ∈ Σ ǫ , we define the operator acting on string fields via (6) for theD part and by ordinary differentiation with respect toζ. It is easy to see that Q 2 = 0 due to (3) and the facts thatD does not depend onζ and that {dζ, ψαβ} = 0 (cf. [23]). We take this nilpotent operator as the BRST operator of our SFT extended to Σ String fields. We now consider a fermionic (odd) string field A[X, ψ, θ i , ζ,ζ] depending not only on X(σ) and ψ(σ) but also on θ i and on the parameter ζ ∈ Σ ǫ . It is important to realize that θ i and ζ do not depend on σ here but may be considered as zero modes of world-sheet fields. Since the operator Q has the split form (14) it is natural to assume the same splitting of the string field A, where AD gaugesD and A∂ gauges∂. Note also thatD takes values in O(1) and∂ in O(0); therefore, AD and A∂ are O(1) and O(0) valued, respectively. Since dζ is the basis section of the bundleŌ(−2) complex conjugate to O(2) and anticommutes with spinors ψαβ we reason that Aζ is bosonic (even) and takes values inŌ(−2). It is also assumed that a term A ζ dζ is absent in the splitting (15), i.e. A is a string field of the (0,1)-form type (cf. [1] for the B-model argument). Note that by definition the string field AD does not contain dζ and is fermionic (odd).
Cubic action. Having dζ and dθ i we introduce the action where . . . is the same integration over (X, ψ) modes as in (4). Note that dζ acts as integration over Σ ǫ for terms containing dζ and as a contour integral around ζ=0 for other terms. The Lagrangian L in (16) can be split into two parts, where we used the cyclicity under the trace and omitted total derivatives.
It is important to note that L 1 takes values in O(2), which is compensated by the holomorphic measure dζ dθ 1 dθ 2 dθ 3 dθ 4 being O(−2) valued. 4 At the same time, L 2 takes values in O(3) which causes it to drop out of the action by virtue of Cauchy's theorem applied to the ζ contour integral. Thus, the action (16) can be rewritten as Moreover, both forms (16) and (20) of the action lead to the same Chern-Simons-type equation of motion, which decomposes intoD A∂ +∂AD + AD , A∂ = 0 andDAD + A 2D = 0 .
Component analysis. Recall that AD and Aζ take values in the bundles O(1) andŌ(−2), respectively. Together with the fact that the θ i are nilpotent and O(1) valued, this determines the dependence of AD and A∂ = Aζdζ on θ i , ζ andζ. Namely, this dependence has the form (cf. [22]) where ζ α ,ζ α and ν are given in (11) and θ i 1 i 2 ...i k := θ i 1 θ i 2 . . . θ i k . The expansion (24) is defined up to a gauge transformation generated by a group-valued function which may depend on ζ andζ. All string fields appearing in the expansion (24) depend only on X(σ) and ψ(σ). From the properties of AD, A∂ and θ i it follows that the fields with an odd number of spinor indices are fermionic (odd) while those with an even number of spinor indices are bosonic (even). Moreover, due to the symmetry of theζ α products and the skewsymmetry of the θ i products all component fields are automatically symmetric in their spinor indices and antisymmetric in their Latin indices.
Substituting (24) into (22), we obtain the equations 5 φ αij = −∇ α φ ij and χ αβijk = 1 2 ∇ (α χ β)ijk and G αβγijkl = − 1 3 ∇ (α G β)γijkl (25) showing that (φ αij , χ αβijk , G αβγijkl ) is a set of auxiliary fields. The other nontrivial equations following from (22) and (23) after substituting (24) read 6 where we introduced Up to constant field rescalings the equations (26) for ǫ=1 coincide with the equations of motion following from the action (9) proposed by Berkovits and Siegel [17]. In the zero mode sector they reduce to the anti-self-dual N =4 super Yang-Mills equations of motion. Hence, we have established that maximally supersymmetric ASDYM theory can be obtained from the standard cubic SFT for a single string field A after extending the setting to the supertwistor space.

Bosonic truncation of open string field theory
In order to make contact with non-supersymmetric ASDYM theory, we subject our string field A from (15) and (24) to the truncation conditions These conditions imply that A depends only on the combination Obviously, the even nilpotent variable θ belongs to the bundle O(4) and the integration measure in (29) to O(−4).

The properties of the truncated string field
are the same as before the truncation, except for the restricted dependence on the Grassmann variables. The operators Q,D and∂, the actions (16) and (20), the Lagrangians (17)- (19) and the equations of motion (21)-(23) are unchanged. However, the expansion (24) now simplifies to where (see (27)) From the properties of AD and A∂ it follows that the string fields A α and G αβγ are odd and the G αβ are even.
Substituting the expansion (32) into the equations of motion (22) and (23), we recover for A α and G αβ the bosonic string field equations displayed already in (7) and for G αβγ the dependence as expected. The same result occurs when putting to zero in (26) the string fieldsχ i , χ i α and φ ij as the truncation (29) demands. All other equations following from (22) and (23) are satisfied automatically, due to (34) and the Bianchi identities.
Hence, we have proven that the cubic supertwistor SFT defined by the action (16) together with the geometric truncation conditions (29) is equivalent to the Berkovits-Siegel SFT given by the action (4). Moreover, (4) and (9) derive from (16) simply by substituting there the expansion (32) or (24), respectively, and integrating over the Grassmann and twistor variables. All this is similar to the field theory case [1,22] where in the supertwistor reformulation of N =4 ASDYM theory as hCS theory the dependence of all fields on the twistor variable ζ is fixed (up to a gauge transformation) by the topology of the supertwistor space and one can integrate over it, descending from six to four real dimensions.

Conclusions
The basic result of this paper can be summarized in the equations where c is an inessential numerical constant. The first step demands a split A = AD + Aζdζ of the basic (supertwistor) string field. The second step requires integrating over Σ 1|4 ǫ and rescaling the field as in (28). Truncating A to its lowest and highest Grassmann components (the O(θ 0 ) and O(θ 4 ) parts) projects the above action to tr(G αβ F αβ ) , which governs bosonic N =2 open SFT. Finally, reducing to the string zero modes one recovers the twistor description of N =4 and N =0 ASDYM on the field theory level.

Acknowledgements
This work was partially supported by the Deutsche Forschungsgemeinschaft (DFG).

A Appendix: Supertwistor space
The twistor space of R 4,0 . Let us consider the Riemann sphere CP 1 ∼ = S 2 with homogeneous coordinates (µ α ) ∈ C 2 . It can be covered by two patches, such that it is covered by two patchesŨ + andŨ − with coordinates (w + , ζ + ) onŨ + and (w − , ζ − ) onŨ − related by w + = ζ n + w − and ζ + = ζ −1 − onŨ + ∩Ũ − . A global holomorphic section of O(n) exists only for n≥0. Over U ± ⊂ CP 1 it is represented by polynomials p (n) ± in ζ ± of degree n with p (n) Recall that the Riemann sphere parametrizes the space of all translational invariant (constant) complex structures on the Euclidean space R 4,0 , and the space P 3 E := R 4 ×CP 1 is called the twistor space of R 4,0 [24]. As a complex manifold P 3 E is a rank 2 holomorphic vector bundle O(1) ⊕ O(1) over CP 1 : For more details and references see e.g. [24,25,22].
The twistor space of R 2,2 . In the Kleinian space R 2,2 of signature (− − + +) constant complex structures are parametrized by the two-sheeted hyperboloid where H + = ζ + ∈ U + : |ζ + | < 1 ∼ = SU(1, 1)/U (1) and In fact, under the action of the group SU(1,1) the Riemann sphere is decomposed into three orbits, where the boundary of both H + and H − is given by with a 1,2,3 ∈ R and a 2 1 −a 2 2 = 1. The twistor space of R 2,2 is the space P 3 K := R 4 ×H 2 which as a complex manifold coincides with the restriction of the rank 2 holomorphic vector bundle (42) to the bundle over H 2 ⊂ CP 1 . Equivalently, it can be described as a space P 3 K = P 3 E \ T 3 , where T 3 is a real three-dimensional subspace of P 3 E stable under an anti-linear involution (real structure) which can be defined on P 3 E . For more details see e.g. [26].
The vector fieldsv can be taken as a basis of vector fields of type (0,1) on P 3 ǫ .
Supertwistors. Let us now add four odd variables θ i such that θ i θ j + θ j θ i = 0 for i, j = 1, 2, 3, 4 and each θ i takes its value in the line bundle O(1) over CP 1 (cf. [1]). For describing them formally we introduce a Grassmann parity changing operator Π which, when acting on a vector bundle, flips the Grassmann parity of the fibre coordinates. Hence, we consider the bundle Π C 4 ⊗ O(1) → CP 1 which is parametrized by complex variables ζ ± ⊂ U ± ⊂ CP 1 and fibre Grassmann coordinates θ i ± such that θ i + = ζ + θ i − on the intersection of the two patches covering the total space of this vector bundle.
With the Grassmann variables θ i one can introduce the supertwistor space P 3|4 E as a holomorphic vector bundle over CP 1 , namely The supertwistor space P 3|4 K is defined as a restriction of the bundle P 3|4 E → CP 1 to the bundle over the two-sheeted hyperboloid H 2 ⊂ CP 1 .