Type II superstring field theory with cyclic L ∞ structure

We construct a complete type II superstring field theory that includes all the NS-NS, R-NS, NS-R and R-R sectors. As in the open and heterotic superstring cases, the R-NS, NS-R and R-R string fields are constrained by using the picture changing operators. In particular, we use a non-local inverse picture changing operator for the constraint on the R-R string field, which seems to be inevitable due to the compatibility of the extra constraint with the closed string constraints. The natural symplectic form in the restricted Hilbert space gives a non-local kinetic action for the R-R sector, but it correctly provides the propagator expected from the firstquantized formulation. Extending the prescription previously obtained for the heterotic string field theory, we give a construction of general type II superstring products, which realizes a cyclic L∞ structure, and thus provides a gauge invariant action based on the homotopy algebraic formulation. Three typical four-string amplitudes derived from the constructed string field theory are demonstrated to agree with those in the first-quantized formulation. We also give the half-Wess-Zumino-Witten action defined in the medium Hilbert space whose left-moving sector is still restricted to the small Hilbert space.


Introduction
In recent years, there has been some significant progresses in constructing gauge invariant superstring field theories. First, a complete WZW-like action for open superstring including both the Neveu-Schwarz and Ramond sectors, representing space-time bosons and fermions, respectively, has been constructed [1] after several significant developments [2]- [13]. A crucial idea to successfully incorporate the Ramond sector is to impose an extra constraint on the Ramond string field, which can naturally be interpreted to come from the fermionic moduli integration over the super-Riemann surface. Shortly thereafter, this has been extended to an alternative formulation in the small Hilbert space [14], in which the gauge symmetry is beautifully realized using a homotopy algebraic structure, the A ∞ algebra. Several interesting studies, such as on the general structure of the complete WZW-formulation [15], [16], on the space-time supersymmetry [17], [18], and on some generalization toward a heterotic string field theory [19]- [21], have further been done. Then, in a previous paper, the authors extend these constructions to the heterotic string field theory [22]. We have first constructed a gauge invariant action in the small Hilbert space by constructing string interactions realizing a homotopy algebraic structure of closed string, cyclic L ∞ algebra, and then have also given the WZW-like action through a field redefinition.
Independent of these developments, Ashoke Sen has developed a closed superstring field theory applicable to both the heterotic and type II theories [23,24]. He have provided a quantum master action in a rather abstract way by considering string off-shell amplitudes allowing a cell decomposition. In addition, instead of imposing constraint as in the former two formulations, he have introduced an extra string field, which becomes free and decouple from the physical sector, to incorporate the Ramond sector consistently. It has shortly been shown that it can also be extended to the open superstring field theory [25].
Thanks to these developments, we now have three independent formulations of superstring field theory, the homotopy algebraic, the WZW-like and the Sen's formulations. Each of these formulations has advantages and disadvantages, and seems to be complementary each other. So the aim of this paper is to fill the blank still remaining by constructing a complete field theory of the type II superstring based on the homotopy algebraic and WZW-like formulations to provide with a solid foundation to nonperturbative studies of the superstring theories. This paper is organized as follows. In section 2 we summarize how the type II superstring field theory is constructed based on the homotopy algebraic structure for the closed string, the cyclic L ∞ algebra. We impose constraints on the string fields in the R-NS, NS-R and R-R sectors. In the R-R sector, in particular, we introduce non-local inverse picture changing operator, which seems to be inevitable due to the compatibility of the extra constraint with the closed string constraints. We construct the free theory and explain that it provides the correct R-R propagator even though the kinetic term is non-local. We show that it can be replaced with the local action if an extra string field is introduced following the Sen's formulation. Then, it is shown that we can construct a gauge invariant action if the string products have the cyclic L ∞ structure. Such string products are explicitly constructed in section 3. The prescription is an extension of the asymmetric construction proposed in [3] for the NS-NS sector, and is obtained by repeating twice the one proposed for the heterotic string products in [22]. The operators with non-zero picture number are inserted first for the left-moving sector and then for the rightmoving sector, following the procedure used for the heterotic string field theory. We confirm that the action constructed in this way actually reproduces typical four-point amplitudes in section 4. We will explicitly calculate three on-shell four-point amplitudes with four R-R, two NS-R two R-R and NS-NS R-NS NS-R R-R external states, and show that they agree with those obtained in the first-quantized formulation. In section 5, we attempt to map the action and gauge transformation to those based on the WZW-like formulation. Unfortunately, however, we only obtain a half-WZW-like action defined not in the large Hilbert space but in the medium Hilbert space, the tensor product of the large Hilbert space for the left-moving sector and the small Hilbert space for the right-moving sector. Section 6 is devoted to the summary and discussion. We summarize how the string field is expanded with respect to the ghost zero modes for each sector in appendix A, which is useful to consider the Batalin-Vilkovisky (BV) quantization [26].
2 Type II string field theory in homotopy algebraic formulation We first summarize several basics of the type II string field theory in the homotopy algebraic formulation. The free field theory is given and discussed in some detail. After confirming that the action of the R-R sector provides the propagator used in the first-quantized formulation, we show that it can also be written in the local form by introducing an extra R-R string field following the Sen's formulation. The gauge invariant interacting action can be obtained if we assume the multi-string products with the cyclic L ∞ structure.

String field and constraints
There are four sectors, the NS-NS, R-NS, NS-R and R-R sectors, in the first-quantized Hilbert space of the type II superstring, H , corresponding to the combination of the Neveu-Schwarz and Ramond boundary conditions for the left-and right-moving fermionic coordinates: where H andH in the right-hand sides are the left-moving and the right-moving small Hilbert spaces, respectively. Accordingly, the type II string field Φ has four components, which is Grassmann even and has ghost number 2 . The first and the last components, Φ N S-N S and Φ R-R , have picture number (−1, −1) and (−1/2, −1/2) , respectively, and represent spacetime bosons. The second and the third components, Φ R-N S and Φ N S-R , have picture number (−1/2, −1) and (−1, −1/2) , respectively, and represent space-time fermions. All of these components satisfy the closed string constraints, The first constraint imposes that the string field does not dependent on the ghost-zero mode c − 0 . Therefore, the NS-NS component, in which only the bc ghosts have zero modes, is expanded with respect to the ghost zero mode as As in the open and heterotic superstring field theories [1], [14], [22], we further restrict the dependence of the other components on the βγ ghost zero modes. For the Φ R-N S and Φ N S-R components, we impose respectively, where XY andXȲ are the projection operators defined by using the picture changing operators and their inverses, which satisfy the relations Here G 0 andḠ 0 are the zero modes of the left and right moving (total) superconformal currents, respectively. Note that the inverse picture changing operators in (2.7) are defined so that the additional constraints (2.6) are compatible with the closed string constraints (2.4). Since the picture changing operators X andX are BRST invariant, they can be written as the BRST exact form in the large Hilbert space: The ghost zero-mode dependence of the components Φ R-N S and Φ N S-R is restricted to the form where G = G 0 + 2γ 0 b 0 andḠ =Ḡ 0 + 2γ 0b0 are the ghost zero-mode independent part of G 0 andḠ 0 , respectively. On the other hand, for the Φ R-R component which depends on both the left and right moving βγ zero modes, we cannot simultaneously impose two conditions, due to their noncommutativity: [XY,XȲ ] = 0 . However, we should notice that the choice of inverse changing operators are not unique. There is a possibility to use the non-local operators, which also satisfy as the inverse changing operators [11]. We can impose the conditions 14) which are now compatible with each other, and also with the closed string constraints (2.4). It can be shown that the ghost zero-mode dependence of Φ R-R is restricted by the constraints (2.14) as Here we define ψ R-R so that the expansion (2.15) has a local form, which will be found to be natural shortly.
Here, if we define the operators G and G −1 by the relations (2.8) and (2.13) can collectively be written as are the projection operators onto H N S-N S , H R-N S , H R-R and H R-R components of the Hilbert space H , respectively. It is also useful to define (r = 0, 1) , then we can write G = XX . It should be careful that G −1 is the inverse of G in this sense. Then we can define the projection operator P G = GG −1 and collectively write the extra constraints (2.6) and (2.14) as We call the Hilbert space restricted by the constraints (2.4) and (2.19) the restricted Hilbert space, or frequently simply restricted space, in this paper. On the restricted Hilbert space, the BRST operator acts consistently A natural symplectic form in the restricted Hilbert space is defined as follows. First, the symplectic form in the space restricted by the closed string constraints (2.4) is defined by using the BPZ inner-product as where Φ| is the BPZ conjugate of |Φ . The symbol |Φ| denotes the Grassmann property of string field Φ : |Φ| = 0 or 1 if Φ is Grassmann even or odd, respectively. For later use, we also define symplectic forms ω m and ωm and ω l in the Hilbert spaces, by where i ϕ 1 | and i ϕ 2 | are the BPZ conjugates of |ϕ 1 i and |ϕ 2 i in H i (i = m,m, l) , respectively. The symplectic form ω s , ω m and ωm are related to ω l as . Then a natural symplectic form in the restricted Hilbert space is defined by It has the non-degenerate cross-diagonal form common in each sector ). It should be noted that the ψ R-R in (2.15) is defined so that the non-locality of Y andȲ in the R-R sector disappears in the symplectic form (2.26). In the following, we will see that this cross-diagonal form of the symplectic form Ω in the restricted space provides a free field theory which can properly be quantized via the BV formalism.

Free field theory
Using the symplectic form Ω in the restricted Hilbert space, the free field action and gauge transformation of the type II superstring field theory are given by where the gauge parameter also has four components Then, by multiplying G , we have thanks to (2.17).
The action (2.27) has superficially the same form as that of the bosonic string field theory, and its BV quantization can be carried out in a similar way [27]. The master action S 0 can simply be given by removing the ghost number restriction on Φ in the classical action: is the string field with the ghost number g . The component Φ (2) is equal to the classical string field Φ , sand the others are the space-time ghosts, anti-ghosts and corresponding anti-fields. The BRST transformation, which keeps the master action (2.32) invariant, is obtained by replacing the parameter Λ in the gauge transformation (2.27) by the field Φ as δ B0 Φ = QΦ . (2.33) It is easy to show that the master action (2.32) actually satisfies the BV master equation. Using the fact that the symplectic form Ω has the cross-diagonal form (2.26), an arbitrary variation of the master action can be written as and thus we have The BRST invariance of the action implies that the classical BV master equation holds: The components φ I and ψ I are identified with the fields and anti-fields in the gauge-fixed basis in the BV formulation [28], respectively. 3 The gauge-fixed action in the Siegel gauge is obtained by setting ψ I = 0 .

R-R action
Before incorporating the interactions, let us examine the action of the R-R sector, in more detail since it is characteristic for the type II superstring field theory. For simplicity, we take the Siegel gauge |ψ RR = 0 in this discussion. After integrating out the ghost zero modes, the master action (2.32) and the BRST transformation (2.33) in the R-R sector become Although this action is non-local, the propagator agrees with that obtained in the first-quantized formulation [29].
If one wants to avoid the non-local action, one can replace it with the Sen-like action as an alternative by introducing an extra Grassmann even string fieldΦ R-R , which is restricted by the closed string constraints (2.4) and has ghost number 2 and picture number −3/2 . Then the alternative action is given bỹ The difference from the Sen's original action, however, is that we can rewrite it using the method of completing the square as thanks to the constraint (2.14), where the equivalence is obvious. SinceΦ R-R appears only in the form of XXΦ R-R in the action (2.40), we can restrictΦ R-R by the condition which is dual to the constraint (2.14) on Φ R-R and restrictsΦ R-R to the form of The Sen-like master action in the generalized Siegel gauge ψ R-R =ψ R-R = 0 then becomes after integrating out the ghost zero modes. Although the extra sector is a higher derivative theory, it stays free and is decoupled from the physical sector if the interaction part of the action does not includeΦ R-R .

Including interactions
The interaction of the type II superstring field theory are described by the multi-string products, which make a string field from n string fields Φ 1 , · · · , Φ n . They are graded symmetric under interchange of the n string fields, and must carry proper ghost number and picture number. In addition, since the type II superstring field in this formulation is in the restricted small Hilbert space, the outputs of the string products must also satisfy the constraint (2.19): By using these string products, the action of the type II superstring field theory is given by where L 1 is identified to the BRST operator: if the string products satisfy the L ∞ relations and the cyclicity with respect to the symplectic form Ω : Here the symbol σ denotes the permutation from {1, · · · , n} to {σ(1), · · · , σ(n)} , and ǫ(σ) is the sign factor of the permutation of the string fields from If the set of string products {L n } satisfies these conditions, it is called the string products with the cyclic L ∞ structure or simply the cyclic L ∞ algebra. The problem is how to construct such an L ∞ algebra.
3 Construction of string products with L ∞ structure Let us construct a set of string products realizing a cyclic L ∞ algebra. We use a coalgebraic representation handling an infinite number of string products in the L ∞ algebra collectively. We follow the notation and convention in [22].

Cyclicity, Ramond numbers and picture number deficit
String products describing the interaction of type II superstrings must have proper ghost number and picture number. Since the ghost number structure is the same as that of the bosonic closed string field theory, here it is enough to consider the picture number that the string products should have. Denote a coderivation corresponding to an (n + 2)-string product (n ≥ 0) with picture number (p, p ′ ) as B n+2 . In order to describe the type II superstring interaction, the output string state must have the same picture number as that of the type II superstring field: the picture number of its NS-NS, R-NS, NS-R and R-R components must be equal to (−1, −1) , (−1/2, −1) , (−1, −1/2) and (−1/2, −1, 2) , respectively. The string products are also characterized by their Ramond and cyclic Ramond number defined by Ramond cyclic Ramond number = # of Ramond inputs ∓ # of Ramond outputs , which are also assigned for each of the left-and right-moving sectors. Since we can consider each sector separately let us first consider the left-moving sector. Suppose that 2r of n + 2 inputs are the R states in the left-moving sector. If we assume the conservation of the space-time fermion number the output must be the NS state, and thus from the picture number conservation. Such a string product is characterized by the cyclic Ramond number 2r and the Ramond number 2r . If 2r + 1 of the inputs are the R states, the output is the R state and which is the case characterized by the cyclic Ramond number 2r + 2 and the Ramond number 2r . Both of these equations (3.2) can be solved as n = p + r − 1 . After repeating the same consideration for the right-moving sector, we can find that a candidate coderivation describing the type II superstring interaction is the one respecting the Ramond number: 4 with B 1 ≡ 0 , which we call the string products with no picture number deficit. However, this is not suitable to consider the cyclicity since the Ramond number is not invariant under the cyclic permutation as in (2.50). So, instead we consider the string products, respecting the cyclic Ramond number that is invariant under the permutation. While it becomes easy to consider the cyclicity, this combination of string products B cannot be used as it is since its NS-NS, R-NS, NS-R and R-R components have picture number deficit (0, 0) , (1, 0) , (0, 1) and (1, 1) , respectively. In the heterotic string field theory, similar string products can naturally be appeared in a nonlinear extension of the combination of the operator (one-string product) Q − η [22]. In the type II superstring theory, however, the analogous combination Q − η −η has no counterpart with picture number deficit (1, 1) , and thus we cannot directly extend the prescription in [22] to construct the required L ∞ algebra. We take an alternative way that is an generalization of the asymmetric construction used in [3] to give those restricted into the NS-NS sector.

Construction of string products
The prescription we propose is simply repeating that used on the heterotic string field theory twice: the first time is for getting the correct structure of left-moving sector by inserting X and/or ξ in the bosonic string products, which we assume to be known [30]- [32], and the second time is for getting the correct structure of right-moving sector by insertingX and/orξ in the (heterotic) string products obtained in the first step . We start from the combined coderivation where p and 2r is the picture and cyclic Ramond numbers of the left-moving sector, respectively. This can be decomposed to D and C by picture number deficit as Suppose that B has zero right-moving picture number and is independent of the right-moving Ramond and cyclic Ramond numbers. The left-moving picture number deficit of D is equal to zero and that of C is equal to one. As was shown in [22], the L ∞ relation for the coderivation following from the equations for the generating function We can show that [l, It was also shown in [22] that the equations (3.10) are satisfied if we postulate the the differential equations by introducing (a generating function of) the gauge products represented by a degree even coderivation (3.14) The differential equations (3.13) can recursively be solved as with the initial condition given by using the interacting part of the bosonic products (string products with no non-zero picture number operator insertion) [22]: The operation ξ• in (3.15a) is defined as that inserting ξ cyclically. Then, by construction, all the B(s, t) and λ(s, t) are cyclic with respect to the symplectic form ω m . They provide a cyclic L ∞ algebra (H m , ω m , D −C) . After decomposing this combined L ∞ algebra into D and C , we can obtain a heterotic L ∞ algebra in the small Hilbert space, satisfying [η, L H ] = [η, L H ] = 0 , by similarity transformation generated by the cohomomorphism with b H = BF . This L H has required picture number structure for the left-moving sector but the right-moving picture number is still equal to zero: Here the subscript or superscript 2r after slash is the left-moving Ramond or cyclic Ramond number, respectively. It is easy to see that b H is cyclic with respect to ω m in the same way as in [22]. We repeat the same procedure for the right-moving sector. Let us consider the combined coderivationD which can be decomposed by the right-moving picture number deficit as It is noted that only the right-moving quantum numbers, the picture, Ramond and cyclic Ramond numbers, are specified. Those of the left-moving sector are implicit but determined properly in our construction below. The L ∞ relation of the coderivationD −C , This time we solve the differential equations (3.28) by starting from the initial condition Similarly solving the higher order products recursively, we obtain Note that the factor X for the left-moving sector can still be pulled out after repeating the procedure for the right-moving sector. The initial conditionB (0) (s) = L (0) H (s) fixes the structure of the left-moving picture number as 2r) .
p+r+1 | 2r . We can again show that b is cyclic with respect to ω l in a similar way in [22], and thus the cyclic L ∞ algebra (H, Ω, L) is obtained.

Four-point amplitudes
In this section, we concretely calculate three typical on-shell physical amplitudes with four external strings in a similar way given in [2,22] to demonstrate how the type II string field theory we have constructed reproduces the first-quantized amplitudes. We take the Siegel gauge defined by the conditions From the action (2.47) we can find that the propagators in this gauge are given by which agree with those appeared in the first-quantized formulation [29].

Four-(R-R) amplitude
Let us first consider the case that all the external strings are in the R-R sector. The firstquantized amplitude is given in the form where Φ 1 , · · · , Φ 4 are on-shell physical R-R vertex operators, satisfying QΦ = 0 , in (−1/2, −1/2) picture. The correlator is evaluated in the small Hilbert space on the complex z-plane. It is not necessary to put any picture changing operators at all. Owing to the same moduli structure as the bosonic closed string, we can express this using the bosonic closed string products L B n as is hidden behind the definition of the string product. This amplitude can be regarded as a multi-linear map: where H Q ⊂ H is the subspace of states annihilated by Q . Putting the string fields Φ 1 , · · · , Φ 4 out, we can express (4.4) as by introducing the bilinear map representation ω s | of the symplectic form ω s defined by (4.7) The expression (4.6) can also be written by using the coderivations as Here, From the type II superstring field theory, on the other hand, the four-(R-R) amplitude is calculated as The second equality holds owing to the fact that the string field b (0,0) 2 (Φ 1 , Φ 2 ) satisfies the closed string constraints (2.4). Rewriting by using the coderivations, we find (4.10) Here, the string products without picture number b (0,0) n are equal to the bosonic string products L B n by construction. 5 Hence, the string field theory amplitude (4.10) certainly agrees with the first-quantized amplitude (4.8).

Two-(NS-R)-two-(R-R) amplitude
Next, we consider the amplitude with two NS-R strings and two R-R strings, which is given in the first-quantized formulation by in the similar representation as the four-(R-R) amplitude. Here, A 4 | is a multi-linear map and X 0 = {Q, ξ} . In this case, the amplitude obtained from the type II string field theory is calculated as . (4.13) In our construction given in the previous section, the string products without right-moving picture number b (1,0) n is equal to the heterotic string products (b H ) (1) n = (BF ) (1) n and is given explicitly by where the last qualities follow from the recursion relation (3.15b) with n = 0 . If we further note that the relations (4.14) can be decomposed with respect to the Ramond and cyclic Ramond numbers. In particular, we find Substituting this into the string field theory amplitude (4.13) and then pulling Q out, we can rewrite it as except for the terms vanishing when they hit the sates in H Q . Inserting 1 = [η, ξ] or 1 = [η,ξ] , we can find that the amplitude (4.17) agrees with the first-quantized one: In the second equality we used [η, λ

(NS-NS)-(R-NS)-(NS-R)-(R-R) amplitude
Finally, let us consider the case with four external strings coming from the four different sectors. The first-quantized amplitude is give by as a multi-linear map, Here, P is the projection operator onto Q , both of which contain two left-moving Ramond states and two right-moving Ramond states. The string field theory amplitude in this case is calculated as by decomposing it with respect to the cyclic Ramond and Ramond numbers. Then, the string field theory amplitude (4.22) can be rewritten as  We can repeat the similar procedure for the left-moving sector. Using b H = BF , we have in particular (4.29b) by using (3.15b). Decomposing them with respect to the cyclic Ramond and Ramond numbers, we can find that π (0,0) 1 Thanks to these relations, the amplitude (4.26) can further be rewritten as except for the terms vanishing when they hit the states in H Q . Again, inserting 1 = {η, ξ} = {η,ξ} , the string field theory amplitude eventually becomes

Relation to the WZW-like formulation
So far we have constructed a complete gauge-invariant action for the type II superstring field theory in the small Hilbert space based on the cyclic L ∞ structure. In the open superstring field theory [14] and heterotic string field theory [22], we can map it to a gauge invariant action in the WZW-like formulation through a field redefinition. In this section we consider whether it is possible to construct a complete WZW-like action in a similar way also for the type II superstring field theory.
Here, let us consider the restriction of the construction to the pure NS-NS sector. If we define generating functions by and both of them are closed in the small Hilbert space: The L ∞ relations (5.2) follow from the differential equations . This is nothing but the asymmetric construction proposed in [3], and thus the string and gauge products we constructed reduce in the NS-NS sector to those obtained by their asymmetric construction [3].
Using the fact that L(s, t)| (0,0) satisfies the differential equation (5.5), we can show that the string products restricted in the NS-NS sector L| (0,0) = L(0, 1)| (0,0) can be written in the form of the similarity transformation [14,22] as and thus is given by the one used in the heterotic string field theory [19,22]. 6 The string fields in the other sectors are simply identified in two formulations. We denote the string fields of the R-NS, NS-R and R-R sectors in the half-WZW-like formulation as Ψ ,Ψ and Σ , respectively, to distinguish which formulation they belong to: The half-WZW-like formulation obtained in this way is the dual (in the sense that the roles of Q andη are exchanged) to that given in [12]. It is not the completely WZW-like formulation defined using the (whole) large Hilbert space H l , but we construct here also an action and a gauge transformation to complete the story. First of all, we rewrite the action (2.47) in the WZW-like form by extending the NS-NS string field Φ N S-N S to Φ N S-N S (t) with t ∈ [0, 1] satisfying Φ N S-N S (0) = 0 and Φ N S-N S (1) = Φ N S-N S . Using the cyclicity, we find It is mapped to the half-WZW-like action through the identification (5.9) and (5.11) as Here we defined the associated fields as with d = ∂ t or δ by introducing one coderivationsξ d derived fromξd . We can show that they satisfy the characteristic identities of the associated fields, dGη(V (t)) = π (0,0) 1 Lη(e ∧Gη(V (t)) ∧ B d (V (t))) , (5.15a) The nilpotent linear operator Dη(t) was introduced as for a general string field ϕ ∈ Hm . The gauge transformation π 1 δ(e ∧Φ ) = π 1 L(e ∧Φ ∧ Λ) , There is also an extra gauge invariance in the half-WZW-like formulation under the transformation, because the identification (5.9) is not one-to-one but The identities (5.15) are enough to guarantee that the action (5.13) is invariant under the gauge transformations (5.19) and (5.21) independently with the gauge invariance in the homotopy algebraic formulation.

Summary and discussion
Extending the procedure for constructing the heterotic string field theory, we have constructed the type II superstring field theory with a cyclic L ∞ structure based on the homotopy algebraic formulation. In addition to the closed string constraints, we impose the extra constraints on the string fields in the R-NS, NS-R and R-R sectors. These constraints restrict the dependence of these string fields on the bosonic ghost zero modes, and also make the field anti-field decomposition in the BV quantization obvious. Although the kinetic term of the R-R string field is non-local, it provides the same propagator in the Siegel gauge as that naturally obtained in the first-quantized formulation. Repeating the procedure used in the construction of the heterotic string products, we have constructed the string products for the type II superstring with the cyclic L ∞ structure acting across all the NS-NS, R-NS, NS-R and R-R sectors. We can map the action and the gauge transformation to those in the half-WZW-like formulation defined using the medium Hilbert space although not in the completely WZW-like formulation in the large Hilbert space.
A remaining interesting task is to construct a completely WZW-like action in the large Hilbert space. In the language introduced in [12], the similarity transformation generated by the cohomomorphism (5.7) map the small Hilbert space L ∞ triplet (η,η; L NS,NS ) to the asymmetric (heterotic) one (η, Lη; Q) , while the symmetric triplet (L η , Lη; Q) is necessary to realize the complete WZW-like formulation. In order to realize it and couple the NS-NS action in [33] to the string fields in the other sectors, it seems to be necessary to find a construction which is an extension of the symmetric construction proposed in [3]. Our method used in this paper, which was developed in [22] for constructing the heterotic string field theory, cannot be extended that way so some completely different approach seems to be needed.
Finally, it should be emphasized that the type II superstring field theory has a possibility to provide a solid basis to the AdS/CFT correspondence which is still mysterious and must be proved. We hope that the gauge invariant action we have constructed will help us explore such an interesting possibility.
Its dual space is spanned by their BPZ conjugates The inner-product matrix between two spaces is off-diagonal: Since there are two pairs of fermionic ghost zero modes, (b 0 , c 0 ) and (b 0 ,c 0 ) , in the closed string theory, their Fock representation is four-dimensional where | ↓↓ ∝ | ↓ ⊗ | ↓ . One of the closed string constraints, b − 0 | ↓↓ = 0 . restrict it to the two-dimensional space, We normalize the states so that ↓↓ |c + 0 c − 0 | ↓↓ = 1 . (A.11)

A.1.2 bosonic ghost
There are also bosonic ghosts (β n , γ n ) in the R sector. They satisfy the commutation relation [γ n , β m ] = δ n+m,0 , (A. 12) and the hermite and BPZ conjugate relations 7 In general it is known that they have an infinitely many Fock representations defined on the ground states with the picture number p as [34] β n |p = 0 , for n > −p − In the string field theory, we use two natural representation with p = −1/2 and −3/2 , whose non-zero mode parts are common Fock space obtained by acting (β −n , γ −n ) with n > 0 on the ground state |0 satisfying β n |0 = γ n |0 = 0 , n > 0 . where the state denoted as |φ N S-N S or |ψ N S-N S represents the non-zero mode part of the Fock representation of the string field. This expansion holds independent of whether the ghost number of Φ N S-N S is restricted or not.

A.2.2 R-NS sector
In the R-NS sector there are also the bosonic ghost zero modes (β 0 , γ 0 ) in the left-moving sector. The R-NS string field Φ R-N S restricted by the constraints (2.4) and (2.6) can be expanded as in which the ghost zero-mode dependence can be separated as The states denoted as |φ R-N S or |ψ R-N S is the string field after separating the ghost-zero modes. The zero modes can be integrated out by using the inner-product 0| ⊗ ↓↓ |c + 0 c − 0 δ(γ 0 )| ↓↓ ⊗ |0 = 1 . (A.30)
(A. 36) In order to construct the Sen-type action, we have to introduce an extra string field with picture number (−3/2, −3/2) , whose zero-mode ground state | ↓↓ ⊗ |0 ⊗ |0 is related to | ↓↓ ⊗ |0 ⊗ |0 through the relations It can be shown that XXΦ R-R obtained by changing the picture actually has the form of (2.15):