Type II superstring field theory revisited

We reconstruct a complete type II superstring field theory with L-infinity structure in a symmetric way concerning the left- and right-moving sectors. Based on the new construction, we show again that the tree-level S-matrix agrees with that obtained using the first-quantization method. Not only is this a simple and elegant reconstruction, but it also enables the action to be mapped to that in the WZW-like superstring field theory, which has not yet been constructed and fills the only gap in the WZW-like formulation.

In the previous papers [14,15], we have attempted to construct complete WZW-like actions, including all sectors representing the space-time bosons and fermions, for the heterotic and the type II superstring field theories by mapping from those of the formulation with the L ∞ algebra method. This construction worked well for the heterotic string theory but not for the type II superstring theory. The reason is that we have taken the left-right asymmetric method for constructing the type II string field theory with an L ∞ structure. In this case, we could only construct an action that is a hybrid of the WZW-like action and the action with an L ∞ structure, which we called the half-WZW-like action. Therefore, we first revisit the formulation based on L ∞ structure and provide an alternative to the asymmetric method, the symmetric way to construct the action for the type II superstring field theory. It is another purpose of this paper.
The paper is organized as follows. In section 2, we summarize how to construct a gaugeinvariant action for the type II superstring field theory based on an L ∞ structure. The string field is suitably constrained and the gauge-invariant action is written by using the string products satisfying the L ∞ algebra relations. We propose in section 3 a new construction method of the string products with cyclic L ∞ structure symmetric with respect to the left-and rightmoving sectors. After introducing the coalgebra representation, we suppose L ∞ algebra using the string products with fixed cyclic Ramond numbers. The significant point, which is different from the other cases, is that their RR output part is including the picture-changing operator (PCO) explicitly. This breaks the cyclicity of the algebra but instead suitably decomposes it into three commutative L ∞ algebras (rather than four), two constraint L ∞ algebras and a dynamical L ∞ algebra. A similarity transformation allows us to transform dynamical L ∞ algebra into the one used for writing down the action and the other two into the constraints that impose that the first algebra closes in the small Hilbert space. In order to explicitly construct such cyclic string products, we give differential equations for their generating functional. These differential equations recursively determine the string products with respect to the number of input string fields, with bosonic products as the initial condition. We show that the new action correctly derives the same tree-level physical S-matrix as that calculated using the firstquantization method in section 4. The proof is based on the homological perturbation theory (HPT), which provides the explicit form of the tree-level S-matrix in closed form and makes it possible to demonstrate the agreement. In section 5, we write down a complete WZW-like action for the type II superstring field theory, which we could not construct previously. After summarizing how the NS-NS action with L ∞ structure was rewritten as a WZW-like action through the map between string fields in two formulations, we extend it to all four sectors. It fills the gap in the WZW-like formulation and should be significant to deepen the understanding of the superstring field theory. Section 6 is devoted to the conclusion and discussion. Finally, it contains two appendices. In Appendix A, we define a projected commutator that plays a significant role in constructing the cyclic products with the L ∞ structure. Appendix B contains the s ands expansions of the generating functionals of cyclic three-string products and corresponding gauge products to help for understanding the flow how they are recursively determined.
2 Type II superstring field theory with L ∞ structure Let us recall how the type II superstring field theory with an L ∞ structure has been constructed [10,15]. The first-quantized Hilbert space, H , of type II superstring is composed of four sectors corresponding to the combination of the NS and Ramond boundary conditions for the left-and right-moving fermionic coordinates: Correspondingly, the type II superstring field Φ has four components: which is Grassmann even 1 and has ghost number 2. We take the picture number of each component as (−1, −1) , (−1/2, −1) , (−1, −1/2) and (−1/2, −1/2) , respectively. The NS-NS and R-R components, Φ N S-N S and Φ R-R , represent space-time bosons, and the R-NS and NS-R components, Φ R-N S and Φ N S-R , represent space-time fermions. The string field Φ is restricted by the closed string constraints We call the Hilbert space restricted by constraints (2.3) the closed string Hilbert space. In addition, it is necessary to impose an extra constraint by introducing a projection operator P G = GG −1 with G = π (0,0) + Xπ (1,0) +Xπ (0,1) + XXπ (1,1) , Here, π (a,b) (a, b = 0, 1) is a projection operator onto a component and, we take the PCO's as Note that G satisfies and is almost exact in the large Hilbert space H l : The operator P −3/2 and P −1/2 are the projector onto the states with ghost number −3/2 and −1/2 , respectively. The BRST operator consistently acts on the string field restricted by (2.12) thanks to the property: P G QP G = QP G . Then, the last constraint restricts the dependence of the ghost zero modes of each component of the string field as We call the Hilbert space further restricted by (2.12) as the restricted Hilbert space, H res . The ghost-zero-mode independent components φ i (ψ i ) (i = NS-NS, R-NS, NS-R, R-R) are Grassmann even (odd) and correspond to the fields (anti-fields) in the gauge-fixed basis when they are quantized by the Batalin-Vilkovisky (BV) formalism. The simplest and practical gauge is obtained by setting ψ = 0 , which we call the Siegel-Ramond (SR) gauge and denote its Hilbert space as H SR . A natural symplectic form in the closed string Hilbert space is defined by using the BPZinner 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. A natural symplectic form in the restricted Hilbert space Ω is defined by using ω as Due to the asymmetry of the inner product among sectors, it is nontrivial to make the L ∞ algebra being cyclic across all the sectors. For later use, it is also convenient to introduce the symplectic form ω l in the large Hilbert space. It is related to ω as if we embed Φ 1 , Φ 2 ∈ H into H l as the fields satisfying the constraint ηΦ i =ηΦ i = 0 (i = 1, 2) . We also use the bilinear map representation of symplectic forms defined by and The action of the type II superstring field theory is defined using the string products L n mapping n string fields to a string field as We identify the one-string product as the BRST operator L 1 = Q , and each (H res ) ∧n (n ≥ 2) is the symmetrized tensor product of H res whose element, Φ 1 ∧ · · · ∧ Φ n ∈ (H res ) ∧n , is defined by 20) where σ and ǫ(σ) denote all the permutations of {1, · · · , n} and the sign factor coming from the exchange {Φ 1 , · · · , Φ n } to {Φ σ(1) , · · · , Φ σ(n) } . Note that the string field L n (Φ 1 , · · · , Φ n ) must satisfy the constraints (2.3) and (2.12): (2.21) If the string products are equipped with the cyclic L ∞ structure, that is, satisfy the L ∞ relations and the cyclicity condition, the action of the type II superstring field theory is given by which is invariant under the gauge transformation The gauge parameter Λ has also four components: Λ = Λ N S-N S + Λ R-N S + Λ N S-R + Λ R-R ∈ H . A set of string products satisfying (2.22) and (2.23) defines a cyclic L ∞ algebra (H res , Ω, {L n }) . The nontrivial task is to provide a prescription that gives these string products.

Symmetric construction of string products
In the previous paper [15], we gave a prescription to construct a set of string products with an L ∞ structure required by complete action of the type II superstring field theory by extending the asymmetric construction proposed in [10]. It is sufficient for defining an action and gauge transformation but is slightly complicated and could not be mapped to those in the WZWlike formulation. In this paper, we propose another prescription by extending the symmetric construction in [10].

L ∞ triplet
We use the coalgebra representation. The coderivation L = ∞ n=0 L n is defined as an operator acting on symmetrized tensor algebra We denote the projection onto H ∧n as π n : π n SH = H ∧n . The string products satisfying L ∞ relation (2. We used a matrix notation describing the string products as a diagonal matrix [10]: The coderivation B can be decomposed into four components according to the sector of outputs as: 7d) 3 The identity operator I n acting on H ∧n is given by I n = 1 n! I ∧ · · · ∧ I n = I ⊗ · · · ⊗ I n . 4 In this paper the square bracket [·, ·] denotes the graded commutator.
Using the coderivation B , we can construct three independent L ∞ algebras, (H l , D) , (H l , C) and (H l ,C) , which are called an L ∞ triplet in [13], with We can merge them into an L ∞ algebra, (H l , D − C −C) , as holds for each picture number deficit and can decompose as Note that it is difficult to make this merged L ∞ algebra (H l , D − C −C) cyclic, unlike the heterotic string field theory case, due to the last (R-R) component. Once this L ∞ triplet is constructed, we can transform it into the triplet (η,η; L) closed in H res as by using the cohomomorphism As was shown in [14], if B is cyclic with respect to ω l , L is cyclic with respect to Ω , and thus gives a cyclic L ∞ algebra (H res , Ω, L) used for the action.

Explicit construction
Now, the task is to construct odd coderivation B satisfying the L ∞ relations (3.11) written for B as Here, the square bracket [·, ·] ab with a, b = 1 or 2 denotes the projected commutator defined by projecting onto specific cyclic Ramond numbers, an extension of that introduced in [14] for the heterotic string field theory. In addition, similar bracket with subscript X = X orX is defined by inserting X at the intermediate state 5 . We give an explicit definition of these brackets in Appendix A, in which we also summarize the Jacobi identities they satisfy. Assume that the cyclic L ∞ algebra (H l , ω l , n+1 ) without picture number, which is straightforwardly constructed similar to that of the bosonic string field theory [19][20][21], is known 6 . We define a generating functional which reduces to L (0,0) at (s,s) = (1, 1) . The parameter s ors is counting the left-or rightmoving picture number deficit from B , respectively. We can show that L B (s,s) satisfies since it can be constructed without using ξ orξ. Then we extend B to include those with non-zero picture number deficit from B and define a generating functional by introducing another parameter t counting the (total) picture number. The L ∞ relations (3.14) are extended for B(s,s, t) to The string product B is obtained as B = B(0, 0, 1) , and the relations (3.19) reduce to (3.14) for B . We can show that if B(s,s, t) satisfies the differential equations, the L ∞ relations (3.19) hold. Here, we introduced two degree-even coderivations λ(s,s, t) and and used an abbreviated notation λ(s,s, t) +λ(s,s, t) = (λ +λ)(s,s, t) . (3.22) These degree-even coderivations, λ(s,s, t) andλ(s,s, t) , are called (generating functionals of) gauge products and can be expanded in the parameters as The bracket with subscript, [·, ·] 22 X with X = Ξ orΞ , is the (graded) commutator with X inserted at the intermediate state, whose explicit definition is given in Appendix A. By differentiating (3.19a) by t and using the differential equation (3.20a), we find that and (3.19c) by t and using the differential equations (3.20) we obtain (s,s) as (s,s) as Using these results, the equations (3.30d) are solved for π 1 λ (1,1) (s,s) and π 1λ (1,1) (s,s) as (s,s) as from (3.30c). We illustrate in Fig. 1 the flow of how the 2-string (gauge) products are determined. The string and gauge products with specific cyclic Ramond numbers are obtained as coefficients by expanding in s ands :  Figure 1: The flow of how the 2-string (gauge) products are determined.
The explicit form of each product is found by expanding the solutions (3.31)-(3.34) in s ands .
In particular, the 2-string product satisfying (3.14) is given by The flow of determining each component is given in Fig. 2. Similarly, expanding the 3-string products as   (s,s) as . where · · · denotes the terms including only the products already given in the lower order. We can solve eqs. (s,s) . We give the flow of how the 3-string (gauge) products are determined in Fig. 3. Their expansion in s ands and the flow of how each component, (gauge) products with specific cyclic Ramond number, are determined are given in Appendix B. In principle, we can repeat a similar procedure Before closing this section, we rewrite the (extended) L ∞ -relations (3.19) and the differential equations (3.20) as where π(s,s, t) = π (0,0) + sπ (1,0) +sπ (0,1) + t sX +sX + ss π (1,1) , These alternative forms are convenient for use in the next section [24].

Tree-level S-matrix
Using HPT, we can show that the tree-level S-matrix derived from (super)string field theory agrees with that calculated using the first-quantization method [22][23][24]. We show in this section that the new prescription proposed in the previous section simplifies this proof for the type II superstring field theory. As a preparation, let us define the on-shell subspace, 1) and the BRST invariant projection operator, The homotopy operator Q + of Q satisfying is defined by and provides a Hodge-Kodaira decomposition of H res , It is compatible with Ω : Ω(H p , H u ) = Ω(H u , H u ) = 0 . Under the SR gauge condition ψ = 0 , H t = ∅ , and the Hilbert space of the quantum string field φ is decomposed to the on-shell and off-shell components : The projection, identity, and homotopy operators,P ,Î , and H , satisfy the relationŝ P =Î + HQ + QH , If we perturb Q by L int = ∞ n=0 L n+2 so that (Q + L int ) 2 = 0 , the homological perturbation lemma tells us that chain complexes and chain maps are deformed as Here, we defined (Î − O) −1 by the formal series: For convenience, introduce Σ with The multi-linear representation, defines a map from H p ⊗ SH p to C . The total S-matrix (at the tree level) is given by S| by restricting the external states onto the gauge-fixed states, that is, H p ⊗SH p to H 0 ⊗SH 0 . If we expand Σ in the number of inputs, Σ = ∞ n=0 Σ n+2 , it induces the expansion of the S-matrix in the number of external states: Each term with specific number of external states is further classified by the number of external Ramond states: S n+3 | = 0≤r,r≤(n+3)/2 S n+3 | (2r,2r) = 0≤r,r≤(n+3)/2 ω l |ξξP 0 ⊗ P 0 π 1 Σ n+2 | (2r,2r) . (4.29) For example, S 3 | contains four terms: Since the Hilbert space H 0 still contains unphysical states in general, the physical S-matrix is defined by projecting it onto the physical subspace, that is, by taking physical states defined by the relative BRST cohomology, H Q = Ker Q/Im Q ⊂ H 0 , as external states: The unitarity of the physical S-matrix is guaranteed by the BRST invariance, [Q, S int ] = 0 . By using the relations of the cohomomorphisms π 1F = π 1 I + Sπ 1 BF , π 1î ′ = P 0 π 1 − Q + Gπ 1 BFî ′ (4.33) following from the definitions (3.13) ofF −1 and (4.23) ofî ′ , respectively, we can show that the Dyson-Schwinger (DS) equation for Σ , , (4.34) holds. By expanding Σ in the number of inputs, it becomes the recurrence relation which determine Σ n+2 recursively. We extend it with three parameters s ,s , and t to the extended DS equation, with G(s,s, t) = π (0,0) + (tX + s)π (1,0) + (tX +s)π (0,1) + t 2 XX + t(sX +sX) + ss π (1,1) , (4.38) and ∂ t ∆(s,s, t) = − Q, Q + ∂ t S(t) + (sΞ +sΞ)π (1,1) + (sΞ +sΞ)π (1,1) ∂s∆(s,s, t) = − η , Q + ∂ t S(t) + (sΞ +sΞ)π (1,1) .
where (S B ) phys n+3 | (2r,2r) in the right-hand side is the bosonic physical amplitude defined by Since the physical amplitude is independent of how we insert PCO's, we can appropriately deform and move them to agree with that obtained using the first-quantization method.

Gauge-invariant action in WZW-like formulation
Using an alternative method, symmetric construction, we have constructed the string products (interactions) with L ∞ structure for the type II superstring. The new gauge-invariant action looks different from the previous one but is the same and nothing essentially new. However, a difference appears when we try to map it to the WZW-like action through a field redefinition. Previously, we could only find a map to the half-WZW-like action [15], but the new construction enables us to construct a map to the complete WZW-like action.

WZW-like action for the NS-NS sector
We first summarize the results obtained in [13] on the construction of the WZW-like action for the NS-NS sector.
For the NS-NS sector, generating functional of string products with the L ∞ structure, L N S-N S (s,s, t) , has been constructed by imposing the differential equations [10], where P denotes the path-ordered product from left to right, we can transform the original triplet (η,η; L N S-N S ) and equations (5.2) to the dual L ∞ triplet (L η =ĝηĝ −1 , Lη =ĝηĝ −1 ; Q) and the equations π 1 L η (e ∧π 1ĝ (e ∧Φ NS-NS ) ) = 0 , (5.4a) π 1 Lη(e ∧π 1ĝ (e ∧Φ NS-NS ) ) = 0 , (5.4b) Qπ 1ĝ (e ∧Φ NS-NS ) = 0 , (5.4c) respectively, which characterize the WZW-like formulation. The constraint equations, are identically satisfied by the pure-gauge (functional) string field 11 with the string field V of the WZW-like formulation having the ghost number 0 and the picture number (0, 0) . Therefore, if we identify it gives a map between the string fields in two formulations. By this identification, the last equation (5.2c) becomes the equation of motion of the WZW-like formulation: In order to write down the WZW-like action, it is also necessary to introduce another important functional field called the associated string field, ,ηdV (t)) + Lη(ηV (t), dV (t))) + · · · , (d = ∂ t , δ, Q). (5.9) Here, V (t) is an extension of V by a parameter t ∈ [0, 1] satisfying V (0) = 0 and V (1) = V . The identification (5.7) induces the map where D ξξ is the coderivation derived from π 1 D ξξ = ξξπ 1 D with D = d for d = ∂ t , δ and D = π 1 L N S-N S for d = Q . These maps (5.7) and (5.10) are consistent with the identities characterizing the associated field, where we introduced the nilpotent linear operators D η (t) and Dη(t) as D η (t)ϕ = π 1 L η (e ∧Gηη(V (t)) ∧ ϕ) , Dη(t)ϕ = π 1 Lη(e ∧Gηη(V (t)) ∧ ϕ) , (5.12) for a general string field ϕ ∈ H N S-N S l . Then, we can map the action to the WZW-like action Using the identities (5.11), we can calculate an arbitrary variation of the action as The gauge transformation generated by Λ is mapped from that generated by Λ N S-N S in the formulation with the L ∞ structure with the identification The extra gauge invariances come from the fact that the map (5.7) is not one-to-one due to the invariance of G ηη (V ) under the variation B δ (V ) = D η Ω + DηΩ following from the identity (5.11a).

Complete WZW-like action and gauge invariance
It is straightforward to extend the results on the NS-NS sector to all four sectors. We first note that the similarity transformation byF is trivial except for the NS-NS sector, and thus, the string product L we constructed reduces to in the NS-NS sector. Then, the differential equations (3.20) implies that the generating functional L| (0,0) (s,s, t) = Q + B| (0,0) (s,s, t) of string product in the NS-NS sector satisfies With this identification, the L ∞ triplet (η,η; L) can transform to the dual triplet (L η , Lη;L) , where π 1L = π 1ĝ Lĝ −1 = π 1 Q + Gπ 1b , (5.21) π 1b = π 1ĝ (b − B| (0,0) )ĝ −1 .

Conclusion and discussion
In this paper, we have revisited the type II superstring field theory with L ∞ structure and proposed an alternative method to construct string products, which is symmetric with respect to the left-and right-moving sectors. The symmetric method makes transparent not only the construction of string products but also the proof that the tree-level S-matrix agrees with that calculated using the first-quantization method. Another advantage of the symmetric construction is that it enables us to write down a WZW-like action through a map between the string fields of the two formulations, which was not possible with the previous (asymmetric) construction method. The complete WZW-like action of type II superstring field theory, which was the only missing piece, has now been constructed. We have completed the superstring field theory for all three complementary formulations, which allows us to use a convenient formulation depending on what we are studying. There is another interesting superstring field theory to consider: the (oriented) open-closed superstring field theory. This is the one that should be derived as type II superstring field theory on a non-trivial D-brane background but is worth constructing independently as it helps to calculate non-perturbative effects. In fact, the one based on the formulation with extra free field has already been constructed [25] and used for studying some non-perturbative effect [26][27][28][29][30]. It is interesting to construct it based on the open-closed homotopy algebra (OCHA) [31] and study the relation to the one in the WZW-like formulation.
which reduce to the conventional ones if two projected commutators are the same type. Similar In section 3, we show that how the generating functionals of 3-string and gauge products with specific picture number are determined. In this appendix, we illustrate the flow to be determined the 3-string (gauge) products with specific cyclic Ramond numbers, which are