A∞/L∞ structure and alternative action for WZW-like superstring field theory

We propose new gauge invariant actions for open NS, heterotic NS, and closed NS-NS superstring field theories. They are based on the large Hilbert space, and have Wess-Zumino-Witten-like expressions which are the ℤ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{Z}}}_2 $$\end{document}-reversed versions of the conventional WZW-like actions. On the basis of the procedure proposed in arXiv:1505.01659, we show that our new WZW-like actions are completely equivalent to A∞/L∞ actions proposed in arXiv:1403.0940 respectively.


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Conventional WZW-like form. It was shown in [19] that Berkovits action can be written as a WZW-like form with WZW-like functionals A Q = A Q [Φ] of the dynamical string field Φ defined by The equation of motion is given by the t-independent form η A Q [Φ] = 0, and we can represent gauge transformations using nilpotent operators Q A Q [Φ] and η , where Λ ≡ e Φ Λ B e −Φ and Ω ≡ Ω B are redefined gauge parameters, and We write m 2 (A, B) for the star product of A and B. A significant feature of this WZW-like form of the action is that one can obtain all properties of the action by using not explicit forms of functionals A Q [Φ], A t [Φ] but only specific algebraical relations of these functionals, which we call WZW-like relations: where d = ∂ t , δ, η and the upper index of (−) A denotes the Grassmann parity of A. While the first relation provides the constraint for A Q [Φ], the second relation specifies its derivatives and the properties of the equation of motion. We call A d [Φ] an associated field. The first relation implies that the WZW-like functional A Q [Φ], which we call a puregauge-like field, 5  This construction is extended to more generic case: open NS superstrings with stubs, heterotic NS strings [19], and closed NS-NS strings [21]. As an example, we consider the Berkovits theory with stubs. The starting point is a set of generic bosonic open string products [35] M  However, when we compare this conventional WZW-like action and the A ∞ action given in [7], we notice that there exists a skew: A ∞ theory is characterised by the pair of equations equation for the ingredients of the conventional WZW-like theory, that of A ∞ theory is supplied by η. This skew is just the obstacle to obtain the off-shell equivalence of two theory by the naive way.
Alternative WZW-like form. It is known that Berkovits action can be also written as alternative WZW-like form with alternative WZW-like functionals A η = A η [Φ] of the dynamical string field Φ defined by Then, the equation of motion is given by the t-independent form and we can represent gauge transformations using nilpotent operators D * η and Q , where Λ ≡ Λ B and Ω ≡ e −Φ Ω B e Φ are redefined gauge parameters, and A significant feature of this alternative WZW-like form is that, as the conventional case, one can obtain all properties of the action by using only specific algebraical relations of alternative WZW-like functionals, which we also call WZW-like relations : , (1.4) where d = ∂ t , δ, Q. Therefore, we can say that this type of WZW-like theory also belongs to the category of so-called the WZW-like formulation. While the first relation provides the constraint for A η [Φ], the second relation specifies the properties of the equation of motion.
Note that this type of WZW-like theory, a dual version of the conventional WZW-like theory, is characterized by the pair of equations One can find that there is no skew between this type of WZW-like theory and A ∞ theory. Actually, as demonstrated in [29], by decomposing the NS superstring product M = Q + M 2 + M 3 + . . . given in [6] as M = G −1 Q G and by using a redefined string field JHEP01(2017)022 Therefore, we expect that one can construct WZW-like actions which are off-shell equivalent to A ∞ and L ∞ actions proposed in [7] as this type of WZW-like theories.
Main results of this paper. What is the starting point of this type of WZW-like theory? We can read it from the constraint for . For the Berkovits theory, it is given by the following A ∞ -product: which we call the dual products of M of [6]. We write D η 2 for −m 2 . Then, one can check that η + D η 2 satisfies the A ∞ -relations of η: We write them as follows: which give the starting point of our alternative WZW-like theory. Actually, one can construct the pure-gauge-like field A η [Φ] = (ηe Φ )e −Φ of (1.3) and the action (1.2) using this D η with the initial condition A η [τ = 0; Φ] = 0, where τ ∈ [0, 1] is a real parameter.
In this paper, we show that a nonassociative extended version of this construction gives our new WZW-like theory, which provides the equivalence of A ∞ /L ∞ formulation and WZW-like formulation. The starting point is the following A ∞ products satisfying the A ∞ -relations of η and derivation properties for d = ∂ t , δ, Q : One can construct these products by the dual products of M = G −1 Q G given in [7], namely D η ≡ G η G −1 . Then, using these dual products D η , we propose alternative WZW-like actions which are equivalent to A ∞ actions proposed in [7] as this type of WZW-like theories where now the A η -shifted dual product D η is given by , Ω .
Note that this WZW-like action for generic open NS strings just reduces to the alternative WZW-like form (1.2) of the Berkovits theory when we take the star product. We would like to emphasize that we do not need a specific form of A η [ϕ] or A t [ϕ] as a functional of given dynamical string field ϕ but only their properties (1.6) to show the properties of the action: its variation, equations of motion, gauge invariance, and so on. In appendix D, we will give explicit forms of two realizations of these functionals A η and A d using two different dynamical string fields: Ψ in the small Hilbert space and Φ in the large Hilbert space. Namely, the equivalence of A ∞ and WZW-like actions for open superstring field theory with stubs.
In the above, we take open NS theory as an example to grab a feature of our alternative WZW-like approach and its necessity. The details of the above open NS theory are discussed in appendix D. In the following sections, our main topic is "heterotic NS theory": we consider the L ∞ action and explain how its WZW-like properties arise. On the basis of the WZW-like structure naturally arising from L ∞ actions, we propose new gauge invariant actions for heterotic NS (and NS-NS strings in appendix E), as well as that for open NS strings with stubs which we introduced in (1.5). These actions are Z 2 -reversed versions of the conventional ones [19,21], and we show that they are completely equivalent to the A ∞ /L ∞ actions proposed in [7].
We expect that our new WZW-like actions would provide a first step to construct complete actions for heterotic and type II string field theories. 7 Actually, an action for open superstring field theory including the R sector was constructed [2] by starting with this type of WZW-like action: the R string field couples to the Berkovits theory for the NS sector gauge-invariantly on the basis of (not the conventional but) this type of WZW-like gauge structure. We would like to mention that although we expect that our new WZW-like actions are also equivalent to the conventional WZW-like actions, the all order equivalence of these has not been proven: we will show lower-order equivalence to the conventional WZW-like actions only. This paper is organized as follows. In section 2, after a brief review of the L ∞ formulation, we clarify a WZW-like structure naturally arising from it. We show that the L ∞ action can be written in our (alternative) WZW-like form: the functionals appearing in the JHEP01(2017)022 action satisfy alternative WZW-like relations, the Z 2 -reversed versions of the conventional WZW-like relations given in [19], which guarantees the gauge invariance of the action. The on-shell condition and the gauge transformation of the (alternative) WZW-like action are derived by using only the (alternative) WZW-like relations. We also see how the gauge parameters appearing in the WZW-like form are parameterised by that in the L ∞ form. Then, we conclude the L ∞ action gives one realization of the alternative WZW-like action S η [Φ] parametrized by the dynamical string field Φ in the small Hilbert space. In section 3, we provide another realisation of the alternative WZW-like action using the string field V in the large Hilbert space. The functionals satisfying our WZW-like relations can be defined by the differential equations which are the Z 2 -reversed versions of those given in [19]. Utilizing them, we construct a new gauge invariant action S η [V ] for heterotic NS string field theory. We derive the condition for the equivalence of the new action S η [V ] and the L ∞ action on the basis of the procedure demonstrated in [29]: these are different parameterizations of the same WZW-like structure and action. Then we also derive the relation between two dynamical string fields Φ and V from the equivalence condition. We end with some conclusions and discussions.
Basic facts and definitions of the coalgebraic notation of A ∞ /L ∞ algebras are summarized in appendix A. In appendix B, we derive a formula which is used in section 2.
In appendix C, we consider the trivially embedding of the string field of the L ∞ action belonging to the small Hilbert space into the string field in the large Hilbert space, and show the embedded action can also be written in the WZW-like form. Appendices D and E are devoted to the open NS and the closed NS-NS theories, respectively.

WZW-like structure from the L ∞ formulation
In this section, we clarify a WZW-like structure naturally arising from the L ∞ formulation for NS heterotic string field theory. After a brief review of the L ∞ formulation [7], we show that the L ∞ action can be written in our (alternative) WZW-like form. We see that the functionals appearing in the action satisfy alternative WZW-like relations, the Z 2 -reversed version of the conventional WZW-like relations in [19], which guarantees the gauge invariance of the action.
Preliminaries. The product of n closed strings is described by a multilinear map b n : H ∧n → H with degree 1 naturally induces a map from the symmetrized tensor algebra S(H) = H ∧0 ⊕ H ∧1 ⊕ H ∧2 ⊕ · · · to S(H) itself, called a coderivation. A map b n : H ∧n → H ′ with degree 0 also naturally induces a map from S(H) to S(H ′ ), called a cohomomorphism. Since it is convenient to write the functionals of the string field and the action in terms of them, we briefly introduce the rules of their actions here.

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The coderivation d n : S(H) → S(H) is naturally derived from a map d n : H ∧n → H with degree one. It act on Φ 1 ∧ · · · ∧ Φ N ∈ H ∧N ≥n ⊂ S(H) as A set of degree zero multilinear maps {f n : H ∧n → H ′ } ∞ n=0 naturally induces a cohomorphism f : S(H) → S(H ′ ), which acts on Φ 1 ∧ · · · ∧ Φ n ∈ H ∧n ⊂ S(H) as For definitions and their more details, see appendix A.

Construction of L ∞ -product and L ∞ action
In this subsection, we briefly review the construction of the NS superstring product L and the action S EKS [Φ] for heterotic string field theory in the L ∞ formulation originally given in [7]. The L ∞ relations holds if we define L[τ] as a solution of the differential equations with the initial condition p+2 are called gauge products. We can take any λ [0] as long as they carry correct quantum numbers: the (p + 2)-product λ p+2 carryies ghost number −2(p + 1) and picture number p + 1. The solution for the differential equations is given by the similarity transformation of Q,

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where G is an invertible cohomomorphism defined by the path-ordered exponential of the gauge products λ [0] [τ ] as follows Here ←(→) on P denotes that the operator at later time acts from the right (left). The differential equations hold since the path-ordered exponentials satisfy We may check directly the L ∞ relations: The cyclicity and the η-derivation properties follow from the suitable choice of λ [0] . Second, let us consider the cyclicity. For L to be cyclic, it is sufficient to choose λ [0] to be BPZ-odd, so that G −1 = G † : (2.10) Third, we check η acts as a derivation on L, namely the η-derivation properties A construction of the suitable gauge product λ [0] is given in [7]. It is helpful to consider a series of generating functions if we define L(s, τ ) as a solution for the following differential equations, Here λ(s, τ ) is a series of generating functions for the gauge products which provides η-derivation properties of L[τ] = L [0] [τ ], the s 0 part of (2.14). Hereafter we take τ = 1 and omit the argument: (2.18) Action for heterotic string field theory in the L ∞ formulation. A gauge-invariant action in the L ∞ formulation is constructed by [7], using the above products L = {L k } k≥1 . The dynamical string field Φ in the L ∞ formulation carries ghost number 2 and picture number −1 and belongs to the small Hilbert space H small : ηΦ = 0. The action is written as follows: where L 1 = Q and A, B is the BPZ inner product with the c − 0 = 1 2 (c 0 − c 0 )-insertion. Let us introduce the t-parametrized string field Φ(t) with t ∈ [0, 1] satisfying Φ(0) = 0 and Φ(1) = Φ, which is a path connecting 0 and the string field Φ in the space of string fields. Utilizing this Φ(t), the action can be represented in the following form: Here we use Φ(t) = ηξΦ(t), η-derivation property, and the cyclicity of L in order to obtain the form in which both ∂ t and ξ act on the first slot of the inner product. Note that this t-dependence is topological and it does not appear in the variation of the action, as we will see later. We can represent the action in the coalgebraic notation as follows: where e ∧Φ is the group-like element defined by π 1 is a projector from the symmetrized tensor algebra to the single-state space and ξ t is a coderivation 8 derived from the linear map ξ∂ t : Φ → ξ∂ t Φ. (See appendix A.) 8 We will define ξ d for the more general class of d later. For ∂t, definitions are equivalent.

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The variation of the action can be taken as 24) and then the equation of motion is given by Since L 2 = 0, the action is invariant under the gauge transformation where the gauge parameter λ is in the small Hilbert space and carries ghost number 1 and picture number −1.

Alternative WZW-like form of L ∞ action
The action in the L ∞ formulation can be transformed as See [32] for heterotic strings, and see also [29] for open strings. We find that the functionals appear in the action, and we will find that these functionals play important roles. One can show that, by introducing a certain set of products satisfying L ∞ -relations, satisfy the (alternative) WZW-like relations: which are Z 2 -reversed versions of conventional WZW-like relations in [19]. In this subsection, after defining these L ∞ products (2.30), which we call the dual L ∞ products for EKS's L ∞ products, we confirm a pair of fields (2.28) and (2.29) satisfy the WZW-like relations (2.31) and (2.32). We write L η for this set of dual L ∞ products of (2.30).

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Construction of the dual L ∞ product L η . The dual L ∞ products L η can be constructed using the cohomomorphism G which provides the NS heterotic string products L = G −1 Q G. 9 The product L η is defined as the similarity transformation of η: These L η carry odd degree, and the n-product L η n carries ghost number 3 − 2n and picture number n − 2. L η satisfiy the L ∞ -relations, which follow from its definition: The Q-derivation properties of L η follow from η-derivatioies property of L: The cyclicity of L η follows from that the gauge products are BPZ-odd: The dual L ∞ products L η is the generating function for (2.30). Namely, we define for any states B 1 , . . . , B n ∈ H. In terms of [B 1 , . . . , B n ] η , the dual products satisfy the following L ∞ relations, Q-derivation properties, and cyclicity: where (−) σ is the sign factor of the permutation {B σ(1) , . . . , B σ(n) }.
For any state A ∈ H, the A-shifted products of L η are defined by They also satisfy L ∞ relations if A is a solution for the Maurer-Cartan equation of L η : 9 See also appendix D or section 3.2 of [4] for generic properties of these types of products.

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Let Ψ η be a solution for this Maurer-Cartan equation of L η . In particular we write D η for the Ψ η -shifted 1-product as From the L ∞ -relation of the Ψ η -shifted L η , we find that D η is nilpotent, and that D η acts on the Ψ η -shifted 2-product [B 1 , B 2 ] η Ψη as a derivation, Note that the shifted products are BPS odd, which follows from that of L η , WZW-like relations. Let us confirm a pair of fields (2.28) and (2.29) satisfy the WZWlike relations, which can be represented as follows: The first relation (2.47) directly follows from the fact that Φ belongs to small space: We call Ψ η [Φ(t)] satisfying (2.47) the pure-gauge-like field. The second relation (2.48) can be confirmed similarly. The operator d which we focus on is the derivation on L η . For example, we can take d = Q, ∂ t , or δ. Their derivation property on L η leads to [[ G −1 d G, η]] = 0, and we can define the coderivation ξ d such that (2.50) Note that for the operator d which commutes with G, such as ∂ t and δ, ξ d is a coderivation derived form dξ. Then, utilizing this ξ d , the following relation holds:

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Then the action in the L ∞ formulation can be written in the alternative WZW-like form: (2.52) The variation of the action can be taken easily using the WZW-like relations (2.47) and (2.48), and the gauge invariances also follows from them, which can be seen in a similar (but Z 2 -reversed) manner to those in [19]. We will see them in the next subsection.

Variation of the action S η
Let us take the variation of the action. Note that in the computation here we use only the WZW-like relations (2.47) and (2.48), and we do not necessitate the explicit forms of the functionals Ψ η and Ψ d . Therefore, even if the functionals Ψ η and Ψ d have different forms, the variation of the action can be taken in the same manner as long as they satisfy the WZW-like relations. In this subsection, we write Ψ η (t) for Ψ η [Φ(t)] and so on for brevity. First, consider the variation of the integrand of (2.52), Utilizing the following relation following from (2.48), The second terms of (2.55) and (2.56) are canceled because of the cyclicity of the Ψ η -shifted L η . Then we find that the variation of the integrand of (2.52) becomes a total derivative of t: Integrating over t, the variation of the action is given by where the pure-gauge-like field Ψ η (t) and the associated field Ψ d (t) vanish at t = 0. Then the variation of the action becomes We find that the variation of the action does not depend on t, and therefore t-dependence is topological.

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Equations of motion. One can derive the on-shell condition from the variation of the action in the WZW-like form (2.59), For completion, let us discuss the equivalence of (2.60) and (2.25). The latter (2.25) can be transformed into the following form: is invertible, (2.60) and (2.25) are equivalent: Note that the overall factor π 1 G −1 · ∧ e ∧Ψη comes from the difference of the ξδΦ and Ψ δ .

Gauge transformations
It follows from the nilpotency of Q and D η that the WZW-like action is invariant under the following form of the gauge transformations, where Ω and Λ are gauge parameters belonging to the large Hilbert space, which carry ghost numbers 0 and 0, and picture numbers 1 and 0, respectively. In this subsection, we see how the gauge transformation (2.26) can be represented in the WZW-like form (2.63). Let us consider the associated field Ψ δ [Φ] = π 1 G ξ δ e ∧Φ = π 1 G (ξδΦ) ∧ e ∧Φ , with δΦ being the gauge transformation (2.26) in the L ∞ -fomulation: (2.64) We find Ψ δ [Φ] can be transformed as follows: While the first two terms of (2.65) correspond to (2.63) with the gauge parameters Ω = Ω[λ, Φ] and Λ = Λ[λ, Φ] parameterized by λ,  Trivial gauge transformation. Trivial gauge transformation is a transformation proportional to the equations of motion. Schematically, it is of the following form, and its gauge invariance follows from the symmetric property of µ ji , They are no physical significance, but in general they may appear in the algebra of the nontrivial gauge transformations, and in the context of the Batalin-Vilkovisky quantization [37,38] it is convenient to consider them. 10 In our case, by the almost same computation with [31], when A + B + C = even, cyclicity of the cohomomorphism H is written as follows: The derivation is in appendix B. We take H = G, A = ξλ, which is even, and B = C = π 1 Le ∧Φ , which are odd. The invariance under the trivial gauge transformation (2.68), 11 follows from the symmetric property:

Alternative parameterisation for closed NS string field theory
Let ϕ be a some dynamical string field and t ∈ [0, 1] be a real parameter. We write ϕ(t) for a path satisfying ϕ(t = 0) = 0 and ϕ(1) = ϕ. In section 2.3, we saw that the gauge invariance of the action is provided by only the algebraic relations (2.47)

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are obtained, the gauge invariant action can be constructed as This form of the action is the Z 2 -reversed version of that in [19], which we call the alternative WZW-like form. We found that by taking ϕ = Φ, (2.28), and (2.29), the L ∞ action S EKS [Φ] gives one realisation of this WZW-like action, In this section we provides another realisation of these functionals, which is parameterized by the string field V in the large Hilbert space. We first see the pure-gauge-like and the associated functional fields can be defined by the differential equations, which are the Z 2 -reversed versions of the construction in [19]. Then, the WZW-like action parameterized by the string field V , is given in terms of them. The equivalence of these actions in the different parameterizations is shown by the almost same procedure performed in [29].

Large space parameterisation
Let V be a dynamical string field which belongs to the large Hilbert space and carries ghost number 1 and picture number 0. In this subsection, we provides the another paramaterization of the pure-gauge-like field Ψ η = Ψ η [V ] and the associated fields . A set of differential equations which are the Z 2 -reversed version of those in [19] give these parameterizations so that WZW-like relations (2.47) and (2.48) hold. Utilizing these functionals, a new gauge invariant action for the string field V is constructed in the (alternative) WZW-like form.
To derive the differential equation which defines Ψ d [V ], let us introduce and consider its differentiation by τ : We define the functional field Ψ d [τ ; V ] by the differential equation The variation of the action can be taken in the same manner as that in section 2.3, 16) and the equation of motion can be read off from it, Since D η and Q are nilpotent, the action is invariant under the gauge transformations, where Ω and Λ are gauge parameters belonging to the large Hilbert space, which carry ghost numbers 0 and 0 and picture numbers 1 and 0, respectively.

Equivalence of the actions in the different parameterizations
In this subsection, we show the equivalence of our new action S η [V ] and L ∞ action S EKS [Φ], by identifying the pure-gauge-like functional fields Ψ η [V ] and Ψ η [Φ]. We also derive the relation of two dynamical string fields V and Φ from this identification. Let us consider the identification of the pure-gauge-like fields in the both parametarisations: The D η -exact terms in the associated fields do not affect to the WZW-like relation Recall that there exists the arbitrariness to add D η -exact terms in the associated fields. Besides, since QΨ η is D η -exact, the difference between Ψ t [Φ] and Ψ t [V ] does not contribute to the action. Then, two actions are shown to be equivalent: The correspondence of the gauge parameters is given by ( we can rewrite the action as the following single functional form: Since G is invertible, by acting G −1 and by projecting by π 1 , the condition of the field corresponding relation which provides S η [V ] from the L ∞ -action S EKS [Φ] is obtained: Expanding it in powers of V , it reads where λ   2 (ξΦ, Φ))+3λ If we choose the form of G as given in [7], it reads

Conclusions and discussions
In this paper, we clarified a WZW-like structure naturally arising from the L ∞ formulation: the pure-gauge-like field Ψ η [Φ] ≡ π 1 G(e ∧Φ ) and the associated fields Ψ d [Φ] ≡ π 1 G(ξ d e ∧Φ ) satisfy the alternative WZW-like relations, which are Z 2 -reversed versions of the conventional WZW-like relations given in [19].  [29]. The direct relation between two dynamical string fields Φ and V , a field redefinition, can be derived from this equivalence with partially gauge-fixing of V or trivial uplift of Φ.
Towards the equivalence to the conventional WZW-like action. Although we expect the equivalence between our (alternative) WZW-like action S η [V ] and Berkovits-Okawa-Zwiebach's (conventional) WZW-like action S BOZ [V c ], their relation remains to be understood. We write V for the dynamical NS heterotic string field of the alternative WZW-like theory, and write V c for that of the conventional WZW-like theory as follows: Both string fields V and V c belong to the large Hilbert space. At least, perturbatively, one can check their equivalence: we set , which implies that the first nontrivial order is κ 2 , namely the quartic interaction. Let us check how the equivalence can be shown in this order. It would be crucial that the 3-products L η 3 and L B 3 are made from the same gauge product λ [1] 3 : Utilizing them, the quartic interactions in both actions can be written in term of λ The difference between the quartic interactions comes from the second terms of (4.1) and (4.2): which can be compensated by the following off-shell field redefinition, To prove their all-order equivalence, it would be helpful to characterize the conventional WZW-like action in terms of the gauge products λ BRST operator which plays a key role in conventional WZW-like formulation, can be written as Q G = E −1 V QE V using a linear map E V [32]. It may provide a solution of this problem to understand the relation between two cohomomorphisms E V and G.
Ramond sector of closed superstring fields. The concept of the WZW-like structure based on the dual product L η will play crucial roles in construction of the complete action of heterotic string field theory including both NS and R sectors. As [2], our WZW-like action S η [V ] will provide a good starting point of the construction of complete actions. 12 It is expected that as demonstrated in [4] for open superstrings, the WZW-like structure including the Ramond sector would be written by the L ∞ products L η = G η G −1 which are dual to the L ∞ products of [8]: It is also expected that the action for closed NS-R and R-NS strings can be constructed in the same manner as that for heterotic string. For closed R-R strings, it is not clear whether or how the kinetic term can be constructed with no constraint yet. If the kinetic term can be constructed, it may be possible to construct the complete action of type II string on the basis of the concept of the WZW-like structure arising from the dual product L η .

A Basic facts of A ∞ and L ∞
In this section, we give a short review of coalgebraic description for A ∞ /L ∞ algebras. For more details, see [40][41][42] or some mathematical manuscripts.
Coalgebras: tensor algebra T (H) and its symmetrization S(H). Let C be a set. Suppose that a coproduct ∆ : C → C ⊗ ′ C is defined on C and it is coassociative Then, the pair (C, ∆) is called a coalgebra. We write T (H) for a tensor algebra of a graded vector space H: One can define a coassociative coprduct ∆ :

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with ⊗ ′ ≡ ⊗, where Φ 1 ⊗ . . . ⊗ Φ n ∈ H ⊗n ⊂ T (H). Then, the pair (T (H), ∆) gives a coalgebra. As well as T (H), its symmetrization S(H) also gives a coalgebra. Recall that the symmetrized tensor product ∧ for Φ 1 , Φ 2 ∈ H is defined by and it satisfies the following properties for Φ 1 , . . . , Φ n ∈ H: We write H ∧n for the vector space spanned by n-fold symmetrized tensor Φ 1 ∧ · · · ∧ Φ n , and S(H) for the symmetrization of T (H), which is called the symmetrized tensor algebra S(H): Recall that from a n-fold multilinear product b n of Φ 1 , . . . , Φ n ∈ H, one can define a linear map on H ⊗n , which we write b n : H ⊗n → H, by where Φ 1 ⊗ · · · ⊗ Φ n ∈ H ⊗n . Let A : H ⊗k → H ⊗l and B : H ⊗m → H ⊗n be multilinear maps. One can naturally define a product of these maps A ⊗ B : H ⊗k+m → H ⊗l+n by The symmetric tensor product of two multilinear maps A : H ∧k → H ∧l and B : H ∧m → H ∧n , A ∧ B : H ∧k+m → H ∧l+n , can also be defined naturally by and b n vanishes when acting on H ∧N ≤n . For example, we find that b 1 acts as follows: and acts as b 0 :

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Multilinear maps as a cohomomorphism. Given two coalgebras C, C ′ , a cohomomorphism f : C → C ′ is a map of degree zero satisfying A set of degree zero multilinear maps {f n : H ∧n → H ′ } ∞ n=0 naturally induces a cohomorphism f : S(H) → S(H ′ ), which we denote as f = {f n } ∞ n=0 . Its action on Φ 1 ∧ · · · ∧ Φ n ∈ H ∧n ⊂ S(H) is defined by (A.23) Its explicit actions are given as follows:

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We denote the collection of the multilinear maps {L k } k≥0 also by L. In particular, if L 0 = 0, (H, L) is called an L ∞ -algebra.
In the case of an L ∞ -algebra, the part of (A.31) that correspond to an n-fold multilinear map H ∧n → H is given by We can act it on B 1 ∧ B 2 ∧ . . . ∧ B n ∈ H ∧n to get the L ∞ relations for the multilinear maps {L k }: Projector and group-like element. We can naturally define a projector π 1 : S(H) → H whose action on Φ ∈ S(H) is given by Note that π 1 acts trivially on H and commutes with one-coderivations. Let H 0 be the degree zero part of H. The following exponential map of Φ ∈ H 0 , is One of the important property of a cohomomorphism is its action on the group-like element: We can see that the cohomomorphisms preserves the group-like element: f(e ∧Φ ) = e ∧Φ ′ . Utilizing the projector and the group-like element, the Maurer-Cartan element for an L ∞ -algebra (H, L) is given by The Maurer-Cartan equation for an L ∞ -algebra (H, L) is given by F Φ = 0, which correspond to the on-shell condition in string field theory based on the L ∞ -algebra(H, L).

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B Derivation of (2.71) For cyclic cohomomorphism H, the following relation holds: where π 2 is the projector S(H) → H ∧ H, and ∇ is the product of the tensor algebra (T (H), ∇). Let us consider its action The left hand side vanishes since π 2 is acting on H ∧n≥3 . For the right hand side, the first term of (B.2) becomes Hereafter we assume A + B + C = even. The inner product in the large Hilbert space is and the 1st term and 4th term of (B.3) cancel since a, b = (−) (a+1)(b+1) b, a . Now, gathering the contribution from all terms of (B.2), and utilizing the identities one can obtain the BPZ property for H: Note that in the computation some terms cancel by the symmetric property of the inner product.

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C Embedding to the large Hilbert space of NS closed string In this appendix, we consider the trivial embedding of the NS closed string field Φ in the L ∞ -formulation into the string field V in the large Hilbert space, and see that the trivially embedded action is also WZW-like form. In other words, we provide a parameterization of the pure-gauge-like field and the associated fields by another large space string field V . In particular, we discuss the relation between the gauge transformations in two representations.
We define the trivial embedding by replacing ξΦ to V which is a string field belonging to the large Hilbert space, so that under the partial-gauge-ficing condition ξ V = 0, the trivial solution V = ξΦ reproduce the L ∞ -formulation parameterized by Ψ, the string field in the small Hilbert space. The action for V is given by One can confirm the functionals appearing in the action satisfy the WZW-like relations: The first relation follows from the projection invariance e ∧η V = ηξe ∧η V as (2.49). To check the second relation, consider where we used Since the second term vanishes when acted by D η , the same computation as (2.51) can be applied to show In fact, the embedding procedure ξΦ → V has an ambiguity: one can consider the replacing ξΦ = ξηξΦ → ξη V . However, the difference of Ψ d produced by the difference of the procedure is only D η -exact term as seen in (C.6), which does not affect to the WZWlike relation (−) d dΨ η = D η Ψ d . Besides, since QΨ η is D η -exact, the D η -exact part of Ψ t does not affect the action, that is, the action does not depend on this ambiguity. Here we fix this ambiguity by requiring the invertibility of Ψ d=δ,∂t as functions of d V [31].

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For derivation d which does not commute with G, such as Q, where To check the WZW-like relations, utilizing Since the second term vanishes when D η acts, the same computation as (2.51) can be applied to show Thus, the action is written in the WZW-like form, where ω and λ are the gauge parameters belonging to the small Hilbert space, which carry ghost numbers 1 and 1, and picture numbers 0 and −1, respectively. The minus sign is a convention in which ηδ V becomes δΦ of (2.24). In the following, we will discuss how this gauge symmetry can be written in the WZW-like form: (C.14) More generally, the second term can be written by −π1ξ Q (λ∧e ∧η V ). The representation above is only a choice of ξ d . However, the difference of the choices becomes η-exact term and it can be absorbed into −ηξω.

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The computation here is parallel to (2.65). Since ∆ T is a trivial gauge transformation as in section 2.4, ∆ T [λ, η V ] = π 1 G ξλ ∧ π 1 L(e ∧η V ) ∧ e ∧η V , (C. 16) we conclude that the gauge transformations of the trivial embedded theory (C.13) are written in the WZW-like form (C.14), where the gauge parameters Ω and Λ are parameterized as In particular, one can find that the gauge transformations generated by η and Q do not mix: the gauge parameters λ and ω of (C.13) are related to Λ and Ω of (C.14) respectively, as seen in (C.17) and (C.18).

D Open NS superstrings with stubs
In this section, we construct a new action for generic open NS string field theory, which we call alternative WZW-like action S η . After defining the dual A ∞ products and explaining its WZW-like structure, we give two realizations of this type of WZW-like action using two different dynamical string fields Φ and Ψ in the large and small Hilbert spaces, respectively.
Dual A ∞ -products and derivation properties. Let η be the coderivation constructed from η, which is nilpotent η 2 = 0, and let a be a nilpotent coderivation satisfying aη = −(−) aη ηa and a 2 = 0. Then, we assume that G −1 : (H, a) → (H S , D a ) is an A ∞ -morphism, where H is the large Hilbert space, H S is the small Hilbert space, and D a ≡ G −1 a G. Note that D a is nilpotent: D 2 a = ( G −1 a G)( G −1 a G) = G −1 a 2 G = 0. For example, one can use Q, Q + m 2 | 2 , and so on for a, and various G appearing in [6][7][8] for G. Suppose that the coderivation D a also commutes with η, which means Then, we can introduce a dual A ∞ products D η defined by Note that the pair of nilpotent maps (D η , a) have the same properties as (D a , η): We can quickly find when the A ∞ products D a commutes with the coderivation η as (D.1), its dual A ∞ product D η and coderivation a also satisfies (D.3) as follows

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In this paper, as these coderication a and A ∞ -morphism G, we always use a ≡ Q and a gauge product G given by [7]. Therefore, the original A ∞ product is equal to the Neveu-Schwarz products M of open stings with stubs given [7], and the dual A ∞ products is always given by The symbol D η always denotes (D.4) in the rest. The dual A ∞ products D η consists of a linear map η and a set of multilinear products {D η n } ∞ n=2 . Now the bilinear product D η 2 just equals to the star product −m 2 because we take G of [7]. We write for higher products of D η . Then, the Maurer-Cartan element of D η = η − m 2 + . . . is given by where A is a state of the large Hilbert space H and π 1 is an natural 1-state projection onto H. Hence, the solution of the Maurer-Cartan eqiation D η (1 − A) −1 = 0 is given by a state A η satisfying ηA η − m 2 (A η , A η ) + · · · = 0, or equivalently, Shift of the dual A ∞ products D η . We introduce the A η -shifted products [B 1 , . . . , B n ] η Aη defined by Note that higher shifted products all vanish [B 1 , . . . , B n>2 ] η Aη = 0 when higher products of

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When A η gives a solution on the Maurer-Cartan equation of D η , these A η -shifted products also satisfy A ∞ -relations, which implies that the linear operator D η becomes nilpotent. We find

D.1 Alternative WZW-like relations and action S η [ϕ]
We call a state A η a pure-gauge-like (functional) field when A η satisfies and has ghost-and-picture number (1| − 1), odd Grassmann parity, and even degree. Let d is a derivation operator commuting with η and let (d g |d p ) be its ghost-and-picture number. We call a state A d a associated (functional) field when A d satisfies and has ghost-and-picture number (d g − 1|d p + 1). Let ϕ be a dynamical string field, t ∈ [0, 1] be a real parameter, and ϕ(t) be a path satisfying ϕ(0) = 0 and ϕ(1) = ϕ. Once a pure-gauge-like (functional) field A η = A η [ϕ] and an associated (functional) field A d = A d [ϕ] are constructed as functionals of given dynamical string field ϕ, one can construct a gauge invariant action (D.9) We write A t [ϕ] for the associated field A d [ϕ] of the parameter differential d = ∂ t . One can check that the t-dependence is "topological". Namely, the variation of the action is given by which is independent of the real parameter t. Since QA η = −D η A Q and (D η ) 2 = Q 2 = 0, we find that δS = 0 with the gauge transformations

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Using this coderivation ξ d , we can construct an associated (functional) field A d [Ψ]. Note that the response of d acting on the group-like element of A η [Ψ] = G(1 − Ψ) −1 is given by .
An associated (functional) field of d is given by .
Picking up the relation on H, or equivalently acting π 1 on this relation on T (H), we obtain which is the simplest case of (−) d dA η [Ψ] = π 1 D η -exact term.

Associated field
The τ = 1 value of the functional A d [τ ; Φ] of the dynamical string field Φ, which we write

E NS-NS sector
In this section, starting from the NS-NS superstring L ∞ -products, we clarify a WZW-like structure and give alternative WZW-like form S η η [ϕ] of the action for NS-NS string field theory.
Although it is sufficient to consider the above (E.3a) and (E.3b), it would be helpful to consider their small associated (functional) fields Ψ ηd = Ψ ηd [ϕ] and Ψ d η = Ψ d η [ϕ] which are defined by has the same ghost, left-moving picture, and right-moving picture numbers as d, this Ψ ηd [ϕ] or Ψ d η [ϕ] has the same as "d plus η" or "d plus η" respectively.

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Likewise, we find the first term of the variation becomes ∂ t Ψ δ , QΨ η η plus extra terms: δΨ t , Q Ψ η η = − D η δΨ t , Ψ Q η (E.10) From the third line to the forth line, we used (E.8b) with (E.3c). If and only if the sum of these extra terms vanishes, the action has a topological t-dependence. However, (E.2b) provides the cancellation of these extra terms. Therefore, using ϕ(0) = 0 and ϕ(1) = ϕ, we obtain . As a result, this form of the variation of (E.4) ensures that it is invariant under (E.5) and the equations of motion is given by (E.6) because of (E.3b).