Notes on the Wess-Zumino-Witten-like structure: $L_{\infty }$ triplet and NS-NS superstring field theory

In the NS-NS sector of superstring field theory, there potentially exist three nilpotent generators of gauge transformations and two constraint equations: It makes the gauge algebra of type II theory somewhat complicated. In this paper, we show that every NS-NS actions have their WZW-like forms, and that a triplet of mutually commutative $L_{\infty }$ products completely determines the gauge structure of NS-NS superstring field theory via its WZW-like structure. We give detailed analysis about it and present its characteristic properties by focusing on two NS-NS actions proposed by arXiv:1512.03379 and arXiv:1403.0940.


Introduction
In the previous work [1], we provided analysis of algebraic framework describing gauge invariances of superstring field theories, which we call the Wess-Zumino-Witten-like structure, and showed that there exist (alternative) WZW-like actions which are off-shell equivalent to A ∞ /L ∞ actions given by [2]. In this paper, we focus on the NS-NS sector and present details of analysis and its characteristic properties: Some implicit or missing parts and several important properties which remain unclear in [1] will be clarified. Through these analysis, we will see that a pair of nilpotent products, which we call an L ∞ triplet, induces WZW-like framework and thus ensures the gauge invariances of superstring field theories of [1][2][3][4][5][6][7][8][9][10][11][12]. 1 Formulation of superstring field theory has developed with understandings about how we can obtain gauge-invariant operator insertions into string interactions. Particularly, in [1][2][3][4][5][6][7][8][9], gauge invariant actions are constructed by operator insertions using first two of (ξ(z), η(z); φ(z)), fermionic superconformal ghosts. Insertions of η(z) are very simple because it has conformal weight 1 and is just a (nilpotent) current: On the basis of it, a gauge-invariant action which has a WZW-like form was proposed by Berkovits in an elegant way [3]. However, at the same time, these η-insertions enlarges the gauge symmetry of the theory, and two nilpotent gauge generators appear. Insertions of ξ(z) are rather complicated but also possible: Using it with nonassociative regulators for [13,14], Erler, Konopka, and Sachs constructed an A ∞ action [4]. This theory does not necessitate to extend gauge symmetry. However, to be gauge invariant, a state Φ appearing in the action must satisfy the constraint equation: η(z) Φ = 0 .
In the NS-NS sector, the situation becomes somewhat complicated: There exist three nilpotent generators of gauge transformations, and we have to impose two constraint equations. To see this extended gauge symmetry, let us recall the kinetic term of an NS-NS action, which was given by Berkovits based on his N = 4 topological prescription [5], where Q is the BRST operator and A, B ≡ A|c − 0 |B is the BPZ inner product with c − 0 ≡ 1 2 (c 0 −c 0 )-insertion. An NS-NS string field Ψ is total ghost number 0, left-moving picture number 0, right-moving picture number 0 state in the left-and-right large Hilbert space. 2 We write η,η, ξ, andξ for the zero modes of η(z),η(z), ξ(z), andξ(z), respectively. As one expects from its construction, it is invariant under the gauge transformations δΨ = η Ω +η Ω + Q Λ + . . . , (1.1b) where Ω, Ω, and Λ denote gauge parameter fields. We thus have three nilpotent gauge generators. When we include all interacting terms, three nonlinear extensions of these nilpotent generators 1 Potentially, it goes for [10][11][12] and other earlier proposals. See the footnote 8 in section 5 and section 6. Note that for the NS sector, its WZW-like structure is induced by a pair of commutative (cyclic) A∞/L∞ products. 2 In this paper, we often call the state space whose superconformal ghost sector is spanned by (ξ(z), η(z); φ(z)) and (ξ(z),η(z);φ(z)) as the left-and-right large Hilbert space H. Likewise, we call the state space consists of states belonging to the kernels of both η andη as the small Hilbert space HS. We always impose (b0 −b0)Ψ = (L0 −L0)Ψ = 0 for all closed superstring field Ψ.
appear [1,8,9]. Then, a full action has a Wess-Zumino-Witten-like form. To see constraints, it is helpful to consider the kinetic term 3 of the L ∞ action [2], An NS-NS string field Φ is total ghost number 2, left-moving picture number −1, and rightmoving picture number −1 state satisfying two constraint equations: η Φ = 0 andη Φ = 0. One can find that if and only if Φ satisfies constraints, the action has gauge invariance under δΦ = Q λ + . . . , (1.2b) where the gauge parameter λ also satisfies constraints: η λ = 0 andη λ = 0 . In [2], starting from Zwiebach's bosonic string products [15] and finding appropriate gauge invariant (ξ;ξ)-insertions, they constructed a suitable NS-NS string products which satisfy (cyclic) L ∞ relations, L NS,NS : Q , L 2 ( · , · ) , L 3 ( · , · , · ) , L 4 ( · , · , · , · ) , . . . , and gave a full action whose interacting terms satisfy L ∞ relations. When we include all interactions, to be gauge invariant (or to be cyclic), a state Φ appearing in the L ∞ action must satisfy two constraint equations: η Φ = 0 andη Φ = 0 . From these analysis, we achieve an idea that a triplet of three nilpotent objects determines the gauge structure of the NS-NS theory: By identifying two of them as constraints, one can construct a gauge invariant action.
Actually, on the basis of this idea, one can generalise or rephrase the construction of the L ∞ action as follows. Let ϕ be a dynamical string field. We first consider a state Φ ηη [ϕ], which will be a functional of ϕ, satisfying two constraint equations, Then, using this Φ ηη [ϕ], a gauge invariant action whose on-shell condition is given by can be constructed: All properties we need are derived from constraint equations for Φ ηη [ϕ]. The resultant action has a WZW-like form and one can prove its gauge invariance via a WZW-like manner without using specific properties of ϕ. As we will see in section 5, by taking ϕ = Φ of (1.2a), it reduces to the original L ∞ action of [2]. Namely, L ∞ formulation is completely described by a triplet of L ∞ product (η,η ; L NS,NS ). Likewise, every known actions for NS-NS superstring field theory potentially have their WZW-like forms described by their L ∞ triplets. For the most general form of the WZW-like structure and action, see section 6 and appendix A.
Furthermore, there exist a dual triplet for this (η,η ; L NS,NS ), which has the completely same information about the gauge structure of the NS-NS theory. Using this dual triplet, one can construct alternative WZW-like action, which is our main focus. First, in section 2, we find that the NS-NS superstring product L NS,NS has two dual L ∞ products: We will see that as well as η,η, or L NS,NS , these L ∞ products have nice algebraic properties. Then, one can consider the constraint equations provided by these L η and Lη : Using a state Ψ ηη [ϕ] satisfying these constraint equations, which will be a functional of some dynamical string field ϕ, we construct a gauge invariant action whose on-shell condition is It also has a WZW-like form and one can prove its gauge invariance without details of ϕ, which we explain in section 3. The L ∞ triplet (L η , Lη ; Q) determines this WZW-like structure and action. All necessitated properties can be derived from the constraint equations for Ψ ηη [ϕ], and we give two explicit forms of this key functional Ψ ηη [ϕ] in section 4. As we show in section 5, these WZW-like actions described by (L η , Lη ; Q) and (η,η ; L NS,NS ) are off-shell equivalent, which would be an interesting aspect of the WZW-like structure. Through these analysis, we would like to show that a triplet of mutually commutative L ∞ products completely determines the WZW-like structure of NS-NS superstring field theory, which is our main result.
In section 5, we present detailed properties of our WZW-like action. Firstly, we show that as well as that of the NS sector, our WZW-like action of the NS-NS sector has a single functional form which consists of single functionals Ψ ηη [ϕ] and elementally operators. Secondly, using this single functional form, we prove the equivalence of two constructions given in section 4. Thirdly, we clarify the relation to L ∞ theory: We find that our WZW-like action and the L ∞ action are off-shell equivalent. Then we give a short discussion about off-shell duality of equivalent L ∞ triplets. Finally, we discuss the relation to the earlier WZW-like theory proposed by [9]. With a brief summary of the WZW-like structure, we end with conclusion in section 6. In appendix A, we discuss the WZW-like action based on a general (nonlinear) L ∞ triplet (L c , Lc ; L p ) . We show that as well as other known WZW-like actions, it also satisfies the expected properties.

Two triplets of L ∞
In this section, we present two triplets of mutually commutative L ∞ products. The L ∞ triplet completely determines the WZW-like action: its form, gauge structure and all algebraic properties. As we will see, it gives the most fundamental ingredient of NS-NS superstring field theory because every known actions potentially have the WZW-like form.
We write the graded commutator of two co-derivations D 1 and D 2 as Note that it satisfies Jacobi identity exactly (without L ∞ homotopy terms): Original L ∞ triplet : (η,η ; L NS,NS ) As we explained, the constraint equations and the of-shell condition of the L ∞ action is described by a triplet of mutually commutative L ∞ -products (η,η ; L NS,NS ), which is the first one of two L ∞ triplets. The other L ∞ triplet is its dual and has the completely same information. Before considering its dual, let us recall how this L ∞ product L NS,NS was constructed. In [2], they introduced a generating function L(s,s ; t) for a series of L ∞ products, and required that L(0, 0; 0) ≡ Q and L(1, 1 ; 0) gives Zwiebach's string products of bosonic closed string field theory [15]. To relate this L(s,s ; t) with operator insertions, it is helpful to consider another generating function µ(s,s ; t) which has all information about operator insertions and will implicitly determine the gauge invariance. They called this µ(s,s ; t) as a gauge product. The NS-NS L ∞ products L NS,NS is included in this generating function L(s,s ; t). By imposing or solving the recursive equations, We write L n for the n-th product of L NS,NS as follows, Note that this µ(s,s ; t) has all information about gauge-invariant operator insertions and thus about how to construct the NS-NS products. Once we determine µ(s,s ; t), how to gaugeinvariantly insert ξ,ξ, and picture-changing operators, the NS-NS L ∞ product L NS,NS is given by the t = 1 value solution of the linear differential equation with the initial condition L NS,NS (t = 0) = Q, where µ(t) ≡ µ(s = 0,s = 0; t). Hence, we can solve it by iterated integration (with direction) and have the following expression, For brevity, we write G for this iterated integral with direction and write L NS,NS = G −1 Q G : It is a path-ordered exponential of coderivation µ, and thus a natural cohomomorphism of L ∞ algebras. In this form, L ∞ relations look trivial: (L) 2 = G −1 (Q) 2 G = 0. Using this form, we find two dual L ∞ products for L NS,NS and a dual of the L ∞ triplet (η,η ; L NS,NS ) . Using path-ordered exponential map G, one can obtain these dual L ∞ products as follows, One can quickly find that these products satisfy L ∞ relations (L α ) 2 = G (α) 2 G −1 = 0 because of (α) 2 = 0 for α = η, η , and have Q-derivation properties because of [[ L NS,NS , α ]] = 0 for α = η, η, which will provide nonlinear extensions of constraint equations. Hence, as well as (η,η ; L NS,NS ), the triplet of L ∞ -products (L η , Lη ; Q) is nilpotent and mutually commutative. Note that we found the correspondence of the commutativity: It is owing to an invertible cohomomorphism G, and thus the L ∞ triplet (L η , Lη ; Q) has the completely same information as (η,η ; L NS,NS ). We thus call (L η , Lη ; Q) as the dual L ∞ triplet. When G is cyclic in the BPZ inner product, this correspondence provides the equivalence of WZW-like actions governed by equivalent L ∞ triplets (See section 5.). In this paper, we write the n-th product of L α as follows,

Nilpotent relations and Derivation properties
For later use, we present explicit forms of algebraic relations satisfied by (L η , Lη ; Q) and some details of related properties. The dual L ∞ product L α for α = η, η satisfies L ∞ -relations, (L α ) 2 = 0. In terms of the n-th component, we have where σ runs over all possible permutations and (−) |σ| denotes the sign of the corresponding permutation. Likewise, L α Q + Q L α = 0 implies that we have Q-derivation properties , where the upper index of (−) A means the grading of A, namely, the total ghost number of A. The commutativity L η Lη + Lη L η = 0 provides The lowest relation of (2.3c) is just ηη +η η = 0, which would be very familiar. One can quickly find that the second lowest relation of (2.3c) is given by which is the matching of (crossed) Leibniz rules. Similarly, one can derive any higher relations of (2.3c). It may look a little complicated, but it is powerful and exact.

Maurer-Cartan element and Shifted L ∞
There is a special element of the L ∞ algebra of L α for α = η,η, which we call the Maurer-Cartan element for L α . As we will see, this element plays central role in WZW-like theory: It appears in the constraint equations, in the on-shell condition, and in the WZW-like action. Likewise, we often refer MC Q (A) ≡ QA as the Maurer-Cartan element for Q . There is an natural operation, a shift of the products, in L ∞ algebras. For any state A, the A-shifted products are defined by Note that the Maurer-Cartan element MC L α (A) behaves as the A-shifted 0-th product. One can check that with MC L α (A), the A-shifted products satisfy weak L ∞ relations: It implies that when given state A satisfies the Maurer-Cartan equation MC L α (A) = 0, then the A-shifted products exactly satisfy the L ∞ relations. Similarly, one can consider the shift of (2.3c) and obtain the weakly commuting relations of two A-shifted products: We thus find that two A-shifted products commute if and only if given state A satisfies both of the Maurer-Cartan equations MC L η (A) = 0 and MC Lη (A) = 0 . Using these relations, we prove the gauge invariance of the WZW-like action for NS-NS superstring field theory.

WZW-like action
Once we have a triplet of mutually commutative L ∞ -products (L η , Lη ; Q), by using these to provide constraints or on-shell equations, we can construct a gauge invariant action, which we explain in this section. We would like emphasis that one can achieve the gauge invariance without using detailed properties of a dynamical string field of the theory. All we need are two functional fields and their algebraic relations.

Algebraic Ingredients
In our WZW-like formulation of the NS-NS sector, two L ∞ -products L η and Lη are used to define constraint equations for (functional) fields, the other L ∞ -product Q is used to give the on-shell condition, and their mutual commutativity ensures the gauge invariance.
A functional field Ψ ηη [ϕ] satisfying these constraint equations plays the most important role, which we call a pure-gauge-like (functional) field. With this functional Ψ ηη [ϕ], the commutativity of L ∞ -products induces key algebraic relations, WZW-like relations. They make possible to prove the gauge invariance without details of the dynamical string field ϕ of the theory.

WZW-like functional field
Let Ψ η η = Ψ ηη [ϕ] be a Grassmann even, ghost number 2, left-moving picture number −1, and right-moving picture number −1 state in the left-and-right large Hilbert space: ηη Ψ ηη = 0. We call this Ψ ηη a pure-gauge-like (functional) field when Ψ ηη satisfies the constraint equations: In other words, Ψ ηη [ϕ] gives a solution of the Maurer-Cartan equations for the both dual products (2.1a) and (2.1b). Therefore, two Ψ ηη [ϕ]-shifted products again have L ∞ relations and commute each other. One can define two linear operators D η and Dη acting on any state A by and two bilinear products of any states A and B by Then, as the first identity of (2.4a), one can quickly find that D η and Dη are nilpotent, As the second identity of (2.4a), the bilinear product satisfies Liebniz rules, Likewise, as the first identity of (2.4b), we have the (anti-) commutation relation, and as the second identity of (2.4b), we can find matching of crossed Liebniz rules,

WZW-like relations
Let D be a derivation operator for both L ∞ -products L η and Lη : Namely, holds for any states A 1 , . . . , A n ∈ H . For example, since the BRST operator Q, a partial differential ∂ t with respect to any formal parameter t ∈ R, and the variation δ of the dynamical string field satisfy the Leibniz rule for these L ∞ -products L η and Lη, one can take D = Q, ∂ t , or δ. By acting this D on the constraint equations (3.1a) and (3.1b), we find D η (D Ψ ηη ) = 0 and Dη(D Ψ ηη ) = 0. Nilpotent properties (D η ) 2 = 0 and (Dη) 2 = 0 imply that with some (functional) state Ψ D [ϕ] belonging to the left-and-right large Hilbert space H, we have When the derivation operator D has ghost number g, left-moving picture number p, and right-moving picture p, the associated field Ψ D [ϕ] has the same quantum numbers: Its ghost number is g, left-moving picture number is p, and right-moving picture number is p.
We started with the L ∞ triplet (L η , Lη ; Q) and obtained the above algebraic ingredients by using two of it as constraints of theory. What is the use of the last L ∞ ? As we will see, its Maurer-Cartan equation gives a constraint describing the mass shell with the above Ψ ηη [ϕ] : Note that this (3.4) is also a special case of (3.3). Thus, the above three relations (3.1a), (3.1b), and (3.3) are fundamental, and we often call them as Wess-Zumino-Witten-like relations in NS-NS superstring field theory.

Action, Equations of motion, and Gauge Invariances
Let ϕ be a dynamical NS-NS string field and ϕ(t) be a path satisfying ϕ(0) = 0 and ϕ(1) = ϕ, where t ∈ [0, 1] is a real parameter. Once we obtain Ψ η η [ϕ] and Ψ D [ϕ] as functionals of given dynamical string field ϕ, we can construct a WZW-like action for NS-NS string field theory: the t-differential associated (functional) field. As we will see, using the variational associated (functional) field Ψ D [ϕ] with D = δ, the variation of this action is given by t-independent form: Then, the WZW-like relation (3.3) implies that the gauge transformations are given by The equation of motion is given by t-independent form One can quickly find these facts by using only WZW-like relations, (3.1a), (3.1b), and (3.3), which we explain in the rest. 5

Variation of the action
Let us recall basic properties of L ∞ -products and the BPZ inner product. The inner product A, B includes the c − 0 -insertion. 6 Hence, for D ′ = D η , Dη, or Q, we have 7 and for α = η,η, we can use the following cyclic and symmetric properties: In particular, note that with setting A = Ψ t and B = D η Ψ δ , the relation (3.2d) provides 5 These computations are similar to those of the earlier WZW-like action [9]. 6 In the left-and-right large Hilbert space, the inner product A, B vanishes unless the sum of A's and B's total ghost, left-moving picture, and right-moving picture numbers are 3, −1, and −1, respectively. 7 The prime denotes that we focus only on the BPZ property and we do not require the derivation property.
We prove that when we have WZW-like functional fields Ψ ηη [ϕ] and Ψ D [ϕ] which satisfy (3.3), our NS-NS action S ηη [ϕ] has topological t-dependence of (3.6). We carry out a direct computation of the variation of the action: .
For brevity, we omit ϕ(t)-dependence of functionals: We do not need it in computations. Using (3.8) in addition to (3.2) and (3.3), we find that the second term can be transformed into Ψ δ , ∂ t (QΨ ηη ) plus extra terms: Likewise, we find the first term of the variation becomes ∂ t Ψ δ , QΨ ηη plus extra terms: (3.11b) If and only if the sum of these extra terms vanishes, the action (3.5) has a topological tdependence. However, (3.10) ensure the cancellation of these extra terms, and we find Using ϕ(0) = 0 and ϕ(1) = ϕ, it concludes our proof of (3.6): In summary, for fixed L ∞ triplet (L η , Lη ; Q), we first consider a functional Ψ ηη satisfying constraint equations (3.1a) and (3.1b) defined by two of it, L η and Lη. Next, using this Ψ ηη , we estimate the WZW-like relation (3.3) and derive the other functional Ψ D , which gives a half input of the action. Lastly, using Ψ ηη , we consider the Maurer-Cartan element of the remaining L ∞ product Q, which provides the on-shell condition (3.4) and thus the other half of the action. Then, combining these, we can obtain a gauge invariant WZW-like action (3.5).

Two constructions
As we showed in section 3, when two states Ψ ηη [ϕ] and Ψ D [ϕ] satisfying (3.3) are obtained, one can find the WZW-like action (3.5). Therefore, the construction of actions is equivalent to finding explicit expressions of these functionals in terms of the dynamical string field ϕ.
In this section, we present two different expressions of these Ψ ηη , Ψ D using two different dynamical string fields Φ and Ψ. It gives two different realisations of our WZW-like action, which we call small-space parametrisation S ηη [Φ] and large-space parametrisation S ηη [Ψ].
Through these constructions, we also see that once we have Ψ ηη [ϕ] explicitly as a functional of ϕ, the other functional Ψ D [ϕ] can be derived from Ψ ηη [ϕ]. It would suggest that Ψ ηη is the fundamental ingredient in WZW-like theory, which we will discuss in the next section.

Small-space parametrisation: ϕ = Φ
We write Φ for a NS-NS dynamical string field belonging to the small Hilbert space: ηΦ = 0 andηΦ = 0. This Φ is a Grassmann even, total ghost number 2, left-moving picture number −1, and right-moving picture number −1 state.
As a functional of Φ, the pure-gauge-like field Ψ η η = Ψ ηη [Φ] can be constructed by (4.1) Note that co-homomorphism G preserves the total ghost, left-moving picture, and right-moving picture numbers and this Ψ ηη [Φ] has correct quantum numbers as a pure-gauge-like field. Thus, to show it, we have to check that (4.1) indeed satisfies the constraint equations (3.1a) and (3.1b). Recall that in coalgebraic notation, we can write (3.1a) and (3.1b) as follows: Since Ψ ηη [Φ] is given by using the group-like element, the following relation holds: Because of (2.1a) and (2.1b), one can quickly find that (4.1) satisfies which provides a proof that (4.1) gives a pure-gauge-like (functional) field. In the last equality, we used the properties of the dynamical string fields: η Φ = 0 andη Φ = 0. Thus, in this small-space parametrisation ϕ = Φ, it is the origin of all algebraic relations of WZW-like theory.
Similarly, as functionals of Φ, the associated (functional) field Ψ D = Ψ D [Φ] with D = ∂ t or D = δ can be constructed by and the associated (associated) field Ψ Q [Φ] can be given by where Q ξξ is a coderivation operation which we will define below.
Using the graded commutator of two coderivations D 1 and D 2 , Namely, the co-derivation G −1 D G commutes with both η andη . Hence, because of η-exactness andη-exactness, there exist a coderivation D ξξ such that Note that any derivation D can be lift to the corresponding coderivation, for which we also write D, because it is a linear map. For example, when D = ∂ t and D = δ, the above coderivation D ξξ is just the operations assigning ξξ D on each slot because of D G = G D . Using D ξξ and the properties of the dynamical string field, η Φ = 0 andη Φ = 0, we find Note that with (4.1), the linear operator D α for α = η, η can be written as We thus find that if we define the associated field Ψ D [Φ] by the following functional of Φ, which reduces to (4.2a) and (4.2b), the Wess-Zumino-Witten-like relation (3.3) indeed holds: Large-space parametrisation: ϕ = Ψ We write Ψ for a dynamical NS-NS string field which belongs to the left-and-right large Hilbert space: ηΨ = 0,ηΨ = 0, and ηηΨ = 0. This Ψ has total ghost number 0, left-moving picture number 0, and right-moving picture number 0.

Pure-gauge-like (functional) field Ψ ηη [Ψ]
Let us consider the solution Ψ ηη [τ ; Ψ] of the following differential equation, with the initial condition Ψ ηη [τ = 0; Ψ] = 0, where for any state A ∈ H, we define Note that (4.3) has the same form as the defining equation of a pure gauge field in bosonic string field theory [16], which is the origin of the name pure-gauge-like (functional) field. We check that this Ψ ηη [Ψ] satisfies (3.1a) and (3.1b). For this purpose, we set Because of the initial condition Ψ ηη [0; Ψ] = 0 of (4.3), it satisfies MC L α (0) = 0. Using (4.3) and (2.4a), we obtain the following linear differential equation where (−) |α| denotes −1 for α = η and +1 for α =η. The initial condition MC L α (τ ) = 0 provides that we have MC L α (τ ) = 0 for any τ , which ensures (4.4) indeed satisfies (3.1a) and (3.1b) and gives a proof that (4.4) is a pure-gauge-like (functional) field. By the iterated integral of (4.3), one can quickly find that a few terms of (4.4) are given by In this parametrisation, the properties of the dynamical string field ηη Ψ = 0 makes possible to use Ψ itself just like a gauge parameter of the nilpotent transformations generated by Lη and Lη, and to have a pure-gauge-like field Ψ ηη [Ψ] as a functional of Ψ. (Note that they are not the gauge transformations of our theory; it only reminds us those of other theories.)

Associated (functional) field Ψ D [Ψ]
We consider the following differential equation with the initial condition Ψ D [0; Ψ] = 0 up to D η -exact or Dη-exact terms. An associated (functional) field Ψ D [Ψ] is obtained by the τ = 1 value solution of (4.6), As D η -exacts and Dη-exacts does not affect in the first slot of (3.5), this Ψ D is determined up to these. To prove (4.7) satisfy (3.3), we set Note that iff we prove I(τ ) = 0 for any τ , it implies we have an appropriate associated field Ψ D [Ψ]. Using (3.2) and (4.3), we find From the third equal to the forth equal, we used the following identity: When Ψ D [τ ; Ψ] satisfies (4.6) up to D η -exacts and Dη-exacts, we have which is the same type of differential equations as (4.5), where {A, B} Ψ ηη is defined by The initial condition I(0) = 0 provides that we have I(τ ) = 0 for any τ , which gives a proof that (4.7) satisfies (3.3). For example, one can quickly find a few terms of Ψ t [Ψ(t)] are

On the D η -exacts and Dη-exacts
We found a defining equation (4.6) of Ψ D [Ψ]. Since it is up to D η -exacts and Dη-exacts, one can find another expression. Note that we have the following identity which provides another expression of (4.8): It ensures that as a defining equation of Ψ D [τ ; Ψ], we can also use The difference between (4.6) and (4.9) is just D η -exacts plus Dη-exacts, which does not affect WZW-like relations and the resultant action: It is just the gauge invariance generated by D η and Dη. Note also that since we have D η (τ )Dη(τ ) = −Dη(τ )D η (τ ), one may compute as (4.10) However, we have the following identity Comparing (4.8) and (4.10) with (4.9), we also find These term can appear or vanish in computations of ∂ τ I(τ ) = {Ψ, I(τ )} Ψ ηη (τ ) .

On the small associated fields
We constructed two functionals Ψ ηη [ϕ] and Ψ D [ϕ]. It is sufficient to give a WZW-like action explicitly. However, one can consider small associated (functional) fields defined by The WZW-like relation (3.3) provides that they satisfy the following relations One may prefer these because of the analogy with the NS sector. For example, using (−) D D Ψ ηη = D η Ψ Dη = DηΨ ηD with derivations D 1 and D 2 satisfying D 1 D 2 = (−) D 1 D 2 D 2 D 1 , one can find 8 On the basis of these functionals and relations, one can obtain another check of the gauge invariance of the action. For details in this direction, see appendix E of [1]. In the rest of this section, we explain how one can construct explicit forms of these as functionals of Φ or Ψ.

Small-space parametrisation
It is easy to obtain these in terms of Φ because the analogy with the NS sector exactly works. We find that small associated (functional) fields Ψ D η and Ψ ηD are given by where we used coderivations D ξ and Dξ such that It is consistent with (4.11). Note that D G = G D for D = ∂ t , δ, but Q G = G L NS,NS .

Large-space parametrisation
The situation becomes somewhat complicated in the large-space parametrisation. One can construct small associated (functional) fields Ψ Dη with J η (0) = 0 provides J η (τ ) = 0 for any τ . Likewise, we find Jη(τ ) = 0 for any τ . We can therefore obtain Ψ ηD and Ψ Dη satisfying (4.12) without using Ψ D and (4.11). When we start with Ψ D and (4.6), does D η Ψ D or D η Ψ D of (4.11) satisfy the above differential equation? The answer is yes; it gives correct solutions up to D η -exacts and Dη-exacts: Conversely, when we set D η A = Ψ ηD [Ψ] and start with these differential equations, can we derive the fact that this A satisfies (4.6)? The answer is again yes; we can re-derive (4.6) up to D η -exacts and Dη-exacts. Thus large and small associated fields both work well.
On the D η -exactness and Dη-exactness We can only specify the large associated (functional) field Ψ D up to D η -and Dη-exact terms, and these ambiguities do not contribute in the action. Therefore, in principle, one could set these any values by hand. We have operators F ξ and Fξ defined by which satisfy D η F ξ + F ξ D η = 1 and Dη Fξ + Fξ Dη = 1, respectively. 9 See also [1,[17][18][19]. These F ξ and Fξ consist of the pure-gauge-like (functional) field Ψ ηη [ϕ] and operators L η , Lη, η,η, ξ andξ. Using these pieces, one can construct Ψ D [ϕ] via Ψ ηD [ϕ] and Ψ Dη [ϕ] as follows, This Ψ D quickly satisfies (3.3), and thus, for example, one can check that (4.3) holds up to D η -exacts and Dη-exacts in large-space parametrisation. Note that as well as that of the NS sector, the form of F or F is not unique. In the NS-NS sector, this type of ambiguities of (4.13) can be crossed over between left-moving and right-moving sectors. Although F ξ and Fξ do not exactly commute under the above choice of (4.13) and the equality holds up to D η -exacts or Dη-exacts, we can have the strict commutativity and equality, which we see in the next section.

Properties
Single functional form As we found, two or more types of functional fields Ψ ηη [ϕ], Ψ D [ϕ] appear in the WZW-like action (3.5). Their algebraic relations make computations easy, but, at the same time, give constraints on these functional fields: the existence of many types of (functional) fields satisfying constraint equations would complicate its gauge fixing problem. It is known that in the NS sector, (alternative) WZW-like actions have single functional forms [19]. We show as well as NS actions, our NS-NS action S ηη [ϕ] has a single functional form which consists of the single functional Ψ ηη [ϕ] and elementally operators. It may be helpful in the gauge fixing problem.
Recall that in the left-and-right large Hilbert space H of the NS-NS sector, because of η ξ + ξ η = 1 andηξ +ξη = 1, the η-complex andη-complex are both exact: . . . Furthermore, since ηη +η η = 0, ηξ +ξ η = 0,η ξ + ξη = 0, and ξξ +ξ ξ = 0 hold, we have the direct sum decomposition of the large state space H as follows: Likewise, the existence of (4.13) satisfying D η F ξ + F ξ D η = 1 and Dη Fξ + Fξ Dη = 1 implies that the both D η -complex and Dη-complex are also exact in this large state space H: However, we saw (4.13) do not exactly commute each other. Does there exist a direct sum decomposition using these exact sequences? To achieve this, we consider One can quickly find that as well as (4.13), this F and its inverse F −1 also provide and it makes possible to have the following decompositions of the identity, Furthermore, now, these operators all are constructed from single F, we have which give us the desired direct sum decomposition of the large state space H : Since QΨ ηη = D η F ξ DηFξ(QΨ ηη ) and DηD η Ψ t = ∂ t Ψ ηη , using this F, we find

Equivalence of two constructions
In section 4, we presented two constructions of the WZW-like action. We explain these two actions are equivalent and derive a field redefinition connecting these. By construction, the equivalence of S ηη [Φ] and S ηη [Ψ] follows if we consider the identification It is trivial from the fact that the WZW-like action (3.5) has the single functional form (5.1) which consists of Ψ ηη and elementally operators. Since both actions have the same WZW-like structure, one can impose this identification and solve it as a field relation. See also [1,[19][20][21][22].

Field relation
Note that the identification of states (5.2) provides the identification of their Fock spaces Under the identification (5.2), by acting ∂ t , we have Note that these D η -exact or Dη-exact term does not contribute in the action. We thus consider The ambiguity appearing in (5.3) is completely absorbed into the gauge transformations: Since cohomomorphism G is invertible, we obtain the following field relation By using the WZW-like relation (3.3), it reduces to the following expression which can be directly derived from (5.2).

Relation to L ∞ theory
We write Φ for the small-space dynamical string field considered in section 4, and write Φ ′ for the dynamical string field of the L ∞ action proposed in [2]. As well as Φ, this Φ ′ belongs to the small Hilbert space: η Φ ′ = 0 andη Φ ′ = 0. Recall that using the small-space dynamical string field Φ, we constructed an action We will show that this S ηη [Φ] is exactly off-shell equivalent to the L ∞ action, Let Φ ′ (t) be a path connecting Φ ′ (0) = 0 and Φ ′ (1) = Φ ′ , where t ∈ [0, 1] is a real parameter. We write S L∞ [Φ ′ (t)] for the function given by replacing Φ ′ of (5.4) with Φ ′ (t), which satisfies Using coalgebraic notation and L NS,NS = G −1 Q G, we find In the second equality, we used the fact that G is a cyclic L ∞ -isomorphism compatible with the BPZ inner product. This just gives one realization of our WZW-like action (3.5) in small-space parametrisation. Hence, with the (trivial) identification of the string fields, we obtained a proof that the L ∞ action S L∞ [Φ ′ ] proposed in [2] [1,22] WZW-like reconstruction of L ∞ action In the L ∞ action, the L ∞ triplet is given by (η,η ; L NS,NS ). We thus consider a functional Φ ηη [ϕ] which satisfies two constraint equations defined by η andη, The existence of Φ D is ensured because η-complex andη-complex are both exact in the leftand-right large Hilbert space. Using Φ ηη [ϕ], we can consider the Maurer-Cartan element for the remaining L ∞ products L NS,NS : Note that there also exists an associated field Φ L [ϕ] such that According to our recipe, utilizing these ingredients, we can construct a WZW-like action 10 : One can check this action (5.7) has topological t-dependence and gauge invariance in the WZWlike manner. In particular, since η andη are linear L ∞ products, their shifted products are themselves. Thus, one can compute it with truncated versions of (3.11b) or (3.11a). We notice that if we set ϕ = Φ satisfying η Φ =η Φ = 0, it naturally induces a trivial form of the functional, Φ ηη [Φ] ≡ Φ, because of the triviality of η-andη-cohomology. Similarly, if we use ϕ = Ψ, it also implies Φ ηη [Ψ] ≡ ηη Ψ . While its small-space parametrisation is just the L ∞ action given by [2], its large-space parametrisation is just a trivial up-lift of small-space one.
Off-shell duality of L ∞ triplets As we mentioned, when G is cyclic in the BPZ inner product, (2.2) ensures not only the equivalence of L ∞ triplets but also the off-shell equivalence of resultant WZW-like actions. To see this, it is useful to consider the Maurer-Cartan-like element in the correlation function : Note that the above sum starts from n = 1, namely, two-inputs is the lowest. In the correlation function . . . , the BPZ cyclic property of G is just G( . . . ) = . . . . We thus obtain where MC L (A ′ ) is the Maurer-Cartan element for L NS,NS and A ′ is a state satisfying dual constraints for A . Note that when the state A satisfies MC L η (A) = MC Lη (A) = 0, the state Let us introduce a Grassmann variablet satisfying (t ) 2 = 0, and write . Using a measure factor d ≡ dt · ∂t , we can express the WZW-like action (3.5) as (5.10) 10 The NS-NS actions given by [10,12] also has this kind of WZW-like structure and WZW-like form of the action. Its L∞ triplet is quickly obtained by replacing L NS,NS of (η,η ; L NS,NS ) with the L∞ products appearing the action of [10,12] because of their small-space constraints.
Then, the equality (5.8) of the Maure-Cartan elements in the correlation function concludes the off-shell equivalence between our WZW-like action (3.5) based on the L ∞ triplet (L η , Lη ; Q) and the (WZW-likely extended) L ∞ action (5.7) based on the L ∞ triplet (η,η ; L NS,NS ) . Note that this off-shell equivalence does not necessitate detailed information about dynamical string fields. It is a powerful and significant consequence of the WZW-like structure.
Relation to the earlier WZW-like theory The L ∞ triplet of the earlier WZW-like action is given by (L −,NS ,η ; η). In this WZW-like NS-NS theory of [9], a solution of both Maurer-Cartan equations for L −,NS andη plays the most important role. We write ϕ ′ for a dynamical NS-NS string field and consider a functional For example, one can take D = η, ∂ t , and δ. Once the above pure-gauge-like (functional) field G L is given, we consider which we call the (earlier) WZW-like relation. Here, Ψ ′ D = Ψ ′ D [ϕ ′ ] is a functional of the dynamical string field, which has the same ghost, left-moving-picture, and right-moving-picture numbers as d. We call this Ψ ′ D [ϕ ′ ] satisfying (5.12) as an associated (functional) field. Note that Q G L , the first G L -shifted L −,NS , satisfies Q G Lη +η Q G L = 0 because of (5.11a) and (5.11b).
By its construction, we notice that the situation is parallel to the NS sector of heterotic string field theory [7]: Unfortunately, as [23], we do not have exact off-shell equivalence at all order but only have lower order equivalence. For example, by taking the following nonlinear partially gauge-fixing condition on ϕ ′ = Ψ ′ with the small-space string field Φ, the action (5.13) reduces to the L ∞ action based on their asymmetric construction of [2]. Hence, WZW-like actions (3.5) and (5.12) relate each other via field redefinitions, at least lower order.

Conclusion
We presented that a triplet of mutually commutative L ∞ products (L c , Lc ; L p ) completely determine the gauge structure of the WZW-like action. As we showed, every known NS-NS superstring field theory [1,2,9,10,12] potentially have the following WZW-like structure and WZW-like form of the action, which is one interesting result: By using two of it as constraint equations, One can prove its gauge invariance using the functional Ψ cc [ϕ] and algebraic relations derived from the mutual commutativity of the L ∞ triplet (L c , Lc ; L p ), 11 without using details of the dynamical string field ϕ. Since each know NS-NS action has its WZW-like form, one can say that to study its L ∞ triplet is equivalent to know the gauge structure of NS-NS superstring field theory. In this paper, we focused on two L ∞ triplets (L η , Lη ; Q) and (η,η ; L NS,NS ) which provide the L ∞ action of [2]. Particularly, we presented detailed analysis of the former and proved their off-shell equivalence with several general or exact results. We also discussed the relation to the earlier WZW-like action of [9]. We showed as well as the WZW-like action of the NS sector, our WZW-like action of the NS-NS sector has a single functional form, which may be a new approach to the gauge-fixing problem of WZW-like theory.
A General WZW-like action based on (L c , Lc ; L p ) In section 6, we gave the general WZW-like action based on a general L ∞ triplet (L c , Lc ; L p ) .
We would like to emphasise that the S cc [ϕ] gives a gauge invariant action for any L ∞ triplet (L c , Lc ; L p ) in the completely same way. In general, field redefinitions U drastically change the string vertices and state space in highly nontrivial manner. In terms of L ∞ algebras, it is just described by a L ∞ morphism between two L ∞ triplets, U : (L c , Lc ; L p ) → (L c′ , Lc ′ ; L p′ ). Hence, the general WZW-like action S cc [ϕ] is covariant under any string field redefinitions. Thus, as a gauge theory, it may capture general field theoretical properties of superstrings.