Notes on the Wess-Zumino-Witten-like structure: L ∞ triplet and NS-NS superstring field theory

Journal of High Energy Physics, May 2017

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 ∞ 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 [1] and [2].

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Notes on the Wess-Zumino-Witten-like structure: L ∞ triplet and NS-NS superstring field theory

Received: March Notes on the Wess-Zumino-Witten-like structure: triplet and NS-NS superstring eld theory Open Access 0 1 2 c The Authors. 0 1 2 0 Yukawa Institute for Theoretical Physics, Kyoto University 1 Na Slovance 2 , Prague 8 , Czech Republic 2 Institute of Physics, the Czech Academy of Sciences In the NS-NS sector of superstring eld 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 products completely determines the gauge structure of NS-NS superstring eld theory via its WZW-like structure. We give detailed analysis about it and present its characteristic triplet; and; eld; theory; String Field Theory; Superstrings and Heterotic Strings - properties by focusing on two NS-NS actions proposed by [1] and [2]. Contents 1 Introduction 2 3 4 Two triplets of L1 WZW-like action Two constructions A General WZW-like action based on (Lc; Lc~ ; Lp) Introduction alent to A1=L 1 actions given by [2]. In this paper, we focus on the NS-NS sector and 1 triplet, theories of [1{12].1 rst two extend gauge symmetry. However, to be gauge invariant, a state appearing in the action must satisfy the constraint equation: H (z) = 0 . S[ ] = where Q is the BRST operator and hA; Bi hAjc0 jBi is the BPZ inner product with c~0)-insertion. An NS-NS string eld is total ghost number 0, left-moving denote gauge parameter elds. We thus have three nilpotent gauge form. To see constraints, it is helpful to consider the kinetic term3 of the L 1 action [2], S[ ] = 1, and = 0 and An NS-NS string eld is total ghost number 2, left-moving picture number right-moving picture number 1 state satisfying two constraint equations: = 0. One can nd that if and only if satis es constraints, the action has gauge invariance under = Q where the gauge parameter also satis es constraints: = 0 and ~ = 0 . In [2], starting 1 relations, and gave a full action whose interacting terms satisfy L 1 relations. When we include all interactions, to be gauge invariant (or to be cyclic), a state appearing in the L must satisfy two constraint equations: = 0 and ~ = 0 . From these analysis, we 1 action state space consists of states belonging to the kernels of both always impose (b0 = (L0 = 0 for all closed superstring eld and ~ as the small Hilbert space HS. We 3Note that these two free actions are equivalent each other with linear partial gauge xing or trivial up-lift. For example, recall that of (1.1a) is obtained by an embedding of of (1.2a) such as ~ = . invariant action. 1 action as follows. Let ' be a dynamical string eld. We rst consider a state ~['], which will be a functional of ', satisfying two constraint equations, Then, using this ~['], a gauge invariant action whose on-shell condition is given by ~['] = 0 ; ~['] = 0 : ~['{] = 0 taking ' = of (1.2a), it reduces to the original L is completely described by a triplet of L 1 product ( ; ~ ; LNS;NS). Likewise, every known 1 action of [2]. Namely, L 1 formulation by their L section 6 and appendix A. 2, we nd that the NS-NS superstring product LNS;NS has two dual L 1 products: We will see that as well as , ~, or LNS;NS, these L Then, one can consider the constraint equations provided by these L and L~ : 1 products have nice algebraic properties. = 0 ; ~['{] ~ = 0 : Using a state ~['] satisfying these constraint equations, which will be a functional of some ~['] = 0 : which we explain in section 3. The L 1 triplet (L ; L~ ; Q) determines this WZW-like struc~['], and we give two explicit forms of this key functional ~['] in section 4. As which is our main result. single functional form which consists of single functionals ~['] and elementally operators. given in section 4. Thirdly, we clarify the relation to L 1 theory: we nd that our WZW-like action and the L 1 action are o -shell equivalent. Then we give a short discussion about o -shell duality of equivalent L 1 triplets. Finally, we discuss the relation to the earlier on a general (nonlinear) L 1 triplet (Lc; Lc~ ; Lp) . We show that as well as other known WZW-like actions, it also satis es the expected properties. Two triplets of L1 In this section, we present two triplets of mutually commutative L 1 products. The L We write the graded commutator of two co-derivations D1 and D2 as D1; D2 D1 D2 Note that it satis es Jacobi identity exactly (without L 1 homotopy terms): D1; [[ D2 ; D3 ]] + ( )D1(D2+D3) D1; [[ D2; D3 ]] Original L1 triplet: ( ; ~ ; LNS;NS). the of-shell condition of the L L1-products ( ; ~ ; LNS;NS), which is the As we explained, the constraint equations and 1 action is described by a triplet of mutually commutative rst one of two L 1 triplets. The other L let us recall how this L 1 product LNS;NS was constructed. In [2], they introduced a generating function L(s; s~ ; t) for a series of L 1 products, and required that L(0; 0; 0) determine the gauge invariance. They called this (s; s~ ; t) as a gauge product. The NSsolving the recursive equations, ; (s; s~ ; t) ; ~ ; (s; s~ ; t) ; (s; s~ ; t), and vice versa. This L(s; s~ ; t) is a series of L 1 products with operator insertions satisfying (s; s~ ; t) can be determined by solving the recursive equation, L(s; s~ ; t) ; (s; s~ ; t) ; which ensures L superstring L (s; s~ ; t) , the NS-NS L(s = 0; s~ = 0; t = 1) : We write Ln for the n-th product of LNS;NS as follows, Ln A1 ; : : : ; ; An 1 LNS;NS A1 ^ : : : ^ An : (s; s~ ; t), how 1 product LNS;NS(t) = LNS;NS(t) ; (t) (s = 0; s~ = 0; t). Hence, we LNS;NS = P! exp 1 algebras. In this form, L form, we nd two dual L 1 products for LNS;NS and a dual of the L 1 triplet ( ; ~ ; LNS;NS) . with two L two dual L these dual L 1 products 1 products as follows, Gb ~ Gb 1 One can quickly nd that these products satisfy L because of ( )2 = 0 for = Gb ( Gb 1 Q Gb) G 1 = ( Gb 1 Q Gb) Gb 1 = because of [[ LNS;NS; ]] = 0 for commutativity: = 0 = 0 ; = ; ~) : It is owing to an invertible cohomomorphism Gb, and thus the L 1 triplet (L ; L~ ; Q) has equivalence of WZW-like actions governed by equivalent L this paper, we write the n-th product of L as follows, 1 triplets (See section 5.). In [A1; : : : ; An] := 1 Gb ( = ; ): Nilpotent relations and derivation properties some details of related properties. The dual L 1 product L = ; satis es L permutation. Likewise, L Q + Q L Q A1; : : : ; An + X( )A1+ +Ak 1 A1; : : : ; QAk; : : : ; An = 0 ; where the upper index of ( ) A means the grading of A, namely, the total ghost number of A. The commutativity L L~ + L~ L = 0 provides 2 = 0 : 1; 2= ;e = 0 ; The lowest relation of (2.3c) is just nd that the second lowest relation of (2.3c) is given by A A ; B ~ + ~ A ; B + ~ A ; B A A ; ~ B = 0; algebra of L for = ; ~, MCL (A) zA ; :}:|: ; A{ and in the WZW-like action. Likewise, we often refer MCQ(A) element for Q . There is an natural operation, a shift of the products, in L QA as the Maurer-Cartan 1 algebras. For any state A, the A-shifted products are de ned by One can check that with MCL (A), the A-shifted products satisfy weak L 1 relations: X( )j jh B (1); : : : ; B (k) A; B (k+1); : : : ; B (n) A i MCL (A); B1; : : : ; Bn A: then the A-shifted products exactly satisfy the L 1 relations. Similarly, one can consider 1; 2= ;e 1 B (1) ; : : : ; B (k) A ; B (k+1) ; : : : ; B (n) A MCL (A) ; B1 ; : : : ; Bn A MCL~(A) ; B1 ; : : : ; Bn A WZW-like action we need are two functional elds and their algebraic relations. Algebraic ingredients A functional eld ['] satisfying these constraint equations plays the most important role, which we call a pure-gauge-like (functional) eld. With this functional ~['], the ' of the theory. WZW-like functional eld. Let ~['] be a Grassmann even, ghost number 2, left-moving picture number 1, and right-moving picture number 1 state in the left-andright large Hilbert space: ~ satis es the constraint equations: ~ 6= 0. We call this ~ a pure-gauge-like (functional) eld = 0; { ~ = 0: In other words, ~['] gives a solution of the Maurer-Cartan equations for the both dual products (2.1a) and (2.1b). Therefore, two ~[']-shifted products again have L 1 relations and commute each other. One can de ne two linear operators D and D~ acting on any state A by ( = ; ); and two bilinear products of any states A and B by ( = ; ): + D A ; B A A ; D B = 0: (D )2A = 0 ; ( = ; ): D D~ + D~ D A = 0; D A ; B ~ A A ; D B ~ +D~ A ; B D~ A ; B A A ; D~ B = 0 : L~ : namely, D D A1; : : : ; An X( )D(A1+ +Ak 1) A1; : : : ; D Ak; : : : ; An ( = ; ) dynamical string eld satisfy the Leibniz rule for these L1-products L and L~, one can nd D (D ~) = 0 and D~(D imply that with some (functional) state D['] belonging to the left-and-right large Hilbert space H, we have D D ['] = D D~ D['] ; the WZW-like relation. Note that the existence of the (functional) state D['] is ensured Hilbert space H . We call this D['] satisfying (3.3) as an associated (functional) eld. and right-moving picture p, the associated eld D['] has the same quantum numbers: its We started with the L 1 triplet (L ; L~ ; Q) and obtained the above algebraic ingre? As we ~['] = 0 : relations in NS-NS superstring eld theory. Action, equations of motion, and gauge invariances Let ' be a dynamical NS-NS string functionals of given dynamical string eld ', we can construct a WZW-like action for ['] and D['] as NS-NS string eld theory: S ~['] = t['(t)]; Q ~['(t)] ; 4In section 5, we will see this fact again. t['(t)] denotes eld. As we will see, using the variational associated (functional) eld D['] with D = , the variation of this action is given by t-independent form: S ~['] = ['] = D The equation of motion is given by t-independent form ['] = D D~ Q['] = Q['] = 0: which we explain in the rest.5 Variation of the action D0 A ; B = ( )D0A A ; D0 B ; A ; B ; C = ( )AB B ; A ; C = ( )A(B+C) B ; C ; A = 0 ; ( = ; ~): t and B = D , the relation (3.2d) provides + [D~A; B] + [D A; B]~ + [A; D~B] = 0: We prove that when we have WZW-like functional elds ~['] and D['] which a direct computation of the variation of the action: S ~['] = t['(t)]; Q ~['(t)] + 5These computations are similar to those of the earlier WZW-like action [9]. B's total ghost, left-moving picture, and right-moving picture numbers are 3, 1, respectively. ~)i plus extra terms: ~) = h t; Q D~D = h t; D~D Q = h ~ i + h Q; D~D [ t; D Q; [D~D + [D~ t; D Likewise, we nd the rst term of the variation becomes h@t ~i plus extra terms: ~ = t; D~D + D~[ t; D~D + D~[ t; D~D t; D~D = h@t + D~[D~D t; D~D = @t Q; D~ [D + [ t; D~D + D~[ t; D~D (3:11a) + (3:11b) = @t S ~['] = ['(t)]; Q ~['(t)] = In summary, for xed L 1 triplet (L ; L~ ; Q), we rst consider a functional ~ satisfying constraint equations (3.1a) and (3.1b) de ned by two of it, L and L~. Next, using which gives a half input of the action. Lastly, using ~, we consider the Maurer-Cartan element of the remaining L 1 product Q, which provides the on-shell condition (3.4) and WZW-like action (3.5). As we showed in section 3, when two states D['] satisfying (3.3) are obtained, In this section, we present two di erent expressions of these D using two di erent dynamical string elds . It gives two di erent realisations of our WZW-like action, Through these constructions, we also see that once we have ~['] explicitly as a functional of ', the other functional D['] can be derived from ~[']. It would suggest next section. for a NS-NS dynamical string eld belonging to the small Hilbert space: = 0 = 0. This is a Grassmann even, total ghost number 2, left-moving picture number 1, and right-moving picture number 1 state. Pure-gauge-like (functional) eld As a functional of , the pure-gauge-like ~[ ] can be constructed by 1 Gb e^ moving picture numbers and this ~[ ] has correct quantum numbers as a pure-gaugeequations (3.1a) and (3.1b). ~[ ] is given by using the group-like element, the following relation holds: e^ ~['] = 0 ; ( = ; ~): Because of (2.1a) and (2.1b), one can quickly nd that (4.1) satis es e^ ~[ ] = ( Gb = Gb = 0 ; = ; ~) ; equality, we used the properties of the dynamical string = 0 and ~ = 0. , it is the origin of all algebraic relations of WZW-like theory. Associated (functional) D[ ]. Similarly, as functionals of , the associated (functional) eld D = D[ ] with D = @t or D = can be constructed by and the associated (associated) eld Q[ ] can be given by 1 Gb 1 Gb Q ~ e^ where Q ~ is a coderivation operation which we will de ne below. Recall that ~ satis es the constraint equations (3.1a) and (3.1b), and thus D D satisfying (3.3). One can derive an explicit form of the functional D[ ] from ~[ ] in this manner. Using the graded commutator of two coderivations D1 and D2, D1 ; D2 D1 D2 = ; ~ . Note that I = Gb = 0 ; Gb 1 D Gb = 0 : Namely, the co-derivation Gb 1 D Gb commutes with both -exactness and ~-exactness, there exist a coderivation D ~ such that and ~ . Hence, because of Using D ~ and the properties of the dynamical string eld, = 0 and ~ = 0, we nd ; [[ ~ ; D ~ ]] 1 Gb D ~ (e^ ) ^ e^ 1 Gb(e^ ) ^ e^ 1 Gb(e^ ) : Note that with (4.1), the linear operator D for = ; can be written as = 1L = ; e) : We thus nd that if we de ne the associated eld D[ ] by the following functional of , deed holds: 1 Gb D ~ (e^ ) ; [ ] = D D~ D[ ]: We write Hilbert space: 6= 0, ~ 6= 0, and ~ 6= 0. This has total ghost number 0, left-moving picture number 0, and right-moving picture number 0. Pure-gauge-like (functional) eld the following di erential equation, ~[ ]. Let us consider the solution with the initial condition D ( )A ; ( = ; ~) : A pure-gauge-like (functional) eld ~[ ] is obtained as the = 1 value solution ~[ ; ] = D ( )D~( ) [ = 1; ]: We check that this ~[ ] satis es (3.1a) and (3.1b). For this purpose, we set ( = ; ~): Because of the initial condition and (2.4a), we obtain the following linear di erential equation MCL ( ) = D ( ) @ = ( )j j MCL ( ) ; D~( ) where ( )j j denotes and +1 for and (3.1b) and gives a proof that (4.4) is a pure-gauge-like (functional) eld. By the ~[ ] = ~ In this parametrisation, the properties of the dynamical string eld L~ and L~, and to have a pure-gauge-like eld [ ] as a functional of . (Note that ~ 6= 0 makes possible Associated (functional) eld We consider the following di erential equation D[ ; ] = ( )DD with the initial condition (functional) eld D[ ] is obtained by the = 1 value solution of (4.6), D[ = 1; ] : As D -exacts and D~-exacts does not a ect in the rst slot of (3.5), this D is determined up to these. To prove (4.7) satisfy (3.3), we set D D~ D[ ; ] + ( ) D[ ]. Using (3.2) and (4.3), we nd = n + D D~ @ = D D~ ; D~ D + D D~ @ = D~ ; I( ) + D D~ @ = D D~ + D D~ @ + (D ) = D D~ From the third equal to the forth equal, we used the following identity: D[ ; ] satis es (4.6) up to D -exacts and D~-exacts, we have @ I( ) = D~( )A ; B t[ (t)] = @t (t) + (t); ~ @t (t) On the D -exacts and D~-exacts. We found a de ning equation (4.6) of D[ ]. Since the following identity which provides another expression of (4.8): @ I( ) = It ensures that as a de ning equation of D[ ; ], we can also use + D D~ @ D = D~( )D ( ), one may compute as @ I( ) = @ + D D~ @ However, we have the following identity Comparing (4.8) and (4.10) with (4.9), we also nd 0 = On the small associated We constructed two functionals ~['] and D[']. It is su cient to give a WZW-like D~ D[']: ~['] = 0; ~['] = 0: ( )D1D2D2D1, one can nd8 ~ = D D~ = D~ D1 D2 ~ ( )D1D2D2 D1 ~ ( )D1 = D -exact; = D~-exact: Small-space parametrisation. It is easy to obtain these in terms of because the D are given by 1 Gb D e^ ; D[ ] = 1 Gb D ~ e^ ; where we used coderivations D and D ~ such that D[ ] as the = 1 value solutions, D~[ = 1; ]; D[ = 1; ]; of the following di erential equations D~[ ; ] = D D~( ) D[ ; ] = D D ( ) 8They follow from direct computations D1D2 ~ = ( )D2D1 D with the initial conditions D~[ = 0; ] = 0 and D[ = 0; ] = 0. The minus sign the equation = [D D~ ; = D~ @ We can therefore obtain D~ satisfying (4.12) without using D and (4.11). When we start with D and (4.6), does D D of (4.11) satisfy the above di eren D ( ) D[ ; ] = + D~ D D[ ] and start with these di erential equations, can we On the D -exactness and D~-exactness. We can only specify the large associated (functional) eld D up to D - and D~-exact terms, and these ambiguities do not conhave operators F and Fe ~ de ned by These F as follows, which satisfy D F + F D and Fe ~ consist of the pure-gauge-like (functional) eld and ~. Using these pieces, one can construct ~['] and operators L , D['] via D['] and D~['] = we see in the next section. The author thanks to T.Erler for comments. Single functional form As we found, two or more types of functional elds D['] appear in the WZW-like consists of the single functional ~['] and elementally operators. It may be helpful in the gauge xing problem. ! : : : (exact) ; Furthermore, since = 0, = 0, ~ + ~ = 0, and = 0 hold, we have the direct sum decomposition of the large state space H as follows: H = Likewise, the existence of (4.13) satisfying D F + F = 1 and D~ Fe ~+ Fe ~D~ = 1 ! : : : (exact) ; decomposition using these exact sequences? To achieve this, we consider Fe ( Fe 1 One can quickly nd that as well as (4.13), this F and its inverse F 1 also provide = F D~ = F ~ F D F + F D = 1 ; D~ F~ + F~ D~ = 1 ; ( F Furthermore, now, these operators all are constructed from single F , we have D D~ + D~ D = 0; D F~ + F~ D = 0; D~ F + F D~ = 0; F F~ + F~ F = 0; H = D D~ H F F~ H : Since Q ~ = D F D~F~(Q ~) and D~D t = @t ~, using this F , we nd S ~['] = t['(t)]; Q ~['(t)]; F F~ Q ~['(t)] : It consists of the single functional ~['] and elementary operators L , L~, , , ~, ~, and F [[D; D ]]F + [[D ; F D F ]] . Equivalence of two constructions equivalence of S ~[ ] and S ~[ ] follows if we consider the identi cation ~[ ] = which consists of ~ and elementally operators. Since both actions have the same See also [1, 19{22]. their Fock spaces Field relation. Note that the identi cation of states (5.2) provides the identi cation of e^ ~[ ] = e^ ~[ ] ; Under the identi cation (5.2), by acting @t, we have t[ ] = t[ ] + D -exacts + D~-exacts: ^ t[ (t)] = Gb e^ (t) = e^ ~['] ~['] = e^ ~['] Since cohomomorphism Gb is invertible, we obtain the following eld relation = 1 = 1 which can be directly derived from (5.2). Relation to L1 theory We write 0 for the dynamical string eld of the L 1 action proposed in [2]. As well as belongs to the small Hilbert space: dynamical string eld , we constructed an action We will show that this S [ ] is exactly o -shell equivalent to the L S ~[ ] = 1 Gb @t (t) ^ e^ (t) ; Q 1 Gb e^ (t) SL1 [ 0] = 1 z n=1 (n + 1)! h ~ 0; Ln+1( 0; :}:|: ; {0; 0) : 1 action, 0(t) be a path connecting 0(0) = 0 and 0(1) = 0, where t 2 [0; 1] is a real SL1 [ 0] = SL1 [ 0(t)] = @t 0(t); 1LNS;NS e^ 0(t) : we obtained a proof that the L 1 action SL1 [ 0] proposed in [2] is equivalent to our S ~[ ]. S ~[ ] and S ~[ ] both are equivalent to that of L 1 formulation. See also [1, 22] by ( ; ~ ; LNS;NS). We thus consider a functional WZW-like reconstruction of L1 action. In the L ~['] which satis es two constraint equations de ned by 1 (e^ ~[']) = 1 ~ (e^ ~[']) = ~ ~['] = 0; ~['] = 0: ~['] = ~) = 0 and ~ (D D['], we have the WZW-like relation, and-right large Hilbert space. Using ~['], we can consider the Maurer-Cartan element for the remaining L 1 products LNS;NS : 1 LNS;NS(e ~[']) = Q Note that there also exists an associated eld L['] such that 1 LNS;NS(e^ ~[']) = SL1 ['] = t['(t)]; 1 LNS;NS(e^ ~['(t)]) t['(t)]; ~ L['(t)] : WZW-like manner. In particular, since and ~ are linear L 1 products, their shifted prodWe notice that if we set ' = = ~ of the functional, , because of the triviality of - and ~-cohomology. Similarly, if we use ' = , it also implies ~ . While its small-space parametrisation is just the L small-space one. O -shell duality of L1 triplets. As we mentioned, when Gb is cyclic in the BPZ inner product, (2.2) ensures not only the equivalence of L 1 triplets but also the o -shell appearing the action of [10, 12] because of their small-space constraints. Cartan-like element in the correlation function : MC (A) thus obtain MCQ(A) = MCQ(A) = MCL(A0) ; t~ t[']. Using a measure factor d dt @ t~ , we can express the WZW-like action (3.5) as S ~ = MCQ(A) ; SL1 = MCL(A0) : ~['] + t~ t['], the WZW-likely extended L 1 action (5.7) can be written as (L ; L~ ; Q) and the (WZW-likely extended) L 1 triplet 1 action (5.7) based on the L WZW-like structure. Relation to the earlier WZW-like theory eld. Let GL be a state which has ghost number 2, left-moving picture number 0, and right-moving picture number 1 state in the large Hilbert space. When this GL satis es ;NS = 0; ~ GL = 0; and ~ : namely D L ;NS ( )DL ;NS D = 0 and D ~ ( )D ~ D = 0. For example, one we consider ( )DD GL = 0D['0] is a functional of the dynamical string eld, which has the same ghost, left-moving-picture, and right-movingpicture numbers as d. We call this 0D['0] satisfying (5.12) as an associated (functional) of (5.11a) and (5.11b). In [9], using these GL['0] and D['], a WZW-like action was given by 0t['0(t)]; GL['0(t)] : We write 0t['0(t)] for the associated eld While the 0 in the left-and-right large Hilbert space in [9], if one on its WZW-like structure. 0 with the small-space string eld , 0 = ~ the action (5.13) reduces to the L 1 action based on their asymmetric construction of [2]. lower order. Conclusion We presented that a triplet of mutually commutative L 1 products (Lc; Lc~ ; Lp) completely constraint equations, 1 Lc e^ cc~['] = 1 Lc~ e^ cc~['] = cc~[']; : : : ; cc~['] = 0; cc~[']; : : : ; cc~['] = 0; and introducing a functional cc~['] of some dynamical string eld ' satisfying these conScc~['] = MCLp (A) = t['(t)]; 1Lp e^ cc~['(t)] ; 1 Lp e^ cc~['] = cc~[']; : : : ; cc~['] = 0 : One can prove its gauge invariance using the functional cc~['] and algebraic relations derived from the mutual commutativity of the L 1 triplet (Lc; Lc~ ; Lp),11 without using and ( ; ~ ; LNS;NS) which provide the L form, one can say that to study its L 1 triplet is equivalent to know the gauge structure of NS-NS superstring eld theory. In this paper, we focused on two L 1 triplets (L ; L~ ; Q) 1 action of [2]. Particularly, we presented detailed WZW-like theory. Acknowledgments Republic, under the grant P201/12/G028. General WZW-like action based on (Lc; Lc~ ; Lp) (Lc; Lc~ ; Lp) . In this appendix, we prove that the general WZW-like action, Scc~['] = t['(t)]; 1Lpe^ cc~['(t)] ; is expected from the result of [19]. Actually, with deep insights, one can nd a pair of (nonlinear) A1 sectors [24]. has topological parameter dependence: its variation is given by Scc~['] = is invariant under the gauge transformations generated by Lc , Lc~, and Lp , Then, because of the nilpotency of L 1 triplet (Lc; Lc~ ; Lp) , the general WZW-like action ['] = 1 Lce^ cc~['] any coderivation D commuting with Lc and Lc~ , we nd ( )DD Lc0e^ cc~['] = Lc0e^ cc~['] ^ 1De^ cc~['] = 0 ; Hence, since - and ~-cohomology are trivial, there exist a state DcDc~ D['] 1LcLc~e^ cc~['] where we de ned Dc0A like relation for a general L 1 triplet (Lc; Lc~ ; Lp) , which provides cc~ = @ cc~ = which are mutually commute with Lc and Lc~ , we nd 1 D1 D2 e^ cc~['] = 1 D1 e^ cc~['] ^ 1( )D2Lc Lc~ e^ cc~['] (c0 = c ; c~) : D['] such that = ( )D2 1Lc Lc~ D1e^ cc~['] = ( )D2 1Lc Lc~ e^ cc~['] ^ 1D1 e^ cc~['] ^ D2['] + ( )D2 1Lc Lc~ e^ cc~['] cc~) , (3.2), and (3.9) . Using these, we nd a half of the variation is t; 1 Lp e^ cc~ ^ DcDc~ ; Dc~Dc 1 Lp e^ cc~ ^ t ; 1LcLc~ e^ cc~ ^ DcDc~ Lp ^ t : We notice that these computation can be carried out by replacing Q ~ = D D~ Q t; 1 Lpe^ cc~ = = @t Dc~Dc 1LcLc~ e^ cc~ ^ DcDc~ ; 1LcLc~ e^ cc~ ^ DcDc~ Lp ^ t : Note that this term can be also obtained by replacing Q of (3.11b) with Lp . Hence, we obtain the desired result ['(t)]; 1Lpe^ cc~['(t)] : 1 triplet (Lc; Lc~ ; Lp) in the completely same way. In general, eld rede nitions Ub terms of L 1 algebras, it is just described by an L 1 morphism between two L 1 triplets, under any string eld rede nitions. Thus, as a gauge theory, it may capture general eld theoretical properties of superstrings. Open Access. This article is distributed under the terms of the Creative Commons any medium, provided the original author(s) and source are credited. 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Hiroaki Matsunaga. Notes on the Wess-Zumino-Witten-like structure: L ∞ triplet and NS-NS superstring field theory, Journal of High Energy Physics, 2017, DOI: 10.1007/JHEP05(2017)095