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

Journal of High Energy Physics, Jan 2017

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 \( {\mathrm{\mathbb{Z}}}_2 \)-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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A ∞ /L ∞ structure and alternative action for WZW-like superstring field theory

Received: November structure and alternative action for WZW-like superstring field theory Keiyu Goto 0 1 2 3 6 Hiroaki Matsunaga 0 1 3 4 5 0 Open Access , c The Authors 1 Na Slovance 2 , Prague 8 , Czech Republic 2 Present address: Komazawa , Setagaya-ku, Tokyo 154-0012 , Japan 3 Komaba , Meguro-ku, Tokyo 153-8902 , Japan 4 Institute of Physics, Academy of Sciences of the Czech Republic 5 Yukawa Institute for Theoretical Physics, Kyoto University 6 Institute of Physics, University of Tokyo 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 WessZumino-Witten-like expressions which are the Z2-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. String Field Theory; Superstrings and Heterotic Strings - A∞/L∞ 1 Introduction and summary Construction of L∞-product and L∞ action Alternative WZW-like form of L∞ action Gauge transformations Alternative parameterisation for closed NS string field theory Large space parameterisation Equivalence of the actions in the different parameterizations Conclusions and discussions A Basic facts of A∞ and L∞ Derivation of (2.71) C Embedding to the large Hilbert space of NS closed string D Open NS superstrings with stubs NS-NS sector Alternative Wess-Zumino-Witten-like relations Introduction and summary Recently, the formulation of superstring field theory has progressed gradually: actions for superstring field theories were constructed and their properties have been clarified [1–4]. In addition to these, 1PI effective field theory approach has provided a good insight into self-dual gauge theory and supergravity [5]. One trigger of these developments was given NS string [6], heterotic NS string, and closed NS-NS string [7] were constructed by giving was extended to the cases including the Ramond sector and the equations of motion were provided in the work of [8]. At least at the tree level, these theories reproduce S-matrices of perturbative superstring theory with insertions of picture-changing operators at external lines [9]. One of our aims in this paper is to develop the understanding of the relation would also give the same results from [22, 23].2 Actually, we have clear understandings and the Berkovits WZW-like action [16] is clarified by the works of [29–31], and for the NS and R sectors, the equivalence of the complete action from which the equations of motion given in [8] are derived and the complete action proposed in [2] is provided by the work of [4].3 It would be important to extend these understandings of open strings to the case of closed strings and to construct complete actions for heterotic and type II theories. However, it seems to be difficult to discuss them on the basis of the same procedure4 as [29] and known WZW-like actions [18–21]: it necessitates other insights because of the skew would be helpful to consider their dual versions. Recall that the Berkovits theory is formulated on the large Hilbert space, which is the Berkovits theory is a Grassmann even, ghost number 0, and picture number 0 state dt he−Φ(t)∂teΦ(t), e−Φ(t)ηeΦ(t), e−Φ(t)QeΦ(t) ∗i, is the graded commutator for the star products of A and B. One can find that this action has nonlinear gauge invariances given by for the graded commutator of operators d1 and d2, d1, d2 = d1 d2 − (−)d1d2 d2 d1. 1For other approaches, see also [10–15]. 2For the R sector, see [24–28]. new result given by T. Erler, Y. Okawa and T. Takezaki, JHEP 08 (2016) 012 [arXiv:1602.02582]. motion up to Q-exact terms, which implies the on-shell states match each other. See [32] for more details. Conventional WZW-like form. It was shown in [19] that Berkovits action can be written as a WZW-like form The equation of motion is given by the t-independent form We write m2(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 which we call WZW-like relations: gauge-like field,5 gives a solution of the Maurer-Cartan equation for the bosonic open string M B = Q + m2, which consists of the BRST operator Q and the star product m2. Recall that M B satm2 m2(A, B), C − m2 A, m2(B, C) equation of motion and pure gauge field in bosonic open string field theory respectively. pure gauge in bosonic theory. we symbolically6 write = 0, [[d, M B]] = 0 for the A∞-relations of M B and the d-derivation properties of M B. These properties yield the WZW-like relations, and then, the constraint equation and the equation of motion are Constraint (C-WZW) : E.O.M. (C-WZW) : 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 M sBtub = Q + ms2t + ms3t + ms4t + . . . M sBtub, M sBtub = 0, [[d, M sBtub]] = 0. Using these M sBtub, d, and the NS open string field ϕ, one can construct a pure-gauge-like field AesQt[ϕ] and an associated field Aesdt[ϕ] via the same type of the defining differential equations as those of the theory without stubs. The resultant theory is given by the following action Sstub[ϕ] = and characterized by the pair of equations: Constraint (C-WZW) : Q AeQ[ϕ] + ms2t AeQ[ϕ], AesQt[ϕ] + X msnt zAesQt[ϕ], . . . , AeQ[ϕ] = 0, st st }| st { E.O.M. (C-WZW) : Constraint (A∞) : E.O.M. (A∞) : = 0, 6We use bold fonts for coalgebraic operations or notions. See appendix A. theory by the naive way. Alternative WZW-like form. It is known that Berkovits action can be also written as alternative WZW-like form Then, the equation of motion is given by the t-independent form Dη∗ Ω ≡ η Ω − m2 Aη[Φ], Ω − (−)Ωm2 Ω, Aη[Φ] . 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 : the category of so-called the WZW-like formulation. While the first relation provides the Note that this type of WZW-like theory, a dual version of the conventional WZW-like theory, is characterized by the pair of equations Constraint (A-WZW) : E.O.M. (A-WZW) : M 2 + M 3 + . . . given in [6] as M = Gb−1Q Gb and by using a redefined string field A∞ action into Constraint (A∞) : E.O.M. (A∞) : Therefore, we expect that one can construct WZW-like actions which are off-shell equivalent Main results of this paper. What is the starting point of this type of WZW-like it is given by the following A∞-product: η + D2 satisfies the A∞-relations of η: η2 = 0, η D2η(A, B) − D2η(ηA, B) − (−)AD2η(A, ηB) = η which give the starting point of our alternative WZW-like theory. Actually, one can con In this paper, we show that a nonassociative extended version of this construction Dη = η + D2η + D3η + D4η + . . . = 0, = 0. One can construct these products by the dual products of M = Gb−1Q Gb given in [7], WZW-like theories ηAη[ϕ] − m2(Aη[ϕ], Aη[ϕ]) + X Dnη zAη[ϕ], .}.|. , Aη[ϕ{] = 0, DηΩ ≡ η Ω−m2 Aη[ϕ], Ω −(−)Ωm2(Ω, Aη[ϕ])+ X X(−)signDnη+1 zAη[ϕ], .}.|. , Aη[ϕ{], 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 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 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 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 Z2-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. action can be written in our (alternative) WZW-like form: the functionals appearing in the 7See a new result given by K. Goto and H. Kunitomo, arXiv:1606.07194. cyclic n=2 action satisfy alternative WZW-like relations, the Z2-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 tion 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 Z2-reversed versions of those given parameterizations of the same WZW-like structure and action. Then we also derive the end with some conclusions and discussions. marized in appendix A. In appendix B, we derive a formula which is used in section 2. 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. that the functionals appearing in the action satisfy alternative WZW-like relations, the Z2-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 bn : H∧n 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 dn(Φ1 ∧ · · · ∧ ΦN ) = (dn ∧ IN−n)(Φ1 ∧ · · · ∧ ΦN ) n!(N − n)! dn(Φσ(1), · · · , Φσ(n)) ∧ Φσ(n+1) ∧ · · · ∧ Φσ(N), (2.1) i≤n k1<···<ki e∧f0 ∧ fk1 (Φ1, . . . , Φk1 ) ∧ fk2−k1 (Φk1+1, . . . , Φk2 )∧ · · · ∧ fki−ki−1 (Φki−1+1, . . . , Φn). For definitions and their more details, see appendix A. In this subsection, we briefly review the construction of the NS superstring product L and NS superstring product L. Let us review the construction of the NS string products B = Q + L p=0 τ pλ[p0+] 2 are called gauge products. differential equations is given by the similarity transformation of Q, where Gb is an invertible cohomomorphism defined by the path-ordered exponential of the 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: L2 = Gb−1Q Gb Gb−1QGb = Gb −1Qb QGb = 0. d=0 p=0 to be BPZ-odd, so that Gb−1 = Gb†: L† = ( Gb−1Q Gb)† = − Gb−1QGb = −L. series of generating functions d=0 p=0 n=0 (n + 2)! n=0 (n + 1)! n=0 (n + 2)! 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: dt hπ1(ξt e∧Φ(t)), π1 L(e∧Φ(t)) i, The variation of the action can be taken as and then the equation of motion is given by picture number −1. dt hπ1(ξt e∧Φ(t)), π1 Gb −1 Q Gb(e∧Φ(t)) i dt hπ1 Gb(ξte∧Φ(t)) , π1 Q Gb(e∧Φ(t)) i. 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 (−)dd Ψη = η Ψd + X 1 zΨη, .}.|. , Ψη{, Ψd η, n=1 (n + 1)! which are Z2-reversed versions of conventional WZW-like relations in [19]. In this substructed using the cohomomorphism Gb which provides the NS heterotic string products (Lη)2 = Gb η Gb −1 Gb η Gb−1 = Gb ηη Gb−1 = 0. (Lη)† = (Gb η Gb −1)† = (Gb −1)† η† Gb† = − Gb η Gb −1. [B1, . . . , Bn]η := π1Lηn(B1 ∧ · · · ∧ Bn) k!(n − k)! (−)|σ| [Biσ(1) , . . . , Biσ(k) ]η, Biσ(k+1) , . . . , Biσ(n) Q B1, . . . , Bn η + X(−)B1+···+Bk−1 B1, . . . , QBk, . . . , Bn n=1 (n + 1)! 9See also appendix D or section 3.2 of [4] for generic properties of these types of products. (Dη)2B = −[π1Lη(e∧Ψη ), B]ηΨη = 0, hB1, [B2, · · · , Bn+1]ηΨη i = (−)B1+B2+···+Bn h[B1, · · · , Bn]ηΨη , Bn+1i. 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: η e∧π1 Gb(e∧Φ(t)) = (Gb η Gb−1) Gb e∧Φ(t) = Gb η e∧Φ(t) = 0. The second relation (2.48) can be confirmed similarly. The operator d which we focus (−)dd Gb e∧Φ(t) = (−)d Gb (Gb −1d Gb) e∧Φ(t) = L η π1 Gb ξd e∧Φ(t) ∧ e∧π1 Gb(e∧Φ(t)) . 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 Z2-reversed) manner to those in [19]. We will see them in the next subsection. 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 the variation of the action can be taken in the same manner as long as they satisfy the First, consider the variation of the integrand of (2.52), Utilizing the following relation following from (2.48), η 0 = [[d1, d2]]Ψη = (−)d1+d2 Dη d1Ψd2 − (−)d1d2 d2Ψd1 + (−)d2d1+d2 [Ψd2 , Ψd1 ]Ψη , the first term can be transformed into δΨt(t), QΨη(t) = ∂tΨδ(t) + [Ψδ(t), Ψt(t)]ηΨη(t), QΨη(t) . − QDηΨt(t) − [QΨη(t), Ψt(t)]ηΨη(t), Ψδ(t) . Integrating over t, the variation of the action is given by the variation of the action becomes We find that the variation of the action does not depend on t, and therefore t-dependence 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: = π1 Gb−1Q e∧Ψη[Φ] = π1Gb −1 (QΨη[Φ]) ∧ e∧Ψη[Φ] . (2.61) − π1 Gb L(ξλ ∧ e∧Φ) + ξλ ∧ π1L(e∧Φ) ∧ e∧Φ Gauge transformations the following form of the gauge transformations, 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 Ψδ[Φ] = π1 Gb ξδe∧Φ Ω[λ, Φ] = π1 Gb (ξπ1L(ξλ ∧ e∧Φ)) ∧ e∧Φ , the third term of (2.65), ΔT [λ, Φ] = π1 Gb ξλ ∧ π1L(e∧Φ) ∧ e∧Φ , corresponds to the trivial gauge transformation of the WZW-like action. Thus, the gauge Trivial gauge transformation. Trivial gauge transformation is a transformation proportional to the equations of motion. Schematically, it is of the following form, 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 cyclicity of the cohomomorphism Hb is written as follows: − π1Gb ξλ ∧ π1L(e∧Φ) ∧ e∧Φ , π1 Gb π1L(e∧Φ) ∧ e∧Φ = − π1 Gb ξλ ∧ e∧Φ , π1 Gb π1L(e∧Φ) ∧ π1L(e∧Φ) ∧ e∧Φ = 0, follows from the symmetric property: Alternative parameterisation for closed NS string field theory of the action is provided by only the algebraic relations (2.47) and (2.48). In other words, 10For more detail, see [39]. on-shell equivalence (2.62) guarantees it is trivial gauge transformation. are obtained, the gauge invariant action can be constructed as This form of the action is the Z2-reversed version of that in [19], which we call the 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 Z2-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 paraA set of differential equations which are the Z2-reversed version of those in [19] give these parameterizations so that WZW-like relations (2.47) and (2.48) hold. these functionals, a new gauge invariant action for the string field V is constructed in the (alternative) WZW-like form. = η V + X 1 zΨη[τ ; V ], .}.|. , Ψη[τ ; V{], V η with the initial condition Cartan equation One can check that the differential equation actually provides a solution for the Maurer∂τ I(τ ) = [V, I(τ )]ηΨη + Dη ∂τ Ψd − dV − [V, Ψd]ηΨη . ∂τ Ψd[τ ; V ] = dV + V, Ψd[τ ; V ] Ψη[τ;V ] and (3.10), one can construct a new gauge invariant action as follows: The variation of the action can be taken in the same manner as that in section 2.3, and the equation of motion can be read off from it, ghost numbers 0 and 0 and picture numbers 1 and 0, respectively. BRST operator which plays a key role in conventional WZW-like formulation, can be problem to understand the relation between two cohomomorphisms EV and Gb. Ramond sector of closed superstring fields. The concept of the WZW-like structure of heterotic string field theory including both NS and R sectors. As [2], our WZW-like It is expected that as demonstrated in [4] for open superstrings, the WZW-like structure 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 Acknowledgments The authors would like to thank Hiroshi Kunitomo and Yuji Okawa for discussions and comments. H.M. would like to express his gratitude to his doctor for medical treatments. The work of H.M. was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. Basic facts of A∞ and L∞ more details, see [40–42] or some mathematical manuscripts. Coalgebras: tensor algebra T (H) and its symmetrization S(H). Let C be a set. T (H) = H⊗0 ⊕ H⊗1 ⊕ H⊗2 ⊕ · · · . vector space H: X(Φ1 ⊗ . . . ⊗ Φk) ⊗′ (Φk+1 ⊗ . . . ⊗ Φn) 12See a new result given by K. Goto and H. Kunitomo, arXiv:1606.07194. coalgebra. As well as T (H), its symmetrization S(H) also gives a coalgebra. Recall that Φ1 ∧ Φ2 ≡ Φ1 ⊗ Φ2 + (−)deg(Φ1)deg(Φ2)Φ2 ⊗ Φ1, Φ1 ∧ Φ2 = (−)deg(Φ1)deg(Φ2)Φ2 ∧ Φ1, X(−)σΦσ(1) ⊗ Φσ(2) ⊗ . . . ⊗ Φσ(n). S(H) for the symmetrization of T (H), which is called the symmetrized tensor algebra S(H): S(H) = H∧0 ⊕ H∧1 ⊕ H∧2 ⊕ · · · . X′(−)σ(Φσ(1) ∧ . . . ∧ Φσ(k)) ⊗′ (Φσ(k+1) ∧ . . . ∧ Φσ(n)) In the case of open superstring field theory, H is the state space of open superstrings, which is a Z2-graded vector space, and its grading which we call degree is given by the superstrings. On the other hand, in the case of closed superstring field theory, H is the state space of closed superstrings, which is a Z2-graded vector space, and its grading which Fock space of closed superstrings. Multi-linear maps as a coderivation. A linear operator m : C → C which raise the degree one is called coderivation if it satisfies Multilinear maps with degree 1 and 0 naturally induce the maps from T (H) to T (H) or from S(H) to S(H). They are called a coderivation and a cohomomorphism respectively. linear map on H⊗n, which we write bn : H⊗n → H, by In = Iz ⊗ I ⊗}|. . . ⊗ {I = 1 Iz ∧ I ∧}|. . . ∧ {I . T (H) → T (H) by bnΦ = (bn ∧ IN−n)Φ , Φ ∈ H∧N≥n ⊂ S(H), Φ1 ∧ Φ2 → b1(Φ1) ∧ Φ2 + (−)deg(Φ1)deg(b1)Φ1 ∧ b1(Φ2). In particular, the coderivation b0 : S(H) → S(H) derived from a map b0 : H0 → H is and acts as b0Φ = (b0 ∧ IN )Φ = b0 ∧ Φ , Φ ∈ H∧N ⊂ S(H) and bn vanishes when acting on H⊗N≤n. map bn : H∧n → H by Multilinear maps as a cohomomorphism. morphism bf : C → C′ is a map of degree zero satisfying Given two coalgebras C, C′, a cohomoΦ1 ∧ · · · ∧ Φn ∈ H∧n ⊂ S(H) is defined by H′}n∞=0 naturally induces a i≤n k1<···<ki e∧f0 ∧ fk1 (Φ1, . . . , Φk1 ) ∧ fk2−k1 (Φk1+1, . . . , Φk2 )∧ · · · ∧ fki−ki−1 (Φki−1+1, . . . , Φn). Given the operator On, we can define its BPZ-conjugation On† as follows: Let H be a graded vector space and S(H) be its symmetrized tensor algebra. A weak M1 + M2 + . . . satisfying (M)2 = 0. Its explicit actions are given as follows: → e∧f0 Φ1 ∧ Φ2 → e∧f0 ∧ f1(Φ1) ∧ f1(Φ2) + e∧f0 ∧ f2(Φ1 ∧ Φ2). Mn† = −Mn. (L)2 = 0. L†n = −Ln. The action of a coderivation on a group-like element is given by Projector and group-like element. is called a group-like element. It satisfies L0 = 0, (H, L) is called an L∞-algebra. map H∧n → H is given by Ln · L1 + Ln−1 · L2 + · · · + L2 · Ln−1 + L1 · Ln = 0. 0 = X′(−)σLj (Li(Bσ(1), . . . , Bσ(i)), Bσ(i+1), . . . , Bσ(n)). b0(e∧Φ) = b0 ∧ (e∧Φ) = b1(Φ) ∧ (e∧Φ) = b1(e∧Φ). One of the important property of a cohomomorphism is its action on the group-like Utilizing the projector and the group-like element, the Maurer-Cartan element for an L∞-algebra (H, L) is given by 2 L2(Φ ∧ Φ) + For cyclic cohomomorphism Hb , the following relation holds: (T (H), ∇). Let us consider its action + (−)AB + (−)BC ⊗ B ⊗ ⊗ A ⊗ ⊗ C ⊗ ⊗ A ⊗ ⊗ C ⊗ ⊗ B ⊗ ⊗ A ⊗ ⊗ B ⊗ ⊗ C ⊗ term of (B.2) becomes ⊗ A ⊗ ⊗ B ⊗ ⊗ C ⊗ ⊗′ ⊗ A ⊗ ⊗ A ⊗ ⊗ A ⊗ ⊗ A ⊗ ⊗′ ⊗ B ⊗ ⊗ B ⊗ ⊗ C ⊗ ⊗ B ⊗ ⊗′ ⊗ C ⊗ ⊗ C ⊗ ⊗ B ⊗ ⊗ C ⊗ gathering the contribution from all terms of (B.2), and utilizing the identities ⊗ a ⊗ ⊗ a ⊗ one can obtain the BPZ property for Hb : ⊗ b ⊗ +(−)ab 1 ⊗ b ⊗ ⊗ a ⊗ Note that in the computation some terms cancel by the symmetric property of the inner Embedding to the large Hilbert space of NS closed string 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 Ve . In particular, we discuss the relation between the gauge transformations in two in the small Hilbert space. The action for Ve is given by SEKS[Ve ] = dt h∂tVe (t), π1 L(e∧ηVe (t)) i dt hπ1 Gb (∂tVe (t)) ∧ e∧ηVe (t) , Q π1 Gb(e∧ηVe (t))i. One can confirm the functionals appearing in the action satisfy the WZW-like relations: the second relation, consider Ψd[Ve ] = π1 Gb d(ηξ +ξη)Ve ∧e∧ηVe = π1 Gb dξηVe ∧e∧ηVe +(−)dDηπ1 Gb dξVe ∧e∧ηVe , (C.5) where we used (−)dπ1 Gb dηξVe ∧e∧ηVe = π1 Gbη dξVe ∧ e∧ηVe = π1Lη π1Gb dξVe ∧e∧ηVe ∧Gb e∧ηVe = Dηπ1 Gb dξVe ∧e∧ηVe . (C.6) applied to show (−)dd Ψη[Ve ] = Dη Ψd[Ve ]. does not affect the action, that is, the action does not depend on this ambiguity. Here we For d = Q, it reads Ψd[Ve ] = π1Gb π1 Gb−1dGb (Ve ∧ R e∧ηVe ) ∧ e∧ηVe , n=0 (n + 1)! ΨQ[Ve ] = π1Gb π1L(Ve ∧ R e∧ηVe ) ∧ e∧ηVe . Ψd[Ve ] = π1 Gb π1 Gb−1d Gb(Ve ∧ R e∧ηVe ) ∧ e∧ηVe = π1 Gb π1(−)d[[η, ξd]](Ve ∧ R e∧ηVe ) ∧ e∧ηVe + (−)dDηπ1 Gb π1ξd(Ve ∧ R e∧ηVe ) ∧ e∧ηVe . applied to show (−)dd Ψη[Ve ] = Dη Ψd[Ve ]. Thus, the action is written in the WZW-like form, SEKS[Ve ] = Its variation and the on-shell condition can be taken by parallel computations in section 2.3. Gauge transformations. The variation of the action SEKS[Ve ] in the form of (C.1) can be taken as δSEKS[Ve ] = hδVe , π1 L(e∧ηVe ) i, δVe = −ηξω − π1L(ξλ ∧ e∧ηVe ), and one can find the action SEKS[Ve ] is invariant under the gauge transformations13 gauge symmetry can be written in the WZW-like form: = −π1Lη π1Gb ξω ∧ e∧ηVe ∧ e∧π1Gb (e∧ηVe ) − Qπ1 Gb(ξλ ∧ e∧ηVe ) − ΔT [λ, ηVe ] in section 2.4, ΔT [λ, ηVe ] = π1 Gb ξλ ∧ π1L(e∧ηVe ) ∧ e∧ηVe , we conclude that the gauge transformations of the trivial embedded theory (C.13) are writΩ[ω, Ve ] = −π1 Gb(ξω ∧ e∧ηVe ), Λ[λ, Ve ] = −π1 Gb(ξλ ∧ e∧ηVe ). as seen in (C.17) and (C.18). Open NS superstrings with stubs In this section, we construct a new action for generic open NS string field theory, which we its WZW-like structure, we give two realizations of this type of WZW-like action using two For example, one can use Q, Q + m2|2, and so on for a, and various Gb appearing in [6–8] (Da)2 = 0, = 0. = 0. = (−)aη Gb η Da Gb −1 = (−)aη Gb η Gb−1 a Gb Gb −1 = (−)aηDη a. Neveu-Schwarz products M of open stings with stubs given [7], Da = DQ = M ≡ Gb−1 Q Gb, and the dual A∞ products is always given by Dη ≡ Gb η Gb −1 = η − m2 + . . . . [B1, . . . , Bn]η ≡ Dnη(B1 ⊗ · · · ⊗ Bn) 1 − A 1 − A 1 − A 1 − A 1 − A 1 − A Shift of the dual A∞ Dη = η − m2 + D3η + D4η . . . vanish: Dn>2 = 0. In particular, we write DηB for [B]η: η ⊗ B1 ⊗ ⊗ · · · ⊗ ⊗ Bn ⊗ ⊗ B ⊗ cyclic n=2 ⊗ B ⊗ = 0, ≡ [A, B]ηAη − (−)AB[B, A]ηAη . We call a state Ad a associated (functional) field when Ad satisfies dynamical string field ϕ, one can construct a gauge invariant action check that the t-dependence is “topological”. Namely, the variation of the action is given by In the rest, we give two realisations of this action by using two different dynamiequivalent to that of A∞ formulation. = 0. because it becomes a trivial solution of the Maurer-Cartan equation as follows = Gb η Aη[Ψ] − m2 Aη[Ψ], Aη[Ψ] + X Let d be a coderivation constructed from a derivation d of (−)dd An associated (functional) field of d is given by because one can directly check (−)dd ferential equation and gives a pure-gauge-like (functional) field. For brevity, we introduce = Dη(τ ) Dη(τ ) Φ = − F (τ ), Φ ]]ηAη[τ;Φ]. F (τ ) ≡ η Aη[τ ; Φ] − m2 Aη[τ ; Φ], Aη[τ ; Φ] + X n=1 cyclic Ad[τ ; Φ] = dΦ + Φ, Ad[τ ; Φ] Aη[τ;Φ] satisfies (D.8) and gives an associated (functional) field. To prove this fact, we introduce whose zeros would provide the WZW-like relation that associated (functional) field must Ad[τ ; Φ] − dΦ − Φ, Ad[τ ; Φ] Aη[τ;Φ] . differential equation NS-NS sector Dual L∞ products. Using path-ordered exponential map Gb , the NS-NS superstring [A1, . . . , An]α := π1 Gb α Gb−1(A1 ∧ · · · ∧ An), X(−)|σ| [Aiσ(1) , . . . , Aiσ(k) ]α, Aiσ(k+1) , . . . , Aiσ(n) X(−)|σ| [Aiσ(1) , . . . , Aiσ(k) ]α1 , Aiσ(k+1) , . . . , Aiσ(n) d B1, . . . , Bn α + X(−)d(B1+···+Bk−1) B1, . . . , d Bk, . . . , Bn Alternative Wess-Zumino-Witten-like relations n=1 (n + 1)! DαB ≡ α B + X 1 zΨηηe, .}.|. , Ψηη{e, B α, Although it is sufficient to consider the above (E.3a) and (E.3b), it would be helpful are defined by Alternative WZW-like action Sηηe[ϕ] gauge invariant action whose gauge transformations are given by equation of motion is given by t-independent form One can find these facts by using WZW-like relations (E.3a), (E.3b), and (E.3c) only, which we explain. Note that computations are almost parallel to the conventional WZWlike case [21]. Variation of the action. To derive (E.5) and (E.6) from (E.4), we compute the variation Dηe d1 Ψηd2 − (−)d1d2 d2Ψηd1 − (−)d1 [Ψηd1 , Ψηd2 ]Ψηηe = 0. For brevity, we omit ϕ(t)-dependence of functionals. Note that the inner product i = (−)ABhB, [A, C]αΨηηei for α = η, ηe. Using (E.8a) with (E.3b) + hΨQηe, Dη [Ψtηe, Ψδ]Ψηηe − [Ψt, Ψηδ]Ψηηe i. = −hδ DηΨt , ΨQηei + h[δΨηηe, Ψt]ηΨηηe, ΨQηei = −hδΨηt, ΨQηei + h[DηeΨηδ, Ψt]ηΨηηe, ΨQηei = h∂tΨηδ, ΨQηei + h[Ψηt, Ψηδ]Ψηηe, ΨQηei + h[DηeΨηδ, Ψt]ηΨηηe, ΨQηei η 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) dt h(E.9) + (E.10)i = variant under (E.5) and the equations of motion is given by (E.6) because of (E.3b). and the linear operator Dα for α = η, ηe becomes Dα = π1Lα I ∧ e∧π1 Gb(e∧Φ) , Proofs of properties. Since the group-like element given by (E.11) satisfies η eπ1Gb (e∧Φ(t)) = (Gb η Gb −1) Gb e∧Φ(t) = Gb η e∧Φ(t) = 0, n=1 (n + 1)! = L η π1 Gb ξdΦ(t) ∧ e∧Φ(t) ∧ e∧π1Ge (e∧Φ(t)) , we find that the second Wess-Zumino-Witten-like relation holds: = L η π1 Gb ξQ e∧Φ(t) ∧ e∧π1 Gb(e∧Φ(t)) . and thus, (E.12) gives an appropriate large associated (functional) field: (E.3c) holds. DαA ≡ α A + X 1 zΨηηe[τ ; Ψ], .}.|. , Ψηηe[τ ; Ψ{], A α. A ∈ H. e and then [[Dη, Dηe]]A = − π1LηeΨηηe, A ηΨηηe − π1LηeeΨηηe, A ηΨηηe = 0 holds for any state ∂τ Ψdηe[τ ; Ψ] = d Dηe Ψ + DηeΨ, Ψdηe[τ ; Ψ] ηΨηηe[τ;Ψ], One can also check these satisfy (E.3b) by the same way as NS theory. The minus sign parallel with that of the conventional WZW-like NS-NS theory [21]. Thus one may consider Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 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Keiyu Goto, Hiroaki Matsunaga. A ∞ /L ∞ structure and alternative action for WZW-like superstring field theory, Journal of High Energy Physics, 2017, 22, DOI: 10.1007/JHEP01(2017)022