Comments on complete actions for open superstring field theory

Journal of High Energy Physics, Nov 2016

We clarify a Wess-Zumino-Witten-like structure including Ramond fields and propose one systematic way to construct gauge invariant actions: Wess-Zumino-Witten-like complete action S WZW. We show that Kunitomo-Okawa’s action proposed in arXiv:​1508.​00366 can obtain a topological parameter dependence of Ramond fields and belongs to our WZW-like framework. In this framework, once a WZW-like functional \( {\mathcal{A}}_{\eta }={\mathcal{A}}_{\eta}\left[\Psi \right] \) of a dynamical string field Ψ is constructed, we obtain one realization of S WZW[Ψ] parametrized by Ψ. On the basis of this way, we construct an action \( \tilde{S} \) whose on-shell condition is equivalent to the Ramond equations of motion proposed in arXiv:​1506.​05774. Using these results, we provide the equivalence of two theories: arXiv:​1508.​00366 and arXiv:​1506.​05774.

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Comments on complete actions for open superstring field theory

Received: October Comments on complete actions for open superstring Hiroaki Matsunaga 0 1 2 Witten-like complete action SWZW. 0 1 2 Open Access 0 1 2 c The Authors. 0 1 2 0 Yukawa Institute of Theoretical Physics, Kyoto University 1 Na Slovance 2 , Prague 8 , Czech Republic 2 Institute of Physics, Academy of Sciences of the Czech Republic We clarify a Wess-Zumino-Witten-like structure including Ramond fields and propose one systematic way to construct gauge invariant actions: in arXiv:1508.00366 can obtain a topological parameter dependence of Ramond fields and belongs to our WZW-like framework. In this framework, once a WZW-like functional Aη = Aη[Ψ] of a dynamical string field Ψ is constructed, we obtain one realization of SWZW[Ψ] parametrized by Ψ. On the basis of this way, we construct an action Se whose on-shell condition is equivalent to the Ramond equations of motion proposed in arXiv:1506.05774. Using these results, we provide the equivalence of two theories: arXiv:1508.00366 and arXiv:1506.05774. We show that Kunitomo-Okawa's action proposed; String Field Theory; Superstrings and Heterotic Strings 1 Introduction 1.1 Complete action and topological t-dependence 2 Wess-Zumino-Witten-like complete action WZW-like structure and XY -projection WZW-like complete action Unified notation Single functional form Another parametrization Another parametrization of the WZW-like complete action Equivalence of EKS and KO theories A Basic facts and some identities Introduction Recently, a field theoretical formulation of superstrings has been moved toward its new phase: an action and equations of motion including the Neveu-Schwarz and Ramond sectors were constructed [1, 2]. With recent developments [3–10], we have gradually obtained new and certain understandings of superstring field theories. In the work of [1], a gauge invariant action including the NS and R sectors was constructed without introducing auxiliary Ramond fields or self-dual constraints. They started with the Wess-Zumino-Wittenis given by one extension of WZW-like formulation [12–17], the other one, the Ramond ory [25, 26] in the early days. This procedure was extended to the case including Ramond fields and the Ramond equations of motion was constructed by introducing the concept of Ramond number projections in [2]. 2See [18–22] for other fascinating approaches using Ramond fields in the large Hilbert space. In this paper, we focus on these two important works [1] and [2], and discuss some interesting properties based on Wess-Zumino-Witten-like point of view. Particularly, we investigate the following three topics and obtain some exact results. 1. We show that one can add the t-dependence of Ramond string fields into the complete action proposed in [1] and make the t-dependence of the action “topological”, which leads us a natural idea of Wess-Zumino-Witten-like structure including Ra2. We clarify a Wess-Zumino-Witten-like structure including Ramond fields and propose a Wess-Zumino-Witten-like complete action. Then, it is proved that one can carry out all computation of our action using the properties of pure-gauge-like and associated fields only. The action proposed in [1] gives one realization of our WZW-like complete action. 3. On the basis of this WZW-like framework, we construct an action whose equations of motion gives the Ramond equations of motion proposed in [2]. As well as the action proposed in [1], this action also gives another realization of our WZW-like complete action: different parameterization of the same WZW-like structure and action. These facts provide the equivalence of two (WZW-like) theories [1] and [2] on the basis of the same discussion demonstrated in [3]. Then, we can also read the relation giving a field redefinition of NS and R string fields with a partial gauge fixing or a trivial uplift by the same way used in [3, 4] or [5] for the NS sector of open superstrings without stubs. This paper is organized as follows. First, we introduce a t-dependence of Ramond string fields and transform the complete action proposed in [1] into the form which has topological t-dependence in section 1.1. Then, we clarify a Wess-Zumino-Witten-like structure including Ramond fields. In section 2, we propose a Wess-Zumino-Witten-like complete action. We show that our WZW-like complete action has so-called topological parameter dependence in section 2.1 and is gauge invariant in 2.2. In particular, these properties all can be proved by computations based only on the properties of pure-gauge-like fields and associated fields, which is a key point of our construction. (In other words, to obtain the variation of the action, equations of motion, and gauge invariance, one does NOT need exof [1], which would heavily depend on the set up of theory. See section 2.4 for a linear map reproducing the same equations of motion as that proposed in [2]. For this purpose, it is this action also gives another realization of our WZW-like complete action. Utilizing these facts, we discuss the equivalence of two theories [1] and [2] in section 3.3. We end with some conclusions. Some proofs are in appendix A. Notation of graded commutators. In this paper, we write [[d1, d1]] for the graded commutator of two operators d1 and d2, [[d1, d2]] ≡ d1 d2 − (−)d1d2 d2 d1. [[A, B]]∗ ≡ m2(A, B) − (−)ABm2(B, A). Complete action and topological t-dependence In this section, we use the same notation as [1]. First, we show that one can add a parameter dependence of R string fields into Kunitomo-Okawa’s action, and that the resultant action has topological parameter dependence. Next, from these computations, of the Ramond sector. We end this section by introducing a Wess-Zumino-Witten-like form of Kunitomo-Okawa’s action. State space and XY -restriction. First, we introduce the large and small Hilbert spaces. The large Hilbert space H is the state space whose superconformal ghost sector is Next, we consider the restriction of the state space. Let X be a picture-changing XY X = X, Y XY = Y, QX = XQ, We use this restricted small Hilbert space HR as the state space of the Ramond string See (2.25) of [1]. a Grassmann even and ghost-and-picture number (0|0) state in the large Hilbert space H, is given by is given by where Q is the BRST operator of open superstrings, hA, Bi is the BPZ inner product of and the invertible linear map F , the full action ANS(t), F (t) also satisfies F (t = 0) = 0 and F (t = 1) = F . Note that F has no ghost-andt do not need the explicit form of F . See [1] or appendix A for the explicit form of F . (See also section 2.4.) on the state space. We show that a complete action of open superstring field theory proposed in [1] can be written as S = − we also show that t-dependence of (1.1) is topological the large Hilbert space H. One can check that ∂t 2 one can also check that (1.3) + (1.4) = As a result, our t-integrated form of the action (1.1) becomes − hΨR(t), F (t)ΨR(t)i +h∂tΨR(t), F (t)ΨR(t)i. (1.4) = − hΨR(t), ∂t F (t)ΨR(t) − F (t)∂tΨR(t)i. The second line is the original form used in [1], but we do not use the second line expression to show that the variation of the action (1.1) is given by (1.2). Translation from the second line to the first line is in appendix A. Since the variation of the first term of the first line in (1.5) is = hξY δΨR, QF ΨRi + hAδNS, m2(F ΨR, F ΨR)i, we obtain (1.2) and the action S has topological t-dependence. These states naturally appear in gauge transformations of the action. The action S has tion (1.1) into our Wess-Zumino-Witten-like form. AδNΛSNS = QΛNS + F ΨR, F Ξ[[F ΨR, ΛNS]]∗ ∗, δΛNS ΨR = XηF ΞDηNS[[F ΨR, ΛNS]]∗, Using properties of F , one can check that the R associated field AdR satisfies Utilizing these expressions, the action becomes ∗ − In the work of [1], all computations of the variation of the action, equations of motion, and gauge invariance heavily depend on the explicit form or properties of the linear map from WZW-like properties of the Ramond sector: (1.11), (1.12), and (1.14). Wess-Zumino-Witten-like complete action We first summarize Wess-Zumino-Witten-like relations of the NS sector and the R sector separately, and show that these relations indeed provide the topological parameter dependence of the action. Second, coupling NS and R, we give a Wess-Zumino-Witten-like complete action and prove that the gauge invariance of the action is also derived from the WZW-like relations. Lastly, we introduce a notation unifying separately given results of NS and R sectors, and another form of the action, which we call a single functional form. WZW-like structure and XY -projection Neveu-Schwarz sector. ANS = AdNS[ϕ] is a ghost-and-picture number (dg|dp) state satisfying d By definition of (2.1) and (2.2), one can check that the relation DηNS d1AdN2S − (−)d1d2 d2AdN1S − (−)d1d2 AdN1S, AdN2S ∗ = 0 Utilizing these (functional) fields, an NS action is given by SNS[ϕ] = − by that of ϕ(t). It is known that the variation of the NS action is given which we call the topological parameter dependence of WZW-like action. See [3, 5, 11, 15]. picture number (1| − 21 ) state satisfying Ramond sector. ant action Wess-Zumino-Witten-likely, whose parameter dependence is topological. We propose that an R action is given by SR[ϕ] = − This SR is Wess-Zumino-Witten-like. In other words, SR has topological t-dependence = ∂ = ∂ = ∂ Topological t-dependence of S . First, we consider the variation of the first term of SR. This term consists of two ingredients: dt hξY ∂t(PηAηR), Q(PηAηR)i − h∂t(PηAηR), AηRi . We can quickly find that the first part has topological t-dependence Note however that the variation of the second ingredient provides an extra term Second, we compute the variation of the second term of SR. Using (2.3), (2.7) for = h∂AδNS, m2(AηR, AηR)i + h[[At , AδNS]]∗, m2(AηR, AηR)i + hAtNS, [[AηR, δAηR]]∗i NS As a result, the variation of the first term of SR is given by hAδNS, m2(AηR, AηR)i − hAδNS, [[AηR, ∂tAηR]]∗i − h[[AδNS, AtNS]]∗, m2(AηR, AηR)i + hδAηR, [[AηR, AtNS]]∗i hAδNS, m2(AηR, AηR)i − h[[AηR, AδNS]]∗, [[AηR, AtNS]]∗i + h[[AηR, AδNS]]∗, ∂tAηRi + hδAηR, [[AηR, AtNS]]∗i + hDηNSAδR, [[AηR, AtNS]]∗i + h[[AηR, AδNS]]∗, DηNSAtR]]∗i In particular, from the forth line to the last line, we applied = h[[AηR, AδNS]]∗, DηNSAtRi − h[[AηR, AδNS]]∗, [[AηR, AtNS]]∗i. − h[[AδNS, AtNS]]∗, [[AηR, AηR]]∗i = hAδNS, [[AηR, Aη ]]∗, AtNS ∗i = −hAδNS, [[AηR, At ]]∗, AηNS ∗i R R As a result, the variation of the second term of SR is given by because of the following relation Hence, (2.10) plus (2.11) provides that SR has topological t-dependence (2.9). WZW-like complete action We propose a Wess-Zumino-Witten complete action and show its gauge invariance on the basis of WZW-like relations (2.1)–(2.3) and (2.6)–(2.8). Action and equations of motion. We propose that a Wess-Zumino-Witten-like complete action is given by Swzw[ϕ] ≡ SNS[ϕ] + SR[ϕ] = − and SR have topological t-dependence, the variation of the action SWZW is given by where AηNS/R = AηNS/R[ϕ] and ANS = AδNS[ϕ] are functionals of the dynamical string field t ϕ, which is the end point of the path ϕ(1). We therefore obtain the equations of motion all belong to the large Hilbert space. The action is invariant under two types of gauge AδNΛS = QΛNS + AηR, ΛR ∗, δΛ(PηAηR) = −PηQ ηΛR − AηNS, ΛR AδNΩS = ηΩNS − AηNS, ΩNS ∗, AδNΛS = QΛNS + AηR, ΛR ∗, δΛ(PηAηR) = −PηQ ηΛR − AηNS, ΛR First, we consider the first term of (2.19) with (2.17). This term consists of two −hδΛ(PηAηR), PξAηRi = hPηQ DηNSΛR − AηR, ΛNS = −hDηNSΛR − AηR, ΛNS ∗, QAηRi. Next, we compute the second term of (2.19) with (2.16). = −h AηR, ΛR , DηNSANQSi − hΛNS, AηR, QAηR ∗ = hDηNSΛR, AηR, ANQS ∗i − h AηR, ΛNS ∗, QAηRi. = 0. Hence, we obtain ∗ − −δΩSwzw = hDηNSΩNS, QAηNS + m2(AηR, AηR)i = 0. We introduce a notation which is useful to unify the results of NS and R sectors. Then, the concept of Ramond number projections proposed in [2] naturaly appears. We say Ramond number of the k-product Mk is n when number of R imputs of Mk minus number of R output of Mk equals to n. The symbol Mk|n denotes the k-product projected onto Ramond number n. For example, R number 0 and 2 projection of the star product m2 are It is helpful to specify whether the (output) state A is NS or R. We write A|NS for the NS (output) state and A|R for the R (output) state. For example, for the sum of NS and R Then, we can write as follows: (NS + R)|NS = NS, (NS + R)|R = R. m2(NS + R, NS + R) 0 m2(NS + R, NS + R) 2 NS = m2(NS, NS), NS = m2(R, R), m2(NS + R, NS + R) 0R = m2(NS + R, NS + R) 2R = 0. NS, R ∗, Pure-gauge-like fields and associated fields. We can introduce a pure-gauge-like (functional) field including both NS and R sectors NS ≡ ηAηNS − m2 AηNS, AηNS = 0, which are just the defining equations of NS and R pure-gauge-like fields (2.1) and (2.6) respectively.3 Similarly, we can also define an associated field of d including both sector such that Ad = Ad[ϕ] satisfies While it seems to be trivial for associative open string field theory, it would be highly nontrivial for closed should be cralified. Ad[ϕ] ≡ AdNS[ϕ] + AdR[ϕ] vide the defining equations of NS and R pure-gauge-like (functional) fields (2.1) and (2.6) Action and equations of motion. In this notation, our Wess-Zumino-Witten-like complete action is given by Swzw[ϕ] = − Aδ∗ ≡ ANS + ξY δ(PηAηR) and the equations of motion is given by δ whose role is explained in section 2.4. Note that the projection onto Ramond number 2 which reproduces NS and R equations of motion (2.14) and (2.15) by NS and R out-puts projections respectively. When we consider another parametrization of the action and its relation to the parametrization given in section 1.1, this notation would be useful. Single functional form WZW-like action. 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. In the rest of this section, we show that one can rewrite the WZW-like action into a form which consists We notice that in the first line of (1.5), while the R term consists of a single functional the following definition of F , which is independent4 of our choice of dynamical string fields, F ≡ definition of F , the following relation holds: It implies that we can convert the graded commutator of operators this relation, one can quickly find that as well as the original WZW-like one, this single functional form also has the topological t-dependence. In the following computations, we form of the complete action (2.30). The variation of the NS action is given by which is the extra term, becomes action into a single functional form and reminds us constraints on the state space spanned Another parametrization algebras or coalgebraic computations see, for example, [2–5, 11, 23, 24, 48] or other mathematical manuscripts [32–34]. In the work of [2], the on-shell conditions of superstring field theories are proposed. For open superstring field theory, it is given by = 0, state space H from T (H) = L = 0, = 0. Note also that in general, the cohomomorphism Gb is constructed by the path-ordered Single functional form. Similarly, we can write the following form of the action using Gb ≡ P exp the Ramond equations of motion proposed in [2]. The proofs of required properties are in Parametrization inspired by Ramond equations of motion. We can construct an (1| − 21 satisfy the defining properties of pure-gauge-like fields: satisfy the defining properties of associated fields: which we prove in section 3.2. Once the defining properties (3.7), (3.8), (3.11), and (3.12) AedNS, AedR defined by (3.9), (3.10), we can construct a gauge invariant action on the basis of Wess-Zumino-Witten-like framework proposed in section 2. Consistency with the XY-projection. To apply our WZW-like framework, we need by (3.6) does not satisfy this property. Note, however, that if we can take Gb satisfying automatically satisfy (3.13) because is given by Gb ≡ PeR dt ξM(t) = I + ξ M2 + We would like to emphasize that it does not necessitate the ciclic property of Gb or Mf to construct the Wess-Zumino-Witten-like complete action. We need the ciclic property of impose (3.13) in a consistent way with the definitions of pure-gauge-like fields (3.5), (3.6). Action and gauge invariance. Utilizing pure-gauge-like and associated fields satisfying (3.7), (3.8), (3.11), (3.12), and (3.13), we construct the Wess-Zumino-Witten-like section 2, the variation of the action is given by and the action is invariant under two types of gauge transformations: the gauge transfor∗ − restricted small Hilbert space HR. Equations of motion. Since the action Se has topological t-dependence and its variation is given by (3.16), we obtain the equations of motion Pη QAeηR[Ψe ] = Pη π1 Q + m2|2 Gb = 0, = 0, which is equivalent to (3.1). While the NS out-put of the equations of motion (3.17) is the same as (3.2), the R out-put (3.18) is equal to the small Hilbert space component of (3.3). Kinetic term. It is interesting to compare kinetic terms of (3.15) and (1.1). In the present parametrization of (3.15), the kinetic term of Se is given by Note that the Ramond kinetic term is just equal to that of Kunitomo-Okawa’s action. Similarly, we quickly check that the NS kinetic term is equivalent to that of Kunitomospectrum as that of [1]. For example, one can use Q, Q + m2|2, and so on for a, and various Gb appearing (Da)2 = 0, = 0. = 0. Gb Gb −1 a Gb η Gb −1 = Gb Da η Gb−1 m2|2 and Gb introduced in (3.14), namely, a gauge product Gb given by [2] with the choice 1 − Ae 1 − Ae 1 − Ae 1 − Ae 1 − Ae 1 − Ae which is just equivalent to the condition (3.7) and (3.8) characterizing NS and R purepure-gauge-like fields. NS and R pure-gaugel-like fields. equation consists of NS and R out-puts = 0, = 0, because it becomes a trivial NS state solution of the Maurer-Cartan equation as follows = 0. Note that π1Dη(1 − AeηNS)−1 = 0 is equal to because it becomes a trivial R state solution of the Maurer-Cartan equation as follows = 0. Shift of the dual A∞ ⊗ B1 ⊗ ⊗ · · · ⊗ ⊗ Bn ⊗ Dη ≡ η − m2|0. In particular, we write DηB for [B]ηAη : = 0 because now we consider ⊗ B ⊗ ∗ − ⊗ B ⊗ = 0. NS and R associated fields. Let d be a coderivation constructed from a derivation d of (−)dd because one can directly check (−)dd = (−)dGG−1 d G the derivation operator d. Note that the response of d acting on the group-like element of = (−)dd NS because one can directly check = ηAedR − AeηNS, Aed ∗ − AeηR, AedNS ∗. R Rii − indeed satisfy WZW-like relations (3.7), (3.8), (3.11), and (3.12). Namely, (3.15) is Equivalence of EKS and KO theories by ϕ, we proposed the Wess-Zumino-Witten-like complete action Swzw[ϕ] = − We found that one realization of this WZW-like complete action is given by setting which is just Kunitomo-Okawa’s action proposed in [1]. This is the WZW-like theory action, which was proposed in section 3.1 and checked in section 3.2, is given by setting which reproduces the Ramond equations of motion proposed in [2]. This is the WZW-like and (3.15) have the same spectrum. As a result, we obtain the equivalence of two theories proposed in [1] and [2], which are different parametrizations of (2.30). See also [3, 11]. In other words, since both (3.28) and (3.29) have the same WZW-like structure and in the same way as [3]. Then, the identification of pure-gauge-fields trivially provides the equivalence of two actions (1.1) and (3.15): the single functional form of (2.36) gives the equivalence of two theories. When we use the WZW-like form of (2.30), it seems that (3.30) indirectly gives the equivalence and does not directly give a in [5], if we start with = Aet, One can check that the same logic used for the NS sector in [5] also goes in the case including the R sector because WZW-like relations exist as we explained, which is in appendix A. In this paper, we have clarified a Wess-Zumino-Witten-like structure including Ramond fields and proposed one systematic way to construct gauge invariant actions, which we call some dynamical string field ϕ is constructed, one obtain one realization of our WZW-like complete action Swzw[ϕ] parametrized by ϕ. On the basis of this way, we have constructed an action Se whose on-shell condition is equivalent to the Ramond equations of motion WZW-like structure and action, which implies the equivalence of two theories [1, 2]. Let us conclude by discussing future directions. Closed superstring field theories. It would be interesting to extend the result of [1] to closed superstring field theories [49]. We expect that our idea of WZW-like structure and action also goes in heterotic and type II theories if the kinetic terms are given by the same A∞/L∞ structures are discussed in [11]. Quantization and supermoduli. We would have to quantize the (WZW-like) complete action and clarify its relation with supermoduli of super-Riemann surfaces [35–39] to obtain a better understandings of superstrings from recent developments in field theoretical approach. The Batalin-Vilkovisky formalism [43, 44] is one helpful way to tackle these problems: a quantum master action is necessitated. As a first step, it is important to then the classical Batalin-Vilkovisky quantization is straightforward. A positive answer is now provided in [50] for open superstring field theory without stubs. It would also be helpful to clarify more detailed relations between recent important developments. Acknowledgments The author would like to thank Keiyu Goto, Hiroshi Kunitomo, and Yuji Okawa for comments. The author also would like to express his gratitude to his doctors, nurses, and all the staffs of University Hospital, Kyoto Prefectural University of Medicine, for medical treatments and care during his long hospitalization. This work was supported in part by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists. Basic facts and some identities We summarize important properties of the BPZ inner product and give proofs of some relations which we skipped in the text. BPZ properties. H has the following properties with the BRST operator Q and the Witten’s star product m2: hA, Bi = (−)ABhB, Ai, hA, QBi = −(−)AhQA, Bi, hA, m2(B, C)i = (−)A(B+C)hB, m2(C, A)i. Associated fields in Kunitomo-Okawa theory. In the work of [1], for any state B ∈ H, the linear map F is defined by F B ≡ X B = AηR ≡ F ΨR = F ηξΨR = DηNSF ΞΨR = DηNSF ΞAηR. AdR ≡ F Ξ (−)ddΨR + [[AηR, AdNS]]∗ + η[[d, Ξ]]AηR . The original form of Kunitomo-Okawa’s action. The original form of KunitomoOkawa’s complete action is = hΨR, F ΨRi = − hξY ΨR, ηXF ΨRi. As we explained in section 1.1, this is equal to (1.1). We check that the identification At = ANS + AtR and Aet = AetNS + AetR provide a field redefinition of (ΦNS, ΨR) and (ΦeNS, ΨeR) t INS(t) = INS(t), AtNS(t) ∗, IR(t) = IR(t), AtNS(t) ∗ + INS(t), AtR(t) ∗. IR(t) = IR(t), AtNS(t) ∗ Under the identification At =∼ Aet, we ⊗ At ⊗ Open Access. 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Hiroaki Matsunaga. Comments on complete actions for open superstring field theory, Journal of High Energy Physics, 2016, 115, DOI: 10.1007/JHEP11(2016)115