#### 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.
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.
JHEP 11 (2015) 199 [arXiv:1506.05774] [INSPIRE].
superstring field theory, arXiv:1505.01659 [INSPIRE].
perspective, JHEP 10 (2015) 157 [arXiv:1505.02069] [INSPIRE].
JHEP 02 (2016) 121 [arXiv:1510.00364] [INSPIRE].
theory, arXiv:1506.06657 [INSPIRE].
[arXiv:1507.08250] [INSPIRE].
[arXiv:1504.00609] [INSPIRE].
[arXiv:1508.02481] [INSPIRE].
[arXiv:1508.05387] [INSPIRE].
[hep-th/9912121] [INSPIRE].
[hep-th/0406212] [INSPIRE].
superstring field theory, arXiv:1512.03379 [INSPIRE].
[Erratum ibid. B 459 (1996) 439] [hep-th/9503099] [INSPIRE].
JHEP 11 (2004) 038 [hep-th/0409018] [INSPIRE].
[19] Y. Michishita, A covariant action with a constraint and Feynman rules for fermions in open
superstring field theory, JHEP 01 (2005) 012 [hep-th/0412215] [INSPIRE].
Prog. Theor. Exp. Phys. 2014 (2014) 043B01 [arXiv:1312.7197] [INSPIRE].
JHEP 04 (2014) 150 [arXiv:1312.2948] [INSPIRE].
JHEP 08 (2014) 158 [arXiv:1403.0940] [INSPIRE].
Nucl. Phys. B 314 (1989) 209 [INSPIRE].
theories, Nucl. Phys. B 276 (1986) 366 [INSPIRE].
field theory, Phys. Lett. B 173 (1986) 134 [INSPIRE].
[hep-th/0109100] [INSPIRE].
theory, JHEP 09 (2015) 011 [arXiv:1407.8485] [INSPIRE].
Phys. Lett. B 208 (1988) 416 [INSPIRE].
Illinois J. Math. 34 (1990) 256.
Rev. Mod. Phys. 60 (1988) 917 [INSPIRE].
hep-th/9706033 [INSPIRE].
JHEP 06 (2015) 022 [arXiv:1411.7478] [INSPIRE].
sector, JHEP 08 (2015) 025 [arXiv:1501.00988] [INSPIRE].
Phys. Lett. B 102 (1981) 27 [INSPIRE].
Nucl. Phys. B 390 (1993) 33 [hep-th/9206084] [INSPIRE].
the Ramond sector, arXiv:1606.07194 [INSPIRE].
Nucl. Phys. B 337 (1990) 363 [INSPIRE].
[1] H. Kunitomo and Y. Okawa , Complete action for open superstring field theory, Prog . Theor. Exp. Phys . 2016 ( 2016 ) 023B01 [arXiv:1508 .00366] [INSPIRE].
[2] T. Erler , S. Konopka and I. Sachs , Ramond equations of motion in superstring field theory , [3] T. Erler , Y. Okawa and T. Takezaki , A∞ structure from the Berkovits formulation of open [4] T. Erler , Relating Berkovits and A∞ superstring field theories; small Hilbert space [5] T. Erler , Relating Berkovits and A∞ superstring field theories; large Hilbert space perspective , [6] K. Goto and H. Matsunaga , On-shell equivalence of two formulations for superstring field [7] S. Konopka , The S-matrix of superstring field theory , JHEP 11 ( 2015 ) 187 [8] A. Sen and E. Witten , Filling the gaps with PCO's , JHEP 09 ( 2015 ) 004 [9] A. Sen , Supersymmetry restoration in superstring perturbation theory , JHEP 12 ( 2015 ) 075 [10] A. Sen , BV master action for heterotic and type II string field theories , JHEP 02 ( 2016 ) 087 [11] K. Goto and H. Matsunaga , A∞/L∞ structure and alternative action for WZW-like [12] N. Berkovits , Super-Poincar´e invariant superstring field theory, Nucl . Phys . B 450 ( 1995 ) 90 [13] N. Berkovits , A new approach to superstring field theory, Fortsch . Phys. 48 ( 2000 ) 31 [14] Y. Okawa and B. Zwiebach , Heterotic string field theory , JHEP 07 ( 2004 ) 042 [15] N. Berkovits , Y. Okawa and B. Zwiebach , WZW-like action for heterotic string field theory , [20] H. Kunitomo , The Ramond sector of heterotic string field theory , [21] H. Kunitomo , First-order equations of motion for heterotic string field theory, Prog . Theor. Exp. Phys . 2014 ( 2014 ) 093B07 [arXiv:1407 .0801] [INSPIRE].
[22] H. Kunitomo , Symmetries and Feynman rules for the Ramond sector in open superstring field theory, Prog . Theor. Exp. Phys . 2015 ( 2015 ) 033B11 [arXiv:1412 .5281] [INSPIRE].
[23] T. Erler , S. Konopka and I. Sachs , Resolving Witten 's superstring field theory, [24] T. Erler , S. Konopka and I. Sachs , NS-NS sector of closed superstring field theory , [25] E. Witten , Interacting field theory of open superstrings, Nucl . Phys . B 276 ( 1986 ) 291 [26] C. Wendt , Scattering amplitudes and contact interactions in Witten's superstring field theory , [27] Y. Kazama , A. Neveu , H. Nicolai and P.C. West , Symmetry structures of superstring field [28] H. Terao and S. Uehara , Gauge invariant actions and gauge fixed actions of free superstring [16] H. Matsunaga , Construction of a gauge-invariant action for type II superstring field theory , [17] H. Matsunaga , Nonlinear gauge invariance and WZW-like action for NS-NS superstring field [29] J.P. Yamron , A gauge invariant action for the free Ramond string , [30] T. Kugo and H. Terao , New gauge symmetries in Witten's Ramond string field theory , [31] E. Witten , Noncommutative geometry and string field theory, Nucl . Phys . B 268 ( 1986 ) 253 [18] N. Berkovits , The Ramond sector of open superstring field theory , JHEP 11 ( 2001 ) 047 [32] E. Getzler and J.D.S. Jones , A∞-algebras and the cyclic bar complex , [33] M. Penkava and A.S. Schwarz , A∞ algebras and the cohomology of moduli spaces , Trans.
Amer. Math. Soc. 169 ( 1995 ) 91 [hep-th /9408064] [INSPIRE].
[34] H. Kajiura , Noncommutative homotopy algebras associated with open strings , Rev. Math. Phys . 19 ( 2007 ) 1 [math/0306332] [INSPIRE].
[35] E.P. Verlinde and H.L. Verlinde , Multiloop calculations in covariant superstring theory , [36] E. D'Hoker and D.H. Phong , The geometry of string perturbation theory, Phys . Lett . B 286 ( 1992 ) 256 [hep-th /9202087] [INSPIRE].
[37] R. Saroja and A. Sen , Picture changing operators in closed fermionic string field theory , [38] A. Belopolsky , Picture changing operators in supergeometry and superstring theory , [39] E. Witten , Superstring perturbation theory revisited , arXiv:1209 .5461 [INSPIRE].
[40] B. Jurˇco and K. Muenster , Type II superstring field theory: geometric approach and operadic description , JHEP 04 ( 2013 ) 126 [arXiv:1303.2323] [INSPIRE].
[41] A. Sen , Gauge invariant 1PI effective action for superstring field theory , [42] A. Sen , Gauge invariant 1PI effective superstring field theory: inclusion of the Ramond [43] I.A. Batalin and G.A. Vilkovisky , Gauge algebra and quantization , [44] I.A. Batalin and G.A. Vilkovisky , Quantization of gauge theories with linearly dependent generators , Phys. Rev. D 28 ( 1983 ) 2567 [Erratum ibid . D 30 ( 1984 ) 508] [INSPIRE].
[45] A.S. Schwarz , Geometry of Batalin-Vilkovisky quantization, Commun . Math. Phys. 155 ( 1993 ) 249 [hep-th /9205088] [INSPIRE].
[46] N. Berkovits , Constrained BV description of string field theory , JHEP 03 ( 2012 ) 012 [47] B. Zwiebach , Closed string field theory: quantum action and the BV master equation , [48] M.R. Gaberdiel and B. Zwiebach , Tensor constructions of open string theories . 1 : foundations, Nucl. Phys . B 505 ( 1997 ) 569 [hep-th /9705038] [INSPIRE].
[49] K. Goto and H. Kunitomo , Construction of action for heterotic string field theory including [50] T. Erler , Y. Okawa and T. Takezaki , Complete action for open superstring field theory with [51] C.R. Preitschopf , C.B. Thorn and S.A. Yost , Superstring field theory, [52] I. Ya. Arefeva , P.B. Medvedev and A.P. Zubarev , New representation for string field solves the consistency problem for open superstring field theory, Nucl . Phys . B 341 ( 1990 ) 464