A ∞ /L ∞ structure and alternative action for WZWlike superstring field theory
Received: November
structure and alternative action for WZWlike 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 , Setagayaku, Tokyo 1540012 , Japan
3 Komaba , Meguroku, Tokyo 1538902 , 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 NSNS superstring field theories. They are based on the large Hilbert space, and have WessZuminoWittenlike expressions which are the Z2reversed versions of the conventional WZWlike actions. On the basis of the procedure proposed in arXiv:1505.01659, we show that our new WZWlike 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 WZWlike 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
NSNS sector
Alternative WessZuminoWittenlike 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
selfdual gauge theory and supergravity [5]. One trigger of these developments was given
NS string [6], heterotic NS string, and closed NSNS 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 Smatrices of
perturbative superstring theory with insertions of picturechanging 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 WZWlike 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 WZWlike 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 Qexact terms, which implies the onshell states match each other. See [32] for more details.
Conventional WZWlike form. It was shown in [19] that Berkovits action can be
written as a WZWlike form
The equation of motion is given by the tindependent form
We write m2(A, B) for the star product of A and B. A significant feature of this WZWlike
form of the action is that one can obtain all properties of the action by using not explicit
which we call WZWlike relations:
gaugelike field,5 gives a solution of the MaurerCartan 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 dderivation properties of M B. These properties yield
the WZWlike relations, and then, the constraint equation and the equation of motion are
Constraint (CWZW) :
E.O.M. (CWZW) :
This construction is extended to more generic case: open NS superstrings with stubs,
heterotic NS strings [19], and closed NSNS 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 puregaugelike
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 (CWZW) : Q AeQ[ϕ] + ms2t AeQ[ϕ], AesQt[ϕ] + X msnt zAesQt[ϕ], . . . , AeQ[ϕ] = 0,
st st } st {
E.O.M. (CWZW) :
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 WZWlike form. It is known that Berkovits action can be also written
as alternative WZWlike form
Then, the equation of motion is given by the tindependent form
Dη∗ Ω ≡ η Ω − m2 Aη[Φ], Ω − (−)Ωm2 Ω, Aη[Φ] .
A significant feature of this alternative WZWlike form is that, as the conventional case,
one can obtain all properties of the action by using only specific algebraical relations of
alternative WZWlike functionals, which we also call WZWlike relations :
the category of socalled the WZWlike formulation. While the first relation provides the
Note that this type of WZWlike theory, a dual version of the conventional WZWlike
theory, is characterized by the pair of equations
Constraint (AWZW) :
E.O.M. (AWZW) :
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 WZWlike actions which are offshell equivalent
Main results of this paper.
What is the starting point of this type of WZWlike
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 WZWlike 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],
WZWlike 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 WZWlike action for generic open NS strings just reduces to the alternative
WZWlike 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
WZWlike 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 NSNS strings in appendix E), as well as that for open NS
strings with stubs which we introduced in (1.5). These actions are Z2reversed 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 WZWlike 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 WZWlike action: the R string field couples to the Berkovits theory for the NS
sector gaugeinvariantly on the basis of (not the conventional but) this type of WZWlike
gauge structure. We would like to mention that although we expect that our new WZWlike
actions are also equivalent to the conventional WZWlike actions, the all order equivalence
of these has not been proven: we will show lowerorder equivalence to the conventional
WZWlike actions only.
action can be written in our (alternative) WZWlike 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 WZWlike relations, the Z2reversed versions of the conventional
WZWlike relations given in [19], which guarantees the gauge invariance of the action. The
onshell condition and the gauge transformation of the (alternative) WZWlike action are
derived by using only the (alternative) WZWlike relations. We also see how the gauge
tion 3, we provide another realisation of the alternative WZWlike action using the string
field V in the large Hilbert space. The functionals satisfying our WZWlike relations can
be defined by the differential equations which are the Z2reversed versions of those given
parameterizations of the same WZWlike 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 WZWlike form. Appendices D and
E are devoted to the open NS and the closed NSNS theories, respectively.
that the functionals appearing in the action satisfy alternative WZWlike relations, the
Z2reversed version of the conventional WZWlike 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 pathordered exponential of the
Here ←(→) on P denotes that the operator at later time acts from the right (left). The
differential equations hold since the pathordered 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 BPZodd, 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)!
tdependence 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 Z2reversed versions of conventional WZWlike 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.
WZWlike 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 WZWlike relations (2.47)
and (2.48), and the gauge invariances also follows from them, which can be seen in a
similar (but Z2reversed) 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
WZWlike 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 tdependence
Equations of motion.
One can derive the onshell condition from the variation of the
action in the WZWlike 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 WZWlike 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 WZWlike 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 BatalinVilkovisky
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].
onshell 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 Z2reversed 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 puregaugelike and
the associated functional fields can be defined by the differential equations, which are the
Z2reversed versions of the construction in [19]. Then, the WZWlike 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 Z2reversed version of those in [19] give
these parameterizations so that WZWlike relations (2.47) and (2.48) hold.
these functionals, a new gauge invariant action for the string field V is constructed in the
(alternative) WZWlike 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 WZWlike formulation, can be
problem to understand the relation between two cohomomorphisms EV and Gb.
Ramond sector of closed superstring fields. The concept of the WZWlike structure
of heterotic string field theory including both NS and R sectors. As [2], our WZWlike
It is expected that as demonstrated in [4] for open superstrings, the WZWlike structure
It is also expected that the action for closed NSR and RNS strings can be constructed
in the same manner as that for heterotic string. For closed RR 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 WZWlike 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 Z2graded 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 Z2graded vector space, and its grading which
Fock space of closed superstrings.
Multilinear 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 BPZconjugation 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 grouplike element is given by
Projector and grouplike element.
is called a grouplike 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 grouplike
Utilizing the projector and the grouplike element, the MaurerCartan 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 WZWlike form. In other words, we provide a
parameterization of the puregaugelike 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 WZWlike 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 WZWlike form,
SEKS[Ve ] =
Its variation and the onshell 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 WZWlike 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 WZWlike structure, we give two realizations of this type of WZWlike action using two
For example, one can use Q, Q + m22, 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.
NeveuSchwarz 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 tdependence 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 MaurerCartan 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 puregaugelike (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 WZWlike relation that associated (functional) field must
Ad[τ ; Φ] − dΦ − Φ, Ad[τ ; Φ] Aη[τ;Φ] .
differential equation
NSNS sector
Dual L∞ products.
Using pathordered exponential map Gb , the NSNS 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 WessZuminoWittenlike 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 WZWlike action Sηηe[ϕ]
gauge invariant action
whose gauge transformations are given by
equation of motion is given by tindependent form
One can find these facts by using WZWlike 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 tdependence. 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 grouplike 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 WessZuminoWittenlike 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 WZWlike NSNS theory [21]. Thus one may consider
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[2] H. Kunitomo and Y. Okawa, Complete action for open superstring field theory,
[4] H. Matsunaga, Comments on complete actions for open superstring field theory,
JHEP 11 (2016) 115 [arXiv:1510.06023] [INSPIRE].
[5] A. Sen, Covariant action for type IIB supergravity, JHEP 07 (2016) 017
JHEP 11 (2015) 199 [arXiv:1506.05774] [INSPIRE].
[8] T. Erler, S. Konopka and I. Sachs, Ramond equations of motion in superstring field theory,
Nucl. Phys. B 314 (1989) 209 [INSPIRE].
the consistency problem for open superstring field theory, Nucl. Phys. B 341 (1990) 464
[14] R. Saroja and A. Sen, Picture changing operators in closed fermionic string field theory,
Phys. Lett. B 286 (1992) 256 [hepth/9202087] [INSPIRE].
[16] N. Berkovits, SuperPoincar´e invariant superstring field theory, Nucl. Phys. B 450 (1995) 90
[18] Y. Okawa and B. Zwiebach, Heterotic string field theory, JHEP 07 (2004) 042
[19] N. Berkovits, Y. Okawa and B. Zwiebach, WZWlike action for heterotic string field theory,
[22] N. Berkovits and C.T. Echevarria, Four point amplitude from open superstring field theory,
superstring field theory, arXiv:1505.01659 [INSPIRE].
perspective, JHEP 10 (2015) 157 [arXiv:1505.02069] [INSPIRE].
JHEP 02 (2016) 121 [arXiv:1510.00364] [INSPIRE].
string theory, Nucl. Phys. B 271 (1986) 93 [INSPIRE].
foundations, Nucl. Phys. B 505 (1997) 569 [hepth/9705038] [INSPIRE].
Nucl. Phys. B 390 (1993) 33 [hepth/9206084] [INSPIRE].
Phys. Lett. B 102 (1981) 27 [INSPIRE].
[41] H. Kajiura, Noncommutative homotopy algebras associated with open strings,
[1] A. Sen, Gauge invariant 1PI effective superstring field theory: inclusion of the Ramond [3] A. Sen, BV master action for heterotic and type II string field theories, JHEP 02 (2016) 087 [6] T. Erler, S. Konopka and I. Sachs, Resolving Witten's superstring field theory, [7] T. Erler, S. Konopka and I. Sachs, NSNS sector of closed superstring field theory, [9] S. Konopka, The Smatrix of superstring field theory, JHEP 11 (2015) 187 [10] E. Witten, Interacting field theory of open superstrings, Nucl. Phys. B 276 (1986) 291 [11] C. Wendt, Scattering amplitudes and contact interactions in Witten's superstring field theory, [12] I. Ya. Arefeva, P.B. Medvedev and A.P. Zubarev, New representation for string field solves [13] C.R. Preitschopf, C.B. Thorn and S.A. Yost, Superstring field theory, [15] B. Jurˇco and K. Muenster, Type II superstring field theory: geometric approach and operadic [17] N. Berkovits, A new approach to superstring field theory, Fortsch. Phys. 48 (2000) 31 [20] H. Matsunaga, Construction of a gaugeinvariant action for type II superstring field theory, [21] H. Matsunaga, Nonlinear gauge invariance and WZWlike action for NSNS superstring field [23] Y. Iimori, T. Noumi, Y. Okawa and S. Torii, From the Berkovits formulation to the Witten formulation in open superstring field theory, JHEP 03 (2014) 044 [arXiv:1312.1677] [24] N. Berkovits, The Ramond sector of open superstring field theory, JHEP 11 (2001) 047 [25] Y. Michishita, A covariant action with a constraint and Feynman rules for fermions in open superstring field theory, JHEP 01 (2005) 012 [hepth/0412215] [INSPIRE]. [26] H. Kunitomo, The Ramond sector of heterotic string field theory, Prog. Theor. Exp. Phys. 2014 (2014) 043B01 [arXiv:1312.7197] [INSPIRE]. [27] H. Kunitomo, Firstorder equations of motion for heterotic string field theory, Prog. Theor. Exp. Phys. 2014 (2014) 093B07 [arXiv:1407.0801] [INSPIRE]. [28] 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]. [29] T. Erler, Y. Okawa and T. Takezaki, A∞ structure from the Berkovits formulation of open [30] T. Erler, Relating Berkovits and A∞ superstring field theories; small Hilbert space [31] T. Erler, Relating Berkovits and A∞ superstring field theories; large Hilbert space perspective, [32] K. Goto and H. Matsunaga, Onshell equivalence of two formulations for superstring field [33] D. Friedan, E.J. Martinec and S.H. Shenker, Conformal invariance, supersymmetry and [34] E. Witten, Noncommutative geometry and string field theory, Nucl. Phys. B 268 (1986) 253 [35] M.R. Gaberdiel and B. Zwiebach, Tensor constructions of open string theories. 1: [36] B. Zwiebach, Closed string field theory: quantum action and the BV master equation, [37] I.A. Batalin and G.A. Vilkovisky, Gauge algebra and quantization, [38] 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]. [39] J. Gomis, J. Paris and S. Samuel, Antibracket, antifields and gauge theory quantization, [40] T. Lada and J. Stasheff, Introduction to SH Lie algebras for physicists,