On the BV formalism of open superstring field theory in the large Hilbert space
JHE
On the BV formalism of open superstring field theory
Hiroaki Matsunaga 0 2
Mitsuru Nomura 0 1
Gauge Symmetry
0 Komaba , Meguroku, 1538902 , Japan
1 Institute of Physics, University of Tokyo
2 Slovance 2 , Prague 8 , Czech Republic
We construct several BV master actions for open superstring field theory in the large Hilbert space. First, we show that a naive use of the conventional BV approach breaks down at the third order of the antifield number expansion, although it enables us to define a simple “string antibracket” taking the Darboux form as spacetime antibrackets. This fact implies that in the large Hilbert space, “string fieldsantifields” should be reassembled to obtain master actions in a simple manner. We determine the assembly of the string antifields on the basis of Berkovits' constrained BV approach, and give solutions to the master equation defined by Dirac antibrackets on the constrained string fieldantifield space. It is expected that partial gaugefixing enables us to relate superstring field theories based on the large and small Hilbert spaces directly: reassembling string fieldsantifields is rather natural from this point of view. Finally, inspired by these results, we revisit the conventional BV approach and construct a BV master action based on the minimal set of string fieldsantifields.
String Field Theory; BRST Quantization; Superstrings and Heterotic Strings

HJEP05(218)
Minimal set: string fieldsantifields
Gauge reducibility and string ghosts
String antifields and string antibracket
Conventional BV approach
Naive construction breaks down
On the gauge tensor formulae
How to assemble string antifields
Extra fields and constraints
Nonminimal set: constrained string fieldsantifields
Constrained BV approach
Preliminary: constrained BV for partially gaugefixed theory
Constrained BV master action based on improved constraints
Alternative: modifying extra string fields
Other constrained BV master actions: switching M to η
Conventional BV approach revisited
Orthogonal decomposition
BV master action
Concluding remarks
A Notations and basic identities
1 Introduction
2
3
4
5
6
7
2.1
2.2
3.1
3.2
4.1
4.2
5.1
5.2
5.3
5.4
6.1
6.2
1
Introduction
Reducible gauge field theories require ghosts, ghostsforghosts, and higherghosts as much
as necessary, whose gauge algebras may necessitate equations of motion to be closed
algebras. The BatalinVilkovisky (BV) formalism gives a convenient way to describe such
gauge theories, in which the action is expressed in terms of fields and antifields [1–4]. A
string field is an assembly of such spacetime fields and string field theory is an infinitely
reducible gauge theory, in which the BV master action is expressed in terms of string fields
and string antifields [5–13].
In bosonic string field theory, its classical BV master action is constructed by just
relaxing the ghost number restriction of the classical action — there is a readymade
– 1 –
proposed by Berkovits [14], which is characterized by a string field living in the large
Hilbert space [15] and a WessZuminoWittenlike (WZWlike) action having the large
gauge invariance. Using a real parameter t ∈ [0, 1], this WZWlike action is often written
in the following condensed form
S[Φ] = −
Z 1
0
dt D At[tΦ] , Q Aη[tΦ] E
bpz
1
= − 2
Φ , Q η Φ bpz + · · · ,
(1.1a)
where Q is the BRST operator, η denotes the zeromode of the eta ghost, hA, Bibpz is the
BPZ inner product of A, B in the large Hilbert space, and Aη[Φ] = η Φ + · · · is a nonlinear
functional1 of the dynamical string field Φ defined by a solution of the MaurerCartan
equation
0 ≡ η Aη[Φ] − Aη[Φ] ∗ Aη[Φ] .
The dots denote the nonlinear interacting terms. The symbol ∗ denotes Witten’s associative
star product [5, 6]; using it, a solution of (1.1b) is given by Aη[ΦB] = (η eΦB )e−ΦB . Since
Q and η are nilpotent and graded commutative, it is invariant under the large gauge
transformations
δΦ = Q Λ + η Ω + . . . ,
(1.1b)
(1.1c)
where Λ and Ω are appropriate string fields of gauge parameters. This large gauge
symmetry complicates its gaugefixing problem. A master action for the free theory can be
constructed in a simple manner [16–18]; however, its ghostantifield parts or BV
transformations take somewhat different forms from the kinetic term of (1.1a) or the linear part
of (1.1c), respectively. By contrast, a nonlinear BV master action has proven to be difficult
to find even for perturbative one, and it has remained unsolved problem up until now.
There is a more practical option: one can ignore the large Hilbert space and consider
superstring field theory within the small Hilbert space [19]; iff all fields and their gauge
algebras are strictly restricted to be small, one can apply the readymade BV procedure.2
Quantum aspects of superstring fields have been studied by utilizing such a gaugefixable
formulation. It is however expected that this gaugefixable theory is obtained from
superstring field theory in the large Hilbert space via partial gaugefixing [21–23].3 It seems to be
intuitively clear at the classical level, but its validity has remained unclear because we have
not succeeded to construct a BV master action in the large Hilbert space yet. A lack of
understanding of the BV formalism of superstring field theory in the large Hilbert space is
not just an issue for the Berkovits formulation but also a matter of all other formulations.4
1A functional At[tΦ] = ∂t(tΦ) + · · · is determined by given Aη[Φ]; for example, At[tΦB] = (∂tetΦB )e−tΦB .
2Higher string ghosts and their gauge algebra can be large keeping the dynamical string field small, in
which the readymade procedure is not applicable [20]: it requires the BV formalism in the large Hilbert
space.
3Partial gauge fixing is an operation omitting some part Φη of the dynamical field Φ = Φξ+Φη by hand; at
the same time, corresponding gauge degrees of (1.1c) are appropriately omitted by hand — like gauge fixing.
4For including the Ramond sector, see [24, 25] for the large; see [26, 27] for the small. For closed
tions of superstring field theory are currently established; these all can be embedded into
the large Hilbert space and their gauge structures are understood in a unified manner —
at least, at the classical level. In this process, even for a very trivial embedding of a
gaugefixable theory based on the small Hilbert space, the original gauge symmetry is enlarged
as (1.1c) and the WessZuminoWittenlike gauge structure arises [20, 33–35]. Since the
gaugefixing of WZWlike string field theory has been an unsolved problem, this result
implies that a gaugefixable theory turns into a gaugeunfixable theory after the embedding
— so, the BV master action should exist even in the large Hilbert space. To see it, we
consider the simplest situation, the large A∞/L∞ theory: an action in the large Hilbert space
which is defined by the trivial embedding of a gaugefixable action in the small Hilbert
space [23, 33].
In this paper, we construct BV master actions for open superstring field theory in
the large Hilbert space on the basis of several different approaches. Through these
constructions, we would like to clarify the following questions; “How can we apply the BV
formalism to the large theory?”, “Why does our readymade BV procedure not work in
the large Hilbert space?” and “What should we take into account to treat large gauge
symmetries?”, which are our motivations. In the most of this paper, we focus on the
NeveuSchwarz (NS) sector of open superstring field theory, the large A∞ theory. One can
apply the completely same prescriptions to the Ramond sector or to the large L∞ action
for closed superstring field theory.
As an example of the WZWlike action. The large A∞ theory is the simplest but a
nontrivial example of the WZWlike open string field theory [35]: it is completely described
by a pair of mutually commutative A∞ algebras (η ; M); alternatively, it is also described
by the equivalent A∞ pair (η −
action (1.1a) based on another solution of (1.1b) given by
∗ ; Q). The large A∞ action just equals to the WZWlike
Aη[tΦ] ≡ π1 Gb 1 − t η Φ
where Gb is an A∞ morphism satisfying Gb η = (η − ∗)Gb and Gb M = Q Gb given in [23, 33].
Hence, our construction of BV master actions for the large A∞ theory provides evidence
of BV master actions for the WZWlike formulation; it will be a first step to clarify the
BV formalism of WZWlike superstring field theory in the large Hilbert space.
Organization of the article. To apply the BV formalism, we have to analyse the gauge
reducibility of the theory and find the minimal set of the fieldsantifields, which we first
explain in section 2. We also explain that the conventional BV approach provides an elegant
string field representation of the BV antibracket, which is very useful for perturbative
constructions. In section 3, we show that although one can construct a lower order BV master
action up to the second order of the antifield number, there exists no proper solution at
higher order within the (naive) conventional BV approach. Section 4 is devoted to explain
how we can avoid the nogo result of section 3 by using the constrained BV approach [36].
In this approach, the construction of a master action is equivalent to specify the form of an
– 3 –
(unconstrained) action Sbv and constraints Γb. In addition to it, we have to specify how to
assemble extra string fields ϕex in string field theory. So, we have to find out an appropriate
pair (Sbv, Γb, ϕex) giving a proper solution of the constrained BV master equation, which we
explain in section 5. First, we show that Berkovits’ prescription [37] works well and gives
a correct constrained BV master action for partially gaugefixed superstring field theory
in the large Hilbert space. In order to remove this partiallygaugefixing assumption, one
has to impose further constraints, reassemble the extra string fields, or replace the starting
(unconstrained) BV action. Then, we construct appropriate constrained BV actions in the
large Hilbert space (without the partiallygaugefixing assumption) on the basis of several
different prescriptions. In particular, a constrained BV master action obtained in section
5.4 resembles canonical transformations switching Q and ηgauge symmetries [20], and
these properties may be helpful to see what happens in the large theory. In section 6, we
revisit the conventional BV approach. On the basis of remediations inspired by the results
of the constrained BV approach, we construct a BV master action within the conventional
BV approach. Notation, basic identities, and some elementary facts are in appendix A.
2
Minimal set: string fieldsantifields
In this paper, we clarify how to apply the BV formalism to superstring field theory in the
large Hilbert space by using the large A∞ theory — the simplest example of the WZWlike
formulation. As we will see, the classical action of the large A∞ theory
1
S[Φ] = − 2
Φ , Q η Φ bpz − 3
1
bpz − 4
1
Φ , M2 η Φ, η Φ
Φ, M3 η Φ, η Φ, η Φ
is given (or defined) by the trivial embedding of the small A∞ theory. In this section, we
analyse its gauge reducibility and give the minimal set of string fieldsantifields.
Let Φ be a NeveuSchwarz (NS) open superstring field living in the large Hilbert space,
which carries worldsheet ghost number 0 and picture number 0. The large string field Φ
reduces to a small string field Ψ ≡ η Φ by acting η on it; the small string field Ψ satisfies
η Ψ = 0 and carries worldsheet ghost number 1 and picture number −1, which lives in the
small Hilbert space. We write M = Q + M2 + · · · for the NS open superstring products
given by [19]: the gth product Mg carries worldsheet ghost number 2 − g and picture
number g − 1. As a functional of the small dynamical string field Ψ, the small A∞ action
bpz + · · ·
(2.1a)
S′[Ψ] is given by
S′[Ψ] = − 2
1
Ψ , Q Ψ
bpz − 3
1
Ψ , M2(Ψ, Ψ) bpz − 4
1
Ψ , M3 Ψ, Ψ, Ψ
bpz + · · · .
We write hhηA, ηBiibpz for the BPZ inner product of ηA and ηB in the small Hilbert space,
which equals to the BPZ inner product hA, ηBibpz = −(−)AhηA, Bibpz in the large Hilbert
space. This small A∞ theory is easily gaugefixable iff all string fields of gauge parameters
and their gauge algebras are also restricted to the small Hilbert space: one can construct
its BV master action Sb′v by just relaxing ghost number constraint as Sb′v ≡ S′[ψ] where
ψ carries all spacetime and worldsheet ghost numbers. By contrast, one cannot construct
– 4 –
where M = Q + M2 + M3 + · · · denotes the A∞ superstring products and hA, Bi is the
graded symplectic form — it is the suspended BPZ inner product (A.5), but we call it as
“the BPZ inner product” simply. This action is invariant under the following large gauge
transformations
δΦ = π1 M, Λ−1,0 1 − η Φ
+ η Λ−1,1 ,
1
where [[C, D]] denotes the graded commutator of coderivations C and D; see appendix A.
The gauge symmetry (2.2) has the following gauge reducibility
δg+1 δgΛ−g,p = 0 ,
δgΛ−g,p = π1 M, Λ−g−1,p 1 − η Φ
+ η Λ−g−1,p+1 ,
where Λ−g,p denotes a gth string gaugeparameter field and defining Λ0,0 ≡ Φ may be
helpful. While the glabel runs from 0 to infinity, the plabel runs from 0 to g as shown
by [16–18]. Hence, the large A∞ theory is infinitely reducible just as the Berkovits
theory [14].
as follows
As well as the string field Φ, these string gaugeparameter fields Λ−g,p can be expanded
in terms of spacetime gaugeparameter fields λgr,p and worldsheet bases Z−g,pi; see
apr
pendix A for these bases. The BV formalism implies that when string gaugeparameter
fields are given by Λ−g,p = Pr λ−r g,pZ−g,pi, corresponding string ghost fields are obtained
r
by replacing each spacetime parameter field λ−r g,p with corresponding spacetime ghosts φgr,p
For simplicity, we take coalgebraic and suspended notation; see appendix A. With a real
parameter t ∈ [0, 1], the large A∞ action (2.1a) has the following compact expression
S[Φ] =
Z 1
0
dt D Φ , M
1
1 − t η Φ
E
,
(2.1b)
More precisely, all of the spacetime fields which are coefficients of these string fields (2.4)
are necessitated to fix the gauge symmetries (2.2). In other words, a set of spacetime ghost
fields Amin ≡ { φg,p  0 < g, 0 ≤ p ≤ g ; r ∈ N
′ r
} are required. We write A0 for the set of
r ′
spacetime dynamical fields, A0 ≡ {φ0,0}r∈N. The pair Amin ≡ A0 ⊕Amin of dynamical fields
A0 and these ghosts Amin requires their spacetime antifields A∗min = {(φrg,p)∗ 0 ≤ g, 0 ≤
′
– 5 –
1
o
p ≤ g ; r ∈ N
is given by
} in the BV formalism. Hence, the minimal set of spacetime fieldsantifields
Amin ≡ Amin ⊕ Amin =
∗
n φrg,p , (φrg,p)∗ 0 ≤ g , 0 ≤ p ≤ g ; r ∈ N o
.
On this minimal set, one can define a nondegenerate antibracket
F , G
min ≡
X
X
∂φgr,p ∂(φgr,p)∗ − ∂(φgr,p)∗ ∂φgr,p
,
→
→
←
where ∂∂φF is the leftderivative, ∂∂φF is the rightderivative, and ∂∂φF = (−)φ(F +1) ∂∂φF holds.
One can quickly find (F, G)min = −(−)(F +1)(G+1)(G, F )min in this expression.
String antifields and string antibracket
In the conventional BV approach, for a given string (ghost) field Φ−g,p of (2.3), its string
antifield (Φ−g,p)∗ is introduced by assigning Z−rg∗,pi to each spacetime antifield (φgr,p)∗,
dual bases are uniquely determined for given bases, this type of string antifield seems to
be most natural. See appendix A for details of these bases. The set of string fields (2.4)
and their string antifields defined by (2.7) gives the minimal set of string fieldsantifields
n
AΦmin ≡
Φ−g,p , (Φ−g,p)∗ 0 ≤ g , 0 ≤ p ≤ g .
o
As shown by [17], this type of string antifield (2.7) provides an elegant string field
representation of the BV antibracket (2.6) in the Darboux form,
F , G
min =
X
g,p
F
←
∂
→
∂
←
∂
→
∂
where A · B denotes the BPZ inner product in the large Hilbert space A · B ≡ hA, Bi . Note
that F and G are functionals of (2.8), which can be identified with functionals of (2.5).
One can define stringfield derivatives of a functional F = F [Φα] of string fields Φα by
utilizing its total derivative δF . When a given string field Φα consists of spacetime fields
{φrα}r as Φα = P
r φαZαr i, we require that the total derivative of F = F [Φα] = F [{φrα}r]
r
has the following expression
δΦα ,
→
∂ F
∂Φα
≡
X δφrα ∂φrα F ,
r
→
∂
←
∂ F
∂Φα
, δΦα
≡ F
X
r
←
∂
∂φrα δφrα ,
which provides the stringfi←eld derivatives. Note that the relation of the left and
right→
derivatives ∂∂φF = (−)φ(F +1) ∂∂φF determines that of stringfield derivatives. We assume that
the variations of string fields (2.3) and string antifields (2.7) are given by
δΦ−g,p ≡
X δφrg,p Z−g,p ,
r
δ(Φ−g,p)∗ =
X δ(φrg,p)∗
r ∗
Z−g,p .
r
r
– 6 –
(2.5)
(2.6)
(2.7)
(2.8)
Then, by using the relations (A.4a) and (A.4b), the BV antibracket (2.6) reduces to (2.9).
Let us consider the free action K[Φ] and its gauge variation — the kinetic term of (1.1a)
and the linear part of (1.1c). Its master action Kbv gives the kinetic term of the master
action Sbv. As shown in [17], a master action for the free theory is given by
1
2
X
g
X
which is indeed a functional of string fields (2.3) and string antifields (2.7).
3
Conventional BV approach
In the conventional BV approach, we require the following three properties to obtain a
master action Sbv = Sbv[ϕ, ϕ∗] as a functional of string fields ϕ and string antifields ϕ∗:
i) Regarding the states: the master action Sbv consists of the dynamical string field,
the string ghost fields introduced in (2.3), and the string antifields given by (2.7).
ii) Regarding the operators and products: the master action Sbv consists of the operators
and products which appear in the original action (2.1b) and its gauge symmetry
algebra (2.2), namely, M, η, and the large BPZ inner product only.
iii) The master action Sbv does not include explicit insertions of ξ or M−1: these
operations enable us to remove the above requirement (i) or (ii) effectively.
However, although a perturbative master action Sbv = S(0)+S(1)+S(2)+. . . is obtained
up to the second order, this (naive) conventional BV approach breaks down at the third
order S(3) of the antifield number expansion. There is no solution satisfying the above
three requirements, which we explain in this section. A reader interested in constructing
BV master actions can skip this section; this section is independent of the other sections.
As expected, if one use ξ or M−1 insertions explicitly, a master action can be
constructed.5 It implies that string ghost fields or string antifields are reassembled, or new
products which never appear in the action nor its gauge invariance are used, to obtain the
master action. In section 5, keeping the forms of string ghost fields and the requirement
for operators and products, we construct the master action by just reassembling (physical)
string antifields.
3.1
Naive construction breaks down
We perturbatively solve the master equation using the antifield number expansion. We
write afn[φ] for the antifield number of the spacetime field or antifield φ. It is assigned to the
spacetime antifields only: afn[φ] = 0 if φ is not an antifield. In particular, afn[c] = 0 for c ∈
5Then, some of higher gauge tensors have to include ξ or M−1 explicitly, although it does not appear
in (2.1b) or (2.2) explicitly. The results of section 5 and 6 imply that there may be difference between the
gauge tensors based on string fieldsantifields and those based on spacetime fieldsantifields.
– 7 –
HJEP05(218)
→
∂
∂φg,p
C and a worldsheet basis has no antifield number. The antifield number is additive with
respect to the multiplication afn[φψ] = afn[φ]+afn[ψ], and thus afn[φ]+afn[ ∂∂φ ] = 0 . We find
afn[φg,p] = −afn
= 0 , afn (φg,p)∗ = −afn
= g + 1 ,
where φg,p denotes a gth ghost. A master action Sbv is a functional of all fieldsantifields
appearing in the minimal set and one can expand it with respect to the antifield number
→
∂
∂(φg,p)∗
Sbv = S + X S(a) ,
∞
a=1
∂S(a)
where S(a) denotes the antifield number a part of the master action Sbv, namely, afn[S(a)] =
a . The original action is the antifield number zero part S(0)
dition of the BV formalism. Because of afn ∂(φg,p)∗
antifield number a part of the master equation is given by
= a − g − 1 and afn ∂∂Sφg(a,p)
= a , the
≡ S, which is the initial
con1
2
Sbv , Sbv
(a)
min ≡
a
b
X
X
b=0 g=0 p=0
g
X S(a−[b−g]) ∂
←
→
∂
∂φg,p ∂(φg,p)∗
S(1+b) = 0 .
(3.1a)
Note that (Sbv, Sbv)(mai)n consists of S(0), · · · , S(a+1) because of ∂∂Sφg(a,p) = ∂(φg,p)∗ = 0 for
∂S(a)
a ≤ g . By solving these, one can construct a solution Sbv of the master equation
(mai)n = 0 .
In the conventional BV approach, (3.1a) has the following string field representation
1
2
Sbv , Sbv
(a)
min ≡
a
X
b
X
g
X S(a−[b−g])
b=0 g=0 p=0
←
∂
∂Φ−g,p
,
→
∂
∂(Φ−g,p)∗
Note that the antifield number expansion of Sbv defines the following odd vector field Δ
lowering the antifield number by one, afn[Δ] = −1,
→
Δ ≡
∞
X
g
X S(g) ∂
←
→
∂
→
where the dot denotes the BPZ inner product in the large Hilbert space. The odd vector
→
field Δ acting on S(a+1) is uniquely determined by given lower parts S(0), · · · , S(a). The
first equation (Sbv, Sbv)(m0i)n = 0 reduces to ΔS(1) = 0 because of ∂(∂φSg(,ap))
∗ = 0 for a ≤ g ; a
solution is given by
D
S(1) =
(Φ)∗ , M(Φ−1,0) + η Φ−1,1 ,
E
(3.2)
∞
X
a=0
→
∂
→
– 8 –
S(1+b) = 0 .
(3.1b)
→
where we wrote M(Φ0) ≡ π1
determined by the second equation (S(1), S(1))min + Δ→S(2) = 0 . To be proper, S(2) has to
include Φ−2,p and (Φ−1,p)∗ in addition to Φ−1,p and (Φ)∗. We find a solution
S(2) =
D
(Φ−1,0)∗ , M(Φ−2,0) +
M(Φ−1,0, η Φ−1,0) + η Φ−2,1
1
+ D(Φ−1,1)∗ , M(Φ−2,1) − 2
+ D(Φ)∗ , 1
2
1
4
M Φ−1,0, M(Φ−1,0) + η Φ−2,2
M Φ−2,0, (Φ)∗ +
M Φ−1,0, η Φ−1,0, (Φ)∗ E
,
E
E
(3.3)
The second line includes a double Mterm, and Mterms and ηterms appear in
symmetric manner. We introduce the following graded symmetric function of ninputs
,
which satisfies M(. . . , A, B, . . .) = (−)ABM(. . . , B, A, . . .) .
As (3.2) and (3.3), we would like to construct the next correction S(3) satisfying the
antifield number 2 part of the master equation (3.1a). However, there is no solution based
on the (naive) conventional BV approach: one cannot construct higher S(a≥3) as a
functional of Φ−g,p and (Φ−g,p)∗ unless using projectors acting on string fieldsantifields.6 The
above lower order solutions S(1) and S(2) uniquely determine the following quantity,
F (2)
≡
←∂S(2)
∂Φ
,
→∂S(1)
∂(Φ)∗
+
←∂S(1)
∂Φ
→∂S(2)
∂(Φ)∗
1
+ X
p=0
←∂S(2)
∂Φ−1,p
,
→∂S(2)
∂(Φ−1,p)∗
.
The antifield number 2 part of the master equation (3.1a) is equivalent to
unfortunately. We find that Δ→S(3) has the following form,
The odd vector field Δ acting on S(3) is uniquely determined by S(0), S(1), and S(2). Note
that a proper S(3) must include Φ−3,p and (Φ−2,p)∗. The equation (3.4) should hold for each
pair of string field variables, and one can find solutions for generic pairs; however, the
equation (3.4) has no solution for the pair of string field variable (Φ−1,0, Φ−1,0, Φ−1,0, (Φ−1,1)∗)
F (2) + Δ S(3) = 0 .
(3.4)
Δ→S(3) =
(e.o.m.) ,
→∂S(3)
∂(Φ−2,2)∗
+ η (Φ−1,1)∗,
+
M (Φ−1,1)∗ ,
→∂S(3)
∂(Φ−2,1)∗
+· · · ,
∗
∗
6For a given string field ϕ, we split it as ϕ = ϕ1 + · · · + ϕn; for each splitpart ϕa, we introduce its (split)
string antifield (ϕa) , which may satisfy ϕ
∗ = (ϕ1)∗ + · · · + (ϕn)∗. As we will see in section 5 or 6, the
master action Sbv can be constructed as a functional of these splitparts ϕ1, . . . , ϕn of the string (anti)field,
Sbv = Sbv[ϕa, (ϕa) ]. It is not a functional of the sum ϕ = ϕ1 + · · · + ϕn or ϕ
∗ = (ϕ1)∗ + · · · + (ϕn)∗. So
we need Pa s.t. Paϕ = ϕa.
– 9 –
where the last dots denote the terms which consist of the other pairs of variables. The
corresponding terms of F (2) must be able to be rewritten into the same form to satisfy (3.4).
Unfortunately, we find
F (2) = D (e.o.m.), F
E
+ D η (Φ−1,1)∗, Fη E
+
D
M Φ−1,1)∗ , FM
E
+ D(Φ−1,1)∗, E
E
+ . . . ,
where the dots denote the terms which consist of the other pairs of variables and the explicit
forms of F , FM, Fη, and E are given by
F ≡ − 4
Fη ≡ − 4
1 h
1 h
FM ≡ 4
1
M Φ−1,0, η Φ−1,0, M Φ−1,0, (Φ−1,1)∗
+ M Φ−1,0, M Φ−1,0, η Φ−1,0, (Φ−1,1)∗ i
,
M Φ−1,0, M(Φ−1,0), M(Φ−1,0) + M Φ−1,0, η Φ−1,0, M(Φ−1,0) ,
i
M Φ−1,0, η Φ−1,0, M(Φ−1,0) ,
M Φ−1,0, M(Φ−1,0, η Φ−1,0) .
1
E ≡ 4
The nonzero fourth term, which is extra and breaks (3.4), cannot be absorbed by the first
three terms. Hence, although one can construct a lower order solution Sbv = S+S(1)+S(2)+
O(3), there is no solution for higher S(a>2) based on the (naive) conventional BV approach.
3.2
On the gauge tensor formulae
The BV master equation is a generating function of the identities satisfied by the gauge
tensors — what does the breakdown of (3.4) mean? Let us consider the gauge tensors
arising from (2.2). We write Rβ , Tβαγ, or Eγαδβ for gauge tensors in the sense of
α
Φ γ
δΦ = RαΦ(Λα) , [[δ1, δ2]]Φ = Rγ Tα1α2(Λα1, Λα2) − ∂Φ′ EαΦ2αΦ1(Λα1, Λα2) ,
∂S
′
where R
Φ
≡ R(0,0) and the Greek indices denote appropriate worldsheet ghost and picture
numbers: RαΦ(Λα) = R(−1,0)(Λ−1,0) + R(−1,1)(Λ−1,0). These R, T , and E define the
folΦ Φ
lowing gauge tensors, which include terms corresponding to (Φ−1,0, Φ−1,0, Φ−1,0, (Φ−1,1)∗),
δ
δι
Aαβγ ≡ 3 cyclic
Bαβγ ≡ 3 cyclic
∂Eδι
′ δι δ ′
∂Φ′ RγΦ − Eαβ Tβγ − ∂Φ′ EβΦγι +
∂Rδα
∂Rια
′
∂Φ′ EβΦγδ .
We find the following relation of the onshell Jacobi identity of the gauge transformations
i δ
Rδ Aαβγ =
∂S
∂Φ′ Bαβγ
′
Φ δ
⇐⇒
X
cyclic
[[δ1, δ2]], δ3 Φ = 3
∂S
′
Φ Φ
∂Φ′ Bα3α2α1 Λα1, Λα2, Λα3 .
Then, the master equation (3.4) is equivalent to the existence of the higher gauge tensors
δ δι
Fαβγ and Dαβγ satisfying
δ δ ι
Aαβγ = −Rι Fαβγ +
∂S
′
∂Φ′ DαΦβδγ .
(3.5)
→
The righthand side has the form as ΔS(3) of (3.4); the lefthand side provides the same
kind of extra terms as F (2) of (3.4). Note however that the relation (3.5) should hold
automatically when the set of independent gauge generators is complete. Since these gauge
tensors R, T , E , . . . are naively defined as functionals of string fields ϕ, this result implies
that one should have to consider them as functionals of spacetime fields or rather fine parts
ϕa of total string fields ϕ = ϕ1 + · · · + ϕn. The master action Sbv will consist of these fine
gauge tensors.
4
Nonminimal set: constrained string fieldsantifields
The previous nogo result implies that we should not consider Sbv as a functional of string
fieldsantifields naively. We should consider rather fine parts ϕa of string fieldsantifields
ϕ =
P
a ϕa such as spacetime fields; or equivalently, we have to reassemble the string
fieldsantifields in order to obtain Sbv as a functional of string fieldsantifields themselves.
4.1
How to assemble string antifields
There is no criteria or rule for how to assemble string antifields unlike string ghost fields in
the BV formalism: it just suggests that how or what kind of spacetime ghost fields must
be introduced from the gauge invariance, and one can introduce their spacetime antifields
such that the antibracket takes the Darboux form. It just tells us whether a given master
action, a functional of spacetime fieldsantifields, is proper or not. In general, the string
antifield (Φg,p)∗ for the string field Φg,p can take the following form,
(Φg,p)∗ =
X(φrg,p)∗ g, p ; r ,
g, p ; r
=
r
X(arg,p)h,q Z2+h,−1+q ,
r
h,q
(4.1)
where (arg,p)h,q is some constant. As we saw in section 2, the relation hΦg,p, (Φg,p)∗i = 1
gives the simplest assembly. For generic assembly of string antifields, its “string field
representation” of the antibracket cannot take the Darboux form: when (arg,p)h,p 6= 0 for
h 6= 0 or q 6= 0, we find (F, G)min = P F ∂ aEab →∂bG where ∂a denotes a string (anti)field
←
derivative and Eab is not an orthogonal antisymmetric matrix. In this paper, we consider
the case (arg,p)h,p = 0 for h 6= 0 but may be (arg,p)0,p 6= 0, which depends on the construction.
Then, all components of Eab have the same Grassmann parity, but its string antibracket
may not be Darboux for the plabel.7
In the large Hilbert space, one can split a given state into its η and ξexact components
The new label of φη or φξ denotes that it is a coefficient spacetime field multiplied by an η or
ξexact worldsheet basis, respectively. Inspired by the formal relation hΦg,p, (Φg,p)∗i 6= 0,
7The spacetime antibracket can always take the Darboux form even if its string field representation
as
cannot.
we require that the string antifield (Φ−g,p)∗ for (4.2a) takes the following form
X(φrg,ξp)∗ η ξ g, p ; r +
X(φrg,ηp)∗ ξ η g, p ; r .
r
r
hΦξ, (Φ∗)ξi = 0 . In terms of spacetime fields, we assume
In other words, the ηexact components (Φ∗)η of the string antifield Φ∗ = (Φ∗)η + (Φ∗)ξ
correspond to the ξcomponents Φξ of the string field Φ = Φη + Φξ because of hΦη, (Φ∗)ηi =
(φg,p)∗ η = φgξ,p ∗
,
(φg,p)∗ ξ = φgη,p ∗
.
As will see, this requirement simplifies our analysis and computations.
How should we assemble the string antifields? — To find it, we take the constrained
BV approach, in which a solution of the constrained master equation determines how the
physical string antifields should be assembled. We first consider extra string fields as [37]
and introduce their string antifields as the (naive) conventional BV approach. Then, we
impose appropriate second constraints and consider the Dirac antibracket, which determines
the physical string antifields on the constrained state space — appropriately assembled
string antifields. In other words, we translated finding the assembly of string antifields into
finding out appropriate constraints such that its Dirac antibracket is welldefined and the
constrained master action becomes the generator of appropriate BV transformations. As
will be discussed in 5.3, for example, one can introduce string antifield (Φ−g,p)∗Γ for (4.2a)
via appropriate constraints Γ as
(Φ−g,p)∗ Γ =
X η
r
φrg,ξp ∗
Z1+g,−p + φrg,ηp ∗
r
r
Z1+g,−1−p
,
(4.3)
which gives a master action on the constrained string fieldantifield space. Then, physical
string antifields Φe = Φe[Φ, Φ∗; Γ] are given by functionals of these string fieldsantifields and
satisfy the canonical relation (Φa, Φeb)Γ = δa,b in the Dirac antibracket defined by (4.7).
4.2
Extra fields and constraints
The Berkovits’ constrained BV approach [37] is a specific case of the BV formalism based
on a redundant nonminimal set [36]. As many as the antifields included in the minimal set,
we introduce extra spacetime ghost fields which carry negative spacetime ghost number,
Aex =
n r
φ−1−g,−p 0 ≤ g , 0 ≤ p ≤ g ; r ∈ N o
.
For these extra spacetime ghosts, we introduce their spacetime antifields Ae∗x
{(φ−g,−p)∗ 0 < g, 0 ≤ p ≤ g}. These give a set of extra fieldsantifields Aex = Aex ⊕ Ae∗x .
=
We consider the nonminimal set of fieldsantifields
A = Amin ⊕ Aex =
n r
φg,p , (φrg,p)∗ ; φ−r 1−g,−p , (φ−r 1−g,−p)
∗ 0 ≤ p ≤ g ; r ∈ N o
and define an antibracket acting on this A by
(4.2b)
(4.2c)
We introduce a set of constraint equations Γ which has the same degrees of freedom as a
set of extra fieldsantifields Aex . These Γ split into the first and second class constraints.
For any functions F, G which are invariant under the first class Γ, one can define a
nondegenerate Dirac antibracket on A/Γ using the second class Γ by
−
X
a,b
F , G Γ ≡
F , G
F , Γa
(Γ, Γ)−1
ab Γb , G .
(4.7)
It enables us to consider a master equation on the constraint space A/Γ. See [36] for
details.
In this paper, we just consider extra spacetime ghosts which have the same labels
as original spacetime antifields appearing in the minimal set as [37]. Hence, our
nonminimal set of fieldsantifields (4.5) is twice the size of the minimal set (2.5). We construct
constrained master actions Sbv based on this redundant set (4.5) and constrained
brackets (4.7), instead of (2.5) and (2.6). In string field theory, by using these extra spacetime
fields, we can introduce extra string fields
HJEP05(218)
Φg,−p =
X φ−g,p g, p ; r
r
r
and consider a set of extra string fieldsantifields. In principle, there is no restriction on its
assembly as long as it gives a solution of the constrained BV master equation.
5
Constrained BV approach
Let us consider the ghost string fields Φ−g,p of (2.3), which are naturally determined by the
gauge reducibility. For each spacetime field φg,p of Φ−g,p = P
r
r r
r φg,p Z−g,pi , its spacetime
antifield (φrg,p)∗ is introduced. Now we add extra ghosts and their antifields, and our set
of fieldsantifields (4.5) is nonminimal and twice the size of (2.5). The pair of spacetime
fields and their antifields {φrg,p, (φrg,p)∗}g,p,r defines a nondegenerate antibracket (4.6) on
functions of fieldsantifields.
We consider a set {Φ1+g,−p0 ≤ g, 0 ≤ p ≤ g} of extra string fields, string fields
consisting of extra spacetime ghosts, and assume that as string ghost fields (2.3), these are
assembled as
antifields.
Φ1+g,−p =
X φ−1−g,−p Z1+g,−p .
r r
r
This type of extra string field has the same Grassmann parity or total grading as original
string fields. As will see, {η Φ1+g,−p}g,p will correspond to a half of conventional string
Let Aϕ ≡ {Φ−g,p, Φ1+g,−p}0≤p≤g be the set of all string fields : the dynamical string
field, string ghost fields, and extra string fields. We write ϕ for the sum of all string fields,
ϕ ≡ ϕ− + ϕex ,
ϕ− ≡ Φ +
X
X Φ−g,p , ϕex ≡
where ϕ− denotes the sum of the original string fields and ϕex denotes the sum of the extra
string fields. As proposed by Berkovits [37], we take the following constrained BV action
which has the same form as the original action (2.1b). Clearly, this Sbv[ϕ] is not proper
on (4.5) and satisfies (Sbv, Sbv) = 0 because it consists of fields only. We introduce antifields
into Sbv such that it gives a proper master action by imposing appropriate constraints Γb.
Note that as well as the original action S[Φ], the above action Sbv[ϕ] has a special
property. Recall that in the large Hilbert space, one can decompose the string field Φ−g,p
as (4.2a), in which a spacetime field φgr,pη is multiplied by an ηexact worldsheet basis.
Then, for any pairs of (g, p), we find the following relation
HJEP05(218)
∂φgr,pη Sbv[ϕ] =
D
r
η ξ Z−g,p, M
1
This kind of property plays a crucial role in the constrained BV approach.
In the rest, we often omit the rlabel and use the following short notation for brevity:
(5.3)
(5.4)
(5.5)
η
→
∂
ξ
∗ ξ
η
Z−g,p ≡ η ξ Z−g,p ,
Z−g,p ≡ ξ η Z−g,p .
∗ ∗
Likewise, we often use Z−g,pi ≡ η ξ Z−g,pi and Z−g,pi ≡ ξ η Z−g,pi as (5.5). See appendix
A for their properties.
We write Φ−g,p = φg,p Z−g,pi + φg,p Z−g,pi for the
decomposition (4.2a) of the string field Φ−g,p. Then, one can expand the antibracket (4.6) using
ξ
ξ
←
∂
→
∂
←
∂
∂φgξ,p ∂(φgξ,p)∗
←
∂
→
∂
+
∂φgη,p ∂(φgη,p)∗
.
5.1
Preliminary: constrained BV for partially gaugefixed theory
We write (ϕ)∗ for the sum of all string antifields. Utilizing ϕ of (5.2) and corresponding
(ϕ)∗, we impose the following constraint on the space of string fields
∂
η
∗ η
Γb ≡ (ϕ)∗ − η ϕ .
(5.6a)
constraint Γb with respect to its spacetime ghost number; for g ≥ 0, we find
This Γb provides constraint equations on each string field, which introduce the spacetime
antifields (φrg,p)∗ into the constrained master action Sbv[ϕ]Γb . One can decompose this
1 h
2
∗
Note that the glabel of Γ denotes its spacetime ghost number, and these give equations
for spacetime fieldsantifields on a fixed worldsheet basis. The ξexact components (Φ∗)ξ
of string antifields Φ∗ = (Φ∗)η + (Φ∗)ξ give the first class constraints. The second class
constraints are imposed between the ξexact components Φξ of string fields Φ = Φη + Φ
ξ
and the ηexact components (Φ∗)η = (Φξ)∗ of string antifields Φ∗ = (Φ∗)η + (Φ∗)ξ. Since
our master action (5.12) is invariant under these first class Γ, we focus on the second class
Γ. Independent second class constraints give the following nonzero antibracket
Γ(g1,p) , Γ(g2′,)p′ = δg+g′,−1 δp+p′,0 (−)g+1 η Z−g,p
ξ
(1)
Z−g,p
∗ η (2) .
∗
ξ η Z−g,pi introduced in (5.5). We quickly find that the following matrix
∗
We used the relation (A.7) and short notations Z−g,pi ≡ η ξ Z−g,pi and Z−g,pi ≡
∗ η
∗ ξ
Γ(g1,p) , Γ(g2′,)p′
−1 = −δg+g′,−1 δp+p′,0 Z−g′,p′
ξ
(1) ξ Z−∗gη′,p′
(2)
gives the inverse of (5.7) in the sense of
X
X
h,q h′,q′
Γ(g1,p) , Γ(h1,q′) · Γ(h1,q′) , Γ(h2′′,q)′ −1
· Γ(h2′′,q)′ , Γ(g2′,)p′ =
Γ(g1,p) , Γ(g2′,)p′ ,
where the dot denotes the inner product of two states: Ai(1)·Bi(1) = hA, Bi = hA(1)
·hB(1)
and Ai(1)
· Bi(2) = hA(1)
· hB(2) = 0 . In particular, in the inner product of (5.9), the
matrix (5.8) works as a projector onto fixed worldsheet ghost and picture numbers, and
switches the label of the state space. By taking inner products on both sides of (5.8), we find
(−)
g
Z−g,p
∗ η (1)
·
Γ(−11)−g,−p , Γ(g2′,)p′
∗ η
· Z1+g′,−p′
(2) = −δg,g′ δp,p′ .
−1
(5.10a)
Note that the half bases satisfy hZg∗,pξ, Zhη,pi = (−)ghZg∗,pη, Zhξ,qi = −δg,h δp,q ; see appendix
A. The antibrackets of these second class Γ and Sbv[ϕ] are
(5.7)
(5.8)
(5.9)
(5.10b)
(5.10c)
(5.11)
HJEP05(218)
Sbv[ϕ] , Γ−1−g,−p
∂φξ ∂(φξ)∗ Γg,p = (−)
,
ξ
1 − η ϕ ∂φg,p
∂ϕ E
∗ η
Z−g,p
Note that −hZ1ξ+g,−p η = (−)gh η Z1ξ+g,−p = (−)gphZ2η+g,−p−1 = hZ−∗gη,p holds. Using
these relations (5.10ac), the definition of dual bases (A.3), and defining properties (A.7),
we obtain
Sbv , Sbv Γ ≡
Sbv , Sbv −
Sbv , Γg,p · Γg,p , Γg′,p′ −1
· Γg′,p′ , Sbv
Γg,p , Sbv[ϕ] = −Γg,p ∂(φξ)∗ ∂φξ Sbv[ϕ] = − Z1+g,−p
D
∂ϕ
ξ
∂φ−1−g,−p
, M
1
←
∂
←
∂
→
∂
ξ
→
∂
= − Z1+g,−p
∗ η
Z1+g,−p η ξ M
= (−)
g M
1
1 − η ϕ Z−g,p
ξ
∗ η
Z−g,p .
∗ η
1
gD
M
1
,
=
= −
X
X
g,p g′,p′
D
M
M
1
X
X
g,p g′,p′
1
, ξ M
1
E
= 0 .
The action Sbv and constraint Γb give a solution of the constrained BV master equation.
Here, we used the mutual commutativity [[η, M]] = 0 and the cyclic A∞ relation of M,
D
Note however that, as can be seen in [37], the constrained BV approach based
on (5.1), (5.3), and (5.6a) works well just for a partially gaugefixed theory. Although it
indeed gives a solution of the constrained master equation as (5.11), we find that Sbv[ϕ]Γ
has an undesirable property as a proper master action for the large theory: Φ−g,p=g for
g > 0 behaves as a nontrivial auxiliary ghost field. We write (ϕ−)∗ for the sum of the
(original) string antifields {(Φ−g,p)∗g ≥ 0, 0 ≤ p ≤ g
} corresponding to the (original)
string fields {Φ−g,p  g ≥ 0, 0 ≤ p ≤ g}. On the constraint surface defined by (5.6a), the
above Sbv[ϕ] takes the following form
Sbv[ϕ]Γ =
dt ξ (η ϕ− + (ϕ−)∗), M
1
1 − t (η ϕ− − (ϕ−)∗)
g
X
k + l
+
l
k + l
η ϕ, Mk
D
Mk
1
n−1
X D
m=0
1
The ghost string field Φ−g,p=g for g > 0 does not have its kinetic term. This line of ghosts
does not exist in a partially gaugefixed theory S[Φξ] from the beginning, and then it gives
a correct proper master action. By contrast, the large theory S[Φ] requires this line to
describe the gauge invariance generated by counting Φη of S[Φ = Φξ + Φη] in spite of
S[Φ] = S[Φξ].
We would like to emphasise that this kind of problem (or ambiguity) occurs in every
superstring field theories based on the large Hilbert space, even for the conventional BV
approach, when we focus on the spacetime fields or fine parts ϕa of string fields ϕ =
ϕ1 + · · · + ϕn. Let us consider the kinetic term of (1.1a). Although string fields live
in the large Hilbert space, the kinetic term K[Φ] satisfies the property (5.4), namely,
K[Φ] = K[Φξ] holds for Φ = Φξ + Φη. It implies that the (φ0η,0)∗dependence becomes
irrelevant in the master equation
←
∂
→
∂
Kbv ∂(φ0η,0)∗ ∂φ0η,0 Kbv = 0 .
The master action Kbv can be any functional of spacetime antifields (φ0η,0)∗; for example,
one could choose h(Φη)∗, η Φ−1,1i = 0 . Then, the kinetic term of the master action (2.10),
which was proposed in [17] on the basis of the conventional BV approach, reduces to
1
2
Kbv =
Φξ, Q η Φξ + Φ∗, Q Φ−1,0 + X
where the string antifield is defined by (2.7) and h(Φξ)∗, η Φ−1,1i = 0 . This is nothing
but the kinetic term of a proper BV master action for the partially gaugefixed theory
K = K[Φξ]. As (5.13) is proper for the partially gaugefixed theory K = K[Φξ] but not
appropriate for the large theory K = K[Φ], the above pair (Sbv, ϕ, Γb) just gives a correct
proper BV master action for partially gaugefixed theory in the large Hilbert space.
We would like to construct a constrained BV master action for the large theory, which
we explain in the rest of this section. We investigate appropriate pairs of (Sbv, ϕ, Γb) which
give kinetic terms for these Φ−g,p=g on the basis of three different approaches.
Constrained BV master action based on improved constraints
We show that the kinetic terms of Sbv[ϕ]Γ in (5.12) can be remedied by improving
constraint equations, keeping the form of Sbv[ϕ] and the assembly (5.1) of extra string fields.
The key property is (5.4). Note that for 0 ≤ p ≤ g, the above constraints can be written as
following (nonlinear) constraint equations8
in which (Φη)∗ does not appear. In addition to these, for 0 ≤ p ≤ g, we impose the
γg,p ≡ (Φ1η+g,−p)∗ − ξ M
1
1 − η ϕ 1−g,p−1
,
γ−1−g,−p ≡ (Φgη,p)∗ − ξ M
1
1 − η ϕ 2+g,−1−p
,
(5.14b)
where [A]g,p denotes the components of A which have worldsheet ghost number g and
constraints γ provides the kinetic terms h(Φ−ηg,p)∗, η Φ−ξg,pi in Sbv[ϕ]Γ,γ .
picture number p .
For example, [ϕ]−g,p = Φ−g,p for (5.2).
As will see, the above
Since these constraints are second class, we have to consider the Dirac antibracket based on Γ and γ :
Sbv, Sbv Γ,γ ≡ Sbv, Sbv − Sbv, Γ · Γ , Γ′ −1
· Γ′ , Sbv − Sbv, γ · γ , Γ −1
· Γ , Sbv
− Sbv, Γ · Γ , γ −1
· γ , Sbv − Sbv , γ · γ, γ′ −1
· γ′ , Sbv .
The inverse matrices labeled by γ take complicated forms because of γ’s nonlinearity.
However, by construction of Sbv[ϕ], we do not have to know the explicit form of (γ, Γ)−1,
8These constraints are weaker than the following type of linear constraint,
γe−1−g,−p ≡ δ(φgη,p)∗ − δφ−1−g,−1−p Mη Z1+g,−1−p ,
Mη Zg,pi ≡ π1 M
h
1
⊗ Zg,pi ⊗
1
i
1 − η ϕ g,p
.
Roughly, the constraints (5.14b) are written as δγ−1−g,−p = δ(φgη,p)∗ −Ph,q δφhη,qA(2+g,−1−p) Z2ξ+g,−1−pi,
g−h,p−q
where A(sg,p) denotes that of MηA = Ps,g,p A(sg,p)Zg,pi and has spacetime ghost number s.
Sbv[ϕ]Γ,γ =
dt Dξ (η ϕ−ξ + η ϕeξx), M
=
=
Z 1
0
X
X
g
X
g
In contrast to (5.12), it includes string antifields for Φ−ηg,p and has the kinetic terms for
all original string fields and their string antifields; it is proper for the large theory.
In the constrained BV approach, the BV transformation of ϕ is given by δbvϕ =
(ϕ, Sbv)Γ,γ . By construction of the constraints, δbvϕ has an orthogonally decomposed
form. Note that while (ϕξ, Γ) is the antibracket of ξexact ϕξ and ηexact Γ, (ϕη, γ) is the
antibracket of ηexact ϕη and ξexact γ . We set Ω ≡ ξ (ϕη, γ ) · (γ, Γ)−1 · (Γ, Sbv) using the
unknown inverse. We find that the BV transformation δbvϕ = δbvϕξ + δbvϕη is given by
δbvϕξ =
ϕξ , Sbv Γ,γ = π1 ξ M
δbvϕη =
ϕη , Sbv Γ,γ = η Ω .
(Γ, γ)−1, or (γ, γ′)−1 to solve the master equation. Because of ∂∂φ∗ Sbv[ϕ] = ∂∂φη Sbv[ϕ] = 0
and ∂(φ∂ξ)∗ γ = 0, we find
γg,p , Sbv[ϕ] = −γg,p ∂(φη)∗ ∂φη Sbv[ϕ] = 0 ,
←
∂
→
∂
and thus the constrained master action (5.3) is invariant9 under new constraints γ. Hence,
these new constraints γ give no contribution to (Sbv, Sbv)Γ,γ and we find
Sbv , Sbv Γ,γ = − Sbv , Γ
Γ , Γ′ −1 Γ′ , Sbv
=
Sbv , Sbv Γ = 0 .
On the constrained subspace, we can rewrite the master action as follows,
(5.15)
E
E
.
(5.16a)
(5.16b)
HJEP05(218)
Alternative: modifying extra string fields
There is another option to obtain a proper BV master action for the large theory. Let us
consider the extra string fields of (5.1). We decompose them as follows
η
η
Φ1+g,−p ≡ φ−1−g,−p Z1+g,−p + φ−1−g,−p Z1+g,−p .
ξ
ξ
For these (5.16a), we introduce additional extra string fields Φ¯ assembled as
ξ η η ξ
Φ¯1+g,−1−p ≡ φ−1−g,−p Z1+g,−1−p + φ−1−g,−p Z1+g,−1−p .
While the set of extra spacetime ghost fields is not changed, the assemblies of extra ghost
string fields are modified by considering additional worldsheet bases. For simplicity, we
consider the sum of these extra ghost string fields as follows
′
ϕex ≡
X
∞ g+1
g=0 p=0
X Φ1+g,−p ,
′
Φ1+g,−p ≡ (1 − δp,g+1) Φ1+g,−p + (1 − δp,0) Φ¯ 1+g,−p .
′
9As another option, one could introduce γ as the first class constraints which preserve Sbv[ϕ].
Note that the plabel of Φ′1+g,−p runs from 0 to g + 1 unlike that of Φ1+g,−p . Now, the sum
of all string fields (5.2) is replaced by ϕ′ ≡ ϕ− + ϕ′ex, and the constrained BV master action
Sbv = Sbv[ϕ′] is a functional of ϕ′ . Namely, Sbv is changed via modifying extra ghost string
fields from ϕ to ϕ′ . Since this Sbv[ϕ′] includes kinetic terms for all string fields as follows
Sbv[ϕ′] =
Φ, Q η Φ
1
2
+ X
∞ g+1
X
the resultant Sbv[ϕ′]Γ′ has corresponding terms under appropriate constraints Γ′ . Instead
of (5.6a), for instance, we can impose the same type of second class constraint
where (ϕ′)∗ denotes the sum of (the ηexact parts of) all string antifields
g
(ϕ′)∗ =
Φ∗ + X
X(Φ−g,p)∗ +
X
g+1
X(Φ′1+g,p)
∗
η
.
The above Γb′ provides the same type of second class constraint as (5.6b) for 0 ≤ p ≤ g. By
contrast, for p = g + 1, we set Φ−g,g+1 = (Φ−g,g+1)∗ = 0 and constraints reduce to simple
ones:
2
′
Γg,p = √
1 h(Φ′1+g,−p)∗ − η Φ−g,p i ,
 −{→z 0}
(g<p)
′
Γ−g−1,−p = √
(Φ−g,p)∗ −η Φ′1+g,−p .
i
1 h
2  −→ 0
{z }
(g<p)
These constraints assign (physical) antifields to worldsheet bases labeled by different
picture numbers and the constrained string antifields are given by (4.3). In this case, the
string field representation of the antibracket does not have the Darboux form. Note
however that the spacetime antibracket itself can always take the Darboux form and thus as
we will see, the constrained master equation holds in the same manner as (5.11).
Note that we consider the second class and do not impose the first class constraints on
the extra string fields this time. While the ηacted extra string field is given by
η ξ
η Φ′1+g,−p = φ−1−g,−p + φ−1−g,1−p (−)1+gη Z1+g,−p ,
i
′
the string antifield for the extra string field Φ1+g,−p is given by
′
Φ1+g,−p
∗ = (φ −ξ1−g,−p)∗ + (φ−η1−g,1−p)
∗i
∗ η
Z1+g,−p .
h ξ
h
The second class constraints are imposed on these ηexact states (5.17ab). Let us check
that the pair (Sbv[ϕ′], Γ′) solves the constrained BV master equation. We consider the
antibracket of constraints, whose φξpart is the same as (5.7). We find
′ (2)
Γ′g(,p1) , Γg′,p′ =
Γ(g1,p) , Γ(g2′,)p′ + Γ′g(,p1)
←
∂
→
∂
∂φη ∂(φη)∗ − ∂(φη)∗ ∂φη Γ′g(′,2p)′ .
←
∂
→
∂
(5.17a)
(5.17b)
(5.18)
In particular, the second term of (5.18) vanishes for p = 0. By construction of extra string
fieldsantifields (5.17ab), one can quickly find that (5.18) is given by
Γ′g(,p1) , Γg′,p′ = δg+g′,−1 δp+p′,0(−)g+1 X
′ (2)
ξ
η Z−g,p+k
∗ η
Z−g,p+k
(2)
for 0 < p ≤ g. The k = 0 parts arise from the first part of (5.18); the k = 1 parts arise
from the second part of (5.18). Likewise, the following relation holds for p = g + 1,
k=0,1
′ (2)
Γ′g(,p1) , Γg′,g′+1
∗ η
In the sense of (5.9), the inverse matrix of (5.18) is given by
Γ′g(,1p) , Γg′,p′
′(2) −1
= δg+g′,−1 δp,0 Z−g′,1
ξ
(1) ξ Z−g′,1
∗η
+ δp,g+1 Z−g′,g′
ξ
(1) ξ Z−∗gη′,g′
(2)
− δg+g′,−1 δp+p′,0
X
k=0,1
ξ
Z−g′,p′+k
(1) ξ Z−g′,p′+k
∗η
(2)
.
∗
Note that Zg,p+ki · Zg,p+li = −δk,l holds. The Dirac antibracket is defined by using (5.19).
Let us consider the antibracket of Γg,p and Sbv[ϕ′], whose φξpart is the same as (5.10b).
′
(2)
(5.19)
HJEP05(218)
Γ′g,p , Sbv[ϕ′] =
Γg,p , Sbv[ϕ′] + Γ′g,p ∂φη ∂(φη)∗ − ∂(φη)∗ ∂φη Sbv[ϕ′]
∗ η
=
Γg,p , Sbv[ϕ] − Z1+g,1−p
Z1+g,1−p η ξ M
←
∂
→
∂
1
∂
→
∂
ξ
The second term is not zero because of ∂∂φη Sbv[ϕ′] 6= 0 unlike (5.4) and contracts with
the k = 1 parts of (5.19) only in the Dirac antibracket, which gives the same contribution
as (5.11) after the sum. The first term contracts with the k = 0 parts only and there are no
other contractions in the Dirac antibracket. Hence, we obtain (Sbv, Sbv)Γ′ = (Sbv, Sbv)Γ +
(Sbv, Sbv)Γ = 0 .
5.4
Other constrained BV master actions: switching M to η
Can we simplify the nonlinear constraints (5.14b) by retaking the constrained BV action
Sbv of (5.3)? — It is possible. The construction of the Dirac antibracket suggests that
such a constrained BV action will be obtained by switching a part of (5.3) which generates
nonlinear Mgauge transformations to the terms which generate linear ηgauge
transformations as [20]. We introduce the following extra string fields Ψ2+g,−1−p and their sum ψ,
g
ψ ≡
X
X Ψ2+g,−1−p ,
g≥0 p=0
Ψ2+g,−1−p ≡
X φ−1−g,p Z2+g,−1−p .
(5.20)
It has the same form as the original string antifield (Φ−g,p)∗. Note that the ηexact
component of Ψ2+g,−1−p equals to η Φ1+g,−p , and thus these new extra string fields are
Grassmann odd: (−)
ϕ = (−)ψ+1. We split the sum of all string fields (5.2) into two parts
ϕ = ϕ1 + ϕ2 .
One can consider any splittings as long as ϕ1 include the dynamical field Φ .
functional of these ϕ1, ϕ2 and ψ, we consider the following action,
Z 1
0
1
1 − t η ϕ1
Sbv[ϕ1; ϕ2, ψ] =
dt
ϕ1 , M
+
ψ , η ϕ2 .
(5.21)
It reduces to the original action (2.1b) if we set all extra fields zero. We write S1 for the
first term and S2 for the second term: Sbv[ϕ1; ϕ2, ψ] = S1[ϕ1] + S2[ϕ2, ψ] . Note that the
ηexact components of ψ do not appear in the second term. The variation of Sbv is given by
D
We find that the action Sbv = S1[ϕ1]+S2[ϕ2, ψ] is invariant under the gauge transformations
δϕ1 = π1
M, Λϕ 1 − η ϕ1
1
+ η Ωϕ1 ,
δϕ2 = η Ωϕ2 ,
δψ = η Ωψ ,
where Λϕ, Ωϕa , and Ωψ denote appropriate gauge parameters. Hence, one can obtain the
gauge invariant action (5.21) by replacing a part of (5.3) by S2, in which Mterms turn
into ηterms while keeping the gauge invariance.
We impose the constraint equations
γb = {Γg,p, γg,p}g,p,
Γ−1−g,−p ≡ (Φ −ξg,p)∗ − η Φ1+g,−p ,
γ−1−g,−p ≡ (Φ−ηg,p)∗ − ξ η Ψ2+g,−1−p ,
(5.22)
for each spacetime ghost number. Note that (Γ, γ) = 0 by construction. We find (Γ, Γ)−1 =
ξ and (γ, γ)−1 = η ξ , and thus (S1, S1)Γ = 0 holds as (5.11). Since (S2, S2)γ = 0 holds
as (5.15), we find that Sbv = S1 + S2 gives a solution of the constrained master equation,
Sbv , Sbv Γ,γ =
S1 , S1 Γ +
S2 , S2 γ = 0 .
The relation between (5.3) and (5.21) may be understood as a BV canonical
transformation.10
form
6
Conventional BV approach revisited
We gave several solutions of constrained master equations in the previous section, in which
Sbvγb of (5.21) will be rather plain. We can rewrite the constraints (5.22) into the simple
where (5.20) is extended for all g by using ψη
≡ η ϕ . This simple expression of the
constraints resembles us the conventional BV approach, and it suggests that one could
construct a BV master action Sbv based on the minimal set within the conventional BV
approach.
10In the context of the conventional BV approach for the free theory, this type of Qη switching operation
is just a result of BV canonical transformations. See [20] for details.
(6.1)
Besides reassembling string antifields, to split string fields ϕ = ϕ1 + ϕ2 and to utilise
each ϕa as an argument of Sbv play a crucial role in the constrained BV approach. As we
show in this section, one can construct a conventional BV master action Sbv as a function of
(Φξ)∗ and (Φη)∗, not a function of the sum (Φ)∗ = (Φξ)∗ + (Φη)∗ . While we introduce the
string antifield (Φ−g,p)∗ for the string field Φ−g,p as the usual conventional BV approach,
we consider their ξ or ηexact components separately. Note that (Φ∗)η = (Φξ)∗ and
(Φ∗)ξ = (Φη)∗ because of
(Φ−ξg,p)∗, Φ−ξg′,p′ = δg,g′ δp,p′ ,
(Φ−ηg,p)∗, Φ−ηg′,p′ = δg,g′ δp,p′ .
Orthogonal decomposition
We consider the orthogonal decomposition of the gauge transformation δΦ = δΦξ + δΦη.
By redefining gauge parameters as follows
Λn−e1w,0 ≡ −Λo−ld1,0 ,
Λn−e1w,1 ≡ π1 ξ
M, Λo−ld1,0 1 − η Φ
1
+ Λo−ld1,1 ,
we set the ηexact component of the gauge transformations δΦη linear
δΦξ = π1 ξ
δΦη = η Λn−e1w,1 .
M, η Λn−e1w,0 1 − η Φ
,
1
(6.2)
(6.3a)
(6.3b)
(6.4a)
(6.4b)
As we will see, it enables us to simplify the other higher gauge transformations: except for
the gauge parameters {Λ−g,0}g>0 carrying picture number 0, the ξcomponents of δΛ−g,p
are proportional to the equations of motion and thus become trivial transformations; the
ηexact components of all δΛ−g,p can be linearised by redefining the gauge parameters. We
= ξ T (Λn−e2w,1) + η Λn−e2w,2 ,
+ η π1 ξ
M, Λo−ld2,1 1 − ηΦ
1
+ Λo−ld2,2
1
where T (Λ) ≡ π1 [[M, Λ]], (e.o.m) 1−ηΦ denotes a trivial transformation. Hence, for any
0 < p ≤ g, the higher gauge transformations can be rewritten as follows
δΛn−egw,p = η Λn−egw−1,p+1 .
δΛn−egw,0 = π1 ξ [[M, η Λn−egw−1,0]] 1 − η Φ
1
+ η Λn−egw−1,1 ,
While the orthogonal decomposition δΦ = δΦξ + δΦη makes δΛξ trivial for p > 0,
redefinitions of Λ make δΛη linear. These operations enable us to obtain a simple BV master
action.
Note that partial gauge fixing is an operation omitting Φη and (6.3b) at the classical
level. Then, the line of Λ−g,p=g of (6.4b) does not appear in its higher gauge
transformations. It gives the gauge reducibility of partially gaugefixed superstring field theory in the
large Hilbert space, in which reassembled string fieldsantifields which correspond to (6.2)
will be rather natural.
6.2
Let F be a functional of the minimal set of spacetime fieldsantifields, which may be a
functional of string fields or string antifields. We perturbatively construct Sbv satisfying
δbvF =
F , Sbv min ,
whose nilpotency is our guiding principle. The initial condition is Sbv(0) = S[Φ]. We write
(Φξ)∗ or (Φη)∗ for the string antifield for the ξ or ηexact component of the dynamical string
field Φ = Φξ +Φη, respectively. As we will see, Sbv becomes a functional of (Φξ)∗ and (Φη)∗ .
We require that as well as Φ ∈ Im[ξ] and Φη
ξ
∈ Im[η], their BV transformations satisfy
δbvΦξ =
Φξ, Sbv min =
∂Sbv
∂(Φξ)∗ ∈ Im[ξ] ,
δbvΦη =
Φη, Sbv min =
∂Sbv
∂(Φη)∗ ∈ Im[η] .
Further, we require that the ηexact components of the BV transformations are linear
η
δbvΦ−g,p =
η
Φ−g,p , Sbv min = η Φ−g−1,p+1 .
In other words, we consider redefinitions of gauge parameter fields given in section 6.1
and focus on the gauge algebra of the orthogonally decomposed gauge transformations.
In general, δΦη−g,p could be a nonlinear function of fieldsantifields. This requirement (6.5)
is too restrictive and should be removed to find a more general form of the BV master
action in the large Hilbert space. However, as we will see, this requirement prohibits any
interacting terms of Φ−g,p6=0 or (Φ−g,p6=0)∗, and it enables us to construct a simple BV
master action. We find that stringantifield derivatives of S(1) are given by
δbvΦξ(0) =
Φξ , Sbv 
(0) =
δbvΦη(0) =
Φη , Sbv 
(0) =
∂S(1)
which are determined from the gauge transformations (6.3ab) and their gauge algebra.
Note that (Φξ)∗ is ηexact (Φξ)∗ = Pη(Φξ)∗ and (Φη)∗ is ξexact (Φη)∗ = Pξ(Φη)∗ . These
stringantifield derivatives (6.6ab) determine the antifield number 1 part of Sbv as follows
(6.5)
(6.6a)
(6.6b)
S(1) = D(Φξ)∗, ξ
M, η Φ−1,0 1 − η Φ
1
+ (Φη)∗, η Φ−1,1 .
(6.7)
Note that this S(1) is not a functional of Φ∗ = (Φξ)∗ + (Φη)∗ but a functional of (Φξ)∗ and
(Φη)∗. Clearly, stringfield derivatives of (6.7) become ηexact states as follows
∂S(1)
∂S(1)
∂Φ−ξ1,0
∂Φξ = π1 [[M, η Φ−1,0]], (Φξ)∗
= π1
M, (Φξ)∗
1
In other words, as the original action, a half of the stringfield derivatives vanish:
∂S(1)
∂Φη =
∂S(1)
∂Φ−η1,0
∂S(1)
∂Φ−η1,1
= 0
as
∂S
∂Φη = 0 .
∂S(1)
∂Φ−ξ1,1
= η (Φη)∗ .
By construction of (6.5), a half of the stringantifield derivatives of S(2) are given by
δbvΦη(1) =
∂S(2)
∂(Φη)∗ = 0 ,
δbvΦ−η1,p
(0) =
∂S(2)
∂(Φ−η1,p)∗ = η Φ−2,1+p (p = 0, 1) .
To solving the master equation, the other stringantifield derivatives of S(2) have to take
hh [[M, η Φ−1,0]], η Φ−1,0 + [[M, η Φ−2,0]], (Φξ)∗ii
= 0 .
Note that the requirement (6.5) prohibits not only nonlinear ηtransformations but also
the interacting terms of Φ−1,1 . These derivatives determine the antifield number 2 part of
the master action S(2) satisfying
S(0) + S(1) + S(2) + · · · , S(0) + S(1) + S(2) + · · · min = 0.
Likewise, one can construct S(3), S(4), and higher S(n) on the basis of the antifield number
expansion. These are functionals of Φ−g,p, (Φ −ξg,p)∗, and (Φ−ηg,p)∗ as expected.
Let ϕp be the sum of all string fields carrying worldsheet picture number p, which can
be decomposed as ϕp = ϕξp + ϕp . We write (ϕξp)∗ or (ϕηp)∗ for the string antifield for the
ξη
or ηexact component of ϕp respectively as follows,
ϕp ≡
∞
g=p
X Φ−g,p ,
∞
g=p
(ϕξp)∗ =
X(Φ −ξg,p)∗ ,
(ϕηp)∗ =
X(Φ−ηg,p)∗ .
∞
g=p
The dynamical string field Φ is included in ϕ0 and the sum ϕ of all string fields is given by
ϕ = ϕ0 + P
p>0 ϕp . We find that the BV master action Sbv = Sbv[ ϕ, (ϕξ)∗(ϕη)∗] is
Z 1
0
Sbv =
dt ϕ0 + ξ (ϕξ0)∗, M
1
1 − t η (ϕ0 + ξ (ϕξ0)∗)
+ X D
(ϕηp−1)∗, η ϕpE .
(6.8)
p>0
ϕp>0, (ϕ0η)∗, and (ϕηp)∗. The variation of Sbv takes the following form
While the first term is a functional of ϕ0 and (ϕξ0)∗, the second term is a functional of
Note that Φ−g,p for p > 0 has no interacting term, and thus the third term has no
contraction with the second or fourth term in the master equation. Clearly, our master action (6.8)
satisfies
1
2
Sbv, Sbv min =
→
←
∂ Sbv
∂ϕξ · ∂∂(ϕSξb)v∗ +
←
∂ Sbv
→
∂ Sbv
∂ϕη · ∂(ϕη)∗ = 0 .
While the BV transformations of string fields take the following forms,
δϕ0 =
δϕp =
ϕξ0 + ϕ0 , Sbv min = π1 ξ M
η
ϕξp + ϕηp , Sbv min = η ϕp+1 ,
1
1 − η (ϕ0 + ξ (ϕ0)∗)
+ η ϕ1 ,
the BV tramsformations of string antifields are given by
δ(ϕ0)∗ = (ϕξ0)∗ + (ϕ0η)∗ , Sbv min = π1 M
δ(ϕp)∗ = (ϕξp)∗ + (ϕηp)∗ , Sbv min = η (ϕpη−1)∗ .
1
1 − η (ϕ0 + ξ (ϕ0)∗) ,
(6.9a)
(6.9b)
(6.10a)
(6.10b)
The role of the second term of (6.8) in perturbation theory depends on the gaugefixing
condition: for example, it is integrated out and trivially decouples in the Siegel gauge; however,
it will provide nontrivial contributions to loop amplitudes in the d0gauge. See [17, 18].
7
Concluding remarks
In this paper, we developed the BatalinVilkovisky formalism of superstring field theory
in the large Hilbert space with the goal of understanding of how to construct large master
actions for interacting theory.
We first showed that the constrained BV approach [36]
is well applicable, in which Berkovits’ simple prescription [37] is rather suitable for the
large but partially gaugefixed theory. By modifying its constraints, extra string fields, or
starting unconstrained action, we constructed several constrained BV master actions in the
large Hilbert space. We next showed that the conventional BV approach is also applicable
iff we give up constructing master actions as naive functionals of string fieldsantifields.
We constructed a BV master action as a functional of fine parts {ϕa}an=1 of string
fieldsantifields ϕ = ϕ1 + · · · + ϕn, not a function of string fieldsantifields themselves. It is worth
mentioning that our analysis is quickly applicable to the large theory which is obtained by
embedding [26] or [30], and thus BV master actions for the large A∞ theory including the
Ramond sector or the large L∞ theory are constructed in the completely same manner.
Also, since BV master actions in the large Hilbert space are obtained, one can discuss the
validity of partial gaugefixing now. We conclude with some remarks.
BV formalism in the large Hilbert space.
First, it is worth visiting and connecting
different pairs of (Sbv, Γb, ϕex) to obtain a better understanding of the BV formalism in the
large Hilbert space. While we gave several constrained BV master actions in section 5,
there would exist some canonical transformations connecting these. Next, it is desirable to
find a more general form of the master action in the large Hilbert space on the basis of the
conventional BV approach. Our master action (6.8) has a simple form but is constructed
based on the requirement (6.5), which will be too restrictive.
WZWlike formulation.
Our results give a simple example of constructing BV master
actions for WZWlike superstring field theory (1.1a), which is based on the
parametrisation (1.2). It is important to clarify whether one can construct a BV master action
for other parametrisation of the WZWlike functional (1.1b), such as Aη[Φ] = (ηeΦ)e−Φ.
One may be able to apply constrained BV approach for the Berkovits theory in a similar
manner, in which partial gauge fixing may take a nonlinear form such that (δeΦ)e−Φ =
QΛ + η Ω − [[Aη, Ω]] is orthogonally decomposed as (6.3ab). A procedure which does not
depend on these parametrisations would be necessitated to apply the BV formalism to the
general WZWlike formulation [34, 35].
Gauge tensors’ formulae.
Since our master actions are not naive functionals of string
fieldsantifields, it is interesting to clarify the difference of the gauge tensors based on
spacetime and string fields. It reveals why a readymade BV procedure does not work in
the large Hilbert space from the point of view of the gauge algebra. Then, how “partial
gauge fixing” fixes the gauge would be clarified.
Acknowledgments
The authors would like to thank Mitsuhiro Kato and the organizers of “Strings, Fields,
and Particles 2017 at Komaba”. H.M. also thanks Ted Erler, Hiroshi Kunitomo, Yuji
Okawa, Martin Schnabl, and Shingo Torii.
M.N. is grateful to Yuji Tachikawa. This
research has been supported by the Grant Agency of the Czech Republic, under the grant
P201/12/G028.
A
Notations and basic identities
In this appendix, we explain our notation and some basic identities briefly. See also [11–
13, 20, 33]. In general, a string field Ψ ≡ ψ Z consists of a set of spacetime fields ψ having
spacetime ghost number and a set of worldsheet bases Z which carry worldsheet ghost
and picture numbers. Its state space H is a vector space, which is equipped with the
BPZ inner product or graded symplectic structure. String products M define a linear map
acting on its tensor algebra T (H).
On the BPZ inner product. Let s[ϕ], gh[ϕ], and pc[ϕ] be the spacetime ghost,
worldsheet ghost, and picture numbers of ϕ, respectively. The Grassmann parity of ϕ is the sum
of the spacetime and worldsheet ghost numbers, which we write G[ϕ] ≡ s[ϕ] + gh[ϕ] . The
upper index of (−)ϕ denotes the ϕ’s Grassmann parity (−)
ϕ
≡ (−)G[ϕ] for brevity, if (−)
ϕ
is put on the BPZ inner product. We write G[ω] for the grading of the BPZ inner product;
it is 0 in the large Hilbert space; it is 1 in the small Hilbert space. The BPZ inner product
hϕ1, ϕ2ibpz is graded symmetric and bilinear
Ψ , Φ bpz = (−)Ψ Φ Φ , Ψ bpz ,
ψ Z , Φ bpz = (−)G[ω]G[ψ]ψ
Z , Φ bpz ,
Φ , ψ Z bpz = (−)G[ψ]G[Z] Φ , Z bpz ψ .
We focus on the large Hilbert space. There exists a complete system of BPZ bases satisfying
Zga,p , Zh,q bpz = (−)gh[Zh,q]pc[Zh,q]δa,b δg+h,2 δp+q,−1 ,
b
(A.1a)
(A.1b)
(A.1c)
(A.2)
where the alabel distinguishes different bases carrying the same worldsheet ghost and
picture numbers. Note that the glabel can run over all integer numbers11 because of p+q =
−1 and these bases indeed satisfy (A.1a). By absorbing the sign of (A.2), one can define
Z2−g,−1−p ≡ (−)gh[Zg,p]pc[Zg,p]Zga,p ,
a ∗
(A.3)
which satisfy simpler relations hZga,p, Zhb,∗qibpz = (−)gh[Zg∗,p]hZga,p∗, Zh,qibpz = δa,b δg,h δp,q .
b
These dual bases are useful to give string antifields. We often omit the alabel for simplicity.
The orthogonal relation (A.3) provides simple decompositions of the unit.
We
introduce the symbol δ(r) satisfying
Pr δ(r)
HJEP05(218)
hZga,p, Zf,ribpzhZ2−f,−1−r, Zhb,qibpz = (−)frδa,cδg+f,2δp+r,−1 ·(−)h,qδc,bδf,h,δr,q . By summing
c c
over (f, r) and using the definition of the dual bases (A.3), we find
(−)gh[Zf∗,r]
c ∗ b
Zga,p, Zf,r bpz Zfc,r, Zh,q bpz =
Zga,p, Zh,q bpzδ(c)δ(f )δ(r) ,
b
Zga,p, Zf,r bpz Zfc,∗r, Zh,q bpz =
c b
Zga,p, Zh,q bpzδ(c)δ(f )δ(r) .
b
(A.4a)
(A.4b)
Utilizing bhABib = hA, Bibpz , these relations can be expressed as follows
X
g,p,a
a ∗
Zg,pib bhZg,p = 1 ,
a
On the graded symplectic form.
We write deg[ϕ] for the worldsheet degree of ϕ,
which is defined by deg[ϕ] ≡ gh[ϕ] − 1 . Then, the (total) degree ǫ[ϕ] of ϕ is defined by
ǫ[ϕ] ≡ deg[ϕ] + s[ϕ] . Note that spacetime fields ψ has no worldsheet degree deg[ψ] = 0
and thus its (total) degree is equal to its spacetime ghost number ǫ[ψ] = s[ψ] . Since the
BPZ inner product is graded symmetric, its suspension gives the graded symplectic form
Note that (−)ϕ denotes the ϕ’s Grassmann parity (−)
ϕ
≡ (−)G[ϕ] even if (−)ϕ is put on
the graded symplectic form. By construction, we can find its defining properties
Ψ , Φ
≡ −(−)gh[Ψ]+s[Φ] Ψ , Φ bpz .
Ψ , Φ
ψ Z, Φ
Φ , ψ Z
= −(−)ǫ[Ψ]ǫ[Φ] Φ , Ψ ,
= (−)ǫ[ω]ǫ[ψ]ψ
= (−)ǫ[ψ]ǫ[Z] Φ , Z
Z , Ψ ,
ψ ,
(A.5)
(A.6a)
(A.6b)
(A.6c)
where ǫ[ω] denotes the total degree of the graded symplectic form (A.5); it is 1 in the large
Hilbert space; it is 0 in the small Hilbert space. Since spacetime fields has no worldsheet
degree, we find ψ Z = (−)ǫ[ψ](ǫ[Z]+1)
Z ψ and ψ1 ψ2 = (−)ǫ[ψ1]ǫ[ψ2]ψ2 ψ1 . We focus on the
large Hilbert space. There exists a complete system of symplectic bases satisfying
Zga,p, Zh,q = −(−)gh[Zg,p]pc[Zg,p]δa,bδg+h,2δp+q,−1 .
b
free master action. Because of Bg,p ≡ Zga,p for g < 1 and Bg,p ≡ Zga,∗p for g > 1, it satisfies
a a
11Torii’s subset: in the earlier works [16–18], just a subset Bg,p = {Zg,p, Zg∗,p}g<1 is used to give the
Bga,p, Bhb,q bpz = θ(1 > g) + (−)gθ(g > 1) δa,b δg+h,2 δp+q,−1 ,
where θ(g > 1) is a step function of g. This subset satisfies (A.4a) for 1 > g and (A.4b) for g > 1 .
One can define
their dual bases by minus (A.3) and quickly find hZga,p∗, Zh,qi = −(−)deg[Zg∗,p]hZga,p, Zhb,∗qi =
b
The completeness condition of the symplectic bases is given by
By using the definition of the dual basis (A.3), we get symplectic versions of (A.4a)
and (A.4b). In the sense of hABi = hA, Bi , this relation can be expressed as follows
X
g,p,a
Zg,pi hZga,p∗ = −1 ,
a
X(−)deg[Zg∗,p]Zg,pi hZg,p = 1 .
a ∗ a
g,p,a
Half bases and projectors. We write Zga,pη for a complete symplectic basis in the small
Hilbert space satisfying
b η
Zga,pη , Zh,q
= δa,b δg+h,3 δp+q,−2 ,
which is equivalent to the half basis defined in (5.5). By comparing the large and small
bases via hhZgη,p, Zh,qii ≡ −hξZgη,p, Zh,qi, we find the following convenient relations
η η
Zgη,p ≡ η ξ Zg,p = (−)gp η Zgξ−1,p+1 ,
ξ η
Zg,p ≡ ξ η Zg,p = (−)gp ξ Zg+1,p−1 ,
η
∗ ξ
worldsheet ghost number −g and picture number p :
η
where g and p of (−)gp denote worldsheet ghost and picture numbers. Note that P−g,p ≡
(2) works as the projector onto the η or ξexact components carrying
η
P−g,p ϕ = − Z−g,p
η
Z−∗gξ,p, ϕ = − Z−g,p
η
Z−∗gξ,p, Φ−ηg,p = φ−g,p Z−g,p = Φ−ηg,p
η η
ξ
for ϕ given by (5.2). Likewise, P−g,p ≡ (−)g+1
ξ ∗ η
Z−g,pi(1) hZ−g,p
projector. Because of ξ Pη = ξ (Pη + Pξ) = ξ and Pξ ξ = (Pη + Pξ) ξ = ξ, we find
(2) works as the ξexact
Z−g−1,p+1 Z−g,p = −ξ X
ξ ∗ξ
η
Z−g,p Z−g,p = ξ
∗ξ
g,p
g,p
X(−)gp+g+p
Z−g,p Z−g−1,p+1 = X(−)g+1
ξ ∗ξ
Z−g,p Z−∗gη,p ξ =
ξ
Since η Pξ = η (Pη + Pξ) = η and Pη η = (Pη + Pξ) η = η, we obtain
X(−)(g+1)p
Z1−g,p−1 Z−g,p = η X(−)g+1
η ∗ η
Z−g,p Z−g,p = η
∗ η
X(−)(g+1)p
η
∗ η
Z1−g,p−1 Z−g+1,p−1 = −
Z−g,p Z−g,p η =
∗ ξ
h X
g,p
h X
g,p
η i
P−g,p = ξ Pη = ξ ,
ξ i
P−g,p ξ = Pξ ξ = ξ .
h X
g,p
h X
g,p
ξ i
P−g,p = η Pξ = η ,
η i
P−g,p η = Pη η = η .
g,p
g,p
η
ξ
X
g,p
A string product M = Q + M2 + M3 + · · · consists of the
BRST operator Q, a nonassociative three vertex M2, and higher vertices Mn>2: these
define multilinear maps {Q, Mn}n≥2 acting on the state space H. In particular, they give a
cyclic A∞ algebra for open string field theory. In general, an nlinear map Mn : H
⊗n
→ H
define a linear map Mn : H
⊗m → H
⊗m−n+1 by the following Leibniz rule on tensors
Mn : Ψ1 ⊗ ··· ⊗ Ψm 7−→
X (−)σk Ψ1 ⊗ ··· ⊗ Ψk−1 ⊗ Mn(Ψk , . . . , Ψk+n−1) ⊗ Ψk+n ⊗ ··· ⊗ Ψm
M = P∞
for n ≤ m, where (−)σk = (−)Mn(Ψ1+···+Ψk−1) denotes the grading. Note that a constant
M0 yields a map M0 inserting M0 in the tensor products; (M0)q maps H
of the qdegree homogeneous polynomial of M0 . By defining Mn(Ψ⊗m) ≡ 0 for n > m,
this Mn gives a coderivation on the tensor algebra, Mn : T (H) → T (H) . The sum
m to the coefficient
n=0 Mn also gives a coderivation. We write [[Ck, Dl]] for the graded commutator
Ck , Dl ≡ Ck Dl − (−)CkDlDl Ck ,
algebra. For each H
which is a coderivation given by a map from H
⊗m to H
1
⊗m−k−l+2. We write 1−Ψ ≡
P∞
n=0 Ψ⊗n for the grouplike element of the tensor algebra T (H), where H⊗0 = C and
Ψ⊗0 = 1. A natural projection πn : T (H) → H
⊗n is defined by πn : P∞
k=0 Ψ1 ⊗ ··· ⊗ Ψk 7−→
Ψ1 ⊗ ··· ⊗ Ψn . When a coderivation M is nilpotent, a pair (T (H), M) defines an (weak) A∞
⊗n, the A∞ relation M2 = 0 of M = P∞
n=1 Mn can be written as follows
πn
X
Mk+1 , Mn−k 1 − Ψ
1
= X Mk+1 zΨ, .}.. , Ψ{, Mn−k(zΨ, . . . , Ψ{), zΨ, .}.. , Ψ{ = 0 .
}
Furthermore, with a coderivation Λ−g,p inserting Λ−g,p into the Fock space T (H), we get
π1
M , Λ−g,p 1 − Ψ
The
suspension
of
−(−)ǫ[Ψ0] Mn(Ψ0, Ψ1 . . . , Ψn−1), Ψn
(−)Pkn=1(n−k)(gh[Ψk]+s[Ψk]+1)M n′(Ψ1, . . . , Ψn) .
σi = ∂s(ǫ[Ψ1] + · · · + ǫ[Ψi] + 1).
1
M n′
∞ n−1
n=1 k=0
is
∂sMn(Ψ1, . . . , Ψn) = Pin=−01(−)σiMn(Ψ1, . . . , ∂sΨi+1, . . . , Ψn), where the sign is defined by
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k
n−k−1
= X X Mn zΨ, .}.. , Ψ{, Λ−g,p, zΨ, .}.. , Ψ{ .
given
by
Mn(Ψ1, . . . , Ψn)
Then, we find
Ψ0, Mn(Ψ1 . . . , Ψn)
if M is cyclic.
For a derivation ∂s, we find
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