Gauge Fixing of Modified Cubic Open Superstring Field Theory

0.28 MB·English

by  Maiko Kohriki

YITP-11-98
November, 2011
Gauge Fixing of Modified Cubic
Open Superstring Field Theory
Maiko Kohriki,∗) Taichiro Kugo∗∗) and Hiroshi Kunitomo∗∗∗)
Yukawa Institute for Theoretical Physics, Kyoto University,
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0 Kyoto 606-8502, Japan
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e Abstract
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The gauge-fixing problem in the modified cubic open superstring field theory is

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discussedindetailforboththeRamondandNeveu-Schwarz (NS)sectorsintheBatalin-
v
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Vilkovisky (BV) framework. We prove for the first time that the same form of action as
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9 the classical gauge-invariant one with the ghost-number constraint on the string field
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. relaxed, gives the master action satisfying the BV master equation. This is achieved
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by identifying independent component fields by analyzing the kernel structure of the
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1 inverse picture-changing operator. The explicit gauge-fixing conditions for the com-
:
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ponent fields are discussed. In a kind of b = 0 gauge, we explicitly obtain an NS
i 0
X
propagator that has poles at zeros of the Virasoro operator L .
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r
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∗) E-mail: [email protected]
∗∗) E-mail: [email protected]
∗∗∗) E-mail: [email protected]
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§1. Introduction
Thecovariantopensuperstringfieldtheory(super-SFT)wasfirstconstructedbyWitten1)
in the form of Chern-Simons three-form action similarly to his bosonic string field theory.2)
He wrote the action using string fields in the natural picture, that is, the Neveu-Schwarz
(NS) string of picture number −1 and the Ramond (R) string of picture number −1/2. The
action had a picture-changing operator X(z) located at the midpoint z = i in the cubic
NS interaction term. Later, however, it was pointed out by Wendt3) that the theory suffers
from a severe divergence problem caused by colliding picture-changing operators in the NS
four-string amplitude. This led to the violation of the associativity, and hence, the gauge
invariance of the theory.
To circumvent this above problem, two approaches were proposed. One is the so-called
modified cubic super-SFT, which has the same form of action as Witten’s but is based on
the 0-picture NS string field, and was proposed by Preitschopf, Thorn, and Yost4) (PTY)
and Arefeva, Medvedev, and Zubarev5) (AMZ), independently. Another is Berkovits’ super-
SFT,6) which has the WZW-type nonpolynomial action for the NS sector. The latter ap-
proach is interesting in the sense that there is no need for a picture-changing operator.
However, in this WZW type approach, the construction of the action for the R sector is
difficult and has never been performed in a satisfactory manner.7)
The modified cubic super-SFT approach, on the other hand, simply gives the R sector as
a cubic action and is beset with no problem. Therefore, we study the modified cubic super-
SFT in this paper. The above-mentioned authors who proposed the modified cubic super-
SFT discussed the perturbation theory in their papers so that they must have discussed the
gauge-fixing problem. However, strangely enough, they did not show that their gauge-fixed
action is really BRST-invariant. First, they tacitly assumed that the gauge-fixed action, or
even the Batalin-Vilkovisky (BV) master action for fields and antifields, takes the same form
as the original gauge-invariant action with the understanding that the field ghost number
constraint is relaxed. However, to the best of our knowledge, no one has ever shown that
the ‘BV action’ thus obtained does really satisfy the BV master equation and hence that
the gauge-fixed action obtained by fixing the gauge by setting the suitable set of antifields
equal to zero is really BRST-invariant.
The invariance of the ‘BV action’ under the BRST transformation δ Φ = Q Φ+Φ∗Φ is
B B
trivial by construction but the BV master equation is not because of the presence of inverse
picture-changing operators that have nontrivial kernels and thus are non-invertible. The
presence of nontrivial kernels of inverse picture changing operators introduces an additional
gauge invariance other than the conventional one. This is important for the very consistency
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of the R sector in Witten’s original super-SFT. This is also the same in this modified cubic
super-SFT with the same R sector.
The NS sector is more difficult in which a double-step inverse picture-changing operator
is present in the modified cubic super-SFT case. The kernel becomes much larger than that
in the R case and the construction of the BRST-invariant projection operator becomes much
more complicated accordingly. Because of this complication in the projection operator,∗) no
explicit analysis of component fields of the NS field reduced into the projected space has
been carried out; hence, no explicit proof of the BV master equation has been given. The
validity of the gauge-fixing procedure could be judged only with it.
The purpose of the present paper is to discuss the gauge fixing of the modified cubic
super-SFT, particularly to give an explicit proof that the same form of action as the original
gauge-invariant super-SFT action with the ghost number constraint relaxed satisfies the BV
master equation. This can be achieved by adopting a simple component field form of the
reducedstringfieldsuggesteddirectlyfromtheformoftheinverse picture-changing operator.
This reduced form is not invariant under the BRST transformation, but it can be modified
so as to keep the form.
This paper is organized as follows. We first recall the gauge-invariant action of the
modified cubic open super-SFT in §2. Then, in order to understand the nature of the
problem, we review in §3 the gauge-fixing procedure for the R sector, which was beautifully
performed by us and Masaki Murata. This procedure was reported in the string field theory
workshops held at APCTP in 20098) and, at YITP in 2010,9) but the full report has never
been published. The kernel problem in the NS sector is much more complicated than that in
the R sector, but we give a solution to this problem in §4, as mentioned above, by defining
the simple reduced form of the NS string field for which the double-step picture-changing
operator YY¯ gives a nondegenerate cross-diagonal metric. We show there that it implies
that the action satisfies the BV master equation. In §5, we discuss the gauge fixing for
the NS sector by explicitly examining the component fields in the reduced NS field. We
propose a set of gauge-fixing conditions that gives a BRST-invariant consistent gauge-fixed
action. However, since it is difficult to find a propagator in a closed form in that gauge, we
propose another gauge in §6 using a nonlocal projection operator. It can give a propagator
explicitly, but the explicit component expression becomes much less clear. The gauge is of
the b = 0 type and the propagator has poles at L = 0. A note on the the gauge fixing for
0 0
the total interacting system is given in §7. The final section, §8, is devoted to discussions.
∗) Many trials have also been carried out to find simpler BRST-invariant projection operators or even
otherpossibilitiesofthedouble-stepinversepicture-changingoperatorforwhichprojectionoperatorsbecome
simpler.9)
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We add three appendices. The explicit expressions for the BRST operators and BRST
transformations are given in Appendix A. The definitions and properties of the picture-
changing operators are summarized in Appendix B. We give the component expressions for
the decomposition by nonlocal projection operators for the R sector in Appendix C.
§2. Gauge-invariant action of modified cubic open super-SFT
The modified cubic open super-SFT action based on the 0-picture NS string Φ and
(−1/2)-picture R string Ψ was proposed by PTY4) and AMZ,5) independently, and is given
by
1 1 1
S = YY¯Φ∗Q Φ+ YY¯Φ∗Φ∗Φ+ YΨ ∗Q Ψ + YΦ∗Ψ ∗Ψ
B B
2 3 2
Z Z Z Z
1 1
= hΦ|YY¯Q |Φi+ hV |YY¯ |Φi|Φi|Φi
B NS-NS-NS
2 3
1
− hΨ¯|YQ |Ψi+hV |Y |Φi|Ψi|Ψi. (2.1)
B NS-R-R
2
In the classical action the string fields Φ and Ψ are constrained to possess ghost number
one.∗) The symbol ∗ denotes the conventional Witten’s string product based onthe midpoint
interaction. The operators Y = Y(i) and Y¯ = Y(−i) are the inverse picture-changing
operators inserted at the string midpoint z = i and its mirror point z = −i.∗∗) Note that
hΨ¯| is the Dirac conjugate of |Ψi:
hΨ¯|=hΨ|ψ0, hΨ|= (|Ψi)†, (2.2)
0
where ψµ is the zero mode of the matter operator ψµ(z) in the R sector.
0
This system is invariant under the gauge transformations
δΦ = Q λ+Φ∗λ−λ∗Φ+X¯(Ψ ∗χ−χ∗Ψ), (2.3a)
B
δΨ = Q χ+Φ∗χ−χ∗Φ+Ψ ∗λ−λ∗Ψ, (2.3b)
B
where λ (χ) is the NS (R) string transformation parameter with ghost number zero and
X¯ = X(−i) is the picture-changing operator at the mirror point.
Furthermore, the marked difference from the bosonic string case is that the ‘measures’
YY¯ = Y(i)Y(−i) in the NS sector and Y = Y(i) in the R sector have nontrivial kernels
since
c(z)Y(z) = γ2(z)Y(z) = 0, (2.4)
∗) This corresponds to the requirement that the coefficient fields carry ghost number zero.
∗∗) Thus we adopt the nonchiral operator as the double-step inverse picture-changing operator following
PTY. The conventions of the picture-changing operators are summarized in Appendix B.
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as is clear from the expression
Y(z) = c(z)δ′(γ(z)). (2.5)
Consequently, the kinetic terms of the action (2.1) are also invariant under additional gauge
transformations:10),11)
δΦ = c(i)λ , δΦ = γ2(i)λ , (2.6)
c γ
δΦ = c(−i)λ′, δΦ = γ2(−i)λ′ . (2.7)
c γ
δΨ = c(i)χ , δΨ = γ2(i)χ . (2.8)
c γ
Thisisactuallythesymmetryofthetotalaction(2.1)includingtheinteractiontermsbecause
the ghost fields γ(z) and c(z) vanish at the interaction point:
hV |c(±i) = hV |γ(±i) = 0, (I = NS-NS-NS or NS-R-R) (2.9)
I I
where c(±i) or γ(±i) can be the ghost coordinate of any of the three strings.
Since the additional gauge invariances are different between the R and NS cases, we
discuss the two sectors separately, initially neglecting the Yukawa coupling term ΦΨΨ.
§3. Gauge fixing of Ramond sector
In this section, we discuss the gauge fixing of the gauge symmetries (2.3b) and (2.8) for
the R string. This can be performed beautifully. We will provide a prototype solution to
the problem.
We first fix the additional gauge invariance (2.8) using the BRST-invariant projection
operator P following Ref. 10):
Y
P ≡ X Y, X = [Q ,Θ(β )] = δ(β )F −b δ′(β ), (3.1)
Y 0 0 B 0 0 0 0 0
where F = [Q ,β ]. The choice of X is not unique and has the freedom of changing the
0 B 0 0
gauge for new gauge symmetries. Here, X is the ‘mode version’ of the picture-changing
0
operator satisfying the properties (B.9) in Appendix B.
The R string field Ψ can be generally expanded in the zero modes of the ghost c and
0
the superghost γ as
0
∞
|Ψi = (γ )n|0i ⊗ |↓i⊗|φ i+c |↓i⊗|ψ i (3.2)
0 β n 0 n
Xn=0 (cid:16) (cid:17)
∞
≡ (γ )n |φ ii+c |ψ ii , (3.3)
0 n 0 n
Xn=0 (cid:16) (cid:17)
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where |↓i and |0i are, respectively, the vacua of the reparametrization ghost zero mode b
β 0
and the superghost zero mode β defined by
0
b |↓i = 0, h↓|c |↓i = 1, (3.4)
0 0
β |0i = 0, h0|δ(γ )|0i = 1. (3.5)
0 β β 0 β
The states |φ i and |ψ i are the states in the sector other than the (super)ghost zero modes.
n n
For brevity, we write |0i ⊗|↓i⊗|ϕi as |ϕii, which satisfies
β
hhϕ |c δ(γ )|ϕ ii = hϕ |ϕ i. (3.6)
1 0 0 2 1 2
The projected field has been shown10) to take the same form as the constrained string
field proposed in Ref. 12):
|Ψˆi≡ P Ψ = |φii−(γ +c F)|ψii, (3.7)
Y 0 0
where F = F + 2b γ is the Ramond-Dirac operator (A.2c) with the ghost zero modes
0 0 0
b ,γ removed from F . Henceforth, the hatted field denotes the field whose kernel degrees
0 0 0
of freedom are projected out by the BRST-invariant projection operator P . The Ramond
Y
kinetic term can be rewritten with the projected field as
1 1
S[Ψˆ] = − hΨ¯|PTYQ P |Ψi = − hΨ|ψ0YX YQ X Y |Ψi
2 Y B Y 2 0 0 B 0
1 1
= − hΨ|YX†ψ0YQ X Y |Ψi = − hX YΨ|ψ0YQ X Y |Ψi
2 0 0 B 0 2 0 0 B 0
1 1 ¯
= − hΨˆ|ψ0YQ |Ψˆi= − hΨˆ|YQ |Ψˆi. (3.8)
2 0 B 2 B
Here we have used {Y,ψ0} = 0 and Eq. (B.14).
0
Next, for the conventional Q gauge invariance, we impose the gauge-fixing condition
B
|ψi = 0. (3.9)
Then the projected field Ψˆ satisfies the following two conditions simultaneously:
β |Ψˆi= 0, and b |Ψˆi= 0. (3.10)
0 0
Let us call this the Ramond-Siegel gauge. We introduce the projection operator into the
Ramond-Siegel gauge subspace:
P ≡ b c δ(β )δ(γ ), |Ψˆ i≡ P |Ψˆi= |φii. (3.11)
G 0 0 0 0 k G
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3.1. Iterative gauge-fixing procedure
For the free action, the gauge fixing can be carried out iteratively just as in the bosonic
case.13)–16) Let us start from the gauge-invariant free action:
1 ¯ 1
S = − Ψˆ Y Q Ψˆ = − hΨ¯ |PTY Q P |Ψ i, (3.12)
0 2 (1) 0 B (1) 2 (1) Y 0 B Y (1)
(cid:10) (cid:12) (cid:12) (cid:11)
where |Ψ i is the classical R string field with ghost number one. Note that the inverse
(1) (cid:12) (cid:12)
picture-changing operator Y can be rewritten as Y = c δ′(γ ) between the projection oper-
0 0 0
ators using Eqs. (B.9):
P†YP = P†Y P . (3.13)
Y Y Y 0 Y
The above action (3.12) is invariant under the gauge transformation with the gauge param-
eter of ghost number zero:
δ|Ψˆ i= Q |Λˆ i. (3.14)
(1) B (0)
We can rewrite the gauge transformation (3.14) as the BRST transformation by introducing
the Fadeev-Popov (FP) ghost field|Ψˆ iand replacing the parameter|Λˆ iwith this field as
(0) (0)
δ |Ψˆ i= Q |Ψˆ i. (3.15)
B (1) B (0)
It is convenient to introduce the FP antighost field |Ψˆ i and the Nakanishi-Lautrup (NL)
(2)
field |Bˆ i with the BRST transformations
(2)
δ |Ψˆ i=|Bˆ i, δ |Bˆ i= 0. (3.16)
B (2) (2) B (2)
Taking the Ramond-Siegel gauge condition
(1−P )|Ψˆ i= 0, (3.17)
G (1)
we add the gauge-fixing and the FP ghost terms to the gauge-invariant action (3.12):
¯
ˆ ˆ
S = S −δ Ψ Y (1−P ) Ψ
1 0 B (2) 0 G (1)
= −21 Ψ¯ˆ(1)(cid:16)Y(cid:10)0QB(cid:12)(cid:12)Ψˆ(1) − B¯ˆ((cid:12)(cid:12)2) Y0(cid:11)((cid:17)1−PG) Ψˆ(1) − Ψ¯ˆ(2) PGTY0QB Ψˆ(0) . (3.18)
(cid:10) (cid:12) (cid:12) (cid:11) (cid:10) (cid:12) (cid:12) (cid:11) (cid:10) (cid:12) (cid:12) (cid:11)
Here, we have used the equality
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
Y (1−P ) = PTY , (3.19)
0 G G 0
which can be easily confirmed by the following direct calculations:
Y (1−P ) = Y −c [δ(γ ),β ]b c δ(β )δ(γ )
0 G 0 0 0 0 0 0 0 0
= Y +c β δ(γ ) = c δ(γ )β , (3.20)
0 0 0 0 0 0 0
PTY = c b δ(γ )δ(β )c [δ(γ ),β ] = c δ(γ )β . (3.21)
G 0 0 0 0 0 0 0 0 0 0 0
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This action S is still invariant under the gauge transformation
1
δ|Ψˆ i= Q |Λˆ i. (3.22)
(0) B (−1)
Repeating the same procedure successively, we can obtain the totally gauge-fixed action
as follows. First, we introduce a series of FP ghost fields|Ψˆ i, FP antighost fields|Ψˆ i,
(−g) (2+g)
and NL fields |Bˆ i for g ≥ 0 with the BRST transformations
(2+g)
δ |Ψˆ i= Q |Ψˆ i, δ |Ψˆ i=|Bˆ i,
B (1−g) B (−g) B (2+g) (2+g)
δ |Bˆ i= 0, for g ≥ 0. (3.23)
B (2+g)
ˆ
Imposing the Ramond-Siegel gauge condition (1 − P )|Ψ i = 0 for the FP ghosts and
G (1−g)
the original field, we obtain
∞
1 ¯ ¯
S = − Ψˆ Y Q Ψˆ − δ Ψˆ Y (1−P ) Ψˆ ,
(1) 0 B (1) B (2+g) 0 G (1−g)
2
(cid:10) (cid:12) (cid:12) (cid:11) Xg=0 (cid:16)(cid:10) (cid:12) (cid:12) (cid:11)(cid:17)
1 ¯ (cid:12) (cid:12) (cid:12) (cid:12)
= − Ψˆ Y Q Ψˆ
(1) 0 B (1)
2
∞(cid:10) (cid:12)¯ (cid:12) (cid:11) ¯
− (cid:12)Bˆ (cid:12)Y (1−P ) Ψˆ + Ψˆ PTY Q Ψˆ . (3.24)
(2+g) 0 G (1−g) (2+g) G 0 B (−g)
Xg=0(cid:16)(cid:10) (cid:12) (cid:12) (cid:11) (cid:10) (cid:12) (cid:12) (cid:11)(cid:17)
(cid:12) (cid:12) (cid:12) (cid:12)
The NL fields |Bˆ i serve as gauge-fixing multipliers corresponding to the Ramond-Siegel
(2+g)
gauge conditions for the FP ghost fields |Ψˆ i and the original field |Ψˆ i:
(−g) (1)
(1−P )|Ψˆ i= 0 for g ≥ 0. (3.25)
G (1−g)
In addition, we can see from the action (3.24) that antighost fields are projected by P and
G
satisfy the Ramond-Siegel gauge condition automatically:
(1−P )P |Ψˆ i= 0 for g ≥ 0. (3.26)
G G (2+g)
Integrating out the NL fields |Bˆ i, therefore, we can replace all the string fields |Ψˆ i by
(2+g) (g)
the gauge-fixed fields |Ψˆ i= P |Ψˆ i and obtain
k(g) G (g)
∞
1 ¯ ¯
S = − Ψˆ Y Q Ψˆ − Ψˆ Y Q Ψˆ ,
k(1) 0 B k(1) k(2+g) 0 B k(−g)
2
g=0
(cid:10) (cid:12) (cid:12) (cid:11) X(cid:10) (cid:12) (cid:12) (cid:11)
1 ¯ (cid:12) (cid:12) (cid:12) (cid:12)
= − Ψˆ Y Q Ψˆ , (3.27)
k 0 B k
2
(cid:10) (cid:12) (cid:12) (cid:11)
where (cid:12) (cid:12)
∞
|Ψˆ i≡ |Ψˆ i. (3.28)
k k(g)
g=−∞
X
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This action (3.27) has the same form as the original action, but with the ghost number
constraint relaxed, and is subject to the gauge condition
(1−P )|Ψˆi= 0. (3.29)
G
Because of the equations of motion
P |Bˆ i= P Q |Ψˆ i for g ≥ 0 (3.30)
G (2+g) G B k(1+g)
derived by varying the action (3.27) with respect to |Ψˆ i= (1−P )|Ψˆ i the BRST
⊥(1−g) G (1−g)
transformations (3.23) can be written in a single BRST transformation law:
δ |Ψˆ i= P Q |Ψˆ i. (3.31)
B k G B k
3.2. Batalin-Vilkovisky (BV) action
The iterative procedure discussed in the previous subsection is transparent and straight-
forward but is not applicable in an interacting case. We need to use the BV formalism to fix
the nonlinear gauge symmetry (2.3) of the interacting theory.17),18) Therefore, here, we first
apply the BV formalism in the free R string case and show that it reproduces the results
obtained in the previous subsection.
In the BV formalism, we require an extended action S(ϕ,ϕ∗) to satisfy the BV master
equation (or Zinn-Justin equation):
∂S ∂S
= 0, (3.32)
∂ϕi ∂ϕ∗
i i
X
where ϕ∗ denotes the antifield conjugate to the field ϕi. If we have such an action, then
i
the gauge fixing can be simply performed by setting the antifields ϕ∗ equal to zero. Note
i
that we are adopting the BV formalism in the so-called gauge-fixed basis.∗) The resultant
gauge-fixed action is given by
S (ϕ) = S(ϕ,ϕ∗ = 0), (3.33)
GF
which is invariant under the (true) BRST transformation
∂S
δ ϕi = . (3.34)
B ∂ϕ∗
i (cid:12)ϕ∗=0
(cid:12)
Indeed, it follows from the BV master equation t(cid:12)hat
(cid:12)
∂S ∂S ∂S
δ S (ϕ) = ·δ ϕi = = 0. (3.35)
B GF ∂ϕi B ∂ϕi ∂ϕ∗
Xi (cid:12)(cid:12)ϕ∗=0 "Xi i #ϕ∗=0
(cid:12)
∗) This is because the BV master actio(cid:12)n in this basis has the same form as the classicalgauge-invariant
action, as we will see shortly.
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LetusapplytheBVformalismtothefreeRstringfieldtheory. Intheprevioussubsection,
it was shown that the gauge-fixed action has the same form as the classical gauge-invariant
action without the ghost number constraint. This suggests that the BV master action also
takes the same form:
1 ¯
S = − Ψˆ Y Q Ψˆ , (3.36)
0 B
2
which is invariant under the BRST transfor(cid:10)ma(cid:12)tion (cid:12) (cid:11)
(cid:12) (cid:12)
δ |Ψˆi= Q |Ψˆi. (3.37)
B B
We now show that this invariance implies that the action (3.36) satisfies the BV master
equation (3.35). Using the component form of the projected fields
|Ψˆi= |φii−(γ +c F)|ψii, (3.38)
0 0
we can calculate the inner product with the metric Y as
0
¯
Ψˆ Y Ψˆ = hhφ¯ |−hhψ¯ |γ c δ′(γ ) |φ ii−γ |ψ ii
1 0 2 1 1 0 0 0 2 0 2
¯ ¯
(cid:10) (cid:12) (cid:12) (cid:11) = (cid:0)hhφ |c δ(γ )|ψ ii(cid:1)+hhψ |c(cid:0)δ(γ )|φ ii (cid:1)
1 0 0 2 1 0 0 2
(cid:12) (cid:12)
=hφ¯ |ψ i+hψ¯ |φ i, (3.39)
1 2 1 2
where we have used Eq. (3.6). From Eq. (3.39), we can see that the metric Y is just the
0
cross-diagonal nondegenerate metric between the two independent components |φi and |ψi
for the projected R field |Ψˆi, just like c in the bosonic string case. Therefore, the general
0
variation δS can be written as
δS = − δΨ¯ˆ Y Q Ψˆ = − δΨ¯ˆ Y δ Ψˆ =hδφ¯|δ ψi+hδψ¯|δ φi, (3.40)
0 B 0 B B B
(cid:10) (cid:12) (cid:12) (cid:11) (cid:10) (cid:12) (cid:12) (cid:11)
with (cid:12) (cid:12) (cid:12) (cid:12)
δ |Ψˆi= |δ φii−(γ +c F)|δ ψii. (3.41)
B B 0 0 B
We thus have∗)
∂S ∂S
= δ φ, = δ ψ. (3.42)
B B
∂ψ ∂φ
As a result, the BRST invariance of the master action implies that
∂S ∂S ∂S ∂S
0 = δ S = δ φ+ δ ψ = 2 , (3.43)
B B B
∂φ ∂ψ ∂φ ∂ψ
∗) The fields φ and ψ in Eqs. (3.42) actually represent the coefficient fields appearing in the expansion
of the stringstates |φi and |ψi, respectively. Sincehδφ¯|=hδφ|ψ0 andψ0 is fermionic, these coefficient fields
0 0
φ and ψ have a nonvanishing inner product between those with opposite statistics, as it should be for the
BV field and antifield. Although φ¯ = φTC for the coefficient fields, we take C = 1, which is possible for
GSO-projected chiral fields.
10