Physical States, Factorization and Nonlinear Structures in Two Dimensional Quantum Gravity

Abstract

The nonlinear structures in 2D quantum gravity coupled to the $(q+1,q)$ minimal model are studied in the Liouville theory to clarify the factorization and the physical states. It is confirmed that the dressed primary states outside the minimal table are identified with the gravitational descendants. Using the discrete states of ghost number zero and one we construct the currents and investigate the Ward identities which are identified with the W and the Virasoro constraints. As nontrivial examples we derive the $L_0$, $L_1$ and $W_{-1}^{(3)}$ equations exactly. $L_n$ and $W^{(k)}_n$ equations are also discussed. We then explicitly show the decoupling of the edge states $O_j ~(j=0~ {\rm mod}~ q) $. We consider the interaction theory perturbed by the cosmological constant $O_1$ and the screening charge $S^+ =O_{2q+1}$. The formalism can be easily generalized to potential models other than the screening charge. Comment: 18 pages, LaTex, YITP/U-93-28

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arXiv:hep-th/9309094v1 17 Sep 1993
YITP Uji Research Center
YITP/U-93-28
hep-th/9309094
September 1993
Physical States, Factorization and Nonlinear Structures
in Two Dimensional Quantum Gravity
Ken-ji HAMADA1
Uji Research Center
Yukawa Institute for Theoretical Physics
Kyoto University, Uji 611, Japan
Abstract
The nonlinear structures in 2D quantum gravity coupled to the (q + 1,q) min-
imal model are studied in the Liouville theory to clarify the factorization and the
physical states. It is confirmed that the dressed primary states outside the min-
imal table are identified with the gravitational descendants. Using the discrete
states of ghost number zero and one we construct the currents and investigate the
Ward identities which are identified with the W and the Virasoro constraints. As
nontrivial examples we derive the L0, L1and W(3)
equations are also discussed. We then explicitly show the decoupling of the edge
states Oj (j = 0 mod q). We consider the interaction theory perturbed by the
cosmological constant O1and the screening charge S+= O2q+1. The formalism
can be easily generalized to potential models other than the screening charge.
−1equations exactly. Lnand W(k)
n
1E-mail address: hamada@yisun1.yukawa.kyoto-u.ac.jp

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1Introduction
Since the discovery of the double scaling limit in the matrix models [1, 2, 3],
many efforts have been made to understand 2D quantum gravity. In the matrix
model approach it was shown that the amplitudes obey the nonlinear equations
called the W and the Virasoro constraints [4, 5]. We have tried to investigate the
nonlinear structures in the Liouville theory approach [6–15] in order to clarify the
nature of physical states and factorization properties of amplitudes. There are a
few works [12, 13], but clear arguments have not been found yet.
In the Liouville theory it has been found that there are an infinite number of the
BRST invariant states. Although the discrete states are discussed in detail [14, 15],
the physical role of them are not sufficiently understood. Here we discuss the role
of the discrete states with ghost number zero and one according to the classification
by Bouwknegt, McCarthy and Pilch (BMP). We also consider the dressed primary
states both inside and outside the minimal table. As shown by Kitazawa [11]
the dressed primary states outside the minimal table no longer decouple in the
combined matter-Liouville theory. Furthermore he claimed these fields to be the
gravitational descendants. In this paper, using the discrete states of the ghost
number zero and one, we construct symmetry currents.2Substituting the currents
into the correlation functions of the dressed primary fields we set up the Ward
identities and identify them with the W and the Virasoro constraints.
The paper is organized as follows. In Sect.2 we summarize the various BRST
invariant states of the (q + 1,q) minimal theory coupled to gravity and define the
symmetry currents. The correlation functions are defined along the argument of
Goulian and Li [10]. We consider the interaction theory perturbed by the cosmo-
logical constant O1and the screening charge S+= O2q+1, where another screening
charge S−is not used. In Sect.3 we discuss the factorization properties of ampli-
tudes. The factorization in the Liouville theory is rather similar with that of the
string theory [8, 9]. However there is a crucial difference in the pole structures
of the propagators, which leads to the nonlinear structures in the Liouville theory
approach. In Sect.4 we discuss the Ward identities corresponding to the Virasoro
costraints. As nontrivial examples we derive L0and L1equations exactly. We also
discuss the Lnequations. The Ward identities corresponding to the W constraints
are discussed in Sect.5, where W(3)
briefly discussed. Then we explicitly show that the edge states Oj(j = 0 mod q)
decouple from the expressions. Sect.6 is devoted to discussion. Here we argue the
generalization of the formalism. If we consider the interaction theory perturbed
by the scaling operator Op+q instead of the screening charge O2q+1, we will get
the gravity theory coupled to (p,q) conformal matter. We also comment on the
unitarity problem in the two dimensional quantum gravity.
−1equation is derived exactly. W(k)
n
equations are
2The importance of the BRST invariant discrete states of ghost number zero is emphasized in
the two dimensional string theory [16, 17, 18]. Also in ref. [12, 13] they play an important role.
2

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2BRST invariant states and correlation func-
tions
In this section we summarize the Liouville theory approach to 2D quantum
gravity coupled to the minimal conformal matter with central charge cM= 1−
and establish our notations and conventions. The action for the Liouville-matter
part is
6
q(q+1)
S0=
1
8π
+1
? ?
? ?
ˆ g(ˆ gαβ∂αφ∂βφ + 2iQLˆRφ)
8π
ˆ g(ˆ gαβ∂αϕ∂βϕ + 2iQMˆRϕ) ,(2.1)
where the scalar fields φ and ϕ are the Liouville and the matter fields respectively.
The background charges are
iQL=
2q + 1
?
2q(q + 1)
,QM=
1
?
2q(q + 1)
,(2.2)
where we use the convention that QLis purely imaginary. In terms of modes
i∂φ(z) =
?
n∈Z
αL
nz−n−1
i∂ϕ(z) =
?
n∈Z
αM
nz−n−1,(2.3)
the Virasoro algebras are given by
LL,M
n
=1
2
?
m∈Z
: αL,M
m αL,M
n−m: −(n + 1)QL,MαL,M
n
,(2.4)
where [αL
The physical states are identified with nontrivial cohomology classes of the
BRST operator
n,αL
m] = [αM
n,αM
m] = nδn+m,0.
QBRST=
?
dz
2πic(z)
?
TL(z) + TM(z) +1
2TG(z)
?
.(2.5)
In terms of modes
QBRST= c0L0− b0M +ˆd ,
L0= LL
(2.6)
(2.7)
0+ LM
0+ LG
0,M =
?
(m − n) : c−mc−nbn+m: .
n?=0
n : c−ncn: ,
ˆd =
?
n?=0
c−n(LL
n+ LM
n) −1
2
?
n+m?=0
n,m?=0
(2.8)
By introducing the lightcone-like combinations of the modes α±
n ?= 0 and the generalized momentum variables
P±(n) =
√2
n=
1
√2(αM
n± iαL
n),
1
?
(αM
0− (n + 1)QM) ± i(αL
0− (n + 1)QL)
?
(2.9)
3

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the operatorˆd is rewritten as
ˆd =
?
n?=0
c−n(α−
nP+(n) + α+
nP−(n))
+
?
n+m?=0
n,m?=0
: c−n
?
α+
−mα−
n+m+1
2(m − n)c−mbn+m
?
: .(2.10)
BMP [15] classify the nontrivial cohomology of the BRST charge. If P+(r) ?= 0
or P−(s) ?= 0 for all r,s ∈ Z, r,s ?= 0 and P+(0)P−(0) = 0, then there exist the
dressed primary states with momentums iαL
0= αjand αM
0= βjparametrized by
αj= (2q + 1 − j)QM,βj= (1 + j)QM.(2.11)
The corresponding fields are given by
Oj=
?
d2zVj(z, ¯ z) =
?
d2zeαjφ(z,¯ z)eiβjϕ(z,¯ z). (2.12)
Here we use the unusual convention for βj. The reason why we adopt it will be
mentioned after correlation functions are defined. We identify these fields with the
gravitationl primaries and their descendants as
Onq+k= σn(Ok) ,(k = 1,···,q − 1;n ∈ Z≥0) , (2.13)
where n = 0 states are gravitational primaries. We now exclude the edge states
of k = 0. This identification was first proposed by Kitazawa [11], who showed
that the dressed primary states outside the minimal table fail to decouple in the
combined matter-Liouville theory. In the following section, by deriving the non-
linear structures directly in the Liouville theory approach, we will confirm this
identification and show that the edge states indeed decouple.
Besides the dressed primary states there exist the nontrivial BRST invariant
states called the discrete states at the momentums3
iαL
0= α−(r+s)q−r,αM
0= β(r−s)q+r. (2.14)
In this paper we only cosider the discrete states of r,s < 0. Then there exist the
states of ghost number zero, Br,s(z). For examples we get
B−1,−1(z) = 1 , (2.15)
B−1,−2(z) =
?
?
bc −
?q + 1
2q
?
(∂φ + i∂ϕ)
?
eα3q+1φ(z)eiβq−1ϕ(z), (2.16)
B−2,−1(z) = bc −
q
2(q + 1)(∂φ − i∂ϕ)
?
eα3q+2φ(z)eiβ−q−2ϕ(z). (2.17)
3Note that βr,s = β(r−s)q+r, where βr,s =
β−= −
1
√2(1 + r)β++
1
√2(1 + s)β−, β+ =
?
q+1
q
and
?
q
q+1.
4

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There also exist the primary states Rr,swith the momentums (2.14). For instance
R−1,−1(z) = ∂φ(z) + (2q + 1)i∂ϕ(z) ,
R−1,−2(z) =1
q
(2.18)
?
−q − 3
4√2∂2φ +7q + 3
4√2
i∂2ϕ +q2− 2q − 1
8
?
?
q(q + 1)
?
(∂φ)2− (∂ϕ)2?
−3q2+ 2q + 1
4q(q + 1)
?
∂φi∂ϕeα3q+1φ(z)eiβq−1ϕ(z), (2.19)
R−2,−1(z) =
1
q + 1
?q + 4
4√2∂2φ +7q + 4
4√2
i∂2ϕ −q2+ 4q + 2
8
?
q(q + 1)
?
(∂φ)2− (∂ϕ)2?
−3q2+ 4q + 2
4q(q + 1)
?
∂φi∂ϕ
?
eα3q+2φ(z)eiβ−q−2ϕ(z), (2.20)
where we remove c ghost in the definition of Rr,s.
Combining Rr,sand¯Br,swe can construct the symmetry currents
Wr,s(z, ¯ z) = Rr,s(z)¯Br,s(¯ z) ,r,s ∈ Z−,(2.21)
which satisfy
∂¯ zWr,s(z, ¯ z) = {¯QBRST,[¯b−1,Wr,s(z, ¯ z)]} .(2.22)
Main assertion of this paper is that the Ward identities of the currents
?
d2z∂¯ z≪ W−k,−n−k(z, ¯ z)
?
j∈S
Oj≫g= 0 ,(k = 1,···q − 1; n ∈ Z≥2) (2.23)
are just the W(k+1)
and others are the W constraints.
Let us define the correlation functions of the Liouville theory. We consider the
interaction theory,
S = S0+ µO1− tO2q+1,
where O1is the cosmological constant and O2q+1is nothing but the screening charge
S+= ei√2β+ϕ, β+=(q + 1)/q. Note that we do not use another screening charge
S−= ei√2β−ϕ, β−= −
scaling operators (2.13).
The matter sector has the symmetry under the constant shift ϕ → ϕ+2π/QM
because then the action only shifts by 2πiχ, where χ is the Euler number, such
that e−Sis invariant. Therefore we restrict the range of the zero mode integral of
ϕ within 0 ≤ ϕ0≤ 2π/QM. On the other hand, since QLis purely imaginary, there
is no such a symmetry for the Liouville sector so that we do not restrict the range
of zero mode of φ. After integrating over the zero modes of the Liouville and the
matter fields the correlation functions are expressed as
tn
n!<
n
constraints. The equations for k = 1 is the Virasoro constraints
(2.24)
?
?
q/(q + 1) because it is not included in the definition of the
≪
?
j∈S
Oj≫g= κ−χµsΓ(−s)
α1
2π
QM
?
j∈S
Oj(O1)s(O2q+1)n>g, (2.25)
5