Conformal Field Theory with Two Kinds of Bosonic Fields and Two Linear Dilatons
Davoud Kamani∗
Faculty of Physics, Amirkabir University of Technology (Tehran Polytechnic) P.O.Box: 15875-4413, Tehran, Iran
(Received on 26 June, 2009)
We consider a two-dimensional conformal field theory which contains two kinds of the bosonic degrees of freedom. Two linear dilaton fields enable us to study a more general case. Various properties of the model such as OPEs, central charge, conformal properties of the fields and associated algebras will be studied.
Keywords: String; Linear Dilaton CFT; OPE.
1. INTRODUCTION
Among the various conformal field theories (CFTs) the linear dilaton CFT has some interesting applications in the string theory,e.g.see [1–6]. In this CFT, modification of the energy-momentum tensor was anticipated by a linear dila-ton field. This CFT significantly changes the behavior of the worldsheet theory. In addition, this CFT is a consistent way for reducing the spacetime dimension without compactifica-tion. The linear dilaton CFT also appears as an ingredient in many string backgrounds, critical and non-critical. Accord-ing to its importance we proceed to develop it.
In this paper we consider an action which is generaliza-tion of the bosonic part of the
N
=2 superstring theory. In addition, we introduce two linear dilaton fields to build our model. Thus, we study a conformally invariant field theory in two flat dimensions. Anticipating to the string theory, we refer to these two dimensions as the string worldsheet. Vari-ous OPEs of the model will be calculated. Due to the dilaton fields, some of these OPEs, and also conformal transforma-tions of the worldsheet fieldsXµ(σ,τ)andY µ(σ
,τ)have de-viations from the standard forms. Presence of some param-eters in the central charge enables us to receive a desirable dimension for the spacetime. The algebra of the oscillators reveals that the oscillators ofX-fields do not commute with that ofY-fields.
This paper is organized as follows. In section 2, we shall introduce the action of the model and the linear dilatonic energy-momentum tensor associated to it. In section 3, var-ious OPEs of the model will be studied. In section 4, con-formal transformations of the worldsheet fields will be ob-tained. In section 5, various quantities will be expressed in terms of the oscillators, and two algebras for the model will be obtained. Section 6 is devoted for the conclusions.
2. THE MODEL
We begin with the action of the scalar fieldsXµ(z,z¯)and Yµ(z,z¯)in two dimensions
S = 1
2πα′
Z
d2z(∂zXµ∂¯zXµ+β∂zYµ∂¯zYµ
+ λ(∂zXµ∂z¯Yµ+∂¯zXµ∂zYµ)), (1)
whereµ∈ {0,1, ...,D−1} and β andλ are constants, i.e. independent ofzand ¯z. For the spacetime we consider the flat Minkowski metricηµν=diag(−1,1, ...,1). The special caseβ=1 andλ=0 indicates the bosonic part of the
N
=2 super-conformal field theory. Thus, we say the set{Xµ} de-scribes the spacetime coordinates. In other words,Xµ(σ,τ) is regarded as the embedding of the worldsheet in the space-time. However,Yµ(σ,τ)enters essentially in the same way. So the set {Yµ(σ,τ)} does not describe additional dimen-sions. The conformal invariance of this action,i.e.symmetry under the conformal transformationsz→z′(z)and ¯z→z¯′(z¯), leads to the zero conformal weights for the fieldsXµandYµ. The equations of motion, extracted from the action (1), are
∂z∂z¯Xµ+λ∂z∂¯zYµ=0,
λ∂z∂¯zXµ+β∂z∂¯zYµ=0. (2)
We assume that the determinant of the coefficients of these equations to be nonzeroi.e.,
det
1 λ λ β
=β−λ26=0. (3)
Therefore, we obtain
∂z∂¯zXµ=0,
∂z∂¯zYµ=0. (4)
These imply∂zXµand∂zYµare functions ofz, and∂¯zXµand ∂¯zYµare functions of ¯z.
Tzz′ ≡T′(z) =−α1′(:∂zXµ∂zXµ:+β:∂zYµ∂zYµ:+λ:∂zXµ∂zYµ:+λ:∂zYµ∂zXµ:), Tz¯′z¯≡T˜′(z¯) =−1
α′(:∂z¯X µ∂¯
zXµ:+β:∂¯zYµ∂¯zYµ:+λ:∂¯zXµ∂¯zYµ:+λ:∂¯zYµ∂¯zXµ:),
Tz′z¯=Tzz¯′ =0, (5)
where : : denotes normal ordering.
It is possible to construct a more general CFT with the same action (1), but with different energy-momentum tensor
T(z) =Λi j:∂zXiµ∂zXjµ:+Vµi∂2zX µ i , ˜
T(z¯) =Λi j:∂¯zXiµ∂¯zXjµ:+Vµi∂2z¯X µ i ,
Tz¯z=Tzz¯ =0, (6)
where sum over iand j is assumed withi,j∈ {1,2}. We define
X1µ=Xµ , X µ 2=Y
µ , Vµ1=Vµ , V
2 µ =Uµ,
Λ=−α1′
1 λ λ β
. (7)
The vectorsVµandUµare fixed in the spacetime. For each pair of these vectors we have a CFT. The extra terms in (6)
are total derivatives. Thus, we shall see that they do not affect the status ofT(z)and ˜T(z¯)as generators of conformal trans-formations. The fieldΦ=VµXµin (6) is linear dilaton. In the same wayΦ′=UµYµis a linear field in the Y-space. In fact, by introducing the worldsheet fields{Yµ(σ
,τ)}and defining the energy-momentum tensor (6), we have generalized the linear dilaton CFT. The caseβ=1 andλ=0 decomposes the model to two copies of the linear dilaton CFT.
3. OPERATOR PRODUCT EXPANSIONS (OPES)
3.1. The OPEsX X,XYandYY
We use the path integral formalism to derive operator equations. Since the path integral of a total derivative is zero, we have the equation
0= Z
DX DY δ
δXµ(z,z¯)
e−SXν(w,w¯)...
= Z
DX DYe−S
1
πα′∂z∂¯z[X µ(z,
¯
z) +λYµ(z,z¯)]Xν(w,w¯) +ηµνδ(2)(z−w,z¯−w¯)
...
=h
1
πα′∂z∂¯z[X µ(z,
¯
z) +λYµ(z,z¯)]Xν(w,w¯) +ηµνδ(2)(z−w,z¯−w¯)
...i. (8)
The point(w,w¯)might be coincident with(z,z¯), but the in-sertion “...” is arbitrary, which is away from(z,z¯)and(w,w¯). Arbitraryness of the insertion implies
1
πα′∂z∂¯z[X µ(z,
¯
z) +λYµ(z,z¯)]Xν(w,w¯)
=−ηµνδ(2)(z−w,z¯−w¯), (9)
as an operator equation. In the same way, the equation
Z
DX DY δ
δXµ(z,z¯)
e−SYν(w,w¯)...
=0, (10)
gives the operator equation
∂z∂¯z[Xµ(z,z¯) +λYµ(z,z¯)]Yν(w,w¯) =0. (11)
In the equation (10) changeXµtoYµ, we obtain
1
πα′∂z∂¯z[λXµ(z,z¯) +βYµ(z,z¯)]Yν(w,w¯)
=−ηµνδ(2)(z−w,z¯−w¯). (12)
Similarly, in the first line of (8) replacingXµbyYµleads to
The equations (9), (11), (12) and (13) give the following equations
∂z∂¯zXµ(z,z¯)Xν(w,w¯) =− πβα′
β−λ2η
µνδ(2)(z−w,z¯−w¯) ,
∂z∂¯zYµ(z,z¯)Xν(w,w¯) = πλα′
β−λ2η
µνδ(2)(z−w,z¯−w¯) ,
∂z∂¯zXµ(z,z¯)Yν(w,w¯) = πλα′
β−λ2η µνδ(2)(z
−w,z¯−w¯),
∂z∂¯zYµ(z,z¯)Yν(w,w¯) =− πα′ β−λ2η
µνδ(2)(z−w,z¯−w¯) . (14)
That is, the equations of motion (4) hold except at the co-incident point(z,z¯) = (w,w¯). Define the matrixQi jas in the following
Q= α′
2(β−λ2)
β −λ
−λ 1
. (15)
Thus, the equations (14) can be written in the compact form
∂z∂¯zXiµ(z,z¯)Xjν(w,w¯) =−2πQi jηµνδ(2)(z−w,z¯−w¯). (16) According to this, we have the normal ordered equation
:Xiµ(z,z¯)Xνj(w,w¯):=Xiµ(z,z¯)Xνj(w,w¯) +Qi jηµνln|z−w|2, (17)
where∂z∂¯zln|z−w|2=2πδ(2)(z−w,z¯−w¯)has been used. The equation (16) and (17) indicate the equation of motion
∂z∂z¯:Xiµ(z,z¯)Xνj(w,w¯):=0. (18)
3.2. TheT TOPE
Let
F
be any functional of {Xµ} and {Yµ}. Thus, the generalization of (17) is defined byF
[X,Y] =exp
−1
2Qi j
Z
d2z1d2z2ln|z1−z2|2 δ δXiµ(z1,z¯1)
δ δXjµ(z2,z¯2)
:
F
[X,Y]: . (19)The OPE for any pair of the operators
F
andG
is given by:
F
::G
:=exp
−Qi j Z
d2z1d2z2ln|z1−z2|2 δ δXiFµ(z1,z¯1)
δ δXµGj (z2,z¯2)
:
F G
:, (20)where the functional derivatives act only on the fields
F
orG
, respectively. ForF
=Xiµ(z,z¯) andG
=Xνj(w,w¯)thisreduces to the equation (17), as expected. Using the equation (20), we obtain
:∂zXiµ(z)∂zXjµ(z)::∂wXkν(w)∂wXlν(w):= :∂zXiµ(z)∂zXjµ(z)∂wXkν(w)∂wXlν(w):+
D
(z−w)4(QikQjl+QilQjk)
−(z 1 −w)2
Qik:∂zXµj(z)∂wXlµ(w):+Qil:∂zXµj(z)∂wXkµ(w): +Qjk:∂zXiµ(z)∂wXlµ(w):+Qjl:∂zXiµ(z)∂wXkµ(w):
. (21)
:∂zXiµ(z)∂zXjµ(z)::∂wXkν(w)∂wXlν(w):∼ D
(z−w)4(QikQjl+QilQjk)
−(z 1 −w)2
Qik:∂wXµj(w)∂wXlµ(w):+Qil:∂wXµj(w)∂wXkµ(w): +Qjk:∂wXiµ(w)∂wXlµ(w):+Qjl:∂wXiµ(w)∂wXkµ(w):
−z 1 −w
Qik:∂2wX µ
j(w)∂wXlµ(w):+Qil:∂2wX µ
j(w)∂wXkµ(w): +Qjk:∂2wX
µ
i(w)∂wXlµ(w):+Qjl:∂2wX µ
i(w)∂wXkµ(w):
, (22)
where the non-singular terms have been omitted. For calcu-lating theT TOPE we also need the following OPEs
:∂zXiν(z)∂zXjν(z):∂2wX µ k(w)∼
−(z 2
−w)3(Qik∂wX µ
j(w) +Qjk∂wXiµ(w))
−(z 2
−w)2(Qik∂ 2 wX
µ
j(w) +Qjk∂2wX µ i(w))
− 1
z−w(Qik∂
3 wX
µ
j(w) +Qjk∂3wX µ
i (w)), (23)
∂2zX µ
k(z):∂wXiν(w)∂wXjν(w): ∼(z 2
−w)3(Qki∂wX µ
j(w) +Qk j∂wXiµ(w)), (24)
∂2 zX
µ i(z)∂
2
wXνj(w)∼ 6
(z−w)4η µνQ
i j. (25)
Adding all these together we obtain theT T OPE as in the following
T(z)T(w)∼ c
2(z−w)4−
1
(z−w)2
2Λi jVµk
Qik∂2wX µ
j(w) +Qjk∂2wX µ i(w)
+Λi jΛkl
Qik∂wXjµ(w)∂wXlµ(w) +Qil∂wXjµ(w)∂wXkµ(w) +Qjk∂wXiµ(w)∂wXlµ(w) +Qjl∂wXiµ(w)∂wXkµ(w)
−z 1 −w
Λi jΛkl
Qik∂2wXjµ(w)∂wXlµ(w) +Qil∂2wXjµ(w)∂wXkµ(w) +Qjk∂2wXiµ(w)∂wXlµ(w) +Qjl∂2wXiµ(w)∂wXkµ(w)
+Λi jVµk
Qik∂3wX µ
j(w) +Qjk∂3wX µ i(w)
. (26)
The ˜TT˜OPE also has a similar form in terms of ¯z, ¯wand ˜c. The central charges are given by
c=c˜=2DΛi jΛkl(QikQjl+QilQjk) +12Qi jVµiV µ j,
=2D+ 6α′ β−λ2(βVµV
µ
−2λVµUµ+UµUµ). (27)
Vanishing conformal anomaly relates the parameters of the model. That is, string actually can move in a wide range of the dimensions. However, by adjusting the variablesβ,λ,Vµ andUµ, we can obtain desirable dimension for the spacetime. The caseVµ=Uµ=0 gives the central chargesc=c˜=2D, i.e.,Dfor{Xµ}andDfor{Yµ}. Ifβ=1 andλ=0, the action
(1) reduces to two copies of the free string action, and hence the energy-momentum tensor (6) is modified to two copies of the energy-momentum tensor of the linear dilaton CFT. In this case the central charge also reduces to two copies of the central charge of the linear dilaton CFT
c=cX+cY, cX=D+6α′VµVµ,
Since there isΛQ=−2I2×2, theT T OPE (26), and
simi-larly the ˜TT˜ OPE take the standard forms T(z)T(w)∼ c
2(z−w)4+
2
(z−w)2T(w) +
1
z−w∂wT(w), ˜
T(z¯)T˜(w¯)∼ c˜
2(z¯−w¯)4+
2
(z¯−w¯)2T˜(w¯) +
1 ¯
z−w¯∂w¯T˜(w¯). (29)
According to the central charge terms,T and ˜T are not con-formal tensors. Apart from these terms, (29) is the statement thatT(z)and ˜T(z¯)are conformal fields of the weights (2,0)
and (0,2), respectively.
3.3. The OPEsT X,TY,T X˜ andTY˜
The OPET Xkµis
T(z)Xkµ(w,w¯)∼ 1
(z−w)2V µ i Qik−
1
z−wΛi j[Qik∂wX µ j(w)
+Qjk∂wXiµ(w)]. (30)
Thus, fork=1 andk=2 we obtain
T(z)Xµ(w,w¯)∼ 1
(z−w)2 α′ 2(β−λ2)(βV
µ−λUµ) + 1 z−w∂wX
µ(w) ,
T(z)Yµ(w,w¯)∼ 1
(z−w)2 α′
2(β−λ2)(−λV
µ+Uµ) + 1 z−w∂wY
µ(w)
. (31)
In the same way we have
˜
T(z¯)Xµ(w,w¯)∼( 1
¯ z−w¯)2
α′ 2(β−λ2)(βV
µ−λUµ) + 1 ¯
z−w¯∂w¯X µ(w¯)
, ˜
T(z¯)Yµ(w,w¯)∼ 1
(z¯−w¯)2 α′
2(β−λ2)(−λV
µ+Uµ) + 1 ¯ z−w¯∂w¯Y
µ( ¯
w). (32)
TheU-terms andV-terms imply that Xµ andYµare not conformal tensor operators. Putting away these terms (the square singular terms) of the above OPEs leads to the condi-tions
βVµ−λUµ=0,
−λVµ+Uµ=0. (33) Since we assumed β−λ26=0, we obtain Vµ=Uµ=0. Therefore, (31) and (32) reduce to the OPEsT′X,T′Y, ˜T′X and ˜T′Y. That is, withT′and ˜T′the fieldsXµandYµare con-formal tensors, as expected. However, we shall not consider the case (33).
4. CONFORMAL TRANSFORMATIONS OFXµANDYµ
The infinitesimal conformal transformationsz→z′=z+ εg(z)and ¯z→z¯′=z¯+εg(z)∗imply the currents
j(z) =ig(z)T(z), ˜
j(z¯) =ig(z)∗T˜(z¯). (34)
For any holomorphic function g(z) these are conserved. These currents lead to the Ward identity
δXiµ(w,w¯) =−ε
Resz→wg(z)T(z)Xiµ(w,w¯) +Res¯ z¯→w¯g(z)∗T˜(z¯)Xiµ(w,w¯)
, (35)
where “Res” and “ ¯Res” are coefficients of(z−w)−1and(z¯−
¯
w)−1, respectively. From the OPEs (31) and (32) and the Ward identity (35) we obtain the conformal transformations δXµ(w,w¯) =−ε[g(w)∂wXµ(w) +g(w)∗∂w¯Xµ(w¯)]
− α′ε
2(β−λ2)(βV
µ−λUµ)[∂
δYµ(w,w¯) =−ε[g(w)∂wYµ(w) +g(w)∗∂w¯Yµ(w¯)]
− α′ε
2(β−λ2)(−λV
µ+Uµ)[∂
wg(w) +∂w¯g(w)∗]. (37) Due to the inhomogeneous parts, originated fromVµ,Uµ,β
andλ, the fieldsXµ andYµdo not transform as conformal tensor. These parts also indicate that these transformations are not infinitesimal coordinate transformationsδz=εg(z)
andδz¯=εg(z)∗.
5. MODE EXPANSIONS
Now we express some quantities of the model in terms of the oscillators ofXµandYµ. The OPEs (31) and (32) give
T(z)∂wXµ(w)∼ 1
(z−w)3 α′ β−λ2(βV
µ
−λUµ) + 1 (z−w)2∂wX
µ(w) + 1 z−w∂
2 wXµ(w), ˜
T(z¯)∂wXµ(w)∼0, (38)
T(z)∂wYµ(w)∼ 1
(z−w)3 α′
β−λ2(−λV µ+Uµ)
+ 1
(z−w)2∂wY
µ(w) + 1 z−w∂
2 wYµ(w), ˜
T(z¯)∂wYµ(w)∼0. (39)
Thus, the conformal weights of∂zXµ(z)and∂zYµ(z)are h∂X=1 , h˜∂X=0,
h∂Y=1 , h˜∂Y=0. (40) According to these conformal weights, we obtain the Laurent expansions
∂zXiµ(z) =−i
r
α′ 2
∞
∑
m=−∞αµ(i)m
zm+1,
∂¯zXiµ(z¯) =−i
r
α′ 2
∞
∑
m=−∞˜
αµ(i)m
¯
zm+1. (41) Single-valuedness ofXµandYµimplies that
αµ(i)0=α˜µ(i)0=
r
α′ 2 p
µ
i, (42)
where pµi is the linear momentum. Now integration of the expansions (41) gives the closed string solution
Xiµ(z,z¯) =xµi −iα ′ 2 p
µ iln|z|
2+i
r
α′ 2
∞
∑
m6=01 m
×
αµ
(i)m zm +
˜
αµ(i)m
¯ zm
. (43)
Reality of Xiµ implies that αµ(i†)m =α(µi)(−m) and ˜αµ(i†)m =
˜
αµ(i)(−m).
The expansions (41) also lead to
αµ(i)m=
r 2 α′ I C dz 2πz
m∂ zXiµ(z), ˜
αµ(i)m=−
r 2 α′ I ˜ C
dz¯ 2πz¯
m∂ ¯
zXiµ(z¯), (44) whereCin thez-plane and ˜Cin the ¯z-plane are counterclock-wise. Therefore, by using the OPEs∂zXiµ(z)∂wXνj(w) and ∂¯zXiµ(z¯)∂w¯Xνj(w¯)we obtain
[αµ(i)m,α ν (j)n] = [α˜
µ (i)m,α˜
ν (j)n] =
2
α′mη µνQ
i jδm,−n, [αµ(i)m,α˜
ν (j)n] =0,
[xµi,p ν
j] = 2i
α′η µνQ
i j. (45)
Forλ6=0 we observe that the oscillators ofXµdo not com-mute with the oscillators ofYµ.
In terms of the oscillators the nonzero elements of the energy-momentum tensor find the forms
T(z) =−α′
2ηµνΛi j ∞
∑
m=−∞∞
∑
n=−∞1 zm+n+2:α
µ (i)mα
ν (j)n:+i
r α′ 2V i µ ∞
∑
m=−∞m+1 zm+2α
µ (i)m,
˜
T(z¯) =−α′
2ηµνΛi j ∞
∑
m=−∞∞
∑
n=−∞1 ¯
zm+n+2: ˜α µ (i)mα˜
ν (j)n:+i
r α′ 2V i µ ∞
∑
m=−∞m+1 ¯ zm+2α˜
µ
The Virasoro operators are
Lm= I
C dz 2πiz
m+1T(z)
, ˜
Lm= I
˜ C
dz¯ 2πiz¯
m+1T˜(z¯)
. (47)
In terms of the oscillators they take the forms
Lm=− α′
2Λi jηµν ∞
∑
n=−∞:αµ(i)m−nαν(j)n:+i
r
α′
2(m+1)V i µα
µ (i)m,
˜ Lm=−
α′
2Λi jηµν ∞
∑
n=−∞: ˜αµ(i)m−nα˜ν(j)n:+i
r
α′
2(m+1)V i µα˜
µ
(i)m. (48)
Using the standard methods one can show that the normal ordering constant for all Lm and ˜Lm is zero. According to the equations (45), or the standard form of the OPEsT Tand
˜
TT˜,i.e.the equations (29), the Virasoro algebra also has the standard form
[Lm,Ln] = (m−n)Lm+n+ c 12(m
3
−m)δm,−n, [L˜m,L˜n] = (m−n)L˜m+n+
˜ c 12(m
3−m)δ m,−n,
[Lm,L˜n] =0. (49)
The Hamiltonian of the system is given by
H=L0+L˜0−
c+c˜
24 . (50)
Thus, the equations (27) and (48) express this Hamiltonian in terms of the oscillators and the parameters of the model
H= 1
2α ′(p
1.p1+2λp1.p2+βp2.p2+2iV.p1+2iU.p2)
−α′Λi jηµν ∞
∑
n=1(αµ(i)(−n)αν(j)n+α˜µ(i)(−n)α˜ν(j)n)
− α′
2(β−λ2)(βV.V−2λV.U+U.U)−
D
6, (51) where the symmetry ofηµνandΛi jwere introduced.
For the open string there are
αµ(i)m=α˜µ(i)m , α µ (i)0=α˜
µ (i)0=
√
2α′pµi, (52) and hence the solution is
Xiµ(z,z¯) =xµi−iα′pµiln|z|2+i
r
α′ 2
∞
∑
m6=0αµ(i)m
m
1 zm+
1 ¯ zm
. (53)
The corresponding energy-momentum tensor and Virasoro operators are given by the first equations of (46) and (48). Thus, the associated Virasoro algebra also is described by the first equation of (49).
Note that we imposed the boundary conditions of the closed string and open string on Yµ(σ,τ). However,
Yµ(σ,τ) may be neither closed nor open. Assuming closeness or openness for Yµ(σ,τ), the worldsheet fields
(Xµ(σ,τ) ,Y µ(σ
,τ)) find four configurations: (closed , closed), (open , open), (open , closed) and (closed , open). We considered the first and the second cases. The third and the fourth cases also can be investigated in the same way.
6. CONCLUSIONS AND SUMMARY
We studied a CFT model with two kinds of the bosonic de-grees of freedomXµandYµ, which interact kinetically with each other. For each kind of these fields we introduced a linear dilaton field.
Using the path integral formalism, we obtained the OPEs X X,XY andYY. These OPEs enabled us to introduce a gen-eral definition for the OPEs. We observed that theT T and
˜
TT˜ OPEs of the model have the standard forms. Due to the vectorsVµandUµ, which define the dilatons, the OPEsT X, TY, ˜T Xand ˜TY have deviations from the standard forms of them. The central charge of the model depends on the space-time dimension, the parameters of the theory and the vectors Vµ andUµ. A vanishing conformal anomaly and hence a desirable dimension for the spacetime can be achieved by tuning these variables.
Putting away the interacting terms of the action, the model split into two copies of the linear dilaton CFT. The splitting also occurs for the energy-momentum tensor and hence for the central charge.
Using the conserved currents, associated to the conformal symmetry, the conformal transformations of the fieldsXµand Yµ have been extracted. Therefore, the vectorsVµandUµ and also the parameters of the model indicate thatXµandYµ are not conformal tensors. That is, these transformations are not pure coordinate transformations.
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