- Email: [email protected]
Global conformal invariance in D dimensions and logarithmic correlation functions
12June
1997
PHYSICS ELSEWER
LETTERS B
Physics Letters B 402 (1997) 282-289
Global conformal invariance in D dimensions and logarithmic correlation functions A.M. Ghezelbash a,b,l, V. Karimipour a*c,2 a Institute for Studies in Theoretical Physics and Mathematics, PO. Box 19395-5531. Tehran, Iran b Department of Physics, Alzahra University, Vanak. Tehran 19834. Iran ’ Department of Physics, Sharif University of Technology, PO. Box 19365-9161, Tehran, Iran Received 20 March 1997 Editor: L. Alvarez-GaumC
Abstract We define transformation of multiplets of fields (Jordan cells) under the D-dimensional conformal group, and calculate two and three point functions of fields, which show logarithmic behaviour. We also show how by a formal differentiation procedure, one can obtain n-point function of logarithmic field theory from those of ordinary conformal field theory. @ 1997 Published by Elsevier Science B.V. PACS: 11.25.Hf Keywords: Conformal
field theory
1. Introduction Recently, logarithmic conforrnal field theories (LCFT) have been studied in a series of papers [ l-131, both for their pure theoretical interest concerning the structure and classification of conformal field theories [ l-51 and for their relevance in some physical systems [6-l 1] which are described by non-unitary or non-minimal conformal models. All such works have dealt with two dimensional conformal field theory [ l] relying heavily on the underlying Virasoro algebra, and have described how the appearance of logarithmic singularities is related to the modification of the representation of the Virasoro algebra. In this paper we will try to understand LOT’s from yet another point of view, that’s we consider ddimensional conformal invariance. Although our results are based on a generalization of the basic idea of [ 11, we hope that by changing or simplifying the context of study, we can add a little bit to the understanding of the subject in general. I E-mail address: * E-mail address:
[email protected]. [email protected].
0370-2693/97/$17.00 0 1997 Published PII SO370-2693(97)00459-O
by Elsevier Science B.V. All rights reserved.
A.M. Ghezelbush,V. Kurimipow/ PhysicsLetters3 402 (19971282-289
283
As is well known, one of the basic assumptions of conformal field theory is the existence of a family of operators, called scaling fields, which transform under scaling S : x + x’ = Ax, simply as follows: 4(X) --+ 4/(X’) = A-*+(X),
(1.1)
where A is the scaling weight of 4(x) . It is also assumed that under the conformal group, such fields transform as
(1.2) where d is the dimension of space and ]]%~‘/ax(l is the Jacobian of the transformation. Eq. encompasses Rq. ( 1.l ) defines the transformation of the quasi-primary fields. For future use we Jacobian equals Ad for scaling transformation and l]~ll-~~ for the Inversion transformation I : x -+ being unity for the other elements of the conformal group. Combination of ( 1.2) with the definition of the correlation functions, i.e.:
( 1.2) which note that the x’ = x/llxl12,
of symmetry
(1.3) allows one to determine the two and the three point functions up to a constant and the four point function up to a function of the cross ratio. It is precisely the assumption that scaling fields constitute irreducible representations of the scaling transformation, which imposes power law singularity on the correlation functions. As we will see, relaxing this assumption, one naturally arrives at logarithmic singularities. It also leads to many other peculiarities in the relation between correlation functions. To begin with, we consider a multiplet of fields
(1.4)
and note that under scaling x --+ Ax, the most general form of the transformation of Q is @(x) +
@‘(x’) = A%(X)
(1.5)
where T’ is an arbitrary matrix. More generally, we replace ( 1.5) by
(1.6) where T is an n x n arbitrary matrix. When T is diagonalizable, one arrives at ordinary scaling fields by redefining @, so that all the fields transform as l-dimensional representation. Otherwise, following [ 121 we assume that T has Jordan form
T=
(1.7)
Rewriting T as 9 1 + J, where Jij = 6i_i,j, Eq. (1.6) can be written in the form @(x) +
@‘(x’) =
ll$h,e(x),
(1.8)
284
A.M. Ghezelbash. V. Karimipow/Physics
where AX = 11% JIJ is a lower triangular
Letters B 402 (1997) 282-289
matrix of the form,
(*
(1.9)
(*X)ii=l~
i.e. for N = 2 we have
(1.10) An important point is that the top field ~$1(x) always transform as an ordinary quasi-primary field. A most curious property of the transformation ( 1.8) is that each field &+I transforms as if it is a formal derivative of C#Qwith respect to 9 [ 131
A+1
(x>
(1.11)
= -&&dk(x)*
This formal relation which field $1, essentially means from those of the ordinary The phrase “with due care” Jordan cell.
determines the transformation of all the fields of a Jordan cell from that of the top that with due care, one can determine the correlation functions of the lower fields top fields simply by formal differentiation. We will elucidate this point later on. in the previous statement refers to the two point function of fields within the same
2. The two point functions We already know by standard arguments [ 131 that the two point function belonging to two different Jordan cells (A,, n) and (Ap, m) vanishes, i.e.:
of the top fields &
and 4p
Due to the observation (1.1 l), it follows that the two point function of all the fields of two different Jordan cells with respect to each other vanish. Therefore in this section we calculate the two point function of the fields within the same Jordan cell. As we will see logarithmic conformal symmetry gives many interesting and unexpected results in this case. Let us denote the matrix of two point functions (4i( X)4j (y)) for all +i, 4j E (A, n) by G( X, y), then from rotation and translation symmetries, this matrix should depends only on 1(x - ~11. From scaling symmetry and using (1.2) and (1.3), we will have
(2.2)
AG(Ilx - yll,A’ = A2AG(hllx - yll>, where A = AdJ, and from inversion
Mwlx-
symmetry,
we have
IIX - YII Yllhq = l141-2A llYll-2Acc~).
(2.3)
where A, =
Defining
“X”-2dJ,
A, =
the matrix F as
“y”-2dJ.
(2.4)
A.M. Ghezelbash, V. Karimipour/ Physics Letters B 402 fi997) 282-289
285
we will have from (2.2) and (2.3), AF(Ilx - YlW
=
F(Allx - Yll>T
(2.5)
and
lb - Yll
A,F( /Ix - YlO$
= F( -
)
(2.6)
ll4l IIYII .
For every arbitrary A, we now choose the points x and y such that J/x/J = A$ and J/y/J = A$. It should be of space. From (2.4), we will have A, = Ai = A$, therefore Eq. (2.6) turns into, and A.v noted that in this way by varying A, we can span all the points
AtF( [Ix - yll) (A’)’ = F(Allx - ~11).
(2.7)
Combining (2.5) and (2.7) and using invertibility of A, we arrive at F = AiF(A
(2.8)
by iterating (2.8), we will have F = AFAt-‘,
and by rearranging, we will have
FA’ = AE
(2.9)
Expanding A in terms of power of InA as (dlnA)2J2 2,
A=l+(dlnA)J+
+
.*.
and comparing both sides, we arrive at F(J’)k=
(J)kE
Since (J”)ij =
6i,j+k,
k=1,2 ,...,
n-l.
(2.10)
we will have from (2.10)) (2.11)
Fi,j-k = Fi-k,j,
which means that on each opposite diagonal of the matrix F, all the correlations are equal. Moreover from FJ = JF, one obtains n
c
Filsj,l+l
I=1
=2 S&l+1
ej,
(2.12)
/=I
which means that if j = 1 and 1 < i 5 n, then Fi-l,j = 0, or Ej=O
forj=l
and 1 liln-1.
(2.13)
Combining this with (2.11). we find that all the correlations above the opposite diagonal are zero. In order to find the final form of F, we use Eq. (2.5) again, this time in infinitesimal form, let A = 1 + EYJ+ o(cy*) where LY= dln A, then from AF(x)A’= F(Ax), we have d(JF
+ FJ’) =x$
(2.14)
286
A.M. Ghezelbash, V. Karimipow/Physics
Due to the property
if
F2.n = CIY + ~2,
Ft,n = ~1, with the recursion
dFi n
i> 1
(2.15)
F3,n =
&Y2 +
c2y + c3,
etc.
(2.16)
relations,
= F;_I,n.
dy
we obtain
the new variable y = 2d In x gives
which upon introducing
)
only the last column of F should be found, therefore from (2.14)
(2.11)
d&n XL =2dFi,,-1, dx
Letters B 402 (1997) 282-289
(2.17)
> I
Thus we have arrived at the final form of the matrix F, which is as follows: .. ... ... ...
0 0 0 .. .
F= \
go ...
0
0 go go gr
0
go al-2
g1
g2
.. .
.. . gll
&-I
(2.18)
where each gi is a polynomial of degree i in y, and gi = dgi+l/dy. All the correlations depend on the n constants which remain undetermined. We have checked (as the reader can check for the single II = 2 case) that inversion symmetry puts no further restrictions on the constants ci.
Cl, * . . 7 c,,
3. n-point correlation
functions
The observation ( 1.11) that the transformation properties of the members of a Jordan cell are as if the are formal derivative of the top field in the cell, allows one to determine the correlation functions of all the fields within a single or different Jordan cells, once the correlation function of the top fields are determined. As an example, from ordinary CFT, we know that conformal symmetry completely determines the three point function up to a constant. Let qb,, q5p and & be the top fields of three Jordan cells (A,, E), ( Ap, m) and A,, n) respectively. Therefore, we know that (4a(x)4p(y)4,(z))
=
lIx _ y~/A,+Ap-A,
11~ _
z;t::Ay-Ap
,ly _
z I,A~+A~-A~,
(3.1)
’
A,py in principle depends on the weights Aa, Ap and AY. Denoting by sbLllr3 we will readily find from (3.1) that
where the constant cell (A,,0
=
11~ _
y~/A,+ArA~
11~ _
z
~~A,f%--ha 11~
dA + ~1~_ y~~A~+h%~~x _ z,,:%-A~,,y 3 For simplicity,
_
z
the second fielc of the
JIAP+A+
IIY- ZII ln( _ Z~~A~+Ar-A, (lb - Yll)(llX-
zld’
we have denoted the top field by 4 and the second field by 41, instead of 41 and q52 respectively.
(3.2)
A.M. Ghezelbash, V. Karimipur/
287
Physics Letters B 402 (1997) 2X2-289
where
is a new undetermined constant. For the correlation functions of fields within a single cell, one should then take the limit /?,r --) a in the above formula. It is not difficult to check that this formula satisfies all the requirements demanded by conformal symmetry. No need to say one can continue this procedure of formal differentiation to determine other correlation functions. It may be asked why we have not applied the procedure of formal differentiation for calculation of two point function, but have followed an independent route. The point is that the two point function has the form
which, upon differentiation, yields derivative of the Kronecker symbol or the delta function which is not easy to handle.
4. The operator product expansion In this section we calculate the operator product of the fields in two different Jordan cells, (A,, N,), and (AS, Np). We assume that the operator product of the two top-fields & E (A,, N,) and q$ E ( AB, NP) can be expressed in terms of the fields and their descendants in other Jordan cells,
where the sum is over the Jordan cells of weight (A,,, N,). The field r$y,n stand for the n-th element of a Jordan cell together with its descendants. We now demand that both sides have the same behaviour under conformal transformations, i.e. we demand that (4.1) is equivalent to the following relation ~h(aJ&(Y’)
=
p&l/x’-
y’ll)qb;,“(y’).
(4.2)
n,Y Considering first the scale transformation, we obtain from ( 1.6) ~-A”-Avaw4~(Y)
=
c
C,y,,,(W~ - YII))A-A7@&J(Y).
(4.3)
n*Y,PlN,
Redefining the function C&,(X> as follows
C$&) =
C&f-Ax) AhAp-A,
lb - YII
’
(4.4)
where C&’ s are constant and comparing Eqs. (4.1) with (4.3) leads to the following equation for the f’s
fn(A-‘x)
= &p(x)Apn9 p=l
(4.5)
288
A.M. Ghezelbash, V. Karimipow/Physics
Letters B 402 (1997) 282-289
or in compact matrix form, f( A-lx)
= Pf(X)
)
(4.6)
h. (4.6) is easily solved. It is in fact a recursion relation for f,,‘s due to the triangular solution is N-i f;(x)
Ui+k( -d In X) k
= c
k
form of A. The genera]
?
(4.7)
k=o where the constants fs(x)
= LYE, f2(x)
It is interesting fk(x)
. . , &‘Nare free. For N = 3 for example, we have
at,.
=
= cus(-dlnx)
fi(x)
+ CQ,
to note that the following
= i(Ys(-dInx)2+a2(-dlnx)
+ai.
relation holds among these functions,
d(--ilnx) fk-‘(x)’
5. Conjugation
(4.91
properties
In this section, we will restrict ourselves to two dimensions and derive the hermitian in a Jordan cell. Our main result is encoded in the following formula &tcz>
multiplet
= i-2A$(t),
As a justification
I/*(Z)
for the validity by (@(z>@‘(w>)ij,
(@(z>@‘(w))
of the fields
(5.1)
of the fields have been suppressed.
(4i(z)4.j(w))
conjugate
= $+~-2@-J’)
where & is an N dimensional
$*(I)
(4.8)
of operators, the superscript
As an example,
when N = 2 and 4 =
= i-2A($(i) of (5.1),
t, denotes transpose and the Z dependence E 04
, we have
(5.2)
+2lnZ&i)).
consider
the transformation
z ---f i, with llz 11> llwll and denote
then
= (O~~(z)~,‘(w>~O)
where cc denotes complex conjugation
(5.3)
= (0~R~+‘(w)~+(z>~0)“” and IO) denotes the vacuum. The right hand side of (5.3)
is then equal
to, 1
~oir”-2(A-“~(1~6’(1)z
--2(A--5’)
,qcc
= w-
2(A-J)
ii
= w-2’p-“(~~l~~‘(~~)z-2(A-~‘) W
(Opq
;‘&‘f)
= (wwP”(z)),
which is the desired equality.
Acknowledgements We would like to thank A. Aghamohammadi
for valuable discussions.
IO)
(5.4)
A.M. Ghezelbash, V. Karimipour / Physics Letters B 402 (1997) 282-289
References [ 1] [2] [3] [4]
V. Gurarie, Nucl. Phys. B 410 [FS] (1993) 535. M.R. Gaberdiel and H.G. Kausch, Nucl. Phys. B 447 (1996) 293. A. Bilal and 1.1.Kogan, Nucl. Phys. B 449 (1995) 569. 1.1.Kogan and N.E. Mavromatos, Phys. Lett. B 375 ( 1996) 11 I. [5] J. Ellis, N.E. Mavromatos and D.V. Nanopoulos, hep-th19605046. [ 61 H. Salem, Yale Preprint YCTP-P38-9 1 ( 1991) [7] L. Rozansky and H. Saleur, Nucl. Phys. B 376 (1992) 461. [8] J. Cardy, J. Phys. A 25 (1992) 201. [9] X.G. Wen, Y.S. Wu and Y. Hatsugai, Nucl. Phys. B 422 [FS] (1994) 476. [lo] M.R. Rahimi Tabar and S. Rouhani, hep-th19606143. [ 111 Z. Maassarani and D. Serban, hep-th/9605062. [ 121 M.A.I. Flohr, hit. J. Mod. Phys. A 11 (1996) 4147. [ 131 M.R. Rahimi Tabar, A. Aghamohammadi and M. Khorrami, hep-th/9610168.
289
1997
PHYSICS ELSEWER
LETTERS B
Physics Letters B 402 (1997) 282-289
Global conformal invariance in D dimensions and logarithmic correlation functions A.M. Ghezelbash a,b,l, V. Karimipour a*c,2 a Institute for Studies in Theoretical Physics and Mathematics, PO. Box 19395-5531. Tehran, Iran b Department of Physics, Alzahra University, Vanak. Tehran 19834. Iran ’ Department of Physics, Sharif University of Technology, PO. Box 19365-9161, Tehran, Iran Received 20 March 1997 Editor: L. Alvarez-GaumC
Abstract We define transformation of multiplets of fields (Jordan cells) under the D-dimensional conformal group, and calculate two and three point functions of fields, which show logarithmic behaviour. We also show how by a formal differentiation procedure, one can obtain n-point function of logarithmic field theory from those of ordinary conformal field theory. @ 1997 Published by Elsevier Science B.V. PACS: 11.25.Hf Keywords: Conformal
field theory
1. Introduction Recently, logarithmic conforrnal field theories (LCFT) have been studied in a series of papers [ l-131, both for their pure theoretical interest concerning the structure and classification of conformal field theories [ l-51 and for their relevance in some physical systems [6-l 1] which are described by non-unitary or non-minimal conformal models. All such works have dealt with two dimensional conformal field theory [ l] relying heavily on the underlying Virasoro algebra, and have described how the appearance of logarithmic singularities is related to the modification of the representation of the Virasoro algebra. In this paper we will try to understand LOT’s from yet another point of view, that’s we consider ddimensional conformal invariance. Although our results are based on a generalization of the basic idea of [ 11, we hope that by changing or simplifying the context of study, we can add a little bit to the understanding of the subject in general. I E-mail address: * E-mail address:
[email protected]. [email protected].
0370-2693/97/$17.00 0 1997 Published PII SO370-2693(97)00459-O
by Elsevier Science B.V. All rights reserved.
A.M. Ghezelbush,V. Kurimipow/ PhysicsLetters3 402 (19971282-289
283
As is well known, one of the basic assumptions of conformal field theory is the existence of a family of operators, called scaling fields, which transform under scaling S : x + x’ = Ax, simply as follows: 4(X) --+ 4/(X’) = A-*+(X),
(1.1)
where A is the scaling weight of 4(x) . It is also assumed that under the conformal group, such fields transform as
(1.2) where d is the dimension of space and ]]%~‘/ax(l is the Jacobian of the transformation. Eq. encompasses Rq. ( 1.l ) defines the transformation of the quasi-primary fields. For future use we Jacobian equals Ad for scaling transformation and l]~ll-~~ for the Inversion transformation I : x -+ being unity for the other elements of the conformal group. Combination of ( 1.2) with the definition of the correlation functions, i.e.:
( 1.2) which note that the x’ = x/llxl12,
of symmetry
(1.3) allows one to determine the two and the three point functions up to a constant and the four point function up to a function of the cross ratio. It is precisely the assumption that scaling fields constitute irreducible representations of the scaling transformation, which imposes power law singularity on the correlation functions. As we will see, relaxing this assumption, one naturally arrives at logarithmic singularities. It also leads to many other peculiarities in the relation between correlation functions. To begin with, we consider a multiplet of fields
(1.4)
and note that under scaling x --+ Ax, the most general form of the transformation of Q is @(x) +
@‘(x’) = A%(X)
(1.5)
where T’ is an arbitrary matrix. More generally, we replace ( 1.5) by
(1.6) where T is an n x n arbitrary matrix. When T is diagonalizable, one arrives at ordinary scaling fields by redefining @, so that all the fields transform as l-dimensional representation. Otherwise, following [ 121 we assume that T has Jordan form
T=
(1.7)
Rewriting T as 9 1 + J, where Jij = 6i_i,j, Eq. (1.6) can be written in the form @(x) +
@‘(x’) =
ll$h,e(x),
(1.8)
284
A.M. Ghezelbash. V. Karimipow/Physics
where AX = 11% JIJ is a lower triangular
Letters B 402 (1997) 282-289
matrix of the form,
(*
(1.9)
(*X)ii=l~
i.e. for N = 2 we have
(1.10) An important point is that the top field ~$1(x) always transform as an ordinary quasi-primary field. A most curious property of the transformation ( 1.8) is that each field &+I transforms as if it is a formal derivative of C#Qwith respect to 9 [ 131
A+1
(x>
(1.11)
= -&&dk(x)*
This formal relation which field $1, essentially means from those of the ordinary The phrase “with due care” Jordan cell.
determines the transformation of all the fields of a Jordan cell from that of the top that with due care, one can determine the correlation functions of the lower fields top fields simply by formal differentiation. We will elucidate this point later on. in the previous statement refers to the two point function of fields within the same
2. The two point functions We already know by standard arguments [ 131 that the two point function belonging to two different Jordan cells (A,, n) and (Ap, m) vanishes, i.e.:
of the top fields &
and 4p
Due to the observation (1.1 l), it follows that the two point function of all the fields of two different Jordan cells with respect to each other vanish. Therefore in this section we calculate the two point function of the fields within the same Jordan cell. As we will see logarithmic conformal symmetry gives many interesting and unexpected results in this case. Let us denote the matrix of two point functions (4i( X)4j (y)) for all +i, 4j E (A, n) by G( X, y), then from rotation and translation symmetries, this matrix should depends only on 1(x - ~11. From scaling symmetry and using (1.2) and (1.3), we will have
(2.2)
AG(Ilx - yll,A’ = A2AG(hllx - yll>, where A = AdJ, and from inversion
Mwlx-
symmetry,
we have
IIX - YII Yllhq = l141-2A llYll-2Acc~).
(2.3)
where A, =
Defining
“X”-2dJ,
A, =
the matrix F as
“y”-2dJ.
(2.4)
A.M. Ghezelbash, V. Karimipour/ Physics Letters B 402 fi997) 282-289
285
we will have from (2.2) and (2.3), AF(Ilx - YlW
=
F(Allx - Yll>T
(2.5)
and
lb - Yll
A,F( /Ix - YlO$
= F( -
)
(2.6)
ll4l IIYII .
For every arbitrary A, we now choose the points x and y such that J/x/J = A$ and J/y/J = A$. It should be of space. From (2.4), we will have A, = Ai = A$, therefore Eq. (2.6) turns into, and A.v noted that in this way by varying A, we can span all the points
AtF( [Ix - yll) (A’)’ = F(Allx - ~11).
(2.7)
Combining (2.5) and (2.7) and using invertibility of A, we arrive at F = AiF(A
(2.8)
by iterating (2.8), we will have F = AFAt-‘,
and by rearranging, we will have
FA’ = AE
(2.9)
Expanding A in terms of power of InA as (dlnA)2J2 2,
A=l+(dlnA)J+
+
.*.
and comparing both sides, we arrive at F(J’)k=
(J)kE
Since (J”)ij =
6i,j+k,
k=1,2 ,...,
n-l.
(2.10)
we will have from (2.10)) (2.11)
Fi,j-k = Fi-k,j,
which means that on each opposite diagonal of the matrix F, all the correlations are equal. Moreover from FJ = JF, one obtains n
c
Filsj,l+l
I=1
=2 S&l+1
ej,
(2.12)
/=I
which means that if j = 1 and 1 < i 5 n, then Fi-l,j = 0, or Ej=O
forj=l
and 1 liln-1.
(2.13)
Combining this with (2.11). we find that all the correlations above the opposite diagonal are zero. In order to find the final form of F, we use Eq. (2.5) again, this time in infinitesimal form, let A = 1 + EYJ+ o(cy*) where LY= dln A, then from AF(x)A’= F(Ax), we have d(JF
+ FJ’) =x$
(2.14)
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A.M. Ghezelbash, V. Karimipow/Physics
Due to the property
if
F2.n = CIY + ~2,
Ft,n = ~1, with the recursion
dFi n
i> 1
(2.15)
F3,n =
&Y2 +
c2y + c3,
etc.
(2.16)
relations,
= F;_I,n.
dy
we obtain
the new variable y = 2d In x gives
which upon introducing
)
only the last column of F should be found, therefore from (2.14)
(2.11)
d&n XL =2dFi,,-1, dx
Letters B 402 (1997) 282-289
(2.17)
> I
Thus we have arrived at the final form of the matrix F, which is as follows: .. ... ... ...
0 0 0 .. .
F= \
go ...
0
0 go go gr
0
go al-2
g1
g2
.. .
.. . gll
&-I
(2.18)
where each gi is a polynomial of degree i in y, and gi = dgi+l/dy. All the correlations depend on the n constants which remain undetermined. We have checked (as the reader can check for the single II = 2 case) that inversion symmetry puts no further restrictions on the constants ci.
Cl, * . . 7 c,,
3. n-point correlation
functions
The observation ( 1.11) that the transformation properties of the members of a Jordan cell are as if the are formal derivative of the top field in the cell, allows one to determine the correlation functions of all the fields within a single or different Jordan cells, once the correlation function of the top fields are determined. As an example, from ordinary CFT, we know that conformal symmetry completely determines the three point function up to a constant. Let qb,, q5p and & be the top fields of three Jordan cells (A,, E), ( Ap, m) and A,, n) respectively. Therefore, we know that (4a(x)4p(y)4,(z))
=
lIx _ y~/A,+Ap-A,
11~ _
z;t::Ay-Ap
,ly _
z I,A~+A~-A~,
(3.1)
’
A,py in principle depends on the weights Aa, Ap and AY. Denoting by sbLllr3 we will readily find from (3.1) that
where the constant cell (A,,0
=
11~ _
y~/A,+ArA~
11~ _
z
~~A,f%--ha 11~
dA + ~1~_ y~~A~+h%~~x _ z,,:%-A~,,y 3 For simplicity,
_
z
the second fielc of the
JIAP+A+
IIY- ZII ln( _ Z~~A~+Ar-A, (lb - Yll)(llX-
zld’
we have denoted the top field by 4 and the second field by 41, instead of 41 and q52 respectively.
(3.2)
A.M. Ghezelbash, V. Karimipur/
287
Physics Letters B 402 (1997) 2X2-289
where
is a new undetermined constant. For the correlation functions of fields within a single cell, one should then take the limit /?,r --) a in the above formula. It is not difficult to check that this formula satisfies all the requirements demanded by conformal symmetry. No need to say one can continue this procedure of formal differentiation to determine other correlation functions. It may be asked why we have not applied the procedure of formal differentiation for calculation of two point function, but have followed an independent route. The point is that the two point function has the form
which, upon differentiation, yields derivative of the Kronecker symbol or the delta function which is not easy to handle.
4. The operator product expansion In this section we calculate the operator product of the fields in two different Jordan cells, (A,, N,), and (AS, Np). We assume that the operator product of the two top-fields & E (A,, N,) and q$ E ( AB, NP) can be expressed in terms of the fields and their descendants in other Jordan cells,
where the sum is over the Jordan cells of weight (A,,, N,). The field r$y,n stand for the n-th element of a Jordan cell together with its descendants. We now demand that both sides have the same behaviour under conformal transformations, i.e. we demand that (4.1) is equivalent to the following relation ~h(aJ&(Y’)
=
p&l/x’-
y’ll)qb;,“(y’).
(4.2)
n,Y Considering first the scale transformation, we obtain from ( 1.6) ~-A”-Avaw4~(Y)
=
c
C,y,,,(W~ - YII))A-A7@&J(Y).
(4.3)
n*Y,PlN,
Redefining the function C&,(X> as follows
C$&) =
C&f-Ax) AhAp-A,
lb - YII
’
(4.4)
where C&’ s are constant and comparing Eqs. (4.1) with (4.3) leads to the following equation for the f’s
fn(A-‘x)
= &p(x)Apn9 p=l
(4.5)
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A.M. Ghezelbash, V. Karimipow/Physics
Letters B 402 (1997) 282-289
or in compact matrix form, f( A-lx)
= Pf(X)
)
(4.6)
h. (4.6) is easily solved. It is in fact a recursion relation for f,,‘s due to the triangular solution is N-i f;(x)
Ui+k( -d In X) k
= c
k
form of A. The genera]
?
(4.7)
k=o where the constants fs(x)
= LYE, f2(x)
It is interesting fk(x)
. . , &‘Nare free. For N = 3 for example, we have
at,.
=
= cus(-dlnx)
fi(x)
+ CQ,
to note that the following
= i(Ys(-dInx)2+a2(-dlnx)
+ai.
relation holds among these functions,
d(--ilnx) fk-‘(x)’
5. Conjugation
(4.91
properties
In this section, we will restrict ourselves to two dimensions and derive the hermitian in a Jordan cell. Our main result is encoded in the following formula &tcz>
multiplet
= i-2A$(t),
As a justification
I/*(Z)
for the validity by (@(z>@‘(w>)ij,
(@(z>@‘(w))
of the fields
(5.1)
of the fields have been suppressed.
(4i(z)4.j(w))
conjugate
= $+~-2@-J’)
where & is an N dimensional
$*(I)
(4.8)
of operators, the superscript
As an example,
when N = 2 and 4 =
= i-2A($(i) of (5.1),
t, denotes transpose and the Z dependence E 04
, we have
(5.2)
+2lnZ&i)).
consider
the transformation
z ---f i, with llz 11> llwll and denote
then
= (O~~(z)~,‘(w>~O)
where cc denotes complex conjugation
(5.3)
= (0~R~+‘(w)~+(z>~0)“” and IO) denotes the vacuum. The right hand side of (5.3)
is then equal
to, 1
~oir”-2(A-“~(1~6’(1)z
--2(A--5’)
,qcc
= w-
2(A-J)
ii
= w-2’p-“(~~l~~‘(~~)z-2(A-~‘) W
(Opq
;‘&‘f)
= (wwP”(z)),
which is the desired equality.
Acknowledgements We would like to thank A. Aghamohammadi
for valuable discussions.
IO)
(5.4)
A.M. Ghezelbash, V. Karimipour / Physics Letters B 402 (1997) 282-289
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