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SU(2)k WZW and Zk parafermion models on the torus
Nuclear Physics B343 (1990) 418—449 North-Holland
SU(2)kWZW AND Zk PARAFERMION MODELS ON THE TORUS T. JAYARAMAN International Centerfor Theoretical Physics, 34100 Trieste, Italy
K.S. NARAIN and M.H. SARMADI Theoty Division, CERN, CH-1211 Geneva 23, Switzerland Received 27 November 1989
We study SU(2)k WZW and /Lk parafermion models using a Coulomb-gas formulation related to that of Wakimoto and Zamolodchikov. We discuss the truncation of the Fock modules of the free bosons of the Coulomb-gas system to obtain the parafermion and SU(2)k modules. Explicit expressions are given for the correlators of these models on the torus. A detailed discussion of their modular transformation and their monodromy is also given, the complications being due to the restrictions on the lattice summations.
1. Introduction Among the known two-dimensional conformal field theories, the ones of particular interest are the Wess—Zumino—Witten (WZW) models [1—31,these being the conformal models in which the chiral algebra is generated by a current based on some Lie group G. At present, all the known rational conformal field theories are of the coset type G/H [41,these being the models in which the chiral algebra is constructed from that of G such that its fields commute with those of the subalgebra based on H (H C G). There is even a conjecture that all the rational conformal theories are of coset type. Therefore a full understanding of the WZW models is of importance. Recently Witten [5] has shown a very interesting connection between WZW models and three-dimensional Chern—Simons gauge theories. Subsequent works [6, 7] have made the connection between the states of the Hilbert space of three-dimensional Chern—Simons theory and the conformal blocks of WZW theory more explicit. In particular, in refs. [6, 7] the known differential equations [2] satisfied by the conformal blocks for the case the Riemann surface is a two-sphere have been derived and the known characters for G SU(2) and arbitrary level k [8] have been obtained. In principle, in this approach one should be able to obtain =
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expressions for the correlators on higher-genus surfaces, however, this has not yet been done. In this paper we will follow another approach, namely the Coulomb-gas approach to SU(2)k and the related 7~k parafermion models (PF) [9, 10] (i.e., SU(2)k/U(l)k). The Coulomb-gas approach to minimal conformal models [11—14] and minimal superconformal models [15, 16] has proved to be an efficient one. Here we will use this formulation to study the characters of SU(2)k WZW and ~ parafermion models and write explicit expressions for some correlators on the torus in these models and study their properties. The organization of this paper is as follows. In sect. 2 we will first review the Coulomb-gas formulation of Wakimoto—Zamolodchikov [17, 18] and by introducing cohomologies similar to Felder’s [12], will investigate the structure of the PF modules and give a derivation of the known PF character. In sect. 3 we will give examples of correlation functions, both for PF and SU(2)k systems and study their properties. In particular we will discuss their monodromy which, due to the restrictions on the domain of lattice summations, is intricate. Sect. 4 contains the concluding remarks. In appendix A we discuss some properties of the lattice sums for PF and SU(2)k systems which are useful for the discussions in sect. 3. Appendix B deals with the problem of the modular transformation for these models. Again the restrictions on the domain of lattice summations make it more involved than the usual cases, and we will handle the problem by exploiting the connection with N 2 minimal models whose modular transformation was worked out in ref. [16]. =
2. Coulomb-gas representation of PF and SU(2)k systems and their characters In this section, we will first review the Coulomb-gas approach to SU(2)k WZW and the related 7~k parafermion models and then discuss a derivation of their characters. The SU(2)k currents J ±(z),J3(z) and the energy—momentum tensor T(z) satisfy the operator product expansion (o.p.e.): Ja(z)J
h
(z’)
=
(k/2)~5’~ 2 + (z-z’)
=
T(z)T(z’)
=
where k is an integer, c
2
+
(z-z’)
=
regular,
2T(z’) 4
3k/(k
+
(z-z)
c/2 (z—z’) +
regular,
(z-z) 3Ja(z~)
Ja(z~)
T(z)Ja(zF)
+
fahJc(z~) ,
+
öT(z’) 2
+
(z—z’) 2) and
f~—
(z—z
)
+
3~~=f3
3f
regular,
—=
1.
(2.1)
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As shown by Wakimoto [17] and by Zamolodchikov [18] (see also ref. [19]) these currents can be expressed in terms of one free boson 4’ (with a background charge) and a free (f3, y) system of weight (1, 0): 1
J3(z)=:f3(z)’y(z):—i
~Ik+2 2
J(z)
=
~(:p(z)y(z)2:_iyk+2a4’(z)y(z)+kay(z)),
T(z)
=
—~:84’(z)a4’(z):+ia
(2.2)
24’(z)—p(z)a-y(z), 03
where a 0 2 1/2~/k + 2 and the normalizations a4’(z)04’(z’) and J3(z)y(z’)= —1/(z—z’). Clearlyare thesuch (f3,y)that system contributes —2/(z—z’) c 1~ 2 to the central charge whereas the boson 4’ gives c4, 1 24a~ and therefore the total central charge is c 3k/(k + 2). One can bosonize the (/3, y) system in terms of two bosons (x, ~) in the usual way as: =
=
=
=
—
=
/3
=
—
+iri)]~
~-3XexP[_T~=(X
y= exP[_)=-(x+i77)].
(2.3)
Note that ij comes with imaginary momenta and contributes negatively to the dimensions. Now by making the following change of basis from (x, 4’) to (4’o, 4’1,4’2): ~,
/k+2
U
k /k+2 k+2 k 4’+i
V
i4’2
/k+2 4”4’~V 2
4’~/k~’
~ yIT~
/k+2 2
X’ (2.4)
+
one can rewrite the currents as:
J~(z)
=
y~~(z) exP[~4’o(z)]~
J(z)
=
~~t(z)exP[~4’(z)]
J3(z)
=
(2.5)
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where ‘I’(z) and ~11t(z) are the
1k
i(fk±2
=
421
a4’ 1(z) +ia4’2(z))exP[~4’2(z)]~
~t(z)
i(\/k±2
=
_i34’2(z))exP[_~4’2(z)].
These parafermions each carry a conformal weight 1 tum tensor is now:
2+ (34’~)2~
T(z)
=
‘((a4’)
(2.6)
1/k. The energy—momen-
—
+ia
(34’2)2)
24’ 03
1.
(2.7)
One can construct screening operators which have dimension one and commute with the currents (i.e., their o.p.e. with the currents are regular or if non-regular they are total derivative). In fact, there are three such operators: =
:ö4’2(z)exp[2ia04’1(z)]:,
V~(z) :exp[_} V7~T~1(z) ±~V~4’2(z)]. =
(2.8a,b) One can easily verify the following o.p.e. of these screening operators with 11’ and ~I’~: ~(z)V1(z’)
i~t(z)V1(z’)
=
—
~~2az[1,exp(
~
4’1(z’)
+
~4’2(z))]
+
=
—
~~a~,[1,exp(
~k~2
4’1(z’)
—
~4’2(z))]
+regular,
regular,
regular, ~Itt(z)V÷(z’)
=
iPt(z)V(z’)
=
regular,
~
IP(z)V_(z’)
_~
=
regular,
V_(z’)exP(_ ~4’2(z))]
+
regular.
Therefore these screening charges commute with the currents. If we denote the momenta of V1(z) and V~(z)by e1 and e ± respectively, then they satisfy the relation (k + 2)e1 + e~+e_= 0. Thus these screening operators can be used to
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balance the charges on higher-genus surfaces. In particular for genus-two surface one expects a background charge + 2a0 on the surface which can be screened by the relation (k + 1)e1 +e~+e..=(—2a0,0,0). The parafermion system is described entirely by the two bosons ~ and 4’~.The parafermion primary fields Pim are labelled by two integers I and m; I 0,. k and m —k,.. k 1. In terms of 4~and 4’2 they are expressed as: =
=
. ,
. . ,
—
ii
m for
—l~
1/2. For m > I one can get the primary where C(l, m) [l!/(1 + m/2)!(l m/2)!] fields by the successive application of the parafermion field ~I’ on Pjm. The primary fields of SU(2)k of spin j and J~-eigenvaluem are then simply =
—
=
~ 2i2m(z)exP[i~4’o(z)].
(2.10)
Now let us turn to the structure of parafermion modules. First note that the screening charges Q±= f dzV~(z)and Q1 fdzV1(z) satisfy the relations: =
(2.11)
~)kQQ 0, Q~Qi+ ~ ± 0. (2.12a,b) These relations can be verified by using the o.p.e. of the screening operators. In the last relation, one obtains an integral of a total derivative which consequently vanishes. Denote by Firn the Fock module built on the vertex operators :exp(—il/2Vk+ 24’~+m/2~/k4’2):.Then for Q~we have (see table 1): —
Q+ and for
~
Q1 Q~k+I_1) > Fim
Q~I+I)
F_i_2,m
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TABLE 1 See text for explanation
F, + 2(k + 2). ,n + 2k
Qy
Qy
Q(k*l_/)
F,÷k+2,,,+k
Qy
Q/
2~,
Fim
F2(k+2)_,2,,,,
—I
Qy
(1*1)
Q(,*l)
F,
Q(i*I)
F3(k+2)_1_2,m+k
(k*l—I)
F2(k+2)+,~,
—
Q(/+ I)
I)—!)
F3(k+2)_,_2,,,_k
F, ±2(k + 2), m + 2k
where
Q~is defined
as =
fdz1 Vi(zi)fdz2 V1(z2)
. ..
fdz~V,(z~),
(2.13)
with the choice of contours being the same as Felder’s [12]. The primary fields listed above satisfy the following conditions: =
~
E
=
Im(Q~~°)
0,
(2.14a)
~ Im(Q~Q).
(2.14b)
Consider the parafermion module ~1m which consists of the states obtained from the primary field cIlm by applying equal numbers of ~I’and ‘I”~operators. The states obtained by applying unequal numbers of ~P and ~I”~differing by multiples of k will be discussed below. From eq. (2.14) it is clear that the parafermion module P, is contained in ,~,
Ker(Q÷)n Ker(Q) Im(Q~Q) Since
Q~=Owe
Ker(Q~~)
~
Im(Q~°)
n Ker(Q~~~)
have Im(Q~)cKer(Q~).We now show that the cohomology of
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/
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Ker(Q ±)/Im(Q,)) is trivial on each Fock space. Indeed if there is a state
in some Fock module =
Frs
such that
Q~4’ =
0, then define the state
~~~exp[~iVk+24’1(z)
T ~4’2(z)]
4’(O).
4” as (2.15)
We then have
Q ~4”(0) =
~~dw 1 2iir
=—~—
exp( - +iV~4’i(w) ±~V~4’2(w)) 4”(O) dz 1 1 dw+—~ dw exp(—+i~i~74’1(w)±~V7~4’2(w)) z 2i~ C,, 2iir C0 —~
x exp(~iV~24’i(z) ~ ~/~4’2(z) t/i(O)),
(2.16)
where the contour C~is around z and not 0 whereas the contour C0 is around 0 and not z. The integral f~5dw is zero as Q ~4’(O) 0 and the integral f~dw is just equal to one. Performing the z-integral we get Q ~4”(O) 4’(O). One can further ask whether the cohomology of Q is trivial in Ker Q+. In other words if 4’ E Ker Q~nKer Q then does there exist a state 4” such that 4’ Q~Q_4”? Indeed take the lowest-level state ~/m in Fim. We know that ~ E Ker(Q~)n Ker(Q~). However if ~ is an image of Q+Q_ of some state 4” then 4” E Fi_2(k±2),m.But the weight the lowest-level in On Fi.2(k+2)m is greater 1(mofcannot be in the state image. the other hand forthan all that of ‘im’ therefore other Frs, (r, s) ~(I, m), shown in table 1, one can easily show that the lowest-level states are in fact in the image of Q±Q.Let us assume that this is true for the entire Fock module Frs((r, s) # (1, m)), i.e., =
=
=
Ker(Q+)flKer(Q) Im(Q~Q)
F (r,s)
for
or
—O -
(r,s)=(I+n 1(k+2),m+n2k),
n1>O,
(r,s) =(—I—2+(n1+2)(k+2),m+n2k),
n1~O, (2.17)
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where n1 ~ n21 and n1
+
Ker(Q~Q)
F
Ker(Q~)®Ker(Q..)
n2
E
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425
27/. This implies that
—0 r,s
—
for
(r,s)=(I—(n1+2)(k+2),m+n2k),
n1>~0,
or
(r,s)=(—l—2—n1(k+2),m+n2k),
n1>0, (2.18)
where n1 ~ 1n21 and n1 + n2 E 27/. This can be seen by introducing the following inner product between a state 4’ E Fr s at some level n above the primary field with a state 4” E F_r_2,_s at the level n above the primary field: =
the two-point function (4’(0)4”(l)).
(2.19)
This is a non-degenerate inner product. In other words, for each 4’ there is a 4” such that ~i/i~4”) 0. Now if 4’ E Ker Q+’ then due to the triviality of Coh(Q~),4’ = Q~4” for some 4” E Fr_(k±2)s—k. Since K4”I4’) = (4”~Q+4”)= KQ+4”I4”) * 0 then Q~4” * 0, where we have used the definition of the inner product (2.19), and deformed the contour of Q+ so that it encircles the point 1. (Note that there is no contribution from infinity.) If we choose 4” such that ~4”~4’) * 0 and (4”~4’~ = 0 for 4, E Im(Q~Q.J then it follows by similar arguments that 4” E Ker(Q~Q). Thus Ker(Q~Q)/Ker Q~ ~ Ker Q_(F_T_2 ~) is dual to Ker Q+ ~ Ker Q/Im(Q+Q)(Fr,s) and therefore the triviality of the latter for the range of (r, s) given in eq. (2.17) implies the triviality of the former. Finally, using the ~2 symmetry 4’2~-’ —4’2,Q~~-~Qwe can flip the sign of s to get eq. (2.18). Now we are in a position to calculate the character. Consider
Xi,m
——
IrKer(Q+)nKer(Q) 1m(Q~Q_)
=
I
‘i’
t,q
L~—c/24
(Fim)
Tr(Ker(Q÷)fl Ker(Q_)yF,,~,)~~ ( L0—c/24\ /
—
Tr
~1—2)k±2).,,,
( L1~—c/24\/ ~q
.
~2 2
Ker(Q~)flKer(Q)
Now the first term is ‘I’ r(Ker(Q+)n Ker(Q_)XF,,,,,)~. I q L,,—c/24\/
— —
V’ ~
I~,
—
I/
I q L11—c/24 r Ker Q+(Fj+,,1o~s1,,,,_,,o)’~
•~1~’
n = 1) F,~
=
~
24)
(—1)’~~’Tr1,,+,,0k--21,n*(,,’—,,)k (qLo_C/
n=0 n’=O
(2.21)
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and the second term Tr
24)
F,_ak±2, (qLo_c/ Ker(Q~)Ker(Q_)
=
(
~
1)
—
~‘
Tr F,(,,.±2xk±2),,+ek
n’=O
~
=
(— ~
1)fl+fl’
n
(qLo_C/24)
Ker(Q.)
0 n’ = 0
(2.22) Combining the above expressions we obtain 1
2”sign(n
Xi,m
(—1)
2
71(T)
1)
n1,n2EZL/2 fI
~
n1 ~In2], —n1 >1n21 2— —(m +2n
xq4Uc+2)
(1+1 ±2n1(k+2))
2 2k)
4k
(2.23)
where sign(O) = + 1. This is indeed the known expression for the parafermion character [8,9]. Hence, if the assumption (2.17) is correct then the parafermion module can be identified with Ker Q~fl Ker Q/Im(Q+Q)(Fj,m) and the latter is in Ker(Q 1). The fact that none of the states in the latter is in Im(Q1) follows from assumption (2.17) that KerQ~flKer Q/Im(Q±Q.J(F_l_2+2(k+2) m) is trivial and the commutivity of Q1 with Q We have not been able to prove assumption (2.17), however, as shown above at least the lowest level states in Frs (and hence the entire irreducible parafermion module built on them) are in the Im(Q~Q ). It remains to be proved that this is so for all Ker Q~fl Ker Q in Frs. For k = 1 one can in fact prove that this is so. Let 4’ ~ Frs (with weight zl * 0) be in Ker Q~flKer Q. Define ~.
=
Applying
Q Q ±
-
on
i/i’
2i~ ~dzz e~’~4i(0).
and using the fact that
Q±Q-4”(O)= =
2’~r~ ~
Q ±4’
=
0 we obtain
e~1~)4’(O)
1 ~dzzT(z)4’(0)= 2t’7~-~1 .
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427
Thus every state in Ker Q~flKer Q with ~1* 0 is in Im(Q~QJ. This is as expected for k = 1; the parafermion module consists of only one state, namely, the identity. If for k ~ 2 eq. (2.17) does not hold, then one can proceed to obtain the character in the following way. Consider the modules F12+2p(k+2),m(P ~ 1) and Fi±
2p(k±2)m(P ~ 0) which are connected to Firn by the action of Q1. In table 1, they are the modules on the horizontal line from Fim. For each of these, we compute the trace by summing (with alternating ± signs) over all the modules which are connected to it by the action of Q ± and Q -. This way one obtains a trace over the states in Ker Q~fl Ker Q_ and Frs/Ker ~ Ker Q_, which is
Tr KerQ~flKerQ_k~ ~
L0—c/24\/
—
mr
(~q L,1—c/24
F,., Ker Q~ Ker Q
1
r+1
-
(~1)Jq 1=0 ~
2
=
77(T)
+
2
E (— 1)Jq
~) s
+2)(~i+ 2~k+2)) _k(~i+
2
r+l 2 2 2)(~i±2(k+2)) _k(~J_~)]
(2.24) 12r,s
We
Let us denote the thisfollowing space, which is obtained after the above summation as now consider sequence: Q
—
1k+I_()
Fim
QS’~~—
—
Note that under the action of -
Fim
Q±Q-
-
0 F2(k+2)+im
Q+Q-
-
F4(k+2)+im
Q+Q-
-
>
...
(2.25)
.
Q~Q_we have
Q±Q-
-
F2(k±2)_i_2.m
Q1k+I_!)
F2(k+2)+lm
F2(k+2)_i_2,m
F4(k±2)_i_2,m
0
Q+Q-
-
S
..
Q±Q-
F±_i_~,,~
.•
Therefore, if the trace is taken over the modules in eq. (2.25) with alternating ± signs, then taking_into account the fact that ~—l—2,m ~ F2(k+2)_i_2,Th and that F,2 is dual to Fim, in effect we are taking the trace over the states in KerQ~nKerQ Im(Q~Q_)
Ker(Q~Q) (Frs)
and
KerQ~®KerQ_
But as we showed above, Ker(Q~Q)/Ker Q~® Ker Q_(Fr,S) is dual to Ker Q÷fl Ker Q/Im(Q÷Q_)(F~_2_5).Therefore we can instead of the sequence in eq.
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(2.25) consider the following sequence: Q 1k±I_/)
Q(I*l)
Q(k±l_I)
F_i_2m
where
Frs
Ker
Q~n
Fim
Ker Q/Im(Q+Q)(Fr,s).
and the cohomology of
Q~k+1i)Q~i+1)o
Q(I+I)
F2(k±2)_,2,,fl
Q1
Now
if
...
,
(2.26)
Q~i±1)Q~k±1—)) =
is trivial on the modules in the
sequence (2.26) except for F1,,, then, by taking the trace over the modules in eq. (2.26) with alternating ±signs, we get a trace over ~ which is the parafermion module, and by using eq. (2.24) and making a trivial change of summation variables we get 1’~)
(qLo_C/24)
Tr 1m(Q~ Ker(Q~ —I)) (F,
,e)
1
(
2
=
77(T)
n
(1+1 +2n
4~’~2~
2fl
—
1)
1
2— —(in +2n 2 4k 2k)
1(k+2))
sign(n1)q
1,n2eL/2 °1 2E Z ~I
~I~2I,
f1
>In2t
(2.27) We have so far discussed the spaces ~ One may wonder what happens to the states obtained from ~ m by applying unequal numbers of ~I’ and ~ differing by a multiple of k. These states could be contained in Fim+2flk,fl being a non-zero integer. In ref. [9], (~t’)”and are identified with identity, so one would not expect to obtain any new states. But in the present Coulomb-gas formalism, these states carry different momenta along the 4’2 direction. However, it turns out that there is an isomorphism between F1,,, and F~m+2nk for any n E 7/. Indeed consider a nontrivial state 77 E Fjm. Then 77 is in Ker(QJ and hence there exists a state ~1’ such that Q.jsj’ = ~ Furthermore ~ Q+n’ * 0, because otherwise due to the triviality of the cohomology of Q+ there would exist a state 77” such that = Q+~i”.This in turn would mean that ~1E Im(Q~Q),which is a contradiction. Now we show that ~ E F,m±2k.Certainly ~ ~ Ker Q÷.It also belongs to Ker Q_ as Q~~QQ71i(l)k+lQ77 =0. Finally if ~ =Q+Qñ’, then QJii’— Q~’)= i~ and Q±(~i’ Q~’)= 0 implying that s~E Im(Q~Q_),which is a contradiction. Therefore ~ is a non-trivial state in Fjm+2k~It is easy to see that the map defined above between F1~and F1,,, + 2k is an isomorphism, and that it can be extended to all Fim±2nk successively. This isomorphism allows one to identify (ifr)k = (114)k = 1 in our formalism and work with only F1,,, for example. This also means that all the physical quantities, such as characters and correlation functions, which are computed using F1 m+2nk would be independent of n. In particular, this implies the shift invariance of the domain of the lattice sums which plays a crucial (1~tY~~
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429
role in proving monodromy and modular covariance of correlation functions and which we will discuss in sect. 3. Now let us discuss the computation of the n-point function on the sphere. For this we need to define a dual field at ~ (say). The dual spaces have already been introduced above;
F_,_2,,,,
(
and
KerQ~flKerQ * Im(Q~Q_) (F1~~))
Ker(Q~Q) Ker(Q~)~ Ker(Q) (F_1_2,~). Thus we can consider a four-point function,
where Ker Q~flKer Q Im(Q~Q) Ker(Q +
~
(F,
Q )
) (F
Ker(Q+) ~ Ker(Q4
i=123
, —
(2.28)
).
4, ~~tfl4
To define this correlation function we must introduce a sufficient number of screening operators so that i~+ 1~+ 13 m1
+
—
2’/k+2 m2 + m3
14
—
+
m4
~ (screening charges)
=
0.
2~ One can define the screening contours as in Dotsenko—Fateev [11] or in Felder [12]. The fact that the intermediate state is also a physical state can be seen in the usual way. That the dressed 4’~(0)x 4’2(z) is in Ker Q÷flKer Q_ follows from the commutativity of Q ± with Q1 and Q T• If on the other hand this product contains a state in Im(Q~Q), then it decouples from the above correlation function as can be seen by deforming the Q + and Q contours and pulling them around 4i~and 4’ and using eq. (2.28). In the next section we describe how to compute two-point functions on the torus. The subtlety here is that one must make sure that the trace is taken over -
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/ Parafermion models
only the physical states and this will be ensured by making use of eqs. (2.20), (2.21) and (2.22) (or eq. (2.27) in case eq. (2.17) does not hold). Before ending this section we note that since SU(2)k is the product of PF and U(l)k, using the expression for the character of PF and the character of U(1)k and following the steps sketched in appendix A one obtains the well-known expression [8] for the character of SU(2)k given in eq. (A.1O) below.
3. Correlators in the PF and SU(2)k systems In this section we will use the Coulomb-gas formulation of SU(2)k current algebra and parafermion system of the previous section to derive expressions for their correlation functions on the torus. For simplicity we will only give the correlators of the currents and the correlators of the primary fields of type j = m = ± The former does not require screening, and thus there is no contour integral, whereas the latter requires one screening operator. Consider first the correlation function of two PF currents; (~!~(z 1)Wt(z2)). Using eq. (2.6) for ~ and ~ we get ~-,
~.
2/k
[_(k+1)a~logo1(z~_z2)+(k+2)2a~,_k2a~2
I(0i(:i~2))
_(a~loge1(z1_z2))2_2k3~logo1(z1_z2)a~2}T,,~(0,w2), (3.1)
where w2 = 2(z1
—
z2) and 2
Fim(Wi,W 2)
1
=
71(T) 2
(—
~
n
4~~2~ 1 ±2i~1(k —(l+ ±2))2 1)2uu1
4k
— —(in
±2n2k)
sign(n1)q
1,n2wz/2 ~1 fl1
±1n21,
2~ 7! fl1
Xexp 2i~(ni+
>In/
/+1 m 2(k+l))w12(~~2+~)w2I
(3.2)
In appendix A we discuss some properties of F,,,, which will be useful in what
/
T. Jayaraman et a!.
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431
follows. In deriving eq. (3.1), we have used the following expressions:
2 2a~ 1log 01(z1 —z2)F(O,0)
=
—
4(k
2) d~,F(0,0), (3.3)
+
a4’2z2 exp( - ~4’2(z2)))
(a4’2(zi) exP(~4’2(z1))
2/k =
2F [2~ log 01(z1 —z2)F+ (4/k)(a~log 01(z1 —z2))
(0hz2))
+8a~log0 1(z1_z2)a~2F+4ka~2F~.
(3.4)
Let us check the behavior of the correlator (3.1) under (z1 —z2) —*0, as well as its monodromy under z1 —*z, + r. For (z1 —z2) —*0 it has the expansion (~P(z1)4,(Z2))im
—‘
(z~—z2) —2+2/k X/,m(T) +
k+2 k
2’~(2iir)aTX,~(T)
~
(3.5)
+...,
—z2)
where Xi,m(T) is the character of the h.w.r. i,,,,. This expansion is expected from the following o.p.e.:
=
(z
2T(z 1
+
_Z2)_2+2/k[l
k + 2 (z1 —z2)
and the fact that
+T,
2)
+
...]
(3.6)
using
2hITw2/kr,÷
F~m(0,W2)
.qi/ke
2(o,w2)
(3.7)
we get 2t~P(zi)~t(z +
T)~t(z2)>,,~
=
e_
2)>,,~÷2.
(3.8)
This is expected from the fusion rule (3.9) Since 0 ~ m <2k, the change in F,,.,, given above is correct as long as m + 2 is also in this range. However if m + 2 is outside this range, then one has to shift the lattice summation. For instance, if m = 2k 1 then in the lattice sum (3.2) one needs to write n2 + (m + 2)/2k = (n2 + 1) + 1/2k which means in turn that the cone n21 < In, I is shifted. The difference between the contribution of these two —
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/
Parafermion models
cones is simply the contribution of the lines n1 = —n2 and n1 = n2 + 1 to the lattice sum (3.2). As we will discuss in appendix A, this difference, i.e., the contribution of these two lines is of the form F~(wi,w2)0i(
)
wi+W2 _w1+w2 2 +F~(wi,w2)0i( 2
where F~kw1,w2) satisfy k0~2F~-~ = ±(k+ 2)a~1F~~. However when this difference is plugged into eq. (3.1), its contribution vanishes and one has the shift invariance of the cone, and therefore the correct monodromy. Now let us consider the correlator of SU(2)k current, KJa(zi)Jb(z2))
=
~abG(z1
z2).
(3.10)
3(z) is We evaluate this by using eq. (2.5) for J’~(z).Since the expression for J simpler we first consider the correlator KJ3(z 3(z 3(z) = ~-ika4’ 1)J 2)). Using J 0(z) we simply get 3(z (J
3(z 1)J
28~F,(0,0,0), (3.11) 2))
=
—k/23~ logO(z, —z2)F,(0,0,0) +k
where F,(w,,w 2,w3)
1
=
~ ~
77(T)
+m/2k)w3]
m
As we discuss in appendix A, it is possible to bring this expression for F, to the more useful expression (A.6). Using eq. (A.6) and setting w1 = w2 = w3 = 0, we get 3(z (J
3(z 1)J
2))
=
_~k(3~log 61(z1 —z2) —4h~3~ log 77(T))x,(T) +
(2ii~/3)(k + 2) a~xi(r),
(3.12)
where x,(r) is the character of SU(2)k in the representation j 1
x,(r)
=
77(T)
=
1/2:
2~”~2(k+2)~. 1+1 ~
(2(k
+
2)n
+1+
(3.13)
1)q
It is easy to verify that this correlator has the correct (z 1 z2) 0 expansion and It is clearly invariant under z1 z1 + T, —
the correct monodromy under z1 —p z1 *
+
T*.
—‘
—~
In ref. [201this correlator was first written by these requirements, since they uniquely fix it.
T. Jayaraman etal.
which is expected. Under (z1
/ Parafermion models
433
z2) —o 0, since
—
1
3~log61(z1—z2)—4i~d~log71(T)=
+O(z1
2
—
z2),
(3.14)
(z, —z2) we get 3(z (J
3(z 1)J
k/2
1)>
2 +
(z1
2i~ —~---(k+ 2) 3~xi(T)
+
O(z1 —z2). (3.15)
—z2)
This is in accordance with the following o.p.e. 3k/2 EJa(z)J~~(z)_
a
(z1 —z2)
+(k+2)T(z2)+...
2
(3.16)
and the fact that KT(z)) = 2iir3~Xi(T). For the correlator (J~(z1)J(z2)) of course we expect to get the same expression as eq. (3.12). Let us see how this comes about. Using eq. (2.5) for J~(z)and J(z) we get 2a~ —k23~ KJ~(z1)J(z2))
=
(_(k+ 1)a~log 01(z1 2
— ~
—
(k+2)
—z2) +
2k3~ log 0,(z,
—
log 01(z1 —z2))
2
z 2)
a~2)F,(O~~2, w3)I~3~~2, (3.17)
where w2 = 2(z1 z2) and the limit w3 w2 should be taken. Now using eq. (A.6) for F1(w1, w2, w3) and taking the limit (w3 w2) —÷0 we get —
—~
—
=
(— (k + 1) a~log 01(z1 —z2) + (4iir/3)(k + 2) 3~+ 4i3r(k + 2) 3~log
77(T)
2—4i~-8~ log 0 x4i1Ta~f(zi—z2) (a~log 0,(z1 —z2)) 1(z1 —z2)f(z1 —z2) 2f(z 2g(z ~2 + 4~1—z2) 8ir 1 —z2))x,(r), (3.18) —
—
—
where
2i7n1z~
f(z)=
~(—1)’
~r(r± 1—I) e
(3.19)
r
q
r#0
~+r(r+ I)
g(z)=
~(_~)r1 r+0
2
(1—q)
(3.20)
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/ Parafermion models
T. Jayaraman ci a!.
Comparing this expression with eq. (3.12) we get the following identity: 4i7r8~f(z)—(3~log01(z))2—4iira~log01(z)f(z)_~.2 —8~-2g(z)=a~log0
2f(z)
1(z) —8hr3~log77(T).
+4ir
We have checked that both sides of this identity are invariant under z they both have the expansion 1
1
—~
(3.21) z +T
and
0’~”(O)
From this itthe follows that thisfunction identity of holds. 0~(0) This +O(z2). identity also be we discuss correlation primary fields ofwill SU(2)k to useful simplifywhen the expression. We now consider the correlation function of primary fields. For simplicity we restrict ourselves to examples which involve only one screening operator. For this we consider the parafermion primary fields =
:exp[_ 2Vk+ 2
4’1(z)
±
which correspond_to I = 1, m = ±1 in eq. (2.9). The screening operator l/,(z) = 34’2(z)exp[i/Y~+ 24’1(z)] should be used to screen the charge. We therefore need the following expression: (a4’2(z)
=
(
O,(z,—z2) 0~(0)
where w2 <~i,
exP[-~~-4’2(zI)]exP[_
=
2(z1
—
)
1/2k
-4’2(z2)])
I
01(z,—z)
(_~0zlo~(0(zz))
+2~3w2
Fim(0,W2),
(3.22)
z2). Using this we then simply get
,(z1) cP1 ,_,(z2)), ~ k+1
1
V~
=
~
0,(z1
—z2)
0~(0)
[~~(
61(z, —z)
k(k+2)
01(z1 —z) 0,(z_z2))
~dz
—
oç(0)
2k3w]Fim(Wi~W2)~
k+2
01(z —z2)
k÷2
eç(O) (3.23)
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/ Parafermion
models
435
x
Fig. 1. The s-channel and t-channel contours ~
C~”,C~’~ and ~
where w1 = z1 + z2 2z and w2 = z2. The s-channel contours ~ and ~ and the t-channel ones C~’~ and C~J)are as in ref. [13] and shown in fig. 1. We have checked that this correlator has the correct monodromy and modular transformation properties and in the identity channel the limit (z1 z2) 0 reproduces the character. The check is basically the same as the one given for the minimal models in refs. [13, 16], except for the restriction of the range 0 ~ m <2k and 0 ~ I ~ k. Under Z1-~Z1+T we have —
—
—
2k(k+2) I
exp[_i~( k ~1 +2
—
—*
~)1Fl+I,m÷l(WI,w
2). (3.24)
F,~(w1,w2) -*q
First note that eq. (3.23) vanishes for 1 = 1 and 1 = k + 1. To see this consider 1 = —1 and replace F,,,, by f~,,, in eq. (3.23) where FIm(Wi, w2) is defined with the domain of summation appearing symmetric for n1 > 0 and n1 <0 (i.e., —n2< 1n11 for n20). Thus the difference between F,,,, and F,,,, is just the contribution of the line n2 = n1 which is equal to F,~(w,,W2)01((w1 w2)/2) where F~~(w,,w2)is given in appendix A (eq. (A.3) with 13=0) and satisfies 2k3~F~kW1,w2)= —2(k+2)a~F~kw,,w2).By plugging this difference in eq. (3.23) one finds that the change in the integral is a total derivative and therefore it is irrelevant meaning that one can replace FI~(W1, W2) by F,~(w1,w2)without any change. Now since F,~(—w1,w2)= —f~~(W1,W2),by a change of integration variable z z1 + z2 z, one can see that for both contours CS’~and C~’~ the integral is minus of itself and therefore it vanishes. The same argument also applies to 1 = k + 1. Therefore as far as 1 is concerned the —
—
—~
—
436
/
T. Jayaraman et a!.
Parafermion models
transformation (3.24) is the expected one. For m m’ 2k
=
2k
—
1 since
m+1 2k =1
we need to shift n2 n2 1. We saw in the example of correlators of currents that this shift introduced terms proportional to 01((w1 + w2)/2) and 01((w1 w2)/2) which did not contribute. For the present example their contribution is a total derivative term into the integrand and therefore do not contribute to the correlator either. We conclude that as a result of this shift invariance of the cone we get the correct monodromy also for m = 0 and m = 2k 1. Moreover the transformation (3.24) is in accordance with the fusion rule [3, 10]: —~
—
—
—
min(1+l’,k—l—-l’)
(3.25) n=I’—l,I
The modular transformation property of eq. (3.23) can be easily obtained by using the modular transformation of F,~(W1,W2) derived appendix B. to the previous 2)k which is in closely related We now our example of SU( example of consider parafermion, namely the correlator of the primary field ~j, m with j = and m = ± Using eq. (2.10) for this primary field we get ~.
K~l/ 2I/2(zI)~I/2_I/2(Z2))
1 =
01(z1
z2)
—
0~(0)
~
01(z1
x azlo~(0~
01(z1 z) 0~(0)
2(k±2)
~dz
01(z z2) eç(0)
k+2
—
k+2
—
~)
—z)
—2ka~2F,(w1,w2,w3),
(3.26)
where w1 = z1 + z2 2z, w2 = z2 and the limit w3 w2 should be taken. In fact it is possible to write this correlator in a much simpler form in the following way. Using eq. (A.6) for F, we have —
k+2
~
F,(w1,w2,w3)
where
F,(w1)
—
~
i~
—~
dW
F,(w1)
+
[f(
w1+w2 2
) _f(
—w1+w2 2
/+1
=
~q(k+2xn+1±1/2(k+2))2exp[2i~(n
+
2(k
+
2)
-
/
T Jayaraman ci a!.
Parafermion models
437
and f(z) was defined in eq. (3.19). Also we have W
2ka~F,(w1,w2,w3)
1+W2
~2k3~2[f(
2
)
W1 + W2
+2(k+2){f(
2
W1+W2
2
+ 2i~[f(
WI+W2
2
) )
~W1+W2
—f(
)
)1F1w1 -
2
—W1 + W2
+f(
_f(
2
~W1+W2
—
)ja~1rwi
—
2 ~WI+W2
—
)j~w~~
2
g(
where g(z) is as in eq. (3.20). Now upon using the identity (3.21) eq. (3.26) reduces to
-
01(z1 —z2) 0~(0)
-
2(k+2)
01(z—z2)
01(z1 O~(O)z)
~
X
k+2
—
~dz
log
Xô
01(z1—z)
-
F1(w1).
z
Again we have checked that this expression has the correct monodromy, modular transformation properties as well as the correct (z1 z2) 0 limit. Moreover it reproduces the correct four-point function on sphere upon degeneration q 0. For instance, for the two contours C’j’) and2”~ ~ and shown in fig. x 1, in the limit q —s 0 choosing after making a change of coordinate x = e 1 = 1 we get —
—*
—~
—
(1—x2)
~ —~-— 2(k+2)x~2F(
I
(1
_X2)
~
2F
2(k+2)X~
1
+
2j
1 k+2’
‘
1 k+2
,1
+
2
2j+2 ,2+ k+2
2 k+2
1 —x 2
which are the known expressions [21] for the two blocks in the t-channel when the external fields j1 = and 12 = at the points 1 and x2 are in the total 112 = 0 representation. These two blocks are respectively the identity block (j = 0) and J = 1 block. ~-
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T Jayaraman et a!.
/
Parafermion mode!s
4. Concluding remarks In this paper, we have shown how one can construct two-point functions for 7~k-parafermionand SU(2)k WZW models on the torus, and have shown that they satisfy the correct monodromy and modular transformation. This construction rests on the Coulomb-gas description of these models, and the truncation of extra states in the Fock modules of the free bosons that are not contained in the irreducible modules of the respective chiral algebras, via screening operators. We have argued that these irreducible modules can be identified with the certain cohomology associated with certain BRST operators, and have obtained the known characters. The basic assumption in this has been on the triviality of this cohomology in the Fock modules other than the central one. It would be interesting to prove this assumption directly using the properties of the screening operators. If one can do this then one may try to construct rational conformal field theories directly by specifying screening charges as in ref. [22]. In ref. [22] an approach to construct a finite-dimensional representation of modular and monodromy groups was given but an understanding of the positivity condition is still lacking. If one can prove the triviality of the cohomologies then one might be able to get conditions on the screening charges to ensure the positivity. One may also try to extend this analysis to genus-2 surfaces. There one expects an extra charge +2a 0 which can be balanced by introducing (k + 2)Q1 and one
Q ± each.
However an analysis of the monodromy and modular transformation of such multicontour problems is technically much more complicated. While completing this work, we received an interesting paper by Bernard and Felder [23] in which a detailed investigation of SU(2)k has been carried out in the formulation in terms of (/3, y) system and the boson 4’. Two other related works have also appeared, one by Nemeschansky [24] and the other by Distler and Qui [25]. Two of the screening operators introduced in ref. [24] are total derivative, and therefore one expects every state to be in the kernel of the Q’s constructed from them, i.e., they cannot give any non-trivial information. We do not understand the given in ref. [251since it contains any non-trivial information. QBRST
Q~k+2)and
one expects it also not to give
We would like to thank M. Caselle and M.A. Namazie for discussions.
Appendix A LATTICE SUMS FOR PF AND SU(2)k SYSTEMS
In this appendix we discuss some properties of the latticethe sums, F,~(w1, w2)the of 2)k. First consider lattice sum for the PF system and F,(w1, W2, w3) of SU(
T Jayaraman ci al.
/
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439
PF system: 1 Ti m( W
1, w2)
1+1
2(k+2))
+
—
k (fl2+
~)
n1,n2w7!/2
77(T) ~I
X
(~1)~~ sign(ni)q~+2)(f1
~
2
=
2
— ~I’~2l’~I >
1+1 exP[2i~(ni + 2(k + 1)
m
)~1— 2i~(n2+ ~)W2
(A.1)
.
As we saw in sect. 3, under z1 z1 + r we may get a situation in which the cone In11 > In21 is shifted. Thus let us see what is the change in F,~(w1,w2)when it is defined over the cone In1 I > In2 aI for some non-zero integer a. It is not difficult to see that the difference between the two is the sum over the points ~ for a ~ 1. It is clear from the expression for F,~(W1, w2) that the lines n1 = + /3 1 and n1 = —/3 should give respectively terms proportional to 01(w1 + W2/2) and 01(w1 w2/2). Explicitly, the difference —~
—
—
—
—
n~,n2EL/2,n1 ~ n1~In2—a~,—n1 >In2—a~
n1,n2~7L/2,n1
,,2e1
n,~In2I,n1>Ifl2~
is equal to ~
2
[F~+(Wi~W2)OI(
)
+F~)(wi,w2)0i(
2
)1~
where 2
1±1+m±2k(131)
F~4~(w 1,w2) = —(—1)
2
4(k+2)
(in±2k(/3—I))2 2k
(i±l+m±2k(/3—l))
m+2k(f3—1) 2k
1+1 +m+2k(f3—1) 4
(w 1+w2)
F~(w1,w2) = {F~with (1+1)
8
—
q
1+1 Xexp~,2i~-2(k+2)~
—
(1+1)2
—*
—(1+1) and
w1
—~
—W1}.
,
(A.2) (A.3)
In sect. 3, what was useful for our discussion of monodromy of the correlation function was the fact that 2k0~F,~~kw1,w2) = ±2(k + 2)a,,F~~(w1,w2). We
440
/ Parafermion
T. Jayaraman etal.
models
made use of this when showing that the additional terms do not contribute. Note that for the character x,,~(r)=F,,~(0,0)since 0I(0)= 0 there are no additional terms and therefore it is independent of the shift of the cone. For what follows it will be more convenient to write the summation appearing symmetric for n1 > 0 and n1 <0, i.e., to have the lines, say, In1 I = —n2> 0 included but 1n11 =+n2 ~ 0 excluded. Denote this by F,~(w1,W2). Then the difference between F1~(w1,w2) and F,~(w1,w2) is the contribution of the points n1 =n2 which is
—
F,~(w1,W2)
=
F~(W1,W2)01((W1
—
W2)/2).
(A.4)
2)k: Now consider the lattice sum for SU( 1 F,(w 1,w2,w3)
m
~ ~F,~(w1,w2)qk(n3+~)
=
77(T)
m
2
exp 2i~ n3
m
+
—
2k
w3 (A.5)
It is possible to make a change of summation and rewrite this sum in a form which is simpler. The expression that we are aiming at is the following: 1+1 ~q(k+2)(n+2(k+2))
2
1+1 exP[2i~(n+ 2(k+2)
qi5r(r+1) )Wi]L(_1)T1_qr~2j~
1+1 x{exP~_2i~-((k+2)n+_~-_)~+2i7rr( 1+1 _exP[2i~((k+2)n+ ~)~+2i~r(
2
2
(A.6)
where ~ = (W1 w2)/k. To obtain this, consider f~,(w1,w2, w3) which is defined as in eq. (A.5) but with F,~(w1,w2)replacing F,~(w1,w2)on the right-hand side. Make the change of summation n3 n3 n2, then since the quadratic term in n2 and n3 changes to a linear term in n3, it is possible to do its summation. Moreover since the dependence on m now is only in the form n2 + m/2k we can also do the summation over m. The range of m is 0 ~ m 2k 2 for 1 ~ 27/ and 1 ~ m s~2k 1 for I E 27/ + 1. Define p = kn2 + (m v)/2 where i-’ = 0 for 1 E 27/ and = 1 for I E 27/ + 1, then summing over m we get —kn1
—
/ Parafermion
q~knix
2n3
=
441
1 for n1 <0. Now the summation over p can be explicitly done
—
qkfllX
(A.7)
lqx
~qxP=sign(ni)
where x
models
Collecting all the terms, we have the following sum
+ ~/T.
1+1
~
(
—
1)2fl3q(
2
2)(nI+ 2(k+2)) +kn~—2kn1n3+n3v
n,,n3EZ/2
n3EL
~i
1+1
xexP{2i~ (ni+ —[(1+1)
2(k+2))w1~1~~/2+n3w3]}
—(1+ 1),w1
Now we make the following change of summation variable: n r = 2n3. Obviously n, r E 7/. Then we get 2Xn+
~
00
~
=
n1
—
n3 and
2(k+2)~( — 1)T q~r(r+i±1+0+4n) 1 — qTe2~ 1+1
~q(k+
1+1
Wi+W 2
xexP{2i~(n+ 2(k+2))W1kn~~( —[(1+1)
—*
—(1+ 1),w1
—*
2
)+v~/2
} (A.8)
—w1].
Upon using 2 (qr
e2)2~~++~~V2
~ 2n+(i+v)/
=
1— q’~e2’~
1
1 qrPe2iI~P+
,,~0
-
1
qre2l~
—
it is easy to see that the difference between eq. (A.8) and (A.6) is a term of the form W 1+W2
F~(w1,w2)0i(
2
)
—W1+W2
+ F~(w1,w2)0i(
2
As in the case of PF, for x,(r) = F,(O, 0, 0) this additional term vanishes and as we have seen in the examples considered in sect. 3 this difference does not contribute to the correlation functions.
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/
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Also note that the U(1)-valued character defined by x,(T,
is obtained from T,(w1, W2, well-known expression [8]
fl
by setting
W3)
—
x,(T,~)
where
0rs(~) is
(A.9)
Tr(qL0_~24 e20T~),
=
w1
=
w2
6,+lk+2(
=
0. Using eq. (A.6) we get the
—~)
0~(0) 2iire’~
0~(~)
=
(A.10)
defined as (A.11)
~
2 =
~qt(fl+S/2~I
In deriving eq. (A.10) we made use of the following identity: a’IUi\
lr(r+l)
00
.
1~./ =
q
—2i~e ~
1 _qre2~
This identity can be easily obtained by writing 6~(~) in product form and then doing a partial fraction. Appendix B MODULAR TRANSFORMATION
The main difficulty in the modular transformation of F,,,, is that the lattice sum is over a restricted domain. This was also the problem in showing the monodromy behaviour of F, however, as shown in appendix A and in sect. 3 the extra terms that appear upon shifting the domain either vanish (for zero screening charge case) or are total derivative (for simple screening contour case). We will use this shift invariance to embed F, in a three-dimensional lattice F, which essentially corresponds to correlators for N = 2 minimal series. Even though F,,,, is also a restricted lattice sum, the existence of a null vector in the lattice allows one to carry out the modular transformation explicitly, as described in ref. [16]. However, we will first describe the case W 1 = w2 = 0, i.e., the character x,,~(r). In this case one can directly study the modular transformation of x,,,,, upon using the following identity ,,,~
,~,
2T Ixl>IyI —f dx dy exp =
2T —
~(x2
2)
e2~
1YP2)
sign(x)
—y
sign(p 0,
1)qP~Pi, if IP1I> P21’ if P11
(B 1)
T Jayaraman ci a!.
/ Parafermion
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443
where p1 *p2. Denote the positive light cone (p1> p21) by L~and the negative one (—p1> 1p21) by L_. Note that the integration in (B.1) is fL— fL~ However p1 7~(n and p2 for our problem are Vk + 2 (n1 + (1 + 1)/2(k + 2)) and ~,/ + 2 m/2k) with In1 I > 1n21. It is easy to see that p = (p1, p2) lies in (L~uLj/G0 where G0 is a discrete subgroup of the proper Lorentz group generated by the element k+1
=
g0
.Vk(k+2)
~k(k+2)
k+1
Now we can bring the integration domains L~and L_ to L+/G0 and L/G0 by applying various elements of G0 on (x, y) and reabsorbing it by applying it on p. The effect of applying G0 on p is to expand the domain of p from (L~uLJ/G0 to L~UL_. In fact, since the integral in eq. (B.1) vanishes p is outside 2. Thuswhen we have L~uL_, we can extend the domain of p to the entire R 77(T) Xi m(T)
y2)I~(
dx dy sign(x) exp{_ 2T~
~fL~uL
=
1)2fh e2’~P1~P2).
(B.2) Now the summation
E
(— 1)2fl1 ~
n1,n2EZ/2 ~I
~
can be replaced by
ex~[i~(x
~
—
n1)~(y~—n2)(1
—
e’~~Y~)
and we can do the integrals to get 2x,~(T) 77(T)
2T~k(k+2) 1
{[2(k2 ~
1+1 xexP[i~(nI k + 2
~2
_~)]si~n(ni)
m —
-
n 2~)j(1
—
e1
1n2))}
(B.3)
444
T Jayaraman ci a!.
/ Parafermion
models
Now write n1=2(k+2)ñ1+(l’+l) and n2=2kñ2+m’ where l’=O,...,2k+3 and m’ = 0,. ,2k 1. Note that because of the factor (1 e’~”2”1)), there is a non-zero contribution only if (1’ m’) E 27/. Thus . .
—
—
—
2X,~(T)
1
/
=
T\/k(k + 2)
77(T)
1’,m’ ~
1’—m’E2ZL
—n,>In 2~
2i~r
l’+l 2(k+2))
x{si~n(nl)exP _~((k+2)(nl+
m’
_k(n
(l+1)(I’+l)
2
exp[i~(
2+~)
k+2
2
mm’
-~)]}.(B.4)
To bring the range of 1’ to the standard one, we transform 1’ k + 2 + 1’ and m’ k + m’ mod 2k when 1’ ~ k + 2. Finally, making use of the transformation (1’ + 1) —o —(1 + 1), n1 —ñ1 and the fact that for 1’ = k + 1 the sum vanishes, we obtain —‘
—~
—~
x,,~(T)
=
1 Vk(k + 2)
k l~o
sin(~
m’=-k
(I+1)(l’+l) k+2
1 )e~mm~xim(_
_).
i—mw 21
(B.5) 1”277(T).
Here we have used Unfortunately this‘q(—1/T)=(—IT) technique does not work when
w
and w2 are not zero; the G0 symmetry, used to enlarge the domain of p to L~uL_ is only a symmetry of the spectrum and not of lattice states. Therefore we will proceed by embedding in a three-dimensional lattice corresponding to N = 2 minimal series whose modular transformation was worked out in ref. [16]. First for 0 <1 < k and —(k + 2)
=
1 —F,’~(T;w1,1~2)03 T 77(T)
where 03 is a Jacobi 0-function 13 e2t~t13z, 03(T;
z)
=
~
fl 3
q!’ 00
—W2+~_
k
,
(B.6)
T Jayaraman et a!.
/ Parafermion
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445
and
Fi’m(T;
Wi,
w2)
1
1+1 2
=
E
(
—
1)2n1
2
sign(ni)q~+2)((fh+ 2(k+2))
2 —
(n2+
2(k+2))
77(T)
1+1 xexP[2i~(nl+ 2(k+l))w12(n2+
Here w2 E 7/ and
(k
+
2)w2
—
m
2(k+2))W2
2~/k, and the summation is over n1, n2
E
m 1mw2 2(k+2) + 2kT2
1+1 1mw1 2(k+2) + 2(k+2)T2 ~
(B.7)
.
~-Zwith n1
where
—
gOt.
Note that the domains are shifted by (Im w1)/T2 and (Im w2)/T2 so that the lattice sums are absolutely convergent. Hence F,,,, is given by a three-dimensional lattice sum. By making the change of summation variables n3 = kñ3 + m’ 2n2, where m’=0,...,k— 1, and n2=ñ2—ñ3, the sum becomes —
1
k1 3
=
77(T)
(~
~
m’=O
1+1
1)~fl
sign(n1)
q(k+2)(ni+
2(k+2))
m±2m’
2
_k(h2+
n1,ñ2wl/2
2k
)
2
fl1 ~1~2E7L
~
x exP[2i~(nI X
+E~~fl/
1+1
+ 2(k +
1)
)~1 2i~(ñ2+ —
m+2m’ 2k
)~2]
0~+(k+2)~ k(k+2)/2(~k(k+ 2)~),
(B.8)
where 051(z) was defined in eq. (A.11). In obtaining this, we have used the fact that the n1, ñ2 sum is exactly the one appearing in the parafermion lattice sum Fi~(r;w1, w2) and consequently using the shift invariance of the domain to bring the latter into the standard form In1 + eI > In2 I. Thus 1 =
k—I
I \ ~ Flin+2m.(T;wI,W2)0m+(k+2)in~k(k+2)/2(~k(k 77~T, m’=O
+ 2)~T). (B.9)
F1~(r,w1,w2,~) is essentially the lattice part of the correlator for the N= 2
446
T. Jayaraman ci al.
/ Parafermion
models
minimal series in NS sector. In fact, by setting ~ = 0 (~ just keeps track of the U(1) quantum number) we obtain the expression for the lattice sum given in ref. [16], and whose modular transformation was also worked out. For the sake of completeness we briefly outline the steps involved in evaluating the modular transformation of F’(T; w1, w2). Because of the existence of lightlike vectors in the lattice, the sum over the direction parallel to light ray becomes linear and can be done explicitly.
Thus splitting the region into four parts 1+1 L~~:n1+ 2(k+2)
+
1+1 L~: n1+ 2(k+2)
+
1+1 L_~: n1+ 2(k+2)
+
1+1 n1+ 2(k+2)
+
L_:
Im(w1—~2)
m
and
n2+ 2(k+2)
and
m n2+ 2(k+2) <0,
<0
and
m n2+ 2(k+2)
Im(w1—~2) 2(k+2)T2 <0
and
m n2+ 2(k+2) <0,
~0
2(k+2)T2
Im(w1—~2) 2(k+2)T2
Im(w1—~2) 2(k+2)T2
the sum over ~ and L__ can be done by changing the summation variable n1 = n~+ n2. The quadratic term in n2 disappears and the sum over n2 just gives a denominator. Putting together the contribution from ~ and L__, one gets a sum over n~from to +c~. Similarly by changing n1 =n~—n2 in L~_and L~ and summing over n2 one gets another denominator with unrestricted sum over nç. The result is —~
F,’~(’r;w1, w2, x)
m
2
1+1
00
2
=q4(+2)exp[_2i~ in 2(~~2)w2j~q*2)(f+2(k+2)±x)
1
Xexp 2i1r1
~1 + 2
1+1
k
+
1
~X
w1
I+1
1
~
1+11
—
1
,
(B.10)
+q_(2)(~1~)m/2e_~~2+w1)
where we have introduced x to carry Out the Poisson summation. By making
T. Jayaraman ci a!. / Parafcrmion models
447
Fourier expansion, one gets an integral over x on a real line of the following form
00
1
dx
2
(k+2)y
1 ___________________________ 1 + q +2~m,/2 e~”~2~’~’/’
1
_____________________________
—
1
+ q2~’~”2
e’’2~”P
where y =x + (w 1 —p)/2(k + Now by the residue theorem
and p is an integer labelling the Fourier mode.
2)T
wi—p
f00 dx=f
(~~~\dx+2i~E (residues). ImI ‘,2(k+2)r
—00—j
But the latter integral is zero as the integral is odd under y
—‘
—y. Taking together
all the residues one gets
k+1
—
F/~(T;w1,w2)=
—
,
k+1
.
sin
,,
‘rr(l + 1)(l’ ,
+
mm’
1)
exp —i~r,
,,
I~+~I~_0th(k+2)
lc+~
1’ — rn in 2!
x ex~[_ 2T(k+ 2) (w~ -
~)]
T,~th(-1,
~).
~,
(B.11)
Using eq. (B.6), modular transformation of 03 and eq. (B.9), one gets
—
F,~(T;w1,w2,fl=exp
iTT
2
k1 k+1
w~
w~
k2k+k(k)
k+1
.
sin
X
th’=Oi’=Oth=—(k+2)
2~2
‘n-(I
l)(I’
+
,
,.,
+
+ 1)
/
—i’rrmth
expi !~
+
i’—thw2l
1
w1
—,
—,
W2
—)oth+(k+2)th. k(k+2)/2(~(k
+
‘. 1 2)—;— (B.12)
T. Jayaraman ci a!.
448
/ Parafcrmion models
One can re-express this by transforming 0 appearing in the right-hand side: W~
i7T
-
w~
~
k—i
k+1
‘n-(l+ 1)(l’
k+1
+
1)
sin +
th’=O 1=0 th= —(k+2)
I’ — th
in
2!
1
—i’n-mth Xexp
F,.
th+2th’
w~ w2
~
k-i-2
T
T
T
(th+(k+2)th’)(s+(k+2)s’) ~ exp —2i~-
k+1k1
x
.
s=0 s’=O
+
x0S+(k+2)S.k(k+2)/2(~k(k+ 2)~).
(B.13)
As F,,,, isdefinedfor mmod2k,let m + 2m’ =rmod2k and th +2th’=Fmod2k, one can sum over th’ and th fixing F. This sum vanishes for s * m mod(k + 2) and for s = m mod(k + 2) one gets a factor of (k2’~one + 2). Thus, gets comparing eq. (B.13) with eq. (B.9) and equating various powers of e F,,,,( ~ w i~w 2) 1 =
1~T
jk(k+2) k+l
ki
XEL51fl
exp
~
W~
W~
k+2
‘rr(l + 1)(l’ k+2
+
1)
1
iirmm’ exp
—
k+2
F,,,,
w~ w2 ~‘
(B.14) modulo irrelevant terms that give a total derivative with respect to the screening contour, when inserted in the full correlation function.
References [1] [2] [3] [4] 15]
E. Witten, Commun. Math. Phys. 92 (1984) 455 V.G. Knizhnik and A.B. Zamolodchikov, NucI. Phys. B247 (1984) 83 D. Gepner and E. Witten, NucI. Phys. B278 (1986) 493 P. Goddard, A. Kent and D. Olive, Phys. Lett. B152 (1985) 88 E. Witten, Commun. Math. Phys. Lett. 121 (1989) 351
T Jayaraman ci a!.
/ Parafermion
models
449
16] M. Bos and V.P. Nair, Phys. Lett. B223 (1989) 61; M. Bos and V.P. Nair, Int. J. Mod. Phys. A5 (1990) 959 [7]J.M.F. Labastida and A.V. Ramallo, Phys. Lett. B227 (1989) 92; Phys. Lett. B228 (1989) 214 [8] V.G. Kac and D.H. Peterson, Adv. in Math. 53 (1984) 125 [9] A.B. Zamolodchikov and V.A. Fateev, Soy. Phys.-JETP 62 (1985) 215 110] D. Gepner and Z. Qiu, NucI. Phys. B285 (1987) 423 [11] VS. Dotsenko and V.A. Fateev, NucI. Phys. B240 [FSI2] (1984) 312; B251 [FSI3] (1985) 691 [12] G. Felder, NucI. Phys. B317 (1989) 215 [13] T. Jayaraman and KS. Narain, NucI. Phys. B331 (1990) 629 114] G. Felder, J. Frolich and G. Keller, Braid matrices and structure constants for minimal conformal models, lAS preprint (1989) [15] G. Mossardo, G. Stokov and M. Stanishkov, NucI. Phys. B305 [FS23] (1988) 69 [16] T. Jayaraman, MA. Namazie, K.S. Narain, C. Nunez and M.H. Sarmadi, Nucl. Phys. B336 (1990) 610 [17] M. Wakimoto, Commun. Math. Phys. 104 (1986) 605 118] A.B. Zamolodchikov, unpublished [19] A. Gerasimov, A. Marshakov, A. Morozov, M. Olshanetsky and S. Shatashvili, Wess—Zumino—Witten model as a theory of free bosons, preprints ITEP-89-70, ITEP-89-72, ITEP-89-74 (1989) 120] S.D. Mathur and S. Muki, Phys. Lett. B210 (1988) 133 1 and monodromy [21] A. Tsuchia andof Y. Kanie, Vertex operators in conformal field theory on P representation braid group, preprint M. Caselle and KS. Narain, NucI. Phys. B323 (1989) 673 123] D. Bernard and G. Felder, Commun. Math. Phys. 127 (1990) 145 [24] 0. Nemeschansky, preprint USC-89/012 (1989) [25] J. Distler and Z. Qiu, preprint CLNS-89/911 (1989) [22]
SU(2)kWZW AND Zk PARAFERMION MODELS ON THE TORUS T. JAYARAMAN International Centerfor Theoretical Physics, 34100 Trieste, Italy
K.S. NARAIN and M.H. SARMADI Theoty Division, CERN, CH-1211 Geneva 23, Switzerland Received 27 November 1989
We study SU(2)k WZW and /Lk parafermion models using a Coulomb-gas formulation related to that of Wakimoto and Zamolodchikov. We discuss the truncation of the Fock modules of the free bosons of the Coulomb-gas system to obtain the parafermion and SU(2)k modules. Explicit expressions are given for the correlators of these models on the torus. A detailed discussion of their modular transformation and their monodromy is also given, the complications being due to the restrictions on the lattice summations.
1. Introduction Among the known two-dimensional conformal field theories, the ones of particular interest are the Wess—Zumino—Witten (WZW) models [1—31,these being the conformal models in which the chiral algebra is generated by a current based on some Lie group G. At present, all the known rational conformal field theories are of the coset type G/H [41,these being the models in which the chiral algebra is constructed from that of G such that its fields commute with those of the subalgebra based on H (H C G). There is even a conjecture that all the rational conformal theories are of coset type. Therefore a full understanding of the WZW models is of importance. Recently Witten [5] has shown a very interesting connection between WZW models and three-dimensional Chern—Simons gauge theories. Subsequent works [6, 7] have made the connection between the states of the Hilbert space of three-dimensional Chern—Simons theory and the conformal blocks of WZW theory more explicit. In particular, in refs. [6, 7] the known differential equations [2] satisfied by the conformal blocks for the case the Riemann surface is a two-sphere have been derived and the known characters for G SU(2) and arbitrary level k [8] have been obtained. In principle, in this approach one should be able to obtain =
0550-3213/90/$03.50 © 1990
—
Elsevier Science Publishers B.V. (North-Holland)
T Jayaraman et al.
/ Parafermion
models
419
expressions for the correlators on higher-genus surfaces, however, this has not yet been done. In this paper we will follow another approach, namely the Coulomb-gas approach to SU(2)k and the related 7~k parafermion models (PF) [9, 10] (i.e., SU(2)k/U(l)k). The Coulomb-gas approach to minimal conformal models [11—14] and minimal superconformal models [15, 16] has proved to be an efficient one. Here we will use this formulation to study the characters of SU(2)k WZW and ~ parafermion models and write explicit expressions for some correlators on the torus in these models and study their properties. The organization of this paper is as follows. In sect. 2 we will first review the Coulomb-gas formulation of Wakimoto—Zamolodchikov [17, 18] and by introducing cohomologies similar to Felder’s [12], will investigate the structure of the PF modules and give a derivation of the known PF character. In sect. 3 we will give examples of correlation functions, both for PF and SU(2)k systems and study their properties. In particular we will discuss their monodromy which, due to the restrictions on the domain of lattice summations, is intricate. Sect. 4 contains the concluding remarks. In appendix A we discuss some properties of the lattice sums for PF and SU(2)k systems which are useful for the discussions in sect. 3. Appendix B deals with the problem of the modular transformation for these models. Again the restrictions on the domain of lattice summations make it more involved than the usual cases, and we will handle the problem by exploiting the connection with N 2 minimal models whose modular transformation was worked out in ref. [16]. =
2. Coulomb-gas representation of PF and SU(2)k systems and their characters In this section, we will first review the Coulomb-gas approach to SU(2)k WZW and the related 7~k parafermion models and then discuss a derivation of their characters. The SU(2)k currents J ±(z),J3(z) and the energy—momentum tensor T(z) satisfy the operator product expansion (o.p.e.): Ja(z)J
h
(z’)
=
(k/2)~5’~ 2 + (z-z’)
=
T(z)T(z’)
=
where k is an integer, c
2
+
(z-z’)
=
regular,
2T(z’) 4
3k/(k
+
(z-z)
c/2 (z—z’) +
regular,
(z-z) 3Ja(z~)
Ja(z~)
T(z)Ja(zF)
+
fahJc(z~) ,
+
öT(z’) 2
+
(z—z’) 2) and
f~—
(z—z
)
+
3~~=f3
3f
regular,
—=
1.
(2.1)
420
T Jayaraman et a!.
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Parafermion models
As shown by Wakimoto [17] and by Zamolodchikov [18] (see also ref. [19]) these currents can be expressed in terms of one free boson 4’ (with a background charge) and a free (f3, y) system of weight (1, 0): 1
J3(z)=:f3(z)’y(z):—i
~Ik+2 2
J(z)
=
~(:p(z)y(z)2:_iyk+2a4’(z)y(z)+kay(z)),
T(z)
=
—~:84’(z)a4’(z):+ia
(2.2)
24’(z)—p(z)a-y(z), 03
where a 0 2 1/2~/k + 2 and the normalizations a4’(z)04’(z’) and J3(z)y(z’)= —1/(z—z’). Clearlyare thesuch (f3,y)that system contributes —2/(z—z’) c 1~ 2 to the central charge whereas the boson 4’ gives c4, 1 24a~ and therefore the total central charge is c 3k/(k + 2). One can bosonize the (/3, y) system in terms of two bosons (x, ~) in the usual way as: =
=
=
=
—
=
/3
=
—
+iri)]~
~-3XexP[_T~=(X
y= exP[_)=-(x+i77)].
(2.3)
Note that ij comes with imaginary momenta and contributes negatively to the dimensions. Now by making the following change of basis from (x, 4’) to (4’o, 4’1,4’2): ~,
/k+2
U
k /k+2 k+2 k 4’+i
V
i4’2
/k+2 4”4’~V 2
4’~/k~’
~ yIT~
/k+2 2
X’ (2.4)
+
one can rewrite the currents as:
J~(z)
=
y~~(z) exP[~4’o(z)]~
J(z)
=
~~t(z)exP[~4’(z)]
J3(z)
=
(2.5)
/ Parafermion models parafermion currents,
T Jayaraman et a!.
where ‘I’(z) and ~11t(z) are the
1k
i(fk±2
=
421
a4’ 1(z) +ia4’2(z))exP[~4’2(z)]~
~t(z)
i(\/k±2
=
_i34’2(z))exP[_~4’2(z)].
These parafermions each carry a conformal weight 1 tum tensor is now:
2+ (34’~)2~
T(z)
=
‘((a4’)
(2.6)
1/k. The energy—momen-
—
+ia
(34’2)2)
24’ 03
1.
(2.7)
One can construct screening operators which have dimension one and commute with the currents (i.e., their o.p.e. with the currents are regular or if non-regular they are total derivative). In fact, there are three such operators: =
:ö4’2(z)exp[2ia04’1(z)]:,
V~(z) :exp[_} V7~T~1(z) ±~V~4’2(z)]. =
(2.8a,b) One can easily verify the following o.p.e. of these screening operators with 11’ and ~I’~: ~(z)V1(z’)
i~t(z)V1(z’)
=
—
~~2az[1,exp(
~
4’1(z’)
+
~4’2(z))]
+
=
—
~~a~,[1,exp(
~k~2
4’1(z’)
—
~4’2(z))]
+regular,
regular,
regular, ~Itt(z)V÷(z’)
=
iPt(z)V(z’)
=
regular,
~
IP(z)V_(z’)
_~
=
regular,
V_(z’)exP(_ ~4’2(z))]
+
regular.
Therefore these screening charges commute with the currents. If we denote the momenta of V1(z) and V~(z)by e1 and e ± respectively, then they satisfy the relation (k + 2)e1 + e~+e_= 0. Thus these screening operators can be used to
422
T Jayaraman et a!.
/ Parafermion
models
balance the charges on higher-genus surfaces. In particular for genus-two surface one expects a background charge + 2a0 on the surface which can be screened by the relation (k + 1)e1 +e~+e..=(—2a0,0,0). The parafermion system is described entirely by the two bosons ~ and 4’~.The parafermion primary fields Pim are labelled by two integers I and m; I 0,. k and m —k,.. k 1. In terms of 4~and 4’2 they are expressed as: =
=
. ,
. . ,
—
ii
m for
—l~
1/2. For m > I one can get the primary where C(l, m) [l!/(1 + m/2)!(l m/2)!] fields by the successive application of the parafermion field ~I’ on Pjm. The primary fields of SU(2)k of spin j and J~-eigenvaluem are then simply =
—
=
~ 2i2m(z)exP[i~4’o(z)].
(2.10)
Now let us turn to the structure of parafermion modules. First note that the screening charges Q±= f dzV~(z)and Q1 fdzV1(z) satisfy the relations: =
(2.11)
~)kQQ 0, Q~Qi+ ~ ± 0. (2.12a,b) These relations can be verified by using the o.p.e. of the screening operators. In the last relation, one obtains an integral of a total derivative which consequently vanishes. Denote by Firn the Fock module built on the vertex operators :exp(—il/2Vk+ 24’~+m/2~/k4’2):.Then for Q~we have (see table 1): —
Q+ and for
~
Q1 Q~k+I_1) > Fim
Q~I+I)
F_i_2,m
T. Jayaraman et a!.
/ Parafermion
models
423
TABLE 1 See text for explanation
F, + 2(k + 2). ,n + 2k
Qy
Qy
Q(k*l_/)
F,÷k+2,,,+k
Qy
Q/
2~,
Fim
F2(k+2)_,2,,,,
—I
Qy
(1*1)
Q(,*l)
F,
Q(i*I)
F3(k+2)_1_2,m+k
(k*l—I)
F2(k+2)+,~,
—
Q(/+ I)
I)—!)
F3(k+2)_,_2,,,_k
F, ±2(k + 2), m + 2k
where
Q~is defined
as =
fdz1 Vi(zi)fdz2 V1(z2)
. ..
fdz~V,(z~),
(2.13)
with the choice of contours being the same as Felder’s [12]. The primary fields listed above satisfy the following conditions: =
~
E
=
Im(Q~~°)
0,
(2.14a)
~ Im(Q~Q).
(2.14b)
Consider the parafermion module ~1m which consists of the states obtained from the primary field cIlm by applying equal numbers of ~I’and ‘I”~operators. The states obtained by applying unequal numbers of ~P and ~I”~differing by multiples of k will be discussed below. From eq. (2.14) it is clear that the parafermion module P, is contained in ,~,
Ker(Q÷)n Ker(Q) Im(Q~Q) Since
Q~=Owe
Ker(Q~~)
~
Im(Q~°)
n Ker(Q~~~)
have Im(Q~)cKer(Q~).We now show that the cohomology of
424
T. Jayaraman et a!.
Q ±(i.e., 4’
/
Parafermion models
Ker(Q ±)/Im(Q,)) is trivial on each Fock space. Indeed if there is a state
in some Fock module =
Frs
such that
Q~4’ =
0, then define the state
~~~exp[~iVk+24’1(z)
T ~4’2(z)]
4’(O).
4” as (2.15)
We then have
Q ~4”(0) =
~~dw 1 2iir
=—~—
exp( - +iV~4’i(w) ±~V~4’2(w)) 4”(O) dz 1 1 dw+—~ dw exp(—+i~i~74’1(w)±~V7~4’2(w)) z 2i~ C,, 2iir C0 —~
x exp(~iV~24’i(z) ~ ~/~4’2(z) t/i(O)),
(2.16)
where the contour C~is around z and not 0 whereas the contour C0 is around 0 and not z. The integral f~5dw is zero as Q ~4’(O) 0 and the integral f~dw is just equal to one. Performing the z-integral we get Q ~4”(O) 4’(O). One can further ask whether the cohomology of Q is trivial in Ker Q+. In other words if 4’ E Ker Q~nKer Q then does there exist a state 4” such that 4’ Q~Q_4”? Indeed take the lowest-level state ~/m in Fim. We know that ~ E Ker(Q~)n Ker(Q~). However if ~ is an image of Q+Q_ of some state 4” then 4” E Fi_2(k±2),m.But the weight the lowest-level in On Fi.2(k+2)m is greater 1(mofcannot be in the state image. the other hand forthan all that of ‘im’ therefore other Frs, (r, s) ~(I, m), shown in table 1, one can easily show that the lowest-level states are in fact in the image of Q±Q.Let us assume that this is true for the entire Fock module Frs((r, s) # (1, m)), i.e., =
=
=
Ker(Q+)flKer(Q) Im(Q~Q)
F (r,s)
for
or
—O -
(r,s)=(I+n 1(k+2),m+n2k),
n1>O,
(r,s) =(—I—2+(n1+2)(k+2),m+n2k),
n1~O, (2.17)
T Jayaraman et a!.
where n1 ~ n21 and n1
+
Ker(Q~Q)
F
Ker(Q~)®Ker(Q..)
n2
E
/ Parafermion
models
425
27/. This implies that
—0 r,s
—
for
(r,s)=(I—(n1+2)(k+2),m+n2k),
n1>~0,
or
(r,s)=(—l—2—n1(k+2),m+n2k),
n1>0, (2.18)
where n1 ~ 1n21 and n1 + n2 E 27/. This can be seen by introducing the following inner product between a state 4’ E Fr s at some level n above the primary field with a state 4” E F_r_2,_s at the level n above the primary field: =
the two-point function (4’(0)4”(l)).
(2.19)
This is a non-degenerate inner product. In other words, for each 4’ there is a 4” such that ~i/i~4”) 0. Now if 4’ E Ker Q+’ then due to the triviality of Coh(Q~),4’ = Q~4” for some 4” E Fr_(k±2)s—k. Since K4”I4’) = (4”~Q+4”)= KQ+4”I4”) * 0 then Q~4” * 0, where we have used the definition of the inner product (2.19), and deformed the contour of Q+ so that it encircles the point 1. (Note that there is no contribution from infinity.) If we choose 4” such that ~4”~4’) * 0 and (4”~4’~ = 0 for 4, E Im(Q~Q.J then it follows by similar arguments that 4” E Ker(Q~Q). Thus Ker(Q~Q)/Ker Q~ ~ Ker Q_(F_T_2 ~) is dual to Ker Q+ ~ Ker Q/Im(Q+Q)(Fr,s) and therefore the triviality of the latter for the range of (r, s) given in eq. (2.17) implies the triviality of the former. Finally, using the ~2 symmetry 4’2~-’ —4’2,Q~~-~Qwe can flip the sign of s to get eq. (2.18). Now we are in a position to calculate the character. Consider
Xi,m
——
IrKer(Q+)nKer(Q) 1m(Q~Q_)
=
I
‘i’
t,q
L~—c/24
(Fim)
Tr(Ker(Q÷)fl Ker(Q_)yF,,~,)~~ ( L0—c/24\ /
—
Tr
~1—2)k±2).,,,
( L1~—c/24\/ ~q
.
~2 2
Ker(Q~)flKer(Q)
Now the first term is ‘I’ r(Ker(Q+)n Ker(Q_)XF,,,,,)~. I q L,,—c/24\/
— —
V’ ~
I~,
—
I/
I q L11—c/24 r Ker Q+(Fj+,,1o~s1,,,,_,,o)’~
•~1~’
n = 1) F,~
=
~
24)
(—1)’~~’Tr1,,+,,0k--21,n*(,,’—,,)k (qLo_C/
n=0 n’=O
(2.21)
426
/ Parafermion models
T. Jayaraman eta!.
and the second term Tr
24)
F,_ak±2, (qLo_c/ Ker(Q~)Ker(Q_)
=
(
~
1)
—
~‘
Tr F,(,,.±2xk±2),,+ek
n’=O
~
=
(— ~
1)fl+fl’
n
(qLo_C/24)
Ker(Q.)
0 n’ = 0
(2.22) Combining the above expressions we obtain 1
2”sign(n
Xi,m
(—1)
2
71(T)
1)
n1,n2EZL/2 fI
~
n1 ~In2], —n1 >1n21 2— —(m +2n
xq4Uc+2)
(1+1 ±2n1(k+2))
2 2k)
4k
(2.23)
where sign(O) = + 1. This is indeed the known expression for the parafermion character [8,9]. Hence, if the assumption (2.17) is correct then the parafermion module can be identified with Ker Q~fl Ker Q/Im(Q+Q)(Fj,m) and the latter is in Ker(Q 1). The fact that none of the states in the latter is in Im(Q1) follows from assumption (2.17) that KerQ~flKer Q/Im(Q±Q.J(F_l_2+2(k+2) m) is trivial and the commutivity of Q1 with Q We have not been able to prove assumption (2.17), however, as shown above at least the lowest level states in Frs (and hence the entire irreducible parafermion module built on them) are in the Im(Q~Q ). It remains to be proved that this is so for all Ker Q~fl Ker Q in Frs. For k = 1 one can in fact prove that this is so. Let 4’ ~ Frs (with weight zl * 0) be in Ker Q~flKer Q. Define ~.
=
Applying
Q Q ±
-
on
i/i’
2i~ ~dzz e~’~4i(0).
and using the fact that
Q±Q-4”(O)= =
2’~r~ ~
Q ±4’
=
0 we obtain
e~1~)4’(O)
1 ~dzzT(z)4’(0)= 2t’7~-~1 .
/ Parafermion
T. Jayaraman et a!.
models
427
Thus every state in Ker Q~flKer Q with ~1* 0 is in Im(Q~QJ. This is as expected for k = 1; the parafermion module consists of only one state, namely, the identity. If for k ~ 2 eq. (2.17) does not hold, then one can proceed to obtain the character in the following way. Consider the modules F12+2p(k+2),m(P ~ 1) and Fi±
2p(k±2)m(P ~ 0) which are connected to Firn by the action of Q1. In table 1, they are the modules on the horizontal line from Fim. For each of these, we compute the trace by summing (with alternating ± signs) over all the modules which are connected to it by the action of Q ± and Q -. This way one obtains a trace over the states in Ker Q~fl Ker Q_ and Frs/Ker ~ Ker Q_, which is
Tr KerQ~flKerQ_k~ ~
L0—c/24\/
—
mr
(~q L,1—c/24
F,., Ker Q~ Ker Q
1
r+1
-
(~1)Jq 1=0 ~
2
=
77(T)
+
2
E (— 1)Jq
~) s
+2)(~i+ 2~k+2)) _k(~i+
2
r+l 2 2 2)(~i±2(k+2)) _k(~J_~)]
(2.24) 12r,s
We
Let us denote the thisfollowing space, which is obtained after the above summation as now consider sequence: Q
—
1k+I_()
Fim
QS’~~—
—
Note that under the action of -
Fim
Q±Q-
-
0 F2(k+2)+im
Q+Q-
-
F4(k+2)+im
Q+Q-
-
>
...
(2.25)
.
Q~Q_we have
Q±Q-
-
F2(k±2)_i_2.m
Q1k+I_!)
F2(k+2)+lm
F2(k+2)_i_2,m
F4(k±2)_i_2,m
0
Q+Q-
-
S
..
Q±Q-
F±_i_~,,~
.•
Therefore, if the trace is taken over the modules in eq. (2.25) with alternating ± signs, then taking_into account the fact that ~—l—2,m ~ F2(k+2)_i_2,Th and that F,2 is dual to Fim, in effect we are taking the trace over the states in KerQ~nKerQ Im(Q~Q_)
Ker(Q~Q) (Frs)
and
KerQ~®KerQ_
But as we showed above, Ker(Q~Q)/Ker Q~® Ker Q_(Fr,S) is dual to Ker Q÷fl Ker Q/Im(Q÷Q_)(F~_2_5).Therefore we can instead of the sequence in eq.
T. Jayaraman et a!.
428
/
Parafermion models
(2.25) consider the following sequence: Q 1k±I_/)
Q(I*l)
Q(k±l_I)
F_i_2m
where
Frs
Ker
Q~n
Fim
Ker Q/Im(Q+Q)(Fr,s).
and the cohomology of
Q~k+1i)Q~i+1)o
Q(I+I)
F2(k±2)_,2,,fl
Q1
Now
if
...
,
(2.26)
Q~i±1)Q~k±1—)) =
is trivial on the modules in the
sequence (2.26) except for F1,,, then, by taking the trace over the modules in eq. (2.26) with alternating ±signs, we get a trace over ~ which is the parafermion module, and by using eq. (2.24) and making a trivial change of summation variables we get 1’~)
(qLo_C/24)
Tr 1m(Q~ Ker(Q~ —I)) (F,
,e)
1
(
2
=
77(T)
n
(1+1 +2n
4~’~2~
2fl
—
1)
1
2— —(in +2n 2 4k 2k)
1(k+2))
sign(n1)q
1,n2eL/2 °1 2E Z ~I
~I~2I,
f1
>In2t
(2.27) We have so far discussed the spaces ~ One may wonder what happens to the states obtained from ~ m by applying unequal numbers of ~I’ and ~ differing by a multiple of k. These states could be contained in Fim+2flk,fl being a non-zero integer. In ref. [9], (~t’)”and are identified with identity, so one would not expect to obtain any new states. But in the present Coulomb-gas formalism, these states carry different momenta along the 4’2 direction. However, it turns out that there is an isomorphism between F1,,, and F~m+2nk for any n E 7/. Indeed consider a nontrivial state 77 E Fjm. Then 77 is in Ker(QJ and hence there exists a state ~1’ such that Q.jsj’ = ~ Furthermore ~ Q+n’ * 0, because otherwise due to the triviality of the cohomology of Q+ there would exist a state 77” such that = Q+~i”.This in turn would mean that ~1E Im(Q~Q),which is a contradiction. Now we show that ~ E F,m±2k.Certainly ~ ~ Ker Q÷.It also belongs to Ker Q_ as Q~~QQ71i(l)k+lQ77 =0. Finally if ~ =Q+Qñ’, then QJii’— Q~’)= i~ and Q±(~i’ Q~’)= 0 implying that s~E Im(Q~Q_),which is a contradiction. Therefore ~ is a non-trivial state in Fjm+2k~It is easy to see that the map defined above between F1~and F1,,, + 2k is an isomorphism, and that it can be extended to all Fim±2nk successively. This isomorphism allows one to identify (ifr)k = (114)k = 1 in our formalism and work with only F1,,, for example. This also means that all the physical quantities, such as characters and correlation functions, which are computed using F1 m+2nk would be independent of n. In particular, this implies the shift invariance of the domain of the lattice sums which plays a crucial (1~tY~~
—
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429
role in proving monodromy and modular covariance of correlation functions and which we will discuss in sect. 3. Now let us discuss the computation of the n-point function on the sphere. For this we need to define a dual field at ~ (say). The dual spaces have already been introduced above;
F_,_2,,,,
(
and
KerQ~flKerQ * Im(Q~Q_) (F1~~))
Ker(Q~Q) Ker(Q~)~ Ker(Q) (F_1_2,~). Thus we can consider a four-point function,
where Ker Q~flKer Q Im(Q~Q) Ker(Q +
~
(F,
Q )
) (F
Ker(Q+) ~ Ker(Q4
i=123
, —
(2.28)
).
4, ~~tfl4
To define this correlation function we must introduce a sufficient number of screening operators so that i~+ 1~+ 13 m1
+
—
2’/k+2 m2 + m3
14
—
+
m4
~ (screening charges)
=
0.
2~ One can define the screening contours as in Dotsenko—Fateev [11] or in Felder [12]. The fact that the intermediate state is also a physical state can be seen in the usual way. That the dressed 4’~(0)x 4’2(z) is in Ker Q÷flKer Q_ follows from the commutativity of Q ± with Q1 and Q T• If on the other hand this product contains a state in Im(Q~Q), then it decouples from the above correlation function as can be seen by deforming the Q + and Q contours and pulling them around 4i~and 4’ and using eq. (2.28). In the next section we describe how to compute two-point functions on the torus. The subtlety here is that one must make sure that the trace is taken over -
430
T Jayaraman et a!.
/ Parafermion models
only the physical states and this will be ensured by making use of eqs. (2.20), (2.21) and (2.22) (or eq. (2.27) in case eq. (2.17) does not hold). Before ending this section we note that since SU(2)k is the product of PF and U(l)k, using the expression for the character of PF and the character of U(1)k and following the steps sketched in appendix A one obtains the well-known expression [8] for the character of SU(2)k given in eq. (A.1O) below.
3. Correlators in the PF and SU(2)k systems In this section we will use the Coulomb-gas formulation of SU(2)k current algebra and parafermion system of the previous section to derive expressions for their correlation functions on the torus. For simplicity we will only give the correlators of the currents and the correlators of the primary fields of type j = m = ± The former does not require screening, and thus there is no contour integral, whereas the latter requires one screening operator. Consider first the correlation function of two PF currents; (~!~(z 1)Wt(z2)). Using eq. (2.6) for ~ and ~ we get ~-,
~.
2/k
[_(k+1)a~logo1(z~_z2)+(k+2)2a~,_k2a~2
I(0i(:i~2))
_(a~loge1(z1_z2))2_2k3~logo1(z1_z2)a~2}T,,~(0,w2), (3.1)
where w2 = 2(z1
—
z2) and 2
Fim(Wi,W 2)
1
=
71(T) 2
(—
~
n
4~~2~ 1 ±2i~1(k —(l+ ±2))2 1)2uu1
4k
— —(in
±2n2k)
sign(n1)q
1,n2wz/2 ~1 fl1
±1n21,
2~ 7! fl1
Xexp 2i~(ni+
>In/
/+1 m 2(k+l))w12(~~2+~)w2I
(3.2)
In appendix A we discuss some properties of F,,,, which will be useful in what
/
T. Jayaraman et a!.
Parafermion models
431
follows. In deriving eq. (3.1), we have used the following expressions:
2 2a~ 1log 01(z1 —z2)F(O,0)
=
—
4(k
2) d~,F(0,0), (3.3)
+
a4’2z2 exp( - ~4’2(z2)))
(a4’2(zi) exP(~4’2(z1))
2/k =
2F [2~ log 01(z1 —z2)F+ (4/k)(a~log 01(z1 —z2))
(0hz2))
+8a~log0 1(z1_z2)a~2F+4ka~2F~.
(3.4)
Let us check the behavior of the correlator (3.1) under (z1 —z2) —*0, as well as its monodromy under z1 —*z, + r. For (z1 —z2) —*0 it has the expansion (~P(z1)4,(Z2))im
—‘
(z~—z2) —2+2/k X/,m(T) +
k+2 k
2’~(2iir)aTX,~(T)
~
(3.5)
+...,
—z2)
where Xi,m(T) is the character of the h.w.r. i,,,,. This expansion is expected from the following o.p.e.:
=
(z
2T(z 1
+
_Z2)_2+2/k[l
k + 2 (z1 —z2)
and the fact that
+T,
2)
+
...]
(3.6)
using
2hITw2/kr,÷
F~m(0,W2)
.qi/ke
2(o,w2)
(3.7)
we get 2t~P(zi)~t(z +
T)~t(z2)>,,~
=
e_
2)>,,~÷2.
(3.8)
This is expected from the fusion rule (3.9) Since 0 ~ m <2k, the change in F,,.,, given above is correct as long as m + 2 is also in this range. However if m + 2 is outside this range, then one has to shift the lattice summation. For instance, if m = 2k 1 then in the lattice sum (3.2) one needs to write n2 + (m + 2)/2k = (n2 + 1) + 1/2k which means in turn that the cone n21 < In, I is shifted. The difference between the contribution of these two —
432
T. Jayaraman et a!.
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Parafermion models
cones is simply the contribution of the lines n1 = —n2 and n1 = n2 + 1 to the lattice sum (3.2). As we will discuss in appendix A, this difference, i.e., the contribution of these two lines is of the form F~(wi,w2)0i(
)
wi+W2 _w1+w2 2 +F~(wi,w2)0i( 2
where F~kw1,w2) satisfy k0~2F~-~ = ±(k+ 2)a~1F~~. However when this difference is plugged into eq. (3.1), its contribution vanishes and one has the shift invariance of the cone, and therefore the correct monodromy. Now let us consider the correlator of SU(2)k current, KJa(zi)Jb(z2))
=
~abG(z1
z2).
(3.10)
3(z) is We evaluate this by using eq. (2.5) for J’~(z).Since the expression for J simpler we first consider the correlator KJ3(z 3(z 3(z) = ~-ika4’ 1)J 2)). Using J 0(z) we simply get 3(z (J
3(z 1)J
28~F,(0,0,0), (3.11) 2))
=
—k/23~ logO(z, —z2)F,(0,0,0) +k
where F,(w,,w 2,w3)
1
=
~ ~
77(T)
+m/2k)w3]
m
As we discuss in appendix A, it is possible to bring this expression for F, to the more useful expression (A.6). Using eq. (A.6) and setting w1 = w2 = w3 = 0, we get 3(z (J
3(z 1)J
2))
=
_~k(3~log 61(z1 —z2) —4h~3~ log 77(T))x,(T) +
(2ii~/3)(k + 2) a~xi(r),
(3.12)
where x,(r) is the character of SU(2)k in the representation j 1
x,(r)
=
77(T)
=
1/2:
2~”~2(k+2)~. 1+1 ~
(2(k
+
2)n
+1+
(3.13)
1)q
It is easy to verify that this correlator has the correct (z 1 z2) 0 expansion and It is clearly invariant under z1 z1 + T, —
the correct monodromy under z1 —p z1 *
+
T*.
—‘
—~
In ref. [201this correlator was first written by these requirements, since they uniquely fix it.
T. Jayaraman etal.
which is expected. Under (z1
/ Parafermion models
433
z2) —o 0, since
—
1
3~log61(z1—z2)—4i~d~log71(T)=
+O(z1
2
—
z2),
(3.14)
(z, —z2) we get 3(z (J
3(z 1)J
k/2
1)>
2 +
(z1
2i~ —~---(k+ 2) 3~xi(T)
+
O(z1 —z2). (3.15)
—z2)
This is in accordance with the following o.p.e. 3k/2 EJa(z)J~~(z)_
a
(z1 —z2)
+(k+2)T(z2)+...
2
(3.16)
and the fact that KT(z)) = 2iir3~Xi(T). For the correlator (J~(z1)J(z2)) of course we expect to get the same expression as eq. (3.12). Let us see how this comes about. Using eq. (2.5) for J~(z)and J(z) we get 2a~ —k23~ KJ~(z1)J(z2))
=
(_(k+ 1)a~log 01(z1 2
— ~
—
(k+2)
—z2) +
2k3~ log 0,(z,
—
log 01(z1 —z2))
2
z 2)
a~2)F,(O~~2, w3)I~3~~2, (3.17)
where w2 = 2(z1 z2) and the limit w3 w2 should be taken. Now using eq. (A.6) for F1(w1, w2, w3) and taking the limit (w3 w2) —÷0 we get —
—~
—
=
(— (k + 1) a~log 01(z1 —z2) + (4iir/3)(k + 2) 3~+ 4i3r(k + 2) 3~log
77(T)
2—4i~-8~ log 0 x4i1Ta~f(zi—z2) (a~log 0,(z1 —z2)) 1(z1 —z2)f(z1 —z2) 2f(z 2g(z ~2 + 4~1—z2) 8ir 1 —z2))x,(r), (3.18) —
—
—
where
2i7n1z~
f(z)=
~(—1)’
~r(r± 1—I) e
(3.19)
r
q
r#0
~+r(r+ I)
g(z)=
~(_~)r1 r+0
2
(1—q)
(3.20)
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T. Jayaraman ci a!.
Comparing this expression with eq. (3.12) we get the following identity: 4i7r8~f(z)—(3~log01(z))2—4iira~log01(z)f(z)_~.2 —8~-2g(z)=a~log0
2f(z)
1(z) —8hr3~log77(T).
+4ir
We have checked that both sides of this identity are invariant under z they both have the expansion 1
1
—~
(3.21) z +T
and
0’~”(O)
From this itthe follows that thisfunction identity of holds. 0~(0) This +O(z2). identity also be we discuss correlation primary fields ofwill SU(2)k to useful simplifywhen the expression. We now consider the correlation function of primary fields. For simplicity we restrict ourselves to examples which involve only one screening operator. For this we consider the parafermion primary fields =
:exp[_ 2Vk+ 2
4’1(z)
±
which correspond_to I = 1, m = ±1 in eq. (2.9). The screening operator l/,(z) = 34’2(z)exp[i/Y~+ 24’1(z)] should be used to screen the charge. We therefore need the following expression: (a4’2(z)
=
(
O,(z,—z2) 0~(0)
where w2 <~i,
exP[-~~-4’2(zI)]exP[_
=
2(z1
—
)
1/2k
-4’2(z2)])
I
01(z,—z)
(_~0zlo~(0(zz))
+2~3w2
Fim(0,W2),
(3.22)
z2). Using this we then simply get
,(z1) cP1 ,_,(z2)), ~ k+1
1
V~
=
~
0,(z1
—z2)
0~(0)
[~~(
61(z, —z)
k(k+2)
01(z1 —z) 0,(z_z2))
~dz
—
oç(0)
2k3w]Fim(Wi~W2)~
k+2
01(z —z2)
k÷2
eç(O) (3.23)
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435
x
Fig. 1. The s-channel and t-channel contours ~
C~”,C~’~ and ~
where w1 = z1 + z2 2z and w2 = z2. The s-channel contours ~ and ~ and the t-channel ones C~’~ and C~J)are as in ref. [13] and shown in fig. 1. We have checked that this correlator has the correct monodromy and modular transformation properties and in the identity channel the limit (z1 z2) 0 reproduces the character. The check is basically the same as the one given for the minimal models in refs. [13, 16], except for the restriction of the range 0 ~ m <2k and 0 ~ I ~ k. Under Z1-~Z1+T we have —
—
—
2k(k+2) I
exp[_i~( k ~1 +2
—
—*
~)1Fl+I,m÷l(WI,w
2). (3.24)
F,~(w1,w2) -*q
First note that eq. (3.23) vanishes for 1 = 1 and 1 = k + 1. To see this consider 1 = —1 and replace F,,,, by f~,,, in eq. (3.23) where FIm(Wi, w2) is defined with the domain of summation appearing symmetric for n1 > 0 and n1 <0 (i.e., —n2< 1n11 for n2
—
—~
—
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transformation (3.24) is the expected one. For m m’ 2k
=
2k
—
1 since
m+1 2k =1
we need to shift n2 n2 1. We saw in the example of correlators of currents that this shift introduced terms proportional to 01((w1 + w2)/2) and 01((w1 w2)/2) which did not contribute. For the present example their contribution is a total derivative term into the integrand and therefore do not contribute to the correlator either. We conclude that as a result of this shift invariance of the cone we get the correct monodromy also for m = 0 and m = 2k 1. Moreover the transformation (3.24) is in accordance with the fusion rule [3, 10]: —~
—
—
—
min(1+l’,k—l—-l’)
(3.25) n=I’—l,I
The modular transformation property of eq. (3.23) can be easily obtained by using the modular transformation of F,~(W1,W2) derived appendix B. to the previous 2)k which is in closely related We now our example of SU( example of consider parafermion, namely the correlator of the primary field ~j, m with j = and m = ± Using eq. (2.10) for this primary field we get ~.
K~l/ 2I/2(zI)~I/2_I/2(Z2))
1 =
01(z1
z2)
—
0~(0)
~
01(z1
x azlo~(0~
01(z1 z) 0~(0)
2(k±2)
~dz
01(z z2) eç(0)
k+2
—
k+2
—
~)
—z)
—2ka~2F,(w1,w2,w3),
(3.26)
where w1 = z1 + z2 2z, w2 = z2 and the limit w3 w2 should be taken. In fact it is possible to write this correlator in a much simpler form in the following way. Using eq. (A.6) for F, we have —
k+2
~
F,(w1,w2,w3)
where
F,(w1)
—
~
i~
—~
dW
F,(w1)
+
[f(
w1+w2 2
) _f(
—w1+w2 2
/+1
=
~q(k+2xn+1±1/2(k+2))2exp[2i~(n
+
2(k
+
2)
-
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437
and f(z) was defined in eq. (3.19). Also we have W
2ka~F,(w1,w2,w3)
1+W2
~2k3~2[f(
2
)
W1 + W2
+2(k+2){f(
2
W1+W2
2
+ 2i~[f(
WI+W2
2
) )
~W1+W2
—f(
)
)1F1w1 -
2
—W1 + W2
+f(
_f(
2
~W1+W2
—
)ja~1rwi
—
2 ~WI+W2
—
)j~w~~
2
g(
where g(z) is as in eq. (3.20). Now upon using the identity (3.21) eq. (3.26) reduces to
-
01(z1 —z2) 0~(0)
-
2(k+2)
01(z—z2)
01(z1 O~(O)z)
~
X
k+2
—
~dz
log
Xô
01(z1—z)
-
F1(w1).
z
Again we have checked that this expression has the correct monodromy, modular transformation properties as well as the correct (z1 z2) 0 limit. Moreover it reproduces the correct four-point function on sphere upon degeneration q 0. For instance, for the two contours C’j’) and2”~ ~ and shown in fig. x 1, in the limit q —s 0 choosing after making a change of coordinate x = e 1 = 1 we get —
—*
—~
—
(1—x2)
~ —~-— 2(k+2)x~2F(
I
(1
_X2)
~
2F
2(k+2)X~
1
+
2j
1 k+2’
‘
1 k+2
,1
+
2
2j+2 ,2+ k+2
2 k+2
1 —x 2
which are the known expressions [21] for the two blocks in the t-channel when the external fields j1 = and 12 = at the points 1 and x2 are in the total 112 = 0 representation. These two blocks are respectively the identity block (j = 0) and J = 1 block. ~-
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4. Concluding remarks In this paper, we have shown how one can construct two-point functions for 7~k-parafermionand SU(2)k WZW models on the torus, and have shown that they satisfy the correct monodromy and modular transformation. This construction rests on the Coulomb-gas description of these models, and the truncation of extra states in the Fock modules of the free bosons that are not contained in the irreducible modules of the respective chiral algebras, via screening operators. We have argued that these irreducible modules can be identified with the certain cohomology associated with certain BRST operators, and have obtained the known characters. The basic assumption in this has been on the triviality of this cohomology in the Fock modules other than the central one. It would be interesting to prove this assumption directly using the properties of the screening operators. If one can do this then one may try to construct rational conformal field theories directly by specifying screening charges as in ref. [22]. In ref. [22] an approach to construct a finite-dimensional representation of modular and monodromy groups was given but an understanding of the positivity condition is still lacking. If one can prove the triviality of the cohomologies then one might be able to get conditions on the screening charges to ensure the positivity. One may also try to extend this analysis to genus-2 surfaces. There one expects an extra charge +2a 0 which can be balanced by introducing (k + 2)Q1 and one
Q ± each.
However an analysis of the monodromy and modular transformation of such multicontour problems is technically much more complicated. While completing this work, we received an interesting paper by Bernard and Felder [23] in which a detailed investigation of SU(2)k has been carried out in the formulation in terms of (/3, y) system and the boson 4’. Two other related works have also appeared, one by Nemeschansky [24] and the other by Distler and Qui [25]. Two of the screening operators introduced in ref. [24] are total derivative, and therefore one expects every state to be in the kernel of the Q’s constructed from them, i.e., they cannot give any non-trivial information. We do not understand the given in ref. [251since it contains any non-trivial information. QBRST
Q~k+2)and
one expects it also not to give
We would like to thank M. Caselle and M.A. Namazie for discussions.
Appendix A LATTICE SUMS FOR PF AND SU(2)k SYSTEMS
In this appendix we discuss some properties of the latticethe sums, F,~(w1, w2)the of 2)k. First consider lattice sum for the PF system and F,(w1, W2, w3) of SU(
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PF system: 1 Ti m( W
1, w2)
1+1
2(k+2))
+
—
k (fl2+
~)
n1,n2w7!/2
77(T) ~I
X
(~1)~~ sign(ni)q~+2)(f1
~
2
=
2
— ~I’~2l’~I >
1+1 exP[2i~(ni + 2(k + 1)
m
)~1— 2i~(n2+ ~)W2
(A.1)
.
As we saw in sect. 3, under z1 z1 + r we may get a situation in which the cone In11 > In21 is shifted. Thus let us see what is the change in F,~(w1,w2)when it is defined over the cone In1 I > In2 aI for some non-zero integer a. It is not difficult to see that the difference between the two is the sum over the points ~ for a ~ 1. It is clear from the expression for F,~(W1, w2) that the lines n1 = + /3 1 and n1 = —/3 should give respectively terms proportional to 01(w1 + W2/2) and 01(w1 w2/2). Explicitly, the difference —~
—
—
—
—
n~,n2EL/2,n1 ~ n1~In2—a~,—n1 >In2—a~
n1,n2~7L/2,n1
,,2e1
n,~In2I,n1>Ifl2~
is equal to ~
2
[F~+(Wi~W2)OI(
)
+F~)(wi,w2)0i(
2
)1~
where 2
1±1+m±2k(131)
F~4~(w 1,w2) = —(—1)
2
4(k+2)
(in±2k(/3—I))2 2k
(i±l+m±2k(/3—l))
m+2k(f3—1) 2k
1+1 +m+2k(f3—1) 4
(w 1+w2)
F~(w1,w2) = {F~with (1+1)
8
—
q
1+1 Xexp~,2i~-2(k+2)~
—
(1+1)2
—*
—(1+1) and
w1
—~
—W1}.
,
(A.2) (A.3)
In sect. 3, what was useful for our discussion of monodromy of the correlation function was the fact that 2k0~F,~~kw1,w2) = ±2(k + 2)a,,F~~(w1,w2). We
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made use of this when showing that the additional terms do not contribute. Note that for the character x,,~(r)=F,,~(0,0)since 0I(0)= 0 there are no additional terms and therefore it is independent of the shift of the cone. For what follows it will be more convenient to write the summation appearing symmetric for n1 > 0 and n1 <0, i.e., to have the lines, say, In1 I = —n2> 0 included but 1n11 =+n2 ~ 0 excluded. Denote this by F,~(w1,W2). Then the difference between F1~(w1,w2) and F,~(w1,w2) is the contribution of the points n1 =n2 which is
—
F,~(w1,W2)
=
F~(W1,W2)01((W1
—
W2)/2).
(A.4)
2)k: Now consider the lattice sum for SU( 1 F,(w 1,w2,w3)
m
~ ~F,~(w1,w2)qk(n3+~)
=
77(T)
m
2
exp 2i~ n3
m
+
—
2k
w3 (A.5)
It is possible to make a change of summation and rewrite this sum in a form which is simpler. The expression that we are aiming at is the following: 1+1 ~q(k+2)(n+2(k+2))
2
1+1 exP[2i~(n+ 2(k+2)
qi5r(r+1) )Wi]L(_1)T1_qr~2j~
1+1 x{exP~_2i~-((k+2)n+_~-_)~+2i7rr( 1+1 _exP[2i~((k+2)n+ ~)~+2i~r(
2
2
(A.6)
where ~ = (W1 w2)/k. To obtain this, consider f~,(w1,w2, w3) which is defined as in eq. (A.5) but with F,~(w1,w2)replacing F,~(w1,w2)on the right-hand side. Make the change of summation n3 n3 n2, then since the quadratic term in n2 and n3 changes to a linear term in n3, it is possible to do its summation. Moreover since the dependence on m now is only in the form n2 + m/2k we can also do the summation over m. The range of m is 0 ~ m 2k 2 for 1 ~ 27/ and 1 ~ m s~2k 1 for I E 27/ + 1. Define p = kn2 + (m v)/2 where i-’ = 0 for 1 E 27/ and = 1 for I E 27/ + 1, then summing over m we get —kn1
0 and —
—~
—
~
—
—
—
,-‘
—
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—
/ Parafermion
q~knix
2n3
=
441
1 for n1 <0. Now the summation over p can be explicitly done
—
qkfllX
(A.7)
lqx
~qxP=sign(ni)
where x
models
Collecting all the terms, we have the following sum
+ ~/T.
1+1
~
(
—
1)2fl3q(
2
2)(nI+ 2(k+2)) +kn~—2kn1n3+n3v
n,,n3EZ/2
n3EL
~i
1+1
xexP{2i~ (ni+ —[(1+1)
2(k+2))w1~1~~/2+n3w3]}
—(1+ 1),w1
Now we make the following change of summation variable: n r = 2n3. Obviously n, r E 7/. Then we get 2Xn+
~
00
~
=
n1
—
n3 and
2(k+2)~( — 1)T q~r(r+i±1+0+4n) 1 — qTe2~ 1+1
~q(k+
1+1
Wi+W 2
xexP{2i~(n+ 2(k+2))W1kn~~( —[(1+1)
—*
—(1+ 1),w1
—*
2
)+v~/2
} (A.8)
—w1].
Upon using 2 (qr
e2)2~~++~~V2
~ 2n+(i+v)/
=
1— q’~e2’~
1
1 qrPe2iI~P+
,,~0
-
1
qre2l~
—
it is easy to see that the difference between eq. (A.8) and (A.6) is a term of the form W 1+W2
F~(w1,w2)0i(
2
)
—W1+W2
+ F~(w1,w2)0i(
2
As in the case of PF, for x,(r) = F,(O, 0, 0) this additional term vanishes and as we have seen in the examples considered in sect. 3 this difference does not contribute to the correlation functions.
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Also note that the U(1)-valued character defined by x,(T,
is obtained from T,(w1, W2, well-known expression [8]
fl
by setting
W3)
—
x,(T,~)
where
0rs(~) is
(A.9)
Tr(qL0_~24 e20T~),
=
w1
=
w2
6,+lk+2(
=
0. Using eq. (A.6) we get the
—~)
0~(0) 2iire’~
0~(~)
=
(A.10)
defined as (A.11)
~
2 =
~qt(fl+S/2~I
In deriving eq. (A.10) we made use of the following identity: a’IUi\
lr(r+l)
00
.
1~./ =
q
—2i~e ~
1 _qre2~
This identity can be easily obtained by writing 6~(~) in product form and then doing a partial fraction. Appendix B MODULAR TRANSFORMATION
The main difficulty in the modular transformation of F,,,, is that the lattice sum is over a restricted domain. This was also the problem in showing the monodromy behaviour of F, however, as shown in appendix A and in sect. 3 the extra terms that appear upon shifting the domain either vanish (for zero screening charge case) or are total derivative (for simple screening contour case). We will use this shift invariance to embed F, in a three-dimensional lattice F, which essentially corresponds to correlators for N = 2 minimal series. Even though F,,,, is also a restricted lattice sum, the existence of a null vector in the lattice allows one to carry out the modular transformation explicitly, as described in ref. [16]. However, we will first describe the case W 1 = w2 = 0, i.e., the character x,,~(r). In this case one can directly study the modular transformation of x,,,,, upon using the following identity ,,,~
,~,
2T Ixl>IyI —f dx dy exp =
2T —
~(x2
2)
e2~
1YP2)
sign(x)
—y
sign(p 0,
1)qP~Pi, if IP1I> P21’ if P11
(B 1)
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443
where p1 *p2. Denote the positive light cone (p1> p21) by L~and the negative one (—p1> 1p21) by L_. Note that the integration in (B.1) is fL— fL~ However p1 7~(n and p2 for our problem are Vk + 2 (n1 + (1 + 1)/2(k + 2)) and ~,/ + 2 m/2k) with In1 I > 1n21. It is easy to see that p = (p1, p2) lies in (L~uLj/G0 where G0 is a discrete subgroup of the proper Lorentz group generated by the element k+1
=
g0
.Vk(k+2)
~k(k+2)
k+1
Now we can bring the integration domains L~and L_ to L+/G0 and L/G0 by applying various elements of G0 on (x, y) and reabsorbing it by applying it on p. The effect of applying G0 on p is to expand the domain of p from (L~uLJ/G0 to L~UL_. In fact, since the integral in eq. (B.1) vanishes p is outside 2. Thuswhen we have L~uL_, we can extend the domain of p to the entire R 77(T) Xi m(T)
y2)I~(
dx dy sign(x) exp{_ 2T~
~fL~uL
=
1)2fh e2’~P1~P2).
(B.2) Now the summation
E
(— 1)2fl1 ~
n1,n2EZ/2 ~I
~
can be replaced by
ex~[i~(x
~
—
n1)~(y~—n2)(1
—
e’~~Y~)
and we can do the integrals to get 2x,~(T) 77(T)
2T~k(k+2) 1
{[2(k2 ~
1+1 xexP[i~(nI k + 2
~2
_~)]si~n(ni)
m —
-
n 2~)j(1
—
e1
1n2))}
(B.3)
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Now write n1=2(k+2)ñ1+(l’+l) and n2=2kñ2+m’ where l’=O,...,2k+3 and m’ = 0,. ,2k 1. Note that because of the factor (1 e’~”2”1)), there is a non-zero contribution only if (1’ m’) E 27/. Thus . .
—
—
—
2X,~(T)
1
/
=
T\/k(k + 2)
77(T)
1’,m’ ~
1’—m’E2ZL
—n,>In 2~
2i~r
l’+l 2(k+2))
x{si~n(nl)exP _~((k+2)(nl+
m’
_k(n
(l+1)(I’+l)
2
exp[i~(
2+~)
k+2
2
mm’
-~)]}.(B.4)
To bring the range of 1’ to the standard one, we transform 1’ k + 2 + 1’ and m’ k + m’ mod 2k when 1’ ~ k + 2. Finally, making use of the transformation (1’ + 1) —o —(1 + 1), n1 —ñ1 and the fact that for 1’ = k + 1 the sum vanishes, we obtain —‘
—~
—~
x,,~(T)
=
1 Vk(k + 2)
k l~o
sin(~
m’=-k
(I+1)(l’+l) k+2
1 )e~mm~xim(_
_).
i—mw 21
(B.5) 1”277(T).
Here we have used Unfortunately this‘q(—1/T)=(—IT) technique does not work when
w
and w2 are not zero; the G0 symmetry, used to enlarge the domain of p to L~uL_ is only a symmetry of the spectrum and not of lattice states. Therefore we will proceed by embedding in a three-dimensional lattice corresponding to N = 2 minimal series whose modular transformation was worked out in ref. [16]. First for 0 <1 < k and —(k + 2)
=
1 —F,’~(T;w1,1~2)03 T 77(T)
where 03 is a Jacobi 0-function 13 e2t~t13z, 03(T;
z)
=
~
fl 3
q!’ 00
—W2+~_
k
,
(B.6)
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445
and
Fi’m(T;
Wi,
w2)
1
1+1 2
=
E
(
—
1)2n1
2
sign(ni)q~+2)((fh+ 2(k+2))
2 —
(n2+
2(k+2))
77(T)
1+1 xexP[2i~(nl+ 2(k+l))w12(n2+
Here w2 E 7/ and
(k
+
2)w2
—
m
2(k+2))W2
2~/k, and the summation is over n1, n2
E
m 1mw2 2(k+2) + 2kT2
1+1 1mw1 2(k+2) + 2(k+2)T2 ~
(B.7)
.
~-Zwith n1
where
—
gOt.
Note that the domains are shifted by (Im w1)/T2 and (Im w2)/T2 so that the lattice sums are absolutely convergent. Hence F,,,, is given by a three-dimensional lattice sum. By making the change of summation variables n3 = kñ3 + m’ 2n2, where m’=0,...,k— 1, and n2=ñ2—ñ3, the sum becomes —
1
k1 3
=
77(T)
(~
~
m’=O
1+1
1)~fl
sign(n1)
q(k+2)(ni+
2(k+2))
m±2m’
2
_k(h2+
n1,ñ2wl/2
2k
)
2
fl1 ~1~2E7L
~
x exP[2i~(nI X
+E~~fl/
1+1
+ 2(k +
1)
)~1 2i~(ñ2+ —
m+2m’ 2k
)~2]
0~+(k+2)~ k(k+2)/2(~k(k+ 2)~),
(B.8)
where 051(z) was defined in eq. (A.11). In obtaining this, we have used the fact that the n1, ñ2 sum is exactly the one appearing in the parafermion lattice sum Fi~(r;w1, w2) and consequently using the shift invariance of the domain to bring the latter into the standard form In1 + eI > In2 I. Thus 1 =
k—I
I \ ~ Flin+2m.(T;wI,W2)0m+(k+2)in~k(k+2)/2(~k(k 77~T, m’=O
+ 2)~T). (B.9)
F1~(r,w1,w2,~) is essentially the lattice part of the correlator for the N= 2
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minimal series in NS sector. In fact, by setting ~ = 0 (~ just keeps track of the U(1) quantum number) we obtain the expression for the lattice sum given in ref. [16], and whose modular transformation was also worked out. For the sake of completeness we briefly outline the steps involved in evaluating the modular transformation of F’(T; w1, w2). Because of the existence of lightlike vectors in the lattice, the sum over the direction parallel to light ray becomes linear and can be done explicitly.
Thus splitting the region into four parts 1+1 L~~:n1+ 2(k+2)
+
1+1 L~: n1+ 2(k+2)
+
1+1 L_~: n1+ 2(k+2)
+
1+1 n1+ 2(k+2)
+
L_:
Im(w1—~2)
m
and
n2+ 2(k+2)
and
m n2+ 2(k+2) <0,
<0
and
m n2+ 2(k+2)
Im(w1—~2) 2(k+2)T2 <0
and
m n2+ 2(k+2) <0,
~0
2(k+2)T2
Im(w1—~2) 2(k+2)T2
Im(w1—~2) 2(k+2)T2
the sum over ~ and L__ can be done by changing the summation variable n1 = n~+ n2. The quadratic term in n2 disappears and the sum over n2 just gives a denominator. Putting together the contribution from ~ and L__, one gets a sum over n~from to +c~. Similarly by changing n1 =n~—n2 in L~_and L~ and summing over n2 one gets another denominator with unrestricted sum over nç. The result is —~
F,’~(’r;w1, w2, x)
m
2
1+1
00
2
=q4(+2)exp[_2i~ in 2(~~2)w2j~q*2)(f+2(k+2)±x)
1
Xexp 2i1r1
~1 + 2
1+1
k
+
1
~X
w1
I+1
1
~
1+11
—
1
,
(B.10)
+q_(2)(~1~)m/2e_~~2+w1)
where we have introduced x to carry Out the Poisson summation. By making
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447
Fourier expansion, one gets an integral over x on a real line of the following form
00
1
dx
2
(k+2)y
1 ___________________________ 1 + q +2~m,/2 e~”~2~’~’/’
1
_____________________________
—
1
+ q2~’~”2
e’’2~”P
where y =x + (w 1 —p)/2(k + Now by the residue theorem
and p is an integer labelling the Fourier mode.
2)T
wi—p
f00 dx=f
(~~~\dx+2i~E (residues). ImI ‘,2(k+2)r
—00—j
But the latter integral is zero as the integral is odd under y
—‘
—y. Taking together
all the residues one gets
k+1
—
F/~(T;w1,w2)=
—
,
k+1
.
sin
,,
‘rr(l + 1)(l’ ,
+
mm’
1)
exp —i~r,
,,
I~+~I~_0th(k+2)
lc+~
1’ — rn in 2!
x ex~[_ 2T(k+ 2) (w~ -
~)]
T,~th(-1,
~).
~,
(B.11)
Using eq. (B.6), modular transformation of 03 and eq. (B.9), one gets
—
F,~(T;w1,w2,fl=exp
iTT
2
k1 k+1
w~
w~
k2k+k(k)
k+1
.
sin
X
th’=Oi’=Oth=—(k+2)
2~2
‘n-(I
l)(I’
+
,
,.,
+
+ 1)
/
—i’rrmth
expi !~
+
i’—thw2l
1
w1
—,
—,
W2
—)oth+(k+2)th. k(k+2)/2(~(k
+
‘. 1 2)—;— (B.12)
T. Jayaraman ci a!.
448
/ Parafcrmion models
One can re-express this by transforming 0 appearing in the right-hand side: W~
i7T
-
w~
~
k—i
k+1
‘n-(l+ 1)(l’
k+1
+
1)
sin +
th’=O 1=0 th= —(k+2)
I’ — th
in
2!
1
—i’n-mth Xexp
F,.
th+2th’
w~ w2
~
k-i-2
T
T
T
(th+(k+2)th’)(s+(k+2)s’) ~ exp —2i~-
k+1k1
x
.
s=0 s’=O
+
x0S+(k+2)S.k(k+2)/2(~k(k+ 2)~).
(B.13)
As F,,,, isdefinedfor mmod2k,let m + 2m’ =rmod2k and th +2th’=Fmod2k, one can sum over th’ and th fixing F. This sum vanishes for s * m mod(k + 2) and for s = m mod(k + 2) one gets a factor of (k2’~one + 2). Thus, gets comparing eq. (B.13) with eq. (B.9) and equating various powers of e F,,,,( ~ w i~w 2) 1 =
1~T
jk(k+2) k+l
ki
XEL51fl
exp
~
W~
W~
k+2
‘rr(l + 1)(l’ k+2
+
1)
1
iirmm’ exp
—
k+2
F,,,,
w~ w2 ~‘
(B.14) modulo irrelevant terms that give a total derivative with respect to the screening contour, when inserted in the full correlation function.
References [1] [2] [3] [4] 15]
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T Jayaraman ci a!.
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models
449
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