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Two-point function on a torus for the SU(3) WZW theory
Volume 215, number 3
PHYSICS LETTERS B
22 December 1988
T W O - P O I N T F U N C T I O N ON A TORUS FOR T H E SU(3) W - Z - W THEORY Sudhakar PANDA and Sumathi RAO Institute of Physics, Sachivalaya Marg, Bhubaneswar
751005, India
Received 8 August 1988
We show that for k = 1 SU (3) W - Z - W theory, there are three current blocks for the two-point function on a torus. We obtain a third-order differential equation for the correlator and solve the equation in terms of the 0-function for each current block. The solutions at Z~ ~ Z2 are shown to be consistent with the operator product expansion.
There has been considerable interest in two-dimensional conformal field theories [ 1 ] ever since the string revival [2 ], since these theories are the classical solutions of the string equations of motion. However, these theories have also been studied in their own right and in particular, a class of conformally invariant field theories called the Wess-Zumino-Witten ( W - Z - W ) models have been extensively investigated and shown to be exactly solvable on a plane [ 3-5 ]. More recently the thrust has been to study these theories on higher genus surfaces and the SU (2) W - Z - W model has been solved on Riemann surfaces of genus g>_- 1 [ 6]. It is of considerable interest to extend this work to other models. In this letter, we attempt to solve the SU (3) W - Z - W model on higher genus Riemann surfaces. More specifically, we compute the two-point function of the primary fields of the k = 1 SU (3) W - Z - W model on a torus. After defining the model and our conventions, we review the derivation of the primary fields of the SU (3) W Z - W model, limited by the level k. The fusion rules for the primary fields allow three current blocks for the twopoint function which can be expressed as a sum of the squares of the moduli of these current blocks. We derive the differential equation satisfied by these current blocks. The differential equation is of third order which is consistent since there are three current blocks. Finally, we obtain the general solutions of the differential equation and show that at Z~ ~ Z2, the solutions agree with the operator product expansion (OPE). The SU (3) W - Z - W model [ 3 ] is defined by the following symmetry algebra:
[j~,rt, O,n J1 = l J.¢'ab~lc ~ n + m +½knfiab~n+,,,,o, [Ln,Ja,] [L~, Lm] = (n-m)L,,,+, + ~c(n3-n)d,+m,o,
=
-mJ'~+ . . . .
(1,2) (3)
where n, m are integers and a, b, c take values from 1 to 8. The value of k in a representation is the level of the representation. Furthermore, the Sugawara relation connects the current algebra generators (ja) to the Virasaro generators ( L , ) as follows: L.
=
( C v + k ) - l ' d. an _ m utma
" ,
(4)
where Cv is defined by f abcf dbc= C~fiaa. When f ~b~are the structure constants of the SU (3) algebra, Cv = 3. From eq. ( 1 ), we see that the currents J~ form an SU (3) algebra. The action of these currents on any state [q~ ) is defined by J~ Iq ~ ) = ( 2 a ) ~ [q~a),
(5)
where the matrices 2 ~ satisfy [2", 2 b] = - i f ab~2~ 530
(6) 0370-2693/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
Volume 215, number 3
PHYSICS LETTERS B
22 December 1988
for consistency. For states in the fundamental representation, we shall choose the 2 a as follows: 21=
25=
(0 (Z ii) (00!i) (i000)(i --½
0
0
0
0
0
-}i
0
,
22=
0
--i
,
23=
0
,
26=
0
1 ,
27=
-½
0
½
0
0
0
-½i
,
24=
(i°!) 0
,-~
i , 28=
,
0
0
0
- 1/2x/'3 0
.
1/
(7) Any field ~ in the theory is labelled by its representation 2 under SU (3) and by its 2 3 and 2 s eigenvalues, i.e., we define any state as I~i 3'~8). It was shown in ref. [ 5 ] that not all the representations of the current algebra are allowed as primary fields and in fact the allowed primary fields are limited by the level k. We adapt their argument specifically to the S U ( 3 ) W - Z - W model. From the symmetry algebra defined in eq. (1), we can identify S U ( 2 ) subgroups. Jl - i J ~ , J~-i + l •J _2l , J3-½k form an I-pseudospin subgroup; J~-iJ~, j4_ 1 +iJ5_l, ~Jo 3 8 + ~Jo i 3 - ½k form a Vpseudospin subgroup and Jl6 -1J1 . 7 , J-16 + i J 7- 1 , ~Jo 3 8 - ~Jo 1 3 - ½k form a U-pseudospin subgroup. Let us for definiteness and convenience, choose the I-pseudospin subgroup to perform our further analysis, though the same results can also be derived using either of the other two subgroups as well. Consider the transformation of a primary field ~ ~ 3.; 8) hw that transforms as the highest weight vector (along the/-spin direction ) o f SU ( 3 ) under /-pseudospin. Since it is a primary field (Jl - i j 2 ) I ~
)3''~8)h~ ) = 0 ,
(8)
which implies that ~;~';~"~h ~ is the lowest weight vector of/-pseudospin. Furthermore, (J03 - - ½ k ) [ 0 ~ . )'3'')'8) .... > = ( 2 3 -
½ k ) ] 0 2 (23''~'8)h'w')
(9)
since 2 3 is diagonal. Hence, we get the condition that (23)h.w. -- ½k~<0 •
(10)
for the representation 2 to be a primary field. For k = 1, only 1, 3 and 3 satisfy this criterion. For k = 2, 1, 3, 3, 6, 6 and 8 satisfy the criterion and are allowed as primary fields. Eq. (9) tells us that the multiplicity of the representation 2 is given by 2 3 - ½k. Hence, we can define a null vector as
(Jl_l + iJ2_l )2(-~3--k/2 )+1 I ~(23,28) .... > = 0 .
( 11 )
Any correlator containing the null vector also vanishes and this property is used in the three-point function to derive the fusion rules [ 1,6 ]. For k = 1, with 1, 3 and 3 as the primary fields, the fusion rules are straightforward: 1 × 3 = 3 , 1 × 3 = 3 , 1 × 1 = 1, 3 X 3 = 3 and 3 X 3 = 1. Using these fusion rules, the possible diagrams for a twopoint function on a torus with external fields transforming as a 3 and a 3 are given in figs. 1 and 2 for two possible channels. Each of the diagrams shows the propagation of a definite primary field (and all its secondaries) along the intermediate states. The two-point function corresponding to this is called a current block. The current blocks defined through fig. 1 are periodic (upto a phase) under Z - Z + 1, whereas those defined through fig. 2 are periodic (upto a phase) under ZI~Zz [ 6 ]. In general, the current blocks in fig. 2 will be linear combinations o f the current blocks in fig. 1. Note that in either channel the number of current blocks is three. N o w let us derive the differential equation satisfied by these current blocks. Since there are three of them, we expect a third-order differential equation. Consider two fields transforming as a 3 and a 3 of SU (3) and define their correlator as 531
Volume 215, number 3
PHYSICS LETTERS B
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Fig. 2. Current blocks which are periodic under Z I ~ - + Z 2
Fig. I. independent current blocks which are periodic under
.
Z~Z+ 1. ( q . ( Z , ) ~ ( Z 2 ) ) = 8.aN( (Z~ - Z 2 ) ,
(12)
where N(Z~ - Z 2 ) is the function to be determined. We also d e f i n e f ( Z ~ - Z2 ) by (ja,
~o~(Z ' ) ~ , 8 ( Z 2 ) ) =
(2a)o~#f(Z,
--Z2) •
(13)
To get a relation betweenf(Z~ - Z 2 ) and N(Z~ - Z 2 ) [6], we multiply both sides ofeq. (13) by (,a.a).,.. Then using the Sugawara relation for n = - 1 ,~ L - l O O ~ -_2 ! j a - I j ~07-'.,
(14)
and 2 a 2 b = gIa
ab
- ~ 1i f"
abc
2 c - ~ 1d
abc
2c ,
(15)
as well as dahb= 0, we get
f ( Z , - Z 2 ) = 3~0Z,N ( Z l - Z2) -= 3ON .
(16)
(We suppress the argument Z~ with respect to which the derivative is taken and the argument Z l - Z2 of the functions, henceforth. ) Next, we define (J~_ , J ~ , ~ . ( Z l )da(Z2) ) = 6aba.aF, (Z~ - Z 2 ) + ifab~(2').eF2(Zl - Z 2 ) +da"c(,a.~),~aF3(Z~ - Z 2 ) • This again is analogous to the SU (2) case except that there is a third function F3 to be determined since we can now have the symmetric structure constants d ~bc. To determine F~ in terms of N, we multiply both sides of eq. ( 17 ) b y f ~,a. The RHS is simply 3iF2. To compute the LHS, we use -1
--1 =~1J-2,3" d
(jd2~oe(Zl)~,8(Z2))=
d~
(~_Zl)
2
(Jd(~)~)oe(Zl)~fl(Z2))
.
(18,19)
Ja(~)f~.(Z~ )~p(Z2)) has poles at ¢=Z~ and ¢=Z2 with residues (2a).j~V and -(2a),~gN. Such a function is fully determined on a torus except for an additive constant, which again is determined (J~_~ q.(Z~ ) ~a(Z2) ) = (Z=),~ 30N. We get
'
) ~fl(Z2 ) ) ~__[ S ( ~ _ Z
1) _ S ( ~ _ g 2 )
..]_S(Z1
_Z2)
]x(2a),~N+ 3 (2a),~ 0N
using (20)
where S ( z - a ) = O:ln 0~ ( z - a ) with 0~ =01/2,~/2 as defined in ref. [7]. Hence, we get
F2 = ½go(Zt - Z 2 ) N , where 8a ( Z t - Z2 ) is the Weierstrass ga-function on the torus defined as ~a = a 2 - 0S where 1
S(Z)= ~ + 532
X
n = even
a~Z"-~ .
(21)
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To get another relation between the F ~, we multiply both sides of eq. ( 17 ) by (2 b),,~. Then we use eqs. ( 14 ), (2), (19) and (20) to get the L H S = 3 0 2 N - 2 f a N . On the RHS, we use dabb=o and dabcdabq=SO cq, to get El + 3F2 - 5 gF3. Hence, we obtain the equation Fl + 3 F 2 - ~ F 3 =302N - faN.
(22)
Finally the null vector condition, eq. (11 ) is implemented. For ~ in the 3 representation O~/2'-~/2"/g is the highest weight vector under I-spin and it is easily checked that 2 3 - ½=0. Therefore, (JL, +iJL,)10~/2,- l/2-/'f) = 0 ,
(23)
which implies that ( (j~_~ _ i j 2 1 ) (jl_l + ijz_ t
)01/2,--I/2xf~--1/2.1/2.~)
=0.
(24)
Using eq. ( 17 ), we get 1 1 Fi - ~F2 - gF3 =0.
(25)
From eqs. (21), (22) and (25), we can solve for
F~ = faN-~c32N,
F 3 = - 9 0 2 U + 9 faU.
(26,27)
However, unlike in the SU (2) case here we do not have a sufficient number of conditions at this stage to get a differential equation for N. To get more conditions, we insert one more current operator in the correlator and define
( J ~ J ~ J C , O , ~ ( Z ~ ) O p ( Z 2 ) )=~,~pifabcMt(Zj -Z2)+~o~#dabcM2(Z,-Z2) + {(~abt~qcM3( Zl __Z2 ) dff(~acf~qbM4( ZI _ Z2 ) + t~bct~qaM5( Zl
_
_
Z2 ) .~-dabmdmCqM6( ZI - Z2 )
+ d .... d"bqMT( Zx - Z 2 ) + idaemf "cqMs( Zt - Z 2 ) + idac'f mbqMg( Z, - Z 2 ) + idaqmf mbCMlo(Zl - Z 2 ) } (2q).p • (28) Now we need various conditions relating the Mi to each other and to the functions F, and N. The same tricks are used as in the previous case, except that it gets algebraically more complicated. Consider the correlator
(ja__ l j b , jc_ IJoOot(Zi ) ~p(Z2 ) ) = (~c)otcv ( J a l jb_ t jc_ 10o~'( Z I ) ~p(Z2 ) ) =6 ,,b¢~a#(si3a +~M4q_~Ms+ SM7)+ifabq(aq)ap(M, _ ½ M 4 + ½ M s _ 5 M T _ S M 9
5 N , o)
+dabq()~q)a#(M2 - ½M4 - ~M5 1 - gM6 5 + 1M7 + 3Ms - 3M9 + 3M, o ) .
(29)
The RHS is simplified using the following relations [ 8 ]
famnfbnqfcqm=3-2fabc ' famndbnqfcqm=
3dabc d,~,,,,,fb,,qdcq,,=_5_rf,,b~' damndbnqdcqm_=__½dabc
The LHS can be evaluated as follows. Eqs. (2), (14) and ( 1 ) are used to get LHS = 2 ( L _ l J ~_,J~,fb.(Z, )~a(Zz) ) -2(JL2Jb_2fb,~(Z, ) qtp(Z2 ) )
--2(Jb_2J~_2fb,~(Z, )~a(Z2) ) - 2 i f ~ b q ( J % ¢ . ( Z , )#a(Z2) ) .
(30)
The first term can easily be written in terms of derivatives ofF,. and the fourth term can be evaluated from eqs. (19) and (20) as ½(2q).~(0 fa )N. To evaluate the second and third terms we note that
(j_2j_,Oa(Z1)~B(Z2))a b =
( ~ - d{ -Z,) 2 (Ja({)Jb_,oe,(Z~)~(Z2)) .
(31)
Hence, we need to determine the singularities of (ja({)jb_ ~,~(Z~ )~a(Z2) ) as a function of{. Now 533
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-boo ( Ja(¢)Jb-l~)o~(Zl)(~.a(Z2) }= ~ (~--Z,)--n--'( JaJb-,~)a(Z,)~p(Z2) > 1
1
- ~ - Z , (JSJ~'-' O,~(Z, )~p(Z2) ) + ( ¢ - Z l ) 2 ( J , J - ~ ¢ . ( Z , )q~a(Z2) ) +regular terms. _
_
a
b
(32)
The residue at the single pole at ZI is determined from the residue at Z2 and the Ward identity [4,5 ] as
R, = 3 ( 2 b2a)~eON=aabac,p( ION) + i f abq( 2 q)afl( __ ]ON) + dabq( 2 q)afl( 3ON).
(33)
The residue at the double pole at Z~ is determined as follows:
R2 = (j{jb_ 1~o~(Z1)~,8(Z2) ) =ifabm(j,~(oo,(Z, ) Cp(Z2) ) + ~kcJab((ga(Zl)~7#(Z2)> = if abq(2 q).~N+ ½kd"bcJapN.
( 34 )
Hence, the correlator ( J~ (~) jb_ 1¢ . ( Z~ ) ~p (Z2) ) is fully determined on the torus but for a constant which again is determined using the known correlator ( ja_ ~je_ ~~,~ ( Z~ ) ~p(Z2 ) ). We get
( ja( ¢)jb_, Ca(Z, )~fl(Z2) ) =
[S(¢--Z, ) - 8 ( ¢ - Z 2 ) + S ( Z 1- Z 2 ) ]R, + go (Z, -Z2)R2 +6~hao,aF, + i f ~bq(2q),pF2 + d~bq(2q),aF3 • (35)
Using eqs. (29), (30), ( 31 ) and ( 35 ) and the values for F, in terms of N, we get
4M3 + {M4 + ~gMs+~MT=-303N+2OgoN+ga~N,
(36)
MI - ½M4 + ½Ms - T~2M7-- ~2M9 - ~2Mlo = go ON,
(37)
M2 - ½M4 - ½Ms - -~M6 + ~M7 + 3M8 - 3M9 + ]M,o = - g ~ 3N+ 9~ goN + 12 go ON.
(38)
Another set of three equations can be derived by considering the correlator ja_ ~jb_ ~JL ~Jboq)~,(Z~ )~p(Z2) ). Using eq. ( 1 ) and other correlators we have already evaluated, we get
4M4++M3+ "gM5 ' 7 " + 5 M 6 =-- 303N+ a~goN+-~-ga~N,
(39)
- M , - ½M3 + ½Ms - ~2M6 - ~2M8 + ~ M , o = go 0W- 3 (0~a)W,
(40)
M ~ - ½ M ~ - ½ M ~ + ¼ M ~ - ~ M ~ - ~ M ~ + ~ M 9 - aM, o = - 9~3N+ 12go ~N+-~goN.
(41)
As an obvious extension, a further set of three equations can be derived by considering the correlator ( j ~ ~j b 1JL 1J~ ~ (Zt) Cp((Z2) ), but the equations turn out to be redundant. Hence, we do not include them. A further set of three equations is obtained by multiplying eq. (28) by f aba and using eqs. (18), (31 ) and (35 ), we get
M,=~goON, M4-Ms+ SM7=-9 goON, M9+M,o=-~goON.
(42,43,44)
More conditions are obtained by multiplying eq. (28) by f ~d and f b~aas well. Of the six equations, three are obviously redundant and the remaining three are
M3-Ma+~MT-~M8=-9go~N+~(Ogo)N,
M s - M g + 2 M , o=-~goON,
Ms-M,o=0.
(45,46,47)
Finally, we use the null vector condition to get two more equations because there are two ways in which the null vector can be used. Firstly,
(j3_, (j~_~ _ i j2_' ) ( j ~ , +i j2_, ){2}1/2,-- i/2x/~-- 1/2,-- 1/2x/~> = 0 , which implies that 534
(48)
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22 December 1988
M5 - 2 M ~ + -~Mlo = 0 .
(49)
The null vector condition can also be implemented as < (JLl -iJZ_l)J3_l (Jl_l + i J2_ l )~1/2,- l/2,fig~- l/2,1/2,/T) = 0 ,
(50)
which yields 2Ml +M4 + ~1M 7 -- -~Mlo = 0 .
(51)
The entire set of eq. ( 3 6 ) - ( 5 1 ) is self-consistent and can be solved for the M~ in terms of N which in turn satisfies a differential equation. The solutions for the M~ are
MI=-M4--~oON,
Ms=M9=M~o=-Ms=-3~ON,
9 3 7 M3=--gON+~goON+~OgaN,
M 6 = 2 7 ~.3.M ..
M7=-~ON,
-33.^ ~,ON-9a~N,
(52,53,54)
M2 = - ~O3N+ ~ 9 ON+ ~O~ N, (55,56,57)
and the differential equation satisfied by N is O 3 N - - g4 go 3 N -
~goN= 0.
(58)
This is the differential equation satisfied by the current blocks and as expected it is a third-order differential equation with three solutions for N(Z~ - Z2 ). The differential equation (58) can be solved very easily at Z~ ~ Z2. If we assume a solution of the form N(Z) ~ Z u, we get /t(#-l)(/~-2)
- ~4/ ~ t-~5 =6 o , t~
(59)
which has the s o l u t i o n / l = - - ~ , 4 and 7. Hence at Z~ ~Z2, the solutions are of the form N(Z~-Z2) ~ ( Z ~ Z2 ) - 2/3, (ZI - Z2 ) 4/3 and (Zt - Z2 ) 7/3 We can check that these solutions agree with the OPE. The OPE gives 03(Zl )¢~(Z2) ~ I( Z, -Z2)-a3-a'-4-Ja_lja_,I(
Z , --Z2)2-Js-J'+
f a & ' j a _ , j b _ , J c _ , I ( Z l -- m2)3-d3-d3dl - ....
(60) Since A3= A~ = C3, ~/ ( Cv + k) = ~ where 2 a2 a = C3. s I, the leading behaviour at Z~ ~ Zz is of the form (Z1 Z: ) -2/3, (Z~ - Z 2 )4/3 and (Z~ - Z 2 ) -7/3, in agreement with our explicit solutions. This is an independent check on the correctness of our differential equation. In fact, in ref. [ 9 ], this argument has been turned around and the OPE has been used to derive the differential equation for this theory as well as other rational conformal field theories. Finally, the differential equation may also be solved, exactly. We start with an ansatz which agrees with the OPE and has the right periodicity properties. The ansatz can be justified [ 10] by noting that this theory is equivalent to a theory of two free bosons [ 11 ]. We explicitly show that the ansatz satisfies the differential equation. The ansatz for the solutions is the following: NI (Zt - Z 2 ) =f(Zl - Z 2 ) =- 0 1 / 3 ° ( ( z l --Z2)13z)0_ ~/3o( (Z, - Z 2 ) [ 3"c)
[0, ((z, - z 2 ) 13) ] 2/3
--A,
Na(Z, -Z2) =f(Z, -Z2 + 0 =--A, N3(Z, -Z2) =f(Z, - Z : - r ) -=A,
(61)
where the Oabare defined in ref. [ 7 ]. Using the solutions in the differential equation, we get
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0 3 N - 4 go O N - ~280 g o N f (f_'," _ 2 f 7 -- 02/3 IkY f
0',
01
2 f ' i 0'l' f 0t
20]' 1 0 f ~ 0'~2 10 0'~0]' 301 -~ 3 f 02 + 3 0,2
800'~ 3 27 03
4 f'i 8 3 go~-+9go
0] 01
28 0go) " 27 (62)
Now, let us c o n s i d e r the p r o p e r t i e s o f the f u n c t i o n w i t h i n the large brackets. Firstly, it is p e r i o d i c in Z ~ Z + 1 a n d Z--, Z + z. N a i v e l y 0 i-' a n d ga have poles at Z = O, r a n d 2z; 01/3o - , has a pole at ~5 z + ~ ~ a n d 0 s 11/30 has a pole at iz+ ' L However, we can explicitly check that the function within the large brackets has no poles at any of ~. these points and consequently no poles anywhere. Finally, we can showthat the function vanishes at Z= 0. This is sufficient to prove that the function vanishes everywhere and hence the solutions in eq. (61 ) satisfy the differential equation. In conclusion, we state the main results obtained in this letter. We showed that the number of current blocks for the two-point correlator of the k= I W-Z-W theory on a torus is three and obtained a third-order differential equation for the current blocks usingthe current algebra null vector. The equations were solvedand the solutions were found to be the formf / [ 0 , ]2/3 where t h e f are defined in eq. (61) and were also shown to be consistent with the OPE. Finally, the same method can also be used to computethe four-pointcorrelator on a torus for the same theory. Extensions to the k= 2 theory as well as to the computation of the correlators of the k= I theory on higher genus surfaces are also possible and work is in progress along these lines [ 12]. We w o u l d like to t h a n k Dr. A s h o k e Sen for several useful discussions a n d D i l e e p J a t k a r for bringing ref. [ 8 ] to o u r a t t e n t i o n . O n e o f the a u t h o r s ( S . P . ) w o u l d also like to a c k n o w l e d g e the D e p a r t m e n t o f Science a n d Technology, G o v e r n m e n t o f I n d i a for an a s s o c i a t e s h i p u n d e r the research grant No. 24 ( 3 P - 5 ) / 8 4 - S T P - I I .
References [ 1 ] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [ 2 ] M.B. Green and J.H. Schwarz, Phys. Lett. B 149 (1984) 117. [3] E. Witten, Commun. Math. Phys. 92 (1984) 451. [ 4 ] V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B 247 (1984) 83. [ 5 ] D. Gepner and E. Witten, Nucl. Phys. B 278 (1986) 493. [6] S.D. Mathur, S. Mukhi and A. Sen, Nucl. Phys. B 305 [FS23] (1988 ) 219. [ 7 ] D. Mumford, Tata Lectures on Theta, I, II (Birhh~iuser, Basel, 1983 ). [8] P. Dittner, Commun. Math. Phys. 22 (1971 ) 238. [9] S.D. Mathur, S. Mukhi and A. Sen, TIFR preprint TIFR/TH/88-32. [ 10] A. Sen and S. Mukhi, private communication. [ 11 ] P. Goddard and D. Olive, Intern. J. Mod. Phys. A 1 (1986) 303. [ 12 ] S. Panda and S. Rao, in preparation.
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PHYSICS LETTERS B
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T W O - P O I N T F U N C T I O N ON A TORUS FOR T H E SU(3) W - Z - W THEORY Sudhakar PANDA and Sumathi RAO Institute of Physics, Sachivalaya Marg, Bhubaneswar
751005, India
Received 8 August 1988
We show that for k = 1 SU (3) W - Z - W theory, there are three current blocks for the two-point function on a torus. We obtain a third-order differential equation for the correlator and solve the equation in terms of the 0-function for each current block. The solutions at Z~ ~ Z2 are shown to be consistent with the operator product expansion.
There has been considerable interest in two-dimensional conformal field theories [ 1 ] ever since the string revival [2 ], since these theories are the classical solutions of the string equations of motion. However, these theories have also been studied in their own right and in particular, a class of conformally invariant field theories called the Wess-Zumino-Witten ( W - Z - W ) models have been extensively investigated and shown to be exactly solvable on a plane [ 3-5 ]. More recently the thrust has been to study these theories on higher genus surfaces and the SU (2) W - Z - W model has been solved on Riemann surfaces of genus g>_- 1 [ 6]. It is of considerable interest to extend this work to other models. In this letter, we attempt to solve the SU (3) W - Z - W model on higher genus Riemann surfaces. More specifically, we compute the two-point function of the primary fields of the k = 1 SU (3) W - Z - W model on a torus. After defining the model and our conventions, we review the derivation of the primary fields of the SU (3) W Z - W model, limited by the level k. The fusion rules for the primary fields allow three current blocks for the twopoint function which can be expressed as a sum of the squares of the moduli of these current blocks. We derive the differential equation satisfied by these current blocks. The differential equation is of third order which is consistent since there are three current blocks. Finally, we obtain the general solutions of the differential equation and show that at Z~ ~ Z2, the solutions agree with the operator product expansion (OPE). The SU (3) W - Z - W model [ 3 ] is defined by the following symmetry algebra:
[j~,rt, O,n J1 = l J.¢'ab~lc ~ n + m +½knfiab~n+,,,,o, [Ln,Ja,] [L~, Lm] = (n-m)L,,,+, + ~c(n3-n)d,+m,o,
=
-mJ'~+ . . . .
(1,2) (3)
where n, m are integers and a, b, c take values from 1 to 8. The value of k in a representation is the level of the representation. Furthermore, the Sugawara relation connects the current algebra generators (ja) to the Virasaro generators ( L , ) as follows: L.
=
( C v + k ) - l ' d. an _ m utma
" ,
(4)
where Cv is defined by f abcf dbc= C~fiaa. When f ~b~are the structure constants of the SU (3) algebra, Cv = 3. From eq. ( 1 ), we see that the currents J~ form an SU (3) algebra. The action of these currents on any state [q~ ) is defined by J~ Iq ~ ) = ( 2 a ) ~ [q~a),
(5)
where the matrices 2 ~ satisfy [2", 2 b] = - i f ab~2~ 530
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for consistency. For states in the fundamental representation, we shall choose the 2 a as follows: 21=
25=
(0 (Z ii) (00!i) (i000)(i --½
0
0
0
0
0
-}i
0
,
22=
0
--i
,
23=
0
,
26=
0
1 ,
27=
-½
0
½
0
0
0
-½i
,
24=
(i°!) 0
,-~
i , 28=
,
0
0
0
- 1/2x/'3 0
.
1/
(7) Any field ~ in the theory is labelled by its representation 2 under SU (3) and by its 2 3 and 2 s eigenvalues, i.e., we define any state as I~i 3'~8). It was shown in ref. [ 5 ] that not all the representations of the current algebra are allowed as primary fields and in fact the allowed primary fields are limited by the level k. We adapt their argument specifically to the S U ( 3 ) W - Z - W model. From the symmetry algebra defined in eq. (1), we can identify S U ( 2 ) subgroups. Jl - i J ~ , J~-i + l •J _2l , J3-½k form an I-pseudospin subgroup; J~-iJ~, j4_ 1 +iJ5_l, ~Jo 3 8 + ~Jo i 3 - ½k form a Vpseudospin subgroup and Jl6 -1J1 . 7 , J-16 + i J 7- 1 , ~Jo 3 8 - ~Jo 1 3 - ½k form a U-pseudospin subgroup. Let us for definiteness and convenience, choose the I-pseudospin subgroup to perform our further analysis, though the same results can also be derived using either of the other two subgroups as well. Consider the transformation of a primary field ~ ~ 3.; 8) hw that transforms as the highest weight vector (along the/-spin direction ) o f SU ( 3 ) under /-pseudospin. Since it is a primary field (Jl - i j 2 ) I ~
)3''~8)h~ ) = 0 ,
(8)
which implies that ~;~';~"~h ~ is the lowest weight vector of/-pseudospin. Furthermore, (J03 - - ½ k ) [ 0 ~ . )'3'')'8) .... > = ( 2 3 -
½ k ) ] 0 2 (23''~'8)h'w')
(9)
since 2 3 is diagonal. Hence, we get the condition that (23)h.w. -- ½k~<0 •
(10)
for the representation 2 to be a primary field. For k = 1, only 1, 3 and 3 satisfy this criterion. For k = 2, 1, 3, 3, 6, 6 and 8 satisfy the criterion and are allowed as primary fields. Eq. (9) tells us that the multiplicity of the representation 2 is given by 2 3 - ½k. Hence, we can define a null vector as
(Jl_l + iJ2_l )2(-~3--k/2 )+1 I ~(23,28) .... > = 0 .
( 11 )
Any correlator containing the null vector also vanishes and this property is used in the three-point function to derive the fusion rules [ 1,6 ]. For k = 1, with 1, 3 and 3 as the primary fields, the fusion rules are straightforward: 1 × 3 = 3 , 1 × 3 = 3 , 1 × 1 = 1, 3 X 3 = 3 and 3 X 3 = 1. Using these fusion rules, the possible diagrams for a twopoint function on a torus with external fields transforming as a 3 and a 3 are given in figs. 1 and 2 for two possible channels. Each of the diagrams shows the propagation of a definite primary field (and all its secondaries) along the intermediate states. The two-point function corresponding to this is called a current block. The current blocks defined through fig. 1 are periodic (upto a phase) under Z - Z + 1, whereas those defined through fig. 2 are periodic (upto a phase) under ZI~Zz [ 6 ]. In general, the current blocks in fig. 2 will be linear combinations o f the current blocks in fig. 1. Note that in either channel the number of current blocks is three. N o w let us derive the differential equation satisfied by these current blocks. Since there are three of them, we expect a third-order differential equation. Consider two fields transforming as a 3 and a 3 of SU (3) and define their correlator as 531
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Fig. 2. Current blocks which are periodic under Z I ~ - + Z 2
Fig. I. independent current blocks which are periodic under
.
Z~Z+ 1. ( q . ( Z , ) ~ ( Z 2 ) ) = 8.aN( (Z~ - Z 2 ) ,
(12)
where N(Z~ - Z 2 ) is the function to be determined. We also d e f i n e f ( Z ~ - Z2 ) by (ja,
~o~(Z ' ) ~ , 8 ( Z 2 ) ) =
(2a)o~#f(Z,
--Z2) •
(13)
To get a relation betweenf(Z~ - Z 2 ) and N(Z~ - Z 2 ) [6], we multiply both sides ofeq. (13) by (,a.a).,.. Then using the Sugawara relation for n = - 1 ,~ L - l O O ~ -_2 ! j a - I j ~07-'.,
(14)
and 2 a 2 b = gIa
ab
- ~ 1i f"
abc
2 c - ~ 1d
abc
2c ,
(15)
as well as dahb= 0, we get
f ( Z , - Z 2 ) = 3~0Z,N ( Z l - Z2) -= 3ON .
(16)
(We suppress the argument Z~ with respect to which the derivative is taken and the argument Z l - Z2 of the functions, henceforth. ) Next, we define (J~_ , J ~ , ~ . ( Z l )da(Z2) ) = 6aba.aF, (Z~ - Z 2 ) + ifab~(2').eF2(Zl - Z 2 ) +da"c(,a.~),~aF3(Z~ - Z 2 ) • This again is analogous to the SU (2) case except that there is a third function F3 to be determined since we can now have the symmetric structure constants d ~bc. To determine F~ in terms of N, we multiply both sides of eq. ( 17 ) b y f ~,a. The RHS is simply 3iF2. To compute the LHS, we use -1
--1 =~1J-2,3" d
(jd2~oe(Zl)~,8(Z2))=
d~
(~_Zl)
2
(Jd(~)~)oe(Zl)~fl(Z2))
.
(18,19)
Ja(~)f~.(Z~ )~p(Z2)) has poles at ¢=Z~ and ¢=Z2 with residues (2a).j~V and -(2a),~gN. Such a function is fully determined on a torus except for an additive constant, which again is determined (J~_~ q.(Z~ ) ~a(Z2) ) = (Z=),~ 30N. We get
'
) ~fl(Z2 ) ) ~__[ S ( ~ _ Z
1) _ S ( ~ _ g 2 )
..]_S(Z1
_Z2)
]x(2a),~N+ 3 (2a),~ 0N
using (20)
where S ( z - a ) = O:ln 0~ ( z - a ) with 0~ =01/2,~/2 as defined in ref. [7]. Hence, we get
F2 = ½go(Zt - Z 2 ) N , where 8a ( Z t - Z2 ) is the Weierstrass ga-function on the torus defined as ~a = a 2 - 0S where 1
S(Z)= ~ + 532
X
n = even
a~Z"-~ .
(21)
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To get another relation between the F ~, we multiply both sides of eq. ( 17 ) by (2 b),,~. Then we use eqs. ( 14 ), (2), (19) and (20) to get the L H S = 3 0 2 N - 2 f a N . On the RHS, we use dabb=o and dabcdabq=SO cq, to get El + 3F2 - 5 gF3. Hence, we obtain the equation Fl + 3 F 2 - ~ F 3 =302N - faN.
(22)
Finally the null vector condition, eq. (11 ) is implemented. For ~ in the 3 representation O~/2'-~/2"/g is the highest weight vector under I-spin and it is easily checked that 2 3 - ½=0. Therefore, (JL, +iJL,)10~/2,- l/2-/'f) = 0 ,
(23)
which implies that ( (j~_~ _ i j 2 1 ) (jl_l + ijz_ t
)01/2,--I/2xf~--1/2.1/2.~)
=0.
(24)
Using eq. ( 17 ), we get 1 1 Fi - ~F2 - gF3 =0.
(25)
From eqs. (21), (22) and (25), we can solve for
F~ = faN-~c32N,
F 3 = - 9 0 2 U + 9 faU.
(26,27)
However, unlike in the SU (2) case here we do not have a sufficient number of conditions at this stage to get a differential equation for N. To get more conditions, we insert one more current operator in the correlator and define
( J ~ J ~ J C , O , ~ ( Z ~ ) O p ( Z 2 ) )=~,~pifabcMt(Zj -Z2)+~o~#dabcM2(Z,-Z2) + {(~abt~qcM3( Zl __Z2 ) dff(~acf~qbM4( ZI _ Z2 ) + t~bct~qaM5( Zl
_
_
Z2 ) .~-dabmdmCqM6( ZI - Z2 )
+ d .... d"bqMT( Zx - Z 2 ) + idaemf "cqMs( Zt - Z 2 ) + idac'f mbqMg( Z, - Z 2 ) + idaqmf mbCMlo(Zl - Z 2 ) } (2q).p • (28) Now we need various conditions relating the Mi to each other and to the functions F, and N. The same tricks are used as in the previous case, except that it gets algebraically more complicated. Consider the correlator
(ja__ l j b , jc_ IJoOot(Zi ) ~p(Z2 ) ) = (~c)otcv ( J a l jb_ t jc_ 10o~'( Z I ) ~p(Z2 ) ) =6 ,,b¢~a#(si3a +~M4q_~Ms+ SM7)+ifabq(aq)ap(M, _ ½ M 4 + ½ M s _ 5 M T _ S M 9
5 N , o)
+dabq()~q)a#(M2 - ½M4 - ~M5 1 - gM6 5 + 1M7 + 3Ms - 3M9 + 3M, o ) .
(29)
The RHS is simplified using the following relations [ 8 ]
famnfbnqfcqm=3-2fabc ' famndbnqfcqm=
3dabc d,~,,,,,fb,,qdcq,,=_5_rf,,b~' damndbnqdcqm_=__½dabc
The LHS can be evaluated as follows. Eqs. (2), (14) and ( 1 ) are used to get LHS = 2 ( L _ l J ~_,J~,fb.(Z, )~a(Zz) ) -2(JL2Jb_2fb,~(Z, ) qtp(Z2 ) )
--2(Jb_2J~_2fb,~(Z, )~a(Z2) ) - 2 i f ~ b q ( J % ¢ . ( Z , )#a(Z2) ) .
(30)
The first term can easily be written in terms of derivatives ofF,. and the fourth term can be evaluated from eqs. (19) and (20) as ½(2q).~(0 fa )N. To evaluate the second and third terms we note that
(j_2j_,Oa(Z1)~B(Z2))a b =
( ~ - d{ -Z,) 2 (Ja({)Jb_,oe,(Z~)~(Z2)) .
(31)
Hence, we need to determine the singularities of (ja({)jb_ ~,~(Z~ )~a(Z2) ) as a function of{. Now 533
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22 December 1988
-boo ( Ja(¢)Jb-l~)o~(Zl)(~.a(Z2) }= ~ (~--Z,)--n--'( JaJb-,~)a(Z,)~p(Z2) > 1
1
- ~ - Z , (JSJ~'-' O,~(Z, )~p(Z2) ) + ( ¢ - Z l ) 2 ( J , J - ~ ¢ . ( Z , )q~a(Z2) ) +regular terms. _
_
a
b
(32)
The residue at the single pole at ZI is determined from the residue at Z2 and the Ward identity [4,5 ] as
R, = 3 ( 2 b2a)~eON=aabac,p( ION) + i f abq( 2 q)afl( __ ]ON) + dabq( 2 q)afl( 3ON).
(33)
The residue at the double pole at Z~ is determined as follows:
R2 = (j{jb_ 1~o~(Z1)~,8(Z2) ) =ifabm(j,~(oo,(Z, ) Cp(Z2) ) + ~kcJab((ga(Zl)~7#(Z2)> = if abq(2 q).~N+ ½kd"bcJapN.
( 34 )
Hence, the correlator ( J~ (~) jb_ 1¢ . ( Z~ ) ~p (Z2) ) is fully determined on the torus but for a constant which again is determined using the known correlator ( ja_ ~je_ ~~,~ ( Z~ ) ~p(Z2 ) ). We get
( ja( ¢)jb_, Ca(Z, )~fl(Z2) ) =
[S(¢--Z, ) - 8 ( ¢ - Z 2 ) + S ( Z 1- Z 2 ) ]R, + go (Z, -Z2)R2 +6~hao,aF, + i f ~bq(2q),pF2 + d~bq(2q),aF3 • (35)
Using eqs. (29), (30), ( 31 ) and ( 35 ) and the values for F, in terms of N, we get
4M3 + {M4 + ~gMs+~MT=-303N+2OgoN+ga~N,
(36)
MI - ½M4 + ½Ms - T~2M7-- ~2M9 - ~2Mlo = go ON,
(37)
M2 - ½M4 - ½Ms - -~M6 + ~M7 + 3M8 - 3M9 + ]M,o = - g ~ 3N+ 9~ goN + 12 go ON.
(38)
Another set of three equations can be derived by considering the correlator ja_ ~jb_ ~JL ~Jboq)~,(Z~ )~p(Z2) ). Using eq. ( 1 ) and other correlators we have already evaluated, we get
4M4++M3+ "gM5 ' 7 " + 5 M 6 =-- 303N+ a~goN+-~-ga~N,
(39)
- M , - ½M3 + ½Ms - ~2M6 - ~2M8 + ~ M , o = go 0W- 3 (0~a)W,
(40)
M ~ - ½ M ~ - ½ M ~ + ¼ M ~ - ~ M ~ - ~ M ~ + ~ M 9 - aM, o = - 9~3N+ 12go ~N+-~goN.
(41)
As an obvious extension, a further set of three equations can be derived by considering the correlator ( j ~ ~j b 1JL 1J~ ~ (Zt) Cp((Z2) ), but the equations turn out to be redundant. Hence, we do not include them. A further set of three equations is obtained by multiplying eq. (28) by f aba and using eqs. (18), (31 ) and (35 ), we get
M,=~goON, M4-Ms+ SM7=-9 goON, M9+M,o=-~goON.
(42,43,44)
More conditions are obtained by multiplying eq. (28) by f ~d and f b~aas well. Of the six equations, three are obviously redundant and the remaining three are
M3-Ma+~MT-~M8=-9go~N+~(Ogo)N,
M s - M g + 2 M , o=-~goON,
Ms-M,o=0.
(45,46,47)
Finally, we use the null vector condition to get two more equations because there are two ways in which the null vector can be used. Firstly,
(j3_, (j~_~ _ i j2_' ) ( j ~ , +i j2_, ){2}1/2,-- i/2x/~-- 1/2,-- 1/2x/~> = 0 , which implies that 534
(48)
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PHYSICS LETTERSB
22 December 1988
M5 - 2 M ~ + -~Mlo = 0 .
(49)
The null vector condition can also be implemented as < (JLl -iJZ_l)J3_l (Jl_l + i J2_ l )~1/2,- l/2,fig~- l/2,1/2,/T) = 0 ,
(50)
which yields 2Ml +M4 + ~1M 7 -- -~Mlo = 0 .
(51)
The entire set of eq. ( 3 6 ) - ( 5 1 ) is self-consistent and can be solved for the M~ in terms of N which in turn satisfies a differential equation. The solutions for the M~ are
MI=-M4--~oON,
Ms=M9=M~o=-Ms=-3~ON,
9 3 7 M3=--gON+~goON+~OgaN,
M 6 = 2 7 ~.3.M ..
M7=-~ON,
-33.^ ~,ON-9a~N,
(52,53,54)
M2 = - ~O3N+ ~ 9 ON+ ~O~ N, (55,56,57)
and the differential equation satisfied by N is O 3 N - - g4 go 3 N -
~goN= 0.
(58)
This is the differential equation satisfied by the current blocks and as expected it is a third-order differential equation with three solutions for N(Z~ - Z2 ). The differential equation (58) can be solved very easily at Z~ ~ Z2. If we assume a solution of the form N(Z) ~ Z u, we get /t(#-l)(/~-2)
- ~4/ ~ t-~5 =6 o , t~
(59)
which has the s o l u t i o n / l = - - ~ , 4 and 7. Hence at Z~ ~Z2, the solutions are of the form N(Z~-Z2) ~ ( Z ~ Z2 ) - 2/3, (ZI - Z2 ) 4/3 and (Zt - Z2 ) 7/3 We can check that these solutions agree with the OPE. The OPE gives 03(Zl )¢~(Z2) ~ I( Z, -Z2)-a3-a'-4-Ja_lja_,I(
Z , --Z2)2-Js-J'+
f a & ' j a _ , j b _ , J c _ , I ( Z l -- m2)3-d3-d3dl - ....
(60) Since A3= A~ = C3, ~/ ( Cv + k) = ~ where 2 a2 a = C3. s I, the leading behaviour at Z~ ~ Zz is of the form (Z1 Z: ) -2/3, (Z~ - Z 2 )4/3 and (Z~ - Z 2 ) -7/3, in agreement with our explicit solutions. This is an independent check on the correctness of our differential equation. In fact, in ref. [ 9 ], this argument has been turned around and the OPE has been used to derive the differential equation for this theory as well as other rational conformal field theories. Finally, the differential equation may also be solved, exactly. We start with an ansatz which agrees with the OPE and has the right periodicity properties. The ansatz can be justified [ 10] by noting that this theory is equivalent to a theory of two free bosons [ 11 ]. We explicitly show that the ansatz satisfies the differential equation. The ansatz for the solutions is the following: NI (Zt - Z 2 ) =f(Zl - Z 2 ) =- 0 1 / 3 ° ( ( z l --Z2)13z)0_ ~/3o( (Z, - Z 2 ) [ 3"c)
[0, ((z, - z 2 ) 13) ] 2/3
--A,
Na(Z, -Z2) =f(Z, -Z2 + 0 =--A, N3(Z, -Z2) =f(Z, - Z : - r ) -=A,
(61)
where the Oabare defined in ref. [ 7 ]. Using the solutions in the differential equation, we get
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Volume 215, number 3
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0 3 N - 4 go O N - ~280 g o N f (f_'," _ 2 f 7 -- 02/3 IkY f
0',
01
2 f ' i 0'l' f 0t
20]' 1 0 f ~ 0'~2 10 0'~0]' 301 -~ 3 f 02 + 3 0,2
800'~ 3 27 03
4 f'i 8 3 go~-+9go
0] 01
28 0go) " 27 (62)
Now, let us c o n s i d e r the p r o p e r t i e s o f the f u n c t i o n w i t h i n the large brackets. Firstly, it is p e r i o d i c in Z ~ Z + 1 a n d Z--, Z + z. N a i v e l y 0 i-' a n d ga have poles at Z = O, r a n d 2z; 01/3o - , has a pole at ~5 z + ~ ~ a n d 0 s 11/30 has a pole at iz+ ' L However, we can explicitly check that the function within the large brackets has no poles at any of ~. these points and consequently no poles anywhere. Finally, we can showthat the function vanishes at Z= 0. This is sufficient to prove that the function vanishes everywhere and hence the solutions in eq. (61 ) satisfy the differential equation. In conclusion, we state the main results obtained in this letter. We showed that the number of current blocks for the two-point correlator of the k= I W-Z-W theory on a torus is three and obtained a third-order differential equation for the current blocks usingthe current algebra null vector. The equations were solvedand the solutions were found to be the formf / [ 0 , ]2/3 where t h e f are defined in eq. (61) and were also shown to be consistent with the OPE. Finally, the same method can also be used to computethe four-pointcorrelator on a torus for the same theory. Extensions to the k= 2 theory as well as to the computation of the correlators of the k= I theory on higher genus surfaces are also possible and work is in progress along these lines [ 12]. We w o u l d like to t h a n k Dr. A s h o k e Sen for several useful discussions a n d D i l e e p J a t k a r for bringing ref. [ 8 ] to o u r a t t e n t i o n . O n e o f the a u t h o r s ( S . P . ) w o u l d also like to a c k n o w l e d g e the D e p a r t m e n t o f Science a n d Technology, G o v e r n m e n t o f I n d i a for an a s s o c i a t e s h i p u n d e r the research grant No. 24 ( 3 P - 5 ) / 8 4 - S T P - I I .
References [ 1 ] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Nucl. Phys. B 241 (1984) 333. [ 2 ] M.B. Green and J.H. Schwarz, Phys. Lett. B 149 (1984) 117. [3] E. Witten, Commun. Math. Phys. 92 (1984) 451. [ 4 ] V.G. Knizhnik and A.B. Zamolodchikov, Nucl. Phys. B 247 (1984) 83. [ 5 ] D. Gepner and E. Witten, Nucl. Phys. B 278 (1986) 493. [6] S.D. Mathur, S. Mukhi and A. Sen, Nucl. Phys. B 305 [FS23] (1988 ) 219. [ 7 ] D. Mumford, Tata Lectures on Theta, I, II (Birhh~iuser, Basel, 1983 ). [8] P. Dittner, Commun. Math. Phys. 22 (1971 ) 238. [9] S.D. Mathur, S. Mukhi and A. Sen, TIFR preprint TIFR/TH/88-32. [ 10] A. Sen and S. Mukhi, private communication. [ 11 ] P. Goddard and D. Olive, Intern. J. Mod. Phys. A 1 (1986) 303. [ 12 ] S. Panda and S. Rao, in preparation.
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