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Advanced Studies in Pure Mathematics 16, 1988 Conformal Field Theory and Solvable Lattice Models pp. 297-
Vertex Operators in Conformal Field Theory on p1. and
Monodromy Representations of Braid Group
Dedicated to Professor Hirosi Toda on his 60th birthday
Akihiro Tsuchiya and YukihiroKanie
Contents § 0. Introduction § 1. Affine Lie algebra of type A?> § 2. Vertex Operators (Primary Fields) § 3. Differential Equations of N-point Functions and Composability of Vertex Operators § 4. Commutation Relations of Vertex Operators § 5. Monodromy Representations of Braid Groups Appendix I. Bases of Tensor Products of .§[ 2 -modules Appendix II. Connection Matrix of Reduced Equation References
§:o. Introduction The 2-dimensional conformal field theory was initiated by A. A. Belavin, A.N. Polyakov and A.B. Zamolodchikov [BPZ] and was developed by many physicists, e.g. [DF], [ZF] etc. In the paper [BPZ], the signi- ficance of the primary fields for this theory is pointed out. V.G. Knizhnik and A.B. Zamolodchikov [KZ] developed the theory with current algebra symmetry, proposed the notion of primary fields with gauge symmetry, and gave the differential equations of multipoint correlation functions. Our aim in this paper is to give rigorous foundations to the work of [KZ], and to reformulate and develop the operator formalism in the con- formal field theory on the complex projective line JP1. The space ;If' of operands is taken to be a sum £= L,.1/!o£ 1 of the integrable highest
weight modules £ 1 of the affine Lie algebra g=.0[(2, C)®C[t, t- 1 ]EBCcof
type A1^1 >. We fix the value £ (positive integer) of the central element c of g on the space £. The Virasoro algebra 2' acts on each ;lf'J through
Received March 4, 1987.
298 A. Tsuchiya and Y. Kanie
the Sugawara forms L(m), m e z. For each Xe ~f(2, C), the field operator X(z)= I:mez X(m)z-m-i obeys the equations of motion:
[L(m), X(z)]=z"'(z! +m+l )x(z).
The currents X(z), Xe ~r(2, C) and the energy-momentum tensor T(z)= I:mez L(m)z-m- 2 preserve each g-module :lt'1• Thus each space :It', may be considered as a free theory. In order to introduce operators de- scribing the interactions in the theory, we define the vertex operators due to V.G. Knizhnik and A.B. Zamolodchikov [KZ]. The vertex operators play a central role in this paper. In Section 2, we show the existence and the uniqueness theorem of the vertex operators. In Section 3 we get the differential equations satisfied by N-point functions, which have only regular singularities. The properties of vertex operators are derived from these differential equations (called the fundamental equations). First, we get the convergence of compositions of vertex oper- ators. The commutation relation of vertex operators is equivalently re- phrased in terms of the connection matrix of the fundamental equations, and is calculated explicilty in a special case. The monodromies of the fundamental equations give rise to representations of the braid group B N· We determine explicitly this monodromy representation in a more special case. In fact, it gives an irreducible representation of the Hecke algebra HN(q) of type AN_1, where q=exp (2rr,l=T/(£+2)). Here it is remarkable that the vacuum expectation values of the products of vertex operators provide canonical bases of these representation spaces and the commu- tation relations of vertex operators give a 'factorization' of the monodromy representations.
Fix a positive integer £ for the value of the central element c, and a
half integer j with O-s,2j< £, then there is a unique (up to isomorphisms) irreducible highest weight left g-module :lt' 1 with a highest weight vector uJ(j). The Lie algebra g has a decomposition g=m+E9gE9CcE9m_, where g=~f(2, C)=CFE9CHE9CE and m±=g©C[t± 1 ]t± 1 (see Section 1.1) The subspace VJ={v e :lt'1 ; m+v=O} is an irreducible g-module of highest weight 2j, i.e. of dimension 2j+ l. We can define the corresponding irreducible highest weight right g (or g)-module :lt'j (or VJ) (and fix a highest weight vector u}(j)), and the
nondegenerate bHinear pairing (called vacuum expectation value) (I): :lt'j
X:lt'r~C such that (u}(j)lu 1 (j))=l and (valw)=(vlaw) for any v e
:It'}, a e g, we :It'!. Its restriction on VJX V 1 is also nondegenerate.
Let :It'= I:'l~o :lt' 1 and :lt't = I:3/~o :lt'j. By an operator we mean a linear mapping</): :lt'-.YP, where ff is a completion of :It'. Note that
300 A. Tsuchiya and Y. Kanie
Remark. i) The inequalities j 1+j 2 +j<t and lj1-j 2 i<j<j1+jz imply the conditionsj 1 ,j 2 ,j<£/2.
ii) Nonzero vertex operators of a fixed type v are unique up to a
constant multiple. For each £CG-vertex v= ( /.),we choose and fix a Jzli
nonzero element 'Pve Hom 6 (Vi8l VJi, VJ.)= Hom 6 (VJ.® VA9 VJi, C) (~ C)
and denote by ibv(z) the associated vertex operator of type v with the
initial term ibv,o='Pv•
For each £CG-vertex v= ( /.),introduce the g-module gJ(v) defined lzli by _gl](v)={ib/u; z); u e VJ}: Xibv(u; z)=ibv(Xu; z) (Xe g). We can show that any operators of the form X(C), Xe g, T(C) and vertex operators are composable. The composability of vertex operators is obtained by using the fact that the differential equations of N-point functions have only regular singular points. Introduce the space m(v) of operators on :If as the C-vector space spanned by the set
{ 1 f^ .. ·f^ d(N·.^ -dCi(CN-z)mN,.^ ·(C1-z)m'XN(CN)·^ .. (2,r./=-f)N CN 01
· · -X 1(( 1 )ib(u; z); Ne Z..,0, X, e g, mi e Z (I<i<N), u e v1},
where C/s are contours around Ct-i such that O is outside CN and z is
inside C1• Introduce a §-module structure and an 2-module structure in m(v) defined by
and
X(m)A(z)= bf dC(C-z)mX(C)A(z) e m(v) 2,r - } C
i(m)A(z) bf dC(C-z)m+IT(()A(z) e m(v) 2,r C
for A(z) e m(v), Xe g, me Z, and some contour C around z such that 0 is outside C.
Theorem2 (Theorem 2.9). For each £CG-vertex v, the g-module
mapping ib: V 1 3 u,,.ibv(u;z)_ e gJ(v) is extended to the §-isomorphism of :lf 1 onto m(v). Here we summarize the relations satisfied by vertex operators:
Conformal Field Theory on P^1
Fundamentalrelations for vertex operators Let <!)(z) be a vertex operator of spin j. Then
X(m)</)(u; z)=O X(O)<!)(u;z)=[X(O), <!)(u;z)]=<!)(Xu; z) i(m)<!)(u; z)=O i(O)<!)(u; z)=L1J<!)(u;z)
i(-l)<!)(u; z)=i<!)(u;_ z) az E(- 1y-^2 J+^1 <!)(u;(j); z)=O.
(m;?:l,Xeg,ue VJ); (XE g, u E VJ); (m>l,ue VJ); (u E VJ);
(ue VJ);
301
Remark that the last equation is derived from the structure of the irreducible g-module :YfJ by using Theorem 2. Now we call the vectors \vac)= uo(O)E £ 0 and <vac=u6(0) e xi the Virasoro vacuum. They satisfies the equalities
_X(m)_ vac) _=L(n)_ vac) = <vac\X(m)=<vac\L(n)=O
(XE g, m>O, n;?:-1); (XE g, ms 0, ns 1).
For an N-ple J =UN,. · ·,j (^) 1) of half integers with 0<2jis£, let
Vv(J)= V[N©·. -@v;:, and let v;(J) denote the invariant subspace of
Vv(J) under the diagonal g-action, where v; denotes the dual g-module of V1• Let <!)lzi) be a vertex operator of spin ji (l<isN), then the vacuum expectation value of the composed operator
is considered as a Vv(J)-valued, formal Laurent series on (zN, · · ·, z 1) and
is called an N-point function (of spin J): If <!)lzt) is of type Vt (1 <i<N), N • <,n WN^ (^ ZN )^ '' •'J/1,n^ (^ Z1 ))^ = (^) i=l n^ zi -4(v,)^ L.J'""'^ C^ m (^) N , .. m,ZN -mN^ '' ·Z1 -m, ,
where CmN· .. ,ni E Vv(J) and the sum is taken over integers mk E Z (1 <k <N) with mN>O and m 1 <0. Let ;rt be the g-action on the i-th component of Vv(J) and introduce the operator Qtk defined by
and !Ji=!Jtt is the action of the Casimir element !J=½HH+EF+FE on
Conformal Field Theory on P 1 303
· · ·, m (^) 1) E (Z 20 )N-i with I;k,,ei mk=Li= £-2ji+ I. Introduce the set &,lJ) defined by
&e(J)={JP=(PN, .. ·,Pi,Po);pi E _!_z>OVi=( ji ) E (CG)g, 2 - PiPi-
where (CG)e is the set of all £CG-vertices. For each JP e &e(J), the N- point function
of type lP is a formal Laurent series solution of the joint system E(J) and B(J), moreover
Theorem 4 (Theorem 3.3). i) For any JP e &,lJ), the Laurent series (JJ/zN, · · ·, z (^) 1) is absolutely convergent in the region ~.={(zN, · · ·, z (^) 1) E CN; \zN> · · · >\z1 } and is analytically continued to a multivalued holomorphic function on the mani- fold XN. ii) {{J)JzN, · · ·, z (^) 1); JPe &lJ)} gives a basis of the solution space of the joint system E(J) and B(J).
As a corollary of Theorem 4, we get
Theorem 5 (Theorem 3.4). Let {J)lz;) be the vertex operator of spin ji and ui E Vj, (1::;:i::;:N). Then the sequence {{J)AuN;zN), · · ·, {/}i(u1 ; z1)}
is composable in the region ~.,o={(zN, · · ·, z 1) E CN; \zN> · · · >\z 1\ >O}
and the composed operator (JJN(uN;zN)· · ,(/)i(u1 ; z (^) 1) is analytically continued to a multivalued holomorphic function on the manifold MN= {(z N, · · · ,z (^) 1) E XN; zi*O}. For £CG-vertices v 2 = (i3k)and v 1 = ( 7 /j)' the composed operator (/Jv 2 (w)(/Jv,(z)of the vertex operators (/Jv,(w)and (/Jv,(z)is multi-valued holo- morphic on the manifold M2 • For a quadruple J=(j 4 ,j 3 ,j 2 ,j (^) 1) of half integers with 0<2j 1 ::;:£, introduce the set /e(J) of intermediate edges, defined by
/.(J) = { k e 21 Z; o::;:2k< £, vlk) = ( !ak) e (CG)e, }
vi(k)=(f 1 ) E (CG)i}·
304 A. Tsuchiya and Y, Kanie
Let J =(j 4 ,j 2,j 3 ,j 1), then we get the g-isomorphism T: Vv(J)-Vv(J) defined by
( Tcp) ( U4 ®u2®ua{8)u 1 ) = cp(u 4 @ua®uli9u1)
for cp e Vv(J) and U 4 Q9UzQ9U 3 Q9U 1 e V(J). For an intermediate edge k e Il1), similarly define the .eCG-vertices vz(k)= (jzk) and vi(k)= (/' 1 ) and consider the composed operator
<b., 20 ,i(w)<b., 1 ,lii(z) of the vertex operators <b.,, 0 ,i(w) and <b.. 1 ,iii(z).
Assume that IiJ)*~- For a point (w, z) e 12 ={(z (^) 2, z1) e R 2 ; z 2 >
z 1 >0}, let <Dv,<ki(z)<b.. 1 ckiCw)denote the analytic continuation of the compo-
sition <bv,<ki(w)<b.. 1 ,k,(z) of the vertex operators along the path b(t), where the path b(t)=(7)(t), C(t)) from the point (w, z) e / 2 to the point (z, w) e
l 2 ={(z 2, z1) e R 2 ; z 1 >z 2 >0} on the manifold M 2 is defined by
7J^ (t) - ---^ w+z^ +^ e~J=ti --, w-z
C(t)= w+z -e~J=ti w~z (t e [O, 11).
Then
Proposition6 (Proposition 4.2). i) There exists a constant square
matrix C(J)=(C!(J))ker,<JJ,iier,(J) such that for each intermediate edge k e
IlJ),
ii) Let J =(t,j 3 ,j 2 ,j 1, s), then the braid relation holds:
C(js,jz,j1, s)C(t,ja,j1,jz)CU1,js,jz, s) = C(t,js,j2,j1)C(jz,js,ji, s)C(t,jz,j1,ja)•
Now our fundamental problem is:
FundamentalProblem. Determine the matrix C(J)=(q(J)) for any quadruple J with IiJ)*~-
In Section 4.2, we solve the fundamental problem for the case where j 3 =½ in J. For generalj (^) 3, we can solve it in principle by the fusion rule (see Section 5.4). Now we takej 2 =ja=½- Then the conditions for the nontriviality, V;;'(J)*O, are divided into the following cases:
(D2) 2
306 A. Tsuchiya and Y. Kanie
The braid group BN acts on this space W(N; t) as monodromies. The commutation relation of vertex operators gives a 'factorization' of this monodromy representation (tr1>1,i, W(N; t)). By the explicit formulae of the representation trN,t obtained from Proposition 7, we get
Theorem8 (Theorem 5.2 and Proposition 5.3). Let q=exp( 2 tr-f=1).
.e+
i) The monodromy representation q 814 trN,t of the braid group BN on the space W(N; t) gives an irreducible and unitarizable representation of the group B N· ii) This representation factors through a representation of the Hecke algebra HN(q) of type AN-i· iii) Our representation (q 814 trN,t•W(N; t)) of the Hecke algebra HAq) is equivalent to the representation (1d^2 •8^ +^2 l, Vi^2 •e+2l) constructed by H. Wenzl [W], where ..:l. is a Young diagram l=[N/2+t, N/2-t].
Notations
g=~r(2, C)=CFEBCHEBCE, where F=(? g), H=(6 -?) and E=
(g6) g= g®C [t, t- 1 ]EBCc: the affine Lie algebra of type A Pl g=CH(O)EBCc: the Cartan subalgebra of g X(n)=X®tn for Xe g and n e Z m±=g®t±C[t±], n+=m+EBCE(O), n_=m_Ef)CF(O), +>±=m±Ef)gEf)Cc: subalgebras of g £'=I; Ce,.+ Ce~: the Virasoro algebra nEZ f2=½H 2 +EF+FE e U(g): the Casimir element of g :X(m)Y(n): : the normal ordered product for X(m), Y(n) e g®C[t, 1- 1 ] X(z)= I; X(n)z-n- 1 (z EC*, Xe g): a current nEZ T(z)= I; L(m)z-m- 2 : the energy momentum tensor mEZ .e:the central charge (we fix .e e Z>othroughout the paper)
IC=.e+
VJ, VJ: the irreducible left and right g-modules of spin j for j e ½Z~ore- spectively v;=Hom(VJ, C): the dual (right) g-module of VJ
£ 1 =.l'fi.e), .l'fj=.l'fj(.e): the integrable highest weight left and right §-
modules respectively <I) : VJ X Vr-~C, £j X £ r-+C: the vacuum expectation values 8/2 8/2 A t/2 A 8/ £=I; JfJCf=I; JfJ; £t=I;£}c£t=I; .#J. j=O j=O j=O j=O
Conformal Field Theory on )P^1
V = {v = ( /.); j, j 1, j 2 e !_.z~0}: the set of vertices U lzli 2 Vc={v EV; j1, jz< 1}
(CG)={v EV; Vi- jz~j ~j1+ jz, j 1+ j 2+ j e .Z}: the set of all CG- vertices (CG)c={v e (CG);j 1 + j 2 + j~£}: the set of all £CG-vertices L1J = j2 + j : the conformal dimension of vertex operators of spin j IC
Ll( v) = L1 J: the conformal dimension of a vertex v
J(v)=L1J+L1h-L1Jz for a vertex v
'i"'(v)=Hom 0 (V],®VJ®Vh; C)
<pv e Homg(VJ®VJi, VJo)~'i"'(v): the nonzero element for each v= ( /.) Jz] e (CG) fixed in Appendix I (/)v(z): the Vertex Operator of type V Whose initial term (/)v,O is <pv for each
v= ( /.) e (CG)c (considered as VJ@£' 31 -+£\ 2 )
Jzli (J)(u;z)=W(z)(u®-)= ~ (J)n(u)z-n-J(v): the homogeneous decomposition nEZ of a vertex operator <P(z) of type v Let W = W 1 ® · · · ® WN the tensor product of g-modules Wk, then ;r 1 : the g-action on the i-th component of W L11" = n 1 + n": the diagonal action on the i-th and k-th components of w Qik = f1ri(H)1riH) + 1rJE)1riF) + 1rJF)1riE) J = (j N, • • ·, j (^) 1): an N-ple of half-integers with O~ 2_j 1 ~ £ V(J)= vjN®·. -@V31, v~(J)= v;:,@-. -@v;: v;(J): the space of all g-invariant elements in V~(])
Y'(J)={JP=(PN, .. ·,Pi,Po); V/JP)=( ji ) E (CG),PN=Po=o} P1P1- Y'c(J)={JP=(PN, · · ·, P1,Po) E Y'(J); vi(p) E (CG)c} J = (j (^) 4, j (^) 3, j (^) 2, j 1 ): a quadruple of half integers with O< 2j 1 ~ £
/(J)= {ke ~ .Z; 0<2k~£, vz(k)= (i3k)e (CG),
vi(k)=(/ 1 ) e (CG)}
Jc(J)= {k e ~ .Z; O~2k<£, vz(k) e (CG)c, vi(k) e (CG)i}
p(k)=(v (^) 3 , v 2 (k), vi(k), v (^) 0) E Y'c(J) for k e /c(J), where v 3 =(/ 4 j)
and Vo=(lo)
Conformal Field Theory on P^1
[v]q=-----=--(q:;f=^ q•^ I 1), 1,1 (q= 1): a q-integer (1,1e Z) q-
(;)
LI
---- : the multinomial coefficient for m = (mN• , , , , m 1 ) with mN!,,,m 1! L=I:m,.
§ 1. AffineLie Algebraof type Af!> In this section, we recall facts on the affine Lie algebra g of type Af^1 > (see V.G. Kac's book [Ka]).
1.1) Lie Algebra of type A 1 and its finite-dimensionalmodules Let g = ~((2, C) the Lie algebra of type Ai, that is, g is a Lie algebra spanned by H = (^) (1 0 _^ 0)1, E^ = (0 00 1) and^ F = (0 1 0)0. The subspace lj = CH is a Cartan subalgebra of g. Its dual lj* is spanned by the element a, de- fined by a(H)=2. Put Ba=CE and !:l-a=CF, then g has the root space decomposition
Let ( , ) : g X g-+C be the invariant symmetric bilinear form, defined by (X, Y)=tr XY, where tr means the trace as 2X2-matrices. Then (H, H)=2, (E, F)= I and (H, E)=(H, F)=O. The Casimir element Q of g is defined as
Here we summarize the facts on finite dimensional modules of g:
Proposition1.1. Fix a half integer j e ½Z:?;o· I) i) There exists a unique irreducibleleft g-module V 1 ( called of spin j) with highest weightja. ii) V 1 is of dimension 2j+I and has a basis {ui(m); m=j,j-I, · · ·,
1- j, - j} satisfying the relations
Huim)=2muim) Euim)= -v'U+m+ l)(j-m) uim+ 1) Fut<m)=-v'U+m)(j-m+ 1) utCm-1)
(- j<S,m<j); ( - j<S,m<i); (-j<m<j).
iii) Euij)=O, Puij):;f=O (O<n<2j) and F2^1 + 1 uij)=O. iv) t2=2U2+ j) on v,.
310 A. Tsuchiya and Y. Kanie
II) i) There exists a unique irreducible right g-module VJ ( called of spinj) with highest weight ja. ii) VJ is of dimension 2j+ l and has a basis {u](m); m=j, j-1, · · ·, 1- j, - j} satisfying the relations:
(VJ)
u}(m)H = 2mu}(m) u}(m)E= ,./(j+m)(j-m+ 1) u}(m-1) u}(m)F= ,./(j+m+ l)(j-m)u}(m+ 1)
(-j<m<j); (- j<m~j); (-j<m<j).
iii) u}(j)F=O, u}(j)E"=!=O(O~n<2j) and u}(j)E2i+ 1 =0. iv) D=2(j2+j) on VJ. III) There exists a unique bilinear form (called vacuum expectation value)
<I ): v;x vj~c
such that l) <ua Iv)= <u I av) for any a e g, <u I e VJ and Iv) e VJ, and 2) <u}(m)ju/m'))=om,m'· Moreover this bilinear form is nondegenerate.
1.2) The affine Lie algebra of type A?>
Let § be the affine Lie algebra of type A?>, that is, § is defined by
with the following wmmutation relations:
[X(m), Y(n)]=X, Y+(X, Y)mom+n,oc
and
c e center of §,
where X(n)=X®tn.
(X, Ye g, m, n e Z),
The Lie algebra g is included in § by identifying X with X(O). Intro- duce the subspace g(n) = g®tn of§ for any n e Z, and subalgebras m±= :Z:::n;,,i g(±n), then§ is decomposed into
§ =m+ EBgEBCcEBm_.
The subspace ~ = CH(O)EBCcis a Cartan subalgebra of g. The dual
~* of~ is identified with C 2 ::i (J, μ), by the formulae:
(A, μ)(c)=A and (l, μ)(H)=2μ.
Now we summarize the facts about the integrable highest weight modules of the Lie algebra §.
312 A. Tsuchiya and Y. Kanie
Definition1.4. i) For each Xe g, we define the formal Laurent series
X(z)= I; X(n)z-n- (^1) (z e C*). nez
ii) Energy-momentum tensor; Segal-Sugawara form ([Se] and [Su]))
For z e C*, define
T(z)
that is,
L(m)
(^1) {!_: H(z)H(z): +: E(z)F(z): +: F(z)E(z):} 2(2-t-c) 2 = I; L(m)z-m-z, mez
l I;{!_ ;H(-k)H(m+k):+:E(-k)F(m-t-k):+ 2(2+c) kez 2
+:F(-k)E(m+k):}.
Then we get
Proposition1.5. i) For any j e ½Z., 0 with 2j<£, the operator L(m), me Z, and L'(O) =(3.e/(2+£))id act on .?lfit) and .?lf1(£). ii) For any m, n e Z,
m^8 -m
[L(m), L(n)]=(m-n)L(m+n)+ (^) 12 Om+n,oL'(O).
iii) For each m e Z and X e g,
[L(m), X(z)]=zm(z ~ +m+I)X(z);
[L(m), X(n)]=-nX(m+n) (n e Z).
iv) The modules .?it'lt) and .?If}(£)have the eigenspace decompositions with respect to the operator L(O):
where .?lt'J,a(.e)and ,n1,a(£) are the eigenspaces ofthe eigenvalue L1,+d, and L1,=(P+ j)/(£+2). In particular, .Yt'.1, 0 (£)= Vi and .?lf}. 0 (£)= VJ. Moreover dim .?If,,a(£)= dim .?If}. a(£) <oo.
Conformal Field Theory on P 1 313
v) Jlt'ti.e)J_Jlt'J,a'(.e) unless d=d', and (I) is nondegenerate on .n"L,;( .e) X Jlt'J,i .e). vi) For any Xe g, m e Z and d>O,
and
Jlt'},i.e)X(m), Jlt'},a(.e)L(m) C Jlt'},a+m(.e).
In the following of this paper, we fix an integer .e>I, put tc=.e+2, and omit .e in the notations Jlt'i(.e), Jlt'J,i.e) etc. (Note that Vo=Jlt'o(O)=C.)
§ 2. Vertex Operators (Primary fields) Throughout this paper we fix the value .e (a positive integer) of the central element con the spaces Jlt' and Jlt't, and use the value tc=.e+2 for convenience.
2.1) Field operators Fix a half integer j with 0<2j<.e. Introduce the product topology to the products £1= na;;,o.n"J,d and £J= na;;,o.n"},a, then the vacuum
expectation (I): Jlt'}X .n"r-+C is uniquely extended to continuous bilinear
pairings (I): .n"}X£'r-+C and £}X.n"J---+C, and there is a topological linear isomorphism £J~Hom 0 (.n"1 ; C), where Hom 0 (Jlt'1 ; C) is equipped with the weak topology. The actions of the Lie algebra g on Jlt' 1 and Jlt'} can be extended to these completions. Consider the direct sums of these modules:
Denote by Il 1 be the projection to thej-th component:
Ilt: Jlt'~Jlt' 1, £~£i; J"c"f~Jlt'}, £J~£J,
then IIJ o Ilk=Ilk a IIJ and II; commutes with the action of g. An operator A on Jlt' means a linear mapping A: Jlt'~£, which is equivalent to give a bilinear map A: .;,t'tX Jlt'---+C, and also to give a linear mapping At: Jlt't_~yf,t by the condition that for any (vie .;,t'tand jw) e Jlt',
(vlAw)=(v]AI w)=(vA.) w).
In order to define compositions of operators, fix dual bases {lua,1), · · ·, iua,mc1)}of ~f! 0 Jlt' 1 ,a and {(va,1], · · ·, (ua,,nal} of ~f!o.n"J,a
Conformal Field Theory on P1 315
(V3) [L(m), (l.i(u; z)]=z"'{z ~ +(m+ l)Ll,}(l.i(u; z)
for Xe g, u e V1, me Z and z e C*, where the number Ll 1 =(P+i)/,c is
called the conformal dimension of the vertex operator (l.i(z) and (l.i(u; z): :;/f-+J'f is the operator defined by
(l.i(u;z)(w)=(l.i(z)(u®w) (u e V1, w e :;ff).
Remark. (V2) is the gauge condition for the field (l.i(z) and (V3) means the equations of motion. Introduce sets V and V, defined by
V ={v= ( / .);j,}1,iz e _!Z;;,;0} :::,V,={v= ( / .) e V;j1,i2<.!:}· JzJ1 2.. Jz]i 2
An element v of Vis called a vertex. For a vertex v= ( /.) e V, we Jzl call j 1 an incoming spin, j 2 an outgoing spin and j an outer spin, and set Ll(v)=Lli (=(P+i)/,c) and J(v)=Ll 1 +Llh-Ll (^) 1,.
v:
r
jz i For a vertex v= ( /.) e V,, a vertex operator (l.i(z) of spin j is called Jz] of type v, if (l.i(u;z)=Il 1 ,(l.i(u;z)Il 11 for any u e Vi. Then we get the following (the proof will be given in Section 2.3):
Proposition2.1.
i) Any vertex operator (l.iof type v ( e V,) has a Laurent series ex-
pansion
(l.i(u;z)= I: (l.i,,.(u)z-n-J(v) nez and (l.i,,.(u)satisfies
(n e Z),
that is,
(n e Z).
ii) Introduce a trilinear form <p: VJ.®V 1 ®Vii-+C defined by
316 A. Tsuchiya and .Y. Kanie
~(v, u, w)=(vld>o(u)lw)=(vl(/)(u; z)lw)z^4 <v^1 l,=o
then ~ is g-invariant:
(v E VJ,, w E VJ,),
~(vX, u, w)=~(v, Xu, w)+~(v, u, Xw) (XE g).
iii) A vertex operator (f) of type v is uniquely determined by the form ~ e Hom 0 (VJ,®VAsWJ,, C) de.fined in ii). We call~ the initial term of the vertex operator d>and sometimes denote d>=d>'P.
For each vertex v= ( /.) e V, introduce the space 't'"(v) defined by
lzl
't'"(v)=Hom 0 (VJ.®VJ®VJi, C)=::Hom 9 (VJ®VJi, VJ.).
It is well-known in the s/ 2 -theory that 't'"(v)=C or 0, and 't'"(v)=C, if and only if v satisfies the Clebsch-Gordan condition:
U1-j2l<j<j1+j2 and j1+j2+jeZ.
Call such vertex a CG-vertex and denote by (CG) the set of all CG vertices:
The following is the key lemma for the existence theorem of vertex operators:
Lemma 2.2. For a vertex v= ( /.) e (CG) n V,, take a nonzero ]2] element ~ e 't'"(v). Then the following conditions are equivalent. i) j+ j 1+ j2<t. ii) ~(v, E 1 -^2 Ji+ 1 u, uJ,U))=O for any v e VJ. and u e VJ.
iii) ~(u} 2 (j 2), P- 2 J•+ 1 u, w)=O for any u e VJ and we VJ,·
A vertex v = ( /.) e V, is called an £CG-vertex, if it satisfies one of 12] the conditions (called the £-constrained Clebsh-Gordan condition) in Lemma 2.2 denoted by (CG), the set of all £CG-vertices, i.e.
Remark2.2'. i) The inequalitiesjj 1 -j 2 J<j<j 1 +j 2 andj+j 1 '+j <£ imply the inequalities j,j 1 ,j 2<.ej'J. In particular, outer spins of
£CG-vertices are not greater than t/2.
ii) By the above rem~rk and the proof of Lemma 2.2, one of the
