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2D gravity-matter couplings from quantum group
Nuclear Physics B (Proc. Suppl.) 25A (1992) 122-136 North-Holland
PROCEEDINGS SUPPLEMENTS
2D GRAVITY-MATTER COUPLINGS F R O M Q U A N T U M G R O U P Jean-Loup GERVAIS Laborateire de Physique Th~oriq~e de I'Ecole Normale S~pErieure 24, fie Lhomond, 75231 PARIS CEDEX 05, FRANCE
The quantum group structure of two-dimensional gravity a~ld minimal models is reviewed. The coupling of the latter with the former is discussed, and the three-point functions on the sphere am shown to agree with the results of matrix models. The corresponding co~rno.'*ogicalconsta~lt is defined in the present operator-approach. It does not correspond to a shift of the interacting Liouville-field.
1. Introduction After the early works[l-7], the recent developments of the operator approach to conformal theories took place when its relationship to quantum groups was recognized[8-11,13-15]. In these notes we sun'.marize t.~ ~ main points of this connection, and show how it may be used practically to compute corlelation functions[15]. We will see that, as ill matrix models, coapling gravity with matter leads to remarkable simplifications. 1 b begin with, it is convenient to give a short introduction to q u a n t u m groups per se ( W h a t follows next is a quick s u m m a r y without any attempt at generality or mathematical rigour). We shall take the case which will be directly useful later on, that is, of the q u a n t u m deformation of sl(2). Introduce a deformation parameter h, and associate with ally given variable x the quantity LxJ ~ s i n ( h x ) / s i n h. For simplicity, we shall assume that h / ~ is not, a rational number s~) that, for ally integer N, [NJ vanishes iff N = 0. Introduce three generators J~., and J3, which satisfy [ j + , j _ ] = sin(2hJa) sin/,
-
L2JaJ,
[Ja, J~-] = =l=J±.
(1)
This is the deformation of the Lie algebra sl(2) denoted by Ue(sl(2)). Tile standard Lie algebra is recovered in the limit h ~ 0. Tile novel feature of the deformed commutation relation is t h a t the right-member is non-lir~ear in J3. As a result, and contrary to the standard case, one cannot simply add commuting infinitesimal generators since this does not give a representation of tile algebra. This "addition" is to he made through the so called co-product. Consider two sets of operators J ± , J3, and K+, K s each satisfying 1, and such that [Jr, If m] = 0. Then tile operators A~= = J+e ihK3 + K.~e-ihJs;
Aa - J3 + IC3
(2)
do satisfy Eq. 1, as a simple computation shows. This co-product formula solves the problem of adding spins, but is non-symmetric between the J and h" generators. This crueiM point will be discussed
J.-L. Gervais / Gravlty-matter couplings
123
below, it is easy to obtain all finite-dimensional irreducible representations of 1. Introduce states [J, M), - J < M < J; together with operators J+, Ja such that: J~:]J, M) = ~/LJ :F MJ LJ 4- M + lJ ]J, M 4- 1),
da]J, M) = M [J, M).
(3)
These operators satisfy the commutation relations l, as one ear,fly verifies. Thus, in tile simple case we cunsider (h/Tr non-rational), the representvtions are specified by their spin J such that 2J is a positive integer, and the dimension of tim spin-J representation is 2J + 1. Consider next. quantities ~(J) and UM,"(g') which span irreducible representations of spin J and J', that is, I¢($) J ~ : ~ ) = ~/LJ :F U J [J 4- M + t"J~as~],
• eGO _- ~"~1 ~,¢(J) , .,a~M
(4)
(J) ~/M' (J*) as with similar formulae for aM"(1')-B": construction, the co-product 2 acting on tim product ~M A f.(J).(J')~
, . . ( ~ ) ~ . i h M ' . ( J ' ) a ~-ihM¢(~),.
(~'),
A [e(~)~tJ')~
A~,~c(J).(J')¢~
gives a representation of I. A priori, the ~ and q do not play the same role in the above formula since 2 is not symmetric. One may however re-establish the symmetry if one a.~svmes that ~ and rI do not commute in a special way. lndccd, there exist braiding relations of the type
~(~).(~') M ",M'
~
~j v~N'~V .(~')d~)
(6)
- J < N < J ; -J'~N'<_J'
such that
A/.(d) (J')X
(¢~ ~/M' ) = A(.(J')¢(J)~ "'V,M' - u )
(7)
The matrix (J,J)Mt, , N*NV may be written as
(J,J ,N'N )MM' =((J, M [ ® (J',M'I) R ( [ J , N ) ~ [J',N')), R = e(-2ia%®J~)(1 + Z (l .=,
c2ih~n eihn(n- l )/2 '
LnJ!
e-~"~'(3+)" ~) *'~"~'(d_ )").
(8)
(O)
R is the so-called universal R matrix ~f Uv(sl(2)). By introducing the braiding relations Eq.6, we have thus re-established the symmetry between ~ and rI. The representation generated by the A's only depends upon the values of J and J ' and not on tbe specific ordering. We may thus di~'~, - its decomposition into irreducible representations, and introduce the corresponding Clebch-Gordan coefficients noted (Jz, Mz; J~, MzlJt, J2; J, M~ + hi..,). Tiley are defined as solutions of the equations
~/[d ~ Mt ~: M~] [d + M, 4- M2 + lJ (Jr, hit; J.~, M'2lJt, J~; J, hit + M~) = +
e~M~ ~/LJx ~ M,J Ld 4- M~ + lJ
(Jr, hi, 4- l; dx, M21J~, d~; J, Mt + i ~ + |)
e-laMtx/tJ2~M2JLJ24-M~+l j
(JhM~;J~,Mx:t:IlJ],J~;J,M~+M.~+]),
(10)
which, for h = 0 coincides with the usual equation for sl(2). A straightforward calculation shows thai~, if this last equation holds, the quantities
J.-L. Gervais/ Gravity.matter couplings
124
--_(~) --M --
E
(JI'MI;J2'M2]JI'J2;J'M)~(Jf:)q(~'~)
(11)
MI+M2=M
transform according to a representation Gf spin J of 1. One sees ;.hat, with the braid relation ~, we are back to a situation similar to the one of standard Lie groups. Thus t i m a l g e b r a 1 n a t u r a l l y d e s e r i b e s t h e s y m m e t r y o n n o n - c o m m u t i n g o b j e c t s s a t i s f y i n g 6. As a last general point, let us consider the limit h ---* 0 of the structure just summarized. Of course one goes back to sl(2) and the non-eommutativity disappears since Eq. 9 indeed shows that in the limit R ~ 1 + O(h). One may however apply the ideas of the classical limit of quantum mechanics where commutators behave as Planek's constant times Poisson brackets, llence we let, starting from Eq. 6, {~), .(J')l
_ ,;
t'.(.t)~(.t')
'IM' iP.B. = ~ k ~ ; M
"(a')¢(J)~/tit'~
'IM' - ',U' ~ M / 1 ~ "J
(12)
and one obtains closure under the form
{~g), ~g;l}p.B. =
~
c~,~, ,o, o'"N'",M ,,' ',#, .C"),~') ,~ ,
03)
-s
2. TIIE CLASSICAL LIOUVILLE DYNAMICS. The problem of quantum gravity m tile conformal gauge is equivalent to the quantization of tile LiouviIle action which may be written as
1
/"
/1 04, ~
1 0~.~+e2~)
(14)
where a and r are the local coordinates such that tile metric tensor takes :he form gob = 6a,b C24' (We use Euclidean coordinates). The complex structure is assumed to be such that tile curves with constant a and r are everywhere tangent to the local imaginary and real axis respectively. In this section we discuss the resolution of the corresponding field equations at tile classical level. The action corresponds to a conform,d theory such that exp(2~) is a conformaIly covariant field[12] with weights (I,I) as we shall recall below. Tile corresponding field equation O2~Oa-T + ~02~ = 2e 2~ is conveniently solved using the following theorem. The field ~0r, r) satisfies 15 if and only if #I #i The factor i means that these solutions should be considered in Minkowskyspace-tlme
(15)
3.-L. Gerva/s / Gr~vityomattercouplings i
e-~=-~z E
b (z+)gj(x-);
z±=a~=ir
125 (16)
j=t,2 where fj (resp.(gi), which are functions of a single variable, are solutions of tile s~me SchrSdinger equation
-f~' + T(x+)fj =0,
( resp. -g;' + T(x_)gj).
(17)
The solutions are normalized such that their Wronskians flf~ - f~f2 and glg~ - g~g~ arc equal to one. The proof goes as follows. l) First check that 16 is indeed solution. Taking the Laplacian of the logarithm of the right-hand side gives 02~
+ O~@ ~ -= 40+0_, = - 4 / ( ~ i,~,) ~
(is)
i=1,2 where D~ = (O/Oer=l:iO/Or)[2.Tile numerator has been simplified by means of the Wronskian condition. This is equivalent to 15. 2) Conversely check that any solution of 15 may be put under tile form 16. If 15 holds one deduces 0~=T( i ) = 0;
w i t h T (~') := e~O~e -~I'
(19)
T (+) are thus functions of a single variable. Next tile equation involving T (+) may be rewritten ~s (-O~ + ~ + ) ) e - ~ - 0
(20)
with solution i
e-@ = ~
E
fj(x+)gj(x_);
with - Jj' + T{+)fi = 0
(21)
" - j=t,2 where the g.~ are arbitrary functions of z_. Using the equation 17 that involves T ~-), one finaliyderives the SchrSdinger equation -9} ~ + T(-)gj = 0. Thus the theorem holds with T = T (+) and T = T (-) q.e.d, o O n e m a y deduce from 19 that the potentials of the two SchrSdinger equations coincide with the two chiral components of the stress-energy tensor. ( Thus these equations are the classicalequivalent of the W a r d identitiesthat ensure the decoupling of Virasoro null vectors). For the time being we shall concentrate on one of the two chiral components. Consider for instance the - chiral components which are analytic functions of z = r + io. In a typical situation,a and ~" m a y be taken as coordinatez of a cylinder obtained by conformal mapping from a particular handle of the Riemann surface considertd. ~" plays the role of imagir,ary time and # is a space variable. O n e m a y work at r = 0 without loss of generality. The potential T(a) is periodic with period say 2~r and we are working on the unit circle. A n y two independent solutions of the Schr6dinger equation is suitable. It scents natural at 1~rstsight to diago~alize the monodromy matrix, that is to choose two solu*ions noted ¢j, j = I, 2, that are periodic up to a multiplicative constant #2. It is convenient to introduce ¢i (a) := ln{tb: ) / v / ~ - In d.i, d l
#2 We assume tlmt the monodromy matrix is diagonalizable
126
J.-L. (Jervals / Graviey-rnatter couplings
are suitable normalization constants. The fields ~j are periodic up to additive constants snd have the expansion
(22) n#O
As shown in [2,3], the canonical P.B. structure ot the action 14 is such t h a t the chiral fields ~j satisfy I gr'),~2( I O.2)}p •B . = 2~r6t(~r, - ~r2), {~(°'),~b~(cr2)}p.B. = {~b2(
T/~ = (~)2 + ~'/VT = (~)~ + ~/V~,
'*tq0 (j) ,Po0 ) ,JP.B. = 1
PCo~)= -V(o~),
(23)
(24)
Eq. 24 is an easy consequence of the Schr'ddinger equation 17. In this double free-field representation either field may be used to compute Poisson brackets of dynamical quantities, obtainH~, the same result. The Poisson bracket of ~bl with ~2 is complicated and not very illuminating. One may show, however, that the fields ~l are closed in the sense t h a t they satisfy a Poisson bracket algebra of the form
{~J(¢), ~(¢")} P.B. = ~ a~T(¢) %(u'), vp.,(o) Im
(25)
where ¢ is the sign of cr - cr'. The relevance of this fact to our discussion is that t h i s l a s t r e l a t i o n is e q u i v a l e n t #3 t o t h e P o i s s o n b r a c k e t r e l a t i o n s 13 w i t h J = J t = 1/2 a n d ¢ = 1. Let us now return to the Liouville field itself. At the classical level, it is trivial to take 16 to any integer power 2 J > 0. One gets, J e _ 2 J ~ _ - ~ I1 i ~2.s ~/.., ,~" ' /l ~ J - M r-(~J ) r~-+ l J~~Zt) l -'~'~-p, v
-
(26)
M = - J
where J 4- M run over integer, and where we have let
f7-7~
The notation anticipates t h a t f~(~) and f ~ ) form representations of spin J . This is explicitly realised as follows. Consider the z+ component for example. U~i;~g the fact t h a t the Wronskian condition f~f~ f2f~ = 1, one verifies t h a t the generators of the spin-J represent.vtion span by f ~ ) ' s are given by
I (~) . - f ~ O + 2 J. f ~ f ~ ,
.1('0
f ~ O.- 2 J f t f i ,
1(~)
f~f~O
J(ftf~f~f~): '
(28)
/) is the derivative operator so that, for ;nstance,
l(..J)J~_~) = ~ ( f ~ ) ' - 2] f,f~f~ ~ --: O, expected.
(29)
In general, using 8, one rederives the text-book formulas
4. JM l(.O d.~ ) = ~ / ( J : T : M ) ( J : i : M + ~ t ( I )
, r(J)
s~rt(J)
(30)
#~ a f t e r t a k i n g linear combinations that are the clv.ssical analogues o f t h e ~ fields o f [8,9] and will be d i s c u s s e d in t h e c o m i n g s e c t i o n .
J.-L. Gen'ais / Gravity-matter couplings
127
which display ;he group structure. Calling ~ J ) the generators of the ~c_ components, one sees that (/;J)-~ ~1)) e-2J
(31)
so that the powers of the metric field are group-invariants.
3. TIIE Q U A N T U M
(GROUP)
STRUCTURE.
Let us now come to the quantum case. The basic point of the method[3,4] is to quantize the above classicalstructure in such a way tbat the conformal structure is maintained, lu particular,powers of the metric tensor must bc primary field.d.This is ensurcd by the following result of [3]: O n the unit circle, z = e i#, and for generic % there exist two equivalent free fields:
~bj(o) = q~oj) + p(oJ)O-I- i E e-i"° p~)/n,
j = 1, 2,
(32)
rico
such that
(33) NO)(~b~) ~ + 4".~/',f7 = N(~)(~) 2 + ¢[/V~'.
(34)
N (l) (resp. N (2)) denote normal orderings witl) respect to the modes of ~bl (resp. 4'2). Eq. 34 defines the stress-energy tensor and the coupling constant 7 of th~ quantum theory. The former generates a representation of the Virasoro algebra with central charge C = 3 + 1/7. The chiral family is built up[4,5,7,9] from the following operators
@j = dj N(-i)(e'v/'h]'ff~'), z~j = ~ N(J)(ehV~-~#,),
i = 1, 2,
(35)
z where d~ and ~ are normalization constants. The parameters h and h are determined as the two solutions of the quadra{ic equation ~ - ~r(C - 13)x/6 + rr2 = 0 which ensures[4] that the fields so defined are solutions of the equations h
L0
h
c n>O
n<0
-,,
~,
-~ +(;)(~ n<0
(h Lne-'"'+ ~ + lfi, - C24
(38) n>O
These are operator Schr~dinger-equations equivalent to the deeoupling of Virasoro null-vectors[4,5,7]. They are the quantum versions of Eq. 17. Since there are two possible quantum modifications h and h, there are four solutions. By operator product ~p}, j = 1, 2, and ~ , j = 1, 2, generate two infinite families of chiral fields which are denoted ~b~), - J < m < J, and ~,.J),rn Y_#7 < ff~ < f; respectively,
J.-L. Oervals / Gravity.matter couplings
128
with .~.0/~) 'r_l/S =
oi.0/2)= ~b~,and .~I/2) ~l , *'1/:~ .~I/2)--_ ~2. ~b~), .~,(Y) ~ , are of the type (1, 2J + 1) w - l / ~ ----
~'l, " q / ~
and ( 2J-t- 1,1), respectively, ~n the BPZ classification. For the zero-modes, it is simpler[9] to define the rescaled variables 6
fl)
2f~
1)
~=~
;
~=~-.
(39)
At this point a pedagogical parenthesis may be in order: the hatted and unhatted ~ fields have the same ehirality; if we go to r ~ 0 they are both functions of z_; there are two counterparts ~'(~)(z+) and ~ ) ( z + ) which will be discussed below. Returning to ~,ir main line, we recall that the Hilbert space in whidl the operators ~b and ~ livc, is a direct sum[9-11] of Foek spaces .~'(w) spanned by the harmonic excitations of highest-weight Virasoro states noted Iw, 0). The harmonic excitations are created by the negative modes of the fields ~bj of Eq. 32. The highest-weight states are eigenw~ctors of the quasi momentum ~, and satisfy L , I ~ , 0 ) = 0, n > 0; (Lo - A(w))[w,0) = 0. The corresponding highest weights A0v ) may be rewritten as ~(~)
87
=~(1+~)
-~m
.
(40)
The commutation relations 33 are to be supplemented by the zero-mode ones:
[q(1),p~l)l =
[q(o,) p(o,)]= i.
('II)
The fields ~band ~ shift the quasi momentum p(o~! = _p~2) by a fixed amount For an arbitrary c-number function f one has
~ s(~,) = s(~, + 2,,) ~ J .
~ ) f(=) = s(~ + ~,~ .~/h) ~ ) .
(42)
The fields ~b and ~ together with their products may be naturally restricted to discrete values of m. They thus live in llilbert spaces #4 of the form "H(~0) --
+oo ( ~ Y(mo + n + ~ ~r/h).
(43)
rl,~=--oo
m0 is a constant which is arbitrary so far. The sl(2, C)-invariant vacuum (with highest weight/-~(m0) = 0) corresponds to Wo = I + x/h,[9] but other choices are also appropriate, as we shall see. Next we display the quantum-group ~tructure of the chiral fields. The operators tb and ~ are closed under O.P.E. and braiding. Each family obeys a quantum group symmetry of the Uq(sl(2)) type. However, the fusion coefficients and R--matrix elements depend upon m and titus do not conunute with the ~b's and ~'s. Their explicit form is unusual, therefore. One may exhibit the standard ~,~(sl(2))-quantum-group structure by changing basis to new families. Following my recent work,J9] let us introd~ace ~)(~):=
~ I,;,=)~)(~), -a<,n_J
- S < M < Y;
#4 Mathematically lhey are not really Hilbert spac~ since :i,elr rn~txlc~are not positive definite
(44)
J.-L. Gervais / Gravity-ma~ter coupll.gs
[j,w)~l = ~
elhn,12~-~ eiht(=+ra)((J J- M t -M+m-t)/2)
~)P" LPJ! Q = LQj!LP-Qj!' [ ,
LnJ!- ]'['~ LrJ LrJ r=1
=
J+M ((J+M+m+t)/2);
sin(hr) sinh
(45) (~6~
where the summation runs for (J - M + m - t)/2 integer. The last equation introduces q-deformed fact~rials and binomial coefficients. The other fields ~(~) are defined in exactly the same way replacing h by/~ everywhere. The symbols are the same with hats, e.g. [hi! _= []~=l [rj, [rJ = sin(hr)/sinh, and so ou. In [9,11] the operator algebra of the { fields was completely determined. In particular it was shown '..hat, for x > a > a' > 0, these operators obey the exchange algebra
~(,s)
(J')
,
M (a)~M' (¢)
, ~'Jv
~,
=
(J')
, (J)
(47)
(J.J)MM"~,, (~)~N ('0,
-J
which exactly coincides with the braiding relations of Uq(sl(2)) recalled in section 1 Eqs 6-9. In [11], the short-distance operator-product expansion of the { fields was shown to be of the form Ji +Jz
~c.,,), ,,:.,~), ,, _
9J, ~{(d(~,- ~,)),,~)-Ac.,,)-,,c.,~) J=lJt-$2l
(J,,MGJ2,M~JJz,J2;J, Mt + M2)(~Jt~+M,(a)+
descendants)},
(48)
where d(a - a') =_ 1 - e -I(¢-¢'), and where the Clebch-Gordan coefficie.-,ts appear, A(J} := - h J ( J + 1)/x - J is the Virasoro-weight of ~(~J)(u). Define the quantum group action on the ~ fields by
J3'~ )
=
M'~(MJ),
I I J:(J) J.~(~) = CL.I+MJL .,'+M ~-.J,.,u:~.
(49)
Then the operator-product {(~')(a){(~)(a') gives a representation of the quantum group algebra 1 with the co-product generators of the type 16
J±:=J±®eih~3+e-ihs~®J±,
Ja:=Ja®l+l®J3,
defined
(50)
(ll )
and where the tensor product is so that (A ® B)({(s~')fa){~)(a')) • • . := (A~M. (#)) where each term in the expansion over J transforms according to a representation of spin J. The family of operators just described is what I call the universal conformal family (UCF) associated with Uq(sl(2)). Its fusion and braiding are explicit reallsations of the abstract structure of the no~:coummti,g ~ymbols ~ ) ~/(~) introduced in section 1. The hatted operators similarly form another UCF associated with U~sl(2)). The two families do not commute, eventhough their mixed braiding relations are simple[9]. They are noted U¢(sl(2)) (3 U~sl(2)) where the ® is not a simple tensor-product.
4. TIlE Q U A N T U M
M E T R I C FIELD O P E R A T O R
I f C > 25 h and h are real and tile structure recalled above is directly handy. This is the weak coupling regime which is connected with the classical limit (7 ~ 0). Let us discuss how the powers of the metric are reconstructed. The standard screening charges -¢~± of the Liouville theory[4,0] are such that
130
J.-L Oervais/ Gravity-ma~tercouplin~
~ '
~°=~V
Q, and a0 are introduced so that they will agree with the standard notation, when we couple with matter. Kac's formula may he written as
= -
c ) + Q),
c) =
+ Jo+,
(52)
where 2J and 2 J are positive integers. Thus tile most general Liouville field is to be written as e x p ( - ( J ~ _ + f a + ) ~ ) . A t first we consid e x p ( - J ~ _ & ) , which are direct quantum-analogues of Eq. 26. "Ihere are two eases to distinguish. 1) One may consider, as is most usual, closed surfaces without boundary. Then the natural region is the whole circle 0 < ¢ < 2x. The quantum version of Eq. 26 will involve the fields ~ ) , together with their counterparts ~(~)(z+) whose exchange properties are similar. Concerning the latter one should ~member that they are functions of z', that is, are anti-analytic functions, so that the orientation of the c~,~plex p)ane is reversed. This may be taken into account simply by replacing i by - i in the above formulae for the ~-fields, that is by taking the complex conjugate of all the c-numbers w i t h o u t t a k i n g t h e H c r m l t l a n c o n j u g a t e o f t h e o p e r a t o r s . The appropriate definition of ~M/)0r) is ~.~, ~J)¢, a ,~ :=
~] ([j,~_)~). ,--(.I).. ~p., (or), -.S
- J < M < J;
(53)
where ( [ J , ~ ) ~ ) " is the complex conj.,gate of ( j J , ~ ) ~ ) . There are two basic requirements that determine e x p ( - J ~ _ ~ ) . The fi that ,~ commutes with any other power of the metric at equal ~'. The second o n concerns the Ililbcrt space of states where the physical operator algebra is realized. The point is that, since we took the fields ~(~) and ~ ) to commute, the quasi momenta m and ~ of the left and right movers are unrelated, wlfile periodicity in # requites that they be equal. This last condition is replaced by the requirement that exp(-JQ_4~) leave the subspace of states with ~ = ~ invariant. The l~tter condition de~nes the physical llilbert space ~ p ~ , where it must he possible to restrict the operator-algebra consistently. At r = 0, the appropriate definition is[i3,15] J
e - Y ~ - ~ ( ° ) = ~ e.~ ~
~(d)~a~,,-Mx ~a) t ~ ,. , ,~~ (-1) ~ - ~ cih(.~_SO ,~,x
(54)
M=-] where cl is a normalisation constant. The transformation of the ~ fields is similar to the one of the fields:
J ~~' ) ) = , ,", ~ ", )
7 , ~(~) = ~/"LJ :t: MJ LJ ± M 4- lJ ~,+x. ~)
(55)
Thus we define,
j.,. = j+~-~s:~ + e i ~ ® 7 , , Js = J~ + 73,
(56)
which does give a representation of 1. Then one easily checks that J:b exp(-Jc~_~) = J n e x p ( - J a _ 4 ~ ) = O,
(57)
J.-L. Gerva/s / Orar/ty-maCter coup//n~
131
so that the quantized Liouville field is a quantum-group invariant. Clearly, the quantum case and the classical case recalled in section 2 are remarkably similar. 2) One may also considet gravity with boundary, following [1,2,14]. A typical situation is the half circle 0 < o < ~r. One may set up boundary conditions such that the system remains conformal, albeit with one type of Virasoro generators only. The left- and right-movers become related as is the case for open strings. The appropriate definition uf the .metric becomes[14]:
e-Ja-d~(°)
=
cj E
(J) Iv~M ('1)(¢)~N (.0 (2a -
(`58)
M,N
where
A(J) = (J, MI {e -'hag ~.~ c''('+')j" (J+Y+' c'"°~'-')e '",'/='t- IJ, N ~' M, s
L'J!LsJ!
r,s=o
J
'
(,59)
These fields may be shown to be mutually local and closed by fusion.
`5. T I l E DRESSING B~ GRAVITY In this section we study the dressing of eonformal models with central charge D by the Liouville field with ce~,tral charge C so that C + D = 2O.
(00)
We shall he concerned with the case D < 1, where tile Liouville theory is in its weakly coupled regime C > 25. As is recalled in the appendix A, tile existence of the UCF's is basically a consequence of the operator differential equations 37, 38. The latter are equivalent to the Virasoro Ward-identities that describe the decoupling of null vectors. Thus tile UCF's, with appropriate quantum deformation parameters also describe the matter with D < 1. We will thus have another copy of the quantum-group structure recalled above. It will be distinguished by primes. Thus we let
h'
~
h'
D='-l+O(-~-+~+2)=l+fi(-~-+~+2),
"
with
h'~'-~2 -
,
(81)
and 60 gives h+h' = g+h' = 0
(63)
Of course we choose tile matter and gravity fields to comumte. Using the notation of last section, the complete quantum group structure is thus of the type:
{
[u,(st(2)) ® u,~st(2))lo[~,(st(~)) o ~(st(2))] } ®
{[~_,(st(2)) ® u,_,(st(2)~l®[.~._º(st(2)l ®.~-,(st(2))] },
(64)
J..L. Ocrv~ie /
132
Gravi~y-ma~r coup]ir~s
where the first (second) line displays the quantum-group structure of gravity ( m a t t e r ) According to the above results, the spectrum of weights of the gravity and matter are respectively given by
(65)
The connection with Kac's table will be spelled out in section 5. As we shall see in section 5, this last formulae is consistent with the usual formulae for minimal models. From the standpoint we are taking, the most general matter field is decribed hy an operator of the form exp(-(J'a:.. + f'~+)X), where X(~r, r) is a local field that commutes with the Liouville-field and whose properties are derived from those of@ by continuation to central charges smaller than one. Following our general conventions we let h' = w(a~)2/2, and h' -- x(~.)2/2. The correct screenin~ operator is the field e x p ( - ( J a _ + f a + ) ~ ) with spins J and f that ~ (J, J) + A n (ye ft) __ 1. In this connection, it is an easy consequence of 60 that A ~ ( ~ , _ j ' _ ') + A ~ , ( J ' , Y ) = 1,
,~(-j~-
l,J') + A~(J',~)--
1
(67)
These two choic~ correspond to the existence of two cosmological terms. Indeed, the unity-operator of matter (J~ -- .~' = 0) is dressed by exp(c~+~,) and exp(a_~) respectively with the choices 67. As is usual, we choose the latter as cosmological term so that the spins of gravity and matter fields will be related by second Eq. 67. Thus we shall be concerned with lr~atr:.x elei~cn~s of the operators
v~,.~(~,~) ----e(( )" + l)a_ - J ' a + ) ~ ( a , r ) e - ( J ' a " + ) " a ~ ) X ( a , r ) .
(68)
Concerning matter, choosing J' > 0 and J ' > O gives ~M > 0 and the formulae of last section may be directly used. Concerning gravity, it has been already emphasized[10,11], that the dressing of matter-operators with positive spins requires the use of gravity fields with negative spins. For example, the cosmological term has Y = -1, J" = 0. Thus the gravity-UCF with parameter h must be extended. This problem was overcome in [10,11] as follows. First, the q,ai,tum-group structure has an obvious symmetry in .I -~ - J - 1 with fixed M, s~ that the universal R-matrix and Clebsch-Gordan coefficients are left unchanged. Second, the coefficients IJ, m ) ~ also have a natural continuation to negative spin. F~r postive J, one has ( ~ ) ( m ) is defined in [11]) I-
Y - 1, ~ ) ~ = IJ, m)~ (-l)l+m/[(2isin(h))l+2s A~)(m)]
It follows that there exist fields ~ - ~ ) similar to ~ ) so that J ~ ' - z ) ( o ) oc Z
(69)
with fusion and braiding similar to ~ ) , and fields ~(-J-z)
( - 1 ) ' + m JJ'~v)~/A~)(=) ~bk-'-z)(a)
(?O)
m:-J
Next, the appropriate dei:nition of positive powers of the metric is, by extension of the discussion of the previous section,
J.-L. Gervais / Gravity-matter couplings J
eIJ + 1)~_~(¢,~) = ~ e1_ ,., _ ,
~
133
(_l)j_,~ e 'h(~-'> ~ " - ' ) ( ~ + ) ~ £ ~ - ~ ) ( , _ )
,/~.
(rl)
Af=-J
Indeed, first this field is local since the exchage properties of the fields ~ J - 1 )
and ~(~J-D arc the same
as those of the fields ~ ) , and ~ ) respectively. Second, it may also be re-expressed in the Bloch-wave basis, and this shows that this field leaves the eonditio~ m = ~ invariant. The symmetry between J and - J - ! is the explanation of the continuation performed in [16]. Next consider the Ililbert space in which we are working. The UCF's which appear in 64 live in spaces with highost-weight vectors of the form I~, 0) ® I~ , 0) ~ I~', 0) ® I~ ' , 0).
(72)
(The two UCF's of a product of the type Uq(sl(2)) ~ Uq-'(sl(2)) are realized in the same llilbert space). In the restricted llilbert space, one has m = ~ , and m ~ = ~ ' . Thus we introduce states of the form Its, m') -- Ira, 0) ® [,~, 0) ® Ira', 0} ® I~', 0).
(73)
The next point concerns the physical on-~hell states. They should satisfy the conJition (to + To + L~, + ~o - 2)1=, = 3 = O,
(74)
where the notation is self-explanatory. According to Eq. 40, this is satisfied if h
=-2
h
2
~ ( 1 + ~-) - ~
hi "1
+ ~(
a'.~
h' - - w '2 = 1.
(75)
+ ~;) - 4~-
It follows from Eqs. 36, 60, that h
n 2
~(I+~)
h'
x 2
+~(l+F)
C+D-2
--
2~-I.
(76)
Moreover, according to Eqs. 39, 63, h ' ~ '2 =
h'~' :~ =
-h~'
(77)
2
so that, according to Eq. 63, the on-shell conditiot~ is ~2 = ~ 2 . Tim sign ambibuity is related with the two possibilities of cosmological term. It will be shown below that the choice, which is consistent with the abow: definition of gravity-dressing is m = -~',
or, equivalently,
~ = m ~.
(78)
The coupling constants Ci,2,s appear in the three-point function as follows
I
k,I
It was computed in [15]. The result takes the form
B~;,
134
J.-L. Ge~v~$ / GraviLyomagter coupllngs
where B j, ~., are normalisation-dependant constants. Tile calculation is sgmewhat involved, but the result Ls remarkably simple. Next, we make contact with other approaches to the same problem. For this we first explictly connect the present group-theoretic notations with more standard conventions. Eqs 51, and 61, correspond to the usual notations C --- 1-4- 3Q 2, and D = 1 - 12o02, for tile gravity and matter central charges. The screening charges are given by (we chose c~0 > 0) . ~ = Q =t=~o, . ' . = i(ao=~ Q).
(81)
The dressed vertex operator 68 may be rewritten as
Vj, ~, =--e(( $' + 1)o_J',~+)~ e-(J'~'_ + Y ' ~ ) X = e~(k)~ - ikX/,~_, /~ -- ( ) + I ) ~ - - J'o~+,
(82)
ik = ¢t_(J'¢/_ + Y'd+),
(83)
A simple calculation, using the formulae just given leads to
k = 2J' - 2Y'h/~ =
2Y+ 2(J +
1)h/~,
~(~) + 0 / 2 = (k + ko)/~_,
,~:~=
~.o._
==
I - h / r (84)
According to 52 65, and 66, the weights are given by
For rational theories (D = 1 - 6(p - p'f/pp', p > p' > 0),
h = -'~',= ,¢p'Ip,
?, = - h ' , = ,¢p/p',
(86)
ko = ( p - p ' ) I p ,
A ~ reduces to Kac's table
[(rP-~/)~-(p-p'f], r=2J'+1, s=2d'+1, k+ko=r-~h/,~.
(87)
The momentum k is defined so that it takes rational values for rational theories. For critical bosonic string with Regge slope a', the tachyon vertex is e x p ( i k X J ' ~ ) . Eq. 82 corresponds to a ' = 1/(c~_) 2. With the conventions just introduced, the result Eq. 80 takes the form ¢m,a
~
- (~r(2 + h / . ) r ( ~ -
B(kl)
h/,O/h) ~ -= .4(k,, k~,k~) = 1 1 Jr(f, + i0)} ~'
(88)
where we left out an overall factor which does not depend upon the momenta. Our last task is to re-establish the cosmological constant, in the present approach, it comes out as follows. Our basic guideline was to write down the most general local operators, as was discussed in section 2. We have not yet really done so, since we may multiply the righ'-hand sides of 54 and 71 by p'f=/2 on the left, and by iz'~/2 on the right, without breaking locality. Th;s constant p~ is arbitrary, and will play the role of the cosmological constant. According to 42 one has
/~-='/~¢'("~ "='/~ e
rn m
t'¢
- t""+;"'/h'r'('~ Y) -e ~m m "
(89)
.I.-L. Gcrvais/ Gravity-matter couplings
135
In the Coulomb-gas picture, the power in Pc is equal to the total number of screening charges. We shall agree with this definition if we let
~(-!~,~_ + s~+)~(.,
T) = .~+.~/h /,:/2c-(J~_ +
]~,+),(~, ~) j,U~.
(90)
7";. ',s we have, according to Eq. 83, (p=)
V~ i, j.(¢,~.)
.",
=
,
t,7(.I +t)+s ./hp-~/2~2j,,~.,(a,r)py/z"
(91)
The flr';t factor coincides with tile KPZ-DDK scaling factor[17-19]. In particular, one h ~
v~",~)(o, ~) = t,7'u7 ~/~Vo, o(., ~) u'. /~,
(9~)
which is the expected scaling behaviour of the cosmological constant. The operators p~¢r~l~ induce a translation on tile variable conjugate to the Liouville .mmentum. This Liouville position-operator is proportional to q(0t) which is tile zero mode of tile free field ~1 whose properties were summarized in the section 3. This translatioe wl,ich is actually a global Weyl transformation, is analogous to the translation of the Liouville field i . the work of DDK. This is ~ e ' l by computing next the pc-dependence of the three-pob,t function. One gets immediately A(~.)(/ca, k2, ks) = A(/q, l::, ks) t ~ °+~'(~'+s~'/h).
(93)
The term #~o arises when p-~=12, and tt~/2 hit the left and right vacua respectively. In the DDK discussion[20,19], it comes from the term of the effective action which is linear in the field # 5 One may verify that the power of p~ is equal to uz + 9~lr/h where u2 and ~ are the gravity screening-numbers. This agrees with the usual definition (see, e.g. [18]). Finally we compare 88 with the result of the matrix model. This part follows [18] closely. The twopoint function A(,.)(k, k) is determined by starting from the three-point function with one cosmological term A(u=)(O, k, k). One writes, making use of 92,
~A.o)(k,k) = A.~)(0,k,k), A..)(k,~) = ~ A . o ) ( 0 , k , k ) .
(94)
Similarly, the partition fimction satisfies
ap< Z (~=)
= A(~)(O,O,O),
Z ( . . ) = (Wo)(='o (m)a l)(w'o _ 2)A(~o)(0,0,0).
(9~)
We finally obtain the rescaled three-point function
I)(kl,k~,ks) ~
(.A(p~)(kl, le2,ka)) ~ Z(~) = I-L(/,-t + k0) i]~.A(~.)(kt,k:) (~r/h + 1)(x/h)(~c/h - 1)'
(96)
which agrees with the results of the matrix models. Clearly, the key point in this final verification is that the final expression 88 for the three-point function factorises. On the other hand, 88 vanishes whenever #5 It would not he there on the torus, since one would take a trace. Tiffs agrees with the DDK result where its contribution is shown to be proportional to the Euler characteristic
136
J.-L. Gervals / Gravity-matter couplings
k: + ks = 0 for a n y o f t h e three legs. According to 87, this h a p p e n s for rationa'_ theories, a t t h e border of K a c ' s table, where r -- p ' , a n d s = p. T h u s formula 96 holds only when t h e b r a n c h i n g rules are satisfied.
References
[1] J.-L. Gervais, A. Neveu, Nuc/. Phys. B199, 59 (1982). [2] J.-L. Germs, A. Neveu, Nucl. Phys. B202, 125 (IS82). [3] J.-L. Gervals, A. Neveu, Nud. Phys. B224, 329 (1983). [4] J.-L. Gervnis, A. Neveu NucL Phys. B238, 125 (184.) NucL Phys. B238, 396 (1984). [5] J.-L. Gerv~s, A. N~veu, Nucl. Phys. B25?[FSI4], 59 (1985). [6] J.-L. Gerv~ds, A. Noveu, Phys. Let¢. 151B, 2"/'I (1985). [7] J.-L. Gerv~s, A. Neveu, Nucl. Plays. B2~,-i, 557 (1986). [8] O. Babelon, Phys. LeeL B215, 523 (1988). [9] J.-L. Gervals, Comm. Ma~h. Phys. 130, 257 (I~0). [10] J.-L. Gervais, Phys. Left. B243, 85 (1990). [II] J.-L. Gefvals, Comm. Magh. P/O's. 13b~ 301 (1991). [12] J.-L. Gervnis, B. Saldta, Nud. Phys. B34, 477 (1971). [13] J.-L. Gefvais, "(Jn the algebraic structure of quantum 8r~,ity in two dimer~i~ns" ICTP prepdnt NSF-ITPg0-176, Sept. 1990, Proceedings of the Trieste Cmderence on topological Methods in quantum field theories, World Scientific. [14] E. Cremmer, J.-L. Gerva~s, "The quantum strip: Liouville theory for open strings" LPTENS preprlnt 90/32. Comm. Math. Phys. to be published. [15] J.-L. Gervnis, "Gravlty-matter couplhtg from Liouville theory", LPTENS preprlnt 91/22. [16] M. Goulian, M. Li, "Correlation functions in Liouville theory" S~mta Barbara pveprlnt, UCBTH-90-61, 1990. [17] V. Knlzl'.nlk, A. Poly~ov, A.A. Zamolodchlkov, Mod. Plays. Lett. A 3, 819, 1988. [18] VLS. botsenko, "Three-polnt correlation functions of minimal theories coupled to 2D gravity" PAR-LFTHE preprlnt 91-18, 1991. [19] J. Distler, H. Kawai, Nucl. Phys. B321, 509 (1651). [20] F. David, (3. II. Aemd. Scl. Paris 307, 1051,1988; Mod. Phys. Lett. A 17, 1651, 1988.
PROCEEDINGS SUPPLEMENTS
2D GRAVITY-MATTER COUPLINGS F R O M Q U A N T U M G R O U P Jean-Loup GERVAIS Laborateire de Physique Th~oriq~e de I'Ecole Normale S~pErieure 24, fie Lhomond, 75231 PARIS CEDEX 05, FRANCE
The quantum group structure of two-dimensional gravity a~ld minimal models is reviewed. The coupling of the latter with the former is discussed, and the three-point functions on the sphere am shown to agree with the results of matrix models. The corresponding co~rno.'*ogicalconsta~lt is defined in the present operator-approach. It does not correspond to a shift of the interacting Liouville-field.
1. Introduction After the early works[l-7], the recent developments of the operator approach to conformal theories took place when its relationship to quantum groups was recognized[8-11,13-15]. In these notes we sun'.marize t.~ ~ main points of this connection, and show how it may be used practically to compute corlelation functions[15]. We will see that, as ill matrix models, coapling gravity with matter leads to remarkable simplifications. 1 b begin with, it is convenient to give a short introduction to q u a n t u m groups per se ( W h a t follows next is a quick s u m m a r y without any attempt at generality or mathematical rigour). We shall take the case which will be directly useful later on, that is, of the q u a n t u m deformation of sl(2). Introduce a deformation parameter h, and associate with ally given variable x the quantity LxJ ~ s i n ( h x ) / s i n h. For simplicity, we shall assume that h / ~ is not, a rational number s~) that, for ally integer N, [NJ vanishes iff N = 0. Introduce three generators J~., and J3, which satisfy [ j + , j _ ] = sin(2hJa) sin/,
-
L2JaJ,
[Ja, J~-] = =l=J±.
(1)
This is the deformation of the Lie algebra sl(2) denoted by Ue(sl(2)). Tile standard Lie algebra is recovered in the limit h ~ 0. Tile novel feature of the deformed commutation relation is t h a t the right-member is non-lir~ear in J3. As a result, and contrary to the standard case, one cannot simply add commuting infinitesimal generators since this does not give a representation of tile algebra. This "addition" is to he made through the so called co-product. Consider two sets of operators J ± , J3, and K+, K s each satisfying 1, and such that [Jr, If m] = 0. Then tile operators A~= = J+e ihK3 + K.~e-ihJs;
Aa - J3 + IC3
(2)
do satisfy Eq. 1, as a simple computation shows. This co-product formula solves the problem of adding spins, but is non-symmetric between the J and h" generators. This crueiM point will be discussed
J.-L. Gervais / Gravlty-matter couplings
123
below, it is easy to obtain all finite-dimensional irreducible representations of 1. Introduce states [J, M), - J < M < J; together with operators J+, Ja such that: J~:]J, M) = ~/LJ :F MJ LJ 4- M + lJ ]J, M 4- 1),
da]J, M) = M [J, M).
(3)
These operators satisfy the commutation relations l, as one ear,fly verifies. Thus, in tile simple case we cunsider (h/Tr non-rational), the representvtions are specified by their spin J such that 2J is a positive integer, and the dimension of tim spin-J representation is 2J + 1. Consider next. quantities ~(J) and UM,"(g') which span irreducible representations of spin J and J', that is, I¢($) J ~ : ~ ) = ~/LJ :F U J [J 4- M + t"J~as~],
• eGO _- ~"~1 ~,¢(J) , .,a~M
(4)
(J) ~/M' (J*) as with similar formulae for aM"(1')-B": construction, the co-product 2 acting on tim product ~M A f.(J).(J')~
, . . ( ~ ) ~ . i h M ' . ( J ' ) a ~-ihM¢(~),.
(~'),
A [e(~)~tJ')~
A~,~c(J).(J')¢~
gives a representation of I. A priori, the ~ and q do not play the same role in the above formula since 2 is not symmetric. One may however re-establish the symmetry if one a.~svmes that ~ and rI do not commute in a special way. lndccd, there exist braiding relations of the type
~(~).(~') M ",M'
~
~j v~N'~V .(~')d~)
(6)
- J < N < J ; -J'~N'<_J'
such that
A/.(d) (J')X
(¢~ ~/M' ) = A(.(J')¢(J)~ "'V,M' - u )
(7)
The matrix (J,J)Mt, , N*NV may be written as
(J,J ,N'N )MM' =((J, M [ ® (J',M'I) R ( [ J , N ) ~ [J',N')), R = e(-2ia%®J~)(1 + Z (l .=,
c2ih~n eihn(n- l )/2 '
LnJ!
e-~"~'(3+)" ~) *'~"~'(d_ )").
(8)
(O)
R is the so-called universal R matrix ~f Uv(sl(2)). By introducing the braiding relations Eq.6, we have thus re-established the symmetry between ~ and rI. The representation generated by the A's only depends upon the values of J and J ' and not on tbe specific ordering. We may thus di~'~, - its decomposition into irreducible representations, and introduce the corresponding Clebch-Gordan coefficients noted (Jz, Mz; J~, MzlJt, J2; J, M~ + hi..,). Tiley are defined as solutions of the equations
~/[d ~ Mt ~: M~] [d + M, 4- M2 + lJ (Jr, hit; J.~, M'2lJt, J~; J, hit + M~) = +
e~M~ ~/LJx ~ M,J Ld 4- M~ + lJ
(Jr, hi, 4- l; dx, M21J~, d~; J, Mt + i ~ + |)
e-laMtx/tJ2~M2JLJ24-M~+l j
(JhM~;J~,Mx:t:IlJ],J~;J,M~+M.~+]),
(10)
which, for h = 0 coincides with the usual equation for sl(2). A straightforward calculation shows thai~, if this last equation holds, the quantities
J.-L. Gervais/ Gravity.matter couplings
124
--_(~) --M --
E
(JI'MI;J2'M2]JI'J2;J'M)~(Jf:)q(~'~)
(11)
MI+M2=M
transform according to a representation Gf spin J of 1. One sees ;.hat, with the braid relation ~, we are back to a situation similar to the one of standard Lie groups. Thus t i m a l g e b r a 1 n a t u r a l l y d e s e r i b e s t h e s y m m e t r y o n n o n - c o m m u t i n g o b j e c t s s a t i s f y i n g 6. As a last general point, let us consider the limit h ---* 0 of the structure just summarized. Of course one goes back to sl(2) and the non-eommutativity disappears since Eq. 9 indeed shows that in the limit R ~ 1 + O(h). One may however apply the ideas of the classical limit of quantum mechanics where commutators behave as Planek's constant times Poisson brackets, llence we let, starting from Eq. 6, {~), .(J')l
_ ,;
t'.(.t)~(.t')
'IM' iP.B. = ~ k ~ ; M
"(a')¢(J)~/tit'~
'IM' - ',U' ~ M / 1 ~ "J
(12)
and one obtains closure under the form
{~g), ~g;l}p.B. =
~
c~,~, ,o, o'"N'",M ,,' ',#, .C"),~') ,~ ,
03)
-s
2. TIIE CLASSICAL LIOUVILLE DYNAMICS. The problem of quantum gravity m tile conformal gauge is equivalent to the quantization of tile LiouviIle action which may be written as
1
/"
/1 04, ~
1 0~.~+e2~)
(14)
where a and r are the local coordinates such that tile metric tensor takes :he form gob = 6a,b C24' (We use Euclidean coordinates). The complex structure is assumed to be such that tile curves with constant a and r are everywhere tangent to the local imaginary and real axis respectively. In this section we discuss the resolution of the corresponding field equations at tile classical level. The action corresponds to a conform,d theory such that exp(2~) is a conformaIly covariant field[12] with weights (I,I) as we shall recall below. Tile corresponding field equation O2~Oa-T + ~02~ = 2e 2~ is conveniently solved using the following theorem. The field ~0r, r) satisfies 15 if and only if #I #i The factor i means that these solutions should be considered in Minkowskyspace-tlme
(15)
3.-L. Gerva/s / Gr~vityomattercouplings i
e-~=-~z E
b (z+)gj(x-);
z±=a~=ir
125 (16)
j=t,2 where fj (resp.(gi), which are functions of a single variable, are solutions of tile s~me SchrSdinger equation
-f~' + T(x+)fj =0,
( resp. -g;' + T(x_)gj).
(17)
The solutions are normalized such that their Wronskians flf~ - f~f2 and glg~ - g~g~ arc equal to one. The proof goes as follows. l) First check that 16 is indeed solution. Taking the Laplacian of the logarithm of the right-hand side gives 02~
+ O~@ ~ -= 40+0_, = - 4 / ( ~ i,~,) ~
(is)
i=1,2 where D~ = (O/Oer=l:iO/Or)[2.Tile numerator has been simplified by means of the Wronskian condition. This is equivalent to 15. 2) Conversely check that any solution of 15 may be put under tile form 16. If 15 holds one deduces 0~=T( i ) = 0;
w i t h T (~') := e~O~e -~I'
(19)
T (+) are thus functions of a single variable. Next tile equation involving T (+) may be rewritten ~s (-O~ + ~ + ) ) e - ~ - 0
(20)
with solution i
e-@ = ~
E
fj(x+)gj(x_);
with - Jj' + T{+)fi = 0
(21)
" - j=t,2 where the g.~ are arbitrary functions of z_. Using the equation 17 that involves T ~-), one finaliyderives the SchrSdinger equation -9} ~ + T(-)gj = 0. Thus the theorem holds with T = T (+) and T = T (-) q.e.d, o O n e m a y deduce from 19 that the potentials of the two SchrSdinger equations coincide with the two chiral components of the stress-energy tensor. ( Thus these equations are the classicalequivalent of the W a r d identitiesthat ensure the decoupling of Virasoro null vectors). For the time being we shall concentrate on one of the two chiral components. Consider for instance the - chiral components which are analytic functions of z = r + io. In a typical situation,a and ~" m a y be taken as coordinatez of a cylinder obtained by conformal mapping from a particular handle of the Riemann surface considertd. ~" plays the role of imagir,ary time and # is a space variable. O n e m a y work at r = 0 without loss of generality. The potential T(a) is periodic with period say 2~r and we are working on the unit circle. A n y two independent solutions of the Schr6dinger equation is suitable. It scents natural at 1~rstsight to diago~alize the monodromy matrix, that is to choose two solu*ions noted ¢j, j = I, 2, that are periodic up to a multiplicative constant #2. It is convenient to introduce ¢i (a) := ln{tb: ) / v / ~ - In d.i, d l
#2 We assume tlmt the monodromy matrix is diagonalizable
126
J.-L. (Jervals / Graviey-rnatter couplings
are suitable normalization constants. The fields ~j are periodic up to additive constants snd have the expansion
(22) n#O
As shown in [2,3], the canonical P.B. structure ot the action 14 is such t h a t the chiral fields ~j satisfy I gr'),~2( I O.2)}p •B . = 2~r6t(~r, - ~r2), {~(°'),~b~(cr2)}p.B. = {~b2(
T/~ = (~)2 + ~'/VT = (~)~ + ~/V~,
'*tq0 (j) ,Po0 ) ,JP.B. = 1
PCo~)= -V(o~),
(23)
(24)
Eq. 24 is an easy consequence of the Schr'ddinger equation 17. In this double free-field representation either field may be used to compute Poisson brackets of dynamical quantities, obtainH~, the same result. The Poisson bracket of ~bl with ~2 is complicated and not very illuminating. One may show, however, that the fields ~l are closed in the sense t h a t they satisfy a Poisson bracket algebra of the form
{~J(¢), ~(¢")} P.B. = ~ a~T(¢) %(u'), vp.,(o) Im
(25)
where ¢ is the sign of cr - cr'. The relevance of this fact to our discussion is that t h i s l a s t r e l a t i o n is e q u i v a l e n t #3 t o t h e P o i s s o n b r a c k e t r e l a t i o n s 13 w i t h J = J t = 1/2 a n d ¢ = 1. Let us now return to the Liouville field itself. At the classical level, it is trivial to take 16 to any integer power 2 J > 0. One gets, J e _ 2 J ~ _ - ~ I1 i ~2.s ~/.., ,~" ' /l ~ J - M r-(~J ) r~-+ l J~~Zt) l -'~'~-p, v
-
(26)
M = - J
where J 4- M run over integer, and where we have let
f7-7~
The notation anticipates t h a t f~(~) and f ~ ) form representations of spin J . This is explicitly realised as follows. Consider the z+ component for example. U~i;~g the fact t h a t the Wronskian condition f~f~ f2f~ = 1, one verifies t h a t the generators of the spin-J represent.vtion span by f ~ ) ' s are given by
I (~) . - f ~ O + 2 J. f ~ f ~ ,
.1('0
f ~ O.- 2 J f t f i ,
1(~)
f~f~O
J(ftf~f~f~): '
(28)
/) is the derivative operator so that, for ;nstance,
l(..J)J~_~) = ~ ( f ~ ) ' - 2] f,f~f~ ~ --: O, expected.
(29)
In general, using 8, one rederives the text-book formulas
4. JM l(.O d.~ ) = ~ / ( J : T : M ) ( J : i : M + ~ t ( I )
, r(J)
s~rt(J)
(30)
#~ a f t e r t a k i n g linear combinations that are the clv.ssical analogues o f t h e ~ fields o f [8,9] and will be d i s c u s s e d in t h e c o m i n g s e c t i o n .
J.-L. Gen'ais / Gravity-matter couplings
127
which display ;he group structure. Calling ~ J ) the generators of the ~c_ components, one sees that (/;J)-~ ~1)) e-2J
(31)
so that the powers of the metric field are group-invariants.
3. TIIE Q U A N T U M
(GROUP)
STRUCTURE.
Let us now come to the quantum case. The basic point of the method[3,4] is to quantize the above classicalstructure in such a way tbat the conformal structure is maintained, lu particular,powers of the metric tensor must bc primary field.d.This is ensurcd by the following result of [3]: O n the unit circle, z = e i#, and for generic % there exist two equivalent free fields:
~bj(o) = q~oj) + p(oJ)O-I- i E e-i"° p~)/n,
j = 1, 2,
(32)
rico
such that
(33) NO)(~b~) ~ + 4".~/',f7 = N(~)(~) 2 + ¢[/V~'.
(34)
N (l) (resp. N (2)) denote normal orderings witl) respect to the modes of ~bl (resp. 4'2). Eq. 34 defines the stress-energy tensor and the coupling constant 7 of th~ quantum theory. The former generates a representation of the Virasoro algebra with central charge C = 3 + 1/7. The chiral family is built up[4,5,7,9] from the following operators
@j = dj N(-i)(e'v/'h]'ff~'), z~j = ~ N(J)(ehV~-~#,),
i = 1, 2,
(35)
z where d~ and ~ are normalization constants. The parameters h and h are determined as the two solutions of the quadra{ic equation ~ - ~r(C - 13)x/6 + rr2 = 0 which ensures[4] that the fields so defined are solutions of the equations h
L0
h
c n>O
n<0
-,,
~,
-~ +(;)(~ n<0
(h Lne-'"'+ ~ + lfi, - C24
(38) n>O
These are operator Schr~dinger-equations equivalent to the deeoupling of Virasoro null-vectors[4,5,7]. They are the quantum versions of Eq. 17. Since there are two possible quantum modifications h and h, there are four solutions. By operator product ~p}, j = 1, 2, and ~ , j = 1, 2, generate two infinite families of chiral fields which are denoted ~b~), - J < m < J, and ~,.J),rn Y_#7 < ff~ < f; respectively,
J.-L. Oervals / Gravity.matter couplings
128
with .~.0/~) 'r_l/S =
oi.0/2)= ~b~,and .~I/2) ~l , *'1/:~ .~I/2)--_ ~2. ~b~), .~,(Y) ~ , are of the type (1, 2J + 1) w - l / ~ ----
~'l, " q / ~
and ( 2J-t- 1,1), respectively, ~n the BPZ classification. For the zero-modes, it is simpler[9] to define the rescaled variables 6
fl)
2f~
1)
~=~
;
~=~-.
(39)
At this point a pedagogical parenthesis may be in order: the hatted and unhatted ~ fields have the same ehirality; if we go to r ~ 0 they are both functions of z_; there are two counterparts ~'(~)(z+) and ~ ) ( z + ) which will be discussed below. Returning to ~,ir main line, we recall that the Hilbert space in whidl the operators ~b and ~ livc, is a direct sum[9-11] of Foek spaces .~'(w) spanned by the harmonic excitations of highest-weight Virasoro states noted Iw, 0). The harmonic excitations are created by the negative modes of the fields ~bj of Eq. 32. The highest-weight states are eigenw~ctors of the quasi momentum ~, and satisfy L , I ~ , 0 ) = 0, n > 0; (Lo - A(w))[w,0) = 0. The corresponding highest weights A0v ) may be rewritten as ~(~)
87
=~(1+~)
-~m
.
(40)
The commutation relations 33 are to be supplemented by the zero-mode ones:
[q(1),p~l)l =
[q(o,) p(o,)]= i.
('II)
The fields ~band ~ shift the quasi momentum p(o~! = _p~2) by a fixed amount For an arbitrary c-number function f one has
~ s(~,) = s(~, + 2,,) ~ J .
~ ) f(=) = s(~ + ~,~ .~/h) ~ ) .
(42)
The fields ~b and ~ together with their products may be naturally restricted to discrete values of m. They thus live in llilbert spaces #4 of the form "H(~0) --
+oo ( ~ Y(mo + n + ~ ~r/h).
(43)
rl,~=--oo
m0 is a constant which is arbitrary so far. The sl(2, C)-invariant vacuum (with highest weight/-~(m0) = 0) corresponds to Wo = I + x/h,[9] but other choices are also appropriate, as we shall see. Next we display the quantum-group ~tructure of the chiral fields. The operators tb and ~ are closed under O.P.E. and braiding. Each family obeys a quantum group symmetry of the Uq(sl(2)) type. However, the fusion coefficients and R--matrix elements depend upon m and titus do not conunute with the ~b's and ~'s. Their explicit form is unusual, therefore. One may exhibit the standard ~,~(sl(2))-quantum-group structure by changing basis to new families. Following my recent work,J9] let us introd~ace ~)(~):=
~ I,;,=)~)(~), -a<,n_J
- S < M < Y;
#4 Mathematically lhey are not really Hilbert spac~ since :i,elr rn~txlc~are not positive definite
(44)
J.-L. Gervais / Gravity-ma~ter coupll.gs
[j,w)~l = ~
elhn,12~-~ eiht(=+ra)((J J- M t -M+m-t)/2)
~)P" LPJ! Q = LQj!LP-Qj!' [ ,
LnJ!- ]'['~ LrJ LrJ r=1
=
J+M ((J+M+m+t)/2);
sin(hr) sinh
(45) (~6~
where the summation runs for (J - M + m - t)/2 integer. The last equation introduces q-deformed fact~rials and binomial coefficients. The other fields ~(~) are defined in exactly the same way replacing h by/~ everywhere. The symbols are the same with hats, e.g. [hi! _= []~=l [rj, [rJ = sin(hr)/sinh, and so ou. In [9,11] the operator algebra of the { fields was completely determined. In particular it was shown '..hat, for x > a > a' > 0, these operators obey the exchange algebra
~(,s)
(J')
,
M (a)~M' (¢)
, ~'Jv
~,
=
(J')
, (J)
(47)
(J.J)MM"~,, (~)~N ('0,
-J
which exactly coincides with the braiding relations of Uq(sl(2)) recalled in section 1 Eqs 6-9. In [11], the short-distance operator-product expansion of the { fields was shown to be of the form Ji +Jz
~c.,,), ,,:.,~), ,, _
9J, ~{(d(~,- ~,)),,~)-Ac.,,)-,,c.,~) J=lJt-$2l
(J,,MGJ2,M~JJz,J2;J, Mt + M2)(~Jt~+M,(a)+
descendants)},
(48)
where d(a - a') =_ 1 - e -I(¢-¢'), and where the Clebch-Gordan coefficie.-,ts appear, A(J} := - h J ( J + 1)/x - J is the Virasoro-weight of ~(~J)(u). Define the quantum group action on the ~ fields by
J3'~ )
=
M'~(MJ),
I I J:(J) J.~(~) = CL.I+MJL .,'+M ~-.J,.,u:~.
(49)
Then the operator-product {(~')(a){(~)(a') gives a representation of the quantum group algebra 1 with the co-product generators of the type 16
J±:=J±®eih~3+e-ihs~®J±,
Ja:=Ja®l+l®J3,
defined
(50)
(ll )
and where the tensor product is so that (A ® B)({(s~')fa){~)(a')) • • . := (A~M. (#)) where each term in the expansion over J transforms according to a representation of spin J. The family of operators just described is what I call the universal conformal family (UCF) associated with Uq(sl(2)). Its fusion and braiding are explicit reallsations of the abstract structure of the no~:coummti,g ~ymbols ~ ) ~/(~) introduced in section 1. The hatted operators similarly form another UCF associated with U~sl(2)). The two families do not commute, eventhough their mixed braiding relations are simple[9]. They are noted U¢(sl(2)) (3 U~sl(2)) where the ® is not a simple tensor-product.
4. TIlE Q U A N T U M
M E T R I C FIELD O P E R A T O R
I f C > 25 h and h are real and tile structure recalled above is directly handy. This is the weak coupling regime which is connected with the classical limit (7 ~ 0). Let us discuss how the powers of the metric are reconstructed. The standard screening charges -¢~± of the Liouville theory[4,0] are such that
130
J.-L Oervais/ Gravity-ma~tercouplin~
~ '
~°=~V
Q, and a0 are introduced so that they will agree with the standard notation, when we couple with matter. Kac's formula may he written as
= -
c ) + Q),
c) =
+ Jo+,
(52)
where 2J and 2 J are positive integers. Thus tile most general Liouville field is to be written as e x p ( - ( J ~ _ + f a + ) ~ ) . A t first we consid e x p ( - J ~ _ & ) , which are direct quantum-analogues of Eq. 26. "Ihere are two eases to distinguish. 1) One may consider, as is most usual, closed surfaces without boundary. Then the natural region is the whole circle 0 < ¢ < 2x. The quantum version of Eq. 26 will involve the fields ~ ) , together with their counterparts ~(~)(z+) whose exchange properties are similar. Concerning the latter one should ~member that they are functions of z', that is, are anti-analytic functions, so that the orientation of the c~,~plex p)ane is reversed. This may be taken into account simply by replacing i by - i in the above formulae for the ~-fields, that is by taking the complex conjugate of all the c-numbers w i t h o u t t a k i n g t h e H c r m l t l a n c o n j u g a t e o f t h e o p e r a t o r s . The appropriate definition of ~M/)0r) is ~.~, ~J)¢, a ,~ :=
~] ([j,~_)~). ,--(.I).. ~p., (or), -.S
- J < M < J;
(53)
where ( [ J , ~ ) ~ ) " is the complex conj.,gate of ( j J , ~ ) ~ ) . There are two basic requirements that determine e x p ( - J ~ _ ~ ) . The fi that ,~ commutes with any other power of the metric at equal ~'. The second o n concerns the Ililbcrt space of states where the physical operator algebra is realized. The point is that, since we took the fields ~(~) and ~ ) to commute, the quasi momenta m and ~ of the left and right movers are unrelated, wlfile periodicity in # requites that they be equal. This last condition is replaced by the requirement that exp(-JQ_4~) leave the subspace of states with ~ = ~ invariant. The l~tter condition de~nes the physical llilbert space ~ p ~ , where it must he possible to restrict the operator-algebra consistently. At r = 0, the appropriate definition is[i3,15] J
e - Y ~ - ~ ( ° ) = ~ e.~ ~
~(d)~a~,,-Mx ~a) t ~ ,. , ,~~ (-1) ~ - ~ cih(.~_SO ,~,x
(54)
M=-] where cl is a normalisation constant. The transformation of the ~ fields is similar to the one of the fields:
J ~~' ) ) = , ,", ~ ", )
7 , ~(~) = ~/"LJ :t: MJ LJ ± M 4- lJ ~,+x. ~)
(55)
Thus we define,
j.,. = j+~-~s:~ + e i ~ ® 7 , , Js = J~ + 73,
(56)
which does give a representation of 1. Then one easily checks that J:b exp(-Jc~_~) = J n e x p ( - J a _ 4 ~ ) = O,
(57)
J.-L. Gerva/s / Orar/ty-maCter coup//n~
131
so that the quantized Liouville field is a quantum-group invariant. Clearly, the quantum case and the classical case recalled in section 2 are remarkably similar. 2) One may also considet gravity with boundary, following [1,2,14]. A typical situation is the half circle 0 < o < ~r. One may set up boundary conditions such that the system remains conformal, albeit with one type of Virasoro generators only. The left- and right-movers become related as is the case for open strings. The appropriate definition uf the .metric becomes[14]:
e-Ja-d~(°)
=
cj E
(J) Iv~M ('1)(¢)~N (.0 (2a -
(`58)
M,N
where
A(J) = (J, MI {e -'hag ~.~ c''('+')j" (J+Y+' c'"°~'-')e '",'/='t- IJ, N ~' M, s
L'J!LsJ!
r,s=o
J
'
(,59)
These fields may be shown to be mutually local and closed by fusion.
`5. T I l E DRESSING B~ GRAVITY In this section we study the dressing of eonformal models with central charge D by the Liouville field with ce~,tral charge C so that C + D = 2O.
(00)
We shall he concerned with the case D < 1, where tile Liouville theory is in its weakly coupled regime C > 25. As is recalled in the appendix A, tile existence of the UCF's is basically a consequence of the operator differential equations 37, 38. The latter are equivalent to the Virasoro Ward-identities that describe the decoupling of null vectors. Thus tile UCF's, with appropriate quantum deformation parameters also describe the matter with D < 1. We will thus have another copy of the quantum-group structure recalled above. It will be distinguished by primes. Thus we let
h'
~
h'
D='-l+O(-~-+~+2)=l+fi(-~-+~+2),
"
with
h'~'-~2 -
,
(81)
and 60 gives h+h' = g+h' = 0
(63)
Of course we choose tile matter and gravity fields to comumte. Using the notation of last section, the complete quantum group structure is thus of the type:
{
[u,(st(2)) ® u,~st(2))lo[~,(st(~)) o ~(st(2))] } ®
{[~_,(st(2)) ® u,_,(st(2)~l®[.~._º(st(2)l ®.~-,(st(2))] },
(64)
J..L. Ocrv~ie /
132
Gravi~y-ma~r coup]ir~s
where the first (second) line displays the quantum-group structure of gravity ( m a t t e r ) According to the above results, the spectrum of weights of the gravity and matter are respectively given by
(65)
The connection with Kac's table will be spelled out in section 5. As we shall see in section 5, this last formulae is consistent with the usual formulae for minimal models. From the standpoint we are taking, the most general matter field is decribed hy an operator of the form exp(-(J'a:.. + f'~+)X), where X(~r, r) is a local field that commutes with the Liouville-field and whose properties are derived from those of@ by continuation to central charges smaller than one. Following our general conventions we let h' = w(a~)2/2, and h' -- x(~.)2/2. The correct screenin~ operator is the field e x p ( - ( J a _ + f a + ) ~ ) with spins J and f that ~ (J, J) + A n (ye ft) __ 1. In this connection, it is an easy consequence of 60 that A ~ ( ~ , _ j ' _ ') + A ~ , ( J ' , Y ) = 1,
,~(-j~-
l,J') + A~(J',~)--
1
(67)
These two choic~ correspond to the existence of two cosmological terms. Indeed, the unity-operator of matter (J~ -- .~' = 0) is dressed by exp(c~+~,) and exp(a_~) respectively with the choices 67. As is usual, we choose the latter as cosmological term so that the spins of gravity and matter fields will be related by second Eq. 67. Thus we shall be concerned with lr~atr:.x elei~cn~s of the operators
v~,.~(~,~) ----e(( )" + l)a_ - J ' a + ) ~ ( a , r ) e - ( J ' a " + ) " a ~ ) X ( a , r ) .
(68)
Concerning matter, choosing J' > 0 and J ' > O gives ~M > 0 and the formulae of last section may be directly used. Concerning gravity, it has been already emphasized[10,11], that the dressing of matter-operators with positive spins requires the use of gravity fields with negative spins. For example, the cosmological term has Y = -1, J" = 0. Thus the gravity-UCF with parameter h must be extended. This problem was overcome in [10,11] as follows. First, the q,ai,tum-group structure has an obvious symmetry in .I -~ - J - 1 with fixed M, s~ that the universal R-matrix and Clebsch-Gordan coefficients are left unchanged. Second, the coefficients IJ, m ) ~ also have a natural continuation to negative spin. F~r postive J, one has ( ~ ) ( m ) is defined in [11]) I-
Y - 1, ~ ) ~ = IJ, m)~ (-l)l+m/[(2isin(h))l+2s A~)(m)]
It follows that there exist fields ~ - ~ ) similar to ~ ) so that J ~ ' - z ) ( o ) oc Z
(69)
with fusion and braiding similar to ~ ) , and fields ~(-J-z)
( - 1 ) ' + m JJ'~v)~/A~)(=) ~bk-'-z)(a)
(?O)
m:-J
Next, the appropriate dei:nition of positive powers of the metric is, by extension of the discussion of the previous section,
J.-L. Gervais / Gravity-matter couplings J
eIJ + 1)~_~(¢,~) = ~ e1_ ,., _ ,
~
133
(_l)j_,~ e 'h(~-'> ~ " - ' ) ( ~ + ) ~ £ ~ - ~ ) ( , _ )
,/~.
(rl)
Af=-J
Indeed, first this field is local since the exchage properties of the fields ~ J - 1 )
and ~(~J-D arc the same
as those of the fields ~ ) , and ~ ) respectively. Second, it may also be re-expressed in the Bloch-wave basis, and this shows that this field leaves the eonditio~ m = ~ invariant. The symmetry between J and - J - ! is the explanation of the continuation performed in [16]. Next consider the Ililbert space in which we are working. The UCF's which appear in 64 live in spaces with highost-weight vectors of the form I~, 0) ® I~ , 0) ~ I~', 0) ® I~ ' , 0).
(72)
(The two UCF's of a product of the type Uq(sl(2)) ~ Uq-'(sl(2)) are realized in the same llilbert space). In the restricted llilbert space, one has m = ~ , and m ~ = ~ ' . Thus we introduce states of the form Its, m') -- Ira, 0) ® [,~, 0) ® Ira', 0} ® I~', 0).
(73)
The next point concerns the physical on-~hell states. They should satisfy the conJition (to + To + L~, + ~o - 2)1=, = 3 = O,
(74)
where the notation is self-explanatory. According to Eq. 40, this is satisfied if h
=-2
h
2
~ ( 1 + ~-) - ~
hi "1
+ ~(
a'.~
h' - - w '2 = 1.
(75)
+ ~;) - 4~-
It follows from Eqs. 36, 60, that h
n 2
~(I+~)
h'
x 2
+~(l+F)
C+D-2
--
2~-I.
(76)
Moreover, according to Eqs. 39, 63, h ' ~ '2 =
h'~' :~ =
-h~'
(77)
2
so that, according to Eq. 63, the on-shell conditiot~ is ~2 = ~ 2 . Tim sign ambibuity is related with the two possibilities of cosmological term. It will be shown below that the choice, which is consistent with the abow: definition of gravity-dressing is m = -~',
or, equivalently,
~ = m ~.
(78)
The coupling constants Ci,2,s appear in the three-point function as follows
I
k,I
It was computed in [15]. The result takes the form
B~;,
134
J.-L. Ge~v~$ / GraviLyomagter coupllngs
where B j, ~., are normalisation-dependant constants. Tile calculation is sgmewhat involved, but the result Ls remarkably simple. Next, we make contact with other approaches to the same problem. For this we first explictly connect the present group-theoretic notations with more standard conventions. Eqs 51, and 61, correspond to the usual notations C --- 1-4- 3Q 2, and D = 1 - 12o02, for tile gravity and matter central charges. The screening charges are given by (we chose c~0 > 0) . ~ = Q =t=~o, . ' . = i(ao=~ Q).
(81)
The dressed vertex operator 68 may be rewritten as
Vj, ~, =--e(( $' + 1)o_J',~+)~ e-(J'~'_ + Y ' ~ ) X = e~(k)~ - ikX/,~_, /~ -- ( ) + I ) ~ - - J'o~+,
(82)
ik = ¢t_(J'¢/_ + Y'd+),
(83)
A simple calculation, using the formulae just given leads to
k = 2J' - 2Y'h/~ =
2Y+ 2(J +
1)h/~,
~(~) + 0 / 2 = (k + ko)/~_,
,~:~=
~.o._
==
I - h / r (84)
According to 52 65, and 66, the weights are given by
For rational theories (D = 1 - 6(p - p'f/pp', p > p' > 0),
h = -'~',= ,¢p'Ip,
?, = - h ' , = ,¢p/p',
(86)
ko = ( p - p ' ) I p ,
A ~ reduces to Kac's table
[(rP-~/)~-(p-p'f], r=2J'+1, s=2d'+1, k+ko=r-~h/,~.
(87)
The momentum k is defined so that it takes rational values for rational theories. For critical bosonic string with Regge slope a', the tachyon vertex is e x p ( i k X J ' ~ ) . Eq. 82 corresponds to a ' = 1/(c~_) 2. With the conventions just introduced, the result Eq. 80 takes the form ¢m,a
~
- (~r(2 + h / . ) r ( ~ -
B(kl)
h/,O/h) ~ -= .4(k,, k~,k~) = 1 1 Jr(f, + i0)} ~'
(88)
where we left out an overall factor which does not depend upon the momenta. Our last task is to re-establish the cosmological constant, in the present approach, it comes out as follows. Our basic guideline was to write down the most general local operators, as was discussed in section 2. We have not yet really done so, since we may multiply the righ'-hand sides of 54 and 71 by p'f=/2 on the left, and by iz'~/2 on the right, without breaking locality. Th;s constant p~ is arbitrary, and will play the role of the cosmological constant. According to 42 one has
/~-='/~¢'("~ "='/~ e
rn m
t'¢
- t""+;"'/h'r'('~ Y) -e ~m m "
(89)
.I.-L. Gcrvais/ Gravity-matter couplings
135
In the Coulomb-gas picture, the power in Pc is equal to the total number of screening charges. We shall agree with this definition if we let
~(-!~,~_ + s~+)~(.,
T) = .~+.~/h /,:/2c-(J~_ +
]~,+),(~, ~) j,U~.
(90)
7";. ',s we have, according to Eq. 83, (p=)
V~ i, j.(¢,~.)
.",
=
,
t,7(.I +t)+s ./hp-~/2~2j,,~.,(a,r)py/z"
(91)
The flr';t factor coincides with tile KPZ-DDK scaling factor[17-19]. In particular, one h ~
v~",~)(o, ~) = t,7'u7 ~/~Vo, o(., ~) u'. /~,
(9~)
which is the expected scaling behaviour of the cosmological constant. The operators p~¢r~l~ induce a translation on tile variable conjugate to the Liouville .mmentum. This Liouville position-operator is proportional to q(0t) which is tile zero mode of tile free field ~1 whose properties were summarized in the section 3. This translatioe wl,ich is actually a global Weyl transformation, is analogous to the translation of the Liouville field i . the work of DDK. This is ~ e ' l by computing next the pc-dependence of the three-pob,t function. One gets immediately A(~.)(/ca, k2, ks) = A(/q, l::, ks) t ~ °+~'(~'+s~'/h).
(93)
The term #~o arises when p-~=12, and tt~/2 hit the left and right vacua respectively. In the DDK discussion[20,19], it comes from the term of the effective action which is linear in the field # 5 One may verify that the power of p~ is equal to uz + 9~lr/h where u2 and ~ are the gravity screening-numbers. This agrees with the usual definition (see, e.g. [18]). Finally we compare 88 with the result of the matrix model. This part follows [18] closely. The twopoint function A(,.)(k, k) is determined by starting from the three-point function with one cosmological term A(u=)(O, k, k). One writes, making use of 92,
~A.o)(k,k) = A.~)(0,k,k), A..)(k,~) = ~ A . o ) ( 0 , k , k ) .
(94)
Similarly, the partition fimction satisfies
ap< Z (~=)
= A(~)(O,O,O),
Z ( . . ) = (Wo)(='o (m)a l)(w'o _ 2)A(~o)(0,0,0).
(9~)
We finally obtain the rescaled three-point function
I)(kl,k~,ks) ~
(.A(p~)(kl, le2,ka)) ~ Z(~) = I-L(/,-t + k0) i]~.A(~.)(kt,k:) (~r/h + 1)(x/h)(~c/h - 1)'
(96)
which agrees with the results of the matrix models. Clearly, the key point in this final verification is that the final expression 88 for the three-point function factorises. On the other hand, 88 vanishes whenever #5 It would not he there on the torus, since one would take a trace. Tiffs agrees with the DDK result where its contribution is shown to be proportional to the Euler characteristic
136
J.-L. Gervals / Gravity-matter couplings
k: + ks = 0 for a n y o f t h e three legs. According to 87, this h a p p e n s for rationa'_ theories, a t t h e border of K a c ' s table, where r -- p ' , a n d s = p. T h u s formula 96 holds only when t h e b r a n c h i n g rules are satisfied.
References
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