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CONFORMAL INVARIANCE AND FINITE SIZE EFFECTS IN CRITICAL TWO DIMENSIONAL STATISTICAL MODELS
Claude Itzykson
Service de Physique Théorique, CEN-Saclay, 9H91 Gif-sur-Yvette Cedex, France
Communication présentée à : 9. Sitges conference Sitges (Spain) 26-30 May 1986 1
CONFORMAI, INVARIANCE AND FINITE SIZE EFFECTS IN CRITICAL TWO DIMENSIONAL STATISTICAL MODELS
Claude Itzykson
Service de Physique Théorique, CEN-Saclay, 91191 Gif-sur-Yvette Cedex. France
I. PRELIMINARIES
1. A systematic review of the recent achievements in conformai invariant two-di mensional field theories would already require a volume size. Here we shall discuss some finite size effects, because they reveal important features as noted by Cardy. The computation of the partition function in a finite geometry -a torus- enables one to understand the spectrum of the theory. The later is a compendium of the critical dimensions of the various operators. Two simple examples, namely bosonic or fermionic free fields (respectively the Gaussian and the Ising model) will serve as a pedagogic introduction, although the corresponding computations are not without some subtleties. We then proceed to considerations related to modular invariance. The forthcoming analysis is of course based on the fundamental work by Belavin, Polyakov and Zamolodchikov (quoted as BPZ) and its subsequent development by a number of authors including Dotsenko, Fateev, Friedan Qui and Shenker, Cardy and many others. It also relies on the mathematics of infinite Lie algebras (also known as current algebras) elaborated by Kac, Feigin and Fuchs, Rocha-Caridi... The whole subject is to a large extent similar to the one of string theories. The following presentation is based on a joint paper with J.B. Zuber and a letter written in collaboration with H. Saleur. Let both of them receive here my warmest thanks. A more thorough exposition is to be found in notes for lectures delivered in Marseille.
2. It will be assumed that the reader has some idea of the essentials of two dimensional conformai invariance presented in detail in the beautiful BPZ article. At variance with the renormalization group analysis this work emphasizes directly the massless critical theory and its associated energy-momentum tensor. The later enables one to study tho effect of a local change of coordinates. This is a familiar construct in the context of quantum field theory, but not until now in statistical mechanics. Changing the coordinate system can be interpreted in a passive way -a mere repa- rametrisation- and therefore without effect on the physical content, if not on its mathematical description. It also admits of an active interpretation as a map from 2 domains to domains (with correlative boundary conditions) in which case it is cf
objective significance. The energy-momentum tensor T„.v is the generator of such infi nitesimal changes. Given the corresponding local infinitesimal variations SAlx) of the observables or fields Alxj, the basic (Ward) identity for correlations reads in twc dimensions
v S A &
0 'to^lpMl))^™- (2: given the change of coordinates x^" -* x4* • 6£.*(x.). The factor 2ir is included for convenience. If !D is the full Euclidean plane, requiring invariance under translations, rota tions and dilatations imposes the conditions
£ (A1|x1j...avfcAp|xpj...)
T*v(;) = V(x) (3*
Z* T£U) = 0
Brackets without subscripts refer to the infinite plane and coordinates have beer, assumed Euclidean x2 * x2 • x2,.
Under a general coordinate transformation x -* x + S € |x|, a scalar field behaves
as A(X) -* AJx) • 6A(x)
LA 6A(x| ) » &«(x).ô • Y Ê'6^*1 AU) (4,
where \ is the scaling dimension of A, since the local dilatation factor is 3
DU*6fc) i / z i D(x) oi 1 + - â.&C If &€ decreases fast enough, tnen integrating by parts in (1) results in
t This form of the identity states that correlations of 6 *T^v(xj with observables, of arguments x , vanish except at the points x . This is what is meant when one says that
T^v is conserved
3%v(x) =0 (6)
One can of course write similar expressions in arbitrary dimension d.
Invariance under euclidean and scale transformations entails in this framework invariance under the full global conformai group. In two dimensions it can be identi
fied with the isochronous Lorentz S0(l,3) or equivalently SL(2,C)/Z2 realized as
rational (or Môbius) transformations group on the complex variable z = Xj+ix2
az+b z -» z' = f(z) = , ad-bc = 1 (7) cz*d
3. Specific to two-dimensions is the existence of a much larger set of conformai transformations beyond the global ones. While it remains true that the above Mobius transformations are the only global ones of the completed plane (i.e. the Riemann sphere), it is nevertheless true that an analytic function mapping one-to-one a domain D on a domain f (3)) = D' induces a conformai map B -» D'. Let
z -» z' = f(z) f-analytic in D (1)
be this invertible map. One assumes the existence of primary tensor-like fields which transform according to
AJz.z) dzh dzh = A'(z'.z') dz'hdz'h (2)
The pair (h,hj defines the conformai dimensions or weights associated to A. This con tains both, the tensor character of A (i.e. its behavior under rotations), thus h-h
may be called the spin of A and has to be an integer if A is unaffected by a 2ir rota tion, and the scaling dimension k
\ = h • h (3
In particular a real scalar field will be such that h = h while in general h * h (and the bar over h does not imply complex conjugation).
It is convenient to use z and z as a substitute for Xj and x2, remembering that z and z have to be treated as independent variables. The metric takes the form
ds2 = dzdz g =g = 0 g - = g- = - ZZ 2 2 ZZ ZZ 2 CO
9z 2Ux* * dx2) bz 2\bxx dx2)
From the constraints on T v, it follows that it has only two independent non vanishing components
T = T iT T T ZZ |< ll- 12> -z-z " 5( 11*«12) (5)
With the proviso explained above, the conservation law (2.6) implies that T (T } is zz \ z z; only a function of z (of z). Henceforth we write
T = T(z) T- - = T(Z) (6) zz z z > /
Since there is a constant parallel between properties in z and z, only the z part of the formula (or the z-dependence of fields) will generally be explicited. It is understood that the z miror equation should be added. The Ward identity can be transformed to read
2 (g(zp) ~* hp g«up)J
where the contour "t encircles the points z} Equivalently by Cauchy's theorem this is also an expression for the short distance expansion
( h-p 1 d^
When g(z) in (7) is a polynomial in z of degree at most 2 it corresponds to an infi nitesimal generator of the global conformai group, in which case both sides must vanish. This is equivalent to requiring that T(z) behaves as z'1 at infinity and leads to the following invariance constraints on correlation functions
translations L-> -—
1 scale transformations 2-, zp -— • h
special conformai Z- \zl -— • 2h z 1
In particular this determines the 2~ and 3~point correlations in the plane.
4. For the purpose of normalization we assume that the partition function is the plane is unity. This amounts to say that the free energy per unit area vanishes. Thus
On the other hand the correlation function for two T's (or two T's) has no reason to vanish. Its non vanishing will be the signal of some anomaly as we shall soon see. Contrarily to what is implied in the title of these lectures and other works on massless two -dimensional field theories, the fact is that they are not invariant under general conformai transformations, but, should one say, covariant in a definite way, which we are now going to characterize. As T(z) behaves as z'u for large z, the two-point function takes the form
The constant c will be the same for (T T) and real, under mild reality assumptions or.
the statistical model. It plays a major role in the theory and is called the central charge for reasons which will appear below. Identifying c will, in a large number of cases, be the key to understanding the critical properties. Let us insert (2) in the Ward identity stating the behavior T -» T*6T of T under an "infinitesimal" conformai transformation z -» z' = z*&€ g(z). As a result
<6T(z)> = 6€ <|> —- g(z')
Corresponding to (3«7) we would expect the transformation law for T to be homogeneous.
with hT = 2
6T(z) = Seig(z) ^ • 2g'(z)j T(z) false (V! which would have required <&T{z)> = 0. This is not the case if c * 0, and is therefore the signal mentioned above: T does not behave as a primary field, but rather picks an inhomogeneous part in its transformation law
6r(2) = ScfgU) j^ + 2g,(z)j T(z) + ^ 6€ g-(z) (5)
For a finite transformation, the integrated form of (5) reads
T(z) dz2 = T'(z*) dz'2 • — {z'.z} dz2 (6) where {z',z} is the Schwarzian derivative of z' = f(z) with respect to z, namely
d3f/dz3 d2f/dz2' 3 {z'.z} (7) df/dz 2 df/dz
This derivative vanishes when z' is related to z by a Mobius transformation, hence the added term does not modify the behavior of T under global conformai transformations, but it does so in any other case. Equation (6) expresses therefore an anomaly.
5- The central charge admits a direct physical interpretation as a shift of free energy in a finite geometry, analogous to the Casimir effect of quantum electrodyna mics. The simplest such situation is the one of an infinite strip with periodic boun dary conditions (a cylinder). For this purpose consider the map z ** u from the complex punctured plane to a periodic strip of width L (u ~ u*nL)
z = exp 2iir - (1]
Thus
2 22 T U (2) strip( > * (r) ( We(*> -I*)
SinCe
2*y c_
Under an infinitesimal deformation of the strip, which need not be conformai,
d2u
Sin ZL L |r
Using cartesian coordinates we envision a dilatation of the strip &€j = &€Uj , &€2 = 0, a quasi-conformai transformation. Assume it is meaningful to introduce the free energy per unit longitudinal length F(L), then (4) translates into
<*F(M 2w c &F(L) = &€ L = &€ (5) dL L 12
Since the free energy per unit area vanishes in the plane (L -* oo), then
F(L) - c | i (6)
Thus c can be extracted from the finite size effect of F(L) at criticality. Equiva-
lently, at this point, let AQ be the lowest eigenvalue of the transfer matrix, then ir IT In A = este L • c - L"1 • ... with a universal L"l correction of size c -. This 0 6 6 assumes of course that longitudinal and transverse length are measured in the same unit, hence requires care in an Hamiltonian approach. There are some qualifications to formula (6) when the state with the lowest eigenvalue of the transfer matrix does not correspond to a "vacuum" state, a phenomenon which can (and does) occur for non unitary models (see §15). At any rate this shows that the central charge is within reach in numerical simu lations and has a well defined physical significance.
6. It is possible to shift from the statistical point of view asing sums over Boltzmann weights to an operator one using a transfer matrix or an Hamiltonian. It is most convenient to use a radial quantization, where propagation in "time" really means dilatation as above. To each observable A is associated an operator A, whereupon the Ward identity reads
sî(z) '£e{f|z'|>U| S5I,,,,*,,,,Î<,) -f|«'|<|.| fsïU')S(z)îu'»} (U
Such an equation holds in particular for T itself with the appropriate 6T(z). It is customary to expand T(z) in a Laurent series as
T(z) * ^- (2) -oo zn'2 8
Inserting this expansion in (1) as well as the corresponding one for &T(z) and using Cauchy's theorem, one finds the celebrated Virasoro infinite Lie algebra
[L.LJ = (n-m)L„ • — n[n2-l) & _ . (3) and a similar one with L's in place of L's with the same central charge c; L's and L's commute. If c vanishes this algebra is one of the infinitesimal generators
«. • «'- h ci of the diffeomorphisms of the unit circle (for |z| = 1). And (3) is a central extension (the unique possible central extension according to Gelfand and Fuchs) of this diffeomorphism algebra using an operator (the identity operator implied as a multiple of c) which commutes with every Ln. Hence the name of central charge.
When n and m are restricted to the set +1, 0, -1 one has a closed SL2 algebra (the
complexification of the Lie algebra of SL(2,C) splits into two commuting SL2 algebras, one involving L's, the other L's). This reflects global conformai invariance unaffected by c. The unitarity condition states tha'
L* = L L* = L (5)
This will eventually be assumed except at the very end. The invariant vacuum or ground state is conformai invariant, hence annihilated by
LQ, L+j (similarly with L). On the other hand the expansion (2) would lead to a non-analytic behavior of T(z)|0> as z -• 0 (negative infinite "time") if we were not to require that
\\0> = 0 n £ -1 (6)
and similarly at z -* *oo
<0|Lp = 0 p < 1 (7)
More generally define (again z, h omitted)
|h> = lia Â(z)|0> z»-0 (8) Then (3-8) shows that Ln|h> = 0 n > 0 L0|h> = h|h> (9) The vacuum can then be identified with the state |h=0> corresponding to the unit operator. Starting from the state |h> we get a collection of states by acting on it with L 's. Using the Virasoro commutation relations as well as the above conditions, it is realized that all such states are linear combinations of those of the form L{p}|h> = L.p L.p ...|h> 0 < p, < p2 < ... (10) It is consistent to define a level structure as the eigenvalue of L -h. As L +L 2-t pi Q 0 '0 generate dilatations, this operator defines the Hamiltonian (while L0-L0 generates rotations). 7. In any definite model, states such as (6.10) need not be linearly independent. Also if one insists on unitarity there may exist further constraints. Let us introduce a distinction between the vector space of states of a given model erected on |h> by acting with the representatives of the L's, and the corresponding abstract vector space or Verma module where states such as (6.10) are linearly independent. The former can generally be thought of as a quotient of the later. Subsequent developments in this section will analyze the Verma module. A prime tool in this investigation is the so called contragredient form obtained as follows. Act on (6.10) with a product Lr_i, qx > 0, ZJ qt = Z* px. The result is a scalar multiplying |h>. Denoting this scalar as ({q},{p}) it defines a symmetric matrix. The number of possible states at a given level N « i p( is equal to the number pN of partitions of N. Recall that (p0 = 1) 00 £ p. «" • r-1— ' ph) (l> n M Then the contragredient form is a pN by pN matrix. If one deals with a unitary model and |h> is normalized, then ({q},{p}) can be identified with the corresponding scalar product and must define a positive definite matrix. One can prove the following general property of Verma modules. If at level N the determinant of the contragredient matrix -called the Kac determinant- vanishes, then at some level 1 £ N' £ N, there exists a special state |s> such that L0|s> = (h+N')|s> 10 and Ln|s> = 0 if n > 0. In other words the Verma module contains an h+N' invariant subspace, namely the one erected on |s>. Indeed by hypothesis, there exists a linear combination \a> of states of level N such that L|qj|a> = 0, if 2- q, > N. Let 1 < M < N be the smallest value of 2-, qi with this property. If M = 1 then \a> is such a special state |s>. If not, choose any non zero vector |a'> among the linear combinations of states of the form Lrqi|a>, 2- q; = M-l set N' = N-(M-l) < N, then any h{q). ^ Qj = N* annihilates |cr'> and the process repeats itself until we hit a special vector at some intermediate level between 1 and N. 8. It is an outstanding achievement of the theory to give the Kac determinants in closed form. They will of course be polynomials in c and h. With a finite amount of labor one can work out the first few non trivial cases. The easiest one is at level two where the contragredient form is {2} {1.1} {2} /4h • ^- 6h \ (1) 12 {1.1}V 6h 4h(2h*l) with a determinant 2 det2(c,h) = 2hjl6h • 2(c-5)h + c) (2) This vanishes for h=0 and h = h+, where 5-c ±\l(l-c)(25-c) h±- ïi (3) One can check that u>° MbiTL-'-L-)|h±> '*' is such that L0|s> = (h±*2)|s> Ljs> « 0 n>0 (5) By virtue of (4), fields with dimensions h+ will be such that their correlation functions satisfy differential equations if one insists that the state |s> in fact vanishes (degenerate representations) -a subject which we will not pursue here. To show that the general form is not easily obtained one may mention that seme time elapsed between the statement of the result by Kac and the appearance of a detailed proof due to Feigin and Fuchs. A subsequent proof was also given by Thorn. 12 The result is best described for c * 1,25 using the following paranetrization d\.i to Friedan Qiu and Shenker, which will be of particular interest when we turn t; specific applications. Set m(m+l) it [r(m+l)-sm]2-l r.s ~ i4m(m^l) " •»-•• "•1-s Here m is for the time being not specified, but r and s are integers (not m-r, ••l-s if m is arbitrary). Then N - detN(c,h) = este x II (h-hr J** " (Ti r.s=l, l£rs where PMis again the number of partitions of the integer M. This result completely exhibits the zeroes in h. Verma modules built on h will contain special vectors at r . s level rs. In a particular realization, it will then be consistent to set these special vectors equal to zero -the more so in the unitary case- since then their norm vanishes. One then speaks of degenerate representations. For a fixed value of m (i.e. of c) degenerate representations are then labelled by a pair of positive integers r.s which could range up to infinity, in which case the model would involve an infinite number of primary fields. One is first interested ir. finding those circumstances where only a finite number of primary fields are necessary. For this to occur BFZ show that m must be rational. One can then find twe relatively prime numbers p and p', such that 1 ^ p' < p. and p P-P P-P Thus (P-P')2 c * 1 - 6 PP' (9. h-» = 5S7 [lrp-sp,)'-(rp,)'] and for the primary fields a (10) l Such theories may be then be called properly degenerate. A further significant requirement is unitarity. Elementary considerations show 12 that this restricts the range to c £ 0. h > 0. Friedan Qui and Shenker have ergued that Mhen c £ 1. no further constraints exists beyond those of irreducibility discussed above (in particular for c = 1 reducibility occurs for h = n2/*4 with n an integer): but for c between 0 and 1 unitary forces one to take a integer (or p = a*l. p* * a in (8)). In suaaary the cases which will be analyzed in sections 11 and 111 except at the very end are those degenerate unitary aodels generated by finitely oany primary operators, with a an integer 2 3 (11) 6 c = 1 • (a*l) the possible list of hr % (or hr s) is to be looked aaong the •(•-!)/2 possible val [r(a*l)-sa]2-l h h r.« * .-r.,M-, - •-4a(a+l. .») • l^S The restriction on the range of r and s is such that the prinary fields together with their "descendants", i.e. fields of the fora Le.n\L, - h.h closed algebra under short distance expansions. We shall find an alternative confirma tion of this stateaent in the sequel. 9- In the degenerate representations not all states built on |h> are linearly independent at a given level N. The nuaber of linearly independent states, ds., is therefore samller than p, the nuaber of partitions of N. A way to record these quan tities is to build a generating function called a character (dQ = 1) X(c.h) = Z d,q"'h (1) N=0 a series which converges for |q| < 1 since d, S p,. For the degenerate unitary repre sentations their expressions have h*en given by Rocha-Caridi based on the work of T*igin and Fuchs. For short, the *eger a being fixed, we label characters by the pair [r.s], then —-—-r<2na(BM)T(»*l)-SB)M] 1a(a^l)L J >-*3(q,"p7S? - (s - -s) (2) Recall that (Euler's pentagonal identity) 13 •00 n(3n*l) P(q) « fi (l-qn) = 2 (-1)" q 2 (3) It still requires some work to extract the dN's from (2). However it is obvious that all powers of q differ by integers. is not a This means that Xrr s](q) single valued function of q, but rather has a branch point at the origin. If one sets q = exp 2TTÎT, Il T > 0 (T) «[r..]^") -•""'- ^..](-»") CO 2 The character Xrr gi can be extended as a function on a Z lattice, with the following properties x[r,s] = x[-r.-s] = x[r+2mk,s+2(m+l)k'] = x[m-r,m*l-sJ = ~x[r.-s]= ~x[-r,-s] (5) Using this in conjunction with Poisson's formula, it follows that under the substitu tion T -» T* = -1/T c c , -2iirr— ^- -2inT -r m(m*l) m(m-l) where the real symmetric matrix A of square equal to one (hence A is unitary) reads 8 i/2 .,(r*s)(p+a) rp so A (-1) sin IT — sin IT (7) [r.s],[p.a] = m(m+l) m m+1 Extended on Z2 ® Z2, A has symmetries compatible with (5)- It is important in the sequel to specify under which conditions the difference hf s-hr, ,, is an integer k. The requirement 4k m(m*l) = {(r+r,)(m+l)-(s*s*)m}{(r-r,)(m*l)-(s-s,)m} is equivalent to the statement 2 2 s -r' = 2m[rs-r's']2 mod 4m m even' „ s2-s'2 =0 mod m+1 (8) Ik v2-v'2 * 0 mod m m*l even < 2 2 s -s* = 2(m+l}[rs-r's']2 mod <»(m*l) where [a]2 means the residue class modulo 2. Two particular equivalent solutions are r' = m-r r" = r r even or odd according to m = 0,2 (mod 4) with (9': s* = s s" = m+l-s s even or odd according to m*l = 0,2 (mod 4) such that (m*l-2s)(2r-m) hr.s " K-r.s ' hrs " hr . .. 1 - s = jj <«>} The additional conditions in (9) ensure that this is an integer. II. FREE FIELDS - GAUSSIAN AND ISING MODELS 10. As a first illustration, we look at the Gaussian model described by a real free scalar massless field * = \ (*P)2 (1) where d «V^y = In » r In — (2) r i2 ' Z)2Z12 The traceless energy-momentum tensor 1 T^v = ^ can be brought to the form T = T = 0„9)2 T = T-- = (a- A Wick-ordering subtraction procedure is understood to insure that f a2 1/2 Hence the central charge is c = 1 (6) Note that properly speaking We next would like to see what happens in a finite region with the shape of a parallelogram with periodic boundary conditions (a torus). This amounts to take doubly periodic fields according to a translation lattice generated by tv.o complex numbers u. and u, «Plz+^Wj+n2!^ j = Set w. T = Im T > 0 (8) w. If T is pure imaginary we have a rectangle in which case we may use a transfer matrix formalism with effective Hamiltonian L0 • L0. If T is complex, Im T is coupled to L0 • L0, Re T to "translations" generated by L0 - LQ. In general due to scale inva riance, results are expected to depend not on Uj and to., individually but on their dimensionless ratio T. Furthermore no particular choice of generators of the lattice should be preferred. Provided that nnn22*ni2n2i = 1, •U integers, or = ni .Cir qualify as well. Then T is changed into n22T*n21 T' (9) nU+ni2T This is a restricted class of Mobius transformations which generate the modular group PSL(2,Z). Correspondingly one speaks of modular invariance. Let us look at the partition function, defined formally as Z = DpeyT vJA 6 — J d2x Here T means that integration over space is restricted to a fundamental cell of area A. The extra 6-function ensures the zero mode subtraction necessary to avoid a blow up 16 of Z. This Gaussian integral is readily formally evaluated as a product to the power -1/2 of all non zero eigenvalues of ainus the Laplacian acting on periodic functions. Let us introduce a basis k1 , k2 of the dual lattice (dual over integers) k1 = -i — k2 = i — (11) A A If k stands for any linear combination with integral coefficients of k1 and k2 . then Z = A1/2 |TT' |2*nk|2 >"1; 2 where "w' means product over all the dual lattice except k=0. This is still undefined (i.e. divergent) and requires further subtraction accomplished as follows. Compute first G(s) "2- (12) 2-mOj m.n |m*nT|2s which converges for Re s > 1, but has a simple pole at s = 1, then analytically continue it as a meromorphic function in s. Set Zj = A1/2 exp - G'(0) (13) which coincides with the previous expression for the partition function, were it not for this divergence at s = 1. This procedure has now become known as ^.-function regularization. The suffix Zj is to remind us that c = 1. We shall not spell out here the details of the calculation which has been repeated many times in the context of string theories. Sufficent to be said that it has first been carried out by Kronecker circa 1880 and is presented with great elegance in A. Weil's book on elliptic functions. The result is Z,(q) = ; q = e21TtT (It) (ImT)1/2|n(T)|2 where Dedekind's function r\{r) is 00 T)(T) * e2inT/2k p(e2i*T) = e2111'24 V\ ll-e2l"nT) (15) As expected this only depends on T and apart from (1«T)'1/2 factorises into the pro duct of a function of T (or of e2i1tT = q) and a function of its conjugate. It admits the expansion (IBT)1'2 Z,(q) « (qq)"1/24 S_ p^qV (16) N.N 17 involving a sua over "right moving" and "left moving" modes which may be looked upon as 1 L 1 2 jqq)- '" Trq ° q^o = (qq)" '^ |Xç., .h .a (q) | (17) The spectrum of L0(L0j is over non negative integers, starting at L0 = 0, with •n T 2 6 L :- - ÏϱY' e (18) In Z, w T ^^(L)=â in agreement with our expectations for c = 1. Zt has to be a modular invariant, hence (n r*n \ •1/2 22 21 (nn*n,2T) « t|(r) (19) nn*ni2T where € is a phase, actually a 24-th root of unity. The prefactor (Imr)'1/2 is specific to the case c = 1 (although essential to ensure modular invariance). It will not appear when c < 1, in which case we expect that Zc = (qq)~2Ï I Xc-h(q) X^q) (20) {h.h} with a sum over a set (h.hj corresponding to the primary fields of the model. The Gaussian case is a limiting one, where only the operator I is primary. Thus if we succeed in finding, given c, a combination such as (20) (where some (h.h J may be repeated and some omitted) which is a modular invariant, we have a can didate for the partition function (on a torus) for a corresponding critical model. All coefficients in (20) are either positive integers or zero and the coefficients of (h.n) = (0,0) has to be unity to have a non degenerate vacuum. Finally the factor 18 ( (periodic) strip provided h.h ^ 0. 11. Returning to the Gaussian model, the two point function fulfills (A is here the Laplacian) -A«p(l) where the r.h.s. is modified to account for the zero-mode subtraction. The symmetric real solution is neatly expressed in terms of Jacobi's 6-functions. We use the notations ,2i1tT _ JiJt "V-! 2i1t Z/w. q = e* * = e y = e 00 (l-y1) fi (l-yq^jl-y-U") F(z) = (2) P(q)' •oo n(n*l) — Z (-y)n q 2 P(q)3 -oo Then exponentiating the solution of (1) T12 * exp -2«p(l ) ; (la z12/Uj) OJ, F(z * exp -2v> • Im zJ2/w, ^ -> ImT a symmetric doubly periodic solution. As z12 -* 0 ri2 - Z12Z12 (4) in agreement with the infinite plane result. In particular for k,l,N integers exp -2^ where Dk j(q) stands for N'N 9 1_ (ei(tt-i) oo 2i-nk ^ ' -2ink N-ZÏ 12 N2 n*~ n*— N N N N D i(q) = q l-e q 1-e q k n (6) n=0 N N V By construction D^ j vanishes if both k and I are multiples of N. The modular proper- N'N ties of the correlation function entail with q' = exp -2iTT~l D N k(q') \ ï(fl) °N-k N-Z^) i I (7) N'N N N N' N and -inf l{S-l)\ 2 iTC 12 I N J (e'2 q) = (8) Dk lie Vz i(i) N'N N N kp 2iir— N (-l)P e " Dj, ln)N(q) = D^ ^q) (9) N'N N' N N N An alternative interpretation of these D's is the following (D stands for determi nant). Let us return to the computation of the partition function. But instead of assuming that -2iir- ( A \2* y i !") 2s B nT N N * * N for k and I not both equal to zero, one can show that 20 2 Z- , = (det -A) j = exp - - G^ jfO) = D k k l(9) (12) N N N'N These D's, with their factorized form (6), appear as the building blocks in terms of which one can express the Virasoro characters. We shall give some examples below: This means that underlying the construction of critical models will be some kind of twisted Gaussian (i.e. free boson field) model. 12. The Ising model at criticality is described by a two component free Fermi field. Denoting these components as «J/ and «J<, understood as taking values in a Grassmann algebra, one has the Lagrangian d - a - (1) dz dz With appropriate normalization it follows that in the infinite plane = (*(1)*(2)) » z^- (2) •l 2 •12 The conformai weights of 1 d i - a - - * — * (3; 2 dz An easy calculation using (2) and (3) shows that 1/4 Hence for the Ising model the central charge is c*r m = 3 (5) corresponding to the starting point of the unitary series. The conformai weights are therefore expected to be given in the table 3 1/2 0 2 1/16 1/16 (6) 1 0 1/2 1 2 21 This leads to the following list of real observable [h = h A = 2h [0.0] 0 identity [1/2.1/2] 1 thermal operator € = «l*i» (7 1/16,1/16] 1/8 "spin" a 3t erefore 2ACT == 1/4; since a = 2-2i> and - = 2-2A., a = 0 and v - with a logarithmic divergence for the specific heat. FinallV y scaling again predicts 8 = 1/8 and y = 7/**> All these critical indices stem, as we see, from the identifica tion of the central charge c = 1/2, a kind of major triumph! To confirm this identification let us look at the partition function on a torus as before. Given appropriate boundary conditions Z should be a product of two Pfaffians, one for d/dz, one for b/dz i.e. (det -A)1/2 up to a numerical factor. The delicate point is in the choice of boundary conditions. In Hamiltonian language, taking the trace over fermionic variables would suggest antisymmetric boundary conditions in the "time" direction, while nothing prevents us from using periodic conditions in the "space" direction. In a fancy language cne set of boundary conditons is called a spin structure. We have just seen ((11-10) and (11-12)) that (det -A)1'2 with the above boundary n conditions is | i/2 0(q)|. This element is not a modular invariant. The modular group is generated by the two operations (T) T - T-l (8) (S) T--T-1 While |D]/2 0(q)| is invariant under (T) it is not under (S). Rather the three quanti ties \ i D D l l/2.1/2l IV1/2I I t/2.0 (T) T - T*1 |D0 1/2| |D1/21/2| |D1/2 J (9) (S) T--T-' |D1/2 1/2| |D1/2>0| |D0 1/2| Conclusion: the modular group acts as the permutation group on three objects and one has to consider at once all three spin structures arising from combinations of perio dic and antiperiodic boundary conditions. The case periodic-periodic is set aside, any how the corresponding determinant vanishes as a result of the zero mode. Correspon dingly we infer that 22 Z (10* U2 » ? (l»i/2.1/2(1)1 - |D0.1/2(q)| • |Dl/2>0iq)|) The numerical factor 1/2 is chosen in such a way that the small q behavior is fqqj"l/48 in agreement with c = 1/2. Happily this is the result obtained by Fisher and Ferdinand in 1967 at the criti cal point of the Ising model. We would like to compare this result with an expansion over Virasoro characters. To do so, we observe that D^ j(q) can be written as squares. Precisely rz Dk i(q) - {USk 1Sl 0) q-W24 d^ ^,,2 (ID 2'2 2'2 with oo d n l/2 d l/2.l/2<1> • (W ) o 1/2(q) = I"! (l-q-i/2) (12) oo dl/2.od) = I17'6 ri (l*qn) Satisfying d 1 6 l/2.l/2(D d01/2(q) = q ' (13) d 8 l6d d l/2.l/2d) - !/2.od) * 0.1/2(l) The last relation (13) is of special importance in string theories but this is not the point here. Hence we can rewrite i/48 dl/2.1/2(1) ± d0.1/2(1) d 2 (!«•) *l/2 - H- « * i 1/2,o(Di r It is then recognized that, with the notation Xh(q) for the Virasoro characters corresponding to c * 1/2, we have 00 00 1 2 1 2 (d (q) d fi fl^q»* ' ) • [I /l-q"* / ) X0(q) - 2 i/M/2 * 0,1/2(1)) '- 2 23 2 2 +00 ' (24k+l) -l (24k+7) -l 48 - q 48 P(q) -, q-o oo 00 1 n 1 il X (q) = ~(dV . (q) • d (q)) = fi (l+q ' 1/2 2 1/2.11/2 1/2/ 0>1/2 2 0 )- "('-'-I (15) +00 ' (24k+5)2-l (24k+ll)2-l' 48 - q 48 -!-Z q~0 P(q) -oo 00 Xl/16<*> " dl/2.o(<3) « I1716 P (l-q») 2 2 •00 (24k-2) -l (24k+10) -1.' 48 48 ^ ai/l6 P(q) -00 q-0 Finally 1/48 2 2 2 Zl/2 * (qq)- {|X0(q)| • |x1/2(q)| * |x1/l6(q)| } (16) exhibiting neatly the respective contributions of the three families of states corres ponding to the three primary operators I, £, a. It is nice to see how all this fits together tying statistical mechanics and the Onsager solution to string theories, conformai invariance and abstract group theory. Interesting other finite size effects ought to be computed for the Ising model like the susceptibility or effective «p4 coupling constant. All this requires the knowledge of correlations on the torus or more generally in a finite geometry. III. MODULAR INVARIANT PARTITION FUNCTIONS 13- As was stated above linear combinations of the form (10-20), which are modular invariants are candidates for partitions functions (on a torus) of conformai invariant theories. Consequently there is a motivation to find such invariants built out of characters of the unitary series for each value of the integer m beyond m=3 (Ising case) and relate them to known statistical bona fide models. The main family is provi ded by the integrable systems studied in particular by Baxter. Not being an expert in 2'* this field I will not venture to spell out any detailed correspondence, and refer the reader to the literature. The first steps in this construction where made by Cardy. I will indicate what 1 know of the general case and suggest that a fair amount of work is still required tc get a complete picture. From now on we fix the integer m, hence c = l-6/«(m*l). There are m(m-l) 2 distinct primary characters Xrr s] of the Virasoro algebra. We look for an invariant of the form c 24 Z = (qq)- ' T Nrr>s]>[rrtS.]X[r,s](q) X[r.,s.](q) (1. Such that is a non ne ative (i) Nrr sl.fr'.s'] K integer N[l.l].[l.l] = 1 (ii) Z is invariant under (T) q-e2inq (2; (S) q-q1 , q = e2i1tT q' = e-2iUT" c "24H We have given in (9-4) and (9.6) (9.7) the behavior of q * Xrr si(q) under (T) ar.d (S). Both these transformations induce a unitary transformation in the space spaned by these quantities. Thus a first candidate is 2h *[r,s](9> (3) • w l a direct generalization of the Ising case. This we call the main series (or Ising like series) of models. In particular if m=4 it corresponds to the tricritical Ising model and, as for m=3, it is the only modular invariant for this value of c = 7/10. Using the notation xh(q) we have explicitly 7/240 2 2 2 27,10 = H- {lXo! HX3/2|^|x1/10| Hx3/5l^|x7/10|^|x3/8o| } (t) 1 '1/2,1/2 '0,1/2 — < 2 n n jk i n Dk j n Dk ii k even \ -.n- k odd ^ k even -.- k even -.- 25 ID,,, , J ID. J n \ 2 n" °k 3 n K 2 n k even —.— k odd —.— k even —.— k odd 10 10 10 10 10 10 10*10 D '1/2.0 l i/*.J n r\ 10' 10 k lo-io Note that 9R*Z fi Dfc ^q) »l * J n i i * i - «. 2~ k even -,- n«0 odd N N The first equality in (4) is nothing but • detailed stai t of 1)?. T) suggests an interpretation in terns of feraions and couples with inter-nel*,;*< boundary conditions. Siailar expressions apply in ail but kri rasjbrr cuabersoae.- so we shall not record thea here. They iaply a traction of invariant theories out of free fields, with soae ingenuity While (3) is the aost straightforward solution to the pronlea. it is far frws exhausting all possibilities. A second general series of invariants is imgi 1 1 f V? the following reaarfc. According to (21 Srr si rr- $- ~. can only differ frtm »r~ if h[r,s]"h[r'.s'] is *° integer, we have discussed in section (9) this pnssitilii;- It particular we have the coabinations (9-9). Taking into account the farther <-««•• r»-*». that N[ 1.1],[1,1] has to be unity (or divides the other »"s; this :«**» t* 'k coapleaentary (or Pott's like) series, which we denote Z ' defi a « I.I ( *) and 4p»l 2^-i x (qq)"^ I tr.s]*'[*p»l-r.s] tr.2p-I 2n*l*rthp 1 27 tiî -r»nedaa. Qui and Sbanfcer and Huse have been able to identify soae of the tls z* particular within a class of restricted solid on solid aodels. li? Following an idea of Gehlea i Rittenberg. J-B. Zuber has initiated a study W frustrated partition Onctions which enable one to get soae grasp on the underlying la ifcort this aeounts to study invariants under a subgroup of the nodular has recently developed the sane ideas. -*:&:' Saeerical studies of various transfer natrices, in particular by H. Saleur tStat it is possible to generate states pertaining to operators with that the indices r and s are outside the canonical ainiaal range, i.e. to invariant jubaodules in the degenerate Veraa nodules. There night isiaal theories involving infinitely aany operators and nodular invariants infinite sans. ?i«-j The generalization to the superconforael algebra eight reveal interesting new <*» ptstioM by Fateev and Zaaolodchikov to introduce c fields. can of this list. IS- Statistical node's af interest are by no neans restricted to the unitary He leak he~e at the degenerate non unitary representations. Recall that for a pair «€ relatively peine integers p" < p one has representations with 6(p-p)2 c * 1 PP V.sl*iT[ (2) 1 i r -i p'-l 1 < s < p-1 Hr.s] * hr~.n[p'-r.p-s. ] <2npp'Tp«sp,)^-(p-p')2 1 y iop' - (s - -s) (3) when e~»* > 1. there exist integers lr0.sc] * [ 1.11 in the required range, such that r,e - s,p* * 1. lanca the corresponding diaension is negative. Fields with such ;iaa have growing correlation functions (!) and Anwag the possible aodels one can show that we have the analog of the aain series. with a partition function on the torus 28 2 Zc = (qq) 24 2, X[r,s] where the sum ranges over the set of distinct hrr g-|. The character corresponding to the unit operator does not provide any more the leading behavior as q -* 0 (infinite long strip). Rather one has now Ceff zc So H~ * (6) C ., = c - 24hr_ --1 = 1 eff Lr0.s0j pp. Hence it is the value ceff which has to be used in (5-6) to give the leading correc tion to the free energy. An example according to Cardy is the one of the Lee-Yang edge singularity of the Ising model in an imaginary external field. This corresponds to a unique primary field 0 apart from the identity, which leads to the choice 22 P = 5 P' = 2 c = - — (7) 5 h[l.l] = ° h[1.2] = •' 5 (8) The real dimension of 4>_1/5 _1/5 is - = -2/5. As one approaches the Lee Yang point the singular part of the magnetization B ff behaves as 8ing ~ (H-Hcr ) , where a is gien according to scaling by a. 6-1 -t|!3 . -1/6 (9) d*2_T» d=2 T,=-4/5 This is in good agreement with results from numerical simulations (recall that a * -1/2 for d=l and a = •- for d S 6). It follows that on a torus _ 11 = 2 ^Lee Y.ng [^[M' * |X.1/5I ) (10) 1_ 1 « (qq)"60(|l*q*...|^(qq) ^ l*q2*... h) Here 29 Ceff « I <"> and this is the value that we observed with H. Saleur and J.B. Zuber in a numerical study using finite size scaling. In a joint letter we also discussed the means to discriminate between unitary and non unitary models numerically by studying the spectrum of excitations. 16. In conclusion let me formulate a personnal appreciation. The work of Belavin, Polyakov and Zamolodchikov has been a major breakthrough in the study of critical phenomena. It provides strong ties with the work on string theories and with interes ting advanced mathematics. Of course it is restricted to two-dimensions, and there are unfortunately no signs at present that similar considerations will unable one to cope with three and four-dimensional problems. In two dimensions there are many facets which deserve further study. Some have been sketched above. Let me still mention yet another one. As one approaches a critical point there exists a whole critical domain where continuous field theory applies. Could it be possible to describe it as a deformation of the conformai algebra and relate it to integrable models? BIBLIOGRAPHY The following list is very fragmentary. Additional references will be found in the papers quoted. - A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov J. Stat Phys. 34, 763 (1984) Nucl. Phys. B24l, 333 (1984) quoted as B.P.Z. - D. Friedan, Z. Qui and S. Shenker, Phys. Rev. Lett. 52, 1575 (1984) see also their contribution in the volume - "Vertex operators in Mathematics and Physics" J. Lepowsky et al. eds - Springer (1985) which also contains articles by A. Rocha Caridi and C. Thorn. - V.S. Dotsenko, Nucl. Phys. B235 (FS11) 54 (1984) - V.S. Dotsenko and V.A. Fateev, Nucl. Phys. B240 (FS12) 312 (1984) and Nucl. Phys. B251 (FS13) 691 (1985) - V.A. Fateev, A.B. Zamolodchikov, Landau Institute preprint (1985) - The work of V.G. Kac appeared in Lecture notes in Physics 24. 44l (1979) and the one of B.L. Feigin and D.B. Fucks is summarized in a (long) Moscow preprint and in Funct. Anal, and Appl. 16, 114 (1982) - 12. 24l (1983)- For elliptic functions we used - A. Weil "Elliptic functions according to Eisenstein and Kronecker" Springer 1976 - J.L. Cardy, J. Phys. AT7_ L, 385 (1984) Nucl. Phys. B240 (FS12) 514 (1984) Phys. Rev. Lett. §4, 1354 (1985) Santa Barbara preprint 78 (1986) - H.W. Blote, J.L. Cardy, M.P. Nightingale, Phys. Rev. Lett. 5.6. 742 (1986) 30 - I. Affleck. Phys. Rev. Lett. 56. 7^6 (1986) - A.E. Ferdinand and M.E. Fisher, Phys. Rev. 185, 832 (1967) -G.V. Gehlen, V. Rittenberg. H. Ruegg, preprint Université de Genève 85/O4-U60 - G.V. Gehlen. V. Rittenberg. preprint Bonn University (1986) - D.A. Huse. Phys. Rev. BJO. 3908 (1984) - C. Itzykson, J.B. Zuber. Saclay preprint 86-019 (1986) to appear in Nucl. Phys. B (FS) - C. Itzykson, H. Saleur, J. Zuber, Saclay preprint 86-27 (1986) to appear in Europhysics Letters - C. Itzykson, Marseille lectures