COMMISSARIAT A L'ENERGIE ATOMIQUE
CENTRE D1ETUDES NUCLEAIRES DE SACLAY CEA-CONF —9795 Service de Documentation
F91I91 GIF SUR YVETTE CEDEX Ll
i i <• FROM THE HARMONIC OSCILLATOR TO THE A-D-E CLASSIFICATION j OF CONFORMAL MODELS ITZYKSON C. I CEA Centre d'Etudes Nucléaires de Saclay, 91 - Gif-sur-Yvette (FR). I Service de Physique Théorique ! i
I V
I Communication presentee à : - Meeting onlntegrable models in quantum field I theory and statistical Mechanics
K£tO (JP) Oct 1988 FROM THE HARMONIC OSCILLATOR TO THE A-D-E CLASSIFICATION OF CONFORMAL MODELS
by
C. Itzykson Service de Physique Théorique C.E.N. Saclay F-91191 Gif-sur- Yvette Cedex, France
1. There is a growing feeling that arithmetics will play a significant role in understanding various physical models. This is not a recent discovery. Arithmetic questions arise in a large array of problems ranging from crystallography to complex dynamical systems, from the quantization conditions to applications of group theory and, since this is the subject of the present meeting, in the beautiful interplay between solvable two dimensional statistical models and conformai field theory. This goes perhaps against the opinion that the "queen of mathematics'' is to be safeguarded from any application. On the other hand, some physicists have recently coined the name "recreational mathematics'' to describe these matters. Let the reader decide if this is an appropriate characterization. In this contribution I would like to exhibit a few simple examples drawn from work with several collab- orators, J.M. Luck and E. Aurell for the first two, A. Cappelli and J.B. Zuber for the next one and more recently M. Bauer. It is a nice opportunity to thank them warmly again, as well as the organizers of the Taniguchi meeting in Kyoto and my collègues at the Physics Department of the University of Tokyo for their kind hospitality.
2. THE HARMONIC OSCILLATOR
The harmonic oscillator is one of the simplest integrable system with energy levels
n non negative integer (2.1)
in an appropriate scale. Consider a gas of non interacting fermions allowed to occupy these levels, at equilibrium at temperature T and chemical potential /i. If E and N denote the total *nergy and number of particles, the grand canonical partition function is
provided we set
The magnitude of q is smaller than unity, since eo is positive, hence z extends as an entire function in the [ y variable y. The coefficients of z in a power series expansion in y are the canonical partition functions
Contribution to the proceedings of the Kyoto meeting Saclay SPhT/88-SSS "Integrable models in quantum field theory and statistical Mechanics" October 1988. Submitted to "Advanced Studies in Pure Mathematics" Volume 19. corresponding to a fixed number of particles
The N values of s label the occupied levels arranged according to increasing energy. Each configuration can be considered as an excitation over a ground state, where the first N single particle levels are occupied, with 2n 1 2 minimal energy E0 = io E^To' ( + J = «oW . This means that 2n, +1 = 2« -1 +11., 0 < Ii < I2 < - < eN, and
N=O 0 The last sum is readily performed, first on IN, then on /jv-i resulting in an identity due to Euler
Let us generalize the problem to a gas of fermions and antifermioni with opposite chemical potential so that the new conserved quantity, again denoted N, is the charge, equal to the number of fermions minus the number of antifermions. The partition function reads
+00
E-oo n=0 Since Zs (?) = Z-N (?) it is sufficient to compute this canonical partition function for N > 0. As usual, when dealing with particles and antipartides, it is convenient to introduce negative single particle energy states. In the neutral vacuum all negative one particle states are occupied. Any other Fock state can then be described by enumerating some filled positive energy states (particles) and holes in the negative energy. sea (anti-particles). In a sector with fixed positive charge N, the lowest possible energy is such that the first 2 N one particle states are occupied (Emin = (o# ) and excitations above that level are described in identical terms as those above the ground state, showing that
We can relate any neutral excited state to the vacuum by setting a correspondence between occupied levels of increasing energy as indicated in Figure 1. Neutrality and finite excitation energy ensure that far enough down the ladder, the corresponding levels have equal energy, and
with P1 > T2 > ... > 0. To ensure that E is finite, only a finite number of r's can be different from zero. Hence each partition of the integer E /2t0 is in one to one correspondence with a neutral state of energy E. " Denoting by at the number of r's equal to k, we have
[T?BJ ™ k=lak=0 which is Euler's generating function for the number of partitions. Collecting the above results, we find Jacobi's triple product identity in the form
+00
JV=-oo *=1 2 Fig.l:Correspondence between occupied states in the neutral sector
Professor Andrews called my attention to the fact that he already met this combinatorial proof (Andrews - 1984) which might possibly occur in much earlier work. The next step is to introduce, besides fermions and antifermions, bosons and antibosons with equal chemical potentials in absolute value. The complete partition function
is meromorphic in y in the complex plane with the origin deleted. The last series is only valid for |?| < \y\ < \q\~l- Taking into account the relation
Cauchy's residue theorem applied to E(yq)y~N in an annulus bounded by the circles \y\ = 1 and |y| = \q2\ which encircle once the simple pole at y = q, yields
(2.8) I*"" an identity again due to Jacobi valid in the annulus \q\ < \y\ < |?|~'. Setting y = 1 and using Euler's identity f[(l + qn)(l-q2n-l) = l (2.9) n=l we obtain the remarkable relation
which will be used in the next section to study energy levels in a square box. Returning to the partition function (2.8), we identify the coefficient of y0 in S (W) - make use of the triple product identity as well as (2.9), and finally change q into -q, to get a relation analogous to (2.10) but valid in four dimensions
(2.11)
JV £ O mod 4 where the last expression results from the previous one by expanding the denominators and resuming in the opposite order. The reader will note that none of the above techniques apply to sums of an odd number of squares. For these consult (Weil - 1974). Let me mention two other identities which express similar facts as (2.10) and (2.11) when squares are replaced by equilateral triangles
1+6 2 12 ' - l^l + qN+q™ < - ) T V n,,n3=-oo W=I * +~ 3 a 3 , « V0W 1 + 12 213 A(«) - 2-4 1 = Z^ j-r^v ( J ni.n9.n.,n. A' = 1 N jk 0 mod 3 Of course the above results translate into statements on representations of integers by quadratic forms, which go back to Fermât and Lagrange.
3. FREE PARTICLE IN A BOX
There a few cases in which one can find exact expressions for the quantum energy levels of a particle enclosed in a compact domain of Euclidean space bounded by finitely many (hyper)-planes, but otherwise free. The wave functions are required to vanish on the boundary. If we look for the absence of "diffraction", meaning that eigenfunctions can be written as superpositions of finitely many plane waves, the Schwarz reflection principle implies that the group generated by the reflections in the walls of the box must tile the space. These VVeyl-Coxeter groups (Coxeter - 1973) ate in one to one correspondence with complex semi- simple algebras, one of the themes of this discussion. In this section we shall confine ourselves to rank two algebras, meaning that we deal with two dimensional (integrable) problems. These are of four types, A2 an equilateral triangle, S2 ~ C2 a triangle with angles T/2, T/4, JT/4 obtained by cutting a square along a diagonal, Di ~ A\ x A\ a rectangle, and Gi half of an equilateral triangle with angles (T/2, IT/3, 7T/6). In convenient units, the energies are simply the integral eigen-values of the quadratic Casimir operator. It seems that Lamé was the first around 1850 to obtain the spectrum in the equilateral case. Berry drew attention to the fact that "accidental" degeneracies in such two dimensional cases have a very intriguing pattern (Berry - 1981) first observed by the mathematician Landau. For definiteness, let us look in parallel at the cases of a square or an equilateral triangle with energy levels
Ea = nl + nl ni,n2>0 (3.1a)
E& = n\ + n\ + mns r»i,n2>0 (3-16) We have dropped inessential factors, so that with ni,nj integers, the energies are themselves integers. To recover the spectrum of the triangles one should impose ni > nj for instance. Interchanging ni,nj for ni ^ »2 leads to an obvious degeneracy. The £3 case corresponds to a rectangle which does not exhibit unexpected degeneracies when the ratio of its sides is irrational; when this ratio is rational, this is a (difficult) generalization of (3.1a). One might not suspect at first such spectra as (3.1) to exhibit anything spectacular. Since a large circle with area xEmm contains approximately x/4 EmKt points of the spectrum (3.1a) with integral energies up to ^mMi the asymptotic density of levels, JT/4, slightly smaller than one, would seem to indicate a smooth behaviour. It comes therefore as a surprise to discover that the fraction of integers belonging to the spectrum up to the maximal value Emtx, or equivalently the probability of an integer up to Emul, to belong to the 1 2 spectrum, vanishes like (In Em»)" ' for large Emtx\ The above estimate would have rather suggested that it tends to a value T/4. In fact both spectra (3.1) are extremely wild as we proceed to show, following work done in collaboration with J.M. Luck. Write D(E) for the degeneracy (multiplicity) of level E and define D(E) = D(E) if E is not a square, D (E) = D (E) + 1 if E is a square. It is easier to work analytically with D (E) while, due to the scarcity of squares, this does not affect the asymptotic estimate!) to be obtained below. Corrections to return from D (E) to D (E) could be easily supplied. By EF (E, d) we shall understand the number of positive integers E' < E such that D (E) takes the value d. Let us study the moments of the distribution of degeneracies
) (3.3)
If k = 0, the sum ranges over d > 0. The reason for introducing D(E) rather than D(E) appears when we expand both sides of (2.10) as double series
I]?"3+ £ «"'+"' = f D(E)qE = ££ (,«»+«0 -,W)) (3.4) n=i m ,»3=1 £=1 I=W=O
meaning that each divisor of E of the form 4/± 1 contributes ±1 to D(E). Denote generically by p and r the odd primes equal to +1 and —1 mod.4 respectively. If the prime decomposition of En reads
1 1 Ea = Î'PÏ V¥ -rf 'if ... Pi = 1(4) ri = -l(4) (3.5) then
0 < a; < Qi •' 0 < 4y < j3j This states that odd primes equal to —1 mod.4 must occur in even power for E to belong to the spectrum, in which case D (E) is equal to the number of its odd divisors factorizable into primes equal to 1 mod.4. Conversely from (3.6) one can derive the identity (2.10) showing how arithmetics and elliptic functions come naturally into contact. The study of the degeneracies is thus a variant of the study of the number of divisors, both share the same "unpredictability" in the distribution of primes. Using the same convention in the equilateral triangle case, it follows from (2.12) that a similar formula holds for Z)A (E) provided we replace p and r by primes of the form +1 and —1 mod.3 and expand E& as
E*=Z'PrP?~4'r{' WHl(J) r, = -l(3) (3.7) In spite of being explicit, the formr!a for D(E) is not as informative as one might hope. Following Dirichlet it is useful to introduce the series £ which allow to obtain the asymptotic behaviour of the moments fit (E) defined above. The explicit expression for D (E) enables one to express % («) as Eulerian products, the underlying reason being that the quadratic forms under consideration are norms of integers in simple quadratic fields. Explicitly
=Ï3ÎF n n p prime r prime P s 1(4) r = -l(4) =Î^W n • n <**> p prime v r prime p =1(3) r s -1(3) These expressions involve polynomials Pt (s) such that
PoW = Pi(X) = I P2(X) = (3.10) and in general
To use the Dirichlet series tjt («), obviously analytic for Re 5 large enough, in the estimate of the moments, one may proceed in the classical fashion as follows. First, using a representation of the step function, we express /it (E) as
JBe »=e 2)T with the line Re s = c in the analytic domain. The contour is then displaced to the left until we hit a singularity in ifc (s). The latter occurs at s = 1, as a pole for Jb > 1 or a branch point for k = 0, as follows from (3.9). The contribution from this singularity yields the dominant behaviour of /i* (E) for large E. y Alternatively, and more sloppily, a behaviour /it (E) ~ (ta E) corresponds in n* (a) to a singularity of the type t°° P (m, J. 1 Ï a I an (tn n) n =s a xj- •'I \S "™ 1 ) Whichever way, we need to characterize the rightmost singularity of rfr (s). Consider first ^1 («). Using the notation ((s) for Riemann's function w-£i- n Jf^) we derive from (3.9) UiW = CWIW (3.14) with M*) = U n (TT^gôëiy <3l5a) pI prime r prune n-° p = 1(4) r = -l(4) (3156) IAW = n p prime n (^=£[(3irW-(3^w] r prune "=" • P = 1(3) r = -l(3) In both cases L (s) is analytic for Re « > 0 and 8'2 (3.16) From the known behaviour of C (s) -a pole at « = 1 with unit residue- we conclude that tfi (s) is meromorphic for Re 5 > 0, with a pole at « = 1 of known residue. This leads to
S 2 E—K» /ill0(jE?)~>r/4 It1A (E) ~T/3 ' (3.17) which of course agrees with the Weyl estimates for the leading behaviour of the integrated density of levels. We turn to (to (E), the quantity discussed by Berry. Comparison between 17c W *°d Vi W given by (3.9) yields f =11A') Y^ II (I-!--
r prime r = -l(3) The products converge for Re s > |, hence cannot vanish in this region, so that one can take their square root -(positive on the real axis). From the expression (3.14) we deduce that IJQ (S) has a branch point at s = l, and
l/3 E—>oo /ioAE) ~ (2AJn E)~ /io,A ~ (2>/3i4A/n s)" (3.19) where An and /IA stand for the finite constants 1 2 l r 2 A11= I] I -'" ) ^= II ( ~ ~ ) (3-20) r prime r prime r=-l(4) r=-l(3) As previously mentioned, these results mean that, far from spreading uniformly over the integers, the levels are highly degenerate and leave huge unoccupied holes, so that asymptotically a particular integer has s, vanishing probability of belonging to the spectrum. Ttom the almost constant "density" (3.17), one would be tempted to conclude that D(E) scales like (In E)1'2. This is supported by the computation of Hi (E) but is not true for higher moments. Indeed, with the notation introduced in (3.15)
fl* W =
A W = ï^=7C2 W C (2*)"1 L\ (s) (3.2 Ji) we obtain . , E-*oo n2t,{E)~-tnE p2A(E) ~ ±*n E (3.22) Note the rational coefficients. For larger values of Jb, comes however the surprise. It is advantageous to replace the polynomial Pt (x) of degree it — 1 by
2 1 Qk(X) = (I-X) ^- P4(X) (3.23)
where the prefactor insures that Qk (x) — 1 vanishes like x2 for small x. One then finds
2 1 1 E-* oo ttk (E) ~ ak (in E) "' - k>0 (3.24) with constants at given by
" "W)]"1 n Q-(F-1) <»•**> p prime (r p =1(4n) ) (3-256) p prime P =1(3) The rapid growth of the moments p* (E) as (In E)3 ">-1 is quite remarkable and perhaps unexpected. We refer to our paper (Itzykson and Luck - 1986) for a study of corrections and similar properties in higher dimension. From the point of view of integrable systems, we understand that degeneracies are related to the existence of higher conservation laws. For instance in the case of the equilateral triangle we have a cubic invariant and for the square a quartic one. It would be interesting to investigate these properties for higher rank Lie algebras and understand in more detail the arithmetic behaviour of the Casimir invariants.
4. RATIONAL BILLIARDS
Let us divert for a while from integrable systems to see yet another connection with arithmetics. The simplest system to study as close as possible to an integrable one is a "quantum billiard", which we take again two-dimensional, in the shape of a triangle but such that the group generated by the reflections in the boundaries does not necessarily tile the plane. We assume that the group is "locally finite", by which we mean that the internal angles of the triangle measured in units of r are rational fractions. When some of these fractions in irreducible terms have a numerator larger than unity, we loose the tiling property. It was observed by Richens and Berry that the classical phase space of a free particle bouncing in such a billiard is foliated by compact orientable two dimensional surfaces in a four dimensional phase space (Richens and Berry - 1981). The genus g and Euler characteristics x of these surfaces are given by
X = 2-2ff = ££(i_p,.) (4.1) where the index t takes three values, we choose t = (O, l,oo), the inner angles of the triangle are written ifi = T Vifii with P> a"d Ii coprimes, and Q stands for the lowest common multiple of the denominators ?,-, in fact of two of them, since CI = I (4.2) In general we write (p, q) for the greatest common divisor of p and q and use indifferently the notations asO mod b, a = 0(6) or b \a.
If we require the numerators pi to be unity, we recover the three integrable triangles (genus 1, torii) discussed in the previous section. To derive formula (4.1) one repeats in essence the reasoning of Riemann and Hurwitz by gluing together triangles corresponding to the finitely many possible directions of the particle velocity (generically 2Q).
It follows from the Riemann mapping theorem that, no matter which values the angles
Let us concentrate on the case where the angles ?«/* are rational, and show that one can attach to such a. triangle an algebraic curve such that triangles and curves come into families with interesting automorphisms and/or analytic maps (Aurell and Itcykson - 1988).
I collect first some data in table I using the following notation. Let It = Qlq>. We can then write the angles as (pi = x PiIQ, with Pi = ItPd so that it is sufficient to list Po, Pi, P00 with sum Q such that these integers have no common factor. Up to genus 5 we find the possibilities listed in table I, disregarding a permutation of Po, Pi, Paa- The table suggests that the sum Q = Po + Pi + fto ranges between 2g + l and
Proposition 4.1 : Let Q and g be defined as above, then
(4.3)
The following proof is due to J.M. Luck. The first inequality follows from the definition of g written as
(4.4)
It is only saturated when all U are equal to one, so that Q is then necessarily odd, and we can take as an
9 example the triangle {Fa,Pi,Pm) = {hs,9)- Henceforth we denote a triangle by the symbol {Po,Pi, Poo}-
P0 P1 P1 Jj - g=l 1 1 1 1 T- I g=5 4 4 3 2 1 VJ l 2 2 5 3 2 3 2 1 7 l 1 5 4 2 4 3 3 5 5 1 I . 6 3 1 6 3 2 B=Z 2 2 1 2 7 1 6 J T 1 3 1 1 8 1 l 7 2 2 4 1 1 7 3 2 7 3 1 4 1 3 7 4 1 8 2 1 5 2 3 9 2 1 9 1 1 5 4 1 6 5 4 5 5 2 m 7 3 10 1 1 8=3 3 2 2 9 5 l 8 5 2 3 1 3 10 3 2 10 4 1 4 2 1 8 5 3 10 7 3 5 1 1 8 7 l 10 9 1 3 3 2 9 5 4 11 6 5 5 2 1 9 7 2 11 7 4 6 1 1 9 8 1 11 8 3 4 3 2 11 9 2 5 3 1 11 10 1 6 2 1 J T 5 3 6 5 l 8 3 1 7 4 3 7 5 2 7 6 1 I - Tai/e / - Rational triangles classified according to genus %p to g = 5.
Since the Pi's are without common factor, so are the 4 = (Pi,Q)- For every prime p, there exists at least one U prime to p, and therefore at least a pair Q, ç,- = Q//,- which admits the same maximal power of p as a common divisor. This same power of p divides the product qjqk since it divides the lowest common multiple Q of q, and qt. It follows that Q2 divides 9o9i?oo and hence
ÎO Î1 9oo
10 Let us order the three integers Ii, for instance (0 > Ii > '«• Set q0 = £• = m > 2, where the last inequality says that each angle is smaller than *. It follows from the above that m > /1Z00 and therefore that t\ + la, < m + 1. Consequently
for m a divisor of Q larger or equal to 2. U m = Q, I0 = 1. Since by assumption ig > l\ > /„,, it follows that I0 = I1= I00 and we recover the lower bound. Let us therefore assume 2 < m < Q/2. In this interval the function ^+ m + 1 assumes its maximum ^ + 3 at both ends and to +1\ + too < $ + 3 or equivalently "£, (4 — 1) < Q/2. Inserting this in equation (4.4) yields the second inequality in the proposition. It can only be saturated for Q even, in which case an example is the triangle {l,2g,2g + I)1 which completes the proof. It follows obviously that the number of triangles of fixed genus is finite. Let t denote a complex coordinate in the plane of the triangle and z in the upper half plane Im 1 > 0. Let the vertices of the triangle be mapped respectively on x = 0,1,00 and choose a length scale such thai
The Scbwarz map from the triangle to the upper half plane is then
For definiteness we assume 1] — to real positive, so that the roots in the integrand are such that it tends to a positive value for x' tending to a point on the segment [0,1]. The area of the triangle is then a Veneziano like amplitude T I — 1 T I —
\r~Q~)T \~Q~)r \ Q J When we choose a rational triangle, the derivative
is related to x by an algebraic equation
y4 = zp>+p-(l-x)p-+p° (4.9) univocally associated to the triangle up to permutations of the vertices. The latter are generated by trans- positions whose squares are automorphisms of the corresponding Riemann surface. It is readily checked that the curve (4.9) has a genus g given by formula (4.1) or (4.4) acd that the Riemann surface is paved by 2Q triangles, each one the preimage of the upper or lower half x-plane with the points 0,l,oo marked on the boundary. The desingularized triangulated surface is a model (even a conformai one) of the classical allowed region in phase space. Each point 0,l,oo, is a singular point of the presentation (4.9) in the vicinity of which one can introduce a regularizing parameter Vi for each of the Ii families of 4,- = QIU covering sheets which meet at this point over the x plane. From this one sees that
U(l) = d< = j (4.10) is a holomorphic differential on the curve with 2 (g - 1) zeroes.
11 For g = I, the integrable case, the holomorphic differential is unique up to a constant factor and the corresponding abelian integral, the variable t, plays the rols of global uniformization parameter. The correspondence between integrable triangles and elliptic curves of modular ratio r reads
3 2 3 A2 {1,1,1} y = [X(I-X)) r = e*" 3 B2 {1,1,2} / = [x(l-x)] T = e*'/< (4.11) 6 s iw iT 3 G2 {1,2,3} y = x (1 ~z)* T = t? '*~
2 3 A2 Y +4X = l 2 4 B2 K +4* = l (4.12) 2 3 G2 V -* =l When the genus g is larger than one, a (compact) Riemann surface admits g linearly independent holomorphic differentials and w W is only one of them. It is noteworthy that one can choose a basis of differentials in such a way that each of their integrals maps the upper half x plane on a rational triangle. We call such triangles associates of the original triangle. Let as before Po, Pu P
u) (v) P/ > 0 P0 + P1W + /*> = Q (4.13) which means of course that the set {p/"'} defines a triangle 7<*) with angles r PJv)/Q. Thus 7<*> = 7 and for v admissible Tw is an associated triangle. It is not garanteed that the P/"* have no common factor. When this happens one observes interesting phenomena as we shall see later.
Proposition 4.2 (i) The number of admissible values of v, hence the number of associated triangles is precisely g. (ii) For v admissible let y<"> and «W be defined as
then y(*)is a meromorphic function on the Riemann surface of the curve C, more precisely a rational function y(") of « and y can be defined satisfying (4.14). Furthermore u^ is holomorphic on this surface, its integral maps the upper half plane on the associated triangle and as v ranges over admissible values the g holomorphic differential are linearly independent.
To prove (i) we note that two distinct values of v in the interval [O1Q - 1] cannot lead to the same 1 ordered triplets. Indeed if for V2 > vi, V2 - «i < Q — 2, («2 - «1) P,- = ktQ we would have Pi/Q = tJ J11 in contradiction with the fact that Q a the smallest possible common denominator. If for each », if is positive so is p/fl"*J = Q - p/"* and since
12 one of the sums is Q the other 2Q1 showing that of the pair of values v,Q-v, only one at most is admissible, and v and Q - v are distinct if all P^ are positive. If one P/"' vanishes, we have t>P; = kiQ so that dividing by I,- = {Pi, Q) we conclude that
U - IJ f = S
according to (4.4), thus proving the first part of the proposition. To prove that the functions yW are meromorphic and the differentials wM holomorphic, it is sufficient to use a local desingularizing parameter at x — 0, l,oo. If we write Pf ' — vPi — r{Q for v admissible, we readily obtain yt") as () +l +I The irreducibility of the curve (4.9) entails that no nontrivial polynomial in y of degree smaller than Q with rational coefficients in x vanishes; as a consequence one obtains the independence of the differentials <*>(") using the expression of ifrh Finally the integral of «W maps the upper half «-plane on the triangle TW. : The Riemann surfaces obtained in this process cany a natural triangulation and form a class with interesting algebraic properties. We conclude this section with a few examples. Consider the triangle {113} of genus 2 and corresponding curve |
y8 = [z(l-z)]4 (4.17a) birationally equivalent to the one of its associated triangle {221} namely
y6 = [z(l-z)]3 (4.176) ' as well as its desingularized model y2 + X5 = l (4.17c) all of which exhibit a group of automorphisms of order 10, Z/2Z x Z/5Z. Each triangle, completed by a reflected image in one of its sides, appears as an elementary tile in the famous Penrose tilings of the plane. The Riemann surface with the largest automorphism group in genus 2 occurs for the associated triangles {4,3,1} and {4,1,3} with equivalent curves '
ya = x*(l-xf y8 = z4(l-i)7 (4.18a) both birationally equivalent to y2 = z(l-*4) (4.186) • with automorphism group of order 48, a double covering of the octaedrai group. Using the form (4.18b) j its triangulation in 16 triangles is a double cove of the one of the Riemann sphere of the variable z when j projecting the faces of a regular octaedron wuh \ ertices at 0,oo and the four fourth roots of unity. As a last example consider the triangle {1,4,16} of genus 10 with Q = 21. The admissible values of v are classified as follows j v = l {1,4,16} v = 2 {2,8,11} t, = 3 {3,12,6} • = 7 {7,7,7} I 4 {4,16,1} 8 {8,11,2} 6 {6,3,12} (4.19) I 16 {16,1,4} 11 {11,2,8} 12 {12,6,3} | I In three groups the associated triangles only differ by a cyclic permutation of the vertices, while the last one is invariant under any permutation. 13 For v = 3,6,12, the integers FJ^ have a common factor 3, and, up to permutation, the corresponding triangle is {1,4,3} of genus 3 (and not 10) while for t> = 7 one obtains in irreducible terms the equilateral triangle {1,1,1} of genus 1. This is an example of rational correspondence between algebraic curves of different genus larger than zero. Starting from the irreducible curve
S = IO \17 (4.20)
if we set y = X = x (4.21)
we obtain the elliptic curve 3 - 1 9 = Y = [A (I -X)] (4.22) Thus instead of considering the curve (4.20) as ramified above the Riemann sphere of the variable X = X, we can also consider it as 7-uple covering of the torus (4.22). It can also be considered as a triple covering of the curve of genus 3
7 g-- _ 3 q Y-X(I-XY' — Y" (i - )T'\ 3 X-X' — rx YV' - (4.23) [x{l_x)]i
According to a theorem of de Rranchis, there are at most finitely many holomorphic maps between algebraic curves of genus larger than one. The triangular curves offer a rich zoo of example*. It would of course be great if the above properties afforded a mean to find algebraic relations between the spectra of the associated triangles.
5. THE A-D-E CLASSIFICATION OF CONFORMAL MODELS
As a last and most significant example, I will describe in this section the classification of minimal conformai models and Wess-Zumino Witten models based on the simplest affine algebra A^ (Cappelli, Itzykson, Zuber - 1987). In the concluding remarks of the next section I shall touch on extensions to other rational cases. It is not necessary to recall here the framework of two dimensional conformai field theory, with its Virasoro, Kac-Moody or extended algebras, Ward identities and partition functions. This is covered in other contributions to this volume. Instead I will concentrate on questions of modular invariance, an aspect of the various monodromy problems which have led to amazing connections with many branches of mathematics. In statistical mechanics, one expects the partition function on a torus, expressed as a sesquilinear form in the characters pertaining to whichever extended algebra is relevant for the description of the finitely many primary operators in a rational conformai theory, to be modular invariant (Cardy - 1986). Such constraints are extremely powerful and reveal some unexpected relations, as examplified by our finding of an A-D-E classification for the simplest cases, suggesting ties with finite group theory, algebraic surfaces, rings of invariant polynomials in envelopping algebras of simple Lie algebras... To carry over this program to a larger class of rational models is a challenge which might reveal new features. We will emphasize here some arithmetical aspects. The mathematical classification has its physical counterpart in a better understanding of universality classes of critical theories and could perhaps shed some light on the behaviour in the vicinity of critical points.
5.1 The modular group F = PSL (x, Z) is the group of transformations ar + b T — T1 = (5.1) cr + d with integral coefficients and unit determinant ad — be = 1. A given element stems therefore from a pair of matrices ±A in f = SL (2; Z). The group T leaves the real projective line and the upper half r plane invariant. Henceforth we assume r to have a positive imaginary part.
14 The modular group describes the effect of a change in basis on the ratio r between two generators of a lattice C in the complex plane, with C/£ identified with a torus (or an elliptic curve). It is generated by two elements T T • (5.2a) satisfying S2 = (STf = 1 (5.26) For any k > I, let F& be the invariant subgroup of level k such that A = ±1 mod k, and F* = F/F*, the factor group with a number of elements |F*| given by rt i4 n - (5.3) p prime Pl* 2 The factor 1/2 is omitted when k = 2, so that |F | = 6. For a. pair of coprime integers ku k3, we have [•*!** ~ r*' x F*3. Hence the non trivial part of representation theory corresponds to the case where k is a power of a prime. We note that the center of F* is made of involutions of the form fl, y* = 1 mod Jb. If T denotes the number of odd prime divisors of k and a = 0,1,2 according respectively to Jb ^ 0,4, Jb = 4 or Jb s 0 mod 8, the number of elements in the center is 2*+r. In any representation these elements will belong to the commutant.
5.2 We will need Dedeldnd's function already encountered in section 2
1 g = exp 2tVr Im r > 0 i; ( (-D'?"* - (5.4)
which transforms under the modular group as
T (5.5) as follows from its definition and Poisson's formula. The product representation shows that T}(T) never vanishes for Im r > 0. Let us introduce the characters corresponding to the "degenerate" series of Virasoro highest weight representations (Feigin and Fuchs - 1983, Rocha-Caridi - 1985) with central charge
e— l — ^(P-P7) / 5 a) such that p and p' are coprime positive integers. The heighest (or conformai) weights are given in terms of two integers r and s in the range 0 When p1 and p are successive integers, p* = ro, p=m + l, m>2 one obtains unitary representations (Friedan, Qiu and Shenker 1985-1986). With L0 standing for the generator of dilatations in the Virasoro algebra, we have the following formula for the character XcH (r) = (5.8) 15 Set n = pp' N = 2n = 2pp' (5.9) A = rp — sp' mod N If r and s are chosen as above, a representative A lies in the range 0 < A < n, multiples of p and p' being excluded. Consequently, the number of possibilities, i.e. of independent characters is 5 (p — 1) (p1 - 1). 7 Since p and p' are coprimes, there exist integers ro and «0 such that rop- «op = 1 with /i(ro,»o) negative unless the representation is unitary (a poor man's way of understanding the condition (p — p1)2 = 1 as necessary for unitary, sufficiency then follows from the coset construction described by Olive in this volume). The combination h — (c/24) inserted in (5.8) will insure "good" modular properties and arises physically from the Casimir effect in a bounded torus. It reads 24 24 and is negative in all cases of interest where PP* > 6, thus could be written -cf/24 with é > 0. Define mod N wi = l mod W (5.10) such that s rp+«p* moi N (li.ll) Among positive integers, p and p* are the smallest such that wo A = A mod N is equivalent to A = 0 mod p and WoA = —A mod N is equivalent to A = 0 mod p1 respectively. The requirement on Virasoro representations is that W0 $ ±1 mod N. With these notations the characters read (5.12) and satisfy XA (r) = x-x (r) = XA+Af M = -Xw0A W (513) One observes that Euler's pentagonal identity (5.4) expresses the fact that for pf = 2, p = 3, N = 12. the Virasoro character reduces to unity. The modular transformation properties follow from Poisson 's formula and the behaviour of ij(r) given in (5.5) /A2 1 \ •1) =exp2i>^— -— J XA(T-) (5.14) „ / -i\ 1 v> 2Ja-AA' . 5 XA("T " =7N ^ exP-jv~Xv( VJV l'tnod JV These transformations are compatible with the symmetry properties (5.13). Using Gauss's formula giving the trace of the finite Fourier transform 1 N = I mod 4 (5.15) A mod N ( 0 N = 2 mod 4 as well as the symmetry XA = X-A which insures that St operateAsT i =n 3th emo subspacd 4 e with eigenvalues ±1, it can be checked that S2 = (ST) = 1. That the finite Fourie2+1 r transformAT = 0 s momaked 4s its appearance among the unitary transformations induced by the modular group, ties nicely the subject with quantum mechanics on systems with finitely many states as we shall see below. 16 From the preceding it follows that I = E i** (5.16) A=I is a modular invariant partition function, which we call the principal invariant, and defines a consistent symmetric theory (symmetric between holomorphic and antihobmorphic parts) involving only scalar primary operators. The subscript notation will be justified in the following. The choices \p',p] = [3,4] and [4,5] correspond to the king model (free Majorana fermions) and its tricritical version. Higher unitary models in this series can be understood as higher multicritical versions which from the point of view of a Landau Lagrangian would have an effective potential of the form Xi (5.17) where one recognizes in the product forms the quantities which occur in the identities referred above. The ratio 1W = ^A (518) transforms under T and S as (5.18) which means that, projectively, the action of the modular is represented by the icosahedral group, revealing the isomorphism of the latter with PSL(2,Z/5Z). We shall encounter further similar instances later on. 5.3 In parallel with minimal conformai models it is instructive to study Wess-Zuminc-Witten models, abbreviated as W-Z-W, (Gepner and Witten -1986) based on the group SU2, or rather its affine Lie algebra version A^\ Whenever necessary we use superscripts, conformai or affine, to distinguish the corresponding characters. In the affine case the models are characterized by a level it, a positive integer such that the central charge is 17 and the highest weights can be specified by a lowest integral or half integral angular momentum I, such that 0 < t < |. To stress the analogy with minimal models, we set (5.21) and write the Kac characters (Kac - 1985) (5.22) ™ W J=_oo Satisfying xF-M = X&(r) = -x3-M (5-23) Here the role of the odd involution is played by the reflection A —• —A. The Jb -t-1 independent characters can be chosen to be those with index A = 1,2,..., fc +1. Under modular transformations T +1) = exp SAr (^ - ± • (r) (5.24a) S (5.246) Vmod N and when JV = 4, A = 1, Xj*1' reduces to unity, thank* to Jacobi's identity. In (5.24b) the factor -i is necessary since xf' is odd in the interchange A «—• —A. In both the conformai as well as the affine case, the modular group Tw of level 2JV is represented by multiples of the identity, As a. consequence, both (5.14) and (5.24) generate protective representations (in general reducible) of T2JV. 5.4 The classification problem consists in finding all partition functions of the form T-, f)= E XX (T)SxJfXV M (5.25) where the indices A, A' range in a fundamental domain B of the symmetry properties within integers mod. N, i.e. l A - Z (r, f ) is modular invariant. B - ZA,A> are non negative integers for A and A' in B with a normalization in the affine case Zi,i = 1 I - and in the conformai one Zp-pitf-fi = 1 expressing the unicity of the vacuum state. Correspondingly, the investigation divides itself into two parts, which we call problem A, the easiest, and problem B1 the real arithmetical one. It will be convenient in the sequel to extend the summation in (5.25) to the full range A, A' mod JV with due attention paid to (anti-) symmetry properties. We can now identify Z\,\' with a matrix in an JV-dimensional vector space with a basis labelled by A mod JV. The overall phases in (5.14) and (5.24) are immaterial when we write the condition for modular invariance T+ZT = S+ZS = Z - exp 2»>^«A,A' mod N (5.26) exp2i* 18 Due to the unitarity of the matrices T and S this means that Z belongs to their commutant, so that problem A is nothing but characterizing this commutant, and if possible finding a basis spanned by matrices with integral coefficients. 5.5 Throughout this section JV = 2n is an even (positive) integer. Let S be a positive divisor of n and 2 put S = n/S, a = (S1S) the greatest common divisor of S and S so that a divides n and N. Since S jot and 6/a are coprime, we can find integers p and (5.27a) and define u mod N/a2 through modN/a* (5.27i) It satisfies 2 2 w + ls2p- mod/V/a w-l = 2 = \ ( mod a •(• 0 otherwise commutes with T and, as an immediate computation confirm*, also with S. Interchanging the rolei off and S amounts to replace u by —ui; Qn corresponds to a = 1, u = 1, i.e. Qn = /> while Qi corresponds to a = 1, u = — 1, i.e. (Œi)A|A, = 6\+v,o- More generally Qf and Un/1 are linearly related when operating on the even or odd subspace under A <—• —A. Let us call a (n) the number of divisors of n (equal to 2 if n is a prime) we have (Gepner and Qiu - 1987). Proposition 5.1 : The commutant of the matrices S and T is spanned by the This proposition was stated as a conjecture in an earlier work with A. Cappelli and J.-B. Zuber. To prove it we introduce a technique useful in a broader context, namely, we give an explicit realization of the adjoint action of the modular group of level 2JV on a suitable basis of JV x N matrices related to the Heisenberg group modulo JV. We call this realization "finite quantum mechanics". It is presumably familiar in a mathematical context under the name of finite metaplectic group. Let {|A}}, A mod JV, be an orthonormal basis in an JV-dimensional Hilbert space H. Define the operators Q and P through Q\X) =exp^|A) PIA) = IA+I) " (5.30) N N QP =eXpjj-PQ Q = P = I Obviously Q and P are related by a Fourier transformation. The polynomials in Q and P generate the full algebra 1H x TC of operators on 1H since we can write any operator M in the form (5.31) k.t mod AT This follows by linearity from a check on elements of the form |A) (A'|. In view of (5.30) we can "normal order" any monomial in P1S and Q1S by pushing the P's to the left of the Q's. 19 The operators S and T generate through their adjoint action canonical transformations preserving the commutation relation between P and Q. If we introduce the convenient symbol {*,i} = exp 2i>^P*<3' (5.32) it is readily checked that Tf{fc,/} T = {k,t-k] 5* {k,t}S = {/,-*} (5.33) The monomials {k,l} are periodic in k and I mod.2Ar but are linearly independent ouly mod.N, since We can recast (5.33) as an action of the modular group generated by S and T as er- -CD (5.35) A*{k,t}A k'.f] = M ( * J In fact, since fjjv is represented by the identity, we obtain effectively an action of TiN. Using (5.29) and (5.31) -we find on the one hand (5.36) y,« mod ^ On the other hand to obtain a general element of the commutant one can average the adjoint action of T2N on an arbitrary element of 4H x H', a familiar idea in finite (or compact) group theory. From the decomposition (5.31) it is sufficient to choose as an arbitrary element a monomial PkQl. Thus we set 1 A+PkQ'A (5.37) Ci) «r» The operators Mkl with k and / rood N {M't+(Nt+(IN — M'kt) span the commutant, but they are far from being linearly independent. First we observe that M'kt vanishes unless k and I are both even. To see this, consider the kernel of the map T2lf —• T* given by the eight matrices ( 1 + aN PN \ Cp.y = 0,1 7JV 1 + aN/ Averaging (5.37) over this invariant subgroup should leave Af'k t invariant and multiplies the coefficient of pak+Uqtk+il by <*,P,-r mod.2 20 This is non vanishing if and only if k' = ak + cl and H — bk + dt are both even. The linear transformation is invertible mod.2, hence k and I have also to be even. Thus in (5.37) we may assume both indices even which allows us to write (5.38) rB The structure of the formula has in fact reduced the average to fn and Jb and I are defined mod.n. Fur- thermore for any matrix [ | 6 TF" we find Mk,i proportional to Mat+a,n+4i so that we can restrict ourselves to pick any representative point of the orbit of [k,Q under V to label Af. This is done as follows. Pick first representative) k,t, in the range 1 to n and let d = (k,l). Then we find mi «id mj such that mi Ib + m^t = d and / t/d mi\ \-k/d mjj (5.39) Mkli = exp f-2tir—J The process can be repeated by identifying Mo1^ with Mn%d and introducing 6 = (n, labelled by divisors of n, generate the commutant. They are linearly independent, since disjoint orbits are labelled by distinct divisors. The relation (5.41) relates the two sets {£2«} and [Mf] through a triangular matrix which can be inverted to yield the M's in terms of the Q's, showing that the latter also generate the commutant and are therefore also linearly independent. This proves the proposition. Of course the advantage of using the basis {12;} is that the matrices have integral coefficients in the |A) basis. 5.6 The heart of the matter is problem B. For the sake of completeness I will repeat here the arguments of our paper. The difficulty stems in the conformai or affine case from the oddness property of the characters. We look first at the affine case. From proposition 5.1 we can write a modular invariant partition function (5.42) Recall that x-x = -Xx so that X(n = 0. Let U = {1,2 n- 1} and L = {n + 1 2n- 1} denote a partition of the integers from 1 to TV distinct from 0 and n mod N with L = — U mod N. A fundamental domain B coïncides with U. Since (Clix)\ = — (On/«x)A >nd (OjJ^v = (JΫ)_A._A'I fot each factorization 21 n = 66 we can replace in (5.42) the combination csttt +e«$îj by (c« - cj) fij or (cj - cj) fi$, we choose which ever has a non negative coefficient. Thus Z{r,f)= (5.43) A,V€C/ with Zw in (5.42) such that c( > 0, and c< > 0 implying cn/< = 0. Only fi« and Oi contribute to the coefficient of XiXi which has to be equal to unity. With the above conventions, this implies Cn = 1, c\ — 0. We have chosen matrices Ut with integral non negative entries, we want to find the coefficient c« so that this is also true for Z\\< — Zx1-H. In the following proposition we exhibit all possible solutions for Z\y. Proposition 5.2 (A-D-E classification) Modular invariant affine (-A^1M partition functions with normalized non negative integral coefficients are of the following types , n>2 n even > 6 On + O2 Es Tl =12 fi» + 0a (5.44) E7 r»=18 Ou + «a + «a Es n = 30 We have two infinite series A and D and three exceptional E cases. The index on A, D1 E, is the rank of the corresponding simple and limply laced Lie algebra. Simply laced meant that all fundamental roots are of equal length, or equivalently that the corresponding Oynkin diagramm his no multiple link. This correspondence is illustrated in table II which gives the partition functions in expanded form: the coefficient of each diagonal term fxXxt 1 < A < n — 1 is the multiple of A in the list of Coxeter exponents for the corresponding Lie algebra and n is its Coxeter number. n>2 On A=J n=4p + 2 ^ Îp+2-A +cc-) P > 1 A odd = 1 A «"=• A odd=l V-I 2p-2 n= 4 P E ix> On+ft2 /> > 2 A odd=l A evens:2 2 «=12 Ixi+XT|2 + !xs + xiil "=18 E7 = 30 + Xn + Xw + X2!>|2 + |X7 + Xi3 + Xi7 + X2ï|2 Eg Table II • List of modular invariant affine partition functions for A[ '. 22 It must be mentioned that an independent proof was found by Kato (Kato -1987). Ours proceeds in three steps, lemmas 5.1, 2 and 3. We set aside the case of the obvious invariant (An-i) with cj = 0 except Cn = 1. For each divisor S \n we define a (S) = (S,n/S) and u (S) as before, such that u2 (S) = 1 mod 2JV/a2 (6). Consider in (5.43) the factor Xi (r), it reads Xt (»")+ S c'Xu(f)(r) arising from those <5's such that a (S) = 1 and u (S) prime to JV, which for 1 < S < n are distinct from ±1. It follows that u (S) € U. If on the contrary w(S) € L, c«X«(«).= —cix-u(t) would be a negative contribution, which requies that there exists a 6' with a (6') = 1 and u(S') = —u(S) = u, such that (e$i — ci)xu >s a positive contribution with c«< > c«. But a = 1, u' = —u means S' = S/n and this possibility is excluded by our conventions. We conclude that all u(S)'a such that c« / 0 and a (S) = 1, have to belong to U, the corresponding coefficient being positive integers. If o? |n, the identity mod N/a2 (5.45) ( mod a relating characters at level it = n - 2 and V — n/a3 — 2, thorn that both sides vanish when n = a3, 3 since XA (r, JV = 2) = 0. Hence when n = a , £v (fta)AV XA' = 0, and the corresponding term may be disregarded in (5.42-43). Define amin to be the smallest a (6) for 6 ^ n such that c« > 0. Then Lemma 5.1 (i) amin = 1 or 2 (ii) if amin = 2 (therefore 4 |n) the unique solution is Z = Hn+ CIi. It is useful to represent the integers mod JV on a circle of radius JV/2x as regularly spaced at distance 1. The upper (lower) semi circle represents U (L). For a (S) = a > 0 the points A' = wA+£ N/a as ( varies are the vertices of a regular a—gon. If or > 4, at least two vertices belong to L U {0, n}, one of them being certainly in L. These negative contributions have to be compensanted by positive ones. Consider the factor of Xoml. (r). By definition of amjn only fi/s such that Ct > 0 and a = amin contribute. This factor is <• a_i. + ( N/am-m mod am Suppose first amjn > 4. Each a—gon involves at least two terms in L. The case where only one term is in £ would require amm = 4 and a set of indices u amm +( JV/ormin ranging over 0, n/2, n, 3n/2(16 |n). Since w is invertible mod n/4 = (2iV/a^,in) and u amm = 0 mod n/2 and a fortiori mod n/4, it follows that ajj,in = 16 is a multiple of n, the only possibility being n = a^,in, the case discarded above. Thus each amin~gon has at least two terms in L, and since the coefficient of Xa_j, is unity, so that at best it can only compensate only one of the negative terms, two possibilities of cancellation are open: either the negative terms are compensated by positive ones of the same a—gon or a negative term pertaining to S is compensated by a positive one from a S' jt S contribution. In the first case 2W = O mod N /a^ia, hence 2 = 0 mod N /a^ which implies n = a*,in, a (S) = am;n, an excluded possibility. In the second hypothesis, the corresponding multipliers u and u' have to satisfy u' = —u mod JV /amin, which is also excluded since it would imply SS' = n. We conclude that amjn < 3. If ormin = 3, beyond the two previous possibilities excluded by a reasoning similar to the above one, we find a third possibility with an equilateral triangle having a unique point in L compensated by Xa^, = Xa w^h a coefficient e$ = 1. But then u = — 1 mod AT/9, so that the corresponding contribution is X-3+X-3+W/3+X-3-W/3 and —3±JV/3 arein Uu[O, n}. Thus3+2n/3 > n, meaning 9>n and since 9 \n we find again an excluded case n = Q^n, a (6) = amin- This proves part (i) of the lemma: ormjn — 1 or 2. Assume now amjn = 2 and consider the coefficient of X2 X2+ 51 «* <.o<*)=2 23 If the two points 2u and 2u + n coincide with ±n/2 mod N, we have 4w = n mod N and this is the excluded case n = 4 = 22. The possibility 2u = 0 mod n cannot arise since w2 = 1 mod n. To obtain a positive contribution, the only possibility left is either that terms arising from two distinct «5 and S' compensate each other, this being excluded as above, or when there exists a unique S, a (6) = 2, c« = 1 such that, by choosing the representative mod /V/4, we have 2ui = — 2 mod N and %> + n g U, meaning that the negative term from X2<* >s compensated by X2- Hence S = 2, n = 4Ar, u = — 1 mod 2Jb. The partition function contains then fin + ÎÎ2 plus possibility a sum over S's such that a (S') > 3. For A € U, £A, (Jin + Oj)x^, XA' "a equal to x\ >f A is odd or to Xn-A if A is even. To discuss the possible occurence of other 0*8 we can retrace the steps of part (i), replacing the contribution of Qn by the one of Un + Oj, which amounts to replace xx by Xn-A if A is even in the first contribution to the factor of xa' • The same arguments exclude any a (S) > 3 and prove part (ii) of the lemma. Lemma 5.2 If u is odd the only possibility is On- Assume on the contrary that there exist additional possibilities. Since n is odd, from the previous lemma, amin = 1. Let y be such that 2 < n" and consider the coefficient of Xat- Since n is odd the only contributions are from S'a such that a (S) = 1 (indeed a (S) has to divide both n and V). The coefficient reads From the beginning of the discussion, all w's have to belong to U. Choose representatives in the range 0 < w < n. Let us then show that 2Tw has also to belong to U for any y such that V < n. If for some S, Vw € L, the corresponding negative contribution has to be compensated by some other arising torn u'V (a/ being possibly one). This requires V (w + u') = O mod JV. Let first 7 = 1. Then u + w' is smaller than N and 2 (u + (•>') being smaller than 2N is equal to N, thus w + u' = n. But u and u' are odd by u2 = 1 mod 2AT, hence a contradiction. Thus u < n/2 and the previous argument can be iterated. If 4 < n L we cannot have Au 6 L. Again an u' would be needed for compensation and 4(ui + w') = O mod N. By the previous bound applied to w and w', O < A(u+u') < 27V, so that 2(w + w') = n leading again to a contradiction, hence w < n/4 and so on. Thus for any y such that V < n we have 2Tu> € (7 and a positive representative can be chosen smaller than n 2—'. Take the maximal y such that 27 < n < 27+1 and w such that O < u < n 2~T. Recall that u n/6 = n/S mod N. But 6, a non trivial divisor of the odd number n is larger than 2. Hence n/6 < V (2/6) < 2? and uf < ^V • 2? = n. Thus u^ = J or u = 1, contrary to the assumption that it corresponds to a divisor S > 2. The lemma is proved. The search for further non trivial solutions is thus restricted to n even and «min = 1. We henceforth assume n even and study the effect of multiplying invertible integers mod N belonging to U by u £ 1, u2 = 1 mod 2N (a = 1), u> e U. Invertible integers mod N belong to (Z/ATZ)*, and we set U* = U n (Z/JVZ)*, L* = L n (ZINZ)'. The following is the crucial observation. Lemma 5.3 Let n be even, distinct from 12 and 30, and N = 2n, for each u Ç.U" such that u2 = 1 mod 27V, excluding u = 1 and, if n = 2 mod 4, excluding also n -1, there exists A € U* such that wA € L*. For each such u we associate a factorization of n into SS where (S, S) = 1 and S (S) is the smallest positive integer such that uS = —S mod N (uS = S mod N). The following pairs are excluded by hypothesis (i) {n, 1} and {l,n} corresponding too» = 1 and u = -1 (É U* (ii) {2,|} for n = 2 mod 4(a = l,w = n-l) (iii) {3,4} for n = 12 (W) {3,10}, {5,6} for n = 30. We search a representative A 6 U* in the range 0 < A < n, such that a representative A' of wA is in the range -n < uX < 0. Since (6, Ï) = 1 we look for A and A' in the form with Q < p < 6,0 < p < 6, (i and p prime respectively to S and S. Since n is even one in the pair S, S is even the other odd. The above conditions imply A prime to S and S hence to n = SS, hence to JV = 2n since n is even. Requiring 0 < A < n and —n < A' < 0 yields 7+I - As f and | should be positive irreducible fractions smaller than one. the lower bounds are irrelevant. If a solution (i, p exists, so does a fortiori a solution 1, p. Thus it is sufficient to look for p in the range 0 < p < B prime to 6 such that -s (i) 2 < 6 < S < n = 66. Set p = 1. The inequalities are satisfied since $ (ii) 2 < 6 < 6 < n, 6 even, hence 6 odd of the form 6 = 2fc + 1, k > 1. Set p = k = îfi. Clearly (p, 6) = 1 and since 6 > 6 > 2, the inequalities are satisifed. (iii) 2 < 6 < 6 < n, 6 odd, 6 = 0 mod 4. Choose p = § - 1 for 6 > 4. The case 6 = 4, 6 = 3 is one of the excluded ones. Hence S = 4k, k > I, and p = 2fc — 1 > 1. Any common factor of p and 6 would divide * — 2p = 2, hence p being odd (p, ^) = 1 and p/f < 1. Since 6 odd is larger than 2 we have 6 > 3, so 1/« < 1/3 and l-l/6> 2/3 while p/« = | - ^ is bounded by i - i < p/6 < ±, i.e. i/«<è (iv) 2<6<6 We can now complete the proof of the proposition. We assume n > 2 since n = 2 is a trivial case with a unique Xi — 1- The choice Qn leads to the /!-invariant and is the unique one for n odd (lemma 5.2). We look then for additional terms involving Q's such that w / ±1 when n is even larger or equal to 4. By lemma 5.1 these additional terms are such that amin = 1 or 2. IfOrnUn = 2 the only possibility is Cin + Oj (and 4 |n). Thus we are left with n even and amin = 1- Consider the coefficient of Xx for X CU*. Only the ftj's such that a (S) = I contribute and, if we have a further invariant, some do occur with a positive coefficient (in which case 6 = n/6 does not occur. Let us show that the corresponding multiplier w most be such that uU* = U' (we may include u = 1 corresponding to Qn). Indeed if A € U* is such that wA 6 i', then for positivity an other u' must occur such that u'X € U* and (u + u') X = 0 mod N. Since A is invertible mod N this entails w + w' = 0 mod N or 66' = n which is excluded by hypothesis. Thus we are in a position to apply lemma 5.3. We consider in turn the cases n = 0 or 2 mod 4. (i) n = 0 mod 4. If n ^ 2 all fi{, 6 £ n with amin = 1 are excluded. Then amin = 2, Un + Q1 is the only non trival possibility (D-type) besides Qn (A-type). For n = 12 an additional solution Qi2 + Q3+ Cl2 is found by inspection (Eg type). (ii) n = 2 mod 4. This implies amjn odd therefore equal to one. According to lemma 5.3 if n ^ 30, the only possible additional term with a (6) = 1 is Q^ (u = n — 1). By looking at the coefficient of Xi (i.e. Ev (®n + C2Q2)^' XX' = (1 - C2) X2 it follows that c2 is 0 or 1. Thus besides Qn (1-type), Qn +Q3 (D-type) c we have in this case for n / 30 as only further possibility Qn + Oj + 5Da(«)>3 <^i where all the a's are odd. Let us show that except when n= 18 and n = 30 all coefficients e$'s have to vanish. The proof parallels the one of lemma 5.1. Let crmin > 3 be the lowest possible value among those occuring in the additional terms. Look at the coefficient of x> where the first term contributes xa' + X- „' and only those fij's such mla Him s«l» that oc {6) = amin yield further contributions. For each such term the corresponding indices of x range over the vertices of a regular polygon with a'mia vertices. As in lemma 5.1, indices belonging to L from these 25 polygons cannot be compensated by those belonging to U from the saine or another polygon with the same number of vertices a'min. This leaves as unique possibility a single polygon (and a single (2«) with coefficient Cf = 1, the negative terms of which being compensated by xa' and *„_„' with amin — 3 pr 5, since the polygon bas at most two vertices in L. Let u be the corresponding multiplier. If the polygon has two vertices in L, we must have mod AT n for some Ç mod a'min. Set n = 2 («réin) ?> ^ 18. Subtract both equations, obtaining N/a'min = n — 2a r 801 min + PM f° Re integer p, i.e. ç [(2p +1) Cr^jn — 2J =1. Thus q = 1 and o4in = 3, n = 18 excluding amin = 5. If alun = 3 it is not possible for the equilateral triangle to have a single term in L compensated or m tne roof of by Xn' Xn-a' " P l«»ima S.I. To summariie the only exceptional cases left correspond to n = 18 and n = 3*0, which are readily studied separately with the result quoted in the proposition 5.2 and this concludes its proof. This proposition which settles the case of W.Z.W. models based on SU(2) is the building block for the classification of various other rational conformai theories. I will quote the main application to minimal models but skip some details. In the case of minimal models it is simpler to return to the original notation, introduced at the begining of this section, where the Virasoro characters are labelled by two integers r and $ mod 2pf and 2p respectively, (PiPO — I1 *ith appropriate (anti-symmetries. Then the role of N = 2n in the affine case is played now by the pair <2p,2j/}. Proposition 5.3 (i) Acting on Virasoro characters of the minimal series the commutant of the modular transformations is spanned by tensor products Qf x fi« in an obvious notation where 6'\j/,S\p. (ii) As a result the conformai invariant partition functions of minimal models are specified by a pair of elements in table II, where the role of n is played by p' and p. Since p and p1 are coprime one of them at least is odd the corresponding invariant being of A type. The partition functions are displayed in table HI, assuming for definiteness that p is odd (not necessarily greater than p'). In the unitary case, where we have successive integers m and m + 1, p is m for m odd or m +1 for m even. As a result we have two infinite series of invariants and three pairs of exceptional models. 26 r=l I P-1 2pr-l p > r odd = 1 r «w=1 2 2 >2 ±Ei E |Xr.| + 2|X3,.| + E (Xr.X;.-r.+C.C.) «=1 Lr oddxl r «ven=2 («I, Vl ,PI 18 ^E {iXi. +X17.I2 + lx«. + Xis.12 + |X7. + XU.I2 + |X».|2 +'[(Xs. + Xw.)xï, +e-c.J} (^7, 30 sE^X1'+*"' + *"' + *»''* + !*7' + *18' + *17'+*»»'*} (JSi. Table III - List of invariant partition function$ in term* of conformai characters with p aimmed odd. In the unitary case p = m, p' = m + 1 for m odd andp = m+l,p' = m for m even. ijt Proposition 5.3 has been applied to JV = 1 minimal superconformai models, parafermionic theories... and can be extended to coset models involving the algebras A1 '. 5.7 Attempts have been made to interpret and justify the appearance of an A-D-E classification of A\ ' W-Z-W models, in particular by Nabm, relating it to quaternionie spaces (Nahm -1987) acd/or to link it with other such classifications such as solutions of Yang-Baxter equations of lattice models, finite subgroups of SUs with the Mc Kay correspondence between their irreducible representations and simply laced affine Dynkin diagrams, isolated singularities of algebraic surfaces... For a review of some of these connections see (Slodowy -1983). We shall give some flavour of these interelations without much elaboration but observe that any real insight should at least have the virtue to give a hint on how to extend this classification to W.Z.W. models based on higher rank Lie groups. This does not. seem to be the case when these lines are written. We shall use as an illustration the Ei affine invariant (n = 30) in table II which we rewrite (5.47) Vl =Xl+Xll + Xl»+Xs» »a = X7 + X13 + XI7 + X» The indices run over the E» exponents split into two disjoint sets; taken together there are all integers primes to 30, in other words representatives of (Z/30Z)*. Under the operation T we bave /49 1\ <5-48> »J(T+1) = exp2i*( —-âJViM 27 Since the fifth power T5 acts as the identity multiplied by exp 2iir/12, it is a theorem that for such a low power the group of level five Ps acts trivially on the ratio z (r) = Jj|£j and the image of the modular group is realized as T5 = PSL (2; Z/5Z) acting on z (r) through homographie transformations. This group is isomorphic to the group At of even permutations on five objects and equivalently to the icosahedral group already encountered, with 60 = j5! elements. Thus we find a "natural" correspondence between the Ea invariant and this finite subgroup of three dimensional rotations. With the notations 2iV _w) u = exp ' ~Vi(r) (5.49) 2 2 U = u +u> = 2 cos — = U +U = = 2 COS y 5 we find (5.50) If following Klein one introduces isobaric polynomials in z i.e. polynomials which under modular transfor- mations are multiplied by a power of the denominator (Klein -1956) V = *(*1O-U**-1) E = s30+ 1-522(*S5-*6)-10005 (r10 (5.51) F = *M (185)w of respective degree* 12 (adding oo as a root of V)1 30 and 20, their zeroes on the Riemann sphere are the vertices, the mid edge point* said mid face points of a regular icosahedron centrally projected on this sphere. Furthermore the following relation holds - F3 + 1728 Vs = 0 (5.52) which may be interpreted as a surface in C3 with an isolated singularity at the origin, pointing to a relation with the classification of isolated singularities on algebraic surfaces. Let us show on this example the connection between finite subgroups of SU(2) and Dynkin diagrams of simply laced Lie algebras. For this purpose it is best to return to homogeneous polynomials in 1/1,1/2 which we denote with the same letters by substituting 5/J2V (**), yj»£ (ïi), tf»F (jf) for V{z), E{z), F(z). These new quantities are at most multiplied by a constant to the powers 12, 30 and 20 or its square to the powers 6, 15 and 10 respectively under modular transformations factors which do not affect the relation (5.52). This is due to the fact that on yi and 3/2 T and S act as unitary matrices (generating Vs ) times exp 2ix/12 and exp 2t>/4 respectively. 5 For two variables Jz1 and y2 which do transform according to F one can however find genuine invariant analogs of V1E, F which we call X, Z and V, homogeneous of degrees 12, 30 and 20 in yt and y't and which, absorbing irrelevant factors, satisfy a relation similar to (5.52) i 3 2 X + Y + Z = 0 (5.53) The binary icosahedral group f5 has 9 classes and 9 irreducible representations Ri. It has been recog- nized by Mc Kay that these representations can be made to correspond one to one to a set of fundamental roots of the affine extension £g of the simply laced Lie algebra £«. Similar properties hold of course for other members of the A-D-E family (Mc Kay -1981, Ford and Mc Kay -1981). To exhibit the correspon- dence, plot as nodes of a graph the representations Ri of At • Since this is a subgroup of SUj, it admits an irreducible representation of dimesion 2 (in fact it admits two such inequivalent representations, we pick one of those corresponding to a choice of root of 5). Call Rt this representation. Then by tensoring with Ri RjX Ri = ®fijRj (5.54) 28 The matrix C = 2 - / is the Cartan matrix of E^\ and /y is symmetric, equal to zero or one. We connect the nodes i and j if fy = 1, and obtain the Dynkin diagram of £g (if one deletes the node Ri corresponding to the identity representation, one obtains the diagram for Es)- If XA. (Ca) is the character of Ri evaluated on the class Ca, the non negative eigenvalues of C are 2 — \t (Ca). The eigenvectors are the columns of the character table pertaining to a fixed class Ca. The following table IV gives the characters of this group with u and v as in (5.49). The top row gives the number of elements in each class -the first column the dimension of the representation (as the character on the identity class). The table is completed by the Dynkin diagram. 1 1 30 20 20 12 12 12 12 R1 1 1 1 1 1 1 1 1 1 R? 2 -2 0 -1 1 u v —v —ii 4 2 -2 0 -1 1 V U —U —v R9 3 3 -1 0 0 —U —0 —V —u RI 3 3 -1 0 0 -V —U —U —V R* 4 -4 0 1 -1 -1 -1 1 1 4 4 0 1 1 -1 -1 -1 -1 Z 5 5 1 -1 -1 0 0 0 0 R8 6 -6 0 0 0 1 1 -1 -1 f >3 Ri R» ft S» TMt IV - Character table for the binary ieoitheiral group. AuocMtd Dynkin diagram. From the same data we can easily recapture the fact that the ring of invariant polynomials in yi, y't is generated by three homogeneous polynomials AT1V1^ of degree 2a = 12, 24 = 20, 2c = 30 respectively (the degrees have to be even to take into account the symmetry fyî,fej —• (-J/1. -S^J)- Note that with n = 30 the Coxeter number (of Es) we have a+6+c=n+l a relation also valid for other A-D-E members. Following (Kostant -1984) let Q(t) denote a formal infinite sum over irreducible representations Dj of SU: (it could be replaced by a convergent sum for |?| < 1 and Xj (6) = sin (2j + l)0/sin B substituted for Dj) Q{q) = 0 q«Dj Ij = 0 (555) DU2*Q{q) = where the second relation expresses that D1^ xDj = Dj+1/2®Dj_l/2. Restrict now each representation to a finite subgroup G C SUj (here the binary icosahedral group) such that Dj = ©n;-,jtfii with non negative integral n,,t. Then 2 Q (î) = © Pk (?) Rk Pt (?) = Y1 n,,W > (5.56) k 2>=0 with Pt (q) a generating function for the number of times (n;,*) the representation Rt appears in the decomposition of Dj. In particular Pi (q) will count the invariants. The relation (5.55) can be understood as referring to (tensor) multiplication by Ri the defining representation of G. Thus 29 Using the fact that the matrix / is symmetric and introducing the column vector P(q) = (P\(q),..\ P(O) = (1,0,0,...) we see that The matrix / is diagonal in the character basis (here the characters are real). Expanding P(O) as Pk (O) = St11 = Y1 xaXk (Cc) a and comparing with the orthogonality of characters c x* ( «)xi (C-) = \G\stt we find xa = l^jl, ft(ç) (5<59) -VWi-«x3(c.)+«» and in particular |gal 1 ~4V Applying this formula to the binary icoeahedral group, we find 1 + o30 1 — î80 1 — oîn P 6 6 > W = (1 _ f U) (1 _ ?20) = (1 _ ,11) (1 _ ,20) (1 _ ^)) = (1 _ ?2.) (1 _ ,«) (1 _ ,»«) ( - °) in agreement with the existence of generating polynomials X,y, Z of degrees 12, 20, 30 satisfying a "homo- geneous" equation of degree 60. One can also compute the other Pk'e which we list as ft W" (i-^H?-t») (5"61) Pi(«) = 1 + 930 Pî(«) = 11 P2 («) = 6 10 14 16 20 a4 P3(«) = ? + « + « +? + ï +î (5.62) M«) = 3 9 1113 17 '9« " Ps(«) = Ml) = Note that XX (5-63) and in particular 2P*(?)dim(fc) = —i-^ (5.64) The exponents appearing in the numerators p? ( 30 One can go even further by looking at the ring ofinvariant polynomials in XK^ graded as [Xn,Yn'Zn'] = On1 + 4n2 + cn3 (half the even degree in y[, y2) modulo those polynomials which are multiple of J£, jf, f£, where / is the left hand side of (5.53), / = Xs + Y3 + Z2, [f] = » here 30 (Saito -1986). Clearly we find a finite ring. In the present case one works mod [X*, Y3, Z] that is we have polynomials 1 X Y X2 YX X3 YX2 YX3 grade O 6 10 12 16 18 22 28 (5.65) 1 +grade 1 7 11 13 17 19 23 29 such that the set of (1+grade) is the set of Coxeter exponents (here of Eg). A generating function (for the exponents) is the polynomial _ (,.-!)(,* -l)(j«-l) where we have taken into account that a + 6 + c=n+l. The value 1> (1), the number of exponents, is the rank of the (non affine) Lie algebra (here 8). We reproduce below the data for the exceptional cases a 6 C n exponents tetrahedral E6 3 4 6 12 1,4,5,7,8,11 f56?« 3 3 2 {0 octahedral E 4 6 9 18 X Y + Y + Z 1,5,7,9,11,13,17 '°'> 2 6 icosahedra] E8 6 10 15 30 X + Y + Z 1,7,11,13,17,19,23,29 The connection between singular manifolds and effective potential in a Landau mean field approach is the theme of some recent investigations. 6. EXTENSIONS It is natural to expect that the W-Z-W models are among the basic building blocks of rational conformai theories. This is strongly suggested by the coset construction of Goddard, Kent and Olive. Therefore one would like to extend the previous classification to higher rank Lie groups. The task looks formidable, but one may hope to see the emergence of some general features. In this last section we describe an (easy) extension to SU(N) models at level one and indicate some ways to deal with rank two Lie groups using the arithmetics of quadratic integers. W-Z-W models based on SU(Af) level one involve N characters labelled by an integer A running from 0 to N — 1 -and identified with the number of boxes in a Young tableau reduced to a single column of length at most N — '.. The empty Young tableau is assigned to the character based on the identity representation of SV(N), the vacuum state. The modular transformation properties imply that the character Xx (T) can be considered as an even periodic function of A mod B, Ii T) = X-X(T) (6.1) j T XX(T + 1) a «paw [ \N -—J 1 6 2 5o XA(t --i\r} VBi*J*r-—- 2i*AA xy(T). . ( - ) The corresponding algebra of fusion rules is isomorphes with the algebra of N—th roots of unity. In order to get positive integral invariant partition functions, properly normalized, we can exploit our results of 31 the preceding section, where the formulas were quite similar except for the fact that jY was even and the character odd in A for SU2. Here we set n = N/2 if 21TV or n = JV if V(N. Apart from this minor change the definition of îî«, 6 \N is the same as in (5.29) except that when n = JV = 1 mod 2 the multiplier w satisfies u2 = 1 mod N/a2 instead of mod 2/V/a2. A second caveat (a reflection of "charge conjugation" symmetry) is that while the restricted characters xx M and XAT-A (T) are equal, this ceases to be true if we keep the dependence on (Cartan torus) angles. So we distinguish a partition function of the type J2x x> Xx Z\X'Xx> from Yl\,\i Xx Zx.N-X'Xx', even through as functions of r only, they appear identical. Proposition 6.1 For every divisor 6 of JV, if JV is odd, or of JV/2 if JV is even, the partition function (T,T) = ^ x(r (63) = £ 12 Xa\(T)xauX+( N/a(r) A.A'mod N/a ( mod a is a positive integral invariant partition function for SU(JV) at level 1 properly normalized. A natural conjecture was that these exhaust the consistent SU(JV) models at level 1. P. Degiovanni has informed me that he succeded in actually proving the conjecture to be true. Let us show some non trivial examples apart from the two obvious solution* 6 = n, ^A mod N \Xx (T)f and * = 1 the "conjugate" solution (numerically equal) EA mod jv XX(T)XN-X(T). Consider first SU(9). Then we have an extra invariant 3>,3 = W)|2 (6.4) ^W = Xo(r) + x3(r) + x«(r) = xo(r) + 2x3(r) behaving as r"1) = ^^) (6.5) (6.6) it is a pleasure to check that il> (r) coincides with the cube root of the modular invariant j (r), namely the unique character of Eg ' at level one r/> (r) = ; (r)I/3 = q~1'3 [l + 248? + 4124?2 + ...] (6.7) More generally for p an odd prime, we have (6.8) a mod p 1 For p= 5, i.e. for SU(25) at level 1 the quantity Xo + 2 (xs + Xio) is equal to the modular invariant up to a constant Xo (T) = 2 (xs + Xio) = j (r) - 120 (6.9) '• One could of course multiply the examples. I In conclusion I will report work in progress with M. Bauer for SU(3). Most of what is said applies with I little modification to Gj or SU(5) (and in fact the discussion of general invariants disregarding integrality conditions, extends readily to higher rank Lie groups). The idea is to make use of the fact that the SU(3) 32 wieght lattice can be identified (up to scale) with a ring of quadratic integers with unique prime factor decomposition (class field number unity) and one can therefore hope to use this multiplicative structure as we did for SU(2). For SU(3) we identify the weight lattice with the ring E of integers in u = exp 2«>/6 j = w2 = exp 2:V/3 (6.10) we write any integer in £ as 1 il = m? with m,- € Z. The norm is familiar from the discussion in section 3. Let us consider a model at level Jb. The Coxeter number of SU(3) being 3, it is convenient to trade k for n = Jb+3 n>3 (6.12) A fundamental domain Bn for characters at level Jt in such that the corresponding suffixes A (quadratic integers) fulfill the inequalities A = mi + mjw ; mi >0, m mi + mj < n (6.13) Bn: 3 >o, The central charge is 8Jb (6.14) C=Jb + 3 ) The conformai weight h\ is related to the norm of A through AA-3 AÂ-n A. C (6.15) ~" 3n * 24 3n The root lattice is then the ideal generated by the prime p=l +w (6.16) and the ratio of the weight to the root lattice is the field /3 with three elements, with additive structure Z/3Z. It was called in older times by physicists the triality t (quark has triality 1, anti-quark triality -1). This is 2 E IpE = F3 t = mi - m2 mod 3 t = AA mod 3 (6.17) The relation with Young tableaux with at most two lines and mi + n>2 — 2 boxes is depicted as ( For rai,ra2 > 0 the representation of SU(3) indexed by A has dimension A3-g 2 tr — p* It is extended over all of £ as an odd function under the Weyl group generated by the reflection A —»  and the rotation of 2T/3, A —> j\. The two Casimir invariants, up to normalisation, are Ci (A) = AA = m\ + ml + mimj (6.19) 3 3 C3(A) =A +  =(mi-m2)(m,+2m2)(2mi+m2) 33 Note that charge conjugation A *—^ u\ or mi <—» n%i leaves Ct invariant but changes the sign of Ca- The three integers, obtained by the action of the Weyl group on mi - m^, di = mi — mî rfj = mi + 2m2 = tt(r) (6.23) and Bn appears as a fundamental domain for a Weyl-Coxeter inhomogeneous group (isomorphic to the one j of Aj '). With { a primitive 3n—th root of unity, £ = exp 2t>/3n, the modular transformation properties of the characters read , 1 +Xx 624 S Xk (-r" ) - ' V ^' 'xy (r) ( ) ^^"^ A'mod *T where the expression of S uses the antisymmetry of Xx with respect to W. The restricted characters are invariant under charge conjugation, a property which translates into X-A = -XA- From the above, we recognize that the diagonal sum E2 (6-25 *$(»•• A = E XA (r)XuX (T) (6-256) A6B. A complementary D series can be described as follows (Bernard -1987, Altschfiler, Lacki, Zaugg -1987). The equilateral triangle Bn is invariant by a rotation of 2T/3 around its center, > * A—> Lemma 6.1 (0 t (A) = ±n mod 3 <=*• | (H) t( 34 The structure of a modular invariant D is then given by ! • Proposition 6.2 The D-invariant reads 4"'(^) = ]£ XA (r) ZJPVXA' (i) n = O mod 3 ( *A,A, ' (), '(A)1A' P W p\ •*' 4 = { Z&, {\ 0 I \ 0 otherwise » (ii) n ^ 0 mod 3 In (i) the conditions p \\ , p |A' mean that A and A' belong to the root lattice, while in (ii) no such condition applies. To Zp we may add the conjugate Zpi which may or may not coïncide with ZJJ. For instance when n = 6 using the explicit notation XA = Xmi.m, 6 2 2 Zf (r,f) = 4 ? (r,f ) = |xi,i + Xi,* + X4.i| + 3|Xl,a| while for n = 7 XaI Xs,a we J If we use the fact that as function of r only, two conjugate characters are equal Xm1,m, = Xm3lmii find 2 'I1 that the difference Z^ - Z^ = 2 Ixs.i - X4,2| and indeed '{ n = 7 X2,i - X4,î = 3 ! One knows a few more invariants (Christe and Ravanini -1988). They emerge from embeddings into W-Z-W models for higher rank Lie groups at level 1 except Z^n (Moore and Seiberg -1988) which derives from the ZQ ' invariant using the identity »» = 12 X2,2 + X2.8+X8,î=X4,4+8 The embeddings aie in SU(6) level 1 (n = 18, c = 24), Es level 1 (n = 12, c = 6) and E7 level 1 (n = 24, c = 24) all the values of n being divisors of 24. For lack of a better notation they are referred as Zg invariants (we dont list a possible distinct conjugate) 4 .1||,.| » + 1X2,3+ X6,!!2 +1X3,2+ Xl.fil2 12) 2 2 n = 12 4 = Ixi.i + xio,i + Xi.io + Xs1S + Xs.a + Xi,5| + 2 |x3,3 + Xs,6 + X6,s| (6.28) 2) 2 3 2 4" = 1X1,1 + Xl1IO + XlO1I I + 1X3,3 + XS,8 + X6.s| + 2 |X4,4| + |X2,S + X5.2 + XS.SP + IXl.4 + X7,l + X4.7I2 + 1X4.1 + Xl,7 + X7,4|2 + XM (X2.Î + X2.8 + X8.2) + [xîâ + xTâ + Xal) X4.4 2 n = 24 Zf*> = Ixi.i + Xi.22 + X22.1 + Xs1S + Xi4,s + Xs1M + Xii.u + Xii,2 + X2,ii + X7.7 + Xio,7 + X7,io| 2 + 1X7,1 + Xi,7 + Xi«.i + Xi.is + X7,i« + Xie,7 + Xs,a + Xs1S + Xn.s + Xs1Ii + X«,n 35 Perhaps to DO surprise, for n = 12, if we define z (r) as the ratio , (T) = V2 X3.3 + X3.6+X6.3 Xl1I + XlO1I + Xl1IO + XS1S + X5.Î + X2,S we find that it transforms according to the tetrahedral group generated by T ? while in the case n = 24 the similar ratio z(T) _ X7.1 + X1.7+- Xl1I +XM1I + Xl1M + - transforms according to the octahedral group, generated by T Z(T + 1) =-iz(r) S Z(-T- Could there exist an icosahedral invariant not yet listed? One might dream that other finite groups (perhapi subgroups of SU(S)) might play a role. If one forgets for the moment the integrality (and positivity property) one can easily describe the commutant of the linear transformations (6-28) where the indices are mod Af. Following the technique of finite quantum mechanics of section 5, we introduce an Hilbert space H of dimension 3n2 spanned by an orthonormal basis {\X)} with A mod AT. The operators T and S act in "H. Now E^ as an additive group is isomorphic to its dual, this justifies introducing Zn2 diagonal operators (6.30) = Q"+", ft, v mod fif Further define Zv? operators P11, fi mod Jf such that C with P" |A) = |A +11) Q" P" = ^"+"•'p-'Qi' (6.32) The operators P,Q,£ generate the Heisenberg on EM and any operator on 1H can be expanded as =3^2 E P11Q11Tt(Q-1P-11M) (6.33) Define for each "vector" A = (p, v) it, v G E a symbol {A} = (6.34) 36 consistent from the fact that ppû+ pp.v is a multiple of 3. Thus {A} is not periodic mod Jf as [A] = but only mod 3n. If A = (a,/?) {A+vVA} ^ y-'+W+ff»+^ {A} On the other hand under the adjoint action of the modular group generated by ad T and ad S we have the simple relations I In ill -M: :)) (6.35) • s-1 •M: T» so that S~2 {X)S2 = {-A} and for any element of f A = C 3\-^A JT1 UM=UX) (6.36) The action is the identity if and only if A = I mod 3n so that the image group is G = F3". If we act on [AJ = P"QV mod Jf we expect therefore 1 .4" [Aj A = (B(M)-B(A) [^4] (6<37) with (B(M)-B(H - £(«td*+e*>»+t«(m>+/i»'))) ^i0n we Ji80 „;(.„ for 8nort (Q(JLAJU1 jt J8 readily verified that for any representative of A mod 3n this is invariant if we change ft and/or v by a multiple of Jf, so that it is well defined for X, mod Jf. Let WA. denote the isotropy group of A mod Jf in F3" , we have Lemma 6.2 The adjoint action of the modular group F3" , formula (6.37) is well defined if we make any substitution (H € K\, H' G HXA) (i) AffforA (ii)A — HA (iii) A—y AH' We have [AiJ] = [AJ and \XAH'} = [AyI]. Pick an arbitrary representative of A mod 3n, then XH differs at most from A by a vector proportional to Jf hence (9(MJu is invariant under A —> XH. Now change AH X A into AH' then ^(^ '- ) = (Q[MU'M)(Q(MH1 on tne other hand if A —• HA we have (9(*HA,X) _ £ Q(\HA,\H)çQ(\H£) _ çQ(A^.A)^(Xff,A) from (J) Hence to prQve (j) and (JJ) it J8 sufBcient to show that (Q(*H£) — i for any A mod Jf and H in WA . For convenience A = (jt,v) is written as (ft = Jb + k'p, v = l + Cp) and the condition X1 = X2 mod JV becomes Jb1 = Jb2 mod 3n, t\ = /2 mod Zn, Jb1 = Jb2 mod n, ^1 = I-, mod n. Let H € WA> ^=I I ad — ic = 1 mod n, then * fk t\ /a b\ f k t \ C 3 C 3-U' mod 3n for some pair p, q, mod 3. Written in full, we find (Q(IH,*) - j(*»- where the conclusion follows from equating the determinants of the matrix equality: (kt — kt) = k(f + qn)— (Jb' + pn) I mod 3n, hence kq—tp = 0 mod 3, and the lemma is proved. Let A mod Jf be a fixed representative of an orbit O G E* x E* /f3" of the action of the modular group on E*1 x E^, then we have the following 37 24 Proposition 6.3 For any orbit O we have a unique invariant, up to a phase depending on the represen- tative point AeO, \eo a* - ^ c e r3n ^ I f "**" (6.38) * and this is a basis of linearly independent non vanishing elements of the commutant. Any element of the commutant is a linear combination of such invariants. Two different orbits involve distinct sets of [A], and K\ does not vanish since it can be written as a sum over distinct terms [A, il] ^CiAi)1 picking a representative A in each left coset F3" / If A = \p,v} the determinant A = (/ui> — /if) ^jp^ mod 3n is obviously constant along each orbit, so is 6 the greatest common divisor of p,v and JV. Unfortunately this is not quite sufficient to characterize the orbits. M. Bauer has obtained an explicit description which will be reported elsewhere. To continue would require to find an appropriate basis of the commutant with integral valued matrix elements and then introduce the antisymmetry properties of the characters. Work is actively pursued in this direction. 38 25 r " REFERENCES D. ALTSCHULER, J. LACKI, Ph. ZAUGG, "TAe affine Weyl group and modular invariant partition function" , Phys. Lett. B288. 628 (1987). G.E. ANDREWS, "Generalized Frobenius partitions" , Memoirs of the American Mathematical Society, 49., number 301 (1984). 4 E. AURELL and C. ITZYKSON, "Rational billiards and algebraic curves" , Submitted to J. of Geom. and 4 Phys., special issue dedicated to I.M. GELFAND. D. BERNARD, "String characters from Kac-Moody automorphisms" , Nucl. Phys. B288. 628-648 (1987). M.V. BERRY, "Quantizing a classically ergodic system: Sinai's billiard and KKR method" , Annals of Physics, New York, 13i 163-216 (1981). A. CAPPELLI, C. ITZYKSON and J.-B. ZUBER, TAe A-D-E classification of minimal and A^ conformai invariant theories", Comm. Math. Phys. 13,1-26 (1987). J.L. CARDY, "Operator content of two-dimensional conformait* invariant theories", Nucl. Phys. B270 186-204 (1986). H.S.M. COXETER, "Regular polytopes", Dover, New York (1973). B.L. FEIGIN and D.B. FUCHS, "Verma modules over the Virasoro algebra", Fuuct. Anal. Appl. IZi 241-248 (1983). D. FORD and J. Mc KAY, "Representations and Coxeter graphs" in "The geometric Vein, The Coxeter FestschriffC, Davis éd., Springer (1981). D. FRIEDAN, Z. QUI and S. SHENKER, "Conformai invariance, unitarily and two dimensional critical exponents" in "Verier operators in mathematics and physics", 3. Lepowsky, S. Mandelstam and I. Singer eds., Springer, New York (1985), and "Details of the non-unitarity proof for highest weight representations oftke Virasoro algebra", Comm. Math. Phys. ML, 535-547 (1986). D. GEPNER and Z. QIU, "Modular invariant partition functions for parafermionic field theories", Nucl. Phys. B2S5 (FSlS) 423-453 (1987). D. GEPNER and E. WITTEN, "String theory on group manifolds", Nucl. Phys. B2ZS, 493-549 (1986). C. ITZYKSON and J.-M. LUCK, "Arithmetical degeneracies in simple quantum systems", J. Phys. A9, 211-239 (1986). V. KAC1 "Fnfinite dimensional Lie algebra", Cambridge University Press (1985). A. KATO, "Classification of modular invariant partition functions in too dimensions", Mod. Phys. Lett. A2, 585-600 (1987). F. KLEIN, "TAe icosahedron and the solution of equations of fifth degree", Dover, New York (1956). J. Mc KAY, "Carton matrices, finite groups of quaternions end Kleinian singularities", Proc. Amer. Math. Soc, 153-154 (1981) 39 B. KOSTANT1 "On finite subgroups of SU(S)1 simple Lie algebras and the Mc Kay correspondence", Ptoc, Nat. Acad. Sci. USA1Sl, 5275-5277 (1984). G. MOORE and N. SEIBERG, "Natunlity in conformai field theory", IASSNS HEP/31, Princeton (1988). W. NAHM, "Quantum field theory in one and two dimensions", Duke Math. Joutn. ££, 579-613 (1987). PJ. RICHENS and M.V. BERRY, "Pseudo integrable systems in classical and quantum mechanics", Physica 4 2E, 495-512 (1981). f A. ROGHA-CARIDI, " Vacuum vector représentations of iht Virasoro algebra" in "Vertex operators", op. V dt. (1985). K. SAITO, "Regular systems of weights and associated singularities", Advanced itudies in pure mathematics, g, 479-526 (1986). P. SLODOWY, "Platonic solids, Kleinian singularities and Lie groups", Springer Lecture Notes in Mathe- matics, JM, 102-138 (1983). A. WEIL, "Sur les sommes de trois et tuëire carrés", L'enseignement mathématique, XX1 215-222 (1974). r k i 40