166 Nuclear Physics B (Proc. Suppl.) 18B (1990) 166-179 North-Holland

C O M B I N A T O R I C S OF M A P P I N G CLASS G R O U P S A N D M A T R I X I N T E G R A T I O N

C. Itzykson and J.B. Zuber

Service de Physique Thdorique de Saclay, Laboratoire de l 'Institut de Recherche Fondamentale du Commissariat ~ l'Energie Atomique F-91191 Gif-sur-Yvette Codex

1 - I N T R O D U C T I O N

In the mid-seventies 't Hooft introduced a clever device to keep track of SU(N)-group theoretic factors in the perturbative expansion of gauge field models. The goal was to find an approximation in the large N (planar) limit. A similar approximation for vector val- ued fields singles out one loop graphs, easily handled, and provides an interesting model in a number of prob- lems - for instance critical phenomena. Alas in the case of gauge theories apart from phenomenological appli- cations, the scheme was not very successful since the leading terms still involve the computation of infinitely many perturbative terms. The story is recorded in a 1979 report by S. Coleman entitled "I/N".

What 't Hooft had shown was that by inserting an appropriate factor of N in front of a path integral action, vertices carry a factor N, propagators a fac- tor 1/N. To follow matrix indices (of gauge fields) he found convenient to represent propagators as double lines with opposite orientation which indicate the flow of indices, connected at vertices to cyclically ordered hooks (representing traces of powers of the field), in such a way that they close on F index loops, each one responsible for a factor N by summation over a drunmy index. For a vacuum connected graph with V vertices L propagators (or links) and F index loops the total power of N is therefore

NV-L+F

On the other hand, we can think of this collection of V vertices, L links and F loops with obvious incidence relations, as a two-dimensional complex which inherits a consistent orientation from the one on loops and is connected by definition. This leads to an identification of the combination V - L + F with the Euler charac- teristic X of an orientable compact surface of genus 0 such that X = 2 - 29. As X takes its maximal value

2 for the spherical (or planar, with an added point at infinity) topology, the leading term was called the "pla- nar" approximation, meaning that the corresponding "fat graphs" can be drawn on a sheet of paper without crossings.

Koplik Neveu and Nussinov then suggested a dras- tic reduction to a zero-dimensional toy model (simple m~egrats) m order to mluers~and the purely combina- torial aspects, using techniques from graph theory. By 1978 in a collaboration which included E. Br6zin and G. Parisi then D. Bessis we investigated the zero- and one dimensional problems at leading order, using sad- dle point methods, then the various subleading correc- tions with the help of orthogonal polynomials as sug- gested by Bessis. One must admit that at the time the physical motivation was slim, except that we found the leading approximation quite accurate in simple quan- tum mechanical problems where it relates to the semi- classical approximation.

Interest was revived in these questions in the mid- eighties when it was realized by David, Kazakov, FrShlich among others that the above techniques are very effec- tive in studying two-dimensional field (or statistical) models coupled to a random geometry in the context of a discretized regularized version, then looking at fixed points where a continuous geometry is restored. This is then called "quantum gravity" and generalizes to cou- pled analogues. At first the study was performed at fixed genus then extended recently to a resummation over all genera with surprising and exciting new results by Br6zin and Kazakov, Gross and Migdal, Douglas and Shenker. This resummation relies up to now on the use of orthogonal polynomial methods and a non trivial scaling limit. This subject now widely studied by various groups will not be pursued here. It is notice- able that in a different mathematical context Penner used matrix integration to i!luminete a computation performed by Hater and Zagier pertaining to the topo- logical properties of the mapping class group of Rio- mann surfaces.

0920-5632/91/$3.50 © Elsevier Science Publishers B.V. (North-llolland)

C. Itzykson, J.B. Zaber / Combinatorics of mapping class groups 167

Here we would like to survey some of the combi- natorics involved in these calculations (they are dual to each other in a sense that will made precise as we proceed) and point out a number of connections with the representation theory of linear and permutation groups. These might suggest further developpments.

We should not give the reader the false impres- sion that we are conversant with algebraic topology. For these aspects we rely on the original articles of Harer and Zagier, Penner and Ivanov quoted in the bibliography.

Briefly stated the mapping class group is an infi- nite discrete group desqribing the classes of (continuous or differentiable) one to one maps of a manifold onto itself up to equivalence under those homotopic (contin- uously deformable) to the identity. A familiar example is the modular group for two dimensional genus one tori. These groups are essential in defining fundamen- tal domains of integration over moduli spaces (of com- plex structures) in perturbative string theories. Their explicit form is still poorly understood in general to that even the topology of these fundamental domains is not easy to describe. This justifies an interest in the most global aspects, an example being the virtual Eu- ler characteristic. The reason for the terminology will be commented below.

Returning to matrix integration, it will be shown that one can keep track of perturbative diagmnuns by coding them using pairs of permutations up to overall conjugacy. This idea might have antecedents in the litterature on graphs. We learnt it from J.M. Drouffe who presented it in an appendix in our joint paper with D. Bessis (which can be consulted for general barrio- ground). This technique enables one to exhibit in a neat way the Poincard duality. On the other hand the (Frobenius) duality between the linear and the per- mutation groups is at the heart of the evaluation of some integrals. It would certainly be interesting to de- velop q-analogs of the integration scheme. We men- tion some work of Andrews and Onofri in this direction.

Hermitian matrices with which we deal exclusively can be considered as spanning the Lie algebra of the unitary group. With some adaptation the calculations presented below should be extended to any Lie algebra of a compact Lie group. For orthogonal and symplec- tic groups one can presumably develop a topologogical interpretation (involving possibly non orientable sur- faces).

2 - M A T R I X I N T E G R A T I O N R E V I S I T E D

We will deal with N x N Hermitian matrices de- noted generically M and depending therefore on N 2 real parameters. We set

dM = H dMii H d ReMijdlm M/j i i

168 C. Itzykson, J.B. Zuber / Combinatorics of mapping c/ass groups

= (Mi jMkt ) = ,5it,Sjk

Each graph factorises into connected components, la- belled by an index a. The V = X]Va = ~vk vertices and L = ~La = ½~kvk = n links thus correspond to a set of orientable surfaces with F = ~F~ faces (or index loops) and yield a term N F = I IaN F" in the compu- tation of (re.) . Each component has genus .q. given by 2 - 2 g , = V , , - L ~ + F a . To sum all these contributions, we code the graphs as foUows. Cut the n propagators in 2n hooks attached to the vertices. Label them arbi- trarily from 1 to 2n. Define two permutations a and r on 2n letters as follows. Take the hook labelled j. One of its double lines directed to a vertex re-emerges in the hook labeUed a( j ) . This defines a permutation a whose conjugacy class codes the vertices and is lVt2 u2...k vh ..., hence independent of the arbitrary labelling of the ver- tices. We identify the conjugacy class [a] with the par- tition v, where brackets around a permutation denotes its class. Thus if Sp is the permutation group on p letters, [Sp] its set of conjugacy classes, we have

e & . [oq =_~

On the other hand the contractions joining two hooks (attached to distinct or identical vertices) define a sec- ond permutation r. Clearly r 2 = 1 and r belongs to the class 2 n (n cycles of length two). The number of cycles in a r is the number of index loops or faces by the very definition of a and r. We write

%7\~ / @@D@ @~@Q /'/'~,/'/ /~//.~

d d d ' ~ ,Do-®o

[ar] (1)2(2) (1)2(2) (4) N s N s N

We can now make use of the characters of the permu- tation group S=n indexed by Young tableaux (denoted Y) also in correspondence with partitions (of 2n). Let ][a][ be the number of elements in the class [a] which we also identify with _v, then

(2n)! I[o1 I=1 ~ I = -~nk2,

We write X r([a]) for the value of the (real, integral) character pertaining to the representation Y evaluated on the class [a]. The orthogonality and completeness relations on characters read

xr([-]) Y~. xr ([rllxV'([ar]) = ( 2n )!6r'r' xr ([l =.])

rES=. (2n)! _ x r([d)x Y @'11 = l-F~e[~],[~,]

Y

[~,r] = g_

Thus if a is a fixed representative of the class H, we find

_~et&d r E [2"] [~r]=g__

= E E _~E[&.] reS2.

where in the first expression the sum over r represents the set of all contractions and is split into contributions according to the corresponding power of N. In other words it simply states the Feynman rules in an abstract

but efficient way. As an example consider the average of t rM 4 which is coded as H = {vk = ~4,k} . This is

t rM 4) = tr M M M M + t r M ' M M M + t r M M M M = N 3 +N 3 +N. On the other hand we take a as the cyclic permutation on four letters. We find three possible permutations of the type r as shown below and from the above formula we get the expected result 2N 3 + N

where xY([12"]) is the dimension of the representation Y. Thus using twice the second of these relations

_~6[S~n] Y , Y ' , r e & . (2n)[2 X v ([r])X Y ([2"])X r ' ([arl )x v (#)

We can now perform the free sum over r and obtain

(ta)= Z (t~), (2.2a) _~[&.] -

(t~_), 112"]l x r ([2"1) xr(g)xY(v) N~'~l--gl = (2n)! E x v ([12.'-]) -

-- y

(2.2b)

Thus ( ts) can be obtained as a sum of contributions over classes E, N ~ ' k may be interpreted as t , ( l l ) and 2 ~ . , i s 1 2-~,., • Formula (2.2b) exhibits a symmetry in the interchange _# ~ v reflecting the (Poincar6) duality between vertices and faces of the cell decompositions of

C. Itzykson, J.B. Zuber / Combinatorics of mapping class groups 169

surfaces. Note that ~'Pk (number of faces) plays some- how the role of area and lnN of its conjugate variable. As a check of equation (2.2) setting _v = [12n] we get the obvious result

(( trM) 2=) = Nn(2n - 1)!!

obtained from all possible pairings of valence 1 ver- tices. On the other hand the highest possible power N 2" only occurs, with coefficient unity, in ((trM~) ") corresponding to the contractions of M~s under each trace. Apart from these two extreme cases, the above sum remains difficult to write in closed form although perhaps one could develop a generating function. As a final remark we observe that in spite of the appear- ance of denominators (t~)~_ is, from its very definition

an integer for each N, thus the coefficient of N ~'pk is an integer (a positive one). Afortiori (ta) is an integer (for each integer N).

Let now the characters of the linear group G L ( N ) associated with the same set of Young tableaux (with 2n boxes) be denoted chy. As they are polynomials in the matrix elements, they can be naturally extended to any matrix M, so that it makes sense to consider chv(M). We have then the beautiful Frobenius reci- procity relation

t~,(M) = ~ c h y ( M ) x Y ( v ) (2.3a)

Or, by inverting this relation

1 c h y ( M ) - (2n]! ~ [E [ x V ( v ) t t ' ( M ) (2.3b)

" _~[S2~]

Thus with the help of (2.2) we can obtain the average of a character trough

(chr) = 1 ~ I-~ II ~1 ~r~'"~ 112=1 I xY(- ~) (2n)12 • _~,_~eiS:.l

xY ' ([2=])xv' (~_)X v ' (~_)

y~ ×v'([12-1)

Summing over E and using the orthogonality of char- acters of $2, this is

xY([2=]) xY(I-t) NStt* I~_l (chv) =[ [2"11 xV([12n] ) (2n)!

~e(s2,]

Since, as was noted above, tg_(ll) = N~gk, the last sum is identified from (2.3b) as chr(1) the d;,-~usion of the Y-representation of the linear group. Using ] [2 =] ]= (2n - 1)!!, we conclude that

(chy) = ( 2 n - 1Mt~,Y[I2 nl' Chy( l l ) , " A ~,t J / ' x Y ( [ 1 2 . ] ) (2.5a)

which admits the easy generalization

( e h y ( M A ) ) M = (2n -- l~,WWvYl[2 ~Fh. chy (A) (2.5b) , " A ',t , ] xY( [12n] )

Each factor on the right hand side of (2.5a) is clearly identified: (2n -- 1)!! is the number of contractions in a homogeneous polynomial in M of degree [ Y [= 2n, next xY([2n]) appears as the signature of Gaussian av- erages, finally the last term is the ratio of the dimen- sions of the representations of the linear and permuta- tion groups corresponding to Y. As far as we can tell formula (2.5) is new. It should admit generalizations for other compact Lie groups.

We present below a short table giving ( chy (M) ) for 2n = 2 and 4

2n Y (clay)

2 (2) N ( N + 1) 2

,,r12' N ( N - 1) 2

~(N + 1)(N + 2)(~ + 3) 4 (4) s

- N ( N 2 - 1)(N + 2) ( 3 ) ( 1 ) 8

(2)2 N 2 ( N 2 - 1) 4

(2)(1) 2 - N ( N 2 - 1 ) ( N - 2) 8

t,¢1'4 N(N - 1 ) ( N - 2 ) ( N - 3 ) 8

The reader will note that for positive integral N these values are always integers (sometimes negative). This is true in general, and we digress to give an argument communicated by G. Segal.

The set of characters chv(M) attached to Young tableaux with a given number [ Y [= 2n of boxes forms a basis of symmetric functions of degree 2n of

170 C. Itzykson, J.B. Zuber / Combinatorics of mapping class groups

the eigenvalues A1, A2, ...AN. Another basis, at tached to part i t ion u = 1 vt 2 v2...2n v2" of 2n, is provided by

m_.= l< i t__ 1, thus Y = (1 + q)(1)p with 1 + q + p = 2n. Furthermore

xYP.,([2n]) = ( - 1 ) p

xYp,, ([2"])

chyp., (11) XYp,q (12" )

5

podd (-l)S"~ ( ~ )

2n , p + q + l = 2 n

The constraint p + 1 _< N is automatically taken into account by the vanishing of the combinatorial factor. Thus splitting the sum over p into even and odd parts

(t[2.l) (2n - 1)II

n--1

+

n--1

p=O p ~Z ~ x N - 2p

(1 + z)N+2"-2P -2 ]

= 2i~r x N 1 +

E . - 1 p=0 P

- - 2/71" i ' ~ - ' ; x N 1 (1 "~- X) 2

j ( dx (1 + 2x)"(1 + x) N

Y 2i7r x N

where in the last step we changed variable, setting y =

C. Itzykson, J.B. Zuber / Combinatorics of mapping class groups 171

1 One concludes therefore that 1 +2x"

{t[2n]) = E Nn+l-2ge:g('n) l_

172 C. ltzykson, J.B. Zuber / Combinatorics of mapping class groups

Introduce creation and annihilation operators

A 0 at = A 0 a = ~ + ~ 2 0)t [a,a t] =11 $ = a + a t

The vacuum state Is = 0) is such that a[0) = 0 and

t s i,s) = (a*)" i0 )

Let us once more compute a generating function for arbitrary complex v

aCy, v) = a'-'O

= Z x (° ¢'0+"') ('a*)'[ O) s - ' O

Setting z =e i av , this is

G(y, v) = j~o 2'~ ~dO (O le~aeY(a+a,)eZa, l o )

and from the commutat ion rules

/o 2~ dO ~-+~z+y(z+O = ,

= ¢~ y2, [2~ dO (vei O A- £~e-i#~ 2n = e~2- +~v E (-~n) I. J0 /

n------O o o

=

n=O

We have therefore

S=O

/0 oo _Z2_~_!y2+v~ oo y2n

= dy2e ' , ' ' E ~ . 2 (vg)" n.-.~O

~--~1 2x eV° = 2x eV°( ~ ) = ~ (v~)" ~ - z l~-z

n ~ O

_

,=o--2--' We conclude that

/0 °° _ r / 2x ( I + x ~ 8

and

T(z) o o

,=0 ( 2 n - 1)!!

= I + E 1 - x \ 1 - - - ~ ] 8 = 0

which agrees with our previous result (2.6). (3.2)

4 - T H E V I R T U A L E U L E R C H A R A C T E R I S - T I C O F T H E M A P P I N G C L A S S G R O U P

Given a closed orientable connected surface of ge nus g, one considers the smooth (i.e. continuous, pos- sibly with continuous derivatives, this is not what mat- ters here) one to one orientation preserving maps. These form a group which possesses an invariant subgroup of these maps homotopic to the identity, with the dis- crete mapping class group as the factor group. The one to one maps act on the homologies in particular on the group H1 the only non trivial one. Given a set of generators of H1, the mapping class group transforms them linearly preserving the intersection matrix, i.e. as Sp(2g, 7/,) transformations, and it is asserted that this homomorphism is surjective. In the case g = 1 with Sp(2, 7] 0 ~ SL(2, 7/,,) this homomorphism is an isomorphism so that the mapping class group is a dou- ble covering of the s tandard modular group PSL(2, 77,,).

The modular group acts on the ratio r of two indepen- dent periods of elliptic curves, considered as a complex variable with positive imaginary part. The quotient of the upper half plane by PSL(2, 7/,) is depicted by a fundamental region which is a quadrangle Im r > ' 0, Iv[ >_ 1,. [Re r[ < 1 with the identification of sides through r ~ r + 1 and r * - r -1. This identifica- tion leads to an orbifold, topologically a sphere with a point (at infinity) deleted and Euler characteristic 1 (an oriented surface obtained from the sphere by glu- ing g handles and deleting s disks or s points has Euler characteristic 2 - 2g - s).

Although one can introduce the modular invari- ant function j ( r ) mapping the upper half r-plane mod PSL(2, 7/,) one to one on the complex plane to depict the situation, the smooth differentiable structure in the j-plane is not equivalent to the one in the r-plane at the pre-images of j = 1728 and j = 0 ( r = 3Vr2"-I mod PSL(2,2~) respectively). This is due to modular transformations with fixed points ( r ' = - r -1 , r ' = - ( r + 1)-1). However there exist subgroups of finite index acting without fixed points in the upper half plane. Such is the case of the modular subgroup of level 2, with index 6. If the elliptic curve is rep- resented as y2 = Pa(x), where the right hand side a polynomial of degree four in x, this modular subgroup leaves invariant the cross ratio of the four roots of this

C. Itzykson, J.B. Zuber / Combinatorics of mapping class groups 173

polynomial. This cross ratio is only defined up to per- mutation of the roots under which it assumes 6 values distinct from 0, 1, and oo (which would correspond to a coincidence of two roots and hence depict degeneracies of the topological torus). At these exceptional values j is infinite. Thus we have a six fold smooth covering of the modular space as a sphere with three punctures. This suggests to define the virtual (or orbifold) Eu- ler characteristic of the modular group of genus 1 as l X ( S 2 - { 0 , 1 , o 0 ) ) = - ~ . However this is not quite what is required since PSL(2, ~ ) is only a factor group of SL(2, 77,) being covered twice. This is responsible for an extra division by a factor 2, so with )~ the virtual Euler characteristic, one has

1 x(SL(2, 2Z)) = - 1-2 (4.1)

This slightly varadoxical result is what is generalizable to arbitrary genus and arbitrary number of punctures. One looks for a contractible space where a subgroup of finite index of the corresponding mapping class group acts without fixed points, computes the ordinary Euler characteristics of the corresponding factor space and divides by the index of the subgroup. One shows that the construction is independent of the various arbitrary choices.

However before proceeding to the general case we would like, for pedestrian physicists as some of us are, to rederive the - 1 / 6 for PSL(2, ~ ) in an other more di- rect (but equivalent) way, exposing perhaps some "nai vet~" on our part. As is well known the upper half r-plane can be endowed naturally with an SL(2, ]R) in- variant metric of constant negative curvature, say -1 . A geodesic triangle (meaning a triangle with arcs of geodesics as sides) has area (Tr - ~ - ~ - 3') if ~,/~, 7 denote the (interior) angles at the vertices. The modu- lar fundamental domain consists of two such triangles, each one with angles 0, ~ and 3" On the other hand for curvature - 1 the Gauss-Bonnet formula reads

1

t r i ang les

for a decomposition into geodesic triangles (the for- mula as it stands also holds for the sphere of curvature +1, where it correctly yields )/ -- 2). In the present case a blind application yields

1 ( ~ ~ ) =-g

Of course the use of the formula does not yield the true Euler characteristic (+1) but instead the virtual one because of the orbifold conical points where the dif- ferential structure is not smooth. Should one however

similarly dissect the 6-fold covering discussed above into twelve similar triangles, one would get the correct integral result - 1 . So we see on this "trivial" exam- ple why and how it is much easier to compute virtual characteristics. As emphasized by Penner and as im- plicitely recognized by Hater and Zagier this is natu- rally done in the context of matrix integration.

In the sequel one denotes F~ the mapping class group for genus g and 1 puncture (F~ for s punctures, Fg --- FOg). It acts on the space of conformal equivalence classes of such surfaces but also on cellular complexes defined by arc decompositions of such a surface. A clever choice and some topological analysis lead Harer and Zagier to a specific construction where the n,,mher of 6g -- 3 -- n dimensional cells weighted by the inverse of the order of their isotropy group is ~ and the virtual characteristic is obtained as the finite sum

x = ( 4 . 2 ) 2n

2g~n_6g--3

To define ~g(n), a combinatorial factor, one pro- cedes by intermediate steps as follows. Consider a 2n- gon, label the sides and identify them pair-wise to get an orientable surface of genus g. The number of dis- tinct fashions to do so is called eg(n). Note that after such an identification one gets a connected graph of a matrix theory with V vertices (of varying valence), n links (or propagators) and 1 face (or index loop) such that V-n4-1 = 2 - 2 g , hence n + l - 2 g > 0. So eg(n) is the number of such graphs. Each of these graphs may contain an arbitrary number of vertices of valence 1 (or "tadpoles"). Let pg(n) the number of graphs defined as before but without tadpoles. Since tadpoles may be attached to any of the two "lips" of a propagators one has clearly

eg(n) = ~m ( ~ ) pg(n-- m) (4.3)

Furthermore graphs without tadpoles may still contain vertices of valence 2, of "self-energy" type. Let )tg(n) denote the number of graphs with n-propagators of genus g and without valence 1 or 2 vertices. One has

Pg(n) = Y~ ( n ) A'(n - (4.4) m

In ~g(n) we only count graphs with vertices of valence k larger or equal to 3. For a given graph let Vk denote their number~ We have

V = E V k = n + l - 2 g ~__kVk=2n k>3 k>3

174 C. Itzykson, J.B. Zu ber / Corn binatorics of mapping class groups

Therefore 2n - 3V is a non negative integer, showing that $g(n) is non vanishing only if n satisfies the in- equalities

2g

C. Itzykson, J.B. Zu bet / Combinatorics of mapping class groups 175

The odd ones all vanish except B1 and

1 1 1 B0----1, B1 = - ~ , B 2 - - ~ , B4 =-3-"0...,

B2,, = ( - 1 ) " + ' IB2.[

Therefore

and

, . ( n ) =

(4.6b)

th t/21 = E B, , (~'nn)! (4.7) n--'--O

(2n)!

(n + 1)!(n - 2g)! \ n+l (4.8)

xcoefficient of t 2g in (~,t-'h~]t/2

In order to contrast it with the direct method of Pen- ner, we now copy the remaining calculation of Hazer and Zagier. Define the three generating functions in n with reference to equations (4.3) and (4.4)

E,(z) = ~ ~.(n)~" .~_0

MgCx) = EPg(n)xn (4.9) n>_O

n>_O

One is in fact interested in Lg(x) which is a polynomial in x. We find successively for Ix] small enough

m~O .>0 du ( 1 T u ) zm

m_>OE #o(m)xm f 2i7r u - x(1 + u) 2

- ~ M g ( 1 - 2x ]

Similazly

l _ x g 1_--~

So altogether

1 E ( x(1 + x) ~ (4.10) Lg(x) = (1 + x)(1 + 2x) g k(1 + 2x) 2 ]

Instead of Eg(x) it is more convenient to define an equivalent generating function which codes the quan- tities %(n). For tha t purpose notice that in (4.8) the

( ,_q~_~ "+~ coefficient of t 2a in kt h t/2] is a polynomial in n of degree g which vanishes when n = - 1 , while the prefactor which can be rewritten

(2n)! 1 ( 2n ) ( n - - l ) ! ( n + l ) ! ( . - 2 g ) ! = n + l ( ~ - - f ~ !

involves the rat io (n - 1)!/(n - 2g)! which is also a polynomial in n of degree 2g - 1 so altogether we fred

as a factor of n + 1 a polynomial in n of degree

d = 3g - 1 with n + 1 as a factor. We c.an expand it not in powers of n but equivalently as (n + 1) times a combination of the d ( = 3g - 1) quantities

1, n - - l , (n --1)(n -- 2), ,(n-1)(n-2)...(n-d+l)

which all vanish except for the first when n = 1, and axe of the form

( n - l ) ! ( n - l ) ! ( n - l ) ! ( . - 1)! ' ( . - 2 ) ! '"" ( . - ~):

r' So altogether inserting a factor ~ for convenience

~ ( " ) = . ! ~ ( 2 ~ ) ~ ( ~ - ; ) !

for some coefficients k(r) which depend on g. This leads one to introduce the polynomial

d A~(x) = Z k(r)x" (4.11)

r=l

in terms of which

.>0 .>0 ~- i (2,-), (n - . ) ! d z"(2n)!

d

d T

r=-I ~/1 - 4x

- , / r= -~ ~ k(') ~ r m I Cousequently

and from (4.10) the two polynomials Lg(x) and Kg(x) are related through

LgCx) = A•Cx(1 -}- x)) (4.13) ( l + x )

176 C. Itzykson, J.B. Zuber / Combinatorics of mapping class groups

Since K is a polynomial of degree d = 3g - 1 without constant term, the r.h.s, of this expression is indeed a polynomial in x of degree 6g - 3. From the definitions (4.2) and (4.9) it follows that

x(r~) = - ~ Lg(-X)

1 ~01 d x - - - - ~ x ( l _ x ) K # ( - x ( 1 - x ) )

1 fo ~ d = ~ dz ~-~(-l)r-'kCr)xr-I(1 - x) ~-' r-----I

d = ~(_1)~_,k(~)~!(~ - 11!

r=l (2r)!

One recalls that

2n (n 4 1) E k(r) (2r)! (n - r)! ~g(n) = n + l r=,

2n ) ( n - 1)(n- 2 ) . . . ( n - 2g + 1) \

= n + l / (

xcoeff, of t ~g in \ ~ ]

Comparing with the above we see tha t X (r~) is ob- (-) tained by setting n = 0 in the factor multiplying n + 1 on the r.h.s, of these expressions. Hence

xCr~) = - ( 2 g - I)! x coeff, of t 2g in B2~

= - ( 2 g - 1)! (29)!

t/2 th t /2

One concludes that

the true Euler characteristics e (r~) and e(rg) also ob- tained by Harer and Zagier. Although the trend cannot yet be seen in this table these authors also prove that in both cases of rg and r I the ratio e/x tends to 1 as

g ~oo.

Tab le of vir tual (X) and true (e) Euler characteristic for 1 < g 1 (4.14) 5 - D I R E C T M E T H O D

an elegant result in terms of Riemann's C-function at odd negative integers, where one recalls the functional equation (for integer g)

(-1)g (2g)! ¢(29) ¢(1 - 2g) - (2~)29 g

It can be shown that the virtual characteristic of mod- uli space without any puncture is given by

X (r~) ¢(1 - 2g) × (rg) = 2 - 2----~ = 2 - 2g 9 > 1 (4.15)

while for genus 1 one has x(r,) = x(r]) . This is recorded in the appended short table together with

Interchanging the roles of the marguerite graphs and their duals obtained by gluing the sides of a 2 n - g o n amounts to use the Poincar~ duality discussed in Sec- tion 2. According to formula (2.2) we can write

1 ( } 2 .

El~k -- n + l -- g (5.1)

To obtain Ag(n) defined in the previous section, all that is required is to modify the summation on the r.h.s, by insisting that #1 - / ~ 2 -- 0, since these give the numbers of vertices of valence 1 and 2 respectively.

C. Itzykson, J.B. Zuber / Combinatorics of mapping class groups 177

Therefore the virtual characteristic is

g x(r ) = 2n n

---- ~ ( - 1 ) n+'

n /~, #1 = #2 = 0

Ep~ = n + 1 - 2 g (5.2)

This may now interpreted, as noted by Penner, as the perturbative contribution of order N to the logarithm of a partition function with interaction Lagran&ian

~ k~#~!

Xk-2 k - - ~ t r M

k>3

if we extract the coefficient of

~r~k_>3(k--2)gt;t : . x2n--2(n+l--2g) : X40 -2

Thus finally (as an asymptotic expansion in x --* +0)

z4g-2×(r~) = coeffic, of N in lnZ(x, N) (5.3a) g

1 ~_trM~X ~

f dM e 6>2 %(m,N) = l t r M 2 (5.3b)

f d M e 2

In fact taking the logarithm, to restrict oneself to con- nected graphs, does not affect the coefficient linear in N, since the graphs having only one index loop are necessarily connected.

There is only one little problem. As it stands Z(x, N) is meaningless (except in a term by term per- turbative expansion) since the integrand in the numer- ator

d e t ( 1 - x M ) e

is undefined unless 11 - xM > 0 and moreover diver- gent (the inequality stands for each eigenvalue of M, or equivalently for the corresponding sesquilinear form). Henceforth we assume x > 0. The above disease is familiar to anyone who has tried to derive Sterling's asymptotic formula for Euler's F-function.

Recall that

r(8 + 1) = du e -u+~ hu

Assnming s real, positive, and large, the integrand h~.~ a maximum for u = s, so that changing variable from u to m through

we find

or writing 1

s = ~ z > O X2

1 ~,~ mkz k 1 r l / z . -z-'2 k

~ F = 2 (5.4)

f+_~ dm e- '2-

The analogy with the previous matrix integral is clear and the method for obtaining StirliI~'S formula also. Asymptotically in the perturhative expansion in pow- ers of x, we can drop exponentially small terms by extending the range of integration to - o o < m < A-oo. This shows by the way that the expansion although asymptotic is necessarily divergent. On the other hand it yields immediately the cure to a sensible Z(x, N) without affecting its asymptotic expau~on at the ori- &in. All what is needed is the insert in the integrand of the numerator a factor 0(1 - xM), with 0(y) the step function, equal to zero for y < 0, to 1 for y > 0.

To complete the calculation is now straightfor- ward. We factor the integral over M in t~n~s of an integral over the eigenvalues A0, A1, ---, AN_l, times an integral over the (diagonalizing) unitary matrices, as in Section 3, introducing as a Jacobian the square of a Vandermonde determinant A(A) 2 = l-I~$t(At - At).

The amended Z(x, N) takes therefore the form

zCz, N) = 1 f A 2 ( A ) I I d ~ (2 )N/2 l-I ¢ O~_k~_N--1 0(1 zA~)(1 " "~ ~ - - ~ A k ) z " e =

(5.5) Writing A(A) as a determinant, using the symmetry of the integral under permutations of the arguments Ak, and finally changing the variables of integration from Ak to y~ = 1-z-~, one obtains, from (5.4),

178 C. Itzykson, J.B. Zuber / Combinatorics of mapping class groups

Z ( x , N ) - (eX2) X2 x N ' , / ~ l]~-lp!

detr ~ - + r + s + l ] 0~v',s_~N-- 1

F r o m

r ( ~ + n + l ) -

it follows that

(~2)~+, r( ) II(1 + px 2) 0

Z(~,,N) = (~2)~2

r-l-s

det H (I + px 2) ]O~r, sSlV_ 1 p----0

The last determinant is evaluated as r+s

det H(1 + px2l]o

C. Itzyksoa, J.B. Zuber / ComMnatorics of mapping class groups 179

and/or perhaps relate them to some recent results in 2d-quantum gravity. Also one could try to compute some mean-values of "observables" in this theory and find a suitable interpretation.

Matrix integration is not limited to hermitian ma- trices. One could think of usixtg real symmetric or anti- symmetric matrices, quaternionic matrices, or ones in- volving Grassmannian variables.., to describe moduli spaces of non orientable or supersymmetric surfaces...

Also instead of integrating over non-compact vec- tor spaces one could consider integrals over compact groups with various weights, the more so since we saw in Section 2 how matrix integration is intimately re- lated to group theory. A work by Andrews and Onofri using the heat kernel over the Cartan torus of uni- tary groups, quoted in the bibliography, seems very suggestive in this direction of relations with "quantum groups". Moreover we have further extensions p~ssi- ble to integrals over coupled matrices, or even matrix quantum mechamics. Finally one can wonder about the status of matrix integration in relation with topological properties of moduli spaces. Is this a simple combina- torial trick or is there a deeper natural explanation for its occurrence?

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