---

Combinatorics of the Modular Group II

arXiv:hep-th/9201001v1 31 Dec 1991SPhT/92-001Combinatorics of the Modular Group IIThe Kontsevich integralsC. Itzykson and J.-B.

ZuberService de Physique Th´eorique de Saclay*, F-91191 Gif-sur-Yvette cedex, FranceAbstract We study algebraic aspects of Kontsevich integrals as generating functionsfor intersection theory over moduli space and review the derivation of Virasoro and KdVconstraints.Contents0. Introduction.

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

.11. Intersection numbers.. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .22.

The Kontsevich integral.. . .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .52.1.

The main theorem. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .52.2.

Expansion of Z on characters and Schur functions. .

. .

. .

. .

. .

. .72.3.

Proof of the first part of the Theorem.. . .

. .

. .

. .

. .

. .

. .

. .

133. From Grassmannians to KdV.

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

194. Matrix Airy equation and Virasoro highest weight conditions.. .

. .

. .

. .

. 295.

Genus expansion.. . .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. 316.

Singular behaviour and Painlev´e equation.. . .

. .

. .

. .

. .

. .

. .

. .

387. Generalization to higher degree potentials .

. .

. .

. .

. .

. .

. .

. .

. .

. 411/1992 Submitted to Int.

J. Mod. Phys.

* Laboratoire de la Direction des Sciences de la Mati`ere du Commissariat `a l’Energie Atomique

0. IntroductionThe study of two–dimensional gravity has uncovered a rich mathematical structure includ-ing Virasoro constraints, KdV flows, N = 2 twisted supersymmetry, etc.

The remarkablecontributions of Witten [1] and Kontsevich [2] to its topological interpretation have re-duced an intersection problem on the moduli space of curves to the computation of amatrix integral over N × N hermitian matricesZ(Λ) =RdM exp −trΛM22−i M36RdM exp −tr ΛM22(0.1)While many aspects of this connection have been already discussed by these authors[3][4],our endeavour has been to study this integral in a purely algebraic context, combiningreviews of former work and new results. We apologize to the expert reader who may skipsec.

1 where we sketch Kontsevich’s construction and parts of sec. 3 which present a shortaccount of Sato’s work on τ–functions.

In sec. 2 we study the integral (0.1) as an expansionin powers of the traces trΛ−r.

After taking a suitable large N limit, we prove the crucialproperty that the asymptotic expansion of this integral does not depend on the even tracestrΛ−2r. This result which followed in Kontsevich’s work from topological considerations isderived here in a purely combinatorial fashion.

The arguments although straightforwardare unfortunately rather intricate since they cannot directly apply to the finite N integral,for which only N of the traces are algebraically independent. The integral is thus subjectto several constraints:(i) The equations of the KdV hierarchy pertaining to a differential operator of secondorder follow from Sato’s work and this independence with respect to the even traces.These provide evidence of the equivalence of Kontsevich’s model with the one–matrixmodel considered by Witten in [1].

(ii) The Virasoro highest weight constraints follow from the matrix Airy’s equationsatisfied by the numerator of (0.1) (sec. 4) [3].As for the standard matrix models, a systematic genus expansion is possible, theleading term of which had been obtained by Witten [1] and from another point of view byMakeenko and Semenoff[5] relying on earlier work by Kazakov and Kostov [6] (sec.

5). Ananalysis of the resulting expressions in a certain singular limit (sec.

6) provides effectiveways to resum families of intersection numbers and derive explicit fomulae.This stepintroduces a Painlev´e equation and its perturbations.1

The integral (0.1) admits a generalization in which the cubic potential is replaced bya suitable higher degree polynomial [7][2]. The corresponding topological interpretationpresented in a recent paper [8] involves an intersection theory on a finite covering ofmoduli space.

On the other hand, it is most likely equivalent to the multimatrix integrals.Our previous discussion extends to these cases without any difficulty of principle (sec. 7)although the calculations soon become very cumbersome.1.

Intersection numbers.Witten conjectured in [1] that the logarithm of the partition function of the general one-matrix model [9][10], expressed in terms of suitable deformation parameters ti could beexpanded asln Z =Xk0,...,ki,...⟨τ k00 . .

. τ kii .

. .⟩tk00k0!

. .

. tkiiki!

. .

. (1.1)where the bracketed rational coefficients admitted the following interpretation as intersec-tion numbers.

Let Mg,n be the moduli space of complex dimension 3g −3 + n ≥0 ofalgebraic curves of genus g with n marked points x1, . .

., xn and Mg,n a suitable compact-ification. The cotangent spaces at xi define line bundles Li with first Chern class c1(Li) in-terpreted as 2–forms over Mg,n.

For integral non-negative df’s such that 3g−3+n = Pf dfthe integralZMg,nc1(L1)d1 . .

.c1(Ln)dn(1.2)(independent of the ordering since 2–forms commute and powers are exterior powers) isa rational positive number when one considers Mg,n as an orbifold, i.e. the quotient of acontractible ball (Teichm¨uller space) by the mapping class group.

If ki = #{df = i, i ≥0}(so that Pi≥0 i ki = 3g −3 + n) then Witten’s conjecture was that ⟨τ k00 . .

. τ kii .

. .⟩definedby (1.1) is given by the integral (1.2).

(We have skipped a number of essential technicalitieswhich make the above definitions sensible). An intuitive picture of the line bundles Li overMg,n is not straightforward but at least one can see that ⟨τ 30 ⟩= 1, since M0,3 is a point!Even to find that ⟨τ1⟩=124 from the definition is non trivial.These intersection numbers being topological invariants, Kontsevich has been able toreduce them to more manageable expressions using a cell decomposition of Mg,n inheritedfrom the physicists’ “fat–graph” expansion of Hermitian matrix integrals.

One considersconnected fat (i.e. double-line) graphs with vertices of valency three or more, genus g and2

n faces (dual to the n punctures). One assigns to each (double) edge e a positive length ℓeand to each face f a perimeter pf = Pe⊂f ℓe, where e ⊂f denotes the incidence relations.The set of such fat graphs with assigned ℓ’s is a decorated combinatorial model for Mg,n.Cells have “dimensions” over the reals obtained by counting the number of independentlengths for fixed values of the perimeters, namely E −n where E is the number of edges.If Vp denotes the number of p–valent vertices, one hasE −n −Xp≥3Vp = 2g −22E =Xp≥3pVp ≥3XVp.

(1.3)ThusE −n ≤2(3g −3 + n)(1.4)with equality (top dimension) if and only if all vertices are trivalent.Let k be the number of edges bordering a face f and ℓ1, ℓ2, . .

., ℓk be their successivelengths as we circle around the boundary counterclockwise (the faces come with a positiveorientation), up to cyclic permutation. One introduces the 2–formω =X1≤a

. .

τdn⟩=Z Yfωdff(1.6)the integral being on top-dimensional cells (we omit a discussion of the compatibility oforientations with the complex structure of Mg,n). A generating function is obtained bycomputingZ ∞0Yfdpf e−λf pf Z (P p2f ωf)3g−3+n(3g −3 + n)!

(1.7)3

where the integral sign stands both for the integral over a cell and a sum over cells ofdimension 3g −3 + n weighted by the inverse of the order of their automorphism group(orbifold integration). On the one hand this isXd1+...+dn=3g−3+n⟨τd1 .

. .

τdn⟩ZnY1dpf e−λf pf p2dffdf! (1.8)= 23g−3+nXd1+...+dn=3g−3+n⟨τd1 .

. .

τdn⟩nY1(2df −1)!! λ−(2df+1)f.On the other hand one can proceed to a direct evaluation using1(3g −3 + n)!Yfdpf ∧ XfX1≤a

. .∧dℓE (1.9)up to an overall orientation which is henceforth ignored.

The computation of the aboveJacobian (which depends on the structure of the graph only through g and n) is a delicatematter for which we refer to [2].Inserting this expression into the integral one notesthat each edge of length ℓis shared by two faces f, f ′ ⊃e, the corresponding integralcontributing a factor2λf +λ′f , while the ratio of powers of 2 reads 25g−5+2n/23g−3+n =22g−2+n = 2E−V . By comparison one obtains Kontsevich’s main identity with Γg,n the setof all face-labelled trivalent connected fat–graphs γ of genus g and n facesXΣn1 df =3g−3+n⟨τd1 .

. .

τdn⟩nYf=1(2df −1)! !λ2df +1f=Xγ∈Γg,n2−V|Autγ|Ye∈γ2λf + λf ′(1.10)where as above V ≡Vγ, e denote the edges of the graph and 2/(λf +λf ′) is the propagatorattached to the edge e bordering f and f ′.The right hand side of this expression is suggestive of the Feynman expansion ofthe logarithm of the matrix integral (0.1) over N × N Hermitian matrices (N →∞).Let Λ stand for a diagonal matrix diag (λ0, .

. ., λN−1) and introduce the infinite set t. ={t0, t1, .

. .} defined as1ti(Λ) = −(2i −1)!!

trΛ−2i−1 . (1.11)1 By convention (−1)!!

= 1.4

By summing the above expression over g and n, one hasF(t.(Λ)) =Xn≥1d1,...,dn≥01n!⟨τd1 . .

. τdn⟩td1(Λ) .

. .

tdn(Λ)(1.12)=Xγ∈ΓN i2V1|Autγ|Ye∈γ2λf + λf ′≡ln Z(N)(Λ)where the last equality is in terms of the Feynman graph expansion of the integral (0.1)and PΓN refers to the summation over fat graphs with N faces and all possible distinctassignments of variables λ to their faces. We have noted that (−1)n = iV since the relation2E = 3V for trivalent graphs implies that the number of vertices V is even = 2p and fromV −E + n = 2p −3p + n ≡0 mod 2, it follows that n is of the parity of p. Formula (0.1)leads to a propagator 2/(λf + λf ′), while the coupling at each vertex is i2.

Thus the abovereads (after letting N →∞)ln Z(Λ) =DeP∞1 τdtdE(1.13)in agreement with (1.1). In the next section, we discuss the precise mechanism of theN →∞limit by which from finite Λ and finitely many independent trΛ−(2i+1) infinitelymany independent variables t are generated.The first non trivial graphs in the Feynman expansion yield after some rearrangementF = ln Z = t303!

+ t124 + t30t13! + 124t0t2 + t212+ .

. .

(1.14)2. The Kontsevich integral.2.1.

The main theoremWith the normalized measuredµΛ(M) =dM exp −12trΛM 2RdM exp −12trΛM 2 ,dM ≡NYi=1dMiiYi

or rather its asymptotic expansion in traces of powers of Λ−1 for large N. A generalizationis presented in sec. 7.More precisely the terms Z(N)k(Λ) can be expressed as polynomials with rational co-efficients in the variablesθr = 1r trΛ−r.

(2.3)Then Z(N)kis homogeneous of degree 3k, if we setdeg θr = r.(2.4)Only the first N of the θr are algebraically independent for an N × N matrix Λ. HoweverLemma 1Considered as a function of θ. ≡{θr}, Z(N)k(Λ) is independent of N for3k ≤N and depends only on θr, 1 ≤r ≤3k.This allows one to define unambiguously the seriesZ(θ.) =Xk≥0Zk(θ.)(2.5)whereZk(θ.) = Z(N)k(θ.)

,N ≥3k(2.6)without any further reference to N. The set of variables θ. is denumerable but each Zk(Z0 = 1) depends on finitely many of them.This almost evident lemma follows fromeq. (2.43) below.Theorem 1 (Kontsevich)(i)∂Z∂θ2r(θ.) = 0r ≥1(2.7)(ii) Z(θ.) is a τ-function for the Korteweg-de Vries equation.

Namely iftr = −(2r + 1)!! θ2r+1 = −(2r −1)!!

trΛ−2r−1u = ∂2∂t20ln Z(2.8)then∂u∂t1= ∂∂t0 112∂2u∂t20+ 12u2(2.9a)and more generally∂∂tnu = ∂∂t0Rn+1. (2.9b)6

In (2.9), the Rn denote the Gelfand-Dikii differential polynomials (derivatives are takenwith respect to t0)R2 =u22 + u′′12R3 =u36 + uu′′12 + u′224 + u(4)240R4 =u424 + uu′224 + u2u′′24+ uu(4)240 + u′u′′′120 + (u′′)2160 + u(6)6720. . .

(2.10)Rn =unn! + · · ·computed from(2n + 1)R′n+1 = 14R′′′n + 2uR′n + u′Rn.

(2.11)Proof of the first part of theorem 1 follows in Kontsevich’s approach from the identities(1.10)–(1.13), relying on topological considerations, namely on the introduction of thecombinatorial model for Mg,n. The rest of this section is devoted to a purely algebraicproof of this result.2.2.

Expansion of Z on characters and Schur functionsThe integral (2.1) in the case N = 1, denoted for short z(λ),z(λ) =R ∞−∞dm e−12 λm2+ i6 m3R ∞−∞dm e−12 λm2= λ2π 12 Z ∞−∞dm e−12 λm2+ i6 m3(2.12)admits the asymptotic expansionz(λ) =∞Xk=0ckλ−3k(2.13a)ck =−29k Γ(3k + 12)(2k)!√π(2.13b)and satisfies the differential equation equivalent to Airy’s equationD2 −λ2z(λ) = 0(2.14)D = −e13 λ3λ12 1λ∂∂λe−13 λ3λ−12= λ +12λ2 −1λ∂∂λ . (2.15)7

For N arbitrary, we can make a shift of variable M →M −iΛ in the numerator ofthe integral (2.2.) This givesZ(N)(Λ) =12N(N−1)2(2π)N22Yrλ12rYr

We then integrate over “angles”, i.e. over the unitary groupwith the action M →UMU −1.

The result [11] (which we now realize had been establishedlong before by Harish-Chandra [12]) yieldsZdM exp i trM 36+ MΛ22== (2π)N22ZY0≤r≤N−1dmr(2π)12 ei16 m3r+ 12 mrλ2rY0≤r

.xN−10x01x11. .

.xN−11............x0N−1x1N−1. .

.xN−1N−1=Y0≤r

., xN−1|, with the understanding that in each row one substitutes successivelyx0, x1,· · ·, xN−1 for the variable x. Inserting (2.17) into (2.16), we findZ(N)(Λ) = |D0z, D1z, · · · , DN−1z||λ0, λ1, · · ·, λN−1|,(2.19)with D defined as in (2.15). The asymptotic expansion involves only inverse powers of Λ.We setxr = λ−1rθk = 1kX0≤r≤N−1xkr ,(2.20)and consider henceforth the function z as given by the formal seriesz(x) =∞X0ckx3k.

(2.21)8

We have D2z = λ2z, and one readily sees thatD2kz = λ2kzmod (D2k−1z, · · · , D0z)D2k+1z = λ2k+1zmod (D2kz, · · · , D0z) ,(2.22)withz = 1λDz(2.23)=1 + x312 + x ddxz =∞X0dkx3kdk = 1 + 6k1 −6k ck.As a consequence, we rewrite (2.19) asZ(N)(Λ) =|λ0z, λ1z, λ2z, · · · ||λ0, λ1, λ2, · · ·, λN−1| ,(2.24)where the last term in any row of the upper determinant is λN−1z if N is odd, andλN−1z if N is even.Finally we cancel from numerator and denominator the productλ0 · · · λN−1N−1, and express Z(N) in terms of the variables xr = λ−1rasZ(N) = |xN−1z, xN−2z, · · · ||xN−1, xN−2, · · ·|. (2.25)Then we substitute the expansions of z and z to obtain the seriesZ(N) =X0≤n0,n1,···,nN−1c(0)n0 c(1)n1 .

. .

c(N−1)nN−1|x3n0+N−1, x3n1+N−2, · · ·, x3nN−1||xN−1, xN−2, · · ·, x0|,(2.26)with the convention thatc(2p)n= cnc(2p+1)n= dn. (2.27)When the indices f0 = 3n0, f1 = 3n1, · · ·, fN−1 = 3nN−1 form a non-increasingsequence, we recognize in the above ratio of determinants the symmetric function inx0, · · ·, xN−1 which corresponds to the polynomial character of the general linear groupGL(N) specified by the Young tableau with rows of length f0, f1, · · ·, fN−1:9

lN−1 = f0 + N −1lN−2 = f1 + N −2.........l0 = fN−1.The latter admits a natural extension outside the standard Weyl chamber, as an antisym-metric function of the unordered exponentslN−1 = f0 + N −1, · · · l0 = fN−1 . (2.28)Henceforth we refer to this extension when we write this character aschlN−1,···,l0 = |xlN−1, · · · , xl0||xN−1, · · ·, x0| .

(2.29)We wish to express this quantity in terms of the traces θk = PN−10xkr. This follows fromthe standard identities for characters which we now recall [13].Thinking of X as thediagonal matrix X ≡Λ−1 = diag (x0, x1, · · ·, xN−1) we set s0 = p0 = 1 and for k ≥1sk(X) =tr ∧k Xdet(1 + uX) =NX0uksk(X)(2.30a)pk(X) =tr ⊗ksym X1det(1 −uX) =∞X0ukpk(X)(2.30b)θk =1k trXk.

(2.30c)The quantities sk, 1 ≤k ≤N, are the elementary symmetric functions corresponding tothe vertical Young tableaux up to N lines, whereas the pk’s are the traces of symmetrictensor products, i.e. they correspond to Young tableaux with only one row.We have from the definitionexp∞X1unθn(X) =∞X0ukpk(X) ,(2.31)10

which expresses the pk’s as homogeneous polynomials of degree k of the θn’s (deg θn = n),ignoring the relations among traces. This justifies the definition of (elementary) Schurfunctions pn viaexp∞X1unθn =∞X0ukpk(θ.) ,(2.32)without reference to any N × N matrix, and where now the p’s are functions of the θ’s.When both p and θ refer to the same matrix X we recover the previous definitions (2.30)and (2.31).

Explicitly we writepr(θ.) =Xν1+2ν2+...=rθν11ν1!θν22ν2!

. .

. .

(2.33)Eq. (2.32) entails∂pr(θ.

)∂θk= ∂kpr(θ. )∂θk1= pr−k(θ.) ,(2.34)where pn vanishes for n < 0.When expanding the matrix elements along successivecolumns, Cauchy’s determinental formuladet11 −xrys0≤r,s≤N−1= |xN−1, · · ·, x0| |yN−1, · · ·, y0|Qr,s(1 −xrys)(2.35)yieldsXl0,···,lN−1ylN−10· · · yl0N−1|xlN−1, · · ·, xl0||xN−1, · · ·, x0| = |yN−1, · · ·, y0|Qr,s(1 −xrys) .

(2.36)Therefore if X ≡diag ({xr}), thenchlN−1,···,l0(X) = coeff. of ylN−10· · · yl0N−1 inyN−10det(1−y0X).

. .y00det(1−y0X).........yN−1N−1det(1−yN−1X).

. .y0N−1det(1−yN−1X).

(2.37)Expandinghdet(1−yX)i−1according to (2.30b), we obtain the classical formula (Jacobi-Schur)chlN−1,···,l0(X) = |xlN−1, · · ·, xl0||xN−1, · · ·, x0| =plN−1−(N−1)(X). .

.plN−1−1(X)plN−1(X)plN−2−(N−1)(X). .

.plN−2−1(X)plN−2(X)......pl0−(N−1)(X). .

. pl0−1(X)pl0(X),(2.38)11

valid for any ordered or unordered sequence lN−1, · · ·, l0. Terms along the diagonal readpf0, pf1, · · ·, pfN−1, and indices increase (decrease) by successive units as one moves froma diagonal term to the right (left).

We abbreviate this expression aschN−1+f0,···,fN−1(θ.) =pf0⋆.

. .⋆⋆pf1.

. .⋆............⋆⋆.

. .pfN−1,(2.39)substituting for the elementary Schur polynomials their expressions in terms of the vari-ables θ.. We conclude thatZ(N) =Xn0,···,nN−1c(0)n0 c(1)n1 · · · c(N−1)nN−1p3n0...p3nN−1(2.40)yields an expression of Z(N) in terms of the infinitely many variables θ.

(which can hence-forth be treated as independent). It follows from eq.

(2.40) thatZ(N)k=Xn0+···+nN−1=kc(0)n0 c(1)n1 · · · c(N−1)nN−1p3n0...p3nN−1(2.41)where each character is of degree 3k. This is obvious for the diagonal term and, as onereadily ascertains, holds also for non-diagonal terms.We are now in position to prove the lemma, which is trivially true for Z(N)0= 1.Suppose 0 < 3k ≤N and for a given term in (2.41) let δ be its “depth”, i.e.

the smallestinteger ≤N such that r ≥δ ⇒nr = 0. From (2.39) it follows that the corresponding termreadsc(0)n0 c(1)n1 · · · c(δ−1)nδ−1p3n0...p3nδ−1;n0 + · · · + nδ−1 = k.The last column of the δ×δ determinant readsp3n0+δ−1, · · · , p3nδ−1T where the subscriptsare δ positive integers with a sum equal to 3k + Pδ−10r.

If 3k < δ, this is smaller than thesum of the first δ positive integers. From Dirichlet’s box principle, two of the subscriptsamong 3n0 + δ −1, 3n1 + δ −2, · · ·, 3nδ−1 have to be equal, which results in two identicallines in the determinant which therefore vanishes.

Hence δ has to be smaller than or equalto 3k, showing thatN ≥3k=⇒Z(N)k= Z(3k)k≡Zk(2.42)which concludes the proof and allows a definition of the formal series Z, without referenceto N.12

2.3. Proof of the first part of the Theorem.

(i) The first part of the theorem will be proved for each Zk which we take equalto Z(3k)k. Differentiating each line successively in the determinental characters using thecrucial formula (2.34) we find2r > 3k∂Zk∂θ2r= 0(2.43)2r ≤3k∂Zk∂θ2r= P3k−1s=0 Zk,(s)Zk,(s)= Pn0+···+n3k−1=k c(0)n0 c(1)n1 · · · c(3k−1)n3k−1p3n0...p3ns−2r...p3n3k−1where subscripts in the s-th row of each determinant have been decreased by 2r units. For0 ≤s ≤3k −1 −2r the subscripts in line s and s + 2r only differ by the interchange ofns and ns+2r.

The determinental character is therefore antisymmetric in the interchangeof indices ns and ns+2r whereas in the product of c’s, due to (2.27) . .

. c(s)ns .

. .

c(s+2r)ns+2r . .

.≡. .

. c(s)ns .

. .

c(s)ns+2r . .

. is symmetric in these indices.

As a consequence, Zk,(s) vanishes fors ≤3k −1 −2r and we need only consider terms with s > 3k −1 −2r, i.e. when thederivative acts on one of the last 2r lines and we cannot use the above argument relyingon the periodicity c(r)n= c(r+2)n.(ii) Therefore fix s such that 3k −2r ≤s ≤3k −1.

The only possibly non-vanishingterms in Zk,(s) are those whose depth δ defined as above to be the smallest integer suchthat r ≥δ ⇒nr = 0, satisfies the inequality 3k −2r ≤s ≤δ −1 ≤3k −1. In this casethey readc(0)n0 c(1)n1 · · · c(δ−1)nδ−1p3n0...p3ns−2r...p3nδ−1, nδ−1 > 0, n0 + .

. .

+ nδ−1 = k . (2.44)The δ indices of the p’s in the last column of the determinant before subtracting 2r fromthe indices of the s-th row are all positive integers and have a sum equal to 3k −δ + Pδ1 t,where 0 ≤3k −δ ≤2r −1.

We now make use of the following13

Lemma 2.If from a set of δ positive integers, with sum exceeding the one of the first δ positive integersby an amount ∆≥0, one decreases one by ∆′ > ∆, then in the new sequence two termscoincide or one is a non positive integer.Think of the original set as occupied integral levels. Let r0 +1 be the first unoccupiedone (r0 ≥0) and r1 the greatest occupied one.

It follows from the hypothesis that r1−r0 ≤∆+ 1. If one decreases one element of the set by an amount ∆′ ≥∆+ 1, it thereforebecomes less than or equal to r0, which proves the lemma.Applying this result to the above circumstance (∆= 2r −1, ∆′ = 2r), we deducethat the only possibly non-vanishing terms in Zk,(s), 3k −2r ≤s ≤3k −1 occur when3ns = 2r −(δ −1 −s) with 0 ≤δ −1 −s ≤2r −1.

Thus 2r −(δ −1 −s) takes the possiblevalues 1, . .

., 2r. If r = 1 this is never a multiple of 3.

Thus we have already obtained∂Z∂θ2= 0. (2.45)We can even say more.

Let a be the integral part of (δ −1)/3 and consider in the lastcolumn starting from the bottom the a + 1 positive subscripts3nδ−1, 3(nδ−1−3 + 1), . .

., 3(nδ−1−3a + a) .For the corresponding character to be non-zero, these have to be all distinct. Hence theirsum is larger than or equal to 3 Pa0(α + 1).

The inequality3aXα=0nδ−1−3α + α≥3a+1X0α(2.46)implies that Paα=0 nδ−1−3α ≥a + 1. Should a non-vanishing term arise in Zk,(s), therewould exist an index ns such that from the preceding observation3ns + δ −1 −s = 2r(2.47)with3k −2r ≤s ≤δ −1 .

(2.48)Thus 3ns +δ −1 = 2r +s ≥3k, or equivalently ns +a ≥k. If 2r is not a multiple of 3, i.e.r ̸= 0mod 3 ,(2.49)14

it follows thatδ −1 −s ̸= 0 mod 3 . (2.50)This means that the subscript 3ns+δ−1−s does not belong to the sequence 3nδ−1−3α+3α.Since the sum of all n’s is k we should havek ≥aX0nδ−1−3α + ns ≥a + 1 + ns(2.51)whereas from the above a + 1 + ns ≥k + 1, a contradiction.

Thus in generalr ̸= 0 mod 3∂Z∂θ2r= 0. (2.52)(iii) The remaining cases are those for which we take a derivative with respect to θ6r.We shall need a relation between the coefficients cn, dn which did not play any specificrole until now.

The series z(x) of (2.21) is a solution of the following differential equationin the variable xx4z′′ + 2(2x3 + 1)z′ + 54x2z = 0 ,(2.53)while z(x) of (2.23) satisfiesz(x) =∞X0dnx3n =1 + x32z + x4z′. (2.54)Lemma 3z(x)z(−x) + z(−x)z(x) = 2(2.55)To see this, substitute z in terms of z and compute the derivative with respect to x,making use of the differential equation for both z(x) and z(−x), to find that it vanishes.In terms of the series expansions this reads2nXs=0(−1)sc2n−s ds = 0n > 0.

(2.56)It was convenient to use generalized characters until now, but we can also recast theexpansion of Z in terms of standard characters indexed by Young tableaux Y (f0 ≥f1 ≥. .

. ≥fδ−1 > 0) at the price of having more complicated coefficients.

From eqs. (2.38) and(2.40) this readsZ =XY, |Y |=0,mod 3ξY chY (θ.

)(2.57)15

withchY (θ.) = det{pfi+j−i(θ.

)}0 ≤i, j ≤δ −1(2.58)andξY = det ξi,j(2.59)ξi,j =(0if fi + j −i ̸= 0 mod 3c(j)13 (fi+j−i)if fi + j −i = 0 mod 3.In the above, i (j) is the row (column) index. The diagonal subscripts in chY are no longermultiples of 3, instead running down the diagonal they form a non-decreasing sequence,the last one being positive (and δ being the number of rows of the corresponding Youngtableau).

The quantity Zk is obtained by restricting the sum to |Y | = 3k.When taking a derivative with respect to θ6r, the first contributing term is Z2r. (Recall that terms in Zk are homogeneous of degree 3k).

Let us therefore investigate first∂Z2r∂θ6r . The only Young tableaux such that |Y | = 6r for which ∂chY /∂θ6r ̸= 0 are of the“Fermi-Bose” type (t + 1)(1)s, s + t + 1 = 6rt + 1z}|{s(2.60)in which case ∂chY /∂θ6r = (−1)s. Applying formula (2.57) we get∂Z2r∂θ6r=22r−1Xs=0(−1)sc2r−sd2r−s+1.

. .c(s)2r1d1.

. .c(s)s.........0. .

.1c(s)1+2r−1Xs=0(−1)sd2r−sc2r−s+1. .

.c(s+1)2r1c1. .

.c(s+1)s.........0. . .1c(s+1)1.

(2.61)16

Expanding each of these determinants along the first column (where c and d with negativeindex are set equal to zero), we find∂Z2r∂θ6r=rXj=0(1 −3j)c2j−1∆2r−(2j−1) + 3jc2j∆2r−2j+rXj=13jd2j∆2r−2j −(3j −2)d2j+1∆2r−2j−1,(2.62)where ∆0 = ∆0 = 1 and∆s =d1c2. .

.c(s)s1c1. .

.c(s)s−1.........0. . .1c(s)1∆s =c1d2.

. .c(s+1)s1d1.

. .c(s+1)s−1.........0. .

.1c(s+1)1. (2.63)Taking into account the identity (2.56) satisfied by the coefficients c and d we readily seeby a recursive argument that ∆and ∆reduce to∆s = ds∆s = cs(2.64)Thus∂Z2r∂θ6r= (3r −1)2rXj=0(−1)jcjd2r−j = 0.

(2.65)(iv) It remains finally to examine∂Z2r+k∂θ6rk > 0Let us look at the expression (2.43) with the required modification k →k+2r, 2r →6r. Asbefore when taking derivatives of the row of index s we need only take into account thoseterms for which s ≥3k, the others vanishing due to the antisymmetry of the characters.This being assumed we consider for fixed s a specific term in the sum (2.43) with characterof depth δ > 3k so that we can omit the rows and columns of label larger than δ −1in the computation of the corresponding determinental character.

The labels in the lastcolumn before derivation are 3n0 + δ −1, . .

.,3nδ−1, no pair of them equal. According to aprevious analysis using lemma 2, to get a non-trivially vanishing derivative, the quantity3ns+δ−1−s (s ≥3k) has to equal 6r .

This means that δ−1−s = 3σ, and nδ−1−3σ = 2r−σ.17

Since by definition nδ−1 > 0, from the preceding reasoning we must have Pσ−1α=0 nδ−1−3α ≥σ, if this sum is non-empty (i.e. σ > 0).

ThenσXα=0nδ−1−3α ≥2r ,(2.66)an inequality which remains obviously true when σ = 0, in which case nδ−1 = 2r. Afortioriδ−1Xρ=3knρ ≥2r ,(2.67)since the latter sum includes the previous one (δ −1 −3σ ≥3k).

Since Pδ−10nρ = 2r + kfrom the homogeneity property of Z2r+k, we have the complementary inequality3k−1X0nρ ≤k. (2.68)Both inequalities must in fact be equalities.

Indeed among the integers of the form δ−1−3α(α ≥0) such that δ −1 −3α ≥k, consider the largest one, say β, for which nδ−1−3α > 0.We have β ≥σ and again appealing to a previous reasoningX0≤ρ<δ−1−3βρ=δ−1 mod 3nρ ≥k ,(2.69)since there are at least k integers equal to δ −1 mod 3 between 0 and 3k −1 < δ −1 −3β.Hence(i) Pσα=0 nδ−1−3α has to equal 2r otherwise we would violate homogeneity;(ii) β has to be equal to σ for the same reason (no other nδ−1−3α except those enteringthe previous sum are > 0);(iii) and finally should nρ0, ρ0 ≥3k, ρ0 ̸= δ−1 mod 3, be positive, then again P0≤ρ≤ρ0ρ=ρ0 mod 3 nρwould be larger than k (since the sum includes ρ0 ≥3k), a contradiction. We concludethat we can replace (2.68) and (2.69) by equalities.

This means that we can write∂Z2r+k∂θ6r=Xs≥3kXn0+···+n3k−1=kn3k+...+n6r+3k=2rc(0)n0 · · · c(6r+3k−1)n6r+3k−1p3n0...p3ns−6r...p3n3k+6r−1(2.70)18

Let us group together all contributions corresponding to a fixed choice of indicesn0, . .

. , n3k−1 with sum equal to k.Taking into account the antisymmetry in the last6r rows we see that the analysis reduces to our previous computation of ∂Z2r/∂θ6r.

To beprecise, the column labelled 3k will correspond to coefficients labelled c or d according tok even or odd so that if k is even∂Z2r+k∂θ6r=Xn0+···+n3k−1=kc(0)n0 · · · c(3k−1)n3k−1p3n0...p3n3k−1∂Z2r∂θ6r. (2.71)If k is odd, we get the same formula with ∂Z2r/∂θ6r replaced by the same expression withc’s and d’s interchanged.

In both cases the expression vanishes since c’s and d’s play asymmetric role in the vanishing of ∂Z2r/∂θ6r as it relies on the identity (2.55) invariant inthe interchange of z and z.We have at last fully proved the first part of Kontsevich’s theorem from a purelyalgebraic standpoint∂Zk∂θ2r= 0(2.72)Even though the proof appears a little long, the steps are completely elementary relying onthe second order differential equation satisfied by z and Weyl antisymmetry of characters.Without repeating in detail each step, it will go through in the generalized case consideredin sec. 7.It may also be worth remarking that by retracing the above discussion, this propertyis very likely to imply the (Airy-like) differential equation.

We now turn to the secondpart of the theorem.3. From Grassmannians to KdVExpert readers will have recognized the connection between the expansion (2.57) of Z andSato’s approach to soliton equations and τ- functions.

The latter relies on a clever rein-terpretation of the familiar Pl¨ucker relations of projective geometry in terms of propertiesof associated (pseudo-)differential operators [14]. In more physical terms, this involves thecharacterization of those submanifolds that correspond to pure Slater determinants (asopposed to their linear combinations) in a many body fermionic space.Let us begin with a short review of the subject following Sato.

Let V be a vector spaceof dimension N equipped with a basis e0, · · ·, eN−1. The field of constants is arbitrary but19

one may think of IR or C. A linear subspace generated by m vectors ξ(0), · · ·, ξ(m−1) isintrinsically described by the antisymmetric multivectorξ(0) ∧· · · ∧ξ(m−1) =X0≤l0≤···≤lm−1≤N−1ξl0,···,lm−1el0 ∧· · · ∧elm−1(3.1)with antisymmetric componentsξl0,···,lm−1 = det ξli,j ,0 ≤i, j ≤m −1 ,ξ(k) = ξi,kei(3.2)All relations to be written being homogeneous we may consider the above quantities asbeing homogeneous components of the corresponding m −1 dimensional linear subspacein the projective space PV (N −1) of dimension N −1. A familiar case is the descriptionof lines (m = 2) ¡in projective three space (N = 4).

An (m −1)-dimensional subspace inPV (N −1) depends on m(N −m) parameters2 (four in the above example, for instancethe intersections of the line with two planes) while the number of coordinates in (3.1)(taking into account homogeneity) isNm−1 (i.e. 5 in the example).

They must thereforesatisfy some (non-linear but homogeneous) relations. These are the Pl¨ucker relations.

Inthe aforementioned example this is the classical quadratic relation expressing that thegeometry of lines in the projective 3–dimensional space is equivalent to the geometry ofpoints on a quadric in 5–dimensional projective space.To derive typical Pl¨ucker relations, we demand that a linear combinationη =Xxk ξ(k)(3.3)lies in the subspace generated by the ξ’s, i.e. that0 = η ∧ξ(0) ∧· · · ξ(m−1) =X0≤l0···lm≤N−1eξl0,···,lm el0 ∧· · · ∧elm(3.4)eξl0,···,lm =mXi=0(−1)im−1Xr=0xr ξ(r)li ξl0,···,bli,···,lm .Choose the coefficients xr as the minors in the last line ofξk0,···,km−2,l ≡ξ(0)k0.

. .ξ(m−1)k0......ξ(0)km−2.

. .ξ(m−1)km−2ξ(0)l. .

.ξ(m−1)l=m−1Xr=0xr ξ(r)l(3.5)2 This is m × N, the number of components of the vectors ξ(k), minus m2, the dimension ofthe linear group GL(m) acting linearly on this vector without modifying the subspace.20

which only depend on the choice of k0, · · ·, km−2 and not on the index l. Hence we get thePl¨ucker relations in the formmXi=0ξk0,···,km−2,li ξl0,···,bli,···,lm = 0. (3.6)The reader might have fun to find the relations among these relations and so on.Inany case by turning the argument around these relations do characterize m–dimensionalvector subspaces in V or the (m −1)–dimensional ones in PV which form the (N, m)Grassmannian (obviously not a vector space but rather an intersection of quadrics).For our purposes we will need a generalization of (3.5) which follows from the obser-vation that we could equally well replace η by some combinationη′ =Xk

Leaving the generalcase aside, we also haveXi

Indeed it wasSato’s idea to associate to points of the Grassmannian a τ–function obtained by replacingexterior products of basis vectors by the corresponding antisymmetric generalized Schurfunctions. In our case one can view vector subspaces as those generated by the functionsz, Dz, D2z, · · · so that Kontsevich’s integral appears as a realization of Sato’s idea.

Thetask is now to translate equivalents of the Pl¨ucker relations in terms of Z.It will be easier to state this in the finite N case we started from, since by letting Nbecome arbitrarily large we will recover the required results term by term in the asymptoticseries. Thus we return to formula (2.40) understanding by pn(θ.) the unconstrained Schurfunctions which we recast in the following formZ(N) =∂N−1fN−1∂θN−11.

. .∂fN−1∂θ1fN−1......∂N−1f0∂θN−11.

. .∂f0∂θ1f0,(3.9)21

wherefN−1(θ.) =Xc(0)n p3n+N−1(θ.)f1(θ.) =Xc(1)n p3n+N−2(θ.)...(3.10)f0(θ.) =Xc(N−1)np3n(θ.)

.In the above we have used eq. (2.34) which is meaningful for Schur functions with in-dependent θ arguments.

The precise meaning of (3.10) for unconstrained θ’s is thereforethat it extracts from the complete formula (2.57) only those terms corresponding to Youngtableaux which have at most N rows. By letting N →∞we reach any desired term.The expression (3.9) singles out the variable θ1, the others playing the role of param-eters.

For the time being we simplify the notations by referring to Z(N) as Z until at theend we restore the correct subscript. Looking at (3.9) we see that it takes the form of aWronskian of the components f0, .

. .

, fN−1 of a vector denoted f. It is therefore naturalto attach it to an N–th order differential operator ∆N such that for an arbitrary functionF of θ1 (d ≡ddθ1 )∆NF =NXr=0wr(θ.) dN−rF = Z−1∂NF∂θN1.

. .F∂NfN−1∂θN1.

. .fN−1......∂Nf0∂θN1.

. .f0(3.11)Strictly speaking the coefficients w should also carry the subscript N. In order to obtaina smooth transcription as N →∞, Sato uses rather than the differential operator ∆N anequivalent pseudodifferential operator W defined as∆N = WN dN ,(3.12)WN =NX0wr d−r(3.13)with w0 = 1 andw1 = Z−1 −∂∂θ1Z.

(3.14)Expanded in power series in the θ’s, w1 will have terms of fixed degree independent ofN for N large enough and the remark applies to the successive coefficients wp (infinitely22

many as N →∞) justifying that we drop eventually all reference to N. Equation (3.14)admits a rather neat generalization aswr = 1Z pr−∂∂θ.Z ,(3.15)where∂∂θ.≡∂∂θ1, 12∂∂θ2, . .

., 1k∂∂θk, . .

. (3.16)(which is meaningful since we consider the θ’s as independent variables).

Indeed let usform the generating functionZ−1 Xr≥0yrpr−∂∂θ.Z = Z−1exp −∞X1yrr∂∂θrZ. (3.17)Acting on any column vector of Z, the operator∂∂θr is equivalent to∂r∂θr1 .

This givesZ−1 Xr≥0yrpr−∂∂θ.Z = Z−1 det(1 −yd)dN−1f, . .

. , (1 −yd)f .

(3.18)Let us compare this with the quantityNX0wryr = Z−1 dety0. .

.yNdNf. .

.f . (3.19)The (N +1)×(N +1) determinant on the right hand side can be computed by subtractingthe first column multiplied by y from the second, the second column multipled by y fromthe third and so on, with the cofactor of the only non-vanishing element in the first rowequal to the previous expression proving eq.

(3.15).One is familiar with the commutation relations involving the operators d−rd−ra =Xk≥0(−1)kr + k −1ka(k)d−r−kad−r =Xr≥0r + k −1kd−r−ka(k)(3.20)where a(k) ≡(dka). These formulae enable one to give a meaning, again droping the indexN, to the “dual” pseudo–differential operatorW ∗=Xd−rw∗rw∗r = Z−1pr ∂∂θ.Z(3.21)23

satisfyingW ∗= W −1(3.22)We relegate the (cumbersome) proof of this latter fact which relies on Pl¨ucker formulae toan appendix of this section.The crux of the matter is the basic equation∂W∂θn= QnW −Wdn(3.23)with Qn a normalized differential operator of order n, Qn = dn + . .

., given byQn =WdnW −1+(3.24)the subscript + refers to the differential part of WdnW −1, as follows from the fact that∂W∂θn is of order d−1.Multiplying it on the right by dN, eq. (3.23) is equivalent to∂∆N∂θn= Qn∆N −∆NdnQn =∆Ndn∆−1N+(3.25)where ∆−1N = d−NW −1 .

Again N is assumed much larger than n fixed, in fact as large aswe want. Among these equations one is trivially verified namely the one for n = 1 whereQ1 = d. To prove (3.25) choose Qn as indicated so that the combination Qn∆N −∆Ndnis a differential operator of order N −1 as it should be if the formula is to make sense.It will then hold if both sides agree when operating on N linearly independent functionswhich (see eq.

(3.11)) we naturally choose as being f0, . .

., fN−1 or equivalently on anylinear combination which we denote by f. Thus ∆Nf = 0 and we want to prove that∂∆N∂θnf + ∆Ndnfdθn1= 0. (3.26)But dnfdθn1 through the definition (3.10) is equal to∂f∂θn reducing the above expression to∂∂θn∆Nf= 0.

Thus eqs. (3.25) and (3.24) hold true and yield in general the so-calledKP hierarchy of compatible integrable systems∂2W∂θn1∂θn2 =∂2W∂θn2∂θn1 for the coefficients ofthe pseudo-differential operator W. An equivalent form is as follows.

SetL = WdW −1 = d + O(d−1)Ln = WdnW −1Qn =Ln+ . (3.27)24

Then from (3.23) we have∂W −1∂θn= −W −1Qn + dnW −1∂L∂θn=[Qn, L] ,(3.28)where in writing these formulae we have implicitly taken the N →∞limit. As a conse-quence of (3.28) we have the zero curvature conditions∂Qm∂θn−∂Qn∂θm+ [Qm, Qn] = 0.

(3.29)According to equations (3.15) and (3.21) the coefficients in W and W −1 do not depend onθ2r. Hence the differential operators Q2r ≡L2r+ commute with L as well as any of itspowers[Q2r, Q2r′] = 0.

(3.30)Using the notations of theorem 1, tr = −(2r + 1)!! θ2r+1, u =∂2∂θ21 ln Z =∂2∂t20 ln Z and(3.15), (3.21) and (3.24)Q2 =d2 + 2uQ3 =d3 + 3ud + 32∂u∂θ1≡Q322+(3.31)Setting m = 2 and n = 3 in (3.29) we conclude that the first non trivial equation in thehierarchy reads∂Q2∂θ3= [Q3, Q2](3.32)i.e.∂u∂t1= ∂∂t0 112∂2u∂t20+ 12u2(3.33)as claimed in the second part of Theorem 1.

Higher equations involve∂∂t2 , · · · and are ofthe form (2.9b),(2.10).In fact, the commutation of L and Q2 implies that L2 = Q2, i.e. that L2 is a differentialoperator.To prove this, one may appeal to a lemma [15] that asserts that the spaceof operators that commute with Q2 is spanned by the powers of the pseudodifferentialoperator square root of Q2, with constant coefficients.

ThusL = Q122 +∞Xl=0αlQ122−l. (3.34)25

Both L and Q122 , however, are functionals of Z with the limit d as Z →1. It follows thatall the constants α vanish andL = Q122 .

(3.35)This implies that the KdV flows (3.28) are generated by theQ2r+1 = (L2r+1)+ = (Qr+ 122)+.Notice that the above identity (3.35) means that L, which a priori depends on all thederivatives of Z, is actually a functional of the sole u = ∂2 ln Z/∂θ21.AppendixIt would seem that Pl¨ucker relations have not entered directly the discussion. One placewhere they play a hidden role is in the computation of the inverse W −1 = W ∗.

Of courseif we need only the first few terms as in (3.31) one can obtain them by a direct calculation.For completeness we give a recursive proof of eq. (3.22).

The integer N being fixed westart with the expressions (3.9)–(3.11) and consider f in (3.10) as a column vector functionof independent θ’s with d ≡∂∂θ1 and f (r) ≡∂f∂θr1 . Dropping the index N∆=WdNW =NX0wrd−r(3.36)wr =Z−1pr−∂∂θ.Z = Z−1(−1)rf(N) .

. .df(N−r) .

. .

fusing a shorthand notation for determinants.The kernel of ∆is the finite di-mensional vector space generated by the components of f.We distinguish a flag(f0), (f0, f1), (f0, f1, f2), · · · and associate to it a sequence of determinantsZ(1) = f0Z(2) =f ′1f1f ′0f0 , · · ·, Z(N) ≡Z(3.37)in terms of which one can write a factorized form (the Miura transformation)∆= Z(N)Z(N−1) dZ(N−1)Z(N)· · ·Z(2)Z(1) dZ(1)Z(2)Z(1)d 1Z(1). (3.38)It is clear that applied to Z(1) = f0, ∆gives 0 while if ∆k is the product of the first kfactors starting from the right and if we assume ∆kf0 = ∆kf1 = .

. .

= ∆kfk−1 = 0, then∆kfk = Z(k+1)Z(k) , hence∆k+1fk =Z(k+1)Z(k)d Z(k)Z(k+1)fk = 026

proving the above factorization. The identity to be established is thereforeW −1 = dN∆−1 = dNZ(1)d−11Z(1)Z(2)Z(1) d−1 Z(1)Z(2)· · · Z(N)Z(N−1) d−1 Z(N−1)Z(N)=Xr≥0d−rw∗r(3.39)with coefficients w∗r given byw∗r = Z−1pr ∂∂θ.Z .

(3.40)Upon taking a generating functionXr≥0yrw∗r = Z−1 exp ∞X1ynn∂∂θnZ = Z−111 −y ∂∂θ1f(N−1) . .

.11 −y ∂∂θ1f(3.41)where upper indices on f label derivatives. We have recognized that acting on each columnof Z the shift operator is equivalent toexp∞X1ynn ∂∂θ1n=11 −y ∂∂θ1.If in the above determinant we subtract from the last column the preceding one multipliedby y and so on, we getw∗r = Z−1|f(N−1+r), f(N−2) · · · f|.

(3.42)Comparing (3.36) and (3.39), we see that the determinental numerators have a naturalpictorial description in terms of Young tableaux. The first determinant is a vertical Youngtableau, the second a horizontal one.

This parallels the correspondence between pr(−θ.) =(−1)rsr(θ.) (recall (2.30)) and pr(θ), and the formula to be established is similar to theidentity det(1 −X) det(1 −X)−1 = 1.In any case with this expression for w∗r we return to (3.39) and note that (3.42) saysthat w∗0 = 1 in agreement with (3.39) for every N whereas, should N = 1, W −1 reduces todf0d−1f −10=Xr≥0d−r f (r)0f0(3.43)again in agreement with (3.42).

We therefore assume that (3.42) holds for any r if N ′ < Nand for r′ ≤r when the size of determinants is N, and establish it for w∗N,r+1 reinstating27

the index N. We haveW −1N= dW −1N−1Z(N)Z(N−1) d−1 Z(N−1)Z(N)=Xk≥0d1−kw∗N−1,kZ(N)Z(N−1) d−1 Z(N−1)Z(N)(3.44)=Xr≥0d−r Xk+l=rw∗N−1,kZ(N)Z(N−1)(l) Z(N−1)Z(N) .Writew∗N,r = vN,rZ(N) . (3.45)We havevN,r = Z(N−1) Xk+l=rvN−1,kZ(N)Z(N−1) 2(l).

(3.46)We want to show thatvN,ρ = |f(N−1+ρ), f(N−2) . .

.f|(3.47)assuming it to be true for N ′ < N (where we only keep the components f0, . .

., fN′−1) andalso for ρ ≤r to prove that it holds for r + 1. Take a derivative of the above identityv′N,r = Z(N−1)′Z(N−1) vN,r + Z(N−1) Xk+l=rvN−1,kZ(N)Z(N−1) 2(l+1).

(3.48)The last sum differs from vN,r+1 by the missing term vN−1,r+1Z(N)Z(N−1) . HencevN,r+1 = v′N,r + vN−1,r+1Z(N)Z(N−1)Z(N−1)′Z(N−1) vN,r.

(3.49)Denote the column vectorf0, · · ·, fN−2T by ϕ (of dimension N −1). According to therecursive hypothesis this readsvN,r+1 = |f(N+r), f(N−2), · · ·, f| +γZ(N−1)(3.50)withγ =dϕ(N−1+r,dϕ(N−1), ϕ(N−2) .

. .

, ϕf (N−1+r), f(N−1),df(N−2), · · ·, f−dϕ(N−1+r, ϕ(N−1),dϕ(N−2) . .

. , ϕf (N−1+r),df(N−1), f(N−2), · · ·, f+ϕ(N−1+r,dϕ(N−1),dϕ(N−2) .

. .

, ϕdf (N−1+r), f(N−1), f(N−2), · · ·, f(3.51)28

where in each term the first determinant is (N −1) × (N −1) dimensional, the secondN × N. We have to show that the combination γ vanishes since we wish to prove thatvN,r+1 is the first term of the r.h.s. of (3.50).

One easily checks that γ = 0 if N = 2, sowe henceforth assume N > 2. To reduce the vanishing of γ to one of Pl¨ucker ’s identities,expand the N × N determinants involving f and its derivatives according to its first line.For each term of the form f (k)N−1 the coefficient is a combination of ϕ–determinants whichvanishes by virtue of the Pl¨ucker relations (3.8), completing the proof of formulas (3.21)and (3.22).4.

Matrix Airy equation and Virasoro highest weight conditions.The differential equation (2.14) generalizes to the N–dimensional case as follows. CallY (Λ) the integral appearing in (2.16) for finite NY (Λ) =ZdM exp itrM 36+ MΛ22.

(4.1)The function Y satisfies for each index k0 =ZdMddMkkexp i6(3trMΛ2 + M 3)(4.2)i.e.0 =D Xl, l̸=kMklMlk + M 2kk + λ2kE(4.3)where ⟨.⟩denotes an integral taken with respect to the weight dM exp itrM36 + MΛ22.The insertion of a diagonal factor Mkk can be achieved by acting with the derivativeoperator −i 1λk∂∂λk on Y . To deal with non-diagonal insertions we express the invarianceof the integral Y under an infinitesimal change of variable of the formM →M + iǫ[X, M],withXab = δakδblMkl.

(4.4)The Jacobian is 1 + iǫ(Mll −Mkk), while the term trM 3 is invariant. Thus0 =DMll −Mkk + i2(λ2k −λ2l )MklMlkE(4.5)29

with no summation implied. Inserting this into (4.3) leads to30 = λ2k + ⟨M 2kk⟩−2iXl,l̸=k⟨Mkk −Mll⟩λ2k −λ2l.

(4.6)This yields the matrix Airy equationshλ2k − 1λk∂∂λk2−2Xl,l̸=k1λ2k −λ2l 1λk∂∂λk−1λl∂∂λl iY = 0(4.7)which can be turned into equivalent equations for Z itself1λ2k Xl1λl!2+14λ4k+ 2Xl,l̸=k1λ2k −λ2l 1λk∂∂λk−1λl∂∂λl−2 1 + 1λ2kXl1λk + λl!∂∂λk+ 1λk∂∂λk2#Z = 0 . (4.8)In the limit N →∞we know from sec.

3 that Z admits an expansion in terms of oddtracestn = −(2n −1)!!Xl1λ2n+1l. (4.9)The differential equations (4.7) can be expanded in inverse powers of λk in the form2Xm≥−11(λ2k)m+2 LmZ = 0.

(4.10)ExplicitlyL−1 =12t20 +Xk≥0tk+1∂∂tk−∂∂t03For an alternative derivation one can transform the “equation of motion” (4.3) into a matrixdifferential equation, assuming at first the argument T ≡Λ2 to be an arbitrary (i.e. not necessarilydiagonal) Hermitian matrix.

Recognizing that the integral is invariant under conjugation of T,hence only a function of its eigenvalues {ta} one then uses∂∂Tkl =Xa∂ta∂Tkl∂∂ta =XaMinkl(ta −T)P ′(ta)∂∂tawhere P(x) = det(x −T) and Minkl denotes the (k, l) minor in the corresponding matrix.30

L0 =18 +Xk≥0(2k + 1)tk∂∂tk−3 ∂∂t1(4.11)L1 =Xk≥1(2k + 1)(2k −1)tk−1∂∂tk+ 12∂2∂t20−15 ∂∂t2L2 =Xk≥2(2k + 1)(2k −1)(2k −3)tk−2∂∂tk+ 3∂2∂t0∂t1−105 ∂∂t3. .

. .

. .. .

.Lm =Xk≥m(2k + 1)!! (2(k −m) −1)!

!tk−m∂∂tk+ 12Xk+l=m−1(2k + 1)!! (2l + 1)!

!∂2∂tk∂tl−(2m + 3)! !∂∂tm+1+ t202 δm+1,0 + 18δm,0In eq.

(4.10), each coefficient has to vanish so thatTheorem 2 (Kontsevich)Z satisfies and is determined by the highest weight conditionsLmZ = 0m ≥−1. (4.12)The operators Lm obey (part of) the Virasoro (or rather the Witt) algebra, namely[Lm, Ln] = (m −n)Lm+nm, n ≥−1(4.13)generated by L−1, L0, L1, L2.

Note that only the first two involve first order derivatives.5. Genus expansion.As in standard matrix models there exists a genus expansion for Fln Z = F =Xg≥0Fg(5.1)One way to obtain it is from the Airy system.

One inserts appropriate extra factors ofN and studies the large N limit, paying attention to corrections, according to a methodapplied to leading order by Kazakov and Kostov [6] and revived by Makeenko and Semenoff[5]. In the spirit of our paper we follow a slightly different approach based on the KdV31

equations and Virasoro constraints. In genus g, Fg collects in the expansion (2.1) all termssuch thatFg(t.) =XkiΣi≥0(i−1)ki=3g−3*Yi≥0(τiti)kiki!+(5.2)We setug(t.) = ∂2Fg∂t20.

(5.3)In the KdV hierarchy (eq. (2.9)) the leading term in the semi-classical or genus expansioncorresponding to the term u0 is obtained from power counting by ignoring in the differentialpolynomial Rn all terms involving derivatives.

It then reduces ton ≥0∂u0∂tn= ∂∂t0un+10(n + 1)! .

(5.4)For n = 0 this is vacuous and has to be supplemented by the first Virasoro condition (4.10)(for m = −1) which amounts to∂u0∂t0= 1 +Xk≥0tk+1∂u0∂tk. (5.5)Inserting (5.4) into (5.5) gives∂∂t0u0 −Xn≥0tnun0n!= 0 .

(5.6)DefineIk(u0, t.) =Xp≥0tk+pup0p! .

(5.7)Equation (5.6) suggestsLemma 4u0(t.) satisfies the implicit equationu0 −I0(u0, t.) = 0. (5.8)To check this, multiply (5.5) by un0n!

and use (5.4) to obtain∂u0∂tn= un0n! +Xk≥0tk+1∂∂tkun+10(n + 1)!

(5.9)32

while∂I0(u0, t.)∂tn= un0n! +Xk≥0tk+1∂∂tnuk+10(k + 1)!

. (5.10)Subtracting and using (5.4) again, we find∂∂tnhu0 −I0(u0, t.)i=Xk≥0tk+1un0n!∂∂tku0 −uk0k!∂∂tnu0(5.11)=Xk≥0tk+1 un0n!∂∂t0uk+10(k + 1)!

−uk0k!∂∂t0un+10(n + 1)! != 0 .The difference u0 −I0(u0, t.) is thus a constant.

Since it vanishes at t. = 0 it is identicallyzero, completing the proof. To find F0 we have to integrate twice eq.

(5.8) with respect tot0. From (5.2) the boundary conditions are given by the vanishing of F0 and∂∂t0 F0 whent0 = 0.

This yields with u0(t.) implicitly given by (5.8)F0 = u306 −Xk≥0uk+20k + 2tkk! + 12Xk≥0uk+10k + 1Xa+b=ktaa!tbb!

. (5.12)Remarks(i) As is generally the case if we extend F0 by considering u0 (originally equal to ∂2F0/∂t20)as an auxiliary independent parameter, we find that the stationarity condition∂F0∂u0= 12 (u0 −I0(u0, t.))2 = 0(5.13)yields equation (5.8).

(ii) The expression for F0 is equivalent to the one given by Makeenko and Semenoff[5]using (infinitely many) eigenvalues λkF0 =13Xkλ3k −13Xk(λk −2s)32 −sXk(λ2k −2s)12+ s36 −12Xk,llnpλ2k −2s +pλ2l −2sλk + λl(5.14)with the condition∂F0∂s = 12 s +Xk1pλ2k −2s!= 0(5.15)upon identification of s with u0, of Ip with −(2p −1)!! Pk1(λ2k−2s)p+ 12 and tn is as in (4.9).33

(iii) From (5.12) or (5.8) we can readily find the first few terms in the expansion of F0which up to a factor t30 only involves the combinations tk−10tk.F0 = t303! + t1t303!

+t2t404! + 2t212!t303!+t3t505!

+ 3t1t2t404! + 6t303!t313!+t4t606!

+6t222! + 4t1t3 t505!

+ 24t303!t414! + 12t2t212!t404!+t5t707!

+ (5t1t4 + 10t2t3) t606! + 120t303!t515!

+30t1t222! + 20t3t212!

t505! + 60t2t313!t404!+t6t808!

+20t232! + 6t1t5 + 15t2t4 t707!

+ 720t303!t616! +90t323!

+ 30t4t212! + 60t1t2t3 t606!+120t3t313!

+ 180t212!t222! t505!

+ 360t2t404!t414!+ . .

. (5.16)In genus zero all coefficients are positive integers (as opposed to fractional) due to thesmoother structure of M0,n, (n ≥3).

Indeed we have the obviousLemma 5The class of formal power series in t0, t1, . .

. which vanish at t. = 0 with non-negativeintegral derivatives at the origin is stable under(i) addition(ii) product(iii) compositionTo apply this to u0 (and hence to F0) we remark that the sequencef0 = t0,fn =∞X0f kn−ktkk!

(5.17)has each of its derivatives at the origin which stabilizes to the corresponding one of u0after finitely many steps.To obtain the next terms we split the Virasoro constraints expressed on ln Z as followsm = −1t202 δg,o +Xk≥0tk+1∂Fg∂tk−∂Fg∂t0= 0m = 018δg,1 +Xk≥0(2k + 1)tk∂Fg∂tk−3∂Fg∂t1= 0(5.18)m = 134

Xk≥1(2k + 1)(2k −1)tk−1∂Fg∂tk−15∂Fg∂t2+ 12∂2Fg−1∂t20+ 12Xg1+g2=g∂Fg1∂t0∂Fg2∂t0= 0m = 2Xk≥2(2k + 1)(2k −1)(2k −3)tk−2∂Fg∂tk−105∂Fg∂t3+ 3∂2Fg−1∂t0∂t1+ 3Xg1+g2=g∂Fg1∂t0∂Fg2∂t1= 0 ,while a similar splitting of the KdV equation (2.9) yields∂ug∂t1= ∂∂t0 112∂2ug−1∂t20+ 12Xg1+g2=gug1ug2!. (5.19)For genus one this equation reads∂∂t0 ∂∂t1−u0∂∂t0 ∂F1∂t0−112∂2u0∂t20= 0.

(5.20)We have∂u0∂t0=11 −I1,∂u0∂t1=u01 −I1p ≥1∂Ip∂t0= Ip+11 −I1, ∂∂t1−u0∂∂t0Ip = δp,1hence we can rewrite112∂2u0∂t20= 112I2(1 −I1)3 = 124∂∂I1I2(1 −I1)2= ∂∂I1∂∂t0 124 ln11 −I1. (5.22)If F1 is a function of t. only through I1, I2, .

. ., we can rewrite ∂∂t1−u0∂∂t0 ∂∂t0F1 =∂∂I1∂∂t0F1(5.23)so that equation (5.20) becomes∂∂t0∂∂I1∂∂t0F1 −124 ln11 −I1= 0.

(5.24)This suggests thatF1 = 124 ln11 −I1,(5.25)35

in agreement with the above hypothesis so that eq. (5.19) is satisfied.

A straightforwardcomputation shows that the Virasoro conditions are satisfied, proving (5.25).RemarkIt is not unexpected that the genus one (or “one-loop”) result involves as usual a logarithm.Expanding F124F1 = t1 +t212! + t0t2+2t313!

+ t3t202! + 2t0t1t2+6t414!

+ t4t303! + 4t202!t222!

+ 6t0t2t212! + 3t1t3t202!+24t515!

+ t5t404! + 24t0t2t313!

+ (4t1t4 + 7t2t3) t303! + 16t1t202!t222!

+ 12t3t202!t212!+120t616! + t6t505!

+ 120t0t2t414! +14t232!

+ 5t1t5 + 11t2t4 t404! + 48t303!t323!

+ 60t3t202!t313!+20t4t212! + 35t1t2t3 t303!

+ 80t202!t212!t222!+720t717! + t7t606!

+ 720t0t2t515! + (6t1t6 + 16t2t5 + 25t3t4) t505!

+ 360t3t202!t414!+84t1t232! + 118t3t222!

+ 30t5t212! + 66t1t2t4 t404!+288t1t303!t323!

+120t4t303! + 480t202!t222!

t313! + 210t2t3t212!t303!+ .

. .

(5.26)All intersection numbers are of the form124× (a positive integer) since I1 and −ln(1 −I1)belong to the class of functions referred to in Lemma 5.For higher genus the AnsatzFg =XP2≤k≤3g−2(k−1)lk=3g−3⟨τ l22 τ l33 . .

. τ l3g−23g−2 ⟩1(1 −I1)2(g−1)+PlpIl22l2!Il33l3!

. .

. Il3g−23g−2l3g−2!, (5.27)which is a finite sum of monomials in Ik/(1 −I1)2k+13 , the number of which is p(3g −3)(with p(n) the number of partitions of n), is consistent with the KdV equation (5.19).Inserted into the latter, it allows one to compute the coefficients with the resultF2 =157605I4(1 −I1)3 + 29I3I2(1 −I1)4 + 28I32(1 −I1)5.

(5.28)Hence⟨τ4⟩=11152⟨τ2τ3⟩=295760⟨τ 32 ⟩=7240(5.29)36

in agreement with Witten [1]. The other intersection numbers can be derived by expanding(5.28) in t..For genus 3 we findF3 =1290304035I7(1 −I1)5 + 539I6I2(1 −I1)6 + 1006I5I3(1 −I1)6 + 4284I5I22(1 −I1)7+607I24(1 −I1)6 + 13452 I4I3I2(1 −I1)7 + 22260I4I32(1 −I1)8 + 2915I33(1 −I1)7+43050I23I22(1 −I1)8 + 81060I3I42(1 −I1)9 + 34300I62(1 −I1)10,(5.30)which yields the table⟨τ7⟩=182944⟨τ4τ3τ2⟩=1121241920⟨τ6τ2⟩=77414720⟨τ4τ 32 ⟩=531152⟨τ5τ3⟩=5031451520⟨τ 33 ⟩=58396768⟨τ5τ 22 ⟩=175760⟨τ 23 τ 22 ⟩=2053456⟨τ 24 ⟩=6071451520⟨τ3τ 42 ⟩= 193288⟨τ 62 ⟩= 1225144Table IThere is no difficulty to pursue these computations as far as one wishes.Remarks(i) It follows from Lemma 5 that all series coefficients in (5.27) reexpressed in terms ofQ tlkk /lk!

are non-negative integers up to the finitely many prefactors.All intersectionnumbers of a fixed genus when written as irreducible fractions have therefore a lowestcommon multiplier (l.c.m. )Dg: D0 = 1, D1 = 24, and for g > 1, Dg is the lowestcommon denominator of the finitely many intersection numbers appearing in (5.27), whenwritten as irreducible fractions.We conjecture that for 1 < g′ ≤g, the order of anyautomorphism group of an algebraic curve of genus g′ (bounded by 84(g −1)) divides Dg.Thus D3 = 2903040 is divible by 168 (the order of the largest automorphism group of agenus 3 curve) and by 48 (the same for genus 2).

(ii) The term in Fg, g > 1 which has the highest power of (1 −I1) in the denominator hasthe form⟨τ 3g−32⟩(3g −3)!I3g−32(1 −I1)5g−537

In the next section we develop a formalism to resum these terms as well as subleadingones akin to the “double scaling limit” of standard matrix models [10] (one recognizes thesame string exponents and the same ingredients). At the other extreme the term with thelowest power of 1 −I1 in the denominator is [1]⟨τ3g−2⟩I3g−2(1 −I1)2g−1⟨τ3g−2⟩=1(24)gg!

. (5.31)The last equality (also valid for g = 1) follows from (5.19) by keeping terms with the lowestpower of (1 −I1)−1.

It implies that (24)gg! divides Dg.6.

Singular behaviour and Painlev´e equation.The expression (2.18) exhibits a singular behaviour as I1 →0. In a first step, we can keepin the KdV equation the dominant terms by considering thatI2 ≈constant,Ik ≈0 for k ≥3.

(6.1)This is consistent with the derivatives of the I’s: ∂Ik/∂t0 = Ik+1/(1 −I1) and ∂Ik/∂t1 =u0Ik+1/(1 −I1) for k ≥2. In this approximation, the genus g contribution to the specificheat ug = ∂2Fg/∂t20 is of the formug = ∂2Fg/∂t20 = αgI3(g−1)+22(1 −I1)5(g−1)+4 .

(6.2)We introduce the scaling variablez = (1 −I1)I3/52(6.3)and separate the genus zero contribution by setting u = u0 + eu, eu = Pg≥1 ug. The KdVequation (2.9a) then reads∂eu∂t1=eu11 −I1+ ∂eu∂t0+ 14I22(1 −I1)5 + 112∂3eu∂t30=eu11 −I1+ ∂eu∂I1+ 14I22(1 −I1)5 + 112I21 −I1∂eu∂I13 .

(6.4)A rescalingeu = I−2/52ψ(z)(6.5)38

leads to the equation∂ψ∂z + ψz1 −∂ψ∂z+14z5 −1121z∂∂z3ψ = 0. (6.6)This equation will be generalized below.

In this particular case of the behaviour (6.1), onecan transform it into the Painlev´e equation: we set t = 2−2/5z2, ψ(z) = z + 21/5φ(t) andfind13φ′′ + φ2 −t = 0 . (6.7)which has the asymptotic expansionφ =Xφgt52 (g−1)+2,φ0 = −1 ,(6.8)where the successive terms satisfyφg+1 = 25g2 −124φg + 12gXm=1φg+1−mφm .

(6.9)Hence in this regimeXg≥2F singg=Xg≥2⟨τ 3g−32⟩(3g −3)!I3(g−1)2(1 −I1)5(g−1)⟨τ 3g−32⟩=2g(3g −3)! (5g −5)(5g −3)φg(6.10)g = 2⟨τ 32 ⟩=7240,g = 3⟨τ 62 ⟩= 1225144 ,g = 4⟨τ 92 ⟩= 181687148, .

. .

. (6.11)This discussion may be extended to the regime in which all (or a finite number of)the I’s are retained and tend to zero according to the following scaling lawz = (1 −I1)I3/52vq = Iq(1 −I1)q−2Iq−12q ≥3(6.12)F singg= z−5(g−1)XΣ2≤k≤3g−2(k−1)lk=3g−3⟨τ l22 τ l33 .

. .

τ l3g−23g−2 ⟩3g−2Yq=3vlqqlq!39

The KdV equation is then rephrased ashz ∂∂z +Xq≥3(q −2)vq∂∂vq+ 1iψ =(6.13)= 1z ψ∆+ 25v3ψ −3 + v312z4 +112z5 [∆−4 −85v3][∆−2 −35v3][∆+ 25v3]ψwhere the same change of function as in (6.5) has been carried out, and ∆denotes thedifferential operator∆= (1 + 35v3)z ∂∂z +Xq≥3−vq+1 + (q −2)vq + (q −1)vqv3 ∂∂vq. (6.14)The contribution to a given genus g involves only a finite number of terms in the sum:q ≤3g −2.

Moreover the differential operators in (6.13) respect the grading in powers ofz,ψg(z, v.) = z−5(g−1)−4ψg(v.) ,thus determining the polynomials ψg(v.) recursively. For instance, if only z and v ≡v3are retained, we have for g > 1 (and ∂v ≡∂/∂v)[5(g −1) + 3 −v∂v] ψg(v) =g−1Xg′=1ψg−g′(v) [4 + 2v + (g′ −1)(5 + 3v) −(1 + 2v)v∂v] ψg′(v)+ 112 [8 + 4v + (g −2)(5 + 3v) −(1 + 2v)v∂v] ×(6.15)[6 + 3v + (g −2)(5 + 3v)−(1 + 2v)v∂v] [4 + 2v + (g −2)(5 + 3v) −(1 + 2v)v∂v] ψg−1(v) ,which yields24 ψ1(v) =2 + v1152 ψ2(v) =196 + 352v + 109v2(6.16)82944 ψ3(v) =117600 + 362564v + 324660v2 + 84699v3 + 3043v43981312 ψ4(v) =1906157232 + 7865959024v + 11212604992v2 + 6581090736v3+ 1465796801v4 + 83580341v5.40

From this one extracts the first intersection numbers of the form ⟨τ l22 τ l33 ⟩. One recoversfor g ≤3 results obtained in (5.29) or in Table I, and in genus g = 4 the results⟨τ 92 ⟩=181687148⟨τ 72 τ3⟩=33262671728⟨τ 52 τ 23 ⟩=7284656912⟨τ 32 τ 33 ⟩=432016912⟨τ 72 τ 43 ⟩=134233331776 .Table IIWe hope we have amply demonstrated the practical use of these expansions.7.

Generalization to higher degree potentialsThe cubic potential in the integral (0.1) may be generalized to a potential of degree p + 1,as noticed by several authors [7],[2],[8]. The case p = 2 is the one discussed previously.Let us consider first the one-variable integral analogous to (2.12), also denoted z(λ).

Thenormalizations are adjusted in such a way as to make the quadratic term positive definite,in order to have a well-defined asymptotic expansionz(λ) =R ∞−∞dm eip2+12(p+1)(m+(−i)p+1λ))p+1>linR ∞−∞dm e−p4 m2λp−1,(7.1)where the subscript “> lin” denotes the sum of terms of degree ≥2 in the polynomial. Byconsiderations similar to those of sect.

2.1, it is easy to see that z(λ) admits an asymptoticexpansion in inverse powers of λp+1z(λ) =∞X0ckλ−(p+1)k(7.2)with c0 = 1. It satisfies a differential equation of order p + 1(Dp −λp)z(λ) = 0(7.3)41

D =λp−12 e−p2(p+1) (−λ)p+1 2(−1)p+1 ∂∂λpλ−p−12 ep2(p+1) (−λ)p+1=λ + (−1)ppp −1λp−2λp−1∂∂λ. (7.4)The corresponding matrix integralZ(N)(Λ) =RdM eip2+12(p+1) tr(M+(−i)p+1Λ))p+1>linRdMe−14 trPp−1k=0 MΛkMΛp−1−k(7.5)may then be handled as in equations (2.17)–(2.24).

One considers the set of functions z(j)defined byz(j)(λ) = λ−jDjz(λ),j ≥0 . (7.6)From eq.

(7.3), it follows thatDr(p−1)+jz = λr(p−1)+jz(j)mod (D0z, . .

. , Dr(p−1)+j−1z).Thus the analogue of (2.24) readsZ(N)(Λ) = |λ0z(0), .

. ., λp−1z(p−1), λpz(0), .

. .||λ0, λ1, λ2, · · ·, λN−1|.

(7.7)The p functions z = z(0), . .

., z(p−1) have asymptotic expansionsz(j)(λ) =∞X0c(j)k λ−k(p+1) ,(7.8)with coefficients c(j)k ; in the sequel the latter are regarded as periodic in j of period p:c(j+p)k= c(j)k . One then proceeds as in sec.

2.2, introducing Schur functions, with theresult (analogous to (2.41)) thatZ(N)k(Λ) =Xn0+···+nN−1=kc(0)n0 c(1)n1 · · · c(N−1)nN−1p(p+1)n0...p(p+1)nN−1(7.9)is independent of N for N ≥(p + 1)k.The steps of sec. 2.2 may then be followed sequentially to prove that Zk, computednow for N = (p + 1)k, is independent of the θrp = trΛ−rp.

When differentiating Zk withrespect to θrp, the only non-trivially non vanishing terms are those for which the derivative42

acts on one of the last rp lines, where r is a multiple of p + 1. The discussion of such acase then appeals to the following identity generalizing (2.56)dethωijz(j)(ωiλ)i0≤i,j≤p−1 = constant(7.10)where ω is a p-th root of unity.

This is proved by differentiating the determinant, usingthe relations (7.6) between the functions z(j). This implies a family of identities on thecoefficientsCkp ≡XPni=kpni=i+π(i) mod pǫπ c(0)n0 .

. .c(p−1)np−1 = 0(7.11)where the summation runs over the configurations of indices ni that may be written asindicated, with π a permutation of the p integers 0, .

. ., p −1.On the other hand, as before, the only contributions to∂Z∂θrp(p+1) come from Youngtableaux with a square-rule shape (as in eq.

(2.60)), and one finds that∂Z∂θrp(p+1)=rp−1Xl=0(−1)lrp−1−lXi=02c(i)rp−l∆(i+j+1)l+p−1Xj=1c(i+j)rp−l ∆(i+j+1)l,(7.12)where the ∆’s are determinants generalizing those of (2.63):∆(j)s=c(j)1c(j+1)2. . .c(j+s−1)s1c(j+1)1. .

.c(j+s−1)s−1.........0. . .1c(j+s−1)1.

(7.13)The expression (7.12) may be recast in the form∂Z∂θrp(p+1)=rXs=1p−1Xi=0(−1)(s−1)p+ih p−1−iXj=0(r−s+1)(p+1)−i+p−1Xj=p−iic(j)(r−s+1)p−i∆(j+1)i+(s−1)p . (7.14)It appears that the combination of c and ∆in the summand in (7.14), namely c(j)sp−i∆(j+1)i,is, up to a sign, the coefficient of c(j)sp−i in the constraint Csp of eq.

(7.11)c(j)sp−i∆(j+1)i= (−1)(p−1)(p−2)2+iX′ǫπc(0)n0 . .

. c(p−1)np−1(7.15)with the sum P′ subject to the same constraints as in (7.11) and to nj = sp −i.

Usingthis fact and after some reshuffling, one finds that ∂Z/∂θrp(p+1) is proportional to theconstraint Crp and thus vanishes,∂Z∂θrp(p+1)= (−1)(p−1)(p−2)2r(p + 1)Crp = 0. (7.16)43

The last part of the argument is carried out as in the end of sec. 2.2, thus completing theproof of the independence of Z with respect to the θrp.One then proceeds as in sec.

3, deriving the higher KdV hierarchies associated with adifferential operator Qp of order p depending on p −1 functions, and as in sec. 4, writingthe generalized Airy equation satisfied by Z.

For example, in the case p = 3, we haven−18tj + D3j +Xk,k̸=jDj1tj −tk(Dj −Dk) +1tj −tk(D2j −D2k)+Xk,lj̸=k̸=l̸=k1(tj −tk)(tj −tl)Dj + circ. oZ = 0(7.17)with the following notationstj = λ3j∂j = ∂∂tjDj = e−38 Σkλ4k Yk,lλ2k + λkλl + λ2l 12 ∂je38 Σkλ4k Yk,lλ2k + λkλl + λ2l−12= ∂j + aj(7.18)aj =12λj −13λ2jXk2λj + λkλ2j + λkλj + λ2kFrom this system of equations, the strong reader will be able to extract the expression ofthe generators of the W3 algebra, in a way similar to (4.10), and to calculate the analoguesof the genus expansion and of the singular behavior discussed in sec.

5 and 6 . .

.Acknowledgements.It is a pleasure to acknowledge some inspiring correspondence with M. Kontsevich and tothank M. Bauer for his assistance in algebraic calculations as well as in the elaboration ofLemma 5 and P. Ginsparg for a critical reading of the manuscript.44

References[1]E. Witten, Two dimensional gravity and intersection theory on moduli space, Surv. inDiff.

Geom. 1 (1991) 243-310.[2]M.

Kontsevich, Intersection theory on the moduli space of curves and the matrix Airyfunction, Bonn preprint MPI/91-77.[3]E. Witten, On the Kontsevich model and other models of two dimensional gravity,preprint IASSNS-HEP-91/24[4]E. Witten, The N matrix model and gauged WZW models, preprint IASSNS-HEP-91/26, to appear in Nucl.

Phys. B.[5]Yu.

Makeenko and G. Semenoff, Properties of hermitean matrix model in an externalfield, Mod. Phys.

Lett. A6 (1991) 3455-3466.[6]V.

Kazakov and I. Kostov, unpublished ;I. Kostov, Random surfaces, solvable lattice models and discrete quantum gravity intwo dimensions, Nucl. Phys.

B (Proc. Suppl.) 10A (1989) 295-322.[7]M.

Adler and P. van Moerbeke, The Wp–gravity version of the Witten–Kontsevichmodel, Brandeis preprint, September 1991[8]E. Witten, Algebraic geometry associated with matrix models of two dimensional grav-ity, preprint IASSNS-HEP-91/74.[9]V. Kazakov, The appearance of matter fields from quantum fluctuations of 2D-gravity,Mod.

Phys. Lett.

4A (1989) 2125-2139.[10]E. Br´ezin and V. Kazakov, Exactly solvable field theories of closed strings, Phys.

Lett.236 (1990) 144-150;M. Douglas and S. Shenker, Strings in less than one dimension, Nucl. Phys.

B335(1990) 635-654;D.J. Gross and A. Migdal, Non perturbative two–dimensional quantum gravity, Phys.Rev.

Lett. 64 (1990) 127-130.[11]C.

Itzykson and J.-B. Zuber, The planar approximation II, J.

Math. Phys.

21 (1980)411-421. [12]Harish-Chandra, Differential operators on a semisimple Lie algebra, Amer.

J. Math.79 (1957) 87-120.[13]F.D. Murnaghan, The Theory of Group Representations, The Johns Hopkins Press,Baltimore, 1938 ;H. Weyl, The Classical Groups, Princeton Univ.

Press, Princeton, N.J., 1946.[14]M. Sato, Soliton equations as dynamical systems on an infinite dimensional Grass-mann manifold, RIMS Kokyuroku, 439 (1981) 30-46 ;The KP hierarchy and infinite-dimensional Grassmann manifolds,Proc.

Symp. PureMath.

49 (1989) 51-66;45

M. Sato and Y. Sato, Soliton equations as dynamical systems on an infinite dimen-sional Grassmann manifold, in Non linear PDE in Applied Science, US-Japan Semi-nar, Tokyo 1982, Lecture Notes in Num. Appl.

Anal. 5 (1982) 259-271.[15]V.G.

Drinfeld and V.V. Sokolov, Lie algebras and equations of the Korteweg–de Vriestype , Journ.

Sov. Math.

30 (1985) 1975-2036.46

---

출처: arXiv:9201.001 • 원문 보기