Latex, 9 pagesWe show that Mirzakhani's recursions for the volumes of moduli space of Riemann surfaces are a special case of random matrix loop equations, and therefore we confirm again that Kontsevitch's integral is a generating function for those volumes. As an application, we propose a formula for the Weil-Petersson volume Vol(M_{g,0})

Eynard, Bertrand

Orantin, Nicolas

arXiv

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Weil-Petersson volume of moduli spaces, Mirzakhani’srecursion and matrix modelsBertrand Eynard, Nicolas OrantinTo cite this version:Bertrand Eynard, Nicolas Orantin. Weil-Petersson volume of moduli spaces, Mirzakhani’srecursion and matrix models. SPhT-T07/065. Latex, 9 pages. 2007. <hal-00149179>HAL Id: hal-00149179https://hal.archives-ouvertes.fr/hal-00149179Submitted on 24 May 2007HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.hal-00149179, version 1 - 24 May 2007SPT-07/065Weil-Petersson volume of moduli spaces, Mirzakhani’srecursion and matrix modelsB. Eynard 1, N. Orantin 2Service de Physique The´orique de Saclay,F-91191 Gif-sur-Yvette Cedex, France.AbstractWe prove that Mirzakhani’s recursions for the volumes of moduli space of Riemannsurfaces are a special case of random matrix recursion relations, and therefore weconfirm again that Kontsevich’s integral is a generating function for those volumes.As an application, we propose a formula for the Weil-Petersson volume Vol(Mg,0).1 IntroductionLetVg,n(L1, . . . , Ln) = Vol(Mg,n)=∑d0+...+dn=3g−3+n(n∏i=01di!) 〈κd01 τd1 . . . τdn〉g,nL2d11 . . . L2dnn(1− 1)denote the volume of the moduli space of curves of genus g, with n geodesic boundariesof lengths L1, . . . , Ln, measured with the Weil-Petersson metrics. Using Teichmullerpants decomposition and hyperbolic geometry, M. Mirzakhani [4] has found a recursionrelation among the Vg,n’s, which allows to compute all of them in a recursive manner. Itwas then observed [5] that this recursion relation is equivalent to Virasoro constraints.1E-mail: bertrand.eynard@cea.fr2E-mail: orantin@cea.fr1In fact, Mirzakhani’s recursion relation takes a form [3] which is amazingly similarto the recursion relation obeyed by matrix models correlation functions ([1, 2]) andwhich were indeed initialy derived from loop equations [1], i.e. Virasoro constraints.Here we make this observation more precise, and we prove that after Laplace trans-form, Mirzakhani’s recursion is identical to the recursion of [2] for the Kontsevichintegral with times (Kontsevich’s integral depends only on odd times):Z(tk) =∫dM e−N Tr [M33+ΛM2] , t2k+3 =1NTrΛ−(2k+3) =(2pi)2k (−1)k(2k + 1)!+2δk,0.(1-2)2 Laplace transformDefine the Laplace transforms of the Vg,n’s:W gn(z1, . . . , zn)= 2−mg,n∫ ∞0dL1 . . . dLne−∑i ziLin∏i=1Li Vg,n(L1, . . . , Ln)= 2−mg,n∑d0+...+dn=3g−3+n(n∏i=01di!) 〈κd01 τd1 . . . τdn〉g,n(2d1 + 1)!z2d1+21. . .(2dn + 1)!z2dn+2n(2− 1)where (see [4]) mg,n = δg,1δn,1.Since the Vg,n’s are even polynomials of the Li’s, of degree 2dg,n wheredg,n = dimMg,n = 3g − 3 + n (2-2)the W gn ’s are even polynomials of the 1/zi’s of degree 2dg,n + 2. Let us also define:W 01 = 0 (2-3)W 02 (z1, z2) =1(z1 − z2)2(2-4)anddEu(z) =12(1z − u−1z + u). (2-5)We prove the following theorems:Theorem 2.1 For any 2g − 2 + n+ 1 > 0, the W gn+1 satisfy the recursion relationW gn+1(z,K) = Res u→0πdEu(z)u sin 2πu[∑gh=0∑J⊂K Wh1+|J |(u, J)Wg−h1+n−|J |(−u,K/J)+W g−1n+2(u,−u,K)](2-6)2where the RHS includes all possible W hk , including W01 = 0 and W02 , and whereK = {z1, . . . , zn} (2-7)is a set of n variables.proof:This relation is merely the Laplace transform of Mirzakhani’s recursion. See theappendix for a detailed proof. Corollary 2.1 W gn are the invariants defined in [2] for the curve:{x(z) = z2−2y(z) = sin (2πz)2π= z − 2π23z3 + 2π415z5 − 4π6315z7 + 2π82835z9 + . . .(2-8)which is a special case of Kontsevich’s curve:Z(tk) =∫dM e−N Tr [M33+ΛM2] , tk =1NTr Λ−k =(2pi)k−3 sin (pik/2)(k − 2)!(2-9)For instance we have:lnZ(tk) =∞∑g=0N2−2gW g0 (2-10)(W g0 is often noted −Fg in the litterature).proof:Eq. 2-6 is precisely the definiton of the invariants of [2] for the curve{x(z) = z2−2y(z) = sin (2πz)2π= z − 2π23z3 + 2π415z5 − 4π6315z7 + 2π82835z9 + . . .(2-11)And it was proved in [2] that this curve is a special case of Kontsevich’s curve:{x(z) = z2y(z) = z − 12∑∞j=0 tj+2zj (2-12)which corresponds to the computation of the topological expansion of the Kontsevichintegral:Z(tk) =∫dM e−N Tr [M33+ΛM2] , tk =1NTr Λ−k (2-13)lnZ(tk) = −∞∑g=0N2−2gFg (2-14)3Theorem 2.2 For any 2g − 2 + n > 0 we have:(2g − 2 + n)W gn(K) =14pi2Resu→0(u cos (2piu)−12pisin (2piu))W gn+1(u,K) (2-15)or in inverse Laplace transform:(2g − 2 + n)Vg,n(K) =12ipiV ′g,n+1(K, 2ipi)(2-16)where ′ means the derivative with respect to the n+ 1th variable.proof:This is a mere application of theorem 4.7. in [2], as well as its Laplace transform.In particular with n = 0 we get:Vg,0 = Vol(Mg,0) =12g − 2V ′g,1(2ipi)2ipi(2-17)for instance for g = 2:V2,0 =43pi62160. (2-18)2.1 ExamplesFrom [4] we get:W 03 =1z21z22z23(2-19)W 11 =18z41+pi212z21(2-20)W 04 =1z21z22z23z24(2pi2 + 3(1z21+1z22+1z23+1z24))(2-21)W 12 =1z21z22(pi44+pi22(1z21+1z22) +58z41+58z42+38z21z22)(2-22)W 05 =1z21z22z23z24z25(10pi4 + 18pi2∑i1z2i+ 15∑i1z4i+ 18∑i<j1z2i z2j)(2-23)W 21 =1192z21(29pi8 +338pi65z21+139pi4z41+203pi2z61+3152z81)(2-24)4Those functions are the same as those which appear in section 10.4.1 of [2], for theKontsevich curve with times:t3 − 2 = 1, t5 = −2pi23, t7 =2pi415, t9 = −4pi6315, t11 =2pi82835, . . . (2-25)i.e. the rational curve:EK ={x(z) = z2−2y(z) = sin (2πz)2π= z − 2π23z3 + 2π415z5 − 4π6315z7 + 2π82835z9 + . . .(2-26)It is to be noted that those tk’s are closely related to the βk’s of [5, 3].3 ConclusionWe have shown that, after Laplace transform, Mirzakhani’s recursions are nothing butthe solution of loop equations (i.e. Virasoro constraints) for the Kontsevich integralwith some given set of times. It would be interesting to understand what the invariantsof [2] compute for an arbitrary spectral curve (for instance for other Kontsevich times).AcknowledgementsThis work is partly supported by the Enigma European network MRT-CT-2004-5652,by the ANR project Ge´ome´trie et inte´grabilite´ en physique mathe´matique ANR-05-BLAN-0029-01, by the Enrage European network MRTN-CT-2004-005616, by the Eu-ropean Science foundation through the Misgam program, by the French and Japaneesegovernments through PAI Sakura, by the Quebec government with the FQRNT.Appendix A Laplace transform of the equationsLet us write:LK = {L1, . . . , Ln} (1-1)Hgn(x, y, LK) = xyVg−1,n+2(x, y, LK) +g∑h=0∑J∈KxVh,1+|J |(x, LJ)yVg−h,n+1−|J |(y, LK/J)(1-2)where all the Vh,k terms in the RHS are such that 2h + k − 2 > 0 (i.e. stable curvesonly), as well as their laplace transform:H˜(g)n (z, z′, LK) :=∫ ∞0dx∫ ∞0dye−zxe−z′yHgn(x, y, LK). (1-3)5Mirzakhani’s recursion reads:2LVg,n+1(L,LK) =∫ L0dt∫ ∞0dx∫ ∞0dyK(x+ y, t)Hgn(x, y, LK)+n∑m=1∫ L0dt∫ ∞0dx(K(x, t+ Lm) +K(x, t− Lm))xVg,n−1(x, Lˆm)(1-4)whereK(x, t) =11 + e(x+t2 )+11 + e(x−t2 )(1-5)and Lˆm = LK/{Lm}.Let H˜gn be the Laplace transform of Hgn with respect to x and y.The Laplace transform of the first term in eq.1-4 is:∑ǫ=±1∫ ∞0dL e−zL∫ L0dt∫ ∞0dx∫ ∞0dy11 + ex+y+ǫt2Hgn(x, y, LK)=∑ǫ=±1∫ ∞0dt∫ ∞tdL e−zL∫ ∞0dx∫ ∞0dy11 + ex+y+ǫt2Hgn(x, y, LK)=∑ǫ=±11z∫ ∞0dt e−zt∫ ∞0dx∫ ∞0dy11 + ex+y+ǫt2Hgn(x, y, LK)= −∞∑j=11z∫ ∞0dt e−zt∫ ∞0dx∫ ∞0dy(−1)je−j2(x+y+t) Hgn(x, y, LK)+∞∑j=01z∫ ∞0dx∫ ∞0dy∫ ∞x+ydt e−zt(−1)jej2(x+y−t) Hgn(x, y, LK)−∞∑j=11z∫ ∞0dx∫ ∞0dy∫ x+y0dt e−zt(−1)je−j2(x+y−t) Hgn(x, y, LK)= −∞∑j=11z∫ ∞0dx∫ ∞0dy(−1)jz + j2e−j2(x+y) Hgn(x, y, LK)+∞∑j=01z∫ ∞0dx∫ ∞0dy(−1)jz + j2e−z(x+y) Hgn(x, y, LK)−∞∑j=11z∫ ∞0dx∫ ∞0dy(−1)jz − j2(1− e−(z−j2)(x+y))e−j2(x+y) Hgn(x, y, LK)= −2∞∑j=1(−1)jz2 −(j2)2 H˜gn(j2 , j2 , LK) + 1z2 H˜gn(z, z, LK)+2∞∑j=1(−1)jz2 −(j2)2 H˜gn(z, z, LK)= −2∞∑j=1(−1)jz2 −(j2)2 H˜gn(j2 , j2 , LK) + 2piz sin 2piz H˜gn(z, z, LK)6=(Resu→z+∞∑j=1Resu→± j2)duu− z2piu sin (2piu)H˜gn(u, u, LK)= Resu→0duz − u2piu sin (2piu)H˜gn(u, u, LK)= Resu→02pi duu sin (2piu)dEu(z) H˜gn(u, u, LK)(1− 6)Using the notationR(x, t, Lm) := (K(x, t+ Lm) +K(x, t− Lm)), (1-7)the Laplace transform of the second term in eq.1-4 is:∫ ∞0dLme−zmLm∫ ∞0dL e−zL∫ L0dt∫ ∞0dxR(x, t, Lm)xVg,n−1(x, Lˆm)=1z∫ ∞0dx∫ ∞0dLme−zmLm∫ ∞0dt e−ztR(x, t, Lm)xVg,n−1(x, Lˆm)=1z∫ ∞0dx∫ ∞0dLme−zmLm∫ ∞Lmdt e−z(t−Lm)K(x, t)xVg,n−1(x, Lˆm)+1z∫ ∞0dx∫ ∞0dLme−zmLm∫ ∞−Lmdt e−z(t+Lm)K(x, t)xVg,n−1(x, Lˆm)=1z∫ ∞0dx∫ ∞0dt e−zt∫ t0dLme−(zm−z)Lm K(x, t)xVg,n−1(x, Lˆm)+1z∫ ∞0dx∫ ∞0dt e−zt∫ ∞0dLme−(zm+z)Lm K(x, t)xVg,n−1(x, Lˆm)+1z∫ ∞0dx∫ 0−∞dt e−zt∫ ∞−tdLme−(zm+z)Lm K(x, t)xVg,n−1(x, Lˆm)=1z∫ ∞0dx∫ ∞0dte−zt − e−zmtzm − zK(x, t)xVg,n−1(x, Lˆm)+1z∫ ∞0dx∫ ∞0dte−zt + ezmtzm + zK(x, t)xVg,n−1(x, Lˆm)+1z∫ ∞0dx∫ ∞0dte−zmtzm + zK(x, t)xVg,n−1(x, Lˆm)=1z∫ ∞0dx∫ ∞0dt(e−zt − e−zmtzm − z+e−zt + e−zmtzm + z)K(x, t)xVg,n−1(x, Lˆm)=1z∫ ∞0dx∫ ∞0dt2zme−zt − 2ze−zmt(z2m − z2)11 + ex+t2xVg,n−1(x, Lˆm)+1z∫ ∞0dx∫ x0dt2zme−zt − 2ze−zmt(z2m − z2)11 + ex−t2xVg,n−1(x, Lˆm)+1z∫ ∞0dx∫ ∞xdt2zme−zt − 2ze−zmt(z2m − z2)11 + ex−t2xVg,n−1(x, Lˆm)= −∞∑j=1(−1)jz∫ ∞0dx∫ ∞0dt2zme−zt − 2ze−zmt(z2m − z2)e−j2(x+t) xVg,n−1(x, Lˆm)7−∞∑j=1(−1)jz∫ ∞0dx∫ x0dt2zme−zt − 2ze−zmt(z2m − z2)e−j2(x−t) xVg,n−1(x, Lˆm)+∞∑j=0(−1)jz∫ ∞0dx∫ ∞xdt2zme−zt − 2ze−zmt(z2m − z2)ej2(x−t) xVg,n−1(x, Lˆm)= −∞∑j=1(−1)jz∫ ∞0dx2zmz+ j2− 2zzm+j2(z2m − z2)e−j2x xVg,n−1(x, Lˆm)−∞∑j=1(−1)jz∫ ∞0dx∫ x0dt2zme−j2x−e−zxz− j2− 2z e−j2x−e−zmxzm−j2(z2m − z2)xVg,n−1(x, Lˆm)+∞∑j=0(−1)jz∫ ∞0dx∫ ∞xdt2zme−zxz+ j2− 2ze−zmxzm+j2(z2m − z2)xVg,n−1(x, Lˆm)= −2∞∑j=1(−1)jzz + zm +j2(zm + z)(z +j2)(zm +j2)Wg,n−1(j2, Lˆm)−2∞∑j=1(−1)jzz + zm −j2(zm + z)(z −j2)(zm −j2)Wg,n−1(j2, Lˆm)+2∞∑j=1(−1)jzzm(z − j2)(z2m − z2)Wg,n−1(z, Lˆm)−2∞∑j=1(−1)j1(zm −j2)(z2m − z2)Wg,n−1(zm, Lˆm)+2∞∑j=0(−1)jzzm(z + j2)(z2m − z2)Wg,n−1(z, Lˆm)−2∞∑j=0(−1)j1(zm +j2)(z2m − z2)Wg,n−1(zm, Lˆm)= −4∞∑j=1Resu→±jpidusin 2piu1zz + zm + u(zm + z)(z + u)(zm + u)Wg,n−1(u, Lˆm)+4zmpiz sin (2piz) (z2m − z2)Wg,n−1(z, Lˆm)−4pisin (2pizm) (z2m − z2)Wg,n−1(zm, Lˆm)= −4∞∑j=1Resu→± j2pidusin 2piuzm(z2 − u2)(z2m − u2)Wg,n−1(u, Lˆm)−4 Resu→z,zmpidusin 2piuzm(z2m − u2)(z2 − u2)Wg,n−1(u, Lˆm)= 4 Resu→0pidusin 2piuzm(z2 − u2)(z2m − u2)Wg,n−1(u, Lˆm)= 2 Resu→0pidu2u sin 2piu(1z − u−1z + u) (1zm − u+1zm + u)Wg,n−1(u, Lˆm)8= 4 Resu→0pidu2u sin 2piu(1z − u−1z + u)1zm − uWg,n−1(u, Lˆm)(1− 8)After taking the derivative with respect to zm that gives the expected term:Resu→02piduu sin 2piudEu(z) 2W02 (u, zm)Wg,n−1(u, Lˆm) (1-9)and therefore the Laplace transform of Eq. (1-4) gives the relation Eq. (2-6).References[1] B. Eynard, “Topological expansion for the 1-hermitian matrix model correlationfunctions”, JHEP/024A/0904, hep-th/0407261.[2] B.Eynard, N.Orantin, “Invariants of algebraic curves and topological expansion”,math-ph/0702045.[3] K. Liu, H. Xu, “A simple proof of Mirzakhani’s recursion formula of Weil-Peterssonvolumes”, math.AG/0705.2086.[4] M. Mirzakhani, “Simple geodesics and Weil-Petersson volumes of moduli spacesof bordered Riemann surfaces”, Invent. Math. 167, 179-222 (2007).[5] M. Mulase, B. Safnuk, “Mirzakhani’s recursion relations, Virasoro constraints andthe KdV hierarchy”, math.AG/0101147.9

Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models

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Weil-Petersson volume of moduli spaces, Mirzakhani's recursion and matrix models

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