THE N-POINT FUNCTIONS FOR INTERSECTION NUMBERS ON MODULI SPACES OF CURVES

Transcription

1 THE N-POINT FUNTIONS FOR INTERSETION NUMBERS ON MODULI SPAES OF URVES KEFENG LIU AND HAO XU Abstract. We derive from Witte s KdV equatio a simple formula of the -poit fuctios for itersectio umbers o moduli spaces of curves, geeralizig Dikgraaf s two-poit fuctio ad Zagier s three-poit fuctio. This formula ucovers may ew idetities about itegrals of ψ classes ad provides a elemetary ad more efficiet algorithm to compute itersectio umbers other tha the celebrated Witte-Kotsevich theorem.. Itroductio We deote by M g, the moduli space of stable -poited geus g complex algebraic curves. Let ψ i be the first her class of the lie budle whose fiber over each poited stable curve is the cotaget lie at the i-th marked poit. We adopt Witte s otatio i this paper, τ d τ d g : ψ d ψ d. M g, These itersectio umbers are the correlatio fuctios of two dimesioal topological quatum gravity. I the famous paper [7], Witte made the remarkable coecture proved by Kotsevich [4] that the geeratig fuctio of above itersectio umbers are govered by KdV hierarchy, which provides a recursive way to compute all these itersectio umbers. Witte s coecture was reformulated by Dikgraaf, Verlide, ad Verlide [DVV] i terms of the Virasoro algebra. Defiitio.. We call the followig geeratig fuctio F x,..., x τ d τ d g the -poit fuctio. P g0 d 3g 3+ The -poit fuctio ecodes all iformatio of the correlatio fuctios of two dimesioal topological quatum gravity. Okoukov [6] obtaied a aalytic expressio of the -poit fuctios usig -dimesioal error-fuctio-type itegrals. Brézi ad Hikami [] apply correlatio fuctios of GUE esemble to fid explicit formulae of -poit fuctios. The first key poit is to cosider the followig ormalized -poit fuctio Gx,..., x exp x3 F x,..., x. 4

2 KEFENG LIU AND HAO XU I particular, we have -poit fuctio Gx, Dikgraaf s -poit fuctio x Gx, y k k! xyx + y x + y k +! k 0 ad Zagier s 3-poit fuctio [8] which we leared from Faber, Gx, y, z r,s 0 r!s r x, y, z 4 r r +!! s 8 s r + s +!, where S r x, y, z ad are the homogeeous symmetric polyomials defied by S r x, y, z xyr x + y r+ + yz r y + z r+ + zx r z + x r+ Z[x, y, z], x + y + z x + y + z3 x, y, z x + yy + zz + x x3 + y 3 + z Although two ad three poit fuctios are foud i the early 990 s, it s ot obvious at all that clea explicit formulae of geeral -poit fuctios should exist. Recall that we oly have closed formula of itersectio umbers i geus zero ad oe. Now we state the mai theorem of this ote. Theorem.. For, Gx,..., x r,s 0 r + 3!! 4 s r + s +!! P rx,..., x x,..., x s, where P r ad are homogeeous symmetric polyomials defied by x,..., x x 3 x3, 3 P r x,..., x x x I J I J x i x i x i Gx I Gx J x i 3r+ 3 r G r x I G r r x J, where I, J, {,,..., } ad G g x I deotes the degree 3g + I 3 homogeeous compoet of the ormalized I -poit fuctio Gx k,..., x k I, where k I. Note that the degree 3r + 3 polyomial P r x,..., x Q[x,..., x ] is expressed by ormalized I -poit fuctios Gx I with I <. So we ca recursively obtai a explicit formula of the -poit fuctio F x,..., x exp x3 4 r 0 Gx,..., x, thus we have a elemetary algorithm to calculate all itersectio umbers of ψ classes other tha the celebrated Witte-Kotsevich s theorem [4, 7], which is the oly feasible way kow before to calculate all itersectio umbers of ψ classes.

3 THE N-POINT FUNTIONS FOR INTERSETION NUMBERS 3 Sice P 0 x, y x+y, P rx, y 0 for r > 0, we get Dikgraaf s -poit fuctio. From P r x, y, z r! r r +! xyr x + y r+ + yz r y + z r+ + zx r z + x r+, x + y + z we also easily recover Zagier s 3-poit fuctio obtaied more tha te years ago. There is aother slightly differet formula of -poit fuctios. Whe 3, this has also bee obtaied by Zagier [8]. Theorem.3. For, F x,..., x exp x 3 4 r,s 0 s P r x,..., x x,..., x s 8 s r + s + s! where P r ad are the same polyomials as defied i theorem.. Theorem.3 follows from Theorem. ad the followig lemma. Lemma.4. Let ad r, s 0. The the followig idetity holds, 3 s s 8 s r + s + s! Proof. Let p r + ad fp, s k0 s k0 k 8 k k! r + 3!! 4 s k r + s k +!! k k k!p + s k!!. We have s k p + s + s fp, s k k!p + s k +!! + k k k k!p + s k +!! k0 k0 s+ s p + s + fp, s + + fp, s s+ s +!p!! s s!p!!. So we have the followig idetity fp, s + s+ s+ p + s + s +!p 3!!, which is ust the idetity 3 if s + is replaced by s. I Sectio we give a proof of the mai theorem. Sectio 3 cotais may ew idetities of the itersectio umbers of the ψ classes derived from our formula of the -poit fuctios. I Sectio 4 we briefly discuss other applicatios of the -poit fuctios. Ackowledgemets. The authors would like to thak Professor Sergei Lado, Edward Witte ad Do Zagier for helpful commets ad their iterests i this work. We also wat to thak Professor arel Faber for his woderful Maple programm for calculatig Hodge itegrals ad for commuicatig Zagier s three-poit fuctio to us.

4 4 KEFENG LIU AND HAO XU. Proof of the mai theorem We ca derive from Witte s KdV equatio the followig coefficiet equatio see [3, 7], d + τ d τ 0 + {,...,}I J τ d 4 τ d τ0 4 τ d τ d τ 0 τ di τ0 3 τ di + τ d τ0 τ di τ0 τ di, which is equivalet to the followig differetial equatio of -poit fuctios F x,..., x, x + x x F x,..., x x 4 x 4 + x x F x,..., x + x x i x i 3 + x i x i F x I F x J. I J So i order to prove Theorem., we eed to check that Ex,..., x : satisfies the followig differetial equatio, 4 x x Gx,..., x x Ex,..., x + x + x3 x 4 x I J x + x x 4 x 3 Ex,..., x x i + x i x i Ex I Ex J. The verificatio is straightforward from the defiitio of Gx,..., x i Theorem.. We ow prove the followig iitial value coditio of Gx,..., x, thus coclude the proof of Theorem.. Gx,..., x, 0 x Gx,..., x. Let M r x,..., x : x i 3 x i r G r x I G r r x J. I J r 0

5 For the left had side, we have x LHS THE N-POINT FUNTIONS FOR INTERSETION NUMBERS 5 r +!! 4 s r + s +!! M r + r,s 0 r +!! 4 s r + s +!! M r s + r,s 0 r,s 0 r +!! 4 s r + s +!! + r+s r,s 0 kr x G r x,..., x x,..., x s r +!! 4 s r + s +!! p+qr k +!!r + 3!! 4 s r + s +!!k +!! where i the last equatio we have used chage of variables. While for the right had side, x RHS r + 3!! 4 s r + s +!! M r s. r,s 0 So we eed oly prove the followig combiatorial idetity p + 3!! 4 q r +!! M p q+s M r s, r+s r +!! 4 s r + s +!! + k +!!r + 3!! 4 s r + s +!!k +!! r + 3!! 4 s r + s +!! i.e. kr r+s r +!! r + 3!! + k +!! k +!! kr r + s +!! r + s +!! for all ad r, s 0. It follows easily from the followig idetity p +!! p!! + p +!! p +!! p + 3!! p +!!. It is typical that from the formula of -poit fuctios i Theorem., may assertios about itersectio umbers will be reduced to combiatorial idetities. 3. New properties of the -poit fuctios I this sectio we derive various ew idetities about the itersectio umbers of the ψ classes by usig our simple formula of the -poit fuctios. Lemma 3.. Let. We have the followig recursio relatio for ormalized -poit fuctios G g x,..., x g + P gx,..., x + x,..., x 4g + G g x,..., x. The followig idetity holds x,..., x x x + x x + x,..., x.

6 6 KEFENG LIU AND HAO XU Proof. We have G g x,..., x r+sg g + P gx,..., x + r + 3!! 4 s g +!! P rx,..., x x,..., x s r+sg r + 3!! 4 s+ g +!! P rx,..., x x,..., x s+ g + P gx,..., x + x,..., x 4g + G g x,..., x. The proof of is easy. Let xd, P x,..., x deotes the coefficiet of xd i a polyomial or formal power series P x,..., x. From the iductive structure i the defiitio of - poit fuctios, we have the followig basic properties of -poit fuctios, their proofs are purely combiatorial. First cosider the ormalized + -poit fuctio Gz, x,..., x. Here we use the variable z to distiguish oe poit. We have the followig theorem about the coefficiets of Gz, x,..., x. Theorem 3.. Let g + 0. If k > g +, d 0 ad d 3g + k, the z k z k, G gz, x,..., x, P gz, x,..., x 0, 0. Let d 0, d g ad a #{ d 0}. The z g + z g +, G gz, x,..., x, P gz, x,..., x 4 g d +!!, a 4 g d +!!. 3 Let d 0, d g +, a #{ d 0} ad b #{ d }. The z g 3+, G gz, x,..., x g + g a a 4 g d, +!! z g 3+, P gz, x,..., x ag + g g + a +5a + 3b 3 3b 4 g d. +!!

7 THE N-POINT FUNTIONS FOR INTERSETION NUMBERS 7 Proof. is obvious from theorem.. We ow prove iductively. z g +, P gz, x,..., x z g +, G gz, x,..., ˆx,..., x where a #{ d 0}. z g +, G g z, x,..., x r+sg g + g + r +!! 4 s g +!! P d g P d g P d r xd P 4 g d g d +!!. a 4 g d +!!, a xd 4 r d +!! a xd 4 g d +!! + x 4g + a xd 4 g d +!! + P d g The statemet 3 ca be proved similarly. x s P d g xd 4 g d +!! g + a xd 4 g d +!! Now cosider the ormalized special + -poit fuctio Gy, y, x,..., x. We have the followig theorem about the coefficiets of Gy, y, x,..., x. Theorem 3.3. Let g 0 ad. If k > g, d 0 ad d 3g + k, the y k, x i Gy, x I G y, x J 0, or equivaletly, I Jy + I J 0 x i y + k τ τ0 τ di g τ k τ0 τ di g g 0. If d ad d g +, the y g, + I Jy x i y + x i Gy, x I G y, x J g + +! 4 g g +! d!!.

8 8 KEFENG LIU AND HAO XU or equivaletly, g I J 0 τ τ0 τ di g τ g τ0 τ di g g g + +! 4 g g +! d!!. 3 If d ad d g +, the y g, x i x i Gy, y, x I Gx J 0. or equivaletly, I J 0 I J 0 I J g τ τ g τ0 τ di g τ0 τ di g g 0. 4 If d 0 ad d g +, the g τ τ0 τ di τ g τ0 τ di + τ τ g τ0 τ di τ0 τ di g + + g 0 τ 0 τ τ g τ d τ d g 5 If k > g, d 0 ad d 3g + k, the y k y k, G gy, y, x,..., x, P gy, y, x,..., x 6 If d 0 ad d g +, the y g, P gy, y, x,..., x If moreover we have d, the y g g ++, G gy, y, x,..., x y g 0, 0., G gy, y, x,..., x g +! 4 g g +! d!!. Proof. We first show that ad imply the statemets is obvious, sice for d i, we have τ di , 5 ad the first idetity of 6 follow easily from Theorem.. τ 0.

9 THE N-POINT FUNTIONS FOR INTERSETION NUMBERS 9 Let d d d. We prove the secod idetity of 6 by iductig o d, the maximum idex. g τ g τ τ d τ d g 0 g 0 τ 0 τ g τ τ d +τ d τ d g g +! 4 g g +! d!!d + g +! 4 g g +! d!!, g k 0 k τ g τ τ d +τ d τ dk τ d g g +!d k 4 g g +! d!!d + where we have used 4. I fact the above idetity still holds if there is oly oe d 0. By explicitly writig dow the -poit fuctios, we give a proof of ad i the case, the geeral case ca be proved similarly. Note also that it is easy to prove Theorem 3.3 for g 0 sice G 0 x,..., x x + + x 3 see Lemma 4.3 ad orollary 4.4. It is easy to prove the followig idetity by iductig o g. 0 y g+, + I Jy x i y + g x i G g y, x I G g g y, x J g 0 r!! 4 s g +!! 4 r r +!! xr + x r x + x s 4 r r +!! 4 s s +!! xr x s. r+sg Because we have y g, + I Jy x i y + x i r!! 4 s g +!! 4 r r +!! r+sg g G g y, x I G g g y, x J g 0 sx r + x r x + x s x x + s + s x r + x r x + x s+ +s + r + x r+ + x r+ x + x s+ + r + r x r+ + x r+ x + x s + r + s + x r+ 4 r r +!! 4 s x s+ s + s x r s +!! x s+ r + r + 4 g g!! x + x g+, the proof of is also easy. x r+ x s It is easy to see that the statemets 5 ad 6 of Theorem 3.3 imply the followig idetities of itersectio umbers which we have aouced i [5]. They are related to Faber s itersectio umber coecture.

10 0 KEFENG LIU AND HAO XU orollary 3.4. Let d 0, #{ d 0} ad d g +. The g 0 τ g τ τ d τ d g g +! 4 g g +! d!!. If #{ d 0} ad a #{ d }, the the right had side becomes g +! 4 g g +! d!! g + a g + a. Let k > g, d 0 ad d 3g + k. The k τ k τ τ d τ d g 0. 0 We also have the followig geeralizatio of statemets ad of Theorem 3.3. The proof is similar. Theorem 3.5. Let g 0, ad r, s 0. If k > g + r + s, d 0 ad d 3g + r + s + k, the y k, x i +r Gy, x I G y, x J 0, or equivaletly, I Jy + k I J 0 τ τ +r 0 x i +s y + τ di g τ k τ +s 0 τ di g g 0. If d ad d g +, the y g+r+s, + I Jy x i +s y + x i +r Gy, x I G y, x J I J r g + + r + s +! 4 g g + r + s +! d!!. or equivaletly, g+r+s 0 τ τ +r 0 τ di g τ g+r+s τ +s 0 τ di g g r g + + r + s +! 4 g g + r + s +! d!!.

11 THE N-POINT FUNTIONS FOR INTERSETION NUMBERS 4. Other applicatios of -poit fuctios From the ew idetities i Sectio 3 ad their derivatios, we ca see that the simple formula of -poit fuctios may be used to prove the followig equivalet statemet of the Faber s itersectio umber coecture: oecture 4.. Let d 0 ad d g +. The τ d τ d τ g g τ d τ d τ d +g τ d+ τ d g I J g τ τ di g τ g τ di g g. 0 It is clear that our explicit formula of -poit fuctios should also shed light o the followig coectural idetity as stated i [5] where the case of has bee proved. oecture 4.. Let g, d ad d g. The g 3 +! g+ g 3! d!! τ d τ d τ g g τ d τ d τ d +g 3τ d+ τ d g + I J g 4 τ τ di g τ g 4 τ di g g If we have d 0, #{ d 0} ad a #{ d } i the above coecture, the the left had side becomes 0 g 3 +! g+ g 3! d!! g + + a g + 3 a. We will discuss the relatio of the above coectures with our simple formula of the - poit fuctios i a forthcomig paper. Here as the first step we oly prove two iterestig combiatorial idetities. Lemma 4.3. Let. 5 Assume that if I, the x i I. We have + {,...,}I Jx x i I x + x i J We have I J I,J x i I {,...,}I J x i J x x i I x i J

12 KEFENG LIU AND HAO XU Proof. Let xd 6 be ay moomial of {,...,}I Jx + x i I x + x i J. Sice d, so if d > 0, the their must exist some > such that d 0. The statemet meas that the polyomial 6 does ot cotai x, so we eed oly prove that after substitute x 0 i 6, the resultig polyomial does ot cotai x. {,..., }I J x x + {,..., }I Jx + x i I + x + x i I x + x i J + x + x i J. x i I x + x i J + So statemet follows by iductio. We prove statemet by iductio. Regard the LHS ad RHS of the idetity 5 as polyomials i x with degree, we eed to prove the equality of 5 whe substitute x x i for i.... It s sufficiet to check the case x x. LHS I J I J x + x i I 4 x + x x i I x + x i J + I J I,J x RHS. x i J + x i I + x i I x i J x i J x i I + x i J Note that if a term has power J, the J is assumed. Fially as a iterestig exercise we give a proof of the followig well-kow formula by usig our formula of the -poit fuctios. orollary 4.4. Let 3, d 0 ad d 3. The 3 τ d τ d 0. d,..., d

13 THE N-POINT FUNTIONS FOR INTERSETION NUMBERS 3 Proof. It s equivalet to prove that for 3 x 3 G 0 x,..., x This is ust the Lemma 4.3. x x I J I J x i x i G 0 x I G 0 x J x i I x i J. Refereces [] E. Brézi ad S. Hikami, Vertices from replica i a radom matrix theory, math-ph/ [] R. Dikgraaf, H. Verlide, ad E. Verlide, Topological strigs i d <, Nuclear Phys. B 35 99, [3]. Faber ad R. Padharipade, Logarithmic series ad Hodge itegrals i the tautological rig, with a appedix by Do Zagier, Michiga Math. J. Fulto volume , 5 5. [4] M. Kotsevich, Itersectio theory o the moduli space of curves ad the matrix Airy fuctio. omm. Math. Phys , o., 3. [5] K. Liu ad H. Xu, New properties of the itersectio umbers o moduli spaces of curves, math.ag/ [6] A. Okoukov, Geeratig fuctios for itersectio umbers o moduli spaces of curves, Iterat. Math. Res. Notices, 00, [7] E. Witte, Two-dimesioal gravity ad itersectio theory o moduli space, Surveys i Differetial Geometry, vol., [8] D. Zagier, The three-poit fuctio for M g, upublished. [9] H. Xu, A Maple program to compute itersectio idices by oecture A., available at eter of Mathematical Scieces, Zheiag Uiversity, Hagzhou, Zheiag 3007, hia; Departmet of Mathematics,Uiversity of aliforia at Los Ageles, Los Ageles, A , USA address: liu@math.ucla.edu, liu@cms.zu.edu.c eter of Mathematical Scieces, Zheiag Uiversity, Hagzhou, Zheiag 3007, hia address: haoxu@cms.zu.edu.c