Peking Mathematical Journal https://doi.org/.7/s4543-8-4-4 ORIGINAL ARTICLE Asymptotic of Enumerative Invariants in CP Gang Tian Dongyi Wei Received: 8 March 8 / Revised: 3 July 8 / Accepted: 5 July 8 Peking University 8 Abstract In this paper, we give the asymptotic expansion of n,d and n,d, where 3d + g! n g,d counts the number of genus g curves in CP through 3d + g points in general position and can be identified with certain Gromov Witten invariants. Keywords Asymptotic Gromov Witten invariants Analytic continuation Contour integration Introduction The GW invariants were constructed first by Ruan Tian for semi-positive symplectic manifolds see [4, 6 8] and subsequently for general symplectic manifolds see [, 3] etc.. For any closed symplectic manifold M of dimension n, its GW invariants are given by a family of multi-linear maps: Φ M g,a,k H g,k, Q H M, Q k Q,. where a H M, Z and g,k denotes the moduli of stable curves of genus g and with k marked points. Let g, a, k be the moduli space of stable genus g maps of homology class a and with k marked points and ev is the evaluation map from g, a, k to g,k M k. If f, Σ, x,, x k is a stable map with smooth domain, then ev simply assigns this map to Σ, x,, x k, f x,, f x k in g,k M k. This map can be extended to stable maps where domain may not be smooth. For any β H g,k, Q and α,, α k H M, Q, we have Φ M g,a,k β;α,, α k = ev β α α k, vir g,a,k Gang Tian: Supported partially by Grants from NSF and NSFC. * Dongyi Wei jnwdyi@63.com Beijing International Center for Mathematical Research and School of Mathematical Sciences, Peking University, Beijing 87, China 3 Vol.:3456789
G. Tian, D. Wei where vir g, a, k is the virtual fundamental class of g, a, k. As usual, we consider the generating function F M g w = a,k Φ M g,a,k ;w,, w, w H M, C. k!. The goal of this paper is to study the asymptotic behavior of Φ M in a, k in the g,a,k case of M = CP and g =,. Since H CP, Z =H CP, Z =H 4 CP, Z =Z, we can denote a by d Z and write w = t γ + t γ + t γ, where {γ i } is an integral basis of H CP, C with deg γ i = i. Then F CP is of the form g F CP t g, t, t = ψ g,d k, k, k t k tk tk, d,k k +k +k =k where ψ g,d k, k, k denotes the evaluation of Φ M g,d,k at k of γ, k of γ and k of γ divided by k!k!k!. It is known that ψ g,d k, k, k = whenever d, k > and ψ g,d, k, k =d k ψg,d,, k. Furthermore, ψ g,d,, k = unless k = 3d + g. Thus, we obtain F CP g t, t, t =p + d= ψ g,d,, 3d + g t 3d +g e dt,.3 where p is a polynomial in t, t, t. We denote ψ g,d,, 3d + g by n g,d. Then, 3d + g! n g,d counts the number of genus g curves of degree d in CP through 3d + g points in general position. Theorem. If g =, then there exist x, {a k } R, with a > such that 3 N n,d = e dx a k d k + Od N, N 4. Theorem. If g =, then there exists {a k } R such that N n,d = e dx 48d + a 3 k d k + Od N 3, N 4, 3
Asymptotic of Enumerative Invariants in CP where x is the constant in Theorem.. Corollary. We have lim d d n,d = lim d d n,d = e x..4 This affirms Conjecture in [9] for the case g =,. It is plausible that d ng,d converges to a fixed number independent of g, for all g. In addition,.4 in the case of genus g = was claimed by Di Francesco and Itzykson in [], but they did not give a proof and provided only some heuristic arguments which are far from being a proof in any mathematical rigor. In [9], Zinger verified some of the claims in [] and proposed a proof of.4 in the case of genus g = under a conjectured condition which is still open. Our proof requires substantially new ideas and technical arguments. In addition, it remains to understand the geometric information encoded in the coefficients a k and a in the above expansions. k Consequences of the WDVV Equation In this section, we show some consequences of the WDVV equation. First, we recall for any symplectic manifold M, F M satisfies the WDVV equation: b,c 3 F M t i t j t b η bc 3 F M t k t l t c = b,c 3 F M t i t l t b η bc 3 F M t k t j t c,. where i, j, k, l run over,, L and w = L i= t i γ i for a basis {γ i } of H M, C. For M = CP and g =,.3 becomes F CP t = t t + t t + n,k t 3k e kt, k=. where t =t, t, t. We can further write F CP = t t + t t + t F t + 3 ln t,.3 where F z = n,d e dz. d=.4 3
G. Tian, D. Wei It follows from WDVV equation that 7 + F 3F F = 6F 33F + 54F +F holds. The following lemma is Corollary in [9]..5 Lemma. We have the estimates for n,d : d d 7 4 d n 7,d 3 d 7. 5.6 Proof For the readers convenience, we outline its proof by following [9]. First, we observe: if n d is a sequence of numbers satisfying: n d = d,d,d +d =d f d f d f d n d n d for some f Z R +, then n d = d! d!d! f n d, d. It is proved in [6, 7] using the WDVV equation n,d = d,d,d +d =d Td, d n,d n,d, where Td, d = d d 3d d d + d 3d 33d 3d. It is easy to see f d f d f d Td, d f d f d, f d where f d =d3d 54 and f d =d 5. If we denote by n j the sequence of d numbers determined by the recursive formula for above n d s together with f replaced f j and n j =, then by induction, we can show n d n,d n. Therefore, we have d 3
Asymptotic of Enumerative Invariants in CP d! d n,d d!d! 54 d! d. d!d! 5 The lemma follows from this and Stirling formula for d!. 3 Analytic Continuation of F By Lemma., F z is an analytic function on {z Rez < x } for some x R, and we get a convergent power series F z = n,d e dz, z C with Rez < x. d= This power series diverges if Rez > x. Since n,d, by.4 < F z < F z < F z < F z, z, x and moreover, since 6F >, 33F + 54F > 33F + 54F by.5 7 + F 3F >, z, x. 3. >, F >, also 3. By.4 and 3., the restrictions of F, F and F to, x are real-valued and are increasing functions. By 3. and 3., F < F < F < 7 on, x, in particular, we have F z = d= d n,d e dz < 7, z, x. It follows that d n,d e dx 7. d= Thus, F, F and F can be continuously extended to {z C Rez x }. Moreover, 7 + F 3F = at z = x, otherwise,.5 could be used to compute all derivatives of F at z = x and get a contradiction to the fact that e x is the convergence radius of F q =F z with q = e z. 3
G. Tian, D. Wei Lemma 3. For > sufficiently small, F {z C Rez<x, Imz [,π]} has a continuous extension to D ={z C Rez < x +, Imz [, π]}, moreover, this extension is analytic in the interior of D and has the expansion at x, F x + re iθ = a d r d e iθd, d= F x + + re iθ = a d r d e iθd, d= r <, θ π, r <, π θ π, 3.3 3.4 for some {a i } i= with a = a 3 = and a d C d, for all d >. Proof First, we observe that 3F F = d= d3d n,d e dz. Since n,d >, for any t, π, we have So 7 3F + F x + it. Then, using.5 and the standard theory of ODE, one can show that F can be analytically continued over a neighborhood of each point x + it with t, π. Hence, it suffices to prove the analytic continuation of F around x and x +. Introduce new real variables t, x, y, w such that dt dz = for example, we can take 3F F x + it < 3F F x = 7. 7 + F 3F, x = 9F 9F + F, y = 3F F, w = F, tz = z + 7 7 + F ds, z s 3F z s 7 xz = 3d 3d n,d e dz, yz = wz = d= d3d n,d e dz, d= d n,d e dz. d= 3
Asymptotic of Enumerative Invariants in CP In fact, F, F, F =,, t +t 3 x, t y, t wt + 3 ln t. Then, x, y, w >, and x, y, w, t are strictly increasing on, x. By definition, x, y, w, t for z, in particular, tz has an inverse zt which maps, t onto, x, where t = lim z x tz. Writing.5 in terms of the variable t, we get x 7x + 4y d y 9x + 8y + yw dt = w 3x + 6y + 9w + w. z 7 y + 3w 3.5 Since x, y, w >, it follows from the third equation in 3.5 that dw dt w, so w blows up at finite time and consequently, t <. By the standard theory of ODE, 3.5 has a real analytic solution x, ŷ, ŵ, ẑt in the maximal interval, t for some t R which extends the solution x, y, w, z on, t, that is, xz, yz, wz, z = x, ŷ, ŵ, ẑtz, z, x. Using 3.5, we deduce that ẑ R, x, ŷ, ŵ are strictly increasing positive functions on, t, and lim =+. t t xt+ŷt+ŵt Moreover, it follows from the equations on x, ŷ, ŵ in 3.5 that It implies d x + ŷ + ŵ 39 + 4ŷ + ŵ x + ŷ + ŵ. dt d log x + ŷ + ŵ 4 + ŷ + ŵ. dt Hence, we have 3
G. Tian, D. Wei t t ŷt+ŵt dt = +. 3.6 Observe that y 3w = 3F F = d= d3d n,d e dz > and y 3w is strictly increasing on, x. It follows that ŷ 3ŵt > for t = tz, z, x and ŷ 3ŵt for t. Using 3.5, we have ŷ 3ŵ = 9 x +9 + ŵŷ 3ŵ +ŵŷ. 3.7 Therefore, ŷ 3ŵt > and ŷ 3ŵ t > for t, t. Further, by 3.7, for all t, t, ŷ 3ŵ t ŷ 3ŵt ŵt +ŵt ŷt. Combining this with 3.6, we get lim ŷ 3ŵt = +, t t so there exists a unique t, t such that ŷ 3ĉt =7. By 3.5, this implies that ẑ t = and ẑ t <. Since 7 + F 3F = at z = x, we have ẑ tx =7 + F 3F x =, so t = tx =t, x = ẑt and there is a > such that for t <, ẑt + t =x + b k t k, ŵt + t = k= c k t k, where b <, b k, c k R, c >. Since ŵt =wẑt for t < t, there is a > such that for < z <, wx z = c k z k, where c k R, c 3 <. Since wz =F z, we have
Asymptotic of Enumerative Invariants in CP F x z =F x F x z + 4 c k k + k + 4 z k+4. Therefore 3.3 is true for θ = π with a = F x, a = F x, a = a 3 =, a k = 4 i k c k 4, k 4. kk Since F is periodic in {Re z < x }, 3.4 is also true for θ = π. Thus, we have the real analytic expressions in 3.3 and 3.4 in a neighborhood of θ = π. It implies the analytic continuation of F around x and x +, and consequently, 3.3 and 3.4. Remark 3. The analytic continuation of F in 3.3 and 3.4 is equivalent to the following expansion: F x + z = a d z d, z <, a = a 3 =. d= We also specify the value of z d. The above expansion around x was claimed in [] without a proof. A justification was provided in [9]. Our proof above is different and clearer. In addition, Lemma 3. is stronger than the expansion claimed in []. It states that F can be analytically continued in a neighborhood of {x + it t [, π]}. We need this stronger version in the subsequent arguments. 4 Proof of Theorem. Now, we complete the Proof of Theorem.. The proof is based on the formula 3.3 and 3.4 and contour integration. Fix any,, using contour integration, we have n,d = = x x + + π F z e dz dz F x + t e dx +t dt π F x + + it e dx ++it dt F x + + t e dx ++t dt. 4. 3
G. Tian, D. Wei By the Hölder inequality, we have π π F x + + it e dx++it dt C e dx +, 4. where C = max t π F x + + it < +. It follows from 3.3, 3.4 and the fact that a = a 3 = that = F x + t e dx+t dt = = a k F x + + t e dx ++t dt F x + t F x + + t e dx+t dt a k t k a k t k k e dx+t dt t k e dx +t dt = e dx a k Ak,, d, 4.3 where Ak,, d = t k e dt dt = d k d t k e t dt. Clearly, we have < Ak,, d < d k + t k e t dt = Γk + d k+ and < Ak +,, d <Ak,, d. Fixing an integer N > 3, for 3 k < N, we have < Γ k + d k+ + Ak,, d = t k e t dt d + d k N t N e t dt < d k N Γ N +, d Furthermore, we have 3
Asymptotic of Enumerative Invariants in CP a k k=n k=n k=n Ak,, d a k π a k π N Ak,, d + a k Γk + d k+ N N k N AN,, d+ πd N+ a k Γk + dk+ Ak,, d πd k+ a k d k N ΓN + πd k+ a k ΓN + N a k N k k N ΓN + + = ΓN + πd N+ πd N+ ΓN + a k k N = C, πd N+ 4.4 where C = a k k+ < +. It follows from 4., 4., 4.3 and 4.4 that e dx n,d N a k Γk + d k+ C e d ΓN + + C CN. πd N+ d N+ Hence, n,d = e dx N a k Γk + d k+ + Od N. Since n,d R, and a N ΓN + = lim d + dn+ n,d e dx N a k Γk +, d k+ we have ia N R. Set a k = a k Γk +. In particular, a 3 = a 5 Γ7 = 4i 5 c Γ7 5 = 4c Γ7 5π >. 3
G. Tian, D. Wei Thus we get the asymptotic expansion of n,d as we stated in Theorem.. 5 Proof of Theorem. We will adopt the notations from the last section. First, we define a generating function F z = n,d e dz. d= 5. It follows from the Eguchi Hori Xiong recursion formula see [5, 8] that F satisfies 7 + F 3F F = 8 F 3F + F. 5. By 3.5, ŵ t >, so we have c >, c > and a 5. Since 7 + F 3F x = and a = a 3 =, by 3.3, for any z with z < and arg z π, we have 7 + F 3F x + z = d= 3d + 4d + a 4 d+4 +d + a d+ z d and the expansion of F 3F + F x + z is d + 6d + 4d + a 8 d+6 d= 3d + 4d + 4 a d+4 +d + a d+ z d. Therefore, for any z with z < and arg z π, we have F x + z = d= a d zd, where a =. By 5., 48 F is analytic in the region {Re z < x }. Since 7 3F + F x + it for < t < π, we deduce from 5. that F has an analytic extension to {Re z < x +, Im z π, z x, x + } for some < <. Moreover, from 3.3, 3.4 and 5., we get F x + re iθ = a d rd e iθd, < r <, θ π, d= 5.3 3
Asymptotic of Enumerative Invariants in CP F x + + re iθ = a d rd e iθd, < r <, π θ π. d= 5.4 Fix <<, using contour integration, we have for all < <, dn,d = x + x π = π + + π π π π π F z e dz dz F x + eit e dx + e it e it dt F x + te dx+t dt F x + + it e dx++it dt F x + + t e dx+t dt F x + + eit e dx + e it e it dt. 5.5 It follows from the Hölder inequality that π π F x + + it e dx++it dt C e dx +, 5.6 where C = max t π F x + + it < +. By 5.3 and 5.4, as, we get F x + te dx+t dt = = = a k F x + + t e dx +t dt F x + t F x + + t e dx+t dt a k tk a k tk k e dx+t dt k= k= t k e dx+t dt e dx a k Ak,, d. 5.7 3
G. Tian, D. Wei By the dominated convergence theorem, π π π π π π π Then, arguing in a similar way as we did in the case of n,d, we deduce N dn,d = e dx a + a k Γk + + Od N d k+. Since a =, we have 48 F x + e it e dx + e it e it dt F x + + eit e dx + e it e it dt a e it e dx e it dt π π π a e it e dx e it dt = a e dx. N n,d = e dx 48d + a k Γk + d k+ 3 + Od N 3. Since n,d R, so does ia k. Set a k = a k Γk +. Thus, we get the asymptotic expansion of n,d as we stated in Theorem.. 6 A conjecture In this section, we propose a conjecture on the asymptotic behavior of Φ M in the general cases. g,a.k For simplicity, we restrict F M g to the even part H ev M of H M, C. Let γ,, γ L be an integral basis for H ev M such that each γ i is of pure degree d i and = d < = d d L = n. Write w = t i γ i, then the restriction of F g to H ev M is of the form F M g t,, t L = a,k k + +k L =k ψ g,a k,, k L t k tk L L, 3
Asymptotic of Enumerative Invariants in CP where ψ g,a k,, k L denotes the evaluation of Φ M g,a,k at k of γ,,k L of γ L divided by k! k L!. Note that ψ g,a k,, k L = unless c Ma +3 ng = L k i d i. i= 6. Further, we have: ψ g,a k,, k L = for k > and a and m ψ g,a, k,, k m, k m,, k L = γ i a k i ψ g,a,,, k m,, k L, i= where m is chosen such that d i = for i m and d i for any i m. Our concern is the asymptotic behavior of ψ g,a as c Ma +. More precisely, let {k m l,, k L l} be a sequence of non-negative tuples satisfying 6. l= with la in place of a, and N k i l =l c ij l j + Ol N, as l, N >, i m. j= Conjecture 6. There are d, c and b, b, such that N ψ g,la,,,, k m l,, k L l = l d c l e b i l i + ol N. i= 6. A deeper problem is what geometric information is encoded in those coefficients b, b,, in other words, can one express those b i in terms of geometric quantities of M. We also expect that the l-th root of ψ g,la,,,, k m l,, k L l converges to a fix constant which depends only on the convergent radius of the generating function F M. Acknowledgements We thank the referee for numerous comments which are helpful in improving the exposition of this paper. References. Di Francesco, P., Itzykson, C.: Quantum intersection rings. arxiv :hep-th/94 75. Fukaya, K., Ono, K.: Arnold conjecture and Gromov Witten invariant. Topology 385, 933 48 999 3
G. Tian, D. Wei 3. Li, J., Tian, G.: Virtual moduli cycles and Gromov Witten invariants of general symplectic manifolds, Topics in symplectic 4-manifolds Irvine, CA, 996, 47 83, Int. Press Lect. Ser., I, Int. Press, Cambridge, MA 987 4. Mcduff, D., Salamon, D.: J-holomorphic curves and quantum cohomology. University Lecture Series, vol. 6, pp. viii+7. American Mathematical Society, Providence, RI 994 5. Pandharipande, R.: A geometric construction of Getzler s elliptic relation. Math. Ann. 334, 75 79 999 6. Ruan, Y., Tian, G.: A mathematical theory of quantum cohomology. J. Differ. Geom. 4, 59 367 995 7. Ruan, Y., Tian, G.: A mathematical theory of quantum cohomology. Math. Res. Lett., 69 78 994 8. Ruan, Y., Tian, G.: Higher genus symplectic invariants and sigma models coupled with gravity. Invent. Math. 33, 455 56 997 9. Zinger, A.: On asymptotic behavior of GW-invariants, preprint 3