Periods and generating functions in Algebraic Geometry

Transcription

1 Periods and generating functions in Algebraic Geometry Hossein Movasati IMPA, Instituto de Matemática Pura e Aplicada, Brazil hossein/

2 Abstract In 1991 Candelas-de la Ossa-Green-Parkes predicted the number of rational curves of a fixed degree d in a generic quintic hypersurface using mirror symmetry. Since then there have been many efforts to give mathematical proofs for their result. The introduction of Gromov-Witten invariants n d was one of these efforts. In this talk we recall the original construction of numbers n d using the Picard-Fuchs equation of periods and then calculating the Yukawa coupling. Then we construct an ordinary differential equation in dimension five which is related to the generating function of n d s. The idea of this work comes from the case of elliptic curves, where mirror symmetry takes a simple form, and the corresponding ordinary differential equation is given by Ramanujan relations between Eisenstein series. Generating functions in this case are usually quasi/differential modular forms.

3 Counting

4 Fibonacci numbers: Counting F n = F n 1 + F n 2, F 0 = 0, F 1 = 1. The Fibonacci numbers give the number of pairs of rabbits n months after a single pair begins breeding (and newly born bunnies are assumed to begin breeding when they are two months old), as first described by Leonardo of Pisa (also known as Fibonacci) in his book Liber Abaci (from mathworld). The generating function for Fibonacci numbers: From this we get F = F n x n = n=0 x 1 x x 2 F n = αn β n α β, α, β = 1 2 (1 ± 5). lim n F n F n 1 = lim n F 1 n n = 1 2 (1 + 5)

5 Weierstrass family of elliptic curves

6 Weierstrass family of elliptic curves E t = {(x, y) C 2 y 2 4x 3 + t 2 x + t 3 = 0}, t = (t 2, t 3 ) C 2 \{ = 0}, := 27t3 2 t3 2 Elliptic integrals

7 Weierstrass family of elliptic curves E t = {(x, y) C 2 y 2 4x 3 + t 2 x + t 3 = 0}, t = (t 2, t 3 ) C 2 \{ = 0}, := 27t 2 3 t3 2 δ Elliptic integrals dx y, δ, xdx y, δ H 1(E t, Z)

8 Weierstrass family of elliptic curves E t = {(x, y) C 2 y 2 4x 3 + t 2 x + t 3 = 0}, t = (t 2, t 3 ) C 2 \{ = 0}, := 27t 2 3 t3 2 δ Elliptic integrals dx y, δ, xdx y, δ H 1(E t, Z) Let {δ 1, δ 2 } be a basis of H 1 (E t, Z), 0 with δ 1, δ 2 = 1.

9 From a generalized Weierstrass uniformization theorem it follows: The functions xdx g 1 := ( δ 2 y )( dx δ 2 y ), dx g 2 := t 2 ( δ 2 y )4, g 3 := t 3 ( can be written in terms of the variable z := dx δ 1 y dx δ 2 y δ 2 dx y )6

10 Eisenstein series g k (z) = a k (1 + ( 1) k 4k σ 2k 1 (n)q n ), k = 1, 2, 3, z H, B k n 1 B 1 = 1 6, B 2 = 1 30, B 3 = 1 42,..., σ i(n) := d i, q := e 2πiz d n (a 1, a 2, a 3 ) = ( 2πi 12, 12(2πi 12 )2, 8( 2πi 12 )3 ),

11 1. The theory of modular forms over SL(2, Z): homogeneous polynomials of the ring C[g 2, g 3 ], deg(g 2 ) = 4, deg(g 3 ) = The theory of quasi/differential modular forms over SL(2, Z): homogeneous polynomials of the ring C[g 1, g 2, g 3 ], deg(g 1 ) = 2, deg(g 2 ) = 4, deg(g 3 ) = 6

12 Monodromy The group SL(2, Z) appears as the monodromy group of the family of elliptic curves E t : Fix a point b C 2 \{ = 0}. The image of the monodromy map π 1 (C 2 \{ = 0}, b) Aut(H 1 (E b, Z)) written in the basis δ 1, δ 2 H 1 (E b, Z), δ 1, δ 2 = 1 is the group SL(2, Z).

13 The monodromy group of the family of elliptic curves E t : y 2 = 4(x t 1 )(x t 2 )(x t 3 ), t = (t 1, t 2, t 3 ) C 3 is Γ(d) := {A SL(2, Z) A ( ) 1 0 mod d}, d =

14 The j-function Monstrous moonshine conjecture, j = t 3 2 t t2 3 = g3 2 (z) g g2 3 = q q q q 3 +. We have = MacKay 1978: is the number of dimensions in which the Monster group can be most simply represented. J.H. Conway, S.P. Norton 1979: Monstrous moonshine conjecture R. Borcherds 1992: Solved

15 T. Gannon, Monstrous Moonshine: The first twenty-five years, 2005 T. Gannon, Moonshine beyond the Monster, The Bridge Connecting Algebra, Modular Forms and Physics, Cambridge Monographs on Mathematical Physics, 2006.

16 Representation theory 1. Let F be field of characteristic zero and, V be a F vector space of finite dimension and G be a group. A representation of G in V is a group homomorphism ϕ : G GL(V ). 2. The character of a representation ϕ is: where Tr is the trace. χ V : G F, χ V (g) = Tr(ϕ(g)) 3. Isomorphic representations have the same characters. 4. An irreducible representation of G is completely determined by its character.

17 Representation theory of finite groups 1. The representations of a finite group G are semisimple (completely reducible). 2. The number of irreducible representations is equal to the number of conjugacy classes.

18 Monster group If normal subgroups of a group G are {1} and G then G is called a simple group. In the classification of all finite simple groups there appears 26 sporadic groups. The Monster group M is the largest of the sporadic groups. M = The Monster group has 194 conjugacy classes and so that number of irreducible representations. Dimensions of irreducible representations of M: 1, , , , , , , , , , , , , ,

19 Conway s and Norton s conjecture There is an infinite-dimensional graded M-module V = m=0 V m with dim(v m ) = c m for all m, where q (j(z) 744) = m=0 c mq m such that the McKay-Thompson series of g G T g (q) = 1 q χ Vm (g)q m m=0 is the normalised main modular function for a genus zero subgroup G g of SL(2, R), commensurable with the modular group SL(2, Z). Note that T 1 (q) = q (j(z) 744).

20 Modularity theorem E : y 2 = 4x 3 a 2 x a 3, a 2, a 3 Z, := a a Let p be a prime and N p be the number of solutions of E working modulo p a p (E) := p N p A version of modularity theorem says that there is modlar form of weight 2 associated to some congruence group, namely f = n=0 a nq n, such that for all primes p. a p = a p (E)

21 Taniyama-Shimura conjecture. After A. Weils, R. Taylor, C. Breuil, B. Conrad, F.Diamond: Modulartiy theorem Book: Fred Diamond, Jerry Michael Shurman, A first course in modular forms, 2005.

22 Elliptic curves over finite field Hasse: For an elliptic curve E over a finite field F q we have: #E(F q ) = q + 1 (α + ᾱ), α Q, α = q. Define σ k (q) = E/F q (α k+1 ᾱ k+1 )/(α ᾱ) #Aut Fq (E) and (27t 2 3 t3 2 )( δ 2 dx y )12 = (27g 2 3 g3 2 ) = ( 2πi 12 )6 (q 24q q q q 5 + +τ(n)q n + ) We have σ 10 (p) = τ(p), p prime

23 G. van de Geer, Siegel modular forms,

24 Counting holomorphic maps from curves to an elliptic curve 1. Let E be a complex elliptic curve and let p 1,..., p 2g 2 be distinct points of E, where g The set X g (d) of equivalence classes of holomorphic maps φ : C E of degree d from compact connected smooth complex curves C to E, which have only one double ramification point over each point p i E and no other ramification points, is finite. By the Hurwitz formula the genus of C is equal to g. 3. Define F g := 1 q d. Aut (φ) d 1 [φ] X d (d) 4. R. Dijkgraaf, M. Douglas, D. Zagier, M. Kaneko: F g Q[E 2, E 4, E 6 ], where E k = 1 a k g k are the classical Eisenstein series.

25 Number of rational curves on K 3 surfaces 1. K3 surface: simply connected+trivial canonical bundle 2. Projective K 3 surfaces fall into countable many families F k, k N: a surface in F k admits a k-dimensional linear system L of curves of genus k. Let k = n + g, n, g N. 3. Let N n (g) be the number of geometric genus g curves in L passing through g points (so that n is the number of nodes). 4. Yau-Zaslow (1996), Beauville(1999), Göttsche(1994), Bryan-Leung(1999): For generic X we have n 0 N n (g)q n+g 1 = ( 1 24 DE 2) g /, D = q d/dq

26 Number of rational curves on Calabi-Yau threefolds Consider the following family of quintic hypersurfaces in P 4 : M ψ : Q = 0, ψ C Q := x x x x x 5 5 5ψx 1x 2 x 3 x 4 x 5 and the differential form 5 i=1 ω = ( 1)i x i dx 1 dx 2 dx i 1 dx i+1 dx 5 dq The group G = g 1, g 2, g 3, g 4 acts on M ψ, where for instance g 2 : (x 1, x 2, x 3, x 4, x 5 ) (x 1, ζ 5 x 2, x 3, x 4, ζ 4 5x 5 ) Let W ψ be the desingularization of M ψ /G.

27 For smooth W ψ we have dim C H 3 (W ψ, Q) = 4 and the periods ω, δ H 3 (W z, Z) δ satisfy ( ( z d ) ) 4 5z(5z d dz dz + 1)(5z d dz + 2)(5z d dz + 3)(5z d dz + 4) = 0. where z = ψ 5.

28 We can take a basis δ 0, δ 1, δ 2, δ 3 be a basis of H 3 (W ψ, Q) such that the monodromy around z = 0 is given by We introduce a new coordinate around z = 0: q = e 2πi R δ ω R 1 δ ω 0, The expression ( 5 δ W := 1 ω δ 2 ω δ 0 ω δ 3 ω 2 ( δ 0 ω) ( δ 1 ω δ 0 ω )3 is invariant under the monodromy and hence it can be written in terms of the new variable q. )

29 Candelas-de la Ossa-Green-Parkes (1991): ( W = c q 1 q + q q n jj 3 q j ) 1 q j + where c 1 is some constant. Using Mirror symmetry they predicted that n j is the number of rational curves of degree j in a generic quintic in P 4.

30 1. Clemens conjecture: There exits a finite number of rational curves of a fixed degree in a generic quintic in P Gopakumar-Vafa conjecture: The numbers n j s are positive integers. 3. The precise approach to n j s is done by Gromov-Witten invariants. 4. For a literature on this see M. Kontsevich: Homological Algebra of Mirror Symmetry, 1994 and the references within there.