Counting problems in Number Theory and Physics

Transcription

1 Counting problems in Number Theory and Physics Hossein Movasati IMPA, Instituto de Matemática Pura e Aplicada, Brazil hossein/ Encontro conjunto CBPF-IMPA, 2011

2 A documentary on string theory by Brian Greene:

3 Counting

4 Fibonacci numbers: Counting F n = F n 1 + F n 2, F 0 = 0, F 1 = 1.

5 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).

6 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: F = F n q n = n=0 q 1 q q 2

7 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 q n = n=0 q 1 q q 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)

8 Eisenstein series

9 Eisenstein series E 2k = 1 + ( 1) k 4k σ 2k 1 (n)q n, B k n 1 k = 1, 2, 3, q C, q < 1,

10 Eisenstein series E 2k = 1 + ( 1) k 4k σ 2k 1 (n)q n, B k n 1 k = 1, 2, 3, q C, q < 1, B 1 = 1 6, B 2 = 1 30, B 3 = 1 42,..., σ i (n) := d n d i,

11 1. The theory of modular forms over SL(2, Z): homogeneous polynomials of the ring C[E 4, E 6 ], deg(e 4 ) = 4, deg(e 6 ) = 6.

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

13 1. The theory of modular forms over SL(2, Z): homogeneous polynomials of the ring C[E 4, E 6 ], deg(e 4 ) = 4, deg(e 6 ) = The theory of quasi/differential modular forms over SL(2, Z): homogeneous polynomials of the ring C[E 2, E 4, E 6 ], deg(e 2 ) = 2, deg(e 4 ) = 4, deg(e 6 ) = 6 3. In general, a (quasi) modular form over a subgroup of SL(2, Z) of finite rank is an element in the algebraic closure of C(E 2, E 4, E 6 ).

14 Monstrous moonshine conjecture,

15 The j-function Monstrous moonshine conjecture, E 3 4 j = 1728 E4 3 E 6 2 =

16 The j-function Monstrous moonshine conjecture, E 3 4 j = 1728 E4 3 E 6 2 = q q q q 3 +.

17 The j-function Monstrous moonshine conjecture, E 3 4 j = 1728 E4 3 E 6 2 = q q q q = MacKay 1978: is the number of dimensions in which the Monster group can be most simply represented.

18 The j-function Monstrous moonshine conjecture, E 3 4 j = 1728 E4 3 E 6 2 = q q q q = 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

19 The j-function Monstrous moonshine conjecture, E 3 4 j = 1728 E4 3 E 6 2 = q q q q = 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

20 Monster group

21 Monster group If normal subgroups of a group G are {1} and G then G is called a simple group.

22 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 appear 26 sporadic groups. The Monster group M is the largest of the sporadic groups. M =

23 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 appear 26 sporadic groups. The Monster group M is the largest of the sporadic groups. M = Dimensions of irreducible representations of M: 1, , , , , , , , , , , , , ,

24 Modularity theorem

25 Modularity theorem An elliptic curve over Z: E : y 2 = 4x 3 a 2 x a 3, a 2, a 3 Z, := a2 3 27a2 3 0.

26 An elliptic curve over Z: 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

27 An elliptic curve over Z: 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 modular 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)

28 Example: E : y 2 + y = x 3 x 2

29 Example: E : y 2 + y = x 3 x 2 The corresponding modular form is η(q) 2 η(q 11 ) 12 = q 2q 2 q 3 +2q 4 +q 5 +2q 6 2q 7 2q 9 2q 10 +q 11 2q q 13 +, where

30 Example: E : y 2 + y = x 3 x 2 The corresponding modular form is η(q) 2 η(q 11 ) 12 = q 2q 2 q 3 +2q 4 +q 5 +2q 6 2q 7 2q 9 2q 10 where +q 11 2q q 13 +, η(q) = 1 24 = q 1 24 is the Dedekind eta function and (1 q n ) n=1 = (E 3 4 E 2 6 ).

31 1. Taniyama-Shimura conjecture.

32 1. Taniyama-Shimura conjecture. 2. A. Weils proved for semistable elliptic curves: This was an essential part of the proof of the Fermat last theorem

33 1. Taniyama-Shimura conjecture. 2. A. Weils proved for semistable elliptic curves: This was an essential part of the proof of the Fermat last theorem 3. R. Taylor, C. Breuil, B. Conrad, F.Diamond: Modulartiy theorem

34 Counting holomorphic maps from curves to an elliptic curve

35 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 2.

36 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.

37 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)

38 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 ].

39 For instance, F 3 (q) = F 2 (q) = (10E 3 2 6E 2E 4 4E 6 ), ( 6E E 4 2 E 4 12E 2 2 E E E 3 2 E 6 12E 2 E 4 E 6 + 4E 2 6 ).

40 Number of rational curves on K 3 surfaces

41 Number of rational curves on K 3 surfaces 1. K3 surface: simply connected+trivial canonical bundle

42 Number of rational curves on K 3 surfaces 1. K3 surface: simply connected+trivial canonical bundle 2. In an (n + g)-dimensional linear system L the generic fiber is of genus n + g.

43 Number of rational curves on K 3 surfaces 1. K3 surface: simply connected+trivial canonical bundle 2. In an (n + g)-dimensional linear system L the generic fiber is of genus n + g. 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).

44 Number of rational curves on K 3 surfaces 1. K3 surface: simply connected+trivial canonical bundle 2. In an (n + g)-dimensional linear system L the generic fiber is of genus n + g. 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, L) we have n=0 N n (g)q n = ( 1 E 2 24 q )g 1728q E4 3 E 6 2.

45 For the case g = 0 (counting rational curves):

46 For the case g = 0 (counting rational curves): N n (0)q n = n=0 1728q E 3 4 E 2 6 = q + 324q q q q q 6 + (by definition N 0 (0) = 1).

47 For the case g = 0 (counting rational curves): N n (0)q n = n=0 1728q E 3 4 E 2 6 = q + 324q q q q q 6 + (by definition N 0 (0) = 1). For instance, a smooth quadric X in P 3 is K 3 and for such a generic X the number of planes tangent to X in three points is 3200.

48 Beyond classical modular forms and elliptic curves?!?

49 Clemens conjecture:

50 Clemens conjecture: There exits a finite number of rational curves of a fixed degree in a generic quintic in P 4.

51 Candelas, de la Ossa, Green, Parkes (1991), in the framework of mirror symmetry calculates a quantity Y called Yukawa coupling:

52 Candelas, de la Ossa, Green, Parkes (1991), in the framework of mirror symmetry calculates a quantity Y called Yukawa coupling: Y = q 1 q q q q 3 1 q n dd 3 q d 1 q d +

53 Candelas, de la Ossa, Green, Parkes (1991), in the framework of mirror symmetry calculates a quantity Y called Yukawa coupling: Y = q 1 q q q q 3 1 q n dd 3 q d 1 q d + They claimed that n d is the number of rational curves of degree d in a generic quintic in P 4.

54 The main ingredient of the theory of modular forms attached to mirror quintic Calabi-Yau varieties is a particular solution of the differential equation Ra 1 :

55 The main ingredient of the theory of modular forms attached to mirror quintic Calabi-Yau varieties is a particular solution of the differential equation Ra 1 : ṫ 0 = 1 t 5 (3750t t 0t 3 625t 4 ) ṫ 1 = 1 t 5 ( t t 4 0 t t 0 t 4 + t 1 t 3 ) ṫ 2 = 1 t 5 ( t t 5 0 t t 4 0 t t 2 0 t t 1 t 4 + 2t 2 t 3 ) ṫ 3 = 1 t 5 ( t t 5 0 t t 4 0 t t 3 0 t t 2 t 4 + 3t 2 3 ) ṫ 4 = 1 t 5 (15625t 4 0 t 4 + 5t 3 t 4 ) ṫ 5 = 1 t 5 ( 625t 5 0 t t 4 0 t 5 + 2t 3 t t 4 t 6 ) ṫ 6 = 1 t 5 (9375t 4 0 t t 3 0 t 5 2t 2 t 5 + 3t 3 t 6 )

56 q-expansion:

57 q-expansion: Take ṫ = 5q t q and write each t i as a formal power series in q, t i = n=0 t i,nq n and substitute in the above differential equation. We see that it determines all the coefficients t i,n uniquely with the initial values: t 0,0 = 1 5, t 0,1 = 24, t 4,0 = 0, t 5,0 0

58 1 24 t 0 = q + 175q q q q q q q q 10 + O(q 11 ) t 1 = q + 930q q q q q q q q q 10 + O(q 11 ) 1 50 t 2 = q q q q q q q q q q 10 + O(q 11 )

59 1 5 t 3 = q q q q q q q q q q 10 + O(q 11 ) t 4 = 0 1q q q q q q q q q q 10 + O(q 11 )

60 1 125 t 5 = q + 938q q q q q q q q q 10 + O(q 11 ) t 6 =

61 1 125 t 5 = q + 938q q q q q q q q q 10 + O(q 11 ) t 6 = Conjecture All q-expansions of 1 24 t , t , 1 50 t , 1 5 t 3 6 5, t 4, t , have positive integer coefficients.

62 1. We get the Yukawa coupling calculated by Candelas, de la Ossa, Green, Parkes (1991): 5 11 (t 4 t 5 0 )2 t 3 5 = q 1 q q 2 1 q q 3 1 q n dd 3 q d 1 q d +

63 1. We get the Yukawa coupling calculated by Candelas, de la Ossa, Green, Parkes (1991): 5 11 (t 4 t 5 0 )2 t 3 5 = q 1 q q 2 1 q q 3 1 q n dd 3 q d 1 q d + 2. Using a result of Yamaguchi and Yau (1994) we get also genus g topological string partition functions.

64 Darboux-Halphen-Ramanujan:

65 Darboux-Halphen-Ramanujan: ṫ 1 = t t 2 Ra 2 : ṫ 2 = 4t 1 t 2 6t 3 ṫ 3 = 6t 1 t t2 2 ṫ = 12q q Write each t i as a formal power series in q, t i = n=0 t i,nq n and substitute in the above differential equation. We see that it determines all the coefficients t i,n uniquely with the initial values: t 1,0 = 1, t 1,1 = 24

66 In fact we have explicit formulas for t i. They are the well-known Eisenstein series: where ( t i = a i E 2i = a i 1 + b i d 2i 1 q d ) 1 q d d=1, i = 1, 2, 3, (1) (b 1, b 2, b 3 ) = ( 24, 240, 504), (a 1, a 2, a 3 ) = (1, 12, 8).

67 Mirror quintic Calabi-Yau varieties:

68 Mirror quintic Calabi-Yau varieties: Let W ψ be the variety obtained by the resolution of singularities of the following quotient: {x P 4 Q = 0}/G, Q = x0 5 + x x x x 4 5 5ψx 0x 1 x 2 x 3 x 4 where G is the group G := {(ζ 1, ζ 2,, ζ 5 ) ζi 5 = 1, ζ 1 ζ 2 ζ 3 ζ 4 ζ 5 = 1} acting in a canonical way.