1. Embedded Curves and Gromov-Witten Invariants Eaman Eftekhary Harvard University

2. Gromov-Witten theory M: Calabi-Yau manifold: i.e. complex Kahler manifold with a Kahler form  and with trivial canonical class.

3. Gromov-Witten theory M: Calabi-Yau manifold: i.e. complex Kahler manifold with a Kahler form  and with trivial canonical class. Suppose that  is a second homology class on M and g is a fixed genus

4. Gromov-Witten theory M: Calabi-Yau manifold: i.e. complex Kahler manifold with a Kahler form  and with trivial canonical class. Suppose that  is a second homology class on M and g is a fixed genus Question 1: Can we count holomorphic curves of genus g which represent the homology class  on M?

5. Gromov-Witten theory M: Calabi-Yau manifold: i.e. complex Kahler manifold with a Kahler form  and with trivial canonical class. Suppose that  is a second homology class on M and g is a fixed genus Question 1: Can we count holomorphic curves of genus g which represent the homology class  on M? Question 2: If the answer is “Yes”, what is the number N ,g) of such curves?

6. Example: The Quintic Threefold

7. This is a hypersurface that has trivial canonical class

8. Example: The Quintic Threefold This is a hypersurface has trivial canonical class Any homology class  of M determines a homology class on the projective space, we call it  as well

9. Example: The Quintic Threefold This is a hypersurface has trivial canonical class Any homology class  of M determines a homology class on the projective space, we call it  as well If  denotes the generator of the second homology of the projective space, the only holomorphic curves in the class  are rational curves, i.e. g=0

10. Example: The Quintic Threefold This is a hypersurface has trivial canonical class Any homology class  of M determines a homology class on the projective space, we call it  as well If  denotes the generator of the second homology of the projective space, the only holomorphic curves in the class  are rational curves, i.e. g=0 N  0)=2875, and N  g)=0, g=1,2,3,…

11. Gromov-Witten theory M: Calabi-Yau manifold: i.e. complex Kahler manifold with a Kahler form  and with trivial canonical class. Suppose that  is a second homology class on M and g is a fixed genus Question 1: Can we count holomorphic curves of genus g which represent the homology class  on M? Question 2: If the answer is “Yes”, what is the number N ,g) of such curves?

12. Answer from Algebraic Geometry We expect to have a finite number of solutions for each fixed  and g

13. Answer from Algebraic Geometry We expect to have a finite number of solutions for each fixed  and g Our expectations are not generally satisfied

14. Correct way of thinking about this problem: Instead of thinking about holomorphic curves in M think about maps from Riemann surfaces of genus g to M that represent the homology class  Here the surface comes from the moduli space of Riemann surfaces of genus g 

15. New Problems: The “moduli space” of solutions to this problem may have a “wrong” dimension which does not agree with our initial expectation (i.e. zero). In fact the dimension may be positive

16. New Problems: The “moduli space” of solutions to this problem may have a “wrong” dimension which does not agree with our initial expectation (i.e. zero). In fact the dimension may be positive The moduli space can be non-compact. We will need to attach certain boundary components to the moduli space to compactify

17. Non-compactness What is the limit of a sequence of maps with domain in the moduli space of genus g surfaces, representing a homology class 

18. Non-compactness Example :

19. Example for convergence! xy=  xy= 0

20. Adding curves with nodes In order to be fair to the two components of the limiting image, we define the limit of the sequence to be a map with domain C( 0 ) (the map will then be the identity map)!

21. Adding curves with nodes In general, we need to add all the holomorphic maps from a “nodal” curve of genus g to M which represent the homology class 

22. Adding curves with nodes In general, we need to add all the holomorphic maps from a “nodal” curve of genus g to M which represent the homology class  The set of nodal curves compactifies the moduli space of Riemann surfaces of genus g

23. A nodal curve of genus 10! Each node will locally look like xy= 

24. A nodal curve of genus 10! Each node will locally look like xy=  Each sphere component has at least 3 marked points

25. Moduli space of genus g curves in M Let M be a Calabi-Yau manifold as before and fix g and  Define the moduli space of curves of genus g representing  as follows. Note that the maps considered here are all holomorphic

26. Definition of Gromov-Witten Invariants This moduli space will have several components, some of them in the boundary. It is connected and its “expected dimension” is zero. It has the structure of an algebraic stack if M is a smooth projective variety.

27. Definition of Gromov-Witten Invariants This moduli space will have several components, some of them in the boundary. It is connected and its “expected dimension” is zero. It has the structure of an algebraic stack if M is a smooth projective variety. One may construct a “virtual fundamental class” for this moduli space which is a zero- cycle. The virtual fundamental class lives in the Chaw group with rational coefficients.

28. Definition of GW-Invariants One may integrate the function 1 against this virtual fundamental class to obtain the GW- invariants: N(M,  )( ) is the total Gromov-Witten invariant for the homology class 

29. Example: The Quintic Threefold If  denotes the generator of the second homology of the projective space, the only holomorphic curves in class  are rational curves, i.e. g=0

30. Example: The Quintic Threefold If  denotes the generator of the second homology of the projective space, the only holomorphic curves in class  are rational curves, i.e. g=0 Previously we computed N(b,0)=2875, and N(b,g)=0, g=1,2,3,…

31. Example: The Quintic Threefold If  denotes the generator of the second homology of the projective space, the only holomorphic curves in class  are rational curves, i.e. g=0 Previously we computed N(b,0)=2875, and N(b,g)=0, g=1,2,3,… With the new definition:

32. Example: The Quintic; An Explanation All the holomorphic curves of genus g bigger than 0 representing  have a domain in the boundary of the moduli space of curves of genus g.

33. Example: The Quintic; An Explanation All the holomorphic curves of genus g bigger than 0 representing  have a domain in the boundary of the moduli space of curves of genus g. If f is a rational holomorphic curve with domain C and if  is a curve of genus g obtained from C by attaching extra components (which are disjoint), extend f to  by a constant function on these extra components.

34. Elements of for a line class  On the yellow component, f is defined and on the blue ones it is constant

35. Structure of the moduli space of solutions Such a solution is obtained as follows:

36. Structure of the moduli space of solutions Such a solution is obtained as follows: Fix a rational curve (C,f) and k points on C

37. Structure of the moduli space of solutions Such a solution is obtained as follows: Fix a rational curve (C,f) and k points on C Fix k values for the genus h(i), i=1,…,k

38. Structure of the moduli space of solutions Such a solution is obtained as follows: Fix a rational curve (C,f) and k points on C Fix k values for the genus h(i), i=1,…,k Choose elements of the moduli space of Riemann surfaces of genus h(i) with one marked point (you may choose from the boundary of this moduli space)

39. Structure of the moduli space of solutions Fix k values for the genus h(i), i=1,…,k Choose elements of the moduli space of Riemann surfaces of genus h(i) with one marked point (you may choose from the boundary of this moduli space) Glue the curve of genus h(i) to the i-th marked point on C to obtain a surface of genus g=h(1)+h(2)+…+h(k)

40. Elements of for a line class  In this example k=3 and h(1)=h(2)=1 while h(3)=2.

41. Structure of the moduli space of solutions Glue the curve of genus h(i) to the i-th marked point on C to obtain a surface of genus g=h(1)+h(2)+…+h(k) We get a component of associate with (C,f):

42. Structure of the moduli space of solutions We need to integrate an Euler class over these components, which will produce rational numbers

43. Structure of the moduli space of solutions We need to integrate an Euler class over these components, which will produce rational numbers Adding these rational numbers we obtain a generating function in variable

44. Structure of the moduli space of solutions We need to integrate an Euler class over these components, which will produce rational numbers Adding these rational numbers we obtain a generating function in variable The computation of this generating function is possible: We obtain:

45. Gopakumar-Vafa Conjecture Suppose that M is a Calabi-Yau manifold as before and for any homology class  on M define the generating function N(M,  )( ) as discussed.

46. Gopakumar-Vafa Conjecture Suppose that M is a Calabi-Yau manifold as before and for any homology class  on M define the generating function N(M,  )( ) as discussed. Let the total Gromov-Witten generating function N(M)(q, ) be defined via

47. Gopakumar-Vafa Conjecture Associated with any genus h and any non-zero homology class  on M is an integral invariant n(h,  ), called the Gopakumar-Vafa invariant, such that

48. Comments on Gopakumar-Vafa Conjecture The GV-invariants are defined in the physics sense! There is no mathematical definition available for now!

49. Comments on Gopakumar-Vafa Conjecture The GV-invariants are defined in the physics sense! There is no mathematical definition available for now! The previous relation may be taken as a definition for n(h,  ) (Bryan- Pandharipande). It is then necessary to show that they are integer-values.

50. Efforts for proving the conjecture (Bryan-Pandharipande) Introduced a local version of the conjecture, and showed that a Mubious inversion formula gives GV- invariants in terms of GW-invariants.

51. Efforts for proving the conjecture (Bryan-Pandharipande) Introduced a local version of the conjecture, and showed that a Mubious inversion formula gives GV- invariants in terms of GW-invariants. (Hsono-Saito-Takahashi) Formalized the approach of Gopakumar and Vafa in the language of intersection cohomology, but did not succeed in defining invariants of a Calabi- Yau manifold. They also failed to show a correspondence between GV and GW.

52. Efforts for proving the conjecture (Maulik-Nekrasov-Okounkov-Pandharipande) Suggested a similar correspondence between GW-invariants an some other integral invariants of CY-manifolds: the Donaldson- Thomas invariants.

53. Efforts for proving the conjecture (Maulik-Nekrasov-Okounkov-Pandharipande) Suggested a similar correspondence between GW-invariants an some other integral invariants of CY-manifolds: the Donaldson- Thomas invariants. (Peng) Proved the integrality of the expressions suggested by Bryan- Pandharipande for local toric Calabi-Yau three-folds using localization.

54. Answer via Symplectic Geometry Again we fix a genus g and a homology class  on the Calabi-Yau three-fold M

55. Answer via Symplectic Geometry Again we fix a genus g and a homology class  on the Calabi-Yau three-fold M We look at the space of maps f from a surface C of genus g to M which represent the homology class 

56. Answer via Symplectic Geometry Again we fix a genus g and a homology class  on the Calabi-Yau three-fold M We look at the space of maps f from a surface C of genus g to M which represent the homology class  Instead of holomorphicity, we assume

57. Answer via Symplectic Geometry Here v is a perturbation term, which is basically a homomorphism of type (0,1) from the tangent space of a huge projective space P to the tangent space of M.

58. Answer via Symplectic Geometry Here v is a perturbation term, which is basically a homomorphism of type (0,1) from the tangent space of a huge projective space P to the tangent space of M. P is chosen so that a certain good cover of the universal curve of the moduli space of genus g curves is contained in P.

59. Answer via Symplectic Geometry In the case of a Calabi-Yau three-fold, this new equation will have a finite number of solutions, if v is a generic perturbation term.

60. Answer via Symplectic Geometry In the case of a Calabi-Yau three-fold, this new equation will have a finite number of solutions, if v is a generic perturbation term. There is a natural sign associated with any solution of this perturbed equation. The number of solutions, counted with sign, is independent of the particular v.

61. Answer via Symplectic Geometry However, it is not independent of the good branched cover of the universal curve. There are several such good covers and the number of solutions depends on the degree of this covering

62. Answer via Symplectic Geometry However, it is not independent of the good branched cover of the universal curve. There are several such good covers and the number of solutions depends on the degree of this covering The good news is: If we divide by this degree we obtain an invariant of M and the pair (g, 

63. Some History: This construction is made by several people: Ruan-Tian, Fukaya-Ono, Siebert, etc..

64. Some History This construction is made by several people: Ruan-Tian, Fukaya-Ono, Siebert, etc.. Historically this predates the approach via algebraic geometry.

65. Some History This construction is made by several people: Ruan-Tian, Fukaya-Ono, Siebert, etc.. Historically this predates the approach via algebraic geometry. Later Li-Tian, etc., used similar ideas to construct the virtual fundamental class in the context of algebraic geometry

66. Some History Siebert showed that the number of solutions obtained this way, after dividing by the degree of the good cover, gives the GW-invariants obtained in algebraic geometry using the virtual fundamental class.

67. Some History Siebert showed that the number of solutions obtained this way, after dividing by the degree of the good cover, gives the GW-invariants obtained in algebraic geometry using the virtual fundamental class. These invariants were first called Gromov-Witten invariants coupled with gravity (Ruan-Tian).

68. GV-Conjecture from a symplectic view-point! The construction of GW-invariants is designed for a general complex Kahler manifold of arbitrary dimension n. Generally the moduli space of solutions is not even expected to have dimension zero. The expected (or virtual) dimension is computed via

69. GV-Conjecture from a symplectic view-point! After a perturbation a moduli space of solutions of this exact dimension is constructed, which may be compactified by adding strata of codimension>1

70. GV-Conjecture from a symplectic view-point! After a perturbation a moduli space of solution of this exact dimension is constructed, which may be compactified by adding strata of codimension>1 In this general context, GW-theory is a way of computing certain interesting integrals over the moduli space of solutions.

71. GV-Conjecture from a symplectic view-point! When M is Calabi-Yau and n=3 the virtual dimension is zero and many of the details of the construction become easier.

72. GV-Conjecture from a symplectic view-point! When M is Calabi-Yau and n=3 the virtual dimension is zero and many of the details of the construction become easier. Ionel-Parker announced several times (starting 4 years ago) that one may carry out the construction with a perturbation term v that does not depend on its P coordinate, if instead of all maps f, we only count the embeddings.

73. GV-Conjecture from a symplectic view-point! If this is true, one will obtain an honest integral count of solutions which would behave very similar to the GV-invariants n(h,  ) (but are different from them in general).

74. GV-Conjecture from a symplectic view-point! If this is true, one will obtain an honest integral count of solutions which would behave very similar to the GV-invariants n(h,  ) (but are different from them in general). Unfortunately, there seems to be serious technical problems coming from convergence of embeddings to multiple covering maps that may not be ruled out by these types of perturbations. They have too much symmetry!

75. Analytic Construction; Revisited! Instead of holomorphicity, we assume Integral values is obtained if instead of a good cover of the universal curve, we embed the universal curve itself in P. Thus, we do not need to assume total independence from the P-factor for v

76. Technical Difficulties If C is a curve of genus g with non- trivial automorphism group G, then in the universal cover, C/G will appear instead of C. If  denotes the map from C to C/G, then the equation for f will look like

77. Technical Difficulties A sequence of embeddings f(i), i=1,2,… from curves C(i) of genus g to M may potentially converge to a map f from a surface C as above to M, such that f factors through the quotient map 

78. Technical Difficulties Using the decomposition of tangent spaces according to the characters of G, and Kuranishi-method for describing a neighborhood of a map f of this later type in the moduli space of solutions, one may rule out a convergence of f(i):C(i)  M to f:C  C/G  M, if G is an Abelian group

79. Technical Difficulties The Kuranishi method fails if G is non- Abelian. In fact transversality fails in a bad way! When  is a primitive homology class or when  p  where  is primitive and p is a prime number, then G is forced to be trivial or Z/pZ, which is Abelian.

80. Dealing with the difficulties! Let M,g,  be as before.

81. Dealing with the difficulties! Let M,g,  be as before. Consider the moduli space of all pairs (C,p) where C is a curve of genus g and p is a point on C. Denote this moduli space by M (g,1). Let C (g,1) be the universal curve and embed it in some P

82. Dealing with the difficulties! Let M,g,  be as before. Consider the moduli space of all pairs (C,p) where C is a curve of genus g and p is a point on C. Denote this moduli space by M (g,1). Let C (g,1) be the universal curve and embed it in some P Let B be a sub-manifold of M representing the Poincare dual of  where  d  and  is primitive (Assume for now that such a B exists)

83. Dealing with the difficulties! Consider all the maps f:(C,p)  (M,B) where (C,p) is in M (g,1) and for a fixed section v.

84. Dealing with the difficulties! Consider all the maps f:(C,p)  (M,B) where (C,p) is in M (g,1) and for a fixed section v. The expected dimension of the moduli space of solutions is again 0.

85. Dealing with the difficulties! Consider all the maps f:(C,p)  (M,B) where (C,p) is in M (g,1) and for a fixed section v. The expected dimension of the moduli space of solutions is again 0. The automorphism groups in this case are all cyclic!

86. Final Result M,g,  as before

87. Final Result M,g,  as before M (g,1), C (g,1) as above. C (g,1) is embedded in a projective space P

88. Final Result M,g,  as before M (g,1), C (g,1) as above. C (g,1) is embedded in a projective space P v: perturbation term on P M

89. Final Result Theorem(E.) The number m(g,  ) of embeddings f:(C,p)  (M,B) with (C,p) in M (g,1) and is finite for generic v. This number is independent of the particular choice of v and B.

90. Final Result Theorem(E.) When  is a primitive homology class, we have the following relation: