Embedded Curves and Gromov-Witten Invariants

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 w and with trivial canonical class.

3. Gromov-Witten theory • M: Calabi-Yau manifold: i.e. complex Kahler manifold with a Kahler form w and with trivial canonical class. • Suppose that b 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 w and with trivial canonical class. • Suppose that b 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 b on M?

5. Gromov-Witten theory • M: Calabi-Yau manifold: i.e. complex Kahler manifold with a Kahler form w and with trivial canonical class. • Suppose that b 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 b on M? • Question 2: If the answer is “Yes”, what is the number N(b,g) of such curves?

6. Example: The Quintic Threefold

7. Example: The Quintic Threefold 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 b of M determines a homology class on the projective space, we call it b as well

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

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

11. Gromov-Witten theory • M: Calabi-Yau manifold: i.e. complex Kahler manifold with a Kahler form w and with trivial canonical class. • Suppose that b 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 b on M? • Question 2: If the answer is “Yes”, what is the number N(b,g) of such curves?

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

13. Answer from Algebraic Geometry • We expect to have a finite number of solutions for each fixed b 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 b. 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 b?

18. Non-compactness • Example:

19. Example for convergence! • xy=e • 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 b

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 b • 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=e

24. A nodal curve of genus 10! • Each node will locally look like xy=e • 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 b • Define the moduli space of curves of genus g representing b 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,b)(l) is the total Gromov-Witten invariant for the homology class b

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

30. Example: The Quintic Threefold If b denotes the generator of the second homology of the projective space, the only holomorphic curves in class b 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 b denotes the generator of the second homology of the projective space, the only holomorphic curves in class b 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 b 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 b 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 S is a curve of genus g obtained from C by attaching extra components (which are disjoint), extend f to S by a constant function on these extra components.

34. Elements of for a line class b • 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 b • 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 l

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 l • 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 b on M define the generating function N(M,b)(l) as discussed.

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

47. Gopakumar-Vafa Conjecture • Associated with any genus h and any non-zero homology class a on M is an integral invariant n(h,a), 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,a) (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.