1. Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall
Beilinson-Drinfeld chiral algebra, geometric Langlands program, and open Gromov-Witten invariants
Makoto Sakurai, University of Tokyo, Hongo (School of Science) and Komaba (Graduate School of Mathematical Sciences)
Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall

2. Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall
Main conjecture
The generating function of closed Gromov-Witten invariants for n generic point blowups of CP2 (n=0,1,…,9) is a birational (pseudo)-modular form
We will especially recover the old work of Kontsevich-Manin (1994) and J.Bryan-Leung (1997 on the psudo-modular form) of enumerative geometry of genus 0 for non-toric varieties
Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall

3. Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall
Plan of talk
Main conjecture
Definitions and reviews of past results
Lattice Heisenberg algebra of loops for ADE type Picard group of del Pezzo surfaces
Future problems
Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall

4. Definition of Beilinson-Drinfeld chiral Hecke algebra
It is categorically equivalent to the “factorization algebras” defined globally on the Riemann surface of smooth complex curves
Vertex algebra with only holomorphic part; which is relevant for genus 0, namely when holomorphic anomaly [Bershadsky-Cecotti-Ooguri-Vafa] conjecture of B-model does not occur.
Super-Lie algebraic version exists with quantum deformation [Malikov-Schechtman]
Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall

5. Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall
Relation to the target space and jet space of refined motivic integration
Drinfeld (2003) proposed an analogue of motivic integration (originally by Kontsevich and further developed by Denef-Loeser) of Kapranov-Vasserot, which relates the global gluing of germ of formal arc space of holomorphic maps and vertex algebras. This is the disk amplitude.
Arkhipov and Kapranov (2004) applied this method to compute the genus 0 small quantum cohomology of toric Fano varieties. In terms of A: 2nd homology class and its dominant cone
Algebraically, it is described by the representable functor of formally smooth ind-scheme with toric action S and exceptional locus D
Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall

6. Special case by Hitchin Hamiltonian
Let P be a principal G-bundle over a Riemann surface Σ.BunG and Hitchin fibration;
ADE type Hitchin system was studied by Diaconescu-Donagi-Pantev, which reproduces the Langlands duality by Donagi-Pantev.
There is a conjecture that the local system of dual group is Fourier-Mukai equivalent to the D-modules on the BunG
Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall

7. The geometric Langlands conjecture
E.Frenkel-Gaitsgory (2005) worked on the local geometric Langlands correspondence and affine Kac-Moody algebras
In the case of Hitchin system, this conjecture is written as Db (D-mod (BunG)) = Db (Loc (LG))
Dual torus fibration would be helpful, but not rigorous from the homological sence. But it is useful in the case of 9 point blowups of 12 nodal curves.
Galois representation (fundamental groups) = Automorphic representation (D-modules) = Chiral algebra (Quantum D-scheme)
Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall

8. Definition of open Gromov-Witten invariants (symplectic geometry)
Derived category of Fukaya category [Fukaya-Oh-Ohta-Ono corrections] of Lagrangian submanifolds with Maslov index (disk instanton amplitude) gave us the first standing point to define open Gromov-Witten invariants
It is, however, not sufficient to define the true category with coisotropic submanifolds [Kapustin-Orlov]
[Kapustin-Witten] conjectured that symplectic geometry side (“A-model”) is obtained by the geometric Langlands duality of dual torus fibration from algebraic geometry side (“B-model”).
Open / closed duality, which was studied in the context of matrix models of B-model, is now likely to be described by the homological mirror symmetry. We did not use type IIA / heterotic correspondence. (Still open problem of definition on heterotic model)
Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall

9. Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall
ADE type Picard groups generate loop groups and lattice Heisenberg algebras
Picard groups of del Pezzo surfaces generate (almost) reductive groups, which will make the Picard groupoids of loop Grassmann
This is the so-called lattice Heisenberg algebra. Its relation to the Heisenberg algebra of Eguchi-Kanno is still not understood. Homological Mirror Symmetry of Auroux-Katzarkov-Orlov will be helpful.
Naïve picture
Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall

10. Some comments and supporting facts of the main conjecture
The motivic integration of jet scheme on the target space induces vertex algebras even the case of non-toric varieties.
We can conclude its vertex algebra has the factorization algebra = chiral algebra structure.
If we can prove the D-module over ADE Hitchin system is the same as motivic integration of blowups of CP2, we can conclude the closed Gromov-Witten invariants are automorphic forms.
If we further prove the coherent shaves of blowups of CP2 are dual to the OX module of geometric Langlands dual, we get the open Gromov-Witten invariants of the derived Fukaya category.
Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall

11. Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall
Future problems
Explicit expression of modular forms of partition functions of topological strings
The distinction between modified T-duality of Strominger-Yau-Zaslow by Arinkin-Polishchuk for S-duality and ordinary T-duality
Better understanding on the definition of “heterotic model” by algebraic analysis and algebraic geometry. The analytic continuation of Kaehler parameters should be realized by the Fourier transformation = analogue of Tate’s Fourier transformation of automorphic representation
Sheraton Waikiki Hotel, Oct 31th 2006, JPS Fall