BeilinsonDrinfeld chiral algebra geometric Langlands program and open

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 31 th 2006, JPS Fall 1

Main conjecture n n The generating function of closed Gromov-Witten invariants for n generic point blowups of CP 2 (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 31 th 2006, JPS Fall 2

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

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 31 th 2006, JPS Fall 4

Relation to the target space and jet space of refined motivic integration n 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: 2 nd 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 31 th 2006, JPS Fall 5

Special case by Hitchin Hamiltonian n Let P be a principal G-bundle over a Riemann surface Σ. Bun. G 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 Bun. G Sheraton Waikiki Hotel, Oct 31 th 2006, JPS Fall 6

The geometric Langlands conjecture n n 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 (Dmod (Bun. G)) = 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 31 th 2006, JPS Fall 7

Definition of open Gromov-Witten invariants (symplectic geometry) n n Derived category of Fukaya category [Fukaya-Oh-Ohta-Ono 2000 + 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 (“Amodel”) 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 31 th 2006, JPS Fall 8

n 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 n n 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 31 th 2006, JPS Fall 9

Some comments and supporting facts of the main conjecture n n 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 CP 2, we can conclude the closed Gromov-Witten invariants are automorphic forms. If we further prove the coherent shaves of blowups of CP 2 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 31 th 2006, JPS Fall 10

Future problems n n n 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 31 th 2006, JPS Fall 11