BeilinsonDrinfeld chiral algebra, geometric Langlands program, and open GromovWitten invariants

1
Beilinson-Drinfeld chiral algebra,
geometricLanglands program, and open
Gromov-Witteninvariants

• Makoto Sakurai, University of Tokyo, Hongo

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Main conjecture

• The generating function of closed Gromov-Witten
  invariants for n generic point blowups of CP2
  (n0,1,,9) is a (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

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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

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Definition of chiral 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

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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 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.

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Special case by Hitchin Hamiltonian

• Let P be a principal G-bundle over a Riemann
  surface S.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

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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.

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Definition of open Gromov-Witten invariants
(symplectic geometry)

• Derived category of Fukaya category
  Fukaya-Oh-Ohta-Ono2000 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)

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ADE type Picard groups generate loop groups and
lattice Heisenberg algebras

• Picard groups of del Pezzo surfaces generate
  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.

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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