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

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

• Makoto Sakurai, University of Tokyo, Hongo
  (School of Science) and Komaba (Graduate School
  of Mathematical Sciences)

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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 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 0for non-toric varieties

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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 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

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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)
  ofKapranov-Vasserot, whichrelates 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
  homologyclass 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

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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.
• Galois representation (fundamental groups)
  Automorphic representation (D-modules) Chiral
  algebra (Quantum D-scheme)

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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
  (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.
• Naïvepicture

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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.

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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. The analytic continuation of
  Kaehler parameters should be realized by the
  Fourier transformation analogue of Tates
  Fourier transformation of automorphic
  representation