Duality between open GromovWitten invariants and BeilinsonDrinfeld chiral algebras

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Duality between open Gromov-Witten invariants and
Beilinson-Drinfeld chiral algebras

• Makoto Sakurai,
• Hongo, University of Tokyo,
• To appear at arXiv.orgBased on the poster at
  Strings 2005

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Whats new in my work?

• Beilinson-Drinfeld chiral algebras are
  sophisticated and have a good physical or
  mathematicians interpretation at most in the
  case of affine curve Hithchin system
• Is heterotic (0,2) model with non-affine target
  space are gluing of affine quantum-Hitchin
  systems? Generalization of Riemann-Hilbert
  correspondence between chiral algebra (D-modules)
  and perverse sheaves to non-flag varieties
• Disk amplitude interpretation of mine in the
  previous JPS
• Wittens work (2005) is not mathematically
  rigorous with some ansatz Cech and gerbe no
  matter how it is appealing. We are trying to
  reduce assumptions. Also, non-perturbative
  effect
• Kapranovs proof on small quantum cohomology was
  only on toric Fanos we will extend the story to
  non-toric space, so that we will get a compact
  inner space

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Plan of talk

• Review of past work genus 0 closed
  Gromov-Witten by refined motivic integration of
  (0,2) model
• Closed GW by loop space cohomology (chiral de
  Rham complex) concentrate on del Pezzo surfaces
• Open Gromov-Witten invariants (all genus
  extensions) and derived category approaches
• Conclusion and future directions
• Appendix How to deal with the geometric
  Langlands program in physics?

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1.Review of Beilinson-Drinfeld chiral algebras,
infinite-dimensional sheaves, and Gromov-Witten
invariants

• Makoto Sakurai immature trial to better
  understand quantum integrable system structure of
  Gromov-Witten / topological M-theory with the
  help of mathematicians work
• Topological vertex and geometric transitions via
  Beilinson-Drinfeld chiral algebras (JPS,Sep
  2004)
• Mathematical principles of topological strings
  / M-theory and Hitchin systems (JPS,Mar 2005)
• Duality between open Gromov-Witten invariants
  and Beilinson-Drinfeld chiral algebras (Strings
  2005)

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Definition of Hitchin system / 2d Yang-Mills
theory (generalization of Hitchin)

• Let P be a principal G-bundle over a Riemann
  surface Swith genus g, which satisfies
  self-duality equations
• It is also descibed as the representation of
  fundamental group p1(S) in the gauge group
  (reductive group) G
• Affine curve S is the WZW model (flag manifolds)
  LaszloBeilinson-Bernstein

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Warmup by G/B and definitions and reviews
Malikov-SchechtmanArkhipov-Kapranov

• G/B by loop groups LG.
• HQ small quantum cohomology ring
• , calculation by affine covers and loop
  space Exceptional locus by the toric action
  essentially virtual localization technique
  ,where ?0M
  is the loop space that respects the complex
  structure

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Disk amplitude and 2 dim YM / SUSY Poisson
sigma-modelSakurais interpretation

• M toric, L0M loop spaces as the boundary of
  stable / holomorphic maps from D2 to M
• It should be the supersymmetric sigma-model with
  B-field / gerbes on Riemann surface, which
  produces the q-deformation and the
  infinite-dimensional sheaves of n-th derivatives

• M not necessarily toric, L0M demands refined
  motivic integrationDrinfeldKapranov-Vasserot
• 2D YM q-deformed of free fermion is the section
  at affine coordinate / germ or curve (Laurant
  expansion at a point)
• What about the case when we have two poles at
  z0, ? Cylinder?

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2.Chiral algebra to not necessarily toric del
Pezzo surfaces degree 0 ( ),1,,9 (1/2
K3) compared to Orlov et.al.

• Can write the coordinate transformation for the
  low degree del Pezzo surfaces, define all the
  algebras and transformations of chiral primary
  affine beta-gamma CFT (Cech) with normal ordering
• However, the correction term by E.Witten is ad
  hoc to remove the singularity we need better
  understanding on the F gerbe/ obstruction /
  anomaly? Difficult in a straightforward way.

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More on del Pezzo surfaces

• Auroux-Katzarkov-Orlov (2005) open GW (derived
  Fukaya category side) of del Pezzos explain
  more later Bryan-Leung

• Why not noncommutative 2-tori? because the
  chiral de Rham complex defined by formal loop
  space (motivic integration) is easier in
  projective spaces
• Higher del Pezzo surfaces will be the first
  example beyond Kapranovs proof on toric Fanos

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3.Open Gromov-Witten invariants and derived
category approaches

• All genus extensions?
• BCOV holomorphic anomaly equation / Ray-Singer
  torsion not confirmed other than quintic CY3
• Rather, we would prefer the conjecture of
  Dubrovin on the uniqueness of semisimple
  Frobenius manifolds by the small quantum
  cohomology and Virasoro conjecture.
• Analytic continuation of CY3/LG is analytical
  and LG is not differential geometrical
  algebro-geometry?
• Dijkgraaf et.al.(2005) helpful to understand the
  topological vertex by SL(2,C) Hitchin system.
  Deformed / resolved conifolds by M-theory flops
  in derived categories?
• Auroux-Katzarkov-Orlov (2005) A-branes with
  B-field

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Naïve conceptual picture of talk to be
confirmed, extended, or modified
Flop in B-model Vafa was in A-model
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Why derived categories for physicists (not for
mathematical physics)?

• Birational geometry is essential for the stringy
  invariants / topological string amplitudes.(cf.
  Kontsevichs theorem)
• Cech cohomology Witten 2005 for chiral algebras
  depends on the coordinate description the loop
  space and curve / instanton countings are more
  essential
• Were not sure there exists a quasi-isomorphism
  between derived functor cohomology and Cech
  cohomology of infinite-dimensional sheaves
  (chiral de Rham complex)
• These are best described by several derived
  categories

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4.Conclusion and future direction

• Extended the theory of Beilinson-Drinfeld BD
  with some ansatz and physics interpretations
  still not perfect in the 3-fold case
• How to deal with mathematicians S-duality
  so-called geometric Langlands duality by
  Fourier-Mukai. Is it the F1/D2 brane duality (in
  B-model) D1 brane Lagrangian (in A-model)?
• Where is Fourier-Mukai used in Kapranovs
  definition ?
• Contrary to our ordinary F1/D1 brane duality in
  IIB-model. Need AIB 2 step mirror symmetry of
  Edward Frenkel?
• Will confirm Orlovs Fourier-Mukai equivalence
  between derived categories of formal gerbes
  (B-field) and non-commutative deformation (van
  den Bergh) for del Pezzo
• Other integrable system Gromov-Witten invariants
  by BD
• Curve targets Okounkov-Pandharipande
• Chen-Ruan orbifolds GW E.Frenkel-Szczesny

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5. Appendix How to deal with the geometric
Langlands program in physics?

• S-duality in the BD chiral algebras / heterotic
  model Sakurai
• BD should be useful to determine genus 0 part of
  B-model
• Then, non-twisted string theory demands that the
  open-closed duality should be the S-duality. Can
  be checked in the pseudo-modular form of ½ K3
  (del Pezzo 9) surface
• Algebraic cycles should be the local picture
  after the geometric Langlands (Fourier-Mukai) /
  S-duality.
• Seiberg duality would be also helpful Witten
  2005 talk at Stony brook, because it should be
  a gauge theoretical interpretation of quantum
  Hitchin system