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Duality between open GromovWitten invariants and BeilinsonDrinfeld chiral algebras

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Title: Duality between open GromovWitten invariants and BeilinsonDrinfeld chiral algebras


1
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

2
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

3
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?

4
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)

5
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

6
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

7
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?

8
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.

9
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

10
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

11
Naïve conceptual picture of talk to be
confirmed, extended, or modified
Flop in B-model Vafa was in A-model
12
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

13
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

14
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
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