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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, Univesity of Tokyo, School of
Science, Department of Physics, Eguchi Lab
(makoto_at_hep-th.phys.s.u-tokyo.ac.jp
) http//www5f.biglobe.ne.jp/makotosakurai/
2
Table of contents
  • Motivation and backgrounds
  • General theory
  • Explicit calculation chiral algebras for del
    Pezzo surfaces
  • Extension of topological M-theory G2 holonomy
    construction
  • Conclusion and future direction

3
a)Motivation 1String / Gauge duality
  • Better understanding on the Chern-Simons /
    Seiberg-Witten duality
  • Worldsheet instanton for (0,2) heterotic sigma
    model / Beilinson-Drinfeld chiral algebra

4
Motivation 2Infinite analysis and String theory
  • Folklore Infinite Heisenberg algebra is the
    representation of loop groups LG Hitchin-Segal /
    Segal-Pressley 80s Counterpart of anomaly
    cancellation / elliptic genus W
  • Study on the loop groups inspired Malliavin
    stochastic analysis
  • Loop space Morse-Floer in the Atiyah-Bott-Witten
    localization theorem of A-model (symplectic)
    What is the infinite algebraic geometry?

5
Motivation 3 Geometry and Arithmetic
  • Quantum integrability of topological vertex vs
    Virasoro conjecture
  • Loop spaces and motives K-theory of
    infinite-dimensional sheaves D not clear its
    physical meaning
  • Mirror symmetry, quantum geometric Langlands
    (when the target is a gaug e gruop), and S-duality

6
Why Beilinson-Drinfeld chiral algebras?
  • Short history of Beilinson-Drinfeld (80s-)
  • Malikov-SchechtmanMS et.al. coordinate
    dependent, mimicking chiral rings
  • Kapranov-Vaserot AK A)Coordinate independent
    loops / motives B)Quantum cohomology for toric
    Fanos
  • Beilinson-Drinfeld BD Sheaf-theoretical beyond
    Cech,tensor categories, and mirror symmetry of
    D-modules / geometric Langlands Still difficult
    to do things from first principle
  • Representation theory of affine Kac-Moodys not
    essential for SCFT loop spaces are intrinsic
    define topological observables
  • Whats new in my work S Sakruai
  • Unify stringy topological invariants by infinite
    algebro-geometry
  • Chiral algebras for higher non-toric del Pezzo
    surfaces

7
b)General theoryWarmup by G/B and definitions
and reviews MS
  • G/B by loop groups LG. HQ quantum cohomology
    ring
  • , calculation by affine covers and loop
    space Exceptional locus by the toric action
    ,where ?0M
    is the loop space that respects the complex
    structure

8
Definition of Hitchin system / 2d Yang-Mills
theory (after Hitchin)
  • Let P be a principal G-bundle over a Riemann
    surface S, which satisfies self-duality
    equations
  • It is also descibed as the representation of
    fundamental group p1(S) in the gauge group G
  • Affine curve S is the WZW model (flag manifolds)
    L

9
Disk amplitude and 2 dim YM / SUSY Poisson
sigma-model AKMSSS2 My 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
  • M not necessarily toric, L0M demands refined
    motivic integrationD
  • 2D YM q-deformed of free fermion is the section
    at affine coordinate / germ or curve (Laurant
    expansion at a point)

10
c)Explicit calculation of sheaves of chiral
primaries Sakurai new work
  • Del Pezzo surfaces (k0,...,8)
  • Toric del Pezzo for
  • Non-toric for k gt 3
  • Degree k surface inwith canonical sheaf
    for
  • Reproduces classics Eguchi-Hori-Xiong, but
    different principles of loop spaces and
    localization of loops (not virtual localization
    of A-model)
  • Pull-buck of the homology classes of target space
    M
  • Not previously done by mathematicians, bacause
    their mathods were only in the toric Fano cases

11
Future works in this directions
  • Better definition of all-genus Gromov-Witten
    invariants / the topological vertex DDDHP, but
    we didnt yet derive from the first principle of
    motives
  • Open-closed duality should explain why the
    K-theory of Drinfeld D reproduces the chiral de
    Rham complex (closed Gromov-Witten)SS2
  • Algebro-geometric / categorical proof of
    geometric transitions without using analytic
    continuation SDDDHP
  • We couldnt calculate from coordinate-free (non
    Cech) sheaf cohomology of Drinfeld, which
    requires more algebro-geometry

12
d)Extension of topological M-theoryS
  • Towards the missing link between 2d Hitchin
    systems ((0,2) heterotic) and the 7d Hitchin
    functional (topological M-theory) From
    Hitchin to Hitchin S2 Analogue of
    mysterious duality?
  • The inconsistency betweenJoyces 7d G2 holonomy
    manifoldsand the 7d SU(3) holonomy solutionsof
    Hitchin flow equation
  • Kovalevs construction of G2 holonomy manifolds
    from 2 Fano 3-folds should be useful it could be
    the completion of the CY3 by the Landau-Ginzburg
    phase S

13
Adding Fano 3-folds to M-theory
  • Twisted connected sum Kovalev of Fano 3-folds
    Mukai produces strictly G2 holonomy
    manifolds with asymptotically CY3 cylinder
    boundaries which could be the initial conditions
    for Hitchin flow equation, which should be
    modified
  • It is also preferable from the LG / Fano B-model
    duality Orlov 2005

14
e)Conclusion and future direction
  • Better understanding on the mathematical
    principles of topological strings / M-theory
    quantum Hitchin systems
  • Beautiful theory of infinite dimensional geometry
    reproduces our past results However,
  • Explicit calculation was difficult to perform
    more general target spaces are awaiting for our
    challenges to ease its difficulty

15
References
  • ADE A.Adams,J.Distler,M.ErnebjergTopological
    Heterotic Rings,hep-th/0506263
  • AK S.Arkhipov and M.Kapranov Toric arc
    schemes and quantum cohomology of toric
    varieties, math.AG/0410054
  • BD A.Beilinson and V.Drinfeld Chiral
    algebras, AMS (2004)
  • D V.Drinfeld Infinite-dimensional vector
    bundles in algebraic geometry (an introduction)
    , math.AG/0309155
  • DDDHPDiaconescu,Dijkgraaf,Donagi,Hofman,Pantev
    Geometric transitions and integrable systems,
    hep-th/0506196
  • F Edward Frenkel Mirror symmetry in two
    steps A-I-B, hep-th/0505131

16
References 2
  • L Y.Laszlo Hitchins and WZW connections are
    the same, Journal of Differential Geometry, 49
    (1998) 547-576
  • MS F.Malikov, V.Schechtman "Deformations of
    vertex algebras, quantum cohomology of toric
    varieties, and elliptic genus", CMP 234 (2003),
    no. 1, 77-100
  • S Makoto Sakurai Moduli space of topological
    M-theory and topological chiral algebras, to
    appear
  • S2 Makoto Sakurai Presentations at the Japan
    Physical Society, Sep 2004 Topological vertex
    and geometric transition via Beilinson-Drinfeld
    chiral algebras, Mar 2005 Mathematical
    principles of topological strings / M-theory and
    Hitchin systems
  • W Edward Witten Two-Dimensional Models With
    (0,2) Supersymmety Perturbative Aspects,
    hep-th/0504078
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