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, 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/
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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

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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

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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?

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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

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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

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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

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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

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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)

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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

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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

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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

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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

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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

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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

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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