Mathematical principles of topological strings Mtheory and Hitchin systems

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Mathematical principles of topological strings /
M-theory and Hitchin systems

• Makoto Sakurai
• High Energy Theory Group, Hongo, Univ. of Tokyo

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Plan of Talk

• Motivation
• Moduli theory for topological strings and Hitchin
  system
• Topological M-theory and quantization
• Conclusion and Future direction

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1.Motivation

• Topological A / B models are significant for the
  string dualities
• BPS degeneracy of M2 / M5 branes in (non)compact
  G2 manifolds is still elusive
• Hitchin system of stable Higgs bundles behind the
  Seiberg-Witten theory is well-handled both
  physically and algebro-geometrically
• However, quantization of topological M-theory and
  the missing link between the Hitchin system (not
  Chern-Simons) and topological M-theory is not
  available yet.

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2.Invitation to stringy invariants

• Several stringy invariants
• Gromov-Witten Jun Li as generalized Floer
  cohomology (?Seiberg-Witten)
• Hilbert scheme of points NakajimaGoettsche
• Motivic integration on CY or Poisson manifolds
  Kontsevich
• Elliptic cohomology Witten,Eguchi-Sugawara
• K-theory for topological chiral algebra
  Beilinson-Drinfeld / chiral de Rham complex
  Kapranov,Malikov-Schechtman-Vaintrob

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Topological chiral algebra approach

• Algebro-geometric formulation of Hitchin system
  and vertex algebra (CFT / Irrational CFT)
• Available for toric Calabi-Yaus and flag
  manifolds G / B (BBorel subgroup)
• Could be a Deformation-quantization counterpart
  for the motivic integration (path-integral) of
  topological (sigma) A-model
• It is a infinite dimensional vector bundle
  formalism whose fiber has a K-theory.
• Gromov-Witten is a morphism operad that
  assigns every stable maps
• to the cohomology of the target space X
• N2 topological twisted SCFT (chiral ring of
  Lerche-Warner-Vafa)
• Tensor category (operads of Manin) and chiral
  Hopf algebra which should have something to do
  with categorification of Floer cohomology and
  Kovanov cohomology

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Binding motivic integration to the category of
stringy invariants

• Drinfelds (Kontsevich-Voevodsky) motivic
  integration, which is thought to be a Legendre
  transformation of the closed string worldsheets
• Tensor category formalism of Fukaya category of
  Floer cohomology, which is a counterpart of the
  motivic integration (path integral) of
  topological sigma model
• Knizhnik-Zamolodchikov-Behrend equation and
  equivalence of Hitchins moduli and WZW model
  (also generalized to Poisson sigma models?)
• Hausels example of principal GSL(2) bundle over
  algebraic curves

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Relation between loop groups, motivic integration
and Gromov-Witten invariants
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cf) Relations to Elliptic genus
Kawai-Yamada-Yang
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Open-closed duality for Gromov-Witten invariants
to be studied

• Moduli of closed string Riemann surface can be
  described by open side
• Also, by Rastelli (actually Strebels proof), the
  moduli space of open (with boundary namely h
  holes with radii l1, , lh) Riemann surface can
  be reprodeced from closed Riemann surface moduli,
  where we have the intersection cohomology

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3.Topolgical M-theory

• Complex dimension d1,2,3
• Stable Higgs bundle (instantons) d1
• Stable sheaves and Fourier-Mukai for semi-stable
  sheaves on K3 surfaces d2
• Gromov-Witten for CY 3-folds, and topological
  vertex (toric)
• We will introduce the notion of stable forms G in
  stead of G2 metrics. From 6-dimensional
  viewpoints, G2 metrics look as if they were
  metrics on the solutions space (moduli) to the
  Hamiltonian flow equation

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Moduli space of stable objects

• Geometric quantization of stable Higgs bundles
  (world-sheet theory) and topological chiral
  algebra
• Stable sheaves on K3 surfaces
• Both are related to the metric of instanton
  moduli space by Kobayashi-Hitchin correspondence

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Hamiltonian flow and Kovalevs construction
Hitchin flow eq.
Global condition is not satisfied for CY3 folds

• Kovalevs gluing of two Fano 3-folds (Chern class
  c1 (X) gt 0), with a well-known classification a
  decade ago, successfully produces plenty of
  compact G2 holonomy manifolds beyond Joyce.

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Towards improvements of quantum topological
M-theory

• Kovanovs construction produces asymptotically
  Calabi-Yau 3-folds in the cylinder, which could
  be an initial condition for improved Hitchin
  equation for generalized complex structure
• At least, for the 1-loop order, we have to
  incorporate the stable 3-form O to be the sum of
  1, 3, and 5-forms. (Witten topological B model
  and topological M-theory)

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4.Conclusion and Future direction

• Loop group is the universal object in the
  algebraically completely integrable systems.
• Towards the Missing link between (topological)
  chiral algebra and quantized topological M-theory
  (from Hitchin to Hitchin)
• Geometric Langlands duality for Hitchin system of
  complex Lie groups G and G? as Mirror symmetry
  (Tamas Hausel) and duality between Heisenberg
  (A-model) and W-algebra (B-model)
• S-duality and black hole entropy statistics,
  which could not be treated in this presentation,
  are plausibly the next evolution of stringy
  geometry