Topological vertex and geometric transitions via BeilinsonDrinfeld topological chiral algebra

1
Topological vertex and geometric transitions via
Beilinson-Drinfeld topological chiral algebra

• Makoto Sakurai
• Particle Theory Group, Hongo, the University of
  Tokyo

2
Table of contents

• Brief review of topological vertex and its
  mathematics
• Vertex operators induced from moduli spaces
  (topological chiral algebra)
• A-model lattice Heisenberg algebra
• B-model W-algebra and BRST reduction
• Relations between two moduli spaces Ziv Ran
  versus Simpson

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Chapter 1Review of topological vertices

• The Topological Vertex (hep-th/0305132)
• Aganagic-Klemm-Mariño-Vafa
• Open / Closed string duality
• Through large Nc limit of Chern-Simons gauge
  theory
• One way to define the open Gromov-Witten
  invariants

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Geometric moduli theory

• Localization theorem localization on the moduli
  space
• Duistermaat-Heckman-Atiyah-Bott-Witten
  (Duistermaat-Heckman Invent. Math. 69 (1982)
  259, Witten hep-th/9204083)
• Instanton calculus of moduli of Yang-Mills
• (closed / open) Gromov-Witten invariants are
  defined in terms of algebraic curves

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Vertex operator interpretations of topological
vertices

• A-model Lattice chiral algebra / Heisenberg
  algebra
• Okounkov-Reshetikhin-Vafa (hep-th/0309208)
• Eguchi-Kanno (hep-th/0312234)
• B-model W-algebra
• Aganagic-Dijkgraaf-Klemm-Mariño-Vafa
  (hep-th/0312085)

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Construction of the moduli space via Heisenberg
algebra I
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Chepter 2First definition of Beilinson-Drinfeld
chiral algebra
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Construction of the moduli space via Heisenberg
algebra II

• E.g. GU(1)n, dual Coxeter number h(G)0, central
  charge1 is constructed
• In this A-model case, non-commutative lattice
  chiral algebras (local Picard ind-scheme) and
  boson-fermion correspondence, Sugawara-constructi
  on is required
• Geometric data is the ?-data(?,?,c), ?GG?Z
  bilinear form of lattice,?superline bundle

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The moduli space via W algebra?

• Quantum Drinfeld-Sokolov (BRST) reduction is
  known

• Still under calculation

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Chapter 3Ran space versus moduli of stable sheaf
/ vector bundles

• Ziv Ran space R(X)
• Horizontal section is the vertex operator
• It consists of finite subset of smooth algebraic
  curve X
• Factorization of finite number of stalks(n
  punctures) over the moduli space R(X)

• It is known that Deligne-Mumford compactification
  is necessary for the construction of the moduli
  space of algebraic curve with genus g and n
  punctures

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Equivalence between factorization algebras and
chiral algebras
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Conclusion and future direction

• We do not realize the global structure of the
  moduli spaceChiral homology?

• Direct relation between the reconstructed moduli
  space of Drinfelds vector bundle and its
  relation to Gelfand-Fuks bundle (Chern-Simons
  theory)