Communications in Mathematical Physics

, Volume 261, Issue 2, pp 451–516 | Cite as

Topological Strings and Integrable Hierarchies

  • Mina Aganagic
  • Robbert Dijkgraaf
  • Albrecht Klemm
  • Marcos Mariño
  • Cumrun Vafa


We consider the topological B-model on local Calabi-Yau geometries. We show how one can solve for the amplitudes by using Open image in new window -algebra symmetries which encode the symmetries of holomorphic diffeomorphisms of the Calabi-Yau. In the highly effective fermionic/brane formulation this leads to a free fermion description of the amplitudes. Furthermore we argue that topological strings on Calabi-Yau geometries provide a unifying picture connecting non-critical (super)strings, integrable hierarchies, and various matrix models. In particular we show how the ordinary matrix model, the double scaling limit of matrix models, and Kontsevich-like matrix model are all related and arise from studying branes in specific local Calabi-Yau three-folds. We also show how an A-model topological string on P1 and local toric threefolds (and in particular the topological vertex) can be realized and solved as B-model topological string amplitudes on a Calabi-Yau manifold.


Manifold Matrix Model Topological String Algebra Symmetry Free Fermion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Mina Aganagic
    • 1
  • Robbert Dijkgraaf
    • 2
  • Albrecht Klemm
    • 3
  • Marcos Mariño
    • 4
  • Cumrun Vafa
    • 5
  1. 1.Department of PhysicsUniversity of Washington at SeattleSeattleUSA
  2. 2.Institute for Theoretical Physics & Korteweg-de Vries, Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands
  3. 3.Physics DepartmentUniversity of Wisconsin at MadisonMadisonUSA
  4. 4.Theory DivisionCERNSwitzerland
  5. 5.Jefferson Physical LaboratoryHarvard UniversityCambridgeUSA

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