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Topological Strings and Integrable Hierarchies

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Abstract

We consider the topological B-model on local Calabi-Yau geometries. We show how one can solve for the amplitudes by using -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.

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Authors and Affiliations

  1. Department of Physics, University of Washington at Seattle, Seattle, WA, 98195-1560, USA

    Mina Aganagic

  2. Institute for Theoretical Physics & Korteweg-de Vries, Institute for Mathematics, University of Amsterdam, Amsterdam, XE, 1018, The Netherlands

    Robbert Dijkgraaf

  3. Physics Department, University of Wisconsin at Madison, Madison, WI, 53706-1390, USA

    Albrecht Klemm

  4. Theory Division, CERN, 1211 Geneva 23, Switzerland

    Marcos Mariño

  5. Jefferson Physical Laboratory, Harvard University, Cambridge, MA, 02138, USA

    Cumrun Vafa

Authors
  1. Mina Aganagic
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  2. Robbert Dijkgraaf
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  3. Albrecht Klemm
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  4. Marcos Mariño
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  5. Cumrun Vafa
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Communicated by N.A. Nekrasov

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Aganagic, M., Dijkgraaf, R., Klemm, A. et al. Topological Strings and Integrable Hierarchies. Commun. Math. Phys. 261, 451–516 (2006). https://doi.org/10.1007/s00220-005-1448-9

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