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Crystal Melting and Toric Calabi-Yau Manifolds

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Abstract

We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric Calabi-Yau threefold. The three-dimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram, and the chemical bond is dictated by the arrows of the quiver diagram. BPS states are constructed by removing atoms from the crystal. This generalizes the earlier results on the BPS state counting to an arbitrary non-compact toric Calabi-Yau manifold. We point out that a proper understanding of the relation between the topological string theory and the crystal melting involves the wall crossing in the Donaldson-Thomas theory.

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Authors and Affiliations

  1. California Institute of Technology, 452-48, Pasadena, CA, 91125, USA

    Hirosi Ooguri & Masahito Yamazaki

  2. Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba, 277-8586, Japan

    Hirosi Ooguri & Masahito Yamazaki

  3. Department of Physics, University of Tokyo, Hongo 7-3-1, Tokyo, 113-0033, Japan

    Masahito Yamazaki

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  1. Hirosi Ooguri
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  2. Masahito Yamazaki
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Correspondence to Masahito Yamazaki.

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Communicated by N. A. Nekrasov

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Ooguri, H., Yamazaki, M. Crystal Melting and Toric Calabi-Yau Manifolds. Commun. Math. Phys. 292, 179–199 (2009). https://doi.org/10.1007/s00220-009-0836-y

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