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Wall Crossing as Seen by Matrix Models

Abstract

The number of BPS bound states of D-branes on a Calabi-Yau manifold depends on two sets of data, the BPS charges and the stability conditions. For D0 and D2-branes bound to a single D6-brane wrapping a Calabi-Yau 3-fold X, both are naturally related to the Kähler moduli space \({{\mathcal M}(X)}\) . We construct unitary one-matrix models which count such BPS states for a class of toric Calabi-Yau manifolds at infinite ’t Hooft coupling. The matrix model for the BPS counting on X turns out to give the topological string partition function for another Calabi-Yau manifold Y, whose Kähler moduli space \({{\mathcal M}(Y)}\) contains two copies of \({{\mathcal M}(X)}\) , one related to the BPS charges and another to the stability conditions. The two sets of data are unified in \({{\mathcal M}(Y)}\) . The matrix models have a number of other interesting features. They compute spectral curves and mirror maps relevant to the remodeling conjecture. For finite ’t Hooft coupling they give rise to yet more general geometry \({\widetilde{Y}}\) containing Y.

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Acknowledgements

We thank Mina Aganagic, Vincento Bouchard, Kentaro Hori, and Yan Soibelman for discussions. H. O. and P. S. thank Hermann Nicolai and the Max-Planck-Institut für Gravitationsphysik for hospitality. Our work is supported in part by the DOE grant DE-FG03-92-ER40701. H. O. and M. Y. are also supported in part by the World Premier International Research Center Initiative of MEXT. H. O. is supported in part by JSPS Grant-in-Aid for Scientific Research (C) 20540256 and by the Humboldt Research Award. P. S. acknowledges the support of the European Commission under the Marie-Curie International Outgoing Fellowship Programme and the Foundation for Polish Science. M. Y. is supported in part by the JSPS Research Fellowship for Young Scientists and the Global COE Program for Physical Science Frontier at the University of Tokyo.

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Correspondence to Piotr Sułkowski.

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On leave from University of Amsterdam and Sołtan Institute for Nuclear Studies, Poland.

Communicated by A. Kapustin

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Ooguri, H., Sułkowski, P. & Yamazaki, M. Wall Crossing as Seen by Matrix Models. Commun. Math. Phys. 307, 429 (2011). https://doi.org/10.1007/s00220-011-1330-x

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Keywords

  • Matrix Model
  • Topological String
  • Hooft Coupling
  • Toric Diagram
  • Wall Crossing