Communications in Mathematical Physics

, Volume 301, Issue 2, pp 517–562 | Cite as

Wall-Crossing, Free Fermions and Crystal Melting

  • Piotr SułkowskiEmail author
Open Access


We describe wall-crossing for local, toric Calabi-Yau manifolds without compact four-cycles, in terms of free fermions, vertex operators, and crystal melting. Firstly, to each such manifold we associate two states in the free fermion Hilbert space. The overlap of these states reproduces the BPS partition function corresponding to the non-commutative Donaldson-Thomas invariants, given by the modulus square of the topological string partition function. Secondly, we introduce the wall-crossing operators which represent crossing the walls of marginal stability associated to changes of the B-field through each two-cycle in the manifold. BPS partition functions in non-trivial chambers are given by the expectation values of these operators. Thirdly, we discuss crystal interpretation of such correlators for this whole class of manifolds. We describe evolution of these crystals upon a change of the moduli, and find crystal interpretation of the flop transition and the DT/PT transition. The crystals which we find generalize and unify various other Calabi-Yau crystal models which appeared in literature in recent years.


Partition Function Vertex Operator Topological String Free Fermion Toric Diagram 
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.



I thank Jim Bryan for discussions which inspired this project. I am also grateful to Robbert Dijkgraaf, Albrecht Klemm, Hirosi Ooguri, Yan Soibelman, Balazs Szendroi, Cumrun Vafa, and Masahito Yamazaki for useful conversations. I appreciate the hospitality and inspiring atmosphere of the Focus Week on New Invariants and Wall Crossing organized at IPMU in Tokyo, International Workshop on Mirror Symmetry organized at the University of Bonn, and 7 th Simons Workshop on Mathematics and Physics, as well as the High Energy Theory Group at Harvard University. This research was supported by the DOE grant DE-FG03-92ER40701FG-02, the Humboldt Fellowship, the Foundation for Polish Science, and the European Commission under the Marie-Curie International Outgoing Fellowship Programme. The contents of this publication reflect only the views of the author and not the views of the European Commission.

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Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaUSA
  2. 2.Jefferson Physical LaboratoryHarvard UniversityCambridgeUSA

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