Abstract
We consider wall-crossing phenomena associated to the counting of D2-branes attached to D4-branes wrapping lagrangian cycles in Calabi-Yau manifolds, both from M-theory and matrix model perspective. Firstly, from M-theory viewpoint, we review that open BPS generating functions in various chambers are given by a restriction of the modulus square of the open topological string partition functions. Secondly, we show that these BPS generating functions can be identified with integrands of matrix models, which naturally arise in the free fermion formulation of corresponding crystal models. A parameter specifying a choice of an open BPS chamber has a natural, geometric interpretation in the crystal model. These results extend previously known relations between open topological string amplitudes and matrix models to include chamber dependence.
Similar content being viewed by others
References
R. Gopakumar and C. Vafa, M-theory and topological strings. I, hep-th/9809187 [SPIRES].
R. Gopakumar and C. Vafa, M-theory and topological strings. II, hep-th/9812127 [SPIRES].
H. Ooguri and C. Vafa, Knot invariants and topological strings, Nucl. Phys. B 577 (2000) 419 [hep-th/9912123] [SPIRES].
F. Denef and G.W. Moore, Split states, entropy enigmas, holes and halos, hep-th/0702146 [SPIRES].
M. Kontsevich and Y. Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435.
D.L. Jafferis and G.W. Moore, Wall crossing in local Calabi Yau manifolds, arXiv:0810.4909 [SPIRES].
W.-y. Chuang and D.L. Jafferis, Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions, Commun. Math. Phys. 292 (2009) 285 [arXiv:0810.5072] [SPIRES].
T. Dimofte and S. Gukov, Refined, Motivic and Quantum, Lett. Math. Phys. 91 (2010) 1 [arXiv:0904.1420] [SPIRES].
H. Ooguri and M. Yamazaki, Crystal Melting and Toric Calabi-Yau Manifolds, Commun. Math. Phys. 292 (2009) 179 [arXiv:0811.2801] [SPIRES].
P. Sulkowski, Wall-crossing, free fermions and crystal melting, Commun. Math. Phys. 301 (2011) 517 [arXiv:0910.5485] [SPIRES].
B. Szendroi, Non-commutative Donaldson-Thomas theory and the conifold, Geom. Topol. 12 (2008) 1171 [arXiv:0705.3419] [SPIRES].
B. Young and J . Bryan, Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds, arXiv:0802.3948 [SPIRES].
K. Nagao and H. Nakajima, Counting invariant of perverse coherent sheaves and its wall-crossing, arXiv:0809.2992 [SPIRES].
K. Nagao, Derived categories of small toric Calabi-Yau 3-folds and counting invariants, arXiv:0809.2994.
K. Nagao, Non-commutative Donaldson-Thomas theory and vertex operators, arXiv:0910.5477 [SPIRES].
M. Aganagic, H. Ooguri, C. Vafa and M. Yamazaki, W all Crossing and M-theory, arXiv:0908.1194 [SPIRES].
R. Dijkgraaf and C. Vafa, Matrix models, topological strings and supersymmetric gauge theories, Nucl. Phys. B 644 (2002) 3 [hep-th/0206255] [SPIRES].
M. Aganagic, R. Dijkgraaf, A. Klemm, M. Mariño and C. Vafa, Topological strings and integrable hierarchies, Commun. Math. Phys. 261 (2006) 451 [hep-th/0312085] [SPIRES].
B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, math-ph/0702045 [SPIRES].
V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, Remodeling the B-model, Commun. Math. Phys. 287 (2009) 117 [arXiv:0709.1453] [SPIRES].
H. Ooguri, P. Sulkowski and M. Yamazaki, Wall Crossing As Seen By Matrix Models, arXiv:1005.1293 [SPIRES].
B. Eynard, A Matrix model for plane partitions and TASEP, J. Stat. Mech. (2009) P 10011 [arXiv:0905.0535] [SPIRES].
R. Dijkgraaf, P. Sulkowski and C. Vafa, W all-crossing and open topological string theory, (unpublished).
M. Aganagic and M. Yamazaki, Open BPS Wall Crossing and M-theory, Nucl. Phys. B 834 (2010) 258 [arXiv:0911.5342] [SPIRES].
P. Sulkowski, Refined matrix models from BPS counting, arXiv:1012.3228 [SPIRES].
M. Aganagic and K. Schaeffer, Wall Crossing, Quivers and Crystals, arXiv:1006.2113 [SPIRES].
R.J. Szabo and M. Tierz, Matrix models and stochastic growth in Donaldson-Thomas theory, arXiv:1005.5643 [SPIRES].
K. Nagao, Refined open non-commutative Donaldson-Thomas invariants for small crepant resolutions, arXiv:0907.3784 [SPIRES].
K. Nagao and M. Yamazaki, The Non-commutative Topological Vertex and Wall Crossing Phenomena, arXiv:0910.5479 [SPIRES].
T. Nishinaka and S. Yamaguchi, Wall-crossing of D4-D2-D0 and flop of the conifold, JHEP 09 (2010) 026 [arXiv:1007.2731] [SPIRES].
S. Cecotti and C. Vafa, BPS Wall Crossing and Topological Strings, arXiv:0910.2615 [SPIRES].
N. Saulina and C. Vafa, D-branes as defects in the Calabi-Yau crystal, hep-th/0404246 [SPIRES].
N. Halmagyi, A. Sinkovics and P. Sulkowski, Knot invariants and Calabi-Yau crystals, JHEP 01 (2006) 040 [hep-th/0506230] [SPIRES].
P. Sulkowski, Calabi-Yau crystals in topological string theory, arXiv:0712.2173 [SPIRES].
J. Gomis and T. Okuda, D-branes as a Bubbling Calabi-Yau, JHEP 07 (2007) 005 [arXiv:0704.3080] [SPIRES].
R. Dijkgraaf, C. Vafa and E. Verlinde, M-theory and a topological string duality, hep-th/0602087 [SPIRES].
M. Aganagic and C. Vafa, Mirror symmetry, D-branes and counting holomorphic discs, hep-th/0012041 [SPIRES].
J.M.F. Labastida, M. Mariño and C. Vafa, Knots, links and branes at large-N, JHEP 11 (2000) 007 [hep-th/0010102] [SPIRES].
D. Gaiotto, A. Strominger and X. Yin, New Connections Between 4D and 5D Black Holes, JHEP 02 (2006) 024 [hep-th/0503217] [SPIRES].
M. Aganagic, A. Klemm and C. Vafa, Disk instantons, mirror symmetry and the duality web, Z. Naturforsch. A 57 (2002) 1 [hep-th/0105045] [SPIRES].
M. Mariño and C. Vafa, Framed knots at large-N, hep-th/0108064 [SPIRES].
Author information
Authors and Affiliations
Corresponding author
Additional information
On leave from University of Amsterdam and Soltan Institute for Nuclear Studies Poland.
Rights and permissions
About this article
Cite this article
Sulkowski, P. Wallcrossing, open BPS counting and matrix models. J. High Energ. Phys. 2011, 89 (2011). https://doi.org/10.1007/JHEP03(2011)089
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2011)089