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August 2010 The Non-commutative Topological Vertex and Wall Crossing Phenomena
Kentaro Nagao, Masahito Yamazaki
Adv. Theor. Math. Phys. 14(4): 1147-1181 (August 2010).


We propose a generalization of the topological vertex, which we call the "non-commutative topological vertex." This gives open BPS invariants for a toric Calabi–Yau manifold without compact 4-cycles, where we have $D0/D2/D6$-branes wrapping holomorphic $0/2/6$-cycles, as well as $D2$- branes wrapping disks whose boundaries are on $D4$-branes wrapping noncompact Lagrangian 3-cycles. The vertex is defined combinatorially using the crystal melting model proposed recently, and depends on the value of closed string moduli at infinity. The vertex in one special chamber gives the same answer as that computed by the ordinary topological vertex. We prove an identify expressing the non-commutative topological vertex of a toric Calabi–Yau manifold $X$ as a specialization of the closed BPS partition function of an orbifold of $X$, thus giving a closed expression for our vertex. We also clarify the action of the Weyl group of an affine $A_L$ Lie algebra on chambers, and comment on the generalization of our results to the case of refined BPS invariants.


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Kentaro Nagao. Masahito Yamazaki. "The Non-commutative Topological Vertex and Wall Crossing Phenomena." Adv. Theor. Math. Phys. 14 (4) 1147 - 1181, August 2010.


Published: August 2010
First available in Project Euclid: 10 August 2011

zbMATH: 1229.81250
MathSciNet: MR2821395

Rights: Copyright © 2010 International Press of Boston

Vol.14 • No. 4 • August 2010
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