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In [K Nagao, Refined open non-commutative Donaldson–Thomas theory for small
toric Calabi–Yau 3–folds, Pacific J. Math. (to appear), arXiv:0907.3784], we
introduced a variant of non-commutative Donaldson–Thomas theory in a
combinatorial way, which is related to the topological vertex by a wall-crossing
phenomenon. In this paper, we (1) provide an alternative definition in a geometric
way, (2) show that the two definitions agree with each other and (3) compute
the invariants using the vertex operator method, following [A Okounkov,
N Reshetikhin, C Vafa, Quantum Calabi–Yau and classical crystals, from:
“The unity of mathematics”, Progr. Math., Birkhäuser (2006) 597–618] and
[B Young, Generating functions for colored 3D Young diagrams and the
Donaldson–Thomas invariants of orbifolds, Duke Math. J. 152 (2010) 115–153]. The
stability parameter in the geometric definition determines the order of the
vertex operators and hence we can understand the wall-crossing formula in
non-commutative Donaldson–Thomas theory as the commutator relation of the
vertex operators.
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