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In arXiv:0907.3784, we introduced a variant of non-commutativeDonaldson-Thomas theory in a combinatorial way, which is related withtopological vertex by a wall-crossing phenomenon. In this paper, we (1) providean alternative definition in a geometric way, (2) show that the two definitionsagree with each other and (3) compute the invariants using the vertex operatormethod, following Okounkov-Reshetikhin-Vafa and Young. The stability parameterin the geometric definition determines the order of the vertex operators andhence we can understand the wall-crossing formula in non-commutativeDonaldson-Thomas theory as the commutator relation of the vertex operators.

Non-commutative Donaldson-Thomas theory and vertex operators