Non-commutative Donaldson–Thomas theory and vertex operators

Abstract

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., Birkhauser (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.