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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Non-commutative Donaldson–Thomas theory and vertex operators

Kentaro Nagao

Geometry & Topology 15 (2011) 1509–1543
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., 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.

Keywords
Donaldson–Thomas theory, wall-crossing, vertex operator
Mathematical Subject Classification 2000
Primary: 14N35
References
Publication
Received: 17 November 2009
Revised: 23 April 2011
Accepted: 3 June 2011
Published: 3 August 2011
Proposed: Jim Bryan
Seconded: Richard Thomas, Simon Donaldson
Authors
Kentaro Nagao
RIMS
Kyoto University
Kyoto 606-8502
Japan