Deformed Calabi-Yau completion and its application to DT theory

Abstract: In this talk, we investigate an application of the theory of deformed Calabi--Yau completion to enumerative geometry. The notion of Calabi--Yau completion was first introduced by Keller as a non-commutative analogue of the canonical bundle. In the same paper, he also introduced a deformed version of the Calabi--Yau completion. We will explain that the deformed Calabi--Yau completion is a non-commutative analogue of an affine bundle modeled on the canonical bundle. Combining this observation with a recent work of Bozec--Calaque--Scherotzke, we prove that the moduli space of coherent sheaves on a certain non-compact Calabi--Yau threefold is described as the critical locus inside a smooth moduli space. This description has several applications in Donaldson--Thomas theory including Toda's \chi-independence conjecture of Gopakumar--Vafa invariants for arbitrary local curves. By dimensional reduction, it implies (and extends) Hausel--Thaddeus's cohomological \chi-independence conjecture for Higgs bundles. This talk is based on a joint work with Naruki Masuda and another joint work with Naoki Koseki.