Categorical Donaldson–Thomas Theory for Local Surfaces: ℤ/2-Periodic Version

Abstract

Abstract We prove two kinds of $\mathbb {Z}/2$-periodic Koszul duality equivalences for triangulated categories of matrix factorizations associated with $(-1)$-shifted cotangents over quasi-smooth affine derived schemes. We use this result to define $\mathbb {Z}/2$-periodic version of Donaldson–Thomas categories for local surfaces, whose ${\mathbb {C}}^{\ast }$-equivariant version was introduced and developed in the author’s previous paper. We compare $\mathbb {Z}/2$-periodic DT category with the ${\mathbb {C}}^{\ast }$-equivariant one and deduce wall-crossing equivalences of $\mathbb {Z}/2$-periodic DT categories from those of ${\mathbb {C}}^{\ast }$-equivariant DT categories.