Hyperbolic localization in Donaldson-Thomas theory

Abstract

In this paper we prove a toric localization formula in the cohomological Donaldson-Thomas theory. Consider a (-1)-shifted symplectic algebraic space with a $\mathbb{G}_m$-action leaving the (-1)-shifted symplectic form invariant (typical examples are the moduli space of stable sheaves or complexes of sheaves on a Calabi-Yau threefold with a $\mathbb{G}_m$-invariant Calabi-Yau form or the intersection of two $\mathbb{G}_m$-invariant Lagrangians in a symplectic space with a $\mathbb{G}_m$-invariant symplectic form). In this case we express the restriction of the Donaldson-Thomas perverse sheaf (or monodromic mixed Hodge module) defined by Joyce et al. to the attracting variety as a sum of cohomological shifts of the DT perverse sheaves on the $\mathbb{G}_m$-fixed components. This result can be seen as a (-1)-shifted version of the Bialynicki-Birula decomposition for smooth schemes. We obtain our result from a similar formula for stacks and Halpern-Leistner's Theta-correspondence, at the level of perverse Nori motives, which we use also to derive foundational constructions in DT theory, in particular the Kontsevich-Soibelman wall crossing formula and the construction of the Cohomological Hall Algebra for smooth projective Calabi-Yau threefolds (a similar construction of the CoHA was also done independently by Kinjo, Park, and Safronov in a recent work). This paper subsumes the previous paper "Hyperbolic localization of the Donaldson-Thomas sheaf" from the same author.