Orthosymplectic Donaldson-Thomas theory
Chenjing Bu
TL;DR
The paper develops a comprehensive orthosymplectic Donaldson–Thomas theory for self‑dual objects in self‑dual 3‑Calabi–Yau linear categories. It constructs epsilon motives and DT invariants within a unified motivic framework based on the motivic Hall algebra and a motivic Hall module, and proves wall‑crossing formulas across stability conditions, including self‑dual variants. The theory extends to self‑dual quivers, orthosymplectic complexes on curves and Calabi–Yau threefolds, and motivic Vafa–Witten invariants on surfaces, with explicit algorithms and examples. This framework provides a robust set of tools for counting orthogonal and symplectic structures in a broad enumerative‑geometric setting and links to string‑theoretic contexts (orientifolds) and curvature phenomena in derived categories.
Abstract
We construct and study Donaldson-Thomas invariants counting orthogonal and symplectic objects in linear categories, which are a generalization of the usual Donaldson-Thomas invariants from the structure groups $\mathrm{GL} (n)$ to the groups $\mathrm{O} (n)$ and $\mathrm{Sp} (2n)$, and a special case of the intrinsic Donaldson-Thomas theory developed by the author, Halpern-Leistner, Ibáñez Núñez, and Kinjo. Our invariants are defined using the motivic Hall algebra and its orthosymplectic analogue, the motivic Hall module. We prove wall-crossing formulae for our invariants, which relate the invariants with respect to different stability conditions. As examples, we define Donaldson-Thomas invariants counting orthogonal and symplectic perfect complexes on a Calabi-Yau threefold, and Donaldson-Thomas invariants counting self-dual representations of a self-dual quiver with potential. In the case of quivers, we compute the invariants explicitly in some cases. We also define a motivic version of Vafa-Witten invariants counting orthogonal and symplectic Higgs complexes on a class of algebraic surfaces.