Categorical Wall-Crossing and Shifted Symplectic Geometry of Derived Representation Stacks

This paper investigates the geometry of derived moduli stacks of representations for Lie algebroids over a smooth projective scheme . We establish a robust framework within the language of derived algebraic geometry to construct the derived representation stack for a given Lie algebroid on . Our primary result demonstrates that if the base scheme is endowed with a Calabi-Yau structure of dimension , then the derived stack is naturally equipped with a -shifted symplectic structure. This construction generalizes and unifies several known results for quiver representations and moduli of objects in Calabi-Yau categories. We then explore the behavior of this geometric structure under variations of stability conditions, in the sense of Bridgeland. We introduce Donaldson-Thomas type invariants by integrating a virtual class over the moduli substacks of semistable objects. The central contribution of this work is the derivation of a categorical wall-crossing formula. This formula provides a precise transformation law for the generating functions of these invariants as the stability parameter crosses a wall of marginal stability. The proof combines techniques from derived deformation theory, the theory of stability conditions on triangulated categories, and the geometry of shifted symplectic stacks. We conclude by illustrating the general theory with an application to the Atiyah algebroid associated with a vector bundle on a Calabi-Yau surface, connecting our results to the Vafa-Witten theory of Higgs bundles.