Quivers with analytic potentials

TL;DR

This paper develops the theory of quivers with analytic potentials, establishing foundational calculus results, proving finite determinacy, and exploring Donaldson-Thomas invariants and their transformations under mutations.

Contribution

It introduces the differential calculus for analytic potentials, proves key theorems like inverse function and Moser's trick, and applies these to DT theory and cluster algebra transformations.

Findings

Established the foundations of differential calculus for analytic potentials.

Proved finite determinacy and a Mather-Yau type theorem for these potentials.

Constructed a perverse sheaf of vanishing cycles and derived transformation formulas for DT invariants.

Abstract

Given a quiver $Q$, a formal potential is called analytic if its coefficients are bounded by the terms of a geometric series. As shown by Toda, the potentials appearing in the deformation theory of complexes of coherent sheaves on complex projective Calabi-Yau threefolds are analytic. Our paper consists of two parts. In the first part, we establish the foundations of the differential calculus of quivers with analytic potentials and prove two fundamental results: the inverse function theorem and Moser's trick. As an application, we prove finite determinacy of analytic potentials with finite-dimensional Jacobi algebra, answering a question of Ben Davison. We also prove a Mather-Yau type theorem for analytic potentials with finite-dimensional Jacobi algebra, extending previous work by the first author with Zhou. In the second part, we study Donaldson-Thomas theory of quivers with analytic potentials. First, we construct a canonical perverse sheaf of vanishing cycles on the moduli stack of finite dimensional modules over the Jacobi algebra of an arbitrary iterated mutation via a separation lemma for analytic potentials. Finally, we provide a transformation formula for DT invariants (weighted by the Behrend function) under iterated mutations as a counterpart of Nagao's result on topological DT invariants. Our result leads to a perverse analogue of the $F$-series in cluster algebras. It also yields a Behrend--weighted Caldero--Chapoton formula for what we call anti-cluster algebras.