Enumerative invariants in self-dual categories. I. Motivic invariants

TL;DR

This paper introduces a motivic framework for enumerative invariants counting self-dual objects in categories with orthogonal and symplectic structures, extending classical invariants and providing computational tools.

Contribution

It develops a motivic approach to self-dual invariants, including wall-crossing formulas and algorithms for quiver representations, expanding the scope of enumerative geometry.

Findings

Defined invariants as elements in a ring of motives.

Proved wall-crossing formulas for different stability conditions.

Provided an explicit algorithm and numerical results for quiver invariants.

Abstract

In this series of papers, we propose a theory of enumerative invariants counting self-dual objects in self-dual categories. Ordinary enumerative invariants in abelian categories can be seen as invariants for the structure group $\mathrm{GL} (n)$, and our theory is an extension of this to structure groups $\mathrm{O} (n)$ and $\mathrm{Sp} (2n)$. Examples of our invariants include invariants counting principal orthogonal or symplectic bundles, and invariants counting self-dual quiver representations. In the present paper, we take the motivic approach, and define our invariants as elements in a ring of motives. We also extract numerical invariants by taking Euler characteristics of these elements. We prove wall-crossing formulae relating our invariants for different stability conditions. We also provide an explicit algorithm computing invariants for quiver representations, and we present some numerical results.