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

Chenjing Bu

TL;DR

This work develops a motivic framework for counting self-dual objects in self-dual categories, extending classical GL(n) invariants to $O(n)$ and $Sp(2n)$ settings. It builds motivic Hall algebras and motivic Hall modules, together with Euler-form data and extension bundles, to define and analyze self-dual enumerative invariants, including wall-crossing formulas and no-pole theorems. The theory applies to self-dual quivers and to orthogonal/symplectic vector bundles and coherent-sheaf settings, with explicit algorithms and examples illustrating computations of invariants and their numerical (Euler-characteristic) realizations. The results establish a coherent structure for self-dual Donaldson--Thomas-type invariants and set the stage for future refinements, including Behrend-weighted DT invariants and broader geometric applications. Overall, the paper provides a comprehensive motivic toolkit for type B/C/D enumerative invariants, connecting categorical stability, filtrations, and duality to concrete computable invariants and wall-crossing behavior.

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.

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

TL;DR

This work develops a motivic framework for counting self-dual objects in self-dual categories, extending classical GL(n) invariants to and settings. It builds motivic Hall algebras and motivic Hall modules, together with Euler-form data and extension bundles, to define and analyze self-dual enumerative invariants, including wall-crossing formulas and no-pole theorems. The theory applies to self-dual quivers and to orthogonal/symplectic vector bundles and coherent-sheaf settings, with explicit algorithms and examples illustrating computations of invariants and their numerical (Euler-characteristic) realizations. The results establish a coherent structure for self-dual Donaldson--Thomas-type invariants and set the stage for future refinements, including Behrend-weighted DT invariants and broader geometric applications. Overall, the paper provides a comprehensive motivic toolkit for type B/C/D enumerative invariants, connecting categorical stability, filtrations, and duality to concrete computable invariants and wall-crossing behavior.

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 , and our theory is an extension of this to structure groups and . 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.
Paper Structure (177 sections, 42 theorems, 432 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 177 sections, 42 theorems, 432 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Acknowledgements.
  3. Notations and conventions.
  4. List of conditions.
  5. Exact categories and moduli stacks
  6. Exact categories
  7. Relation to abelian categories.
  8. The Grothendieck monoid.
  9. Quasi-abelian categories
  10. Relation to abelian categories.
  11. Categories of filtrations
  12. The class monoid.
  13. Projection functors.
  14. Stability conditions
  15. Remark.
  16. ...and 162 more sections

Key Result

Lemma 2.2.4

Let $\mathcal{A}$ be a $\mathbb{K}$-linear quasi-abelian category, $E \in \mathcal{A}$ an object, $n \geqslant 1$ an integer, and $\iota \colon \mathbb{K}^n \hookrightarrow \mathrm{End} (E)$ an inclusion of $\mathbb{K}$-algebras, where $\mathbb{K}^n$ has component-wise multiplication. Then, there ex up to isomorphisms, such that for each $i$, the $i$-th basis element $e_i \in \mathbb{K}^n$ acts on

Figures (1)

  • Figure 6.1: The quiver-induced dual operation

Theorems & Definitions (134)

  • Remark 2.1.5
  • Definition 2.2.2
  • Lemma 2.2.4
  • Proof 1
  • Definition 2.3.1
  • Theorem 2.4.3
  • Proof 2
  • Theorem 2.4.4
  • Proof 3
  • Remark 2.6.6
  • ...and 124 more