Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Motivic and Arithmetic Refinements of Donaldson-Thomas invariants

Felipe Espreafico, Johannes Walcher

TL;DR

The paper investigates how arithmetic refinements of Donaldson-Thomas invariants, valued in the Grothendieck-Witt ring $GW(k)$, relate to motivic DT invariants in the Grothendieck ring $\mathcal{M}_k$, using the compactly supported $\mathbb{A}^1$-Euler characteristic $\chi_c^{\mathbb{A}^1}$ to pass between these worlds. It develops the foundational framework: motivic DT invariants via $K_0(\mathrm{Var}(k))$, the motivic nearby class $S_f$, and the virtual class of a critical locus, together with the $\hat{\mu}$-equivariant refinements; and it introduces the $\mathbb{A}^1$-algebra of quadratic forms through $GW(k)$, Chow–Witt theory, and EKL classes for local refinements. The paper then performs two concrete tests: degree-zero DT invariants of $\mathbb{A}^3$ (via the Hilbert scheme as a critical locus) and refined Gopakumar–Vafa invariants at the Castelnuovo bound, deriving explicit motivic and arithmetic formulas that specialize to the classical MacMahon function in the complex case and to symmetric variants in the real case. The results illuminate a tight connection between motivic and arithmetic refinements, suggest a physical interpretation in terms of BPS state counting, and provide a general method to predict arithmetic counts over arbitrary fields from known real/complex data.

Abstract

In recent years, a version of enumerative geometry over arbitrary fields has been developed and studied by Kass-Wickelgren, Levine, and others, in which the counts obtained are not integers but quadratic forms. Aiming to understand the relation to other "refined invariants", and especially their possible interpretation in quantum theory, we explain how to obtain a quadratic version of Donaldson-Thomas invariants from the motivic invariants defined in the work of Kontsevich and Soibelman and pose some questions. We calculate these invariants in a few simple examples that provide standard tests for these questions, including degree zero invariants of $\mathbb A^3$ and higher-genus Gopakumar-Vafa invariants recently studied by Liu and Ruan. The comparison with known real and complex counts plays a central role throughout.

On Motivic and Arithmetic Refinements of Donaldson-Thomas invariants

TL;DR

The paper investigates how arithmetic refinements of Donaldson-Thomas invariants, valued in the Grothendieck-Witt ring , relate to motivic DT invariants in the Grothendieck ring , using the compactly supported -Euler characteristic to pass between these worlds. It develops the foundational framework: motivic DT invariants via , the motivic nearby class , and the virtual class of a critical locus, together with the -equivariant refinements; and it introduces the -algebra of quadratic forms through , Chow–Witt theory, and EKL classes for local refinements. The paper then performs two concrete tests: degree-zero DT invariants of (via the Hilbert scheme as a critical locus) and refined Gopakumar–Vafa invariants at the Castelnuovo bound, deriving explicit motivic and arithmetic formulas that specialize to the classical MacMahon function in the complex case and to symmetric variants in the real case. The results illuminate a tight connection between motivic and arithmetic refinements, suggest a physical interpretation in terms of BPS state counting, and provide a general method to predict arithmetic counts over arbitrary fields from known real/complex data.

Abstract

In recent years, a version of enumerative geometry over arbitrary fields has been developed and studied by Kass-Wickelgren, Levine, and others, in which the counts obtained are not integers but quadratic forms. Aiming to understand the relation to other "refined invariants", and especially their possible interpretation in quantum theory, we explain how to obtain a quadratic version of Donaldson-Thomas invariants from the motivic invariants defined in the work of Kontsevich and Soibelman and pose some questions. We calculate these invariants in a few simple examples that provide standard tests for these questions, including degree zero invariants of and higher-genus Gopakumar-Vafa invariants recently studied by Liu and Ruan. The comparison with known real and complex counts plays a central role throughout.
Paper Structure (14 sections, 6 theorems, 57 equations)

This paper contains 14 sections, 6 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Motivic DT invariants
  3. The Grothendieck ring of varieties
  4. Donaldson-Thomas invariants
  5. The motivic nearby class
  6. Virtual class of a critical locus
  7. A1-Enumerative Geometry
  8. Rings of Quadratic forms
  9. Chow-Witt groups and Euler Characteristics
  10. Local A1-degree and Eisenbud-Khimshiashvili-Levine classes
  11. Degree zero invariants of A3
  12. The Hilbert Scheme of points of the affine space as a critical locus
  13. The virtual classes of the Hilbert Scheme of Points
  14. Real and refined Gopakumar-Vafa invariants at the Castelnuovo bound

Key Result

Proposition 2.4.3

Let $f:X\to \mathbb C$ be a regular morphism on a smooth quasi-projective variety. Let $Z$ be the critical locus of $f$. Assume that there exists an action of a connected complex torus on $X$ such that $f$ is equivariant with respect to a primitive character. If there is a one parameter subgroup $\m where $[X_1]$ is the class of the fibre $f^{-1}(1)$.

Theorems & Definitions (32)

  • Definition 2.1.1
  • Remark 2.1.2
  • Remark 2.1.3
  • Example 2.1.4
  • Definition 2.3.1
  • Definition 2.3.2
  • Definition 2.3.3
  • Example 2.3.4
  • Definition 2.4.1
  • Example 2.4.2
  • ...and 22 more