On Motivic and Arithmetic Refinements of Donaldson-Thomas invariants

TL;DR

This paper develops a quadratic refinement of Donaldson-Thomas invariants within a motivic framework, exploring their relation to other refined invariants and their potential quantum interpretations, supported by explicit examples.

Contribution

It introduces a quadratic version of Donaldson-Thomas invariants derived from motivic invariants and investigates their properties through concrete calculations.

Findings

Computed quadratic DT invariants for simple cases like degree zero in a3^3

Compared new invariants with real and complex counts, revealing their relationships

Posed open questions about the interpretation and applications of these refined invariants

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.