On motivic and arithmetic refinements of Donaldson-Thomas invariants

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 A3 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.

Keywords

DT invariants, Hilbert scheme

2010 Mathematics Subject Classification

14N35, 81T30

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Received 28 July 2023

Accepted 19 March 2024

Published 7 June 2024