Aiming to understand the relation to other “refined invariants”, and especially their possible interpretation in quantum theory, we explain how to obtain a quadratic, A¹-version of Donaldson-Thomas invariants from the motivic refinements first introduced in Kontsevich-Soibelman. Following ideas from Behrend, Bryan and Szendroi, we provide predictions for these invariants in a few simple examples, mainly the computation of DT invariants of A³.

Our main goal is to draw relationships with the literature, including works of Levine, Denef and Loser, Azouri, Pepin-Lehaulleur, Srinivas among others. We begin with a brief introduction to A¹-enumerative geometry and in the end, we pose some further questions on possible extensions of our definitions. This is joint work with Johannes Walcher. We also comment on joint work in progress with Ran Azouri.