Low degree motivic Donaldson-Thomas invariants of the three-dimensional projective space

Levine has constructed motivic analogues of virtual fundamental classes, living in cohomology of Witt sheaves. We use this to define motivic Donaldson-Thomas invariants ${\tilde{I}}_n$ for ${\mathbb{P}}^3$ over $\mathbb{R}$. We show that for $n$ odd, ${\tilde{I}}_n=0$ and we compute ${\tilde{I}}_2=10,{\tilde{I}}_4=25$ and ${\tilde{I}}_6=-50$. We then make a conjecture about the general case, which could be a motivic analogue of a classical theorem of Maulik-Nekrasov-Okounkov-Pandharipande. The results presented here also form a chapter in the authors thesis, which was submitted on May 30’th, 2023.