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

Abstract

We use Levine's motivic analogues of virtual fundamental classes to define motivic Donaldson-Thomas invariants ${\tilde{I}}_n$ for $P^3$ over $R$. 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 “is” a motivic analogue of a classical theorem of Maulik-Nekrasov-Okounkov-Pandharipande.