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

TL;DR

This paper defines and computes low-degree motivic Donaldson-Thomas invariants for the three-dimensional projective space over the reals, revealing specific values and proposing a conjecture for the general case.

Contribution

It introduces motivic Donaldson-Thomas invariants for $ ext{P}^3$ over $ ext{R}$ and computes explicit low-degree cases, proposing a new conjecture in the field.

Findings

$ ilde{I}_n=0$ for odd $n$

Computed $ ilde{I}_2=10$, $ ilde{I}_4=25$, $ ilde{I}_6=-50$

Proposed a conjecture for the general case

Abstract

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.