Quadratic Donaldson-Thomas invariants for $({\mathbb{P}}^1{)}^3$ and some other smooth proper toric threefolds

Abstract

Using virtual localization in Witt sheaf cohomology, we show that the generating series of quadratic Donaldson-Thomas invariants of $({\mathbb{P}}^1{)}^3$, valued in the Witt ring of $\mathbb{R}$, $W(\mathbb{R})\cong \mathbb{Z}$, is equal to $M(q^2{)}^{-8}$, where $M(q)$ is the MacMahon function. This confirms a modified version of a conjecture of Viergever. We also show that a localized version of this conjecture holds for certain iterated blow-ups of $({\mathbb{P}}^1{)}^3$ and other related smooth proper toric varieties.

In the introduction, we altered our main conjecture slightly, deleting a condition for the precise value of the constant $\epsilon \in \{\pm 1\}$, and added comments about the vanishing of the odd quadratic DT invariants