Degree-zero Donaldson–Thomas formula for Hilbert schemes of points

Let $X$ be a smooth 3-fold and let $\mathscr{L}_1,\mathscr{L}_2$ be line bundles on $X$ satisfying

$$\mathscr{L}_1\otimes\mathscr{L}_2=\mathscr{K}_X.$$

Let $\operatorname{Hilb}(X,\mathrm{points})=\bigsqcup_{n\geq 0}\operatorname{Hilb}(X,n)$ be the Hilbert scheme of points, and let $\widetilde{\mathscr{O}}_{\mathrm{vir}}$ be the specified virtual structure sheaf. Degree-zero Donaldson–Thomas conjecture.

$$\chi\left(\operatorname{Hilb}(X,\mathrm{points}),\widetilde{\mathscr{O}}_{\mathrm{vir}}\right)=\mathsf{S}^{\bullet}\chi\left(X,\frac{q\mathscr{L}_1\left(TX+\mathscr{K}_X-T^*X-\mathscr{K}_X^*\right)}{(1-q\mathscr{L}_1)(1-q\mathscr{L}_2^{-1})}\right).$$

This is presented as a reformulation of a conjecture from the cited work on degree-zero Donaldson–Thomas invariants. It predicts a closed symmetric-algebra expression for the K-theoretic invariant of points on $X$, but is not proved in the supplied text.