The DT–PT virtual structure sheaf localization conjecture

Let $M$ be the relevant moduli space with obstruction theory $cmathbb{E}^\bullet$, equipped with an action of the torus $\mathsf{A}$ and a further $\mathbb{C}^\times$-action, and let $M^{\mathsf{A}}$ and $M^{\mathsf{A}\times\mathbb{C}^\times}$ denote the corresponding fixed loci. Write $\iota\colon M^{\mathsf{A}}\hookrightarrow M$ for the inclusion, let $\widehat{\mathcal{O}}^\mathrm{vir}_M$ be the Oh–Thomas virtual structure sheaf, and let $\widehat{\mathcal{O}}^{\mathrm{vir},\mathrm{BF}}$ and the virtual Euler classes in the formula be the Behrend–Fantechi-style localized classes associated with the induced obstruction theories. A canonical halving is a class $\mathbb{E}_{1/2}$ satisfying

$$\mathbb{E}_{1/2}+\mathbb{E}_{1/2}^\vee=\mathbb{E}^\bullet\big|_{M^{\mathsf{A}\times\mathbb{C}^\times}}.$$

The DT–PT virtual structure sheaf localization conjecture. There exists a canonical halving $\mathbb{E}_{1/2}$, given explicitly in the source's Definition~, such that

$$\left(\iota_*\right)^{-1}\widehat{\mathcal{O}}^\mathrm{vir}_M=\left[\frac{\widehat{\mathcal{O}}^{\mathrm{vir},\mathrm{BF}}_{M^{\mathsf{A}\times\mathbb{C}^\times}}}{\widehat{\mathsf{e}}_{\mathbb{C}^\times}(\mathcal{N}^{\mathrm{vir},\mathrm{BF}})}\frac{\widehat{\mathsf{e}}_{\mathbb{C}^\times}(\mathcal{T}^{\mathrm{vir},\mathrm{BF}}_{M^\mathsf{A}})}{\widehat{\mathsf{e}}_{\mathbb{C}^\times}(\mathbb{E}_{1/2}^{\mathsf{A}\text{-fix}})}\frac{1}{\widehat{\mathsf{e}}_{\mathsf{A}\times\mathbb{C}^\times}(\mathbb{E}_{1/2}^{\mathsf{A}\text{-mov}})}\right]\Bigg|_{\mathsf{A}}.$$

The second fraction is a sign $\pm1$ on each connected component $F\subset M^{\mathsf{A}\times\mathbb{C}^\times}$, and the restriction appearing in the conjectural formula is required to be well-defined. The claim proposes an explicit relation between the Oh–Thomas virtual structure sheaf and the localized Behrend–Fantechi cycle on fixed loci; the corresponding restriction for the Behrend–Fantechi localization is well-defined by properness of $M^{\mathsf{A}}$, while well-definedness of the restriction in the displayed conjectural identity remains part of the conjecture.