The DT–PT₀ topological vertex correspondence

Fix plane partitions $\boldsymbol{\mu}=\{\mu_a\}_{a=1}^{4}$ corresponding to $(\mathbb C^*)^3$-fixed subschemes of $\mathbb C^3$ of dimension at most one, viewed as asymptotic cross-sections along the four coordinate axes. Let $\mathsf V^{\mathrm{DT}}_{\mu_1\mu_2\mu_3\mu_4}(q)$ and $\mathsf V^{\mathrm{PT}_0}_{\mu_1\mu_2\mu_3\mu_4}(q)$ denote the corresponding $K$-theoretic topological vertices, with square-root choices at fixed points. DT–PT₀ topological vertex correspondence. There exists a choice of $\pm\sqrt{\cdot}$ at the $(\mathbb C^*)^4$-fixed points such that

$$\frac{\mathsf V^{\mathrm{DT}}_{\mu_1\mu_2\mu_3\mu_4}(q)}{\mathsf V^{\mathrm{DT}}_{\varnothing\varnothing\varnothing\varnothing}(q)}=\mathsf V^{\mathrm{PT}_0}_{\mu_1\mu_2\mu_3\mu_4}(q).$$

The numerical version was proved combinatorially by Joyce–Wang–You, and the full $K$-theoretic version was established by Kiem–Li–Toda; hence this conjecture is resolved.