The equivariant DT–PT₀ correspondence

Let $X$ be a Calabi–Yau 4-fold with an algebraic torus $T$ acting while preserving the Calabi–Yau volume form, and let $L$ be a $T$-equivariant line bundle on $X$. For $\gamma\in H_c^4(X,\mathbb Q)$ and $\beta\in H_c^6(X,\mathbb Q)$, suppose that $\curly P_{\gamma,\beta,n}^{(q)}(X)^T$ consists of isolated reduced points for every $n$ and $q\in\{-1,0\}$. Equivariant DT–PT₀ correspondence. There are choices of square roots at all $T$-fixed points such that

$$\frac{\sum_n\langle\!\langle L\rangle\!\rangle_{X,\gamma,\beta,n}^{\mathrm{DT}}q^n}{\sum_n\langle\!\langle L\rangle\!\rangle_{X,0,0,n}^{\mathrm{DT}}q^n}=\sum_n\langle\!\langle L\rangle\!\rangle_{X,\gamma,\beta,n}^{\mathrm{PT}_0}q^n$$

as an identity of Laurent series in $q$ with coefficients in $\mathbb Q(t^{1/2},y^{1/2})$. This is the equivariant refinement of the DT–PT₀ correspondence under isolated-reduced-fixed-point hypotheses; no general resolution is stated.