The critical COHA–deformed W-algebra isomorphism for Calabi–Yau threefolds

Let $M$ be a Calabi–Yau threefold, let $H^*(M)$ denote its cohomology, and let $W^+(M)$ be the algebra generated by $T_n(\lambda)$ for $n{\geqslant} 0$ and $\lambda\in H^*(M)$, subject to the relations (b) and (e)–(g) of Definition of the source, with $s_2$ replaced by $c_2(TM)$ and $c_1\Delta_S$ replaced by $\Delta_M$. Let $\mathbf{H}^{\mathrm{crit}}_0(M)=H^*_{\mathrm{crit}}(\mathcal{C}oh^0(M))$ be the critical COHA of the stack of finite-length sheaves on $M$. Critical COHA–deformed W-algebra conjecture. There is an isomorphism

$$\mathbf{H}^{\mathrm{crit}}_0(M)\simeq W^+(M).$$

This predicts that the critical COHA of any Calabi–Yau threefold admits a presentation by the corresponding deformed $W$-algebra, extending the surface result discussed in the paper. The statement is presented as an expectation, and no proof or resolution is supplied here.