Vertex algebras from divisors on Calabi-Yau threefolds

We construct vertex algebras $\mathbb{V}(Y,S)$ from divisors $S$ on toric Calabi-Yau threefolds $Y$, satisfying conjectures of Gaiotto-Rapcak and Feigin-Gukov, as the kernel of screening operators on lattice vertex algebras determined by the GKM graph of $Y$ and a filtration on $\mathcal{O}_S$. We prove that there are representations of $\mathbb{V}(Y,S)$ on the homology groups of various moduli spaces of coherent sheaves on $Y$ supported on $S$ constructed in a companion paper with Rapcak, defined by certain Hecke modifications of these sheaves along points and curve classes in the divisor $S$. This generalizes the common mathematical formulation of a conjecture of Alday-Gaiotto-Tachikawa, the special case in which $Y=\mathbb{C}^3$ and $S=r[\mathbb{C}^2]$, to toric threefolds and divisors as proposed by Gaiotto-Rapcak. We outline an approach to the general conjecture and prove many special cases and partial results using tools developed in the companion paper, following the proof of the original conjecture by Schiffmann-Vasserot and its generalization to divisors in $\mathbb{C}^3$ by Rapcak-Soibelman-Yang-Zhao. The vertex algebras $\mathbb{V}(Y,S)$ conjecturally include $W$-superalgebras $ W_{f_0,f_1}^\kappa(\mathfrak{gl}_{m|n})$ and genus zero class $\mathcal{S}$ chiral algebras $\mathbb{V}^{\mathcal{S}}_{\text{Gl}_m;f_1,...,f_k}$, each for general nilpotents $f_i$. By definition, this implies the existence of a family of compatible free field realizations of these vertex algebras, relevant to their parabolic induction and inverse quantum Hamiltonian reduction. We prove these conjectures in the examples of lowest non-trivial rank for each case, and outline the proof in general for some cases.