The ADE instanton quantum field theory action conjecture

Let $X_M$ be the resolution of the $A_{M-1}$ singularity, and let $\mathscr O_{c,\lambda_1,\dots,\lambda_{M-1}}(X_M)$ be its family of deformation-quantizations. Let $U^{QFT}_\hbar(\mathfrak{gl}_N\otimes\mathscr O_{c,\lambda_1,\dots,\lambda_{M-1}}(X_M))$ be the deformation obtained by Koszul duality from the five-dimensional field theory. Let $X_N$ be the resolution of the $A_{N-1}$ singularity. ADE instanton action conjecture. The algebra

$$U^{QFT}_\hbar(\mathfrak{gl}_N\otimes\mathscr O_{c,\lambda_1,\dots,\lambda_{M-1}}(X_M))$$

acts on the equivariant cohomology of quasi-maps from $\mathbb P^1$ to the moduli of rank-$M$ instantons on $X_N$. The parameters $\hbar$ and $c$ are equivariant parameters for rotations of $\mathbb C\times X_N$ preserving the Calabi–Yau structure, while the $\lambda_i$ rotate the framing at infinity. This conjecture predicts a broader ADE-level action generalizing the preceding categorified Donaldson–Thomas constructions; its validity is left open in the supplied text.