Elliptic dynamical quantum supergroup actions for type A quiver varieties

Let $Q$ be a finite or affine type A quiver with nodes decorated as above, and let $\mathfrak g_Q$ be the Lie superalgebra associated with the corresponding Kac–Dynkin diagram. For a generic stability parameter $\zeta$ and dimension vectors $\mathbf v,\mathbf w\in\mathbb N^{Q_0}$, let $\mathcal M^{\zeta}(\mathbf v,\mathbf w)$ be the corresponding quiver variety. Define

$$\mathsf E_{\mathsf T}(\mathcal M^{\zeta}(\mathbf w))=\bigsqcup_{\mathbf v}\mathsf E_{\mathsf T}(\mathcal M^{\zeta}(\mathbf v,\mathbf w)),$$ $$K_{\mathsf T}(\mathcal M^{\zeta}(\mathbf w))=\bigoplus_{\mathbf v}K_{\mathsf T}(\mathcal M^{\zeta}(\mathbf v,\mathbf w)),$$

and

$$H_{\mathsf T}(\mathcal M^{\zeta}(\mathbf w))=\bigoplus_{\mathbf v}H_{\mathsf T}(\mathcal M^{\zeta}(\mathbf v,\mathbf w)).$$

Type A quiver-variety action conjecture. There exists an action of the elliptic dynamical quantum supergroup $E_{\hbar,\tau}(\mathfrak g_Q)$ on $\mathsf E_{\mathsf T}(\mathcal M^{\zeta}(\mathbf w))$, an action of the quantum affine superalgebra $\mathscr U_{\hbar}(\hat{\mathfrak g}_Q)$ on $K_{\mathsf T}(\mathcal M^{\zeta}(\mathbf w))$, and an action of the super Yangian $\mathsf Y_{\hbar}(\mathfrak g_Q)$ on $H_{\mathsf T}(\mathcal M^{\zeta}(\mathbf w))$. Moreover, all these actions factor through the corresponding Maulik–Okounkov quantum groups constructed via stable envelopes for $\mathcal M^{\zeta}(\mathbf w)$. The conjecture proposes a uniform quantum-group action framework for finite and affine type A quiver varieties, extending the stable-envelope constructions across elliptic, K-theoretic, and cohomological settings.