The level-$(0,-|\boldsymbol w|)$ geometric action conjecture

Let $\boldsymbol w=(w_0,\ldots,w_{N-1})$ and let $\mathcal M(\boldsymbol v,\boldsymbol w)$ be the quiver variety of type $A^{(1)}_{N-1}$. Let $\mathcal F^{(0,-1)(k)}_{\boldsymbol u^{(k)}}$ be the indicated level-$(0,-1)$ representations of $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$, let $\mathrm E_\mathrm T(\mathcal M(\boldsymbol v,\boldsymbol w))$ denote equivariant elliptic cohomology, and let $|\boldsymbol\lambda\rangle_{\boldsymbol u}$ and $[\boldsymbol\lambda]$ denote the corresponding tensor-product basis and $\mathrm T$-fixed-point class.

Level-$(0,-|\boldsymbol w|)$ geometric action conjecture. The tensor product representation

$$\mathcal F^{(0,-1)(N-1)}_{\boldsymbol u^{(N-1)}}\widetilde\otimes\cdots\widetilde\otimes\mathcal F^{(0,-1)(0)}_{\boldsymbol u^{(0)}}$$

of $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$ is equivalent to an expected level-$(0,-|\boldsymbol w|)$ geometric action of the same algebra on

$$\bigoplus_{\boldsymbol v}\mathrm E_\mathrm T(\mathcal M(\boldsymbol v,\boldsymbol w)),$$

under the identification of $|\boldsymbol\lambda\rangle_{\boldsymbol u}$ with the $\mathrm T$-fixed-point class $[\boldsymbol\lambda]$.

This conjecture extends the single-framing representation to arbitrary framing vector $\boldsymbol w$. It is presented as following from the preceding conjecture and the vertex-operator construction; no resolution is stated.