Affine opers and conformal affine Toda

For $\mathfrak g$ a Kac-Moody algebra of affine type, we show that there is an $\text{Aut}\, \mathcal O$-equivariant identification between $\text{Fun}\,\text{Op}_{\mathfrak g}(D)$, the functions on space ${\mathfrak g}$-opers disc, and $W\subset \pi_0$, intersection kernels screenings inside vacuum Fock module $\pi_0$. This kernel $W$ generated by two states: conformal vector, state $\delta_{-1}\left|0\right>$. We latter endows $\pi_0$ with canonical notion translation $T^{\text{(aff)}}$, use it to define densities in integrals motion classical Conformal Affine Toda field theory. The $\text{Aut}\,\mathcal O$-action defines bundle $\Pi$ over $\mathbb P^1$ fibre product bundles $\Pi \otimes \Omega^j$, where $\Omega^j$ are tensor powers bundle, come endowed one-parameter family holomorphic connections, $\nabla^{\text{(aff)}} - \alpha T^{\text{(aff)}}$, $\alpha\in \mathbb C$. global sections $[\mathbf v_j dt^{j+1} ] \in H^1(\mathbb P^1, \Pi\otimes \Omega^j,\nabla^{\text{(aff)}})$ de Rham cohomology $\nabla^{\mathrm{(aff)}}$. Any choice g}$-Miura oper $\chi$ gives connection $\nabla^{\mathrm{(aff)}}_\chi$ $\Omega^j$. Using coinvariants, map $\mathsf F_\chi$ from \Omega^j$ F_\chi \nabla^{\text{(aff)}} = \nabla^{\text{(aff)}}_\chi \mathsf F_\chi$, so descends well-defined cohomologies. Under this map, classes ]$ sent $H^1(\mathbb \Omega^j,\nabla^{\text{(aff)}}_\chi)$ defined g}$-oper underlying $\chi$.