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 algebra of functions on the space of $\mathfrak{g}$-opers on the disc, and $W\subset {\pi }_0$, the intersection of kernels of screenings inside a vacuum Fock module ${\pi }_0$. This kernel $W$ is generated by two states: a conformal vector, and a state ${\delta }_{-1}\mid 0>$. We show that the latter endows ${\pi }_0$ with a canonical notion of translation $T^{\text{(aff)}}$, and use it to define the densities in ${\pi }_0$ of integrals of motion of classical Conformal Affine Toda field theory. The $\text{Aut}\mathcal{O}$-action defines a bundle $\mathrm{\Pi }$ over ${\mathbb{P}}^1$ with fibre ${\pi }_0$. We show that the product bundles $\mathrm{\Pi }\otimes {\mathrm{\Omega }}^j$, where ${\mathrm{\Omega }}^j$ are tensor powers of the canonical bundle, come endowed with a one-parameter family of holomorphic connections, $\nabla^{$\text{(aff)}$} - \alpha T^{$\text{(aff)}$}$,$\alpha\in \mathbb C$. The integrals of motion of Conformal Affine Toda define global sections $[\mathbf v_j dt^{j+1} ] \in H^1(\mathbb P^1, \Pi\otimes \Omega^j,\nabla^{$\text{(aff)}$})$ of the de Rham cohomology of ${\mathrm{\nabla }}^{(\mathrm{a}\mathrm{f}\mathrm{f})}$. Any choice of $\mathfrak{g}$-Miura oper $\chi$ gives a connection ${\mathrm{\nabla }}_{\chi }^{(\mathrm{a}\mathrm{f}\mathrm{f})}$ on ${\mathrm{\Omega }}^j$. Using coinvariants, we define a map ${\mathsf{F}}_{\chi }$ from sections of $\mathrm{\Pi }\otimes {\mathrm{\Omega }}^j$ to sections of ${\mathrm{\Omega }}^j$. We show that $\mathsf F_\chi \nabla^{$\text{(aff)}$} = \nabla^{$\text{(aff)}$}\chi \mathsf F\chi$,sothat$\mathsf F_\chi$ descends to a well-defined map of cohomologies. Under this map, the classes $[\mathbf v_j dt^{j+1} ]$aresenttotheclassesin$H^1(\mathbb P^1, \Omega^j,\nabla^{$\text{(aff)}$}_\chi)$definedbythe${\mathfrak g}$-oper underlying $\chi$.