Affine opers and conformal affine Toda

Abstract

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 $\delta_{-1}\left|0\right>$. 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 $\Pi$ over $\mathbb P^1$ with fibre $\pi_0$. We show that the product bundles $\Pi \otimes \Omega^j$, where $\Omega^j$ 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 $\nabla^{\mathrm{(aff)}}$. Any choice of ${\mathfrak g}$-Miura oper $\chi$ gives a connection $\nabla^{\mathrm{(aff)}}_\chi$ on $\Omega^j$. Using coinvariants, we define a map $\mathsf F_\chi$ from sections of $\Pi \otimes \Omega^j$ to sections of $\Omega^j$. We show that $\mathsf F_\chi \nabla^{\text{(aff)}} = \nabla^{\text{(aff)}}_\chi \mathsf F_\chi$, so that $\mathsf F_\chi$ descends to a well-defined map of cohomologies. Under this map, the classes $[\mathbf v_j dt^{j+1} ]$ are sent to the classes in $H^1(\mathbb P^1, \Omega^j,\nabla^{\text{(aff)}}_\chi)$ defined by the ${\mathfrak g}$-oper underlying $\chi$.