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 ${\delta }_{-1}|0⟩$. 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 $\Pi$ 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 $[{\mathbf{v}}_{j}d{t}^{j+1}]\in H^1({\mathbb{P}}^1,\Pi \otimes {\Omega }^j,{\nabla }^{\text{(aff)}})$ of the de Rham cohomology of ${\nabla }^{(\mathrm{aff})}$. Any choice of $\mathfrak{g}$-Miura oper $\chi$ gives a connection ${\nabla }_{\chi }^{(\mathrm{aff})}$ 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 ${\mathsf{F}}_{\chi }{\nabla }^{\text{(aff)}}={\nabla }_{\chi }^{\text{(aff)}}{\mathsf{F}}_{\chi }$, so that ${\mathsf{F}}_{\chi }$ descends to a well-defined map of cohomologies. Under this map, the classes $[{\mathbf{v}}_{j}d{t}^{j+1}]$ are sent to the classes in $H^1({\mathbb{P}}^1,{\Omega }^j,{\nabla }_{\chi }^{\text{(aff)}})$ defined by the $\mathfrak{g}$-oper underlying $\chi$.

to appear in the Journal of the London Mathematical Society; the Lie algebra g and its Langlands dual are exchanged relative to earlier arxiv version; introduction reorganized following helpful suggestions of the referee

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Cited by in corpus (1)

• Higher current algebras, homotopy Manin triples, and a rectilinear adelic complex