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 $\chi$.
Highlights
The Aut O-action defines a bundle Π over P1 with fibre π0
Π0 is the vacuum Fock module for a system of rank(g) free bosons and its subspace W is defined to be the intersection of the kernels of certain screening operators associated to the simple roots of g. (Details are in Section 2.) It turns out that W is a Poisson vertex algebra generated by rank(g) generators
The generators of W can be seen as the integrals of motion of the classical Toda field theory associated to g, whose Hamiltonian H in the light-cone formalism is nothing but the sum of the screening operators [17]
Summary
Our first objective is to recall the definitions of the objects in the embedding (1): the vacuum Fock module π0 and the screening operators which act on it. The Fock module π0 is a → 0 limit of a one-parameter family of vertex algebras π0. It is useful to start with this whole family, because doing so provides a natural way of understanding the semi-classical structures on π0, as we shall see . Free fields Let h be a finite-dimensional vector space over C, equipped with a non-degenerate symmetric bilinear form κ(·|·). We pick a basis {bi}di=im h of h and let {bi}di=im h ⊂ h be its dual basis with respect to the form κ(·|·):. We shall regard h as a commutative Lie algebra. Let be a parameter and denote by hthe central extension of the loop algebra h((t)), by a one-dimensional centre C1, defined by the cocycle res κ(f |dg). Hhas commutation relations [bim, bjn] = 1mδn+m,0 κ bi|bj ,.
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