Affine opers and conformal affine Toda

• Published in: Journal of the London Mathematical Society Vol. 104; no. 5; pp. 2148 - 2207
• Main Author: Young, Charles A. S.
• Format: Journal Article
• Language: English
• Published: 01-12-2021
• Subjects: 17B67 (secondary); 17B69; 81R10; 81R12 (primary)
• ISSN: 0024-6107; 1469-7750
• DOI: 10.1112/jlms.12494

For g a Kac–Moody algebra of affine type, we show that there is an AutO‐equivariant identification between FunOpg(D), the algebra of functions on the space of g‐opers on the disc, and W⊂π0, the intersection of kernels of screenings inside a vacuum Fock module π0. This kernel W is generated by two states: a conformal vector and a state δ−1|0>. We show that the latter endows π0 with a canonical notion of translation T(aff), and use it to define the densities in π0 of integrals of motion of classical Conformal Affine Toda field theory.
The AutO‐action defines a bundle Π over P1 with fibre π0. We show that the product bundles Π⊗Ωj, where Ωj are tensor powers of the canonical bundle, come endowed with a one‐parameter family of holomorphic connections, ∇(aff)−αT(aff), α∈C. The integrals of motion of Conformal Affine Toda define global sections [vjdtj+1]∈H1(P1,Π⊗Ωj,∇(aff)) of the de Rham cohomology of ∇(aff).
Any choice of g‐Miura oper χ gives a connection ∇χ(aff) on Ωj. Using coinvariants, we define a map Fχ from sections of Π⊗Ωj to sections of Ωj. We show that Fχ∇(aff)=∇χ(aff)Fχ, so that Fχ descends to a well‐defined map of cohomologies. Under this map, the classes [vjdtj+1] are sent to the classes in H1(P1,Ωj,∇χ(aff)) defined by the g‐oper underlying χ.