Affine opers and conformal affine Toda

Abstract

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