Free Field Realisation of the Chiral Universal Centraliser

In the TQFT formalism of Moore-Tachikawa for describing Higgs branches of theories of class $$, the space associated to the unpunctured sphere in type $$ is the universal centraliser $$, where $$. In more physical terms, this space arises as the Coulomb branch of pure $$ gauge theory in three dimensions with gauge group $$, the Langlands dual. In the analogous formalism for describing chiral algebras of class $$, the vertex algebra associated to the sphere has been dubbed the chiral universal centraliser. In this paper, we construct an open, symplectic embedding from a cover of the Kostant-Toda lattice of type $$ to the universal centraliser of G-extending a classic result of Kostant. Using this embedding and some observations on the Poisson algebraic structure of $$, we propose a free field realisation of the chiral universal centraliser for any simple group G. We exploit this realisation to develop free field realisations of chiral algebras of class $$ of type $$ for theories of genus zero with $$ punctures. These realisations make generalised S-duality completely manifest, and the generalisation to $$ punctures is conceptually clear, though technically burdensome.