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 . In this paper, we construct an open, symplectic embedding from a cover of the Kostant-Toda lattice of type to the universal centraliser of -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 . 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 -duality completely manifest, and the generalisation to punctures is conceptually clear, though technically burdensome.