Free Field Realisation of the Chiral Universal Centraliser

In the TQFT formalism of Moore–Tachikawa for describing Higgs branches of theories of class [Formula: see text] , the space associated to the unpunctured sphere in type [Formula: see text] is the universal centraliser [Formula: see text] , where [Formula: see text] . In more physical terms, this space arises as the Coulomb branch of pure [Formula: see text] gauge theory in three dimensions with gauge group [Formula: see text] , the Langlands dual. In the analogous formalism for describing chiral algebras of class [Formula: see text] , 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 [Formula: see text] to the universal centraliser of G—extending a classic result of Kostant. Using this embedding and some observations on the Poisson algebraic structure of [Formula: see text] , 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 [Formula: see text] of type [Formula: see text] for theories of genus zero with [Formula: see text] punctures. These realisations make generalised S-duality completely manifest, and the generalisation to [Formula: see text] punctures is conceptually clear, though technically burdensome.