In the TQFT formalism of Moore-Tachikawa for describing Higgs branches of theories of class $\mathcal{ S}$, the space associated to the unpunctured sphere in type $\mathfrak{ g}$ is the universal centraliser $\mathfrak{ Z}_G$, where $\mathfrak{ g}=Lie(G)$. In more physical terms, this space arises as the Coulomb branch of pure $\mathcal{ N}=4$ gauge theory in three dimensions with gauge group $\check{ G}$, the Langlands dual. In the analogous formalism for describing chiral algebras of class $\mathcal{ S}$, the vertex algebra associated to the sphere has been dubbed the \emph{ chiral universal centraliser}. In this paper, we construct an open, symplectic embedding from a cover of the Kostant-Toda lattice of type $\mathfrak{ g}$ to the universal centraliser of $G$, extending a classic result of Kostant. Using this embedding and some observations on the Poisson algebraic structure of $\mathfrak{ Z}_G$, 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 $\mathcal{ S}$ of type $\mathfrak{ a}_1$ for theories of genus zero with up to six punctures. These realisations make generalised $S$-duality completely manifest, and the generalisation to more than six punctures is conceptually clear, though technically burdensome.
Categories: hep-th math-ph math.MP math.QA math.RT math.SG