Free Field Realisation of the Chiral Universal Centraliser.

In the TQFT formalism of Moore-Tachikawa for describing Higgs branches of theories of class S , the space associated to the unpunctured sphere in type g is the universal centraliser Z G , where g = L i e ( G ) . In more physical terms, this space arises as the Coulomb branch of pure N = 4 gauge theory in three dimensions with gauge group G ˇ , the Langlands dual. In the analogous formalism for describing chiral algebras of class S , 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 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 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 S of type a 1 for theories of genus zero with n = 1 , … , 6 punctures. These realisations make generalised S -duality completely manifest, and the generalisation to n ⩾ 7 punctures is conceptually clear, though technically burdensome.