Free Field Realisation of the Chiral Universal Centraliser

Abstract

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 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 n=1,ldots ,6 punctures. These realisations make generalised S-duality completely manifest, and the generalisation to ngeqslant 7 punctures is conceptually clear, though technically burdensome.