Free field realisation and 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 \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.

arXiv comment50 pages, one appendix; v2 corresponds to journal version

References in corpus (19)

• Infinite Chiral Symmetry in Four Dimensions
• The Vacuum Structure and Spectrum of N=2 Supersymmetric SU(N) Gauge Theory
• Integrable systems and supersymmetric gauge theory
• Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, II
• Vertex operator algebras, Higgs branches, and modular differential equations
• Chiral de Rham complex
• The Relation between Quantum W algebras and Lie algebras
• Chiral algebras of class S
• Vertex Operator Algebras and 3d N=4 gauge theories
• Joseph ideals and lisse minimal W-algebras
• Abelianizing vertex algebras
• Free Field Realizations from the Higgs Branch
• Logarithmic W-algebras and Argyres-Douglas theories at higher rank
• VOAs and rank-two instanton SCFTs
• Wakimoto realizations of current algebras: an explicit construction
• Localization of affine W-algebras
• A geometric free field realisation for the genus-two class $\mathcal{S}$ theory of type ${\mathfrak{a}}_1$
• Kostant-Toda lattices and the universal centralizer
• Twisted chiral algebras of class $\mathcal{S}$ and mixed Feigin-Frenkel gluing

Cited by in corpus (1)

• Affine $\mathcal{W}$-algebras and Miura maps from 3d $\mathcal{N}=4$ non-Abelian quiver gauge theories