A geometric free field realisation for the genus-two class $\mathcal{S}$ theory of type ${\mathfrak{a}}_1$

Abstract

We present a free field realisation for the vertex operator algebra associated to the genus-two, class $\mathcal{S}$ superconformal field theory of type ${\mathfrak{a}}_1$. The free field realisation is in the style of recent work by the authors, and is formulated in terms of a one-dimensional isotropic lattice vertex algebra along with two pairs of symplectic fermions. Our realisation makes manifest an enhanced $\mathrm{USp}(4)$ outer automorphism group of the VOA that is inherited from the symplectic fermion system. This extends an $\mathrm{SU}(2)$ outer automorphism that has been observed in recent work of Kiyoshige and Nishinaka and significantly simplifies the structure of the algebra. Along the way, we also produce a realisation of the generic subregular Drinfel'd-Sokolov $\mathcal{W}$ algebra of type $c_2$ in terms of the generic principle $\mathcal{W}$ algebra of type ${\mathfrak{c}}_2$ and a one-dimensional isotropic lattice vertex algebra.

arXiv comment11 pages; small edits for clarity in v2; to appear in Physical Review D

References in corpus (15)

• N=2 dualities
• Infinite Chiral Symmetry in Four Dimensions
• Mirrors of 3d Sicilian theories
• Vertex operator algebras, Higgs branches, and modular differential equations
• Vertex Operator Algebras and 3d N=4 gauge theories
• VOAs labelled by complex reflection groups and 4d SCFTs
• Tri-vertices and SU(2)'s
• Free Field Realizations from the Higgs Branch
• SCFT/VOA correspondence via $\Omega$-deformation
• Chiral algebras from Ω-deformation
• Flavor symmetries and unitarity bounds in $\mathcal{N}=2$ SCFTs
• VOAs and rank-two instanton SCFTs
• Orbifolds of symplectic fermion algebras
• Universal two-parameter even spin ${\mathcal{W}}_{\infty }$-algebra
• The Chiral Algebra of Genus Two Class $\mathcal{S}$ Theory

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• Schur Indices of Class $\mathcal{S}$ and Quasimodular Forms
• Surface defects, flavored modular differential equations and modularity
• Defects, modular differential equations, and free field realization of N = 4 VOAs
• Free field realisation of boundary vertex algebras for Abelian gauge theories in three dimensions
• Affine $\mathcal{W}$-algebras and Miura maps from 3d $\mathcal{N}=4$ non-Abelian quiver gauge theories
• Free field realisation and the chiral universal centraliser
• Orthosymplectic Feigin-Semikhatov duality