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.

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

References in corpus (2)

• Mirrors of 3d Sicilian theories
• Tri-vertices and SU(2)'s

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