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 ${\rm USp}(4)$ outer automorphism group of the VOA that is inherited from the symplectic fermion system. This extends an ${\rm SU(2)}$ 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 $\mathcal{c}_2$ in terms of the generic principle $\mathcal{W}$ algebra of type $\mathfrak{c}_2$ and a one-dimensional isotropic lattice vertex algebra.
Highlights
To any four-dimensional N 1⁄4 2 superconformal field theory (SCFT) one may canonically associate a vertex operatoralgebra (VOA) by restriction to the cohomology of a particular conformal supercharge [1], or equivalently, by introducing a certain Ω background that deforms the holomorphic-topological twist of the theory [2,3]
We present an interesting new instance of such a free field realization, this time for the vertex operator (super)algebra (VOA) associated to the class S theory of type a1 for a genus-two surface with no punctures
The most prominent feature of the construction is the manifestation of the large outer automorphism group USp(4)
Summary
To any four-dimensional N 1⁄4 2 superconformal field theory (SCFT) one may canonically associate a vertex operator (super)algebra (VOA) by restriction to the cohomology of a particular conformal supercharge [1], or equivalently, by introducing a certain Ω background that deforms the holomorphic-topological twist of the theory [2,3]. We present an interesting new instance of such a free field realization, this time for the VOA associated to the class S theory of type a1 for a genus-two surface with no punctures. Our realization makes manifest a (surprisingly large) USp(4) outer automorphism symmetry that extends the previously identified SU(2) and is inherited directly from the symplectic fermions We use this to give a more compact presentation of the VOA in terms of the OPEs of strong generators than that of [10]. IV we elaborate on a number of technical aspects of our construction These include the existence of a sub-VOA of type W1⁄21; 2; 2; 2 that is realized in terms of the lattice bosons and a W1⁄22; 4 subalgebra of the symplectic fermions. V we describe the canonical R filtration on our free field vertex algebra
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