S-duality for the large $N=4$ superconformal algebra

Abstract

We prove some conjectures about vertex algebras which emerge in gauge theory constructions associated to the geometric Langlands program. In particular, we present the conjectural kernel vertex algebra for the $S{T}^{2}S$ duality transformation in $SU(2)$ gauge theory. We find a surprising coincidence, which gives a powerful hint about the nature of the corresponding duality wall. Concretely, we determine the branching rules for the small $N=4$ superconformal algebra at central charge $-9$ as well as for the generic large $N=4$ superconformal algebra at central charge $-6$. Moreover we obtain the affine vertex superalgebra of $\mathfrak{osp}(1|2)$ and the $N=1$ superconformal algebra times a free fermion as Quantum Hamiltonian reductions of the large $N=4$ superconformal algebras at $c=-6$.

arXiv commentRelationship between large and small N=4 superconformal VOAs is clarified

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Cited by in corpus (14)

• Gluing vertex algebras
• The vertex algebras ${\mathcal{R}}^{(p)}$ and ${\mathcal{V}}^{(p)}$
• Tensor categories of affine Lie algebras beyond admissible levels
• Direct limit completions of vertex tensor categories
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• Correspondences of categories for subregular W-algebras and principal W-superalgebras
• $N=4$ superconformal algebras and diagonal cosets
• Affine Laumon spaces and iterated W-algebras
• Higher rank FZZ-dualities
• Quasi-lisse extension of affine ${\mathfrak{sl}}_2$ à la Feigin--Tipunin
• $N=1$ super Virasoro tensor categories
• Invariant subalgebras of the small $\mathcal{N}=4$ superconformal algebra
• FZZ-triality and large $\mathcal{N}=4$ super Liouville theory
• Duality via convolution of W-algebras