Remarks on Associated Varieties and Minimal Tension Holography

We comment on certain analogies that have recently emerged between path integral localisation phenomena in minimal tension string theory on $\mathrm{AdS}_3\times S^3\times T^4$ and associated varieties of boundary vertex algebras of 3d $\mathcal{N}=4$ theories. We give evidence for the fact that the path integral localisation relies on an intriguing relation between the conformal boundary of $\mathrm{AdS}_3$ and the associated variety of $V_1(\mathfrak{psl}(2|2))$, (the complexification of) the affine algebra that enters the world-sheet description of the string theory. We explain the origin of certain operators (related to ``secret representations") that have appeared in this description. Based on the expectation that similar statements will hold for $\mathrm{AdS}_5\times S^5$, we explicitly write down free field realisations of $V_1(\mathfrak{psl}(4|4))$ that may hopefully be useful to study the world-sheet dual of free $\mathcal{N}=4$ SYM.