Remarks on Associated Varieties and Minimal Tension Holography

TL;DR

This paper explores the connections between path integral localization in minimal tension string theory on AdS3 and associated varieties of boundary vertex algebras, providing new insights into the algebraic structures underlying holography.

Contribution

It proposes a relation between AdS3 boundary phenomena and associated varieties of affine superalgebras, and introduces free field realizations for studying AdS5/CFT4 duality.

Findings

Path integral localization relates to associated varieties of boundary vertex algebras.

Explicit free field realizations of V_1(psl(4|4)) are constructed.

Evidence suggests similar structures may extend to AdS5/CFT4 correspondence.

Abstract

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.