$\text{AdS}_4$ Holography and the Hilbert Scheme

We elucidate a holographic relationship between the enumerative geometry of the Hilbert scheme of $N$ points in the plane $\mathbb{C}^2$, with $N$ large, and the entropy of certain magnetically charged black holes with $\text{AdS}_4$ asymptotics. Specifically, we demonstrate how the entropy functional arises from the asymptotics of 't Hooft and Wilson line operators in a 3d $\mathcal{N}= 4$ gauge theory. The gauge-Bethe correspondence allows us to interpret this calculation in terms of the enumerative geometry of the Hilbert scheme and thereby conjecture that the entropy is saturated by expectation values of certain natural operators in the quantum $K$-theory ring acting on the localised $K$-theory of the Hilbert scheme. We give numerical evidence that the large $N$ limit is saturated by contributions from a certain vacuum/fixed point on the Hilbert scheme, associated to a particular triangular-shaped Young diagram, by evolving solutions to the Bethe equations numerically at finite (but large) $N$ towards the classical limit. We thus conjecture a concrete geometric holographic dual of the so-called gravitational/Cardy block.