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

TL;DR

The paper builds a geometric holographic bridge between the Hilbert scheme of N points in C^2 and magnetically charged AdS_4 black holes by embedding the Hilbert scheme in a 3d N=4 gauge theory framework and the gauge-Bethe correspondence. It shows that the black hole entropy functional arises from large-N, q→1 asymptotics of twisted indices and hemisphere vertex functions, interpreted as expectation values in the quantum K-theory ring QK_T(X_N). A dominant vacuum corresponding to a triangular Young diagram saturates the large-N entropy, providing a precise geometric dual to the gravitational/Cardy block via a factorised, line-operator enriched QK_T(X_N) picture. Numerically, finite-N solutions evolve to ζ→0 in a way that matches the triangular partition, supporting the holographic identification and suggesting a concrete geometric dual for the Cardy block in this AdS_4 holographic setup.

Abstract

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.

Figures (11)

• Figure 1: Partitions are specified by their parts $\lambda=(\lambda_1,\lambda_2,\ldots)$. The transpose partition is denoted $\lambda^{\vee}$. Partitions can be written as Young diagrams in $\mathbb{Z}^2$ with boxes labelled by $s=(i,j) \in \lambda$, where $(i,j)$ run over the rows and columns respectively. The arm and leg lengths of $s \in \lambda$ are defined as $a_{\lambda}(s) = \lambda_{i} - j$, $l_{\lambda}(s) = \lambda^\vee_j-i$. The hook and the content of a box $s$ are $h_{\lambda}(s) = a_{\lambda}(s) + l_{\lambda}(s) +1$, $c_{\lambda}(s) = j-i$,
• Figure 2: The factorisation setup. The path integral on $S^2 \times S^1$ in the presence of a line operator is sliced along a $S^1 \times S^1$ boundary with a complete set of states $|\lambda \rangle \langle \lambda |$ inserted.
• Figure 3: The $\mathcal{N}=4$ ADHM quiver.
• Figure 4: The triangular partitions of $N = 10$ (left) and $N = 13$ (right), as defined in \ref{['def triangular partitions']}.
• Figure 5: Plots of the density $\rho(t)$, where the discrete points are extracted from the numerical solutions at $N=2,10,25$, while the continuous curve is the large $N$ solution given in \ref{['crit density']}. We have chosen $\Delta_1 = \frac{\pi^2}{4}$, $\Delta_2=e$, $\Delta_C=\frac{\pi}{9}$, which are in the allowed ranges \ref{['chem pt def']}, \ref{['delta_C range']}. One can observe that the agreement between the numerical solution and the analytic prediction is improving with increasing $N$.

Theorems & Definitions (2)

• Conjecture 1
• Conjecture 2