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

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.

46 pages, 11 figures

References in corpus (37)

• N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals
• Embedding AdS Black Holes in Ten and Eleven Dimensions
• Towards the F-Theorem: N=2 Field Theories on the Three-Sphere
• Supersymmetric vacua and Bethe ansatz
• Black hole microstates in AdS${}_4$ from supersymmetric localization
• Vortex Counting and Lagrangian 3-manifolds
• A topologically twisted index for three-dimensional supersymmetric theories
• Quantum integrability and supersymmetric vacua
• Multi-Matrix Models and Tri-Sasaki Einstein Spaces
• Supersymmetric AdS_4 black holes and attractors
• Nonperturbative Tests of Three-Dimensional Dualities
• Comments on twisted indices in 3d supersymmetric gauge theories
• Exact microstate counting for dyonic black holes in AdS4
• Flow equations and attractors for black holes in N = 2 U(1) gauged supergravity
• Static supersymmetric black holes in AdS_4 with spherical symmetry
• Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory
• Higgs branch localization in three dimensions
• Large $N$ matrix models for 3d $\mathcal{N}=2$ theories: twisted index, free energy and black holes
• On the completeness of solutions of Bethe's equations
• Large $N$ topologically twisted index: necklace quivers, dualities, and Sasaki-Einstein spaces
• Gluing gravitational blocks for AdS black holes
• Static BPS Black Holes in AdS4 with General Dyonic Charges
• Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and partition functions on Seifert manifolds: A review
• Universal 3d Cardy Block and Black Hole Entropy
• Kirwan surjectivity for quiver varieties
• Static BPS black holes in U(1) gauged supergravity
• Integrals of motion from quantum toroidal algebras
• Boundaries, Vermas, and Factorisation
• Twisted Indices of 3d $\mathcal{N}=4$ Gauge Theories and Enumerative Geometry of Quasi-Maps
• Quantum K-theory of Quiver Varieties and Many-Body Systems
• 3d $\mathcal{N}=4$ Gauge Theories on an Elliptic Curve
• Microstates of accelerating and supersymmetric AdS${}_4$ black holes from the spindle index
• Algebraic geometry and Bethe ansatz (I) the quotient ring for BAE
• Topological Correlators and Surface Defects from Equivariant Cohomology
• Blocks and Vortices in the 3d ADHM Quiver Gauge Theory
• Characters of tangent spaces at torus fixed points and $3d$-mirror symmetry
• ADHM Wilson line defect indices

Cited by in corpus (1)

• Abelian dualities and line defect indices for 3d gauge theories