Emergent AdS Geometry and Black Hole Thermodynamics from Functional Renormalization Group

Abstract: We present a non-perturbative holographic dual description for the (O(N)) vector model in (d)-dimensional Euclidean space within the functional renormalization group (FRG) framework. By continuously iterating Wilsonian RG transformations, the extra-dimensional scale coordinate is identified as the radial direction of an emergent ((d+1))-dimensional bulk spacetime. We construct a bidirectional holographic dictionary that maps non-perturbative fluctuations directly into the emergent bulk metric warping factors. Under the massless critical configuration, the emergent gravitational vacuum spontaneously organizes into a stable, regular Anti-de Sitter ((\text{AdS}{d+1})) geometry without coordinate singularities, satisfying all foundational local energy conditions. Near the thermal horizon, by systematically eliminating the conical deficit singularity, we rigorously prove that the semiclassical Hawking temperature identically matches the boundary field theory temperature ((T_H \equiv T)). Finally, we show that the near-horizon thermodynamic potentials exactly satisfy the First Law of Black Hole Thermodynamics, spontaneously generating the Bekenstein-Hawking area law ((S{\text{horizon}} = \frac{N}{4}\mathcal{A})) from a first-principles, bottom-up derivation.

• The paper demonstrates that nonperturbative functional RG flows dynamically generate an emergent AdS geometry and black hole horizons with exact thermodynamic correspondence.
• It establishes a detailed bulk-boundary dictionary by mapping RG scale evolution to the bulk radial coordinate and computing curvature invariants directly from quantum fields.
• The study verifies that the emergent bulk energy-momentum tensor satisfies local energy conditions and reproduces the Bekenstein-Hawking area law from first principles.

Emergent AdS Geometry and Black Hole Thermodynamics from Functional Renormalization Group

Overview and Context

This work presents a non-perturbative, first-principles construction of holographic duality within the context of the $O(N)$ vector model in $d$-dimensional Euclidean space, exploiting the functional renormalization group (FRG) to dynamically generate emergent Anti-de Sitter ($\text{AdS}$) geometries and black hole thermodynamics. Unlike prior approaches reliant upon static or truncated backgrounds, this framework establishes a systematic, continuous mapping between RG scale evolution in the boundary theory and the radial coordinate of the dual bulk, allowing a full translation of non-perturbative boundary fluctuations into dynamically backreacted gravitational metrics.

The central results demonstrate that the boundary RG flow organizes the bulk geometry into a regular $\text{AdS}_{d+1}$ space at criticality and, at finite temperature, produces black-hole-like horizons with thermodynamic properties in rigorous correspondence with the boundary theory.

Constructing the Holographic Dictionary via Functional RG

The approach begins with the quantum $O(N)$ vector model, recast by a Hubbard-Stratonovich transformation into a form accommodating large-$N$ analysis. Sequential, infinitesimal Wilsonian RG steps are iterated to integrate out high-energy modes at each scale, yielding effective bulk dynamics for an auxiliary collective field $\varphi(x,z)$ across the emergent RG coordinate $z$.

The resulting partition function captures all orders of scale evolution and includes Jacobian contributions from field decompositions. Bulk effective dynamics for $\varphi(x,z)$ are codified in a second-order action whose Lagrangian density $\mathcal{L}_\text{bulk}$ is intrinsically non-perturbative, containing both the field and RG-induced quantum corrections.

Crucially, the boundary (UV and IR) conditions for the collective field are derived from the stationarity of the effective action, directly coupling the bulk profile to microscopic boundary data. This ensures the emergent geometry is not arbitrary but strictly dictated by the physical RG flow of the underlying theory.

Emergent Bulk Geometry and Curvature Properties

By analyzing normal mode fluctuations about the saddle-point of $d$0, the linearized bulk equation aligns with the Laplace-Beltrami operator in a $d$1-dimensional Riemannian metric. This enables identification of the emergent metric components $d$2 (radial) and $d$3 (transverse) as explicit functionals of the boundary Green's functions $d$4 and $d$5. The mapping is bidirectional: all geometric curvature invariants can be expressed in terms of the original quantum fields, completing a precise RG/geometry dictionary.

The emergent Ricci scalar $d$6, Ricci tensor components, and their evolution along the bulk are computed for both the gapped ($d$7) and critical ($d$8) regimes:

Figure 1: Evolution of the emergent bulk curvature invariants—$d$9, $\text{AdS}$0, and $\text{AdS}$1—as a function of the holographic radial coordinate $\text{AdS}$2 for both gapped and gapless phases.

In the gapped phase, as the bulk approaches the IR, the matter sector freezes out and the curvature exhibits a singular IR endpoint, geometrically ending the spacetime. Conversely, at criticality, the non-linear RG effects combine to restore maximal symmetry; the geometry becomes that of a smooth $\text{AdS}$3 space with constant negative curvature, exemplified by $\text{AdS}$4.

Bulk Energy-Momentum Tensor and Local Energy Conditions

The study constructs the emergent energy-momentum tensor $\text{AdS}$5 from the Einstein tensor of the induced metric. The tensor’s radial and transverse pressure components are analyzed via the effective fluid variables $\text{AdS}$6, $\text{AdS}$7, and $\text{AdS}$8. Numerical evaluation confirms that, throughout both gapped and gapless cases, the emergent geometry consistently satisfies weakness, strong, and dominant energy conditions. The effective matter content from the RG flow thus produces physically viable, attractive bulk gravity.

Figure 2: Numerical verification that the emergent bulk fluid variables satisfy all local (WEC/SEC/DEC) energy conditions along $\text{AdS}$9 for both the gapped and critical phases.

Black Hole Thermodynamics and Holographic Consistency

At finite temperature, the IR cutoff of the RG flow becomes the location of an emergent bulk horizon. The framework provides:

• An explicit mapping $\text{AdS}_{d+1}$0 relating the RG scale to boundary temperature.
• Proof that the horizon is at infinite proper distance, as expected for a black hole throat.
• Near-horizon ($\text{AdS}_{d+1}$1) analysis yields a Rindler metric. Demanding regularity (avoiding a conical singularity) gives the semiclassical Hawking temperature, which exactly matches the boundary theory temperature, $\text{AdS}_{d+1}$2, across all regimes without subleading anomalies.
• The horizon area $\text{AdS}_{d+1}$3, computed via the spatial metric determinant, enters in the entropy expression.

The entropy and free energy functionals, expressed both in terms of boundary and geometric (bulk) variables, satisfy the relevant thermodynamic identities exactly. The Bekenstein-Hawking area law, $\text{AdS}_{d+1}$4, is realized from microscopic field theory data rather than imposed externally:

$\text{AdS}_{d+1}$5

The parameter $\text{AdS}_{d+1}$6 relating $\text{AdS}_{d+1}$7 to $\text{AdS}_{d+1}$8 is not free; it is fixed analytically by matching UV and IR data, closing the duality with no arbitrary parameters.

Implications and Future Directions

This work establishes the dynamic emergence of $\text{AdS}_{d+1}$9 geometry and black hole thermodynamics from first-principles RG analysis, resolving limitations of earlier kinematic or truncated holographic models which could not capture horizon physics or produced spurious scaling anomalies in the infrared. Key contributions include:

• A mathematically explicit, bidirectional RG/geometry dictionary.
• Nonperturbative, dynamically backreacted horizon formation with fully consistent bulk thermodynamics.
• Exact, anomaly-free identification of boundary and bulk temperatures.
• Derivation of the area law for entropy directly from field theory.

From a practical perspective, this construction supplies a robust, scalable procedure to map boundary RG flows to bulk gravitational physics, opening clear routes for application to more general QFTs. From a theoretical standpoint, the approach clarifies the underpinning mechanism for the emergence of classical gravity and black hole thermodynamics in holographic duality.

Anticipated extensions include generalization to large-$O(N)$0 matrix and gauge theories, enabling direct first-principles derivations of dual bulk gravitons and higher-spin gauge fields, as well as explorations in supersymmetric sectors to test the universality and exactness of the mechanism. The analytic control over UV/IR matching may also advance understanding in RG monotonicity theorems, quantum information perspectives on holography, and nontrivial checks against supersymmetric localization.

Conclusion

This work delivers a technically complete demonstration that non-perturbative functional RG flows in a large-$O(N)$1 vector model dynamically generate a bulk $O(N)$2 geometry at criticality and an emergent black hole with consistent thermodynamics at finite temperature. The bulk metric, curvature, and thermodynamic properties are all directly constructed from boundary field theory quantities with no free parameters, culminating in an exact realization of the Bekenstein-Hawking entropy law. This realization sets a new standard for the rigor and scope of holographic dual constructions rooted in quantum field theoretic renormalization group dynamics.