Holographic Renormalization Program

Updated 19 August 2025

Holographic Renormalization Program is a systematic framework that removes divergences in gauge/gravity dualities using boundary counterterms and geometric techniques.
It employs recursive and Hamilton–Jacobi methods to interpret the radial coordinate as an emergent renormalization group scale, linking bulk dynamics to field theory observables.
The program extends to non-conformal, disordered, and non-local systems, incorporating advanced tools like stochastic quantization and machine learning analogues for robust RG flow analysis.

The Holographic Renormalization Program is a geometric and algorithmic procedure developed to render finite, regulate, and define field-theoretic data in gauge/gravity dualities, most notably AdS/CFT and its generalizations. This framework systematically removes divergences from gravitational actions and observables by introducing boundary counterterms, establishes a bulk interpretation of renormalization group (RG) flow, and matches the physical data of strongly coupled field theories to bulk dynamics, even in the presence of complex phenomena such as disorder, non-conformal asymptotics, or non-local observables.

1. Foundations and Motivations

Holographic renormalization arose to address the divergence structure present in on-shell gravitational actions and correlators evaluated on asymptotically locally AdS (AlAdS) spacetimes. These divergences are the gravitational dual of ultraviolet (UV) divergences in the boundary field theory. The standard approach is to introduce a radial cutoff at finite proper distance (e.g., at $z=\epsilon$ or $r=\text{const}$ in FG coordinates), compute the action and observables, then add covariant local counterterms on the cutoff hypersurface to cancel divergences as the cutoff is removed.

Key objectives include:

Ensuring the on-shell action and one-point functions are finite after renormalization.
Ensuring a well-defined variational principle with Dirichlet (or, in special circumstances, more general) boundary conditions.
Establishing a holographic dictionary linking bulk quantities (metric, fields, their canonical momenta) to field theory data (sources, expectation values, Ward identities, anomalies).

This approach has now evolved to accommodate non-AdS asymptotics, non-conformal branes, spacetime disorder, RG flows between fixed points, and non-local observables, incorporating advanced techniques such as the Hamilton–Jacobi (HJ) formalism, functional renormalization, stochastic analogies, and machine learning analogues.

2. Algorithmic and Hamilton–Jacobi Approaches

A central development is the use of the radial Hamilton–Jacobi equation, which views the radial coordinate as an emergent scale or “RG time”:

The on-shell action $S_\mathrm{on\text{-}shell}[\gamma_{ij}, \{\phi_I\}, r]$ is treated as a Hamilton’s principal functional, depending on boundary data $\gamma_{ij}, \phi_I$ on the hypersurface $r$ .
The Hamilton–Jacobi equation $\partial_r S_\mathrm{on\text{-}shell} + H[\gamma_{ij}, \phi_I; \pi^{ij}, \pi_{\phi_I}] = 0$ encodes the dynamics of the theory, with the canonical momenta given by $\pi^{ij} = \delta S_\mathrm{on\text{-}shell}/\delta \gamma_{ij}$ and $\pi_{\phi_I} = \delta S_\mathrm{on\text{-}shell}/\delta \phi_I$ .
$S_\mathrm{on\text{-}shell}$ and its divergent part (the counterterms $S_\mathrm{ct}$ ) are constructed as a covariant derivative expansion in boundary geometrical and field invariants:

$U[\gamma_{ij}, \phi, r] = U_{(0)} + U_{(2)} + U_{(4)} + \cdots$

where $U_{(2n)}$ contains terms with $2n$ derivatives or field powers.

Recursive and systematic solution methods (see (Papadimitriou, 2011, Elvang et al., 2016, Ma et al., 2022)) solve for the generating functionals and the counterterms, either order-by-order in the derivative expansion or by generating exact ansatz forms in terms of radial scalings and scaling parameters. These approaches are extended efficiently to models with multiple scalars, marginal or relevant deformations, and more general backgrounds, including non-conformal branes and Improved Holographic QCD (Papadimitriou, 2011, Korpas, 2022).

3. Counterterms: Construction and Generalizations

Counterterms are built from both intrinsic (boundary-induced) and, in some contexts, extrinsic (radial derivative) quantities:

Intrinsic counterterms are covariant expressions in curvature invariants, scalar fields, and their derivatives on the cutoff hypersurface. This construction directly ensures the Dirichlet problem is well posed for boundary values.
The extrinsic holographic renormalization method (Anastasiou et al., 24 May 2024) constructs counterterms from radial derivatives of fields (e.g., $\partial_r \Phi, \partial_r^2 \Phi, \ldots$ for a scalar). By exploiting the Fefferman–Graham (FG) expansion and the structure of asymptotically AdS spacetimes, the extrinsic approach is shown to be fully equivalent to the intrinsic construction provided all non-normalizable modes are fixed as boundary data. In the massless scalar case, this procedure is maximally efficient due to the structural simplification of the expansion.

In gravitational cases, analogous constructions—such as the Kounterterm method—employ boundary terms involving the extrinsic curvature and its combinations, thus bridging “extrinsic” and “intrinsic” approaches.

4. Specialized Contexts: Disorder, Non-Conformal Branes, Non-Local Observables

Disordered Holographic Systems and Functional Flows

For strongly correlated systems with quenched disorder, standard RG tracks a few running couplings. In holography, an entire function-valued distribution $P_V[W(x)]$ tracks the disorder, whose flow with the radial (holographic energy) scale is encoded by convolution with the bulk-to-boundary propagator $G$ . The flow equations geometrize the infinite set of running couplings and systematically determine disorder-averaged observables and their thermodynamic impact, validated by the Harris criterion (Adams et al., 2011).

Non-Conformal and Ricci-Flat Backgrounds

For non-conformal branes (e.g., D $p$ -branes, $p<5$ ), the recursive HJ approach is applied using specialized expansions to accommodate non-AdS asymptotics (Papadimitriou, 2011, Korpas, 2022). In Ricci-flat spacetimes, the program extends the AdS/CFT correspondence to codimension-2 boundaries at null infinity, with renormalized actions, stress tensors, and anomalies constructed via adapted FG expansions and boundary counterterms (Costa, 2012).

Non-Local and Spinning Observables

Renormalization of non-local rotating observables, such as energies for spinning string configurations, may employ either direct holographic subtraction of divergent worldsheet boundary terms or a mass subtraction scheme (where the energy of a spinning trailing string, with or without drag, is subtracted from that of the bound state). These schemes yield distinct renormalized energies and are sensitive to symmetry breaking and background anisotropy (Giantsos et al., 2022).

5. Quantum Information, Complexity, and Machine Learning Analogues

The renormalization of holographic entanglement entropy (HEE) and subregion complexity (HSC) leverages counterterms localized both on the cutoff surface and, for universal divergences, on the intersection (codimension-2 boundary) with the Ryu–Takayanagi (RT) minimal surface (Taylor et al., 2016, Jang et al., 2020). Notably, in some dimensions and operator scaling regimes, it is impossible to covariantly cancel all divergences in HSC using local curvature invariants, indicating limitations intrinsic to the dual field theory's properties (Jang et al., 2020).

Machine learning analogues, particularly deep neural network architectures such as RBMs, are mapped onto RG flows, with the hierarchical coarse-graining by network layers identified as a functional RG process. Entanglement structure emerges as an “area law” dividing hidden and visible layers, analogous to minimal surface prescriptions for holographic entropy (Howard, 2018).

6. Holographic RG, Wilsonian and Quantum RG, and Stochastic Analogs

Wilsonian RG approaches in AdS/CFT replace the identification of running couplings with the value of the bulk field at a finite cutoff surface by defining the effective IR theory after explicitly integrating out UV degrees of freedom (Balasubramanian et al., 2012). The appearance of non-local multi-trace operators (e.g., double-trace terms with non-local kernels) is a universal feature of Wilsonian “integrating out” in AdS, encoding the gluing of UV and IR data and the entanglement of field theory energy scales.

The Quantum RG framework rigorously maps the RG beta functions of the underlying quantum field theory to the equations of motion for dynamical sources in the bulk, demonstrating that when beta functions are gradients, the bulk action possesses scale-reversal symmetry; in the special case where only the energy-momentum tensor survives, an Einstein–Hilbert gravity emerges as the IR holographic description (Lee, 2013).

Recent work demonstrates a direct correspondence between the RG flow of multi-trace deformations in holographic theories and the Langevin/Fokker–Planck evolution of Stochastic Quantization, through the identification of AdS radial position with stochastic “time” and the on-shell action with the Euclidean stochastic action (Oh, 2021).

7. Supersymmetry, Anomalies, and Extensions

In supersymmetric contexts, the compatibility of standard holographic renormalization with the BPS structure of dual field theories depends sensitively on spacetime dimension. In four dimensions, standard counterterms suffice to match supersymmetric Ward identities and localization results; in five-dimensional supergravity, novel finite counterterms, not of the standard diffeomorphism- and gauge-invariant form, must be added to ensure invariance under deformations of the boundary geometry and to reproduce the expected BPS partition functions and charge relations (Genolini et al., 2016).

The interplay between boundary conditions (fixing all non-normalizable modes) and consistency of the variational principle is critical when employing advanced (e.g., extrinsic) renormalization prescriptions. The correct choice ensures finiteness, compatibility with the GKPW prescription, and the integrity of the holographic dictionary (Anastasiou et al., 24 May 2024).

Table: Central Methods in Holographic Renormalization

Method/Framework	Key Feature / Output	Applicability & Key References
Fefferman–Graham (FG)	Coordinate expansion; boundary counterterms	AdS backgrounds, standard models
Hamilton–Jacobi (HJ)	Recursive derivative expansion, ODEs for CTs	General backgrounds, multiple fields
Wilsonian RG	Integration over UV/IR, non-local multi-trace	Multi-trace deformations, IR effective
Extrinsic prescription	CTs from radial derivatives of fields	Scalar fields, efficient in massless
Kounterterm method	CTs involve extrinsic curvature	AdS gravity, higher-derivative actions
Quantum RG	Bulk as dynamical RG flow, gradient flows	Large $N$ matrix field theory
Stochastic/RG mapping	Langevin, Fokker–Planck ↔ holographic Wilsonian	Multi-trace flows in AdS/CFT
Deep Learning	Neural layers as coarse-graining steps	Entanglement, area laws, “EHM”

In summary, the Holographic Renormalization Program comprises a set of algorithmic and geometric techniques that regulate gravitational actions, establish the field theory–bulk correspondence, reproduce field theory anomalies and Ward identities, and generalize to highly nontrivial settings: disordered, non-conformal, or non-local systems. The framework now encompasses advanced recursive algorithms, alternative extrinsic formulations, explicit stochastic and machine learning analogies, and special handling required for supersymmetric sectors and information-theoretic observables. This consolidated approach enables a highly systematic and model-independent definition of the holographic dictionary for a broad class of dual gauge/gravity theories.