Holographic Renormalization Group

• Holographic Renormalization Group is a framework that maps the renormalization flow of a quantum field theory onto the radial evolution of a dual gravitational system.
• It geometrizes scale transformations by linking the bulk radial coordinate with the effective field theory cutoff and the integration of high-energy modes.
• The approach facilitates precise computations of anomalies, critical exponents, and phase transitions through superpotential methods and higher-curvature corrections.

The Holographic Renormalization Group (RG) is a theoretical framework that realizes the renormalization group flow of a quantum field theory as geometric evolution in a higher-dimensional gravitational (bulk) theory. This construction is central to the AdS/CFT correspondence and its generalizations, encoding scale dependence and critical phenomena of boundary field theories into radial evolution equations for bulk fields. The holographic RG formalism unifies gravitational and field-theoretical notions of scale, facilitates calculations of anomalies and critical data, and provides a geometric setting to identify universality classes and the nature of irreversibility in RG flows.

1. Geometric Implementation of the RG in Holography

Holographic RG identifies the additional (radial) coordinate in an asymptotically AdS (or more general bulk) spacetime with the energy scale or RG “time” of the dual boundary theory. The field-theoretic notion of integrating out short-distance (high-energy) modes is mapped to integrating inward along the bulk radial direction, with each hypersurface at fixed radial coordinate corresponding to an effective theory with a specified UV cutoff.

The gravitational metric is usually expressed in ADM or Fefferman-Graham form: $$ds^2 = N^2(r, x)\, dr^2 + G_{\mu\nu}(r, x)\left(dx^\mu + N^\mu(r,x) dr\right)\left(dx^\nu + N^\nu(r,x) dr\right).$$ In Einstein gravity, full spacetime diffeomorphism invariance implies a symmetry between radial translations and spacetime transformations; the holographic RG treats the radial direction as the generator of scale transformations in the field theory (Nakayama, 2012).

When the theory is restricted (e.g., to foliation-preserving diffeomorphisms), the allowed coordinate transformations preserve the foliation by constant-r hypersurfaces, and the only nontrivial scaling symmetry corresponds to dilatations: $$r \to r + \text{const}, \quad x^\mu \to e^{-\text{const}}x^\mu.$$ Thus, the holographic RG in this context geometrizes scale rather than full conformal transformations, and the associated flows correspond to scale, but not necessarily conformal, RG transformations.

2. Holographic RG Flows, Superpotentials, and Bulk Equations

The RG flow in the field theory is mapped onto a gravitational boundary value problem, with the boundary conditions or sources for bulk fields directly related to running couplings and dynamically generated operator expectation values.

In higher-curvature or extended gravity (e.g., Lovelock, quasi-topological), as well as standard Einstein-scalar systems, the ‘domain wall’ or ‘superpotential’ method encodes beta functions and central functions. For a scalar-gravity model, in ‘domain wall’ coordinates, the fields obey first-order flow equations: $$A'(r) = -\frac{1}{2(d-1)}W(\phi), \quad \phi'(r) = W'(\phi),$$ with $A(r)$ defined via $ds^2 = dr^2 + e^{2A(r)}\eta_{\mu\nu}dx^\mu dx^\nu$, and $W(\phi)$ is a superpotential fixed by the equation

$$[W'(\phi)]^2 - \frac{d}{2(d-1)} W(\phi)^2 = 2V(\phi).$$

The RG ‘beta function’ for the running coupling is identified as

$$\beta_\phi = \frac{d\phi}{dA} = -2(d-1)\frac{W'(\phi)}{W(\phi)}.$$

This formalism provides a precise mapping from the RG flow of couplings in the field theory into the radial flow of the corresponding bulk fields (Bourdier et al., 2013, Sotkov et al., 2012).

RG fixed points correspond to extrema of $W(\phi)$ or $V(\phi)$, and critical exponents and scaling data are read off from linearized solutions around these points. Near marginal or nearly-marginal operators, the form of $V(\phi)$ and $W(\phi)$ determine whether the RG flow is perturbatively relevant, irrelevant, or exhibits nontrivial scaling.

3. Trace Anomaly, Consistency Conditions, and Breaking of Conformal Invariance

The process of holographic renormalization allows for the calculation of anomalies in the boundary QFT. Using the Fefferman–Graham expansion, the near-boundary asymptotics of the bulk metric,

$$ds^2 = l^2\left( \frac{d\rho^2}{4\rho^2} + \frac{1}{\rho} g_{\mu\nu}(\rho,x) dx^\mu dx^\nu \right),$$

with $$g_{\mu\nu}(\rho,x) = g^{(0)}_{\mu\nu}(x) + \ldots + \rho^{d/2}\left[ g^{(d/2)}_{\mu\nu}(x) + \log\rho\, h^{(d/2)}_{\mu\nu}(x)\right] +...$$, enables the systematic extraction of the holographic trace anomaly as the coefficient of $\log\rho$ divergences in the regulated on-shell action.

For models with reduced symmetry, such as foliation-preserving gravity, the trace anomaly generically includes terms forbidden by the Wess–Zumino consistency condition, such as $R^2$ in $d=4$. Explicitly (Nakayama, 2012),

$$\langle T^\mu_\mu \rangle = c\left[ E_4 - \mathrm{Weyl}^2 \right] - \left(2X-1\right)\,\alpha\, R^2,$$

where $E_4$ is the Euler density, and $X=1$ corresponds to the Einstein limit. The presence of the $R^2$ term signals that the dual field theory is only scale invariant and is not fully conformal, violating the Wess–Zumino consistency conditions.

This construction reveals the interplay between the bulk symmetries (or their breaking) and the structure of anomalies in the dual, and connects closely to field-theoretic results indicating that unitary scale-invariant but non-conformal theories are highly constrained or pathological in certain dimensions.

4. Holographic Dictionary: Central Functions, a/c–Theorems, and Physical Constraints

The RG flow equations in the bulk permit the definition of scale-dependent central functions (e.g., $a(\sigma)$, $c(\sigma)$) along the flow (Sotkov et al., 2012, Bourdier et al., 2013). These are obtained from the on-shell action or the linear response to background geometry and are constructed as explicit functions of the superpotential and bulk curvature couplings (for theories with extended gravity, such as Gauss–Bonnet or quasi-topological models): $$c(\sigma) \sim (L_k/l_{pl})^{d-2}\left[1-2f-3f^2\right], \quad f \sim L^2 W(\sigma)^2/(d-2)^2,$$ and likewise for $a(\sigma)$ with further prefactor modifications.

The monotonicity (i.e., $da/dl < 0$) and positivity of these central functions along the flow provide a holographic realization of Zamolodchikov-type $c$–theorems. Constraints from the positivity of energy flux (e.g., the requirement that energy one-point functions at null infinity are positive) translate to inequalities involving the effective vacuum scale $f$. These constraints partition the allowed space of RG flows, determining the possible critical behavior and controlling the presence or absence of physical (unitary) phases.

This framework reveals multiple classes of phase transitions, including second-order flows, infinite-order transitions (BKT or Miransky type where the beta function has a higher-order zero), and transitions to gapped (massive) phases signaled by the termination or change of sign of the $a$- or $c$-function. Detailed phase structure thus emerges in direct terms of the holographically dual gravity theory and the choice of superpotential.

5. Extensions: Higher Curvature Corrections, Bounce Solutions, and Beyond Conformal Geometry

When higher-derivative curvature corrections are included (e.g., quadratic terms parameterized by $\kappa_1$, $\kappa_2$ or more general Lovelock couplings), the RG flow is governed by more intricate first- or even higher-order equations for the superpotential. These corrections affect the admissible range of $W(\phi)$, leading to both upper and lower bounds. Furthermore, for positive higher-curvature couplings, singular ‘walls’ can arise in configuration space, leading to “bounce” solutions where the flow reverses before reaching the singularity. Such bounces are reflected in the structure of the dual RG flow, possibly indicating new endpoints or obstructions in the space of flows (Ghodsi et al., 2021).

These richer geometries modify holographic $c$-functions and can induce nontrivial effects on entanglement and Rényi entropies in the dual theory, while also affecting the structure and monotonicity of central functions along the RG trajectory.

6. Wilsonian and Functional RG: Effective Actions, Scheme Dependence, and Locality

Wilsonian interpretations of holographic RG have been pursued through both functional path integral methods and exact RG equations (Balasubramanian et al., 2012, Sathiapalan et al., 2017). In these formulations, the bulk fields are split at a finite radial cutoff, integrating out UV (large energy/momentum) modes to produce effective, non-fluctuating data at the cutoff surface. The standard identification of the running coupling as the value of the bulk field at the cutoff is correct only in specific renormalization schemes and only in the classical (large $N$) limit; otherwise, the running couplings are scheme-dependent and may not satisfy the bulk equations of motion everywhere.

The induced effective action generally contains nonlocal multi-trace operators due to finite propagation time in the bulk, but these become effectively local after coarse-graining over scales much larger than the cutoff. Thus, RG flows retain locality at large scales, and exact mapping between QFT and gravity schemes is possible if the proper identification of sources and couplings is made, especially using dimensional regularization in both bulk and boundary (Bzowski et al., 2019).

7. Summary Table: Foliation-Preserving Versus Full Diffeomorphism Gravity in Holographic RG

Feature	Full Diffeo-Invariant Gravity	Foliation-Preserving Gravity
Bulk symmetry group	All spacetime diffeomorphisms	Foliation-preserving diffeos
Special conformal invariance	Present	Absent
Dual field theory symmetry	Scale + conformal + Poincaré	Scale + Poincaré
Holographic trace anomaly (d=4)	Euler – Weyl²	Euler – Weyl² + R² (for X ≠ 1)
Wess–Zumino consistency	Satisfied	Violated (by R² term)
Central function monotonicity	c-theorem (monotonic)	May not exist (pathologies)
Phase structure	Conformal/Massless/IR fixed	Scale-invariant only possible

The data support the conclusion that holographic RG provides a geometric, nonperturbative encoding of renormalization group flows, anomalies, and central data of a wide class of boundary field theories. The detailed structure of the bulk theory—including its symmetry content, curvature corrections, and renormalization scheme—fully controls the universality properties and physical viability of the dual field-theoretic description.

References: (Nakayama, 2012, Sotkov et al., 2012, Balasubramanian et al., 2012, Bourdier et al., 2013, Ghodsi et al., 2021, Rajagopal et al., 2015, Bzowski et al., 2019)