Holographic RG Flows in Higher-Curvature Gravity

• Holographic RG flows are a framework that geometrically maps renormalization group evolution to domain wall solutions in higher-dimensional gravity.
• They encode the running of couplings and trace anomalies in dual QFTs by employing superpotential methods and first-order formalisms with higher-derivative corrections.
• The approach leverages Lovelock couplings and cubic quasi-topological terms to classify universality classes, critical phenomena, and phase transitions in holographic models.

Holographic Renormalization Group (RG) flows provide a geometric realization of renormalization group evolution in strongly coupled quantum field theories (QFTs), via gravitational duals constructed in extensions of Einstein gravity, such as Gauss–Bonnet and cubic quasi-topological gravity. In these settings, the radial coordinate of the bulk theory encodes the energy scale of the dual QFT, and the resulting gravitational solutions capture both the running of couplings and the evolution of conformal data, including trace anomalies and central charges. The framework naturally accommodates higher-curvature corrections and exploits methods—most notably the superpotential or first-order domain wall formalism—that encode the full nonperturbative dynamics of the RG flow.

1. Domain Wall Solutions and Holographic Setup

Renormalization group flows are realized via flat domain wall (DW) solutions of the form

$$\mathrm{d}s_d^2 = \mathrm{d}r^2 + e^{2A(r)} \eta_{ij}\,\mathrm{d}x^i \mathrm{d}x^j,$$

where $A(r)$ is a warp factor and %%%%1%%%% is interpreted as an RG scale. The matter sector consists of a bulk scalar field $\sigma(r)$ representing the running coupling of the dual QFT. Extended gravity models, including those with Gauss–Bonnet and cubic quasi-topological (QTG) terms, are characterized by higher-derivative corrections—controlled by Lovelock coupling constants—that modify the DW equations via algebraic “correction factors.” Explicitly, the first-order DW equations arising from cubic QTG corrections are

$$\sigma'(r) = \frac{2}{\kappa} W'(\sigma) C_0(W), \quad A'(r) = -\frac{\kappa}{d-2} W(\sigma),$$

with

$$C_0(W) = 1 - \frac{2 \kappa^2 L^2 W^2}{(d-2)^2} - \frac{3 \kappa^4 L^4 W^4}{(d-2)^4}.$$

Here, $W(\sigma)$ is the superpotential, $\kappa$ the gravitational coupling, and $L$ a characteristic AdS radius. Boundary conditions at $r \to \pm\infty$ fix $e^{A(r)}$ to yield distinct AdS asymptotics, corresponding to different CFT fixed points in the dual theory—encoding the entire RG trajectory as an interpolation between vacua.

2. Holographic Beta-Functions and Trace Anomalies

The RG scale is identified as $l = -A(r)$, so the beta-function associated to the running scalar is

$$\frac{d\sigma}{dl} = -\beta(\sigma) = \frac{2(d-2)}{\kappa^2} \frac{W'(\sigma)}{W(\sigma)}\, C_0(W),$$

directly encoding all higher-curvature contributions through $C_0(W)$. The trace anomalies—specifically central functions $a(l)$ and $c(l)$—are constructed as explicit functions of the superpotential and its derivatives. For the c-function in cubic QTG: $$c(\sigma) = \left( \frac{d-2}{-l_{pl} \kappa W(\sigma)}\right)^{d-2} \left(1 - 2f(\sigma) - 3f^2(\sigma)\right), \quad f(\sigma) = \frac{L^2 \kappa^2 W^2(\sigma)}{(d-2)^2}.$$ In Einstein gravity (vanishing higher-curvature contributions), $a = c$. These expressions enable a precise RG evolution of trace anomalies and encode all dependence on both Lovelock couplings and the detailed form of the matter superpotential.

3. Critical Phenomena, Universality Classes, and Topological Vacua

RG fixed points arise at zeros of the beta-function, i.e., when $W'(\sigma^*) = 0$ or when $C_0(W)=0$ (the latter signal “topological” vacua that are possible due to higher-curvature corrections). Linearization near fixed points yields

$$\beta(\sigma) \approx -s\, (\sigma-\sigma^*), \quad \Delta = d-1 - s,$$

where $$s = -\left.\frac{d\beta}{d\sigma}\right|_{\sigma^*}$$, and $\Delta$ is the scaling dimension of the corresponding perturbing operator. The Breitenlohner–Freedman bound,

$$m^2(\sigma^*) = \frac{\kappa^2 W^2(\sigma^*)}{(d-2)^2} \, s(s-d+1),$$

links the stability of gravity vacua to the unitarity of the dual CFT. Different RG universality classes are thus characterized by their critical exponents $s$, which in turn depend on both the superpotential's analytic properties and Lovelock couplings. When the Lovelock correction factor $C_0(W)$ vanishes, the solutions approach “topological” vacua, leading to nontrivial behavior such as phase transitions of different orders.

4. Lovelock Couplings, Superpotential Engineering, and Phase Structure

The higher-derivative corrections modify the structure of the dual QFT via two mechanisms:

• The algebraic correction factor $C_0(W)$ introduces constraints on the allowed values of the superpotential (e.g., positivity and ghost-freeness require $C_0(W)>0$), restricting the admissible domain wall solutions.
• The form of the superpotential ($W(\sigma)$), for instance, a quartic “Higgs-like” form $W(\sigma) = -B[(\sigma^2 - x_0)^2 + D]$, determines both the location and the nature (massless vs. massive) of the RG phases. Parameters such as the gap $D$ and the separation of extrema control the relative hierarchy between the UV and IR AdS radii and, consequently, the possible existence of massive or massless flows.

As a result, the combined parameter space of Lovelock couplings and superpotential features supports a variety of physical behaviors, including:

• Massless phases: Domain wall solution interpolates between regular AdS vacua (fixed points at extrema of $W$), yielding a diverging correlation length in the UV and vanishing correlation length in the IR.
• Massive phases: Associated with domain wall “chains” which terminate at singularities, where the correlation length remains finite and a mass gap appears.

Phase transitions (including infinite order, BKT-type) are geometric in origin, arising as boundaries where two or more domain wall solutions coincide.

5. Central Charges, a/c-Theorems, and Monotonicity Constraints

Within the generalized framework, both central charges $a(\sigma)$ and $c(\sigma)$ become explicit functions of the running bulk scalar field—the RG “coupling.” The c-function and a-function may differ in value due to higher-curvature corrections: $$c(\sigma) = \left( \frac{d-2}{-l_{pl} \kappa W(\sigma)} \right)^{d-2} (1-2f(\sigma) - 3f^2(\sigma)),$$

$$a(\sigma) = \left( \frac{d-2}{-l_{pl} \kappa W(\sigma)} \right)^{d-2} \left(1-2\frac{d-2}{d-4} f(\sigma)-3\frac{d-2}{d-6}f^2(\sigma) \right),$$

and reduce to the standard Einstein values for small $f(\sigma)$.

An exact holographic a-theorem is established,

$\frac{da}{dl} = -\beta(\sigma)^2\, g(\sigma) < 0,$

with $g(\sigma)$ a positive-definite metric. Thus, a decreases monotonically along any RG flow, and a_UV > a_IR. Under further constraints and for allowed values of $f(\sigma)$, a c-theorem for the c-function may also hold; however, for general Lovelock couplings, both positivity and monotonicity require nontrivial inequalities to be satisfied, reflecting that energy flux positivity and absence of ghosts must be imposed throughout the RG evolution.

6. Energy Flux Positivity and Physical Admissibility

The requirement of positive energy fluxes at null infinity translates into precise constraints on the three-point coefficients $t_2$, $t_4$, and, hence, on the ratio $a/c$ via

$t_2 = \frac{(d-1)(d-2)}{d-3}(1-a/c).$

Holographic calculation of the energy fluxes for all polarizations leads to bounds such as

$-\frac{d-2}{d-4} \leq t_2 \leq \frac{d-1}{2}.$

These inequalities restrict the allowed region in parameter space (notably, the range of $f(\sigma)$) where the underlying QFT remains unitary and causal, and are necessary for the validity of the holographic c and a-theorems.

7. Summary Table of Core Quantities

Quantity	Definition / Formula	Physical Significance
DW Metric Ansatz	$$\mathrm{d}s^2 = \mathrm{d}r^2 + e^{2A(r)} \eta_{ij} \mathrm{d}x^i \mathrm{d}x^j$$	RG “scale-factor” geometry
Beta-function	$$\beta(\sigma) = -\frac{d\sigma}{dl} = \frac{2(d-2)}{\kappa^2}\frac{W'(\sigma)}{W(\sigma)}\, C_0(W)$$	RG running of coupling in dual QFT
c-function	$$c(\sigma) = \left(\frac{d-2}{-l_{pl}\kappa W} \right)^{d-2} (1-2f-3f^2)$$	Holographic central charge (trace anomaly coefficient)
a-theorem	$\frac{da}{dl} = -\beta(\sigma)^2 g(\sigma) < 0$	Monotonic decrease of a-function
Energy flux bounds	$- \frac{d-2}{d-4} \leq t_2 \leq \frac{d-1}{2}$	Unitarity/causality of boundary QFT

The Lovelock–type models thus yield a highly constrained holographic representation of nonperturbative RG flows, encoding intricate information about operator spectra, central charges, and critical phenomena. The interplay of higher-derivative couplings, superpotential structure, and flux positivity enables a systematic classification of universality classes, massless and massive phases, and phase transitions—including cases with $a \neq c$. This elucidates a comprehensive mapping between gravitational domain wall dynamics and the full RG data of strongly-coupled QFTs in diverse dimensions, providing powerful tools for both model-building and analytic paper of critical phenomena in holographic duals (Sotkov et al., 2012).