Holographic RG Flows and Domain-Wall Geometries

A Poincaré-invariant domain wall geometrizes one boundary RG trajectory when its scalar profiles, quantization branches, state, and regularity are specified. The warp factor provides a useful scale coordinate, scalar gradients define beta functions along that solution, and the null energy condition can produce monotone quantities. Radial position is nevertheless not a unique Wilsonian momentum cutoff, and a regular classical flow is not by itself a complete field-theory RG construction.

Required background. Radial Hamilton–Jacobi flow supplies radial evolution. Renormalized one-point functions separates source and response branches. Helpful background. Conformal perturbation theory and beta functions and Wilsonian theory space provide the boundary meanings being compared.

First application. Solve a simple Einstein-scalar domain wall near a critical point and extract the linearized beta function associated with the scalar mass.

Einstein–scalar domain walls

Use Lorentzian signature $(+,-,\ldots,-)$ on the $d$-dimensional slices and action

$$S=\frac{1}{2\kappa^2}\int\mathrm d^{d+1}x\sqrt{\lvert g\rvert} \left[R+\frac12G_{ab}(\phi)\partial_M\phi^a\partial^M\phi^b -V(\phi)\right].$$

For

$$\mathrm ds^2=e^{2A(r)}\eta_{ij}\mathrm dx^i\mathrm dx^j-\mathrm dr^2, \qquad \phi^a=\phi^a(r),$$

the Einstein equations include

$$A''=-\frac{1}{2(d-1)}G_{ab}\phi'^a\phi'^b,$$

and a first-order constraint relating $A'^2$, $\phi'^2$, and $V$. With positive scalar target metric, $A''\le0$. The sign is checked by pure AdS, for which $\phi'=0$ and $A'=1/L$ in a radial coordinate increasing toward the ultraviolet.

If a local superpotential $W(\phi)$ exists such that

$$V=\frac12G^{ab}\partial_aW\partial_bW -\frac{d}{4(d-1)}W^2,$$

then a branch of solutions obeys

$$\phi'^a=G^{ab}\partial_bW, \qquad A'=-\frac{W}{2(d-1)},$$

after choosing the compatible radial orientation. This first-order representation is useful but neither unique nor globally guaranteed.

Beta functions from the scale factor

Along a monotonic segment of the solution, define

$$\beta^a_{\mathrm{geom}} \equiv\frac{\mathrm d\phi^a}{\mathrm dA} =-2(d-1)G^{ab}\partial_b\log W.$$

Near an AdS critical point, a scalar with $m^2L^2=\Delta(\Delta-d)$ has two branches. The source-driven branch behaves as

$$\beta_{\mathrm{geom}}\simeq-(d-\Delta)\phi,$$

while a response-driven branch can have different leading behavior. Identifying $\phi$ with a renormalized coupling and $A$ with $\log\mu$ requires a declared source scheme. A normalizable condensate profile is not automatically a running coupling.

The radial Hamilton–Jacobi functional gives a more precise comparison: its local part generates beta-like source flow, while the finite nonlocal part carries expectation values. Finite counterterms reparameterize the couplings and transform beta functions as vector fields on theory space.

Fixed points, endpoints, and regularity

At a critical point $\partial_aV=0$, the geometry approaches AdS and the boundary theory approaches a conformal fixed point if the full dictionary and stability conditions hold. An interior endpoint may instead be another AdS region, a cap, a horizon, or a singularity. A finite warp factor or potential alone does not classify it.

For singular domain walls, a commonly used necessary diagnostic is that the scalar potential remain bounded above along the solution so the singularity can arise as a limit of regular finite-temperature geometries Gubser 2000. This “good singularity” test is not a theorem of acceptable quantum-gravity completion; fluctuations, uplift, string corrections, and boundary observables still require checks.

A monotonic geometric quantity

For Einstein matter satisfying the null energy condition, define schematically

$$a(r)=\frac{C_d}{\kappa^2[A'(r)]^{d-1}},$$

with $C_d$ chosen so that $a$ matches the appropriate central coefficient at an AdS fixed point. Then

$$a'(r) =-\frac{(d-1)C_dA''}{\kappa^2[A']^d}\ge0$$

toward increasing $r$ when $A'>0$. Thus $a$ decreases from ultraviolet to infrared. Higher-curvature gravity, violations of the relevant energy condition, or nonmonotonic $A$ require a different argument.

As an explicit application, linearize $V$ about a critical point and solve the scalar equation. Extracting $\beta\simeq-(d-\Delta)\phi$ from the source branch and the fixed-point value of $a$ checks the mass-dimension relation, radial orientation, and central-charge normalization together.

Evidence boundary

A smooth domain wall with correct ultraviolet falloffs demonstrates a classical bulk solution and a candidate RG interpretation. It does not prove that the infrared endpoint defines a unitary CFT, that the truncation is consistent in a top-down theory, or that radial integration equals Wilsonian elimination of high-momentum modes. Those are separate spectral, uplift, and cutoff-action tests.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• de Boer, J., Verlinde, E., and Verlinde, H. “On the Holographic Renormalization Group.” Journal of High Energy Physics 2000, 003 (2000). DOI. arXiv.
• Freedman, D. Z., Gubser, S. S., Pilch, K., and Warner, N. P. “Renormalization Group Flows from Holography—Supersymmetry and a c-Theorem.” Advances in Theoretical and Mathematical Physics 3 (1999): 363–417. DOI. arXiv.
• Gubser, S. S. “Curvature Singularities: The Good, the Bad, and the Naked.” Advances in Theoretical and Mathematical Physics 4 (2000): 679–745. DOI. arXiv.