Holographic Bulk Dual Action

• Holographic Bulk Dual Action is a framework that encodes the renormalization group flow and boundary dynamics of quantum field theories via a bulk path integral.
• It leverages functional RG and Hamilton-Jacobi methods to derive explicit beta-functions and map boundary conditions into emergent gravitational dynamics.
• The formulation interconnects bulk and surface terms, translating boundary anomalies and entropy into classical saddle-point solutions and quantum corrections.

A holographic bulk dual action is the explicit bulk path integral representation whose classical saddle, quantum corrections, and boundary conditions encode the dynamics, symmetries, and renormalization group (RG) flow of an associated boundary QFT, such as a conformal field theory (CFT). Its structure renders the RG flow manifest in the bulk spacetime, often with the radial coordinate $z$ interpreted as the energy scale/renormalization parameter. This action is central to AdS/CFT duality, emergent gravity, and functional RG prescriptions in quantum field theory.

1. Formulation via Functional RG and Holography

Dual holography constructs the bulk action by reformulating the functional RG evolution of boundary probability distributions $P_\Lambda[\varphi(x)]$ as a Fokker-Planck-type equation in the cutoff scale $\Lambda$ ($r \equiv \ln \Lambda$). The Polchinski RG flow is recast into a path integral over an extended $(x, r)$ domain with Lagrange multipliers $\pi(x,r)$ that enforce boundary-to-bulk Langevin-type dynamics. The solution yields a bulk action

$$Z = \int D\varphi D\pi\; e^{-S_{\rm bulk}[\varphi,\pi]} \,,$$

where $S_{\rm bulk}$ is derived by integrating over both $x$ and $r$ domains with explicit kernel $C_\Lambda(x,y,r)$ and RG drift potentials $V_\Lambda[\varphi]$ (Kim et al., 8 Nov 2025). In the semiclassical limit, the action leads to a Hamilton-Jacobi equation for the on-shell action, directly mirroring the local Callan-Symanzik RG equation.

2. Generalized Bulk Action Structure and Explicit RG Flow

The holographic bulk dual action generates RG flow equations for all bulk fields and encodes gradient-flow $\beta$-functions. For a generic background metric $\gamma_{ij}(x,r)$ and scalar field $\varphi(x,r)$, the general form is

$$S_{\rm bulk} = \int_0^R dr \int_{\Sigma_r} d^d x\; \Big\{ \pi^{ij}(\partial_r \gamma_{ij} - \beta_{ij}) + \pi_\varphi(\partial_r \varphi - \beta_\varphi) - 2\kappa^2 \sqrt{\gamma} \bigl( \pi^{ij}\mathcal{G}_{ijkl}\pi^{kl} + \tfrac{1}{2}\pi_\varphi^2 \bigr) - V_{\rm eff}[\gamma,\varphi] \Big\}$$

with boundary term $$+\int_{\Sigma_R}d^d x\;V_{\rm eff}[\gamma,\varphi]_{r=R}$$, and gradient-flow functions

$$\beta_{ij} = \frac{1}{\sqrt{\gamma}} \frac{\delta V_{\rm eff}}{\delta \gamma^{ij}},\qquad \beta_\varphi = \frac{1}{\sqrt{\gamma}} \frac{\delta V_{\rm eff}}{\delta \varphi}.$$

Here, $\mathcal{G}_{ijkl}$ is the DeWitt supermetric, $V_{\rm eff}$ includes curvature, kinetic, and potential terms. This structure interpolates between purely holographic AdS/CFT forms and explicit functional RG boundary flows (Kim et al., 8 Nov 2025).

3. Decomposition into Bulk and Surface Terms

For gravitational theories, the action decomposes into bulk and surface contributions with a precise holographic relationship. The Einstein-Hilbert (EH) and Lanczos-Lovelock (LL) actions admit (Kolekar et al., 2010)

$$S_{EH/LL} = \int_M d^Dx\; \Bigl( L_{\rm bulk} + \partial_c X^c \Bigr)$$

where $L_{\rm bulk}$ contains $\Gamma\Gamma$-type kinetic terms, and the surface term $\partial_c X^c$ yields Wald entropy on a black hole horizon. There is a holographic identity:

$$\left( \frac{D}{2} - m \right) L_{\rm sur}^{(m)} = - \partial_i \Big[ g_{ab} \frac{\delta L_{\rm bulk}^{(m)}}{\delta(\partial_i g_{ab})} + \partial_j g_{ab} \frac{\partial L_{\rm bulk}^{(m)}}{\partial(\partial_i \partial_j g_{ab})} \Big]$$

which interrelates the dynamical content of both sectors. The decomposition supports the emergent gravity paradigm, with the bulk term encoding energy density (ADM Hamiltonian density) and the surface term encoding entropy (Noether charge/Wald entropy).

4. Hamilton–Jacobi Structure and Boundary Correspondence

In the saddle-point (large-$N$ or semiclassical) limit, the canonical momenta are set by $$\pi(x,r) = \delta \mathcal{S} / \delta \varphi(x,r)$$, and the Hamilton-Jacobi equation becomes

$$\mathcal{H} \left( \varphi, \frac{\delta \mathcal{S}}{\delta \varphi} \right) + \partial_r \mathcal{S} = 0$$

where functional derivatives act with respect to all bulk fields. This is equivalent to the boundary RG equation, e.g. local Callan-Symanzik, governing $\mathcal{S}$. The identification of the bulk path integral measure

$$Z_{\rm CFT}(g_{UV}, \varphi_{UV}) = \int D\gamma D\pi D\varphi D\pi_\varphi \; e^{-S_{\rm bulk}}$$

with fixed UV boundary data, and the correspondence between bulk on-shell action and boundary effective action, formalizes the AdS/CFT dictionary for RG flows (Kim et al., 8 Nov 2025).

5. Comparison with Other Duality Frameworks

The holographic bulk dual action generalizes across formulations:

• In higher-spin/bilocal holography, AdS actions for collective fields arise from changes of variables in vector models, with locality and gauge invariance preserved under appropriate coordinate/combinatoric maps (Koch et al., 2024).
• The Hamiltonian for composite/bilocal fields in the SYK and related models appears as a function of bi-local order parameter fields (self-energy, Green's function, polarization), governed by non-linear second-order flow equations manifestly encoding $1/N$ corrections and RG dynamics, with consistent UV/IR boundary conditions (Choun et al., 2024).
• In emergent gravity and thermodynamic analogies, the bulk action's decomposition (energy/entropy) mirrors Legendre transforms between thermodynamic potentials, and the holographic identity functions as an analog of $\delta S / \delta E = 1/T$ (Kolekar et al., 2010).
• For non-AdS or flat-space duals, locality of interactions and correspondence with boundary correlators depend sensitively on the background geometry; only in AdS does the ERG→bulk construction yield fully local couplings for all spins (Sathiapalan, 2024).

6. Operational Role and Physical Significance

The holographic bulk dual action can be interpreted as the generator for all RG-resummed observables in the boundary theory, with the radial coordinate $z$ or $r$ operating as the emergent scale parameter. Saddle-point solutions yield bulk equations of motion directly encapsulating the beta-functions and non-perturbative RG flows. The inclusion of Weyl anomaly terms in the bulk action realizes the boundary conformal anomaly. The path integral formulation naturally incorporates quantum fluctuations via boundary sources and supports derivations of Schwinger-Keldysh/real-time effective actions by gluing mixed-signature bulk geometries (Boer et al., 2018).

7. Key Formulae and Theoretical Identities

Feature	Representative Formula	Reference
Bulk action with RG flow	$S_{\rm bulk}[\gamma,\pi;\varphi,\pi_\varphi]$	(Kim et al., 8 Nov 2025)
Fokker-Planck equation	$\frac{d}{d\ln\Lambda} P_\Lambda[\varphi] = ...$	(Kim et al., 8 Nov 2025)
Bulk-surface holographic id	$(D/2-m) L_{\rm sur}^{(m)} = -\partial_i[ ... ]$	(Kolekar et al., 2010)
Hamilton-Jacobi eqn	$$\mathcal{H}(\varphi, \delta\mathcal{S}/\delta\varphi) + \partial_r \mathcal{S}=0$$	(Kim et al., 8 Nov 2025)
Gradient-flow $\beta$	$$\beta_{ij} = \frac{1}{\sqrt{\gamma}}\frac{\delta V_{\rm eff}}{\delta\gamma^{ij}}$$	(Kim et al., 8 Nov 2025)
Thermodynamic analogy	$A_E = \beta E - S = -\beta F$	(Kolekar et al., 2010)

This formalism provides a unified, rigorous framework for mapping boundary RG trajectories, anomalies, operator dimensions, and correlators to explicit bulk spacetime field equations and action functionals. The explicit $\beta$-functions encode the running dynamics, while the path integral structure supports computation of all boundary observables via bulk computations. The holographic bulk dual action thus forms the cornerstone of modern functional RG-based approaches to quantum gravity and strongly coupled QFTs.