Freelance Holography Program

• Freelance Holography is a framework that generalizes AdS/CFT by allowing arbitrary boundary conditions on finite timelike hypersurfaces.
• It utilizes a one-parameter family of renormalized boundary terms to produce finite on-shell actions and introduces multi-trace deformations in the dual theory.
• The method connects radial flow in the bulk to emergent induced gravity, paving the way for duals of non-conformal and non-AdS theories.

The Freelance Holography Program is a framework within theoretical physics that systematically extends the AdS/CFT correspondence by relaxing both the position and the nature of spacetime boundaries on which the holographic duality is formulated. Rather than being restricted to asymptotic AdS infinity with Dirichlet boundary conditions, the formalism permits the construction of gauge/gravity dualities on arbitrary timelike hypersurfaces in the bulk, with arbitrary (renormalized) boundary conditions for the gravitational fields. This generalization leads directly to a one-parameter family of renormalized boundary conditions that allow for finite on-shell actions, and connects the radial evolution in the bulk to a deformation flow in the boundary theory—a structure which, in a number of instances, is reminiscent of T𝑇̄-type multitrace deformations. Critically, the program establishes the emergence of induced gravity and enables the exploration of holography beyond the conventional paradigm (Taghiloo, 24 Sep 2025).

1. Extension of Holographic Duality to Arbitrary Boundaries

The core advance of the Freelance Holography Program is the redefinition of the holographic dictionary so that it holds not only at conformal infinity but across all timelike hypersurfaces Σ[r] within AdS or other bulk geometries. In standard AdS/CFT, the dual conformal field theory is defined on the boundary at infinity, with the bulk’s asymptotic data (typically Dirichlet boundary values) acting as fixed sources for dual operators. Within the freelance framework, instead, the dual boundary theory can be associated with finite radial locations, i.e., “cutoff surfaces”, inside the bulk. Here, the radial coordinate is reinterpreted as a deformation parameter in both the gravitational theory and its boundary dual.

Mathematically, given a bulk manifold 𝓜 and a surface Σ[r], the bulk action is modified to include arbitrary boundary terms,

$$S^{W}_{\text{bulk}}[𝒥;\mathcal{M}] = S^{D}_{\text{bulk}}[𝒥;\mathcal{M}] + \int_\Sigma n_\mu W^\mu$$

with the dual boundary action

$$S^W_{\text{bdry}}[𝒥; \Sigma] = S^D_{\text{bdry}}[𝒥; \Sigma] + W[\Sigma],$$

where $W[\Sigma]$ encodes the freedom in boundary conditions parameterized by $w$ (Taghiloo, 24 Sep 2025).

2. Arbitrary and Renormalized Boundary Conditions

Standard AdS holography restricts boundary data to Dirichlet-type, fixing the leading falloffs of bulk fields. The Freelance Holography Program systematically “accommodates arbitrary boundary conditions” by supplementing the action with appropriate boundary or counterterms, such that the variational principle is well-posed and the on-shell action is finite even as the boundary is taken to infinity. Boundary conditions may thus be Dirichlet, Neumann, mixed, or more general, enabling a broad class of dual QFT deformations.

The program constructs a one-parameter family of such renormalized boundary conditions, labeled by $w$, such that

$S^W_{\text{bulk}}[𝒥; 𝓜] = S^D_{\text{bulk}}[𝒥; 𝓜] + W[\Sigma],$

ensures a finite on-shell action after suitable renormalization. In contrast to conventional holographic renormalization—which typically cancels divergences post hoc—the program integrates the renormalization directly into the selection of boundary conditions (Taghiloo, 24 Sep 2025).

3. Evolution Equations and Deformation Flow

An essential consequence of the generalization to arbitrary boundaries is that the on-shell bulk action (or free energy) satisfies a radial flow equation. As the radial cutoff is moved inward, or as the boundary is deformed, the boundary action $\hat{S}_{\rm bdry}(r)$ evolves according to a flow equation that is quadratic in the canonical momenta conjugate to boundary fields,

$$r\frac{dS_{\mathrm{bdry}}}{dr} = -\ell \int_\Sigma \sqrt{-h} \, \mathcal{T},$$

where $\mathcal{T}$ is a quadratic operator built from the Brown–York energy–momentum tensor (or corresponding conjugate momenta) (Taghiloo, 24 Sep 2025).

In the context of gravity, the flow may take the form

$$r\frac{dS_{\text{bdry}}}{dr} = -\ell \int_\Sigma \sqrt{-h}\left[T^{ab} T_{ab} - \frac{1}{d-1}T^2\right],$$

where $T^{ab}$ is the boundary stress tensor and $T$ its trace. These flows generalize the T𝑇̄ deformation in two dimensions to higher-dimensional and more general settings.

4. Multi-Trace Deformations and Induced Gravity

Allowing arbitrary (particularly mixed or Neumann) boundary conditions for bulk fields is formally equivalent to introducing multi-trace deformations in the dual boundary theory. This is encoded in the altered holographic dictionary:

$$\mathcal{J}^i = \frac{\delta W}{\delta \mathcal{O}_i},$$

where $\mathcal{J}^i$ is the boundary source and $\mathcal{O}_i$ the dual operator (Parvizi et al., 12 Mar 2025, Taghiloo, 24 Sep 2025). As a result, gravity on the finite boundary surface can be interpreted as “induced” by this flow, i.e., emergent as a dynamical effect from the multitrace deformation of the dual boundary field theory. The boundary theory thus acquires new local dynamics, potentially becoming non-conformal or even non-relativistic depending on the deformation trajectory.

A notable mathematical result is that the combination of arbitrary boundary conditions and the flow of the boundary action automatically yields a finite renormalized effective action, in contrast to the divergent actions encountered in pure Dirichlet holography (Taghiloo, 24 Sep 2025).

5. Mathematical Structures and Formalism

The formalism is underpinned by the covariant phase space approach. Freedoms in defining the symplectic structure—termed $W$, $Y$, and $Z$ freedoms—allow for the consistent tuning of boundary conditions and the corresponding dual theories (Parvizi et al., 12 Mar 2025). Schematically, the symplectic potential decomposition reads:

$$\Theta_{\text{bulk}} = \Theta^{(D)}_{\text{bulk}} + \delta W_{\text{bulk}} + dY_{\text{bulk}}.$$

The crucial implication is that $W$-freedom ties directly to physical boundary conditions, $Y$-freedom determines the on-shell symplectic form of the dual gauge theory, and $Z$-freedom governs possible corner contributions when boundaries intersect.

Boundary conditions may be imposed as:

• Dirichlet: $J$ fixed, $\delta J = 0$.
• Neumann: canonical momentum fixed.
• Mixed/conformal: a linear combination fixed or, in certain cases, only the conformal class fixed.

The evolution of the boundary theory under radial flow is governed by

$$\frac{d}{dr}\,\hat{S}_{\rm bdry}(r) = \mathcal{S}_{\rm deform}(r),$$

where $\mathcal{S}_{\rm deform}(r)$ encodes the generating deformation related to the specific $W$-term choice.

6. Consequences and Research Directions

The Freelance Holography Program provides several far-reaching consequences:

• Gravity on a finite cutoff can be interpreted as an induced, non-fundamental phenomenon, dynamically encoded via the RG flow of the dual theory.
• The approach offers a direct method for constructing duals of non-conformal (e.g., massive, Lifshitz, or non-relativistic) and even non-relativistic field theories by appropriate choice of $W$ and boundary terms.
• It facilitates exploration of the UV/IR correspondence, holographic renormalization group, the emergence of soft hair and gravitational memory, and more broadly promotes systematic paper of background independence in holography.
• The flexible treatment of boundaries opens potential physical applications in black hole thermodynamics, the paper of null or near-horizon boundaries, the membrane paradigm, and possible future generalizations to spacetimes that are not asymptotically AdS.

A plausible implication is that this boundary-agnostic formulation could serve as a foundational structure for future non-AdS, or even non-gravitational, holographic dualities.

7. Schematic Table of Key Formal Ingredients

Mathematical Structure	Standard AdS/CFT	Freelance Holography Program
Boundary location	Asymptotic infinity	Arbitrary hypersurface Σ[r]
Boundary conditions	Dirichlet	Arbitrary (Dirichlet, Neumann, mixed)
On-shell action divergence	Requires counterterms	Renormalized by 1-param boundary term
Holographic dictionary	Fixed	Modulated by $W$-freedom
RG/flow equation	Fixed source theory	Quadratic deformation flow

Conclusion

The Freelance Holography Program generalizes the gauge/gravity (AdS/CFT) correspondence to arbitrary boundaries and arbitrary, renormalized boundary conditions, made precise within the covariant phase space formalism. By constructing deformed, finite boundary actions and tracking the radial flow of boundary conditions—often mapping to multi-trace or T𝑇̄-like deformations in the dual theory—the framework makes unambiguous the emergence of induced gravity and the possibility of boundary theory engineering across a broad spectrum of applications. This perspective opens systematic new territory in holography, quantum gravity, and the paper of emergent spacetime (Taghiloo, 24 Sep 2025, Parvizi et al., 12 Mar 2025, Parvizi et al., 12 Mar 2025).