AdS/CFT Equivalence Transformation

• AdS/CFT equivalence transformation is a set of mappings and renormalizations that relates boundary quantum field observables and bulk gravitational dynamics.
• The approach unifies different operator dictionaries, linking bulk path integrals with boundary correlation functions while ensuring consistent renormalization.
• Analytic continuation and composite operator mixing are critical for extending the correspondence to de Sitter spaces and incorporating dynamical gravity.

The AdS/CFT equivalence transformation encompasses a set of correspondences, redefinitions, and mathematical identities that map observables, dynamics, and operator structures between quantum field theories defined on the boundary of Anti-de Sitter (AdS) space and gravitational or higher-dimensional field theories in the bulk. While the original AdS/CFT correspondence is a duality between a (d+1)-dimensional quantum gravitational theory in AdS and a d-dimensional conformal field theory (CFT) on its boundary, the equivalence transformation goes deeper, relating functional calculi, renormalization procedures, path integrals, and wavefunctions, thus providing explicit procedural maps between different but formally equivalent representations of the underlying physics.

1. Operator Correspondence: Differentiate vs. Extrapolate Dictionaries

Two main operator dictionaries are employed for computing CFT correlators from AdS bulk data:

• Differentiate (GKPW) Dictionary: The CFT correlators are obtained by differentiating the bulk partition function with respect to the boundary source coupling, typically denoted β(x). The GKPW prescription enforces the identification $Z_{\text{bulk}}[\beta] = Z_{\text{CFT}}[\beta]$ and computes correlators as functional derivatives:

$$\langle \mathcal{O}(x_1)\dots\mathcal{O}(x_n) \rangle = \left.\frac{\delta^n Z_{\text{bulk}}[\beta]}{\delta\beta(x_1)\dots\delta\beta(x_n)}\right|_{\beta=0}$$

• Extrapolate (BDHM) Dictionary: Here, one computes bulk correlators of the field $\phi(x,z)$ and takes the scaling limit as $z \to 0$, rescaled by $z^{-\Delta}$, to extract CFT correlators:

$$\langle \mathcal{O}(x_1)\dots\mathcal{O}(x_n) \rangle = \lim_{z\rightarrow 0} z^{-n\Delta} \langle \phi(x_1, z)\dots\phi(x_n, z)\rangle$$

These two dictionaries can be unified at the level of the bulk path integral by splitting the bulk functional into UV and IR wavefunctions (over a surface at $z = \ell$): $\Psi_{\text{IR}}[\tilde{\phi}; \ell]$ (for $z > \ell$) and $$\Psi_{\text{UV}}[\beta, \tilde{\phi}; \epsilon, \ell]$$ (for $\epsilon < z < \ell$), so that

$$Z_{\text{bulk}}[\beta] = \int D\tilde{\phi} \; \Psi_{\text{IR}}[\tilde{\phi}; \ell]\Psi_{\text{UV}}[\beta, \tilde{\phi}; \epsilon, \ell]$$

Differentiating $Z_{\text{bulk}}[\beta]$ with respect to $\beta$ introduces insertions of $\tilde{\phi}(x)$ with a scheme-dependent factor. Careful renormalization of composite operators is required to ensure complete equivalence between the dictionaries in AdS (Harlow et al., 2011).

2. Role of Renormalization and Composite Operator Mixing

In interacting bulk theories, the UV wavefunction $\Psi_{\text{UV}}$ develops non-linear dependence on boundary data, and functional differentiation produces not only the linear term (corresponding to single-trace operator insertions) but also terms involving multi-trace and composite operators. Typical expansions take the form

$$(1/\Psi_{\text{UV}})\frac{\delta \Psi_{\text{UV}}}{\delta\beta(x)}\bigg|_{\beta=0} = c_1\tilde{\phi}(x) + c_2\tilde{\phi}^2(x) + c_3\tilde{\phi}^3(x) + \cdots$$

This mixing arises from the short-distance singularities of composite operators in perturbation theory, e.g., when $\phi^3$ operators contain terms proportional to $\phi$ itself. The proper renormalization scheme subtracts these contributions to ensure only the desired leading terms survive, maintaining the equivalence between extrapolate and differentiate dictionaries even in the presence of interactions (Harlow et al., 2011).

3. Analytic Continuation and Differences for dS/CFT

Analytic continuation from Euclidean AdS to Lorentzian de Sitter (dS) is performed via $z \mapsto -iT$ and $L_{\text{AdS}} \mapsto iL_{\text{dS}}$, mapping the AdS metric and wavefunctions to their dS analogues. The IR AdS wavefunction analytically continues directly to the dS wavefunction with Euclidean initial conditions:

$$\Psi_{\text{dS}}[\tilde{\phi}, T] = \Psi_{\text{IR,AdS}}[\tilde{\phi}, \ell \rightarrow -iT, L_{\text{AdS}} \rightarrow iL_{\text{dS}}]$$

A critical distinction arises in the computation of expectation values: in AdS, expectation values are computed as overlaps of two wavefunctions (UV and IR), whereas in dS, probabilistic averages employ $|\Psi|^2$. The local terms that fix boundary conditions in AdS cancel upon squaring in dS, leading to inequivalent dictionaries: the "differentiate" dictionary in dS yields conformal weights $$\delta = \frac{d}{2} + \frac{1}{2}\sqrt{d^2 - 4m^2}$$, while "extrapolate" prescriptions mix dimensions $\delta$ and $d-\delta$ (Harlow et al., 2011).

4. Inclusion of Dynamical Gravity and Wheeler–DeWitt Constraints

When including dynamical bulk gravity, the gravitational path integral must be performed over both matter and metric configurations with boundary data fixed on $z = 0$. The total partition function is given by

$$Z[\phi_0, h_{ij}] = \int_{g|_{z=0} = h_{ij}} \mathcal{D}g_{\mu\nu}\mathcal{D}\phi \; e^{-S[g, \phi]}$$

A partial gauge-fixing enables a natural "radial" slicing, and the wavefunction $\Psi_{\text{UV}}$ (now a functional of the induced metric and matter data) satisfies a Schrödinger-like (Wheeler–DeWitt) equation in geodesic distance $L$. At $L=0$, the wavefunction becomes sharply peaked on the reference boundary metric. The global partition function is again expressed as an overlap of UV and IR wavefunctions:

$$Z[h_0] = \int Dh \, \Psi_{\text{UV}}[h_0, h; L] \Psi_{\text{IR}}[h]$$

This construction persists for de Sitter with necessary adjustments for Lorentzian signature and analytic continuation. Subtleties in contour integration and Stokes phenomena persist at nonperturbative level, but in perturbative effective field theory, the analytic continuation carries over (Harlow et al., 2011).

5. Wavefunction Approaches and the Emergence of Conformal Ward Identities

The partition function and wavefunction approaches make explicit the emergence of conformal structure from bulk AdS gravity. For instance, the solution $\Psi(Y)$ of the radial Wheeler–DeWitt equation exhibits the asymptotic behavior

$$\Psi(Y) = e^{i S^{(0)} + i S^{(1)} + \cdots} Z_+(Y) + e^{-i S^{(0)} - i S^{(1)} + \cdots} Z_-(Y)$$

where $S^{(0)}$ is the classical on-shell action (including necessary counterterms) and $Z_+(Y)$ satisfies a (possibly anomalous) conformal Ward identity. In the $p \rightarrow 0$ limit (boundary limit), quantum corrections are subleading, and the asymptotics match the partition function of a conformal field theory on the boundary (Cianfrani et al., 2013).

6. Unified Perspective and Implications

The AdS/CFT equivalence transformation demonstrates that—at the level of renormalized functional integrals and expectation values—multiple distinct "dictionaries" (GKPW, BDHM, operator/wavefunction, partition function, path integral with dynamical gravity) yield equivalent predictions for correlators in AdS, provided composite operator renormalization is properly implemented. This equivalence is specific to AdS and is broken or altered in dS due to the different structure of quantum averages.

The explicit mapping at the level of wavefunctions, path integrals, Hamilton–Jacobi and renormalization group flows (cf. (Radicevic, 2011)), and the operator formalism (Terashima, 2017), as well as the necessary inclusion of counterterms and composite operator mixings, collectively underpin the structural robustness of the AdS/CFT correspondence and clarify the precise sense in which bulk gravitational and boundary CFT theories are physically and mathematically equivalent under the duality.

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Table: Comparison of "Differentiate" and "Extrapolate" Dictionaries

Bulk–Boundary Map	Prescription	Equivalence in AdS?
Differentiate (GKPW)	$\delta/\delta \beta$ of $Z_{\text{bulk}}$	Yes (requires renormalization)
Extrapolate (BDHM)	$z^{-\Delta} \phi(x,z)_{z \to 0}$	Yes (after subtraction)
Differentiate (dS)	$\delta/\delta \tilde\phi$ of $\Psi[\tilde\phi]$ (then square)	No (distinct spectrum)
Extrapolate (dS)	$T^{-\delta} \phi(x,T)_{T \to 0}$	No (mixture of dimensions)

• Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT (Harlow et al., 2011)
• Wheeler–De Witt equation and AdS/CFT correspondence (Cianfrani et al., 2013)
• Connecting the Holographic and Wilsonian Renormalization Groups (Radicevic, 2011)
• AdS/CFT Correspondence in Operator Formalism (Terashima, 2017)

These results provide a rigorous and unified understanding of the AdS/CFT equivalence transformation, its necessary technical refinements (especially renormalization of composite operators), its analytic continuation and breakdown in dS/CFT, and its robustness upon inclusion of dynamical gravity.