Holographic Reformulation and AdS3 Dual

• Holographic reformulation is the method of mapping strongly coupled quantum dynamics in CFT2 to weakly coupled gravitational phenomena in AdS3 using conformal symmetry and eikonal phase shifts.
• The framework reconstructs the Regge limit behavior and CFT OPE data through perturbative expansions, inversion formulas, and impact-parameter representations in various geometries.
• It extends to higher-spin, boundary/defect CFTs, and RG flows, demonstrating robust modular structure and bootstrap consistency in AdS3/CFT2 dualities.

A holographic reformulation refers to expressing strongly coupled quantum phenomena in a conformal field theory (CFT) in terms of weakly coupled gravitational or geometric data in a higher-dimensional anti-de Sitter (AdS) space. In the AdS$_3$/CFT$_2$ correspondence, the well-understood properties of two-dimensional conformal field theories and classical three-dimensional gravity/supergravity, often with additional brane or string-theoretic structure, provide an exact dictionary that equates CFT operator correlators, OPE data, and RG flows with geometric and dynamical properties of AdS$_3$ backgrounds. The AdS$_3$ dual framework is especially powerful due to the enhanced symmetry—in particular the infinite-dimensional Virasoro algebra(s) and their extensions—which severely constrain the dynamics on both sides.

1. Holographic Reconstruction of Regge Correlators

In the limit of large CFT central charge $c$ and operator dimensions, the Regge (high-energy, fixed impact parameter) limit of CFT$_2$ four-point correlators is controlled by the propagation of null geodesics in bulk AdS$_3$ geometries. Specifically, for heavy-heavy-light-light (HHLL) correlators, where "heavy" operators $O_H$ of dimension $\sim O(c)$ source a classical background and "light" operators $O_L$ with $h_L,\bar{h}_L\ll c$ probe this background, the four-point function

$$G(z,\bar z)=\langle O_H(\infty) O_L(1) \bar O_L(z,\bar z) \bar O_H(0) \rangle$$

reduces in the Regge regime to an impact-parameter representation built from a bulk eikonal phase shift $\delta(b,s)$. This phase shift encodes the time–angle delay accumulated by the null probe, and exponentiates to reconstruct the full Regge limit of the CFT correlator via

$$G_{\rm Regge}(\sigma,\eta) \approx 1 + 2i \int_0^\infty db\, b\, J_0(b\sigma) e^{-i\pi\Delta_L + 2i\delta(b,s)},$$

where $\sigma\to0$ corresponds to high energies, $b$ is the AdS$_3$ impact parameter, $J_0$ is the relevant Bessel function, and the factor $e^{-i\pi\Delta_L}$ implements the contour monodromy in the cross-ratio plane (Ceplak et al., 2021).

The bulk eikonal phase admits a perturbative expansion in the parameter $\mu=24\bar h_H/c$: $$\delta(b,s) = \sum_{k=1}^\infty \mu^k \delta_k(b,s)$$ with each $\delta_k(b,s)$ an elementary function arising from the specifics of the background geometry. For example, in conical defect AdS$_3$ ($\alpha = \sqrt{1-\mu}$)

$$\delta(b,s) = \pi s (1-b)(\alpha^{-1}-1) = \pi s (1-b)[\mu + \tfrac34 \mu^2 + \tfrac{5}{8} \mu^3 + \cdots]$$

leading to simple expressions for the bulk-to-boundary OPE data.

2. All-Orders Matching to OPE Data

The OPE data for heavy–light multi-trace operators of arbitrary spin in the cross-channel can be related, to all orders in $\mu$, to the bulk eikonal phase. For double-trace operators of spin $J$, the anomalous dimensions $\gamma_{n,\bar n}$ satisfy

$$\gamma_{n,\bar n} = -\frac1\pi\, \text{Res}_{h\to n+\bar n+2h_L} \int_0^\infty db\, P_b(n,\bar n)\, \delta(b,s)$$

where $P_b(n,\bar n)$ is a calculable Bessel-function kernel, and the residue extracts the appropriate OPE pole. This inversion formula directly connects the expansion of the Regge correlator in powers of $\mu$ to the CFT anomalous dimensions via explicit elementary integrals (Ceplak et al., 2021).

Explicit evaluation in conical defect backgrounds, as well as in smooth D1-D5 microstate (superstratum) geometries, confirms this all-order mapping. For instance, for conical defects,

$$\gamma_{n,\bar n} = -\frac{1}{\pi} \delta_1(n,\bar n) + \cdots = 2\,\min(n,\bar n)(\alpha^{-1} - 1)$$

holds exactly.

3. Impact-Parameter Representation and Modular Structure

The ability to reconstruct holographic four-point correlators from the bulk phase shift relies critically on the modular structure of 2D CFTs and the simplification inherent in AdS$_3$ gravity. In particular, the impact-parameter (or "partial wave") representation reflects the factorization of CFT correlators into conformal blocks, themselves dominated in the Regge limit by strongly coupled exchanges that sum into an eikonal exponentiation. This resummation is interpreted as the semiclassical propagation of the probe in a nontrivial background geometry (e.g., conical defect, BTZ black hole, D1–D5 microstate) (Ceplak et al., 2021, Giusto et al., 2018).

This representation clarifies how semiclassical bulk dynamics encode nontrivial OPE data and anomalous dimensions at strong coupling, in perfect accord with bootstrap constraints and large-$c$ Virasoro symmetry.

4. Explicit Microstate Geometries and the Black-Hole Regime

The holographic Regge limit admits exact constructions in two classes:

• Conical Defect Geometries: Parameterized by $\mu<1$, these provide completely solvable backgrounds where all order-$\mu$ phase-shift coefficients $\delta_k(b,s)$ are known, and the full CFT OPE data is reconstructed via the inversion formula.
• Superstratum (D1–D5 Microstate) Geometries: For a class of smooth geometries with explicit known four-point correlators (polylogarithmic or Fourier-sum form in the correlator), the impact-parameter technique reconstructs the OPE data, including anomalous dimensions and OPE coefficients, at every order in $\mu$. These cases are particularly relevant for studying nontrivial black-hole microstates and their CFT avatars (Ceplak et al., 2021).

In both cases, the high-energy probe regime fully reorganizes the operator expansion into geometric (phase shift) data, illuminating the interplay between semiclassical bulk geometry and CFT spectrum.

5. Generalizations and Ambit of Holographic Reformulation

These holographic reformulation techniques extend naturally to a wide class of AdS$_3$/CFT$_2$ setups:

• Spin-3 and Higher-Spin Holography: The entire boundary/bulk correspondence for 3D spin-3 gravity (with $\mathcal{W}_3$-augmented CFT) is recast as propagation in an eight-dimensional group-manifold, with phase shift and OPE matching determined by the associated Casimir eigenvalues, and with black holes corresponding to nontrivial connections in the extended symmetry algebra (Nakayama et al., 2019).
• Boundary and Defect CFTs: Quotient constructions and AdS/BCFT frameworks geometrically realize boundary and interface CFTs, with holographic data determined by global symmetries and fixed-point brane loci (Shashi, 2020).
• Non-Orientable and Topological Holography: Recent work extends the duality to non-orientable AdS$_3$ backgrounds and chiral boundary CFTs with simplified spectrum and modular properties (Pathak et al., 18 Apr 2024).
• Holographic RG Flows and Deformations: Bulk transitions (e.g., vacuum bubble nucleation, domain walls) map to RG flows and relevant deformations in the dual CFT, with entanglement entropy serving as a $c$-function (Antonelli et al., 2018).
• General Covariant Entanglement Quantities: Entanglement wedge cross-section and related geometric constructs encode odd entanglement entropy and negativity for general time-dependent states, with direct large-$c$ matching to CFT calculations (Biswas et al., 2023, Basak et al., 2021).

6. Rigorous Structure and Bootstrap Consistency

A key aspect of these holographic reformulations is the unique (crossing/OPE-consistency constrained) analytic structure of AdS$_3$/CFT$_2$ correlators. The holographic (bulk) calculation trivially matches all known CFT bootstrap results—uniquely fixing, for example, all terms in the analytic expression of the holographic four-point function of single-trace operators at order $O(1/N)$ (Giusto et al., 2018). The needing of no arbitrary contact terms or undetermined quartic couplings is not a coincidence, but follows from the modular-invariant and highly constrained nature of the dual field theory.

7. Conclusion

The holographic reformulation and AdS$_3$ dual construction translate intricate CFT$_2$ dynamics at large central charge into geometric eikonal phase data and explicit bulk propagation in controlled backgrounds. The impact-parameter and inversion formalism enables all-order resummation and precisely bridges CFT Regge/OPE data, anomalous dimensions, and microstate structure with semiclassical gravitational dynamics, permitting direct computation of black-hole spectra, correlation functions, and dynamical probes deep in the regime of strongly coupled quantum gravity (Ceplak et al., 2021). These techniques form the foundation for further developments in higher-spin, defect/interface, and non-orientable holography, and underlie the geometric picture of emergent gravitational dynamics in strongly coupled two-dimensional CFTs.