Holographic Analytic Continuation

• Holographic analytic continuation is a set of methods that adapts standard analytic continuation to complex holographic scenarios with defect insertions and nontrivial boundary conditions.
• It employs geometric reflection, image techniques, and integral transforms to overcome limitations of standard Wick rotation in spacetimes with mixed signatures.
• The approach integrates advanced tools like Lefschetz thimble decomposition and supersymmetric localization to consistently reconstruct bulk observables from boundary data.

Holographic analytic continuation is a collection of methodologies and prescriptions that make analytic continuation compatible with holographic dualities, especially in settings where straightforward Wick rotation or analytic continuation either fails or introduces ambiguities due to nontrivial spacetime identifications, defect insertions, altered boundary conditions, or complex gauge-theoretic structures. In such contexts, new geometric, integral, or cycle-based procedures are employed to yield correlators, partition functions, and superconformal indices appropriate to the Lorentzian, Euclidean, or mixed signature regimes, consistently reconstructing bulk observables from boundary CFT or gauge theory data.

1. Limitations of Standard Analytic Continuation in Holography

The typical Wick rotation and analytic continuation of correlators, which is effective in pure AdS backgrounds, encounters obstructions in many holographic scenarios. For instance, in Lorentzian AdS₃ with moving defects, global identifications mix time and angular coordinates, precluding the existence of a Euclidean section respecting such identifications (Aref'eva et al., 2016). Additionally, in Lorentzian signature, geodesic approximations only connect spacelike-separated boundary points, as no timelike geodesics reach the boundary; timelike geodesics bounce in the interior. These features obstruct naive analytic continuation and call for generalized prescriptions.

In Liouville theory, the path integral produces meromorphic correlators in parameters (Liouville momenta), but the semiclassical analysis reveals the necessity to sum over families of complex and even multivalued solutions to maintain analytic continuation outside the classical "physical" domain (Harlow et al., 2011). The picture is further complicated when physical regions of classical solutions are absent, requiring a reformulation in terms of Chern-Simons theory and thimble decomposition.

2. Image and Reflection Methods for Defect Spacetimes

Aref’eva–Khramtsov–Tikhanovskaya’s improved image–geodesic prescription enables the analytic continuation of two-point functions and entanglement entropy in Lorentzian AdS₃ spacetimes deformed by point defects (Aref'eva et al., 2016). The construction involves:

• Defining forward and backward image points under conical defect identifications and their inverses.
• Writing the boundary two-point function as a sum over images, each weighted by regulated geodesic length and phase:

$$G^A_Δ(a,b)=G_{Δ,0}(a,b)\,Θ_0(a,b) + \sum_{n=1}^\infty G^A_{Δ,n}(a,b^{*n})\,Θ_+(a,b^{*n})+\sum_{n=1}^\infty G^A_{Δ,n}(a,b^{\#n})\,Θ_−(a,b^{\#n})$$

• For timelike separations, replacing the naive geodesic length (complex) with the real length of a reflected spacelike geodesic between $a^R$ and $b^{*n}$, augmented by a phase factor $e^{-i\pi\Delta\,\text{sign}[\sin(t_a-t_b^{*n})]}$ (Wightman function).
• This approach provides a universal prescription valid for arbitrary time intervals, covering cases where Euclidean continuation is unavailable or ambiguous.

In static orbifold cases ($\Delta=2\pi/N$), where analytic continuation is possible, this method reproduces the standard GKPW result, demonstrating consistency across regimes.

3. Integral Transforms and Kernel Analytic Continuation

The inverse Gel'fand–Graev–Radon (GGR) transform generalizes the HKLL bulk reconstruction for both Euclidean AdS and Lorentzian de Sitter spacetimes (Bhowmick et al., 2019):

• For Euclidean AdS$_{n}$, bulk fields are reconstructed by a boundary integral with a kernel $$K_E(z,\vec{x}|\tilde{x})=(z/(z^2+|\vec{x}-\tilde{x}|^2))^{\Delta}$$ and analytic normalization $C_E(n,\Delta)$.
• Analytic continuation to Lorentzian de Sitter is obtained by substituting $z_E\to i\,z_{d}$ and adjusting the kernel and normalization accordingly.
• In Lorentzian AdS with time-like boundary patches, the same GGR kernel is used, but the domain of boundary integration is restricted to the causal interior (light cone) consistent with causality.
• For quotiented spacetimes like BTZ black holes, the prescription maps the boundary periodicity into the integration measure without altering the kernel.

This unified kernel approach demonstrates that analytic continuation can be consistently implemented at the level of bulk–boundary integral transforms, provided the domains and phases are properly regulated.

4. Analytic Continuation and Lefschetz Thimble Decomposition in Liouville/Chern-Simons Theory

Liouville theory correlators, particularly the three-point DOZZ formula, are meromorphic in the Liouville momenta, and analytic continuation involves deforming complex integration cycles in the path integral (Harlow et al., 2011). The semiclassical limit in various regions corresponds to sums over real, complex, or multivalued saddle solutions, classified by their monodromy data.

• Outside the classical physical region, complex and multivalued solutions must be incorporated, and the path integral expands into a sum over Lefschetz thimbles $\mathcal{J}_\rho$, each attached to a critical point.
• The reformulation via SL(2,ℂ) Chern–Simons theory provides a geometric framework: each flat connection with fixed puncture monodromies corresponds to a saddle, and phase differences among thimbles induce Stokes jumps, reproducing analytic and nonanalytic behavior of the DOZZ formula under continuation.
• This construction extends holographically: the analytic continuation of Liouville correlators is realized by summing Chern–Simons partition functions over Lefschetz thimbles weighted by monodromy defect data.

5. Analytic Continuation of Dimensions and Supersymmetric Localization

In sphere partition functions and supersymmetric localization for gauge theories, analytic continuation in dimension $d$ is implemented by expressing fluctuation determinants and partition functions in closed forms analytic in $d$ (Gorantis et al., 2017):

• Determinants for multiplets admit expressions in terms of Gamma functions and infinite products—these are then continued in $d$ to arbitrary complex values.
• In $\mathcal{N}=1$ gauge theory, continuation from $d=3$ to $d=4$ yields perturbative partition functions and the correct one-loop $\beta$-function coefficients.
• Mass deformations for $\mathcal{N}=1^*$ theory are incorporated by shifting multiplet masses in the localization formulas, and analytic continuation in $d$ leads to a free energy expansion whose real part matches exactly the holographic supergravity predictions term-by-term.
• This demonstrates that analytic continuation in $d$ within localization frameworks is coherent with holographic dualities, providing precise matches for physical observables.

6. Analytic Continuation in Matrix Integral Indices and Giant Graviton Expansions

In the computation of superconformal indices via unitary-matrix integrals—particularly in the context of giant graviton expansions—analytic continuation is invoked to resolve contour integration ambiguities (Ezroura et al., 2024):

• The giant graviton index $\hat{Z}_m(q)$ is computed as a contour integral, with possible ambiguity in the "pole selection" when fugacities are inverted ($q \to q^{-1}$).
• Murthy's kernel expansion rigorously underpins the identification of the correct analytic continuation: the prescription is to deform the contour so that the set of poles is preserved, or equivalently, to analytically continue the known finite-N closed formula.
• In practice, the wrapped-D3-brane index is defined by analytic continuation of the finite-N answer, ensuring exact matching with the world-volume dynamics in the dual AdS geometry.
• This principle generalizes to indices and partition functions in other contexts, such as lens-space indices, 3D block integrals, and topologically twisted indices.

7. Synthesis and Universal Prescriptions

The aforementioned methodologies collectively highlight a universal feature: analytic continuation in holography is not a naive formal replacement, but an operation that demands careful geometric, combinatorial, or integral prescriptions adapted to the physical spacetime, boundary identifications, and gauge-theoretic content.

• Reflection and image methods ensure real-valued geodesic data and correct phases in defect backgrounds.
• Integral transforms require domain and normalization adjustments to preserve causality and kernel invariance.
• Lefschetz thimble decompositions account for multivaluedness and nontrivial analytic structure in meromorphic correlators.
• Analytic continuation of partition functions and indices, especially in matrix-integral contexts, is informed by contour deformation or closed-form analytic continuation to maintain consistency with physical brane expansions.

A plausible implication is that this collection of analytic continuation techniques represents the systematic resolution to ambiguities and obstructions that arise in holographic dualities, ensuring the preservation of physicality, causality, and exact matching with bulk gravitational/string dynamics. All cited approaches have been explicitly demonstrated to provide agreement between field-theoretic and holographic computations in their respective domains (Aref'eva et al., 2016, Bhowmick et al., 2019, Harlow et al., 2011, Gorantis et al., 2017, Ezroura et al., 2024).