Real-Time Finite-Temperature Holography

• Real-time finite-temperature holography is an approach that uses gauge/gravity duality to study thermal and dynamical properties of strongly coupled field theories via gravitational duals.
• It extends traditional Euclidean holography to include Schwinger–Keldysh contours, integrating black brane geometries with real-time dynamics and thermal state preparation.
• The framework provides actionable insights into transport coefficients, IR scaling, and fluctuation-dissipation relations through analytic continuation, numerical matching, and effective field theory methods.

Real-time finite-temperature holography is an approach within gauge/gravity duality that enables the analysis of non-equilibrium and thermal quantum field theories through their gravitational duals, emphasizing the calculation of real-time observables at finite temperature. This methodology extends traditional Euclidean holographic techniques to Schwinger–Keldysh (SK) or real-time contours, facilitating the computation of dynamical response, dissipation, transport, and fluctuation phenomena in strongly coupled systems. It encompasses a diverse range of physical contexts, including charged black brane solutions, non-relativistic backgrounds, real-time correlation functions, aging systems, fluctuation-dissipation relations, and entanglement in the presence of thermal effects.

1. Holographic Real-Time Formalism and Boundary Conditions

The real-time holographic prescription is implemented by constructing bulk geometries tailored to the boundary SK contour, often involving piecewise Euclidean and Lorentzian segments. For a CFT at finite temperature, the contour typically closes after a Euclidean time segment of length β (inverse temperature), followed by Lorentzian evolution. The dual bulk geometry is built by gluing Euclidean black-hole "caps" (which prepare the thermal state) to Lorentzian exterior regions of AdS black holes (encoding real-time dynamics), with enforced C¹ matching of metric and canonical momenta across junctions (Botta-Cantcheff et al., 2018).

Boundary conditions must be imposed such that the induced metric and scalar fields (and their conjugate momenta) are continuous across segment interfaces. This construction geometrically encodes the process known as the Unruh trick: analytic continuation and thermal Boltzmann factors are realized as global features of the gluing procedure. The Hartle–Hawking–Maldacena (HHM) or Thermofield Double (TFD) state naturally arises as the initial "thermal" state of the real-time path integral.

2. Near-Extremal Solutions and Power-Law Scaling Geometries

A central class of models involves charged black brane solutions in Einstein–Maxwell–scalar (EMS) gravity, characterized by power-law scaling and finite temperature. In the extremal limit, these backgrounds possess vanishing entropy, yielding unique zero-temperature ground states. For physical regularization, an "emblackening factor" is introduced (such as $U(r) = C_1 r^\beta [1 - (r_h/r)^\omega]$), specifying a horizon at $r = r_h$. Finite temperature arises from periodic identification in Euclidean continuation, with temperature given by $T = (C_1 \omega / 4\pi) r_h^{\beta - 1}$. Entropy vanishes as $T \to 0$ for $\beta > 1$ (Perlmutter, 2010).

The holographic dictionary relates such bulk geometries to IR scaling phases of gauge theories at finite density, capturing universal behaviors such as entropy scaling and transport properties in the far-infrared.

3. Domain Wall Holography and Nonconformal Vacua

Domain wall holography (DW/QFT) generalizes standard AdS/CFT by using bulk vacua that are not asymptotically AdS, but domain wall-type backgrounds characterized by metrics of the form ($$ds^2 = (A r)^{\gamma'} \eta_{\mu\nu} dx^\mu dx^\nu + dr^2 / (A r)^{\gamma'}$$). The scalar field typically exhibits logarithmic running, and both the potential and gauge coupling are taken as exponentials of the scalar. Domain wall backgrounds are conformally related to AdS via Weyl rescaling ($\Omega(r) = (A r)^{\gamma' - 2}$), underpinning the generalized scale invariance found in certain strongly coupled field theories, and establishing holographic mapping even when the boundary theory is not truly conformal (in the usual sense) (Perlmutter, 2010).

This framework recovers universal IR behaviors (e.g., Lifshitz-type scaling, scaling entropies) observed in nonconformal D-brane constructions and applies even when the bulk potential admits AdS critical points, provided a single exponential dominates within the relevant regime.

4. Real-Time Correlators and Finite-Temperature Propagators

Real-time correlators, especially of two-point and higher-point functions, encode essential dynamical information such as response, transport, and dissipative phenomena. In holography, these are obtained by solving bulk wave equations (such as the Klein–Gordon equation for scalars) with appropriate boundary conditions set by the real-time contour. The thermal bulk-to-boundary propagators are constructed via analytic continuation (e.g., from Euclidean to retarded Green's functions) (Barnes et al., 2010). Thermal correlators satisfy the Kubo–Martin–Schwinger (KMS) condition, which is manifest in the boundary correlators as fluctuation-dissipation relations (FDRs).

The holographic SK prescription also allows for the systematic application of diagrammatic rules, such as Veltman's circling, to assign retarded, advanced, time-ordered, or Wightman boundary conditions at bulk vertices. These assignments are preserved by the gravitational computation and guarantee correct spectral and causal structure in the boundary correlators.

5. Generalized Scale Invariance, Lifshitz Physics, and Aging Systems

Generalized scale invariance arises in settings where the traditional conformal symmetry is broken (e.g., due to running scalars), but a residual scaling invariance is preserved after appropriate conformal transformation (the "conformal frame"). In domain wall backgrounds, the IR metric transforms into a Lifshitz-like geometry featuring a dynamical exponent $z > 1$, which governs critical phenomena at low energy (Perlmutter, 2010). Aging systems, as modeled holographically by explicit time-dependent geometries (such as AdS in light-cone coordinates with aging invariance), exhibit two-time correlators breaking time translation invariance and displaying power-law decay, slower relaxation at older waiting times, and scaling symmetry governing non-equilibrium dynamics (Hyun et al., 2011).

These features provide holographic access to out-of-equilibrium systems and universality classes beyond the reach of equilibrium AdS/CFT.

6. Thermalization, Transport, and Fluctuation-Dissipation Relations

Transport coefficients (conductivity, viscosity, Hall response) are encoded holographically through linear perturbations in the bulk and extraction of real-time retarded correlators. For example, phase transitions in charged dilatonic black branes are signaled by condensation of scalar operators, and the conductivity displays non-monotonic dependence on temperature and frequency (such as Drude peaks and synchrotron resonances). In the zero-temperature limit, universal power-law suppression of DC conductivity ($\sigma_{DC} \sim T^2$), and optical conductivity scaling ($\text{Re}[\sigma(\omega)] \sim \omega^2$), arises from the near-horizon analysis of Schrödinger-like equations governing fluctuations (Cadoni et al., 2011).

Holographic formulations at finite density incorporate dissipation (infalling quasi-normal modes) and fluctuations (Hawking radiation), yielding nonlinear FDRs via SK and KMS relations. Diagrammatic computation of n-point functions via Witten diagrams in the bulk exterior region systematically recovers fluctuation-dissipation theorems at arbitrary density and temperature (Sharma, 29 Jan 2025).

7. Numerical and Effective Field Theory Context

Global interpolating solutions between IR scaling regimes and UV completions are typically constructed via shooting methods, matching expansions near horizon and boundary, and validating the analytic scaling ansätze (Perlmutter, 2010). Effective field theory principles motivate the applicability of domain wall holography even when the full UV completion is unknown; universal IR scaling is attributed to a dominant single exponential in the bulk potential. In practice, numerical trial parameters ($D=4, \eta=1/2, \alpha=1, V_0=6, r_h=1$) match UV boundary constants to the predicted domain wall forms, confirming interpolation and supporting universality.

Table: Key Elements and Example Equations

Element	Description	Example Formula
Near-extremal black brane	Finite T charged solution; horizon shields singularity	$U(r)=C_1 r^\beta [1-(r_h/r)^\omega]$
Domain wall metric	Nonconformal holography; conformally related to AdS	$$ds^2=(A r)^{\gamma'} \eta_{\mu\nu} dx^\mu dx^\nu+dr^2/(A r)^{\gamma'}$$
Conformal frame transformation	Reveals generalized scale invariance / Lifshitz symmetry	$$\tilde{ds}^2 = \Omega(r) ds^2, \Omega(r) = (A r)^{\gamma'-2}$$
Optical conductivity at low T	Universal scaling at horizon	$\text{Re}[\sigma(\omega)] \sim \omega^2$
Effective field theory relation	D-brane inspired running coupling	$g_{eff}^2(E) = g_{YM}^2 N E^{p-3}$

References to Key Papers and Context

• Domain wall holography and global interpolation: (Perlmutter, 2010)
• Real-time correlators, SK prescription, nonrelativistic backgrounds: (Barnes et al., 2010, Botta-Cantcheff et al., 2018, Arias et al., 2023)
• Aging holography and non-equilibrium correlators: (Hyun et al., 2011)
• Nonlinear fluctuation-dissipation relations at finite density: (Sharma, 29 Jan 2025)
• Charged dilatonic black brane transport and phase transitions: (Cadoni et al., 2011)
• Context for effective field theory and nonconformal D-branes: (Perlmutter, 2010)

Summary

Real-time finite-temperature holography constitutes a robust framework for analyzing the dynamics, transport, and universal scaling behaviors of strongly coupled quantum field theories out of equilibrium. Through precise construction of bulk geometries, implementation of real-time boundary conditions, analytic and numerical matching, and diagrammatic evaluation of response functions, it provides deep insight into IR scaling solutions, generalized scale invariance, fluctuation-dissipation relations, and phase transitions in both conformal and nonconformal regimes, with broad applicability to condensed matter, QCD, and statistical physics.