Gravitational Schwinger-Keldysh Geometry

• Gravitational Schwinger-Keldysh Geometry is a framework that unites real-time path integral methods with holography to describe dissipation, response, and chaos in gravitational systems.
• It employs BRST symmetries and thermal cohomology to enforce gauge invariance and capture fluctuation-dissipation dynamics in strongly-coupled quantum fields.
• The approach constructs mixed-signature bulk geometries that model hydrodynamic effective actions, stochastic noise, and out-of-time-ordered correlator behavior.

Gravitational Schwinger-Keldysh Geometry is an overview of real-time quantum field theoretic techniques, topological and cohomological structures, and gravitational (often black hole or cosmological) backgrounds designed to capture the dynamics, fluctuations, and dissipation of quantum systems in and out of equilibrium when gravity is present or dualized via holography. Originating in the need to describe real-time correlation functions and effective dynamics of strongly-coupled systems, this framework integrates closed-time path (Schwinger-Keldysh) formalism, thermal equivariant cohomology, and doubled (or complexified) gravitational geometries. The resulting structure provides the correct language for dissipation, hydrodynamic response, the emergence of soft modes, and chaos (scrambling) in gravitational systems.

1. Real-Time Doubling and Path Integral Contour in Gravity

At the foundation of gravitational Schwinger-Keldysh geometry is the doubling of degrees of freedom to construct a generating functional for real-time correlators. The formalism operates on a contour in the complex time plane (the Schwinger-Keldysh or "Keldysh" contour), which evolves the system forward with a set of sources $J_R$ and backward with another set $J_L$:

$$Z_{\text{SK}}[J_R, J_L] = \mathrm{Tr}\left[ U[J_R]\, \rho_0\, U^\dagger[J_L] \right]$$

In gravitational (specifically, holographic) settings, this doubling is geometrized by "filling in" the contour in the bulk with corresponding spacetime segments—typically a combination of Lorentzian and Euclidean black hole geometries. For example, a mixed-signature spacetime may be built by gluing two Lorentzian patches (dual to forward/backward evolution) to a Euclidean "cap" that encodes the thermal initial state (Boer et al., 2018, Glorioso et al., 2018, Pantelidou et al., 2022, Haehl et al., 14 Oct 2024). The process is schematically depicted as:

Field Theory	Holographic Dual
SK timefold	Glued Lorentzian/Euclidean BH
$J_R$, $J_L$	Two boundaries/segments
KMS periodicity	Euclidean time circle

The mixed-signature (complexified) bulk geometry precisely mirrors the SK contour, with the "cut" and "glue" operations defining both the statistical ensemble and the real-time evolution.

2. BRST Topology, Thermal Cohomology, and Extended Supersymmetry

The doubling of fields in the SK formalism leads naturally to underlying topological (BRST-like) symmetries. From an algebraic viewpoint, each observable becomes part of a quadruplet—a SK supermultiplet—naturally extended in a superspace labeled by two Grassmann-odd coordinates $(\theta, \bar\theta)$. The superalgebra has nilpotent charges $Q$, $\bar Q$, and, in thermal contexts, additional KMS supercharges and derivations (Haehl et al., 2016, Geracie et al., 2017, Jensen et al., 2018):

$Q^2 = \bar Q^2 = 0, \quad \{Q, \bar Q\} \sim L$

where $L$ generates infinitesimal (thermal) KMS translations. Thermal equivariant cohomology recasts the SK-KMS algebra in geometric terms, identifying physical observables as basic forms (horizontal and invariant) under these supersymmetries. In the cohomological language, these symmetries ensure that causal, fluctuation-dissipation, and unitarity constraints survive renormalization and organize low-energy effective actions.

The BRST extension is crucial in gravity/gauge contexts because the dynamical variables (metric, connection) possess gauge redundancies. The cohomological formalism not only enforces the Slavnov-Taylor identities in non-equilibrium gauge theories (Czajka et al., 2014), but also underpins anomaly-related transport (via inflow and Chern-Simons terms) and consistency of the operator product expansion and difference operators (Geracie et al., 2017, Jensen et al., 2018).

3. Bulk Geometries: Mixed-Signature and Complexification

To implement the SK formalism in gravitational settings, especially in holography, one builds a bulk geometry whose topology is dictated by the SK contour. For a system at finite temperature, the gravitational dual is an eternal AdS black hole whose two boundaries represent the doubled degrees of freedom. These are joined by Euclidean "caps" (cigar-like geometries), which encode the KMS periodicity (Boer et al., 2018, Glorioso et al., 2018, Pantelidou et al., 2022, Haehl et al., 14 Oct 2024).

An explicit construction frequently uses complexified radial coordinates (mock tortoise coordinates $\zeta$) to "wrap" around the black hole horizon. The analytic continuation involves circling the horizon in the complex $r$-plane, matching boundary conditions for fields on both R/L segments. For a bulk field $\phi$, solutions take the schematic form:

$$\phi(\zeta, \omega, \mathbf{k}) = G_{\text{in}}(\omega, \mathbf{k}, \zeta)\left[(1+n_\omega)J_R - n_\omega J_L\right] - G_{\text{in}}(-\omega, \mathbf{k}, \zeta)\, e^{\beta \omega (1-\zeta)}(J_R - J_L)$$

where $G_{\text{in}}$ is the ingoing bulk-to-boundary propagator, and $n_\omega$ is the thermal occupation factor (Jana et al., 2020, Haehl et al., 14 Oct 2024).

This prescription is equivalent, up to matching conditions, to imposing retarded (ingoing) conditions at the horizon on a single Lorentzian BH geometry (Pantelidou et al., 2022).

4. Effective Actions, Dissipation, and Fluctuations

The gravitational SK framework yields real-time effective actions encoding dissipative and stochastic hydrodynamics. In holographic models, infrared (near-horizon) dynamics are isolated, and effective actions for soft (hydrodynamic) modes are written in terms of "average" ($r$) and "difference" ($a$) fields—paralleling the r/a basis of SK field theory:

$$S_{\text{eff}} = \int \left( \chi B_{r,t}B_{a,t} - \sigma B_{a,x}\partial_t B_{r,x} + i\sigma T B_{a,x}^2 + \cdots \right)$$

Dissipation arises via imaginary (noise) terms, enforcing generalized fluctuation-dissipation theorems (Boer et al., 2018, Bu et al., 2020). All-order expansions in frequency/momentum are possible, either via analytic gradient expansions or numeric solutions to coupled bulk ODEs. The colored (momentum-dependent) nature of the noise reflects the fact that the gravitational "environment" (e.g., the black hole) is responsible for statistical fluctuations and memory effects (Bu et al., 2020).

An immediate application is the derivation of diffusion constants, conductivities, and stochastic constitutive relations for conserved U(1) currents from the holographic dual—where all transport coefficient functions are accessible both analytically (in the hydrodynamic limit) and numerically at finite momenta (Bu et al., 2020).

5. Gauge and Diffeomorphism Invariance, Ghost Sectors, and Transversality

A critical structural feature of Schwinger-Keldysh geometry in gravity/gauge theory is the systematic enforcement of gauge or diffeomorphism invariance, which ensures the cancellation of unphysical (longitudinal/pure gauge) modes. This is done via the explicit inclusion of Faddeev-Popov ghosts in the SK generating functional (Czajka et al., 2014), leading to Slavnov-Taylor identities such as

$$- p^\mu\, \mathcal{D}_{\mu\nu}^{ab}(p) = p_\nu\, \Delta_{ab}(-p)$$

where $\mathcal{D}(p)$ and $\Delta(p)$ are gluon and ghost propagators respectively. These identities guarantee, for example, the transversality ($k_\mu \Pi^{\mu\nu}(k)=0$) of the polarization tensor, both in equilibrium and non-equilibrium, a requirement for physical, gauge-invariant observables.

The Hilbert space construction for the SK-BRST symmetry involves quadrupling and ghost-dressing of operators, necessary for consistent OPEs and unitarity of the effective theory—even in gravitational systems (Geracie et al., 2017, Haehl et al., 2016).

6. Quantum Information, Scrambling, and Out-of-Time-Ordered Correlators

Gravitational Schwinger-Keldysh geometry underpins the analysis of chaos, scrambling, and information flow in black holes. Out-of-time-ordered correlators (OTOCs)—whose evaluation requires extended SK (k-OTO) contours—probe the exponential growth of commutators and the spread of perturbations ("scrambling"):

$$\text{OTOC}(T, x) = \frac{\mathrm{Tr}(V^\dagger W^\dagger(T, x)\, V\, W(T, x)\, \rho_\beta)}{\mathrm{Tr}(V^\dagger V\, \rho_\beta)\, \mathrm{Tr}(W^\dagger W\, \rho_\beta)}$$

In gravity duals, these correlators are computed by evaluating scattering in black hole geometries perturbed by shockwaves, yielding an eikonal phase associated with Lyapunov growth:

$$\delta_\text{grav}(s, b) \propto G_N s\, e^{T - \mu b}$$

where $s$ is the center-of-mass energy, $b$ the impact parameter, and $\mu$ a scale set by the BH geometry (Haehl et al., 14 Oct 2024). In the low-energy limit of AdS$_2$ (and SYK-like models), the relevant degrees of freedom are Schwarzian "scramblon" modes, whose fluctuation effective actions encode the maximal Lyapunov exponent and the structure of gravitational chaos.

7. Boundary Dynamics, Soft Modes, and Null Geodesic Motion

The gravitational SK geometry is sensitive to boundary conditions and the dynamics of "edge" or "soft" modes. For spacetimes with boundaries (e.g., black hole horizons, AdS asymptotics), large diffeomorphisms non-trivial at the boundary carve out a space of gravitational vacua parameterized by boundary data. The effective (induced) metric on this vacuum space is pseudo-Riemannian and is fully determined by the boundary (Kutluk et al., 2019):

$$\langle \zeta_1, \zeta_2 \rangle_h = \oint_{\partial M} \sqrt{k}\, d^2 y\, \left( \zeta_1^a D^\perp \zeta_{2,a} - K_{ab} \zeta_1^a \zeta_2^b \right)$$

The Hamiltonian constraint enforces null geodesics on this space, imposing light-like (tensionless) motion for the evolutionary dynamics of boundary or soft (Goldstone) modes. This structure helps in organizing IR (memory) effects, understanding asymptotic symmetries, and analyzing the vacuum degeneracy in quantum gravity.

8. Minimal Length Scales and Quantum Gravitational Observables

The SK (in-in) formalism is also instrumental in clarifying the minimum length issue in quantum gravity (Casadio et al., 2020). Causal, in-in expectation values of the proper distance vanish smoothly as points coincide,

$$\lim_{x \to y} \langle ds^2 \rangle_\text{in-in} = 0,$$

indicating no minimum geometric length, even when including all-order and non-perturbative corrections. In contrast, in-out amplitudes (as in S-matrix processes) can feature a nonzero minimal scale, of order the Planck length, due to the regularization brought in by Feynman (as opposed to retarded) propagators in scattering observables.

9. Diagrammatics, Hydrodynamic Expansions, and Transport

Diagrammatic rules and effective actions in gravitational SK geometry generalize to contexts such as cosmological perturbations (Chen et al., 2017), fluid dynamics (Boer et al., 2018, Bu et al., 2020), and string-theoretic perturbation theory (Horava et al., 2020). In cosmology, the SK diagrammatic method provides a systematic expansion for the calculation of in-in (expectation value) correlators, even in the presence of derivative couplings, and enables tractable computation of signatures such as the quantum primordial standard clock.

The hydrodynamic effective actions derived from holography, expressed in terms of r/a variables, reproduce dissipative constitutive relations with fluctuation-induced colored noise and nonlocal memory kernels, making the SK formalism the preferred language for modern hydrodynamic and transport phenomena at strong coupling.

---

Gravitational Schwinger-Keldysh geometry thus forms a unifying framework, blending cohomological symmetry, thermal/dissipative dynamics, holography, effective field theory, and real-time quantum information features. Its structure is central for describing dissipation, response, anomaly-induced transport, hydrodynamic effective theories, and chaos in quantum gravitational systems, providing both precise computational prescriptions and a robust symmetry-protected underpinning for the organizing principles of non-equilibrium gravity.