CGL Contour in Holography and Dissipative Dynamics

• The Crossley–Glorioso–Liu contour is a holographic realization of Schwinger–Keldysh doubling, using analytic continuation of the bulk radial coordinate to geometrically encode field-theory duplicates.
• It manifests both as a horizon-cap prescription and an analytically continued bulk contour, effectively incorporating thermal KMS periodicity and causal response into holographic calculations.
• The construction underpins derivations of exact thermal correlators and dissipative hydrodynamic actions by linking bulk gravitational dynamics with Einstein constraints and fluctuation–dissipation relations.

Searching arXiv for papers on the Crossley–Glorioso–Liu contour and related holographic Schwinger–Keldysh constructions. The Crossley–Glorioso–Liu contour is a holographic implementation of the Schwinger–Keldysh real-time contour in which the bulk radial coordinate is analytically continued so that the doubled field-theory degrees of freedom are realized geometrically rather than introduced only at the level of the boundary effective theory. In the recent literature it appears in two closely related forms: as a horizon-cap prescription that glues two copies of a black-brane exterior through a small contour around the horizon, and as an analytically continued bulk contour with two asymptotic boundaries joined near the horizon. In both forms, its function is to encode dissipation, Kubo–Martin–Schwinger periodicity, and causal response in holography; in particular, it has been used to derive exact thermal Schwinger–Keldysh correlators for probe scalars and a dissipative hydrodynamic action for the stress tensor (Banerjee et al., 2022, Blake et al., 1 Jul 2026).

1. Definition and geometric realization

The defining statement of the construction is that the Schwinger–Keldysh doubling is implemented by a complexified radial contour in the bulk. One formulation describes the prescription as deriving Schwinger–Keldysh actions “from solving bulk dynamics on a contour in which the bulk radial direction is suitably analytically continued,” while another presents it as a “horizon cap” that realizes the thermal contour in the bulk by circling the horizon in the complexified radial plane (Blake et al., 1 Jul 2026, Banerjee et al., 2022).

In the equilibrium scalar construction, the contour does not use an eternal black hole with two asymptotic boundaries. Instead, it takes two copies of the same black-brane exterior, corresponding to the forward and backward Schwinger–Keldysh legs, and glues them through a complex radial contour looping around the horizon. The second copy has reversed time orientation so that the glued geometry has a single orientation. The horizon cap is a small clockwise contour in the complexified radial plane around the horizon $r=r_h$, obtained by analytically continuing from one exterior to the other along a small circle of radius $\epsilon$ (Banerjee et al., 2022).

In the dissipative stress-tensor construction, by contrast, the analytically continued CGL contour is described as having two asymptotic boundaries. The two straight branches of the contour are said to capture the two exteriors of the maximally extended black-hole spacetime, and the corresponding asymptotic source data $g_{\mu\nu}^{1(s)}$ and $g_{\mu\nu}^{2(s)}$ couple to the two Schwinger–Keldysh copies. The matching between branches is implemented by analytic continuation near the horizon, for example through

$\log(r-r_0)\rightarrow \log(r-r_0)-2\pi i .$

The contour used there is explicitly the analytic continuation of the radial coordinate shown in the paper’s Figure 1, rather than an arbitrary deformation (Blake et al., 1 Jul 2026).

A plausible implication is that the invariant content of the CGL contour is not a unique global spacetime topology, but the horizon-centered analytic continuation that implements branch doubling and real-time thermal physics.

2. Schwinger–Keldysh structure and horizon analyticity

The CGL construction organizes real-time observables in the standard Schwinger–Keldysh basis. In the stress-tensor analysis, the doubled hydrodynamic fields and sources are written in average/difference form as

$$\xi_{\mu}^{+} = \frac{\xi_{\mu}^1 + \xi_{\mu}^2}{2}, \qquad \xi_{\mu}^{-} = \xi_{\mu}^1 - \xi_{\mu}^2,$$

$$\delta g_{\mu \nu}^{+ (s)} = \frac{\delta g_{\mu \nu}^{1 (s)} + \delta g_{\mu \nu}^{2 (s)}}{2}, \qquad \delta g_{\mu \nu}^{- (s)} = \delta g_{\mu \nu}^{1 (s)} - \delta g_{\mu \nu}^{2 (s)} .$$

This is the standard average/difference organization of dissipative Schwinger–Keldysh EFTs (Blake et al., 1 Jul 2026).

In the scalar equilibrium derivation, the thermal KMS factor arises from the continuation of the outgoing mode around the horizon cap. If $p$ multiplies the ingoing mode and $q$ the outgoing mode, the cap continuation imposes

$$p_2(\omega,\mathbf{k})=p_1(\omega,\mathbf{k}), \qquad q_2(\omega,\mathbf{k})=e^{\beta\omega} q_1(\omega,\mathbf{k}) .$$

The corresponding bulk mode matrix is

$\epsilon$0

which directly reproduces the thermal Schwinger–Keldysh matrix factorization and therefore the KMS relation

$\epsilon$1

The same derivation yields the Son–Starinets identifications

$\epsilon$2

so that retarded response is the ingoing solution and advanced response is the outgoing solution, at arbitrary frequency and momentum for a minimally coupled probe scalar (Banerjee et al., 2022).

A central structural point is that the on-shell action receives contributions only from ingoing–outgoing cross terms. The paper shows that $\epsilon$3 and $\epsilon$4, leaving only the mixed term. This matches the Schwinger–Keldysh normalization condition $\epsilon$5 and explains why the horizon-cap continuation reproduces the full thermal matrix rather than only its retarded component (Banerjee et al., 2022).

3. Relative diffeomorphisms and the dissipative hydrodynamic action

In the stress-tensor construction, the hydrodynamic degrees of freedom are not added by hand at the boundary; they are realized as relative diffeomorphisms between the black-hole horizon and the two asymptotic boundaries of the CGL contour. In the transverse channel one has

$\epsilon$6

while in the longitudinal sector

$\epsilon$7

with an analogous integral expression for $\epsilon$8. These fields are therefore bulk geometric data measuring horizon-to-boundary relative diffeomorphisms on each Schwinger–Keldysh branch (Blake et al., 1 Jul 2026).

The action depends on the hydrodynamic fields and sources only through diffeomorphism-invariant combinations. In the transverse sector,

$\epsilon$9

and the longitudinal channel is organized analogously through $g_{\mu\nu}^{1(s)}$0, $g_{\mu\nu}^{1(s)}$1, $g_{\mu\nu}^{1(s)}$2, and $g_{\mu\nu}^{1(s)}$3. The nonlinear presentation rewrites the theory in terms of induced-metric variables $g_{\mu\nu}^{1(s)}$4, which furnish a manifestly diffeomorphism-invariant form of the hydrodynamic action (Blake et al., 1 Jul 2026).

The derivation proceeds by expanding the bulk action around equilibrium,

$g_{\mu\nu}^{1(s)}$5

solving the radial dynamical Einstein equations, and deliberately leaving some constraint equations unimposed. Those remaining constraints become the equations of motion of the hydrodynamic fields. The paper states this as the key off-shell statement that the hydrodynamic equations of motion coincide with the Einstein constraint equations left off shell. The resulting action is valid to quadratic order in perturbations about thermal equilibrium and to first order in the hydrodynamic derivative expansion (Blake et al., 1 Jul 2026).

Dissipation enters through the contour at the horizon. The near-horizon analytic continuation and continuity conditions on logarithmically singular modes generate explicitly $g_{\mu\nu}^{1(s)}$6-weighted difference-sector terms. In the longitudinal channel, for example,

$g_{\mu\nu}^{1(s)}$7

and the transverse sector contains the corresponding term proportional to $g_{\mu\nu}^{1(s)}$8. These are the dissipative Schwinger–Keldysh terms generated by horizon matching rather than imposed phenomenologically (Blake et al., 1 Jul 2026).

4. Correlation functions in equilibrium and nonequilibrium

At the level of generating functionals, the relevant Schwinger–Keldysh structure is

$g_{\mu\nu}^{1(s)}$9

with the analogous metric-source expression used in the hydrodynamic stress-tensor analysis. This is the canonical real-time organization into response and symmetric correlators (Blake et al., 1 Jul 2026).

For the equilibrium probe-scalar problem, the horizon-cap prescription is shown to reproduce the exact thermal Schwinger–Keldysh correlator matrix at arbitrary frequency and momentum. The result is stronger than a low-frequency check: KMS periodicity and the ingoing boundary condition emerge from the contour construction itself. In the language of the paper, the prescription is consistent with the Kubo–Martin–Schwinger periodicity and with the Son–Starinets retarded prescription for all $g_{\mu\nu}^{2(s)}$0 in the scalar sector (Banerjee et al., 2022).

For nonequilibrium Bjorken flow, the contour is generalized by a Weyl rescaling and bulk diffeomorphism that make the late-time geometry resemble a static black brane in appropriate variables. In the rescaled variables, the horizon has constant surface gravity and area at late time, although no timelike Killing vector emerges. The resulting correlators admit what the paper calls a generalized bilocal thermal structure to all orders in the hydrodynamic large-proper-time expansion. In the perfect-fluid limit, the Schwinger–Keldysh correlators with space-time reparametrized arguments become thermal at an appropriate temperature (Banerjee et al., 2022).

The stress-tensor action provides the complementary result for gravitational fluctuations in AdS$g_{\mu\nu}^{2(s)}$1. After integrating out the doubled hydrodynamic fields, the resulting generating functional reproduces the known hydrodynamic Green’s functions for the stress tensor. The paper explicitly verifies consistency with hydrodynamic expectations and with the fluctuation–dissipation relation

$g_{\mu\nu}^{2(s)}$2

This establishes that the CGL contour is not merely a formal doubling device but a calculational framework for dissipative real-time observables (Blake et al., 1 Jul 2026).

5. Horizon boundary conditions, preserved symmetries, and chaos-related structure

Because the two branches of the contour are stitched together at the horizon, horizon boundary conditions are a constitutive part of the construction. In the stress-tensor analysis, the preferred conditions are

$g_{\mu\nu}^{2(s)}$3

together with continuity of $g_{\mu\nu}^{2(s)}$4 and $g_{\mu\nu}^{2(s)}$5 after analytic continuation around the horizon. Appendix C is cited for the statement that, with these conditions, the horizon variation vanishes and the equations of motion of the hydrodynamic action coincide with the Einstein constraints left unimposed (Blake et al., 1 Jul 2026).

The radial-gauge-preserving diffeomorphisms that preserve these boundary conditions induce specific shifts of the hydrodynamic fields $g_{\mu\nu}^{2(s)}$6. Among these are exponentially decaying and exponentially growing terms. The paper explicitly relates these horizon symmetries to conjectured hydrodynamic symmetries responsible for many-body quantum chaos, but also states that the exponentially growing mode obtained under the preferred conditions has a profile different from that proposed by Knysh–Liu–Pinzani-Fokeeva (Blake et al., 1 Jul 2026).

The scalar Bjorken-flow analysis exhibits an allied horizon sensitivity in a different language. There the horizon cap must be pinned to the nonequilibrium event horizon rather than to the apparent horizon. The reason given is regularity: cancellation of singular terms in the large-proper-time expansion fixes residual gauge freedom so that the cap remains attached to the event horizon order by order. The apparent horizon differs from the event horizon beginning at second order, and pinning the cap to the apparent horizon would fail the required analyticity and regularity conditions (Banerjee et al., 2022).

Alternative horizon boundary conditions were also studied in the stress-tensor setting: $g_{\mu\nu}^{2(s)}$7 These are precisely the conditions whose preserving diffeomorphisms reproduce the horizon symmetries of Knysh:2024asf. However, the paper emphasizes that for these alternative conditions the variational principle is not manifestly consistent, so one cannot straightforwardly conclude that they are symmetries of the full hydrodynamic action beyond the order checked (Blake et al., 1 Jul 2026).

6. Conceptual scope, variants, and unresolved points

Several recurrent misunderstandings are explicitly corrected by the recent literature. First, the CGL contour is not synonymous with one immutable bulk geometry. One paper formulates it without an eternal black hole with two asymptotic boundaries, using instead two copies of a single exterior glued by a horizon cap, while another uses an analytically continued contour with two asymptotic boundaries and interprets its straight branches as the two exteriors of a maximally extended black hole (Banerjee et al., 2022, Blake et al., 1 Jul 2026).

Second, the contour is not an arbitrary Schwinger–Keldysh deformation. In the dissipative hydrodynamic construction it is the specific analytic continuation of the radial coordinate around the horizon branch point, and the novelty lies not in redefining the contour but in using it to derive a stress-tensor dissipative action from gravity (Blake et al., 1 Jul 2026).

Third, one should not attribute to the stress-tensor paper an explicit derivation of dynamical KMS symmetry. The paper states that such a formulation would be desirable and that a future, more abstract treatment may allow a more precise relation to the CGL formulation and a more direct KMS verification. What it does establish is the fluctuation–dissipation relation at the order computed. Likewise, it does not formulate the result explicitly in terms of an entropy-current symmetry (Blake et al., 1 Jul 2026).

The nonequilibrium scalar generalization also has clearly stated limits. Its exact all-$g_{\mu\nu}^{2(s)}$8 proof applies to a minimally coupled probe scalar in equilibrium, while the Bjorken-flow extension is controlled in the hydrodynamic late-time expansion rather than in arbitrary far-from-equilibrium dynamics. The paper further proposes, rather than fully derives, that the Stokes data governing the trans-series completion of the hydrodynamic correlator expansion should be functions of bilocal separations and should encode information about initial conditions and quantum fluctuations behind the horizon cap (Banerjee et al., 2022).

Taken together, these results define the Crossley–Glorioso–Liu contour as a horizon-analyticity prescription for real-time holography. Its operational content is the geometric realization of Schwinger–Keldysh doubling, the emergence of KMS factors from analytic continuation of outgoing modes, the emergence of retarded causality from ingoing regularity, and, in the gravitational stress-tensor sector, the identification of hydrodynamic variables with relative diffeomorphisms between horizon and boundary (Banerjee et al., 2022, Blake et al., 1 Jul 2026).