A Holographic prescription for generalized Schwinger-Keldysh contours

TL;DR

The paper addresses the challenge of computing real-time thermal correlators with arbitrary operator ordering, including out-of-time-order correlators (OTOCs), in holography. It introduces a holographic prescription based on a multi-fold Schwinger-Keldysh contour dual to a bulk geometry grSK_n, built from multiple AdS-black-hole segments glued along horizons and constrained by unitarity and the KMS condition. For the four-fold case, it provides a concrete analysis of a probe scalar: contact diagrams vanish for OTOCs, while exchange diagrams yield nontrivial, factorized radial integrals that encode the OTOC structure, and it outlines a general conjecture for higher-order observables with partial proofs. The framework offers a systematic route to compute fully real-time gravitational dynamics, including higher-point correlators and chaotic signatures, and opens avenues for thermal bootstrap and extensions to conserved currents and more general states.

Abstract

We provide a holographic prescription to compute real-time thermal correlators with arbitrary operator ordering. In field theory, these correlation functions are captured by a multi-fold Schwinger-Keldysh time contour. We propose a holographic dual for these contours, which generalizes the gravitational Schwinger-Keldysh geometry previously advocated in the literature. Our geometry consists of multiple AdS-black holes glued together at the future and past horizons, with matching conditions determined by unitarity and the KMS condition. As a proof of concept, we solve for a probe scalar field in this geometry and compute bulk-bulk and bulk-boundary propagators, in terms of which we evaluate the 4-point functions at tree-level. We show that in perturbation theory, the lowest-order diagrams that contribute non-trivially to the out-of-time order four-point function are exchange diagrams which explore the full four-fold geometry. Furthermore, these diagrams reduce to a simple factorized expression. We propose a conjecture on the structure of higher order observables and provide a partial proof by studying a subset of the contributing diagrams.

Figures (27)

• Figure 1: The Schwinger-Keldysh contour for a thermal state $\rho_{\beta}$. We include a segment of Euclidean time evolution, preparing the state, and the endpoints are identified as $\tau \sim \tau+\beta$.
• Figure 2: $4-$fold contour for the thermal state.
• Figure 3: Two equivalent SK correlators, related by unitarity.
• Figure 4: Penrose diagrams for two copies of the domain of outer communication of the AdS black hole. The gravitational Schwinger-Keldysh geometry is obtained by identifying the horizons (blue) and joining the regions $t=0$ (dashed, red) to an Euclidean black hole geometry.
• Figure 5: Hankel contour describing the complexified radial coordinate.