Replica-symmetric saddle existence and stability in time-dependent Lorentzian settings

Establish the existence and stability of semiclassical replica-symmetric (\u2124_n-symmetric) saddles for Lorentzian replica path integrals in fully general time-dependent geometries, clarifying their dependence on spacetime details and on the Schwinger–Keldysh contour prescription.

Background

Many derivations of entanglement entropy via the replica trick assume that the dominant gravitational or field-theoretic replica geometries respect \u2124_n symmetry. In Lorentzian, time-dependent contexts (e.g., quenches, expanding cosmologies, evaporating black holes), such an assumption becomes nontrivial and its validity has not been proven in full generality.

The authors explicitly rely on this assumption while acknowledging that the existence and stability of these saddles have not been established for fully dynamical real-time backgrounds. Resolving this issue would solidify the foundations of Lorentzian replica constructions and clarify when replica-symmetric saddles correctly dominate.

References

The existence and stability of such saddles in fully general time-dependent geometries is not proven and may depend on the details of the spacetime and contour prescription.

— Replica Trick in Time-Dependent Geometries  (2601.08756 - Irakleous, 13 Jan 2026) in Section 2, footnote to the assumption of semiclassical dominance of replica-symmetric saddles

Other open problems include massive embeddings, magnetic fields, the low-temperature finite-density scaling limit, a fully controlled Lorentzian probe-replica derivation and a direct study of disorder-driven long-range entanglement.

— Density-driven reversal of flavour-entanglement anisotropy in the D3-D7 holographic conductor  (2609.26641 - Okabayashi, 22 Sep 2026) in Section 6.5, “Scope and limitations”