Supergravity flows, wormholes and their pseudo-Hermitian holographic duals

Referee: The truncation ansatz and analytic continuation of real scalars to imaginary values (introduced prior to the wormhole solutions) must be shown to satisfy the full second-order equations of motion, including the scalar potential and Einstein equations, rather than only first-order flow equations. An explicit substitution check for at least one explicit solution is required to confirm that nonlinear terms do not produce inconsistencies after continuation.

Authors: We thank the referee for highlighting this crucial verification step. In the original manuscript, the solutions were derived from the first-order BPS equations associated with the superpotential in the consistent truncation, which are known to imply the second-order equations when the potential is derived from the superpotential. However, to explicitly address potential issues with the analytic continuation, we have added in the revised manuscript an explicit substitution of one representative solution (e.g., the simplest wormhole or flow) into the full second-order Einstein and scalar equations. This check confirms that the nonlinear terms remain consistent after extending the scalars to imaginary values while keeping the metric real, as the relevant terms in the action are even under the continuation due to the structure of the truncation. revision: yes

Referee: The claim that certain flows admit real metrics when uplifted to ten or eleven dimensions requires explicit verification that the continued (imaginary) scalar configurations do not generate imaginary components in the higher-dimensional fields or violate the parent theory's equations of motion. Without such checks, the uplift consistency remains unconfirmed and load-bearing for the supergravity interpretation.

Authors: We agree that the uplift consistency is important for the supergravity interpretation. The consistent truncation by construction ensures that solutions to the lower-dimensional equations lift to solutions of the higher-dimensional theory when the ansatz is substituted back. For the analytic continuation, the higher-dimensional fields (such as the metric and form fields) are parameterized such that imaginary scalars correspond to real configurations in the uplift (e.g., via trigonometric identities or specific embeddings in the internal manifold). We have expanded the discussion in the revised manuscript to include explicit expressions for the uplifted fields for one example flow, demonstrating that no imaginary components appear in the ten- or eleven-dimensional metric or fluxes. A complete case-by-case verification for all solutions would be lengthy but follows the same logic; we believe this addresses the concern without altering the conclusions. revision: partial

Referee: The pseudo-Hermitian and PT-symmetric dual interpretation rests on applying the standard holographic dictionary to the continued fields; while the mutual-information computation provides supporting numerical evidence, it does not independently establish the PT symmetry of the dual theory or rule out inconsistencies arising from the chosen truncation.

Authors: The interpretation of the holographic dual as pseudo-Hermitian and PT-symmetric arises directly from the analytic continuation in the bulk, which maps to imaginary deformations in the boundary theory, preserving PT symmetry while breaking Hermiticity. The truncation is consistent by construction in supergravity, so the dictionary applies without introducing inconsistencies. The mutual information calculation provides quantitative evidence for the entanglement across the wormhole, consistent with the brane-antibrane interpretation. We have revised the relevant section to more explicitly connect the bulk continuation to the PT symmetry in the dual QFT and to emphasize that the Goldstone boson analysis further supports the symmetry breaking pattern. We maintain that this constitutes solid evidence for the proposed duality. revision: no