Nonlinear completion of the projection-covariant endpoint dictionary

Derive a unique nonlinear completion, directly from the reduced action, of the projection-covariant prescription identifying quantum-extremal-surface and defect-extremal-surface endpoint locations beyond linear order in the brane fluctuation.

Background

The paper compares quantum extremal surface endpoints in the partially reduced JT description with defect-extremal-surface endpoints in the higher-dimensional AdS3/BCFT2 geometry. At finite brane angle, the raw AdS2-slicing coordinates of the perturbed endpoints differ at first order. The authors resolve this discrepancy at linear order by using a background projection onto the reduced coordinate, interpreted as a tangential or Stückelberg compensation, which restores agreement between the endpoint locations through O(Φr).

This construction is explicitly presented only as a linearized prescription. The unresolved problem is to derive a unique nonlinear projection map from the reduced action, thereby determining how the endpoint dictionary and associated finite-angle corrections should be extended beyond linear order. This would clarify the origin of possible finite-cutoff, nonlocal, or additional Stückelberg terms generated by integrating out the transverse AdS wedge.