Boundary constraints and geometric realizations in holographic TT̄-deformed BCFT

Overview

We study the intrinsic $T\overline{T }$ deformation of boundary conformal field theories from the viewpoint of boundary Ward identities and holographic realizations. We formulate the deformation through the mixed asymptotic variational principle in AdS$_{3}$, which gives the standard quadratic trace relation for the renormalized stress tensor without taking a finite cutoff surface as the definition of the theory.For the stress-tensor/displacement sector of a BCFT without an independent boundary stress tensor, the reflective condition T$_{tx|Σ}$ = 0 implies a pointwise reduction of the quadratic stress-tensor data. In a semiclassical factorized state, and with boundary contact terms treated separately, this gives an algebraic constraint relating the boundary energy-density and displacement one-point functions. Suppressing expectation brackets in this formula, we find ${T}_{tt}=-\frac{\mathcal{D}}{1-\pi \lambda \mathcal{D}}.$ This relation should be understood as a local boundary constraint following from the trace Ward identity, not as a definition of a separate boundary term in the full integrated $T\overline{T }$ flow.On the holographic side we analyze two cutoff realizations in AdS$_{3}$/BCFT$_{2}$. Type A uses a rigid radial cutoff, while Type B uses an AdS$_{2}$ cutoff surface. In both cases the deformation coupling is fixed by Brown-York trace matching; for Type B the dictionary depends on the Weyl frame used to normalize the induced AdS$_{2}$ metric. At zero and finite temperature we compare the RT results with standard BCFT entropy formulas evaluated on the same induced geometries and with the same endpoint prescriptions. These are internal RT/BCFT consistency checks, not independent Brown-York derivations or tests of a separate boundary evolution law.

Context

The intrinsic deformation is formulated through a mixed asymptotic variational principle. A reflective boundary is considered in the stress-tensor/displacement sector without an independent boundary stress tensor.

Main result

In a semiclassical factorized state, with boundary contact terms treated separately, the trace Ward identity gives a local algebraic constraint between boundary energy density and displacement one-point functions. Type A and Type B cutoff realizations are compared with matched Weyl frames, induced geometries and endpoint prescriptions. The entropy comparisons check internal RT/BCFT consistency.

Why it matters

Clarifies the regime in which boundary Ward constraints and matched cutoff geometries describe a TT̄-deformed BCFT. The results are local algebraic constraints and internal entropy consistency checks.

Citation

@article{Deng:260107509,
  author = {Deng, Feiyu},
  title = {Boundary constraints and geometric realizations in holographic TT̄-deformed BCFT},
  year = {2026},
  eprint = {2601.07509},
  archivePrefix = {arXiv},
  primaryClass = {hep-th},
  journal = {JHEP},
  volume = {09},
  pages = {249},
  doi = {10.1007/JHEP09(2026)249}
}