Boundary constraints and geometric realizations in holographic TT ̄-deformed BCFT

A bstract We study the intrinsic TT ̄ 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 Tₜₜ=-D/1-π λ 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 TT ̄ 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.