Boundary flow and geometric realization in holographic $T\bar T$-deformed BCFT

TL;DR

This paper investigates the $Tar T$ deformation of BCFTs from a field-theoretic and holographic perspective, deriving exact relations and boundary actions, and analyzing two holographic models to understand boundary effects.

Contribution

It provides an intrinsic, boundary-localized formulation of $Tar T$ deformation in BCFTs and establishes a holographic dual analysis with two distinct realizations.

Findings

Derived exact quadratic stress tensor relation without cutoff.

Obtained a boundary action reorganizing boundary data.

Identified differences between holographic models in boundary displacement.

Abstract

We study the $T\bar T$ deformation of boundary conformal field theories (BCFTs) from an intrinsic field-theoretic perspective. Formulating the deformation as a modification of the asymptotic variational principle in AdS$_3$, we obtain the exact quadratic trace relation for the stress tensor without introducing a finite radial cutoff, which we take as the fundamental definition of the deformed theory. When restricted to a BCFT without independent boundary degrees of freedom, the intrinsic $T\bar T$ deformation becomes genuinely boundary-localized. Imposing reflective boundary conditions collapses the bulk composite operator to a universal one-dimensional irrelevant flow governed entirely by the displacement operator. We integrate this flow in closed form and derive an induced boundary action, showing that the deformation reorganizes existing boundary data without introducing new boundary degrees of freedom. We further establish a precise equivalence between a fixed-boundary description and a moving-boundary description, interpreted as a reparametrization of the variational problem rather than physical boundary dynamics. On the holographic side, we analyze two inequivalent realizations in AdS$_3$/BCFT$_2$, referred to as Type~A and Type~B. In Type~A, a rigid cutoff surface intersects the end-of-the-world brane at finite position, leading to an apparent boundary displacement. In Type~B, the cutoff surface is asymptotically AdS$_2$, so that the BCFT boundary is geometrically pinned and the displacement operator vanishes identically. Using entanglement entropy at zero and finite temperature, we disentangle universal consequences of the intrinsic boundary-localized flow from features that depend on the holographic implementation.