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

Feiyu Deng

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.

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

Abstract

We study the 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, 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 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/BCFT, 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, 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.
Paper Structure (25 sections, 151 equations, 8 figures)

This paper contains 25 sections, 151 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Conceptual summary and scope.
  3. Intrinsic boundary formulation of the TTbar deformation and its holographic realizations
  4. BCFT kinematics and boundary Ward identities
  5. Stress tensor.
  6. Ward identities.
  7. Displacement operator
  8. Intrinsic definition of the $T\bar{T}$ deformation in AdS$_3$
  9. Two equivalent holographic pictures
  10. Boundary-localized flow induced by the TTbar deformation
  11. Restriction of the TTbar operator to the boundary
  12. Boundary trace relation and solution in terms of the displacement operator
  13. Boundary-localized flow equation
  14. Why the boundary-localized $T\bar{T}$ flow is governed by the displacement operator
  15. Fixed--moving boundary equivalence
  16. ...and 10 more sections

Figures (8)

  • Figure 1: Bulk picture of the Type A deformation in pure AdS$_3$.
  • Figure 2: RT surface for the Type A deformation in pure AdS$_3$.
  • Figure 3: Bulk picture of the Type A deformation in the BTZ background.
  • Figure 4: RT surface for the Type A deformation in the BTZ case.
  • Figure 5: Bulk picture of the Type B deformation in the pure AdS case. The bulk region is the slab $\rho_0\le\rho\le\rho_c$ in AdS$_2$ slicing. The Dirichlet surface at $\rho=\rho_c$ has an induced AdS$_2$ metric and intersects the EOW brane only at the asymptotic boundary $y\to0$, so the BCFT boundary remains fixed.
  • ...and 3 more figures