Permeable conformal walls and holography

TL;DR

The paper develops a comprehensive framework for permeable domain walls between 2D CFTs that preserve Virasoro symmetry, using folding and boundary-state methods to study transmission, reflection, and the Casimir energy ${\cal E}=-\frac{1}{8\pi d}{\rm Li}_2({\cal R}^2)$ with ${\cal R}=\frac{1-\lambda^2}{1+\lambda^2}$. It extends from a simple free scalar to supersymmetric cases and Kähler targets, deriving half-superfield constructions that preserve supersymmetry across interfaces and generalizing to holomorphic branes in product manifolds. The work then applies these ideas to holographic AdS$_3$ setups (NS5/F1) where domain walls separate CFTs with different central charges and moduli, performing a supergravity calculation of the Casimir energy for a stretched $(p,q)$ string and relating it to moduli-flow pictures in symmetric-product orbifolds; fundamental vs. D-string junctions yield qualitatively different behaviors, illuminating the limits of gravity/CFT comparisons. The discussion connects domain walls to Nakajima algebras on instanton moduli spaces, suggesting rich algebraic structures and potential extensions to multi-wall junctions and condensed-matter interfaces. Overall, the paper provides a versatile toolkit for analyzing permeable CFT interfaces and their holographic avatars, with implications for both field theory and string theory.

Abstract

We study conformal field theories in two dimensions separated by domain walls, which preserve at least one Virasoro algebra. We develop tools to study such domain walls, extending and clarifying the concept of `folding' discussed in the condensed-matter literature. We analyze the conditions for unbroken supersymmetry, and discuss the holographic duals in AdS3 when they exist. One of the interesting observables is the Casimir energy between a wall and an anti-wall. When these separate free scalar field theories with different target-space radii, the Casimir energy is given by the dilogarithm function of the reflection probability. The walls with holographic duals in AdS3 separate two sigma models, whose target spaces are moduli spaces of Yang-Mills instantons on T4 or K3. In the supergravity limit, the Casimir energy is computable as classical energy of a brane that connects the walls through AdS3. We compare this result with expectations from the sigma-model point of view.

Figures (8)

• Figure 1: Conformal Ward identities are obtained by inserting $\ \oint_C [T f(z)dz - {\bar{T}}{ f}(\bar{z})d{\bar{z}}]$ in correlation functions. In deforming the contour from $C_1$ to $C_2$ we pick up contributions from the broken-line segments. These cancel out provided $T-{\bar{T}}$ is continuous. The crosses in the figure stand for local field insertions.
• Figure 2: The moduli space of gluing matrices, $M(\vartheta)$ on the left and $M^\prime(\vartheta)$ on the right, where $\vartheta = {\rm arctan}\lambda \in [-\pi/2,\pi/2]$. Perfectly-reflecting walls are labeled by the two boundary conditions, Dirichlet (D) or Neumann (N), on either side of the defect. Totally-transmitting defects are labeled by the periodicity properties of $(\partial_-\phi,\partial_+\phi)$ when $x$ is compactified on a circle.
• Figure 3: The incoming and outgoing waves can be related by the matrix $S$ .
• Figure 4: The region of rescaled radius ($r_2=\lambda r_1$) bounded by a defect and an anti-defect. Time runs in the upward direction. The interfaces feel an attractive Casimir force.
• Figure 5: Folding the plane along the defect line leads to a description of the permeable defects as regular D-branes in a two-dimensional target space.