Symmetries of perturbed conformal field theories

TL;DR

The paper investigates how symmetries of conformal field theories behave under exactly marginal bulk perturbations, focusing on current-current deformations Φ=Jar{J}. It derives an all-orders criterion for bulk chiral generators to survive, showing the preserved bulk symmetry is $ ext{A}_{ ext{inv}}= ext{A}$ with $J_0 S=0$, and analyzes how boundary gluing automorphisms can be adjusted to maintain bulk symmetries, leading to a boundary spectrum that forms twisted representations of $ ext{A}_{ ext{inv}}$. It provides concrete examples in Gepner models, WZW models, diagonal torus branes, and $ ext{SU}(2) imes ext{SU}(2)$ permutation branes, and discusses matrix-factorisation perspectives for complex structure deformations. The results imply branes always preserve as much symmetry as possible, given the bulk deformation, and illuminate open-string moduli and boundary theories in perturbed backgrounds. Overall, the work clarifies the interplay between bulk and boundary symmetries under marginal deformations and offers precise criteria and illustrative examples for their persistence and structure.

Abstract

The symmetries of perturbed conformal field theories are analysed. We explain which generators of the chiral algebras of a bulk theory survive a perturbation by an exactly marginal bulk field. We also study the behaviour of D-branes under current-current bulk deformations. We find that the branes always continue to preserve as much symmetry as they possibly can, i.e. as much as is preserved in the bulk. We illustrate these findings with several examples, including permutation branes in WZW models and B-type D-branes in Gepner models.

Figures (3)

• Figure 1: An illustration of the gluing of the left- and right-moving chiral algebras: By the gluing condition \ref{['basic']}, the subalgebra $\bar{\mathcal{A}}_{\text{inv}}$ is glued to $\omega(\mathcal{A}_{\text{inv}})$, whereas $\mathcal{A}_{\text{inv}}$ is glued to $\omega^{-1} (\bar{\mathcal{A}}_{\text{inv}})$. After the deformation, only the fields in $\mathcal{A}_{\text{inv}}$ and $\bar{\mathcal{A}}_{\text{inv}}$ stay chiral, which means that it only makes sense to glue the fields in $\mathcal{A}_{\text{c}}$ and $\omega^{-1} (\bar{\mathcal{A}}_{\text{c}})$.
• Figure 2: When one of the radii in the two-torus is deformed, the brane continues to stretch diagonally and its inclination changes.
• Figure 3: The diagonal brane with length $L_{b}$ and a string with length $L_{s}$ that winds perpendicular to the brane around the torus.