Factorizing Defects from Generalized Pinning Fields

TL;DR

This work provides a rigorous framework for generalized pinning field defects in conformal field theories, defining UV defects via ${\cal D}_h({\mathcal{O}})=\left[e^{h \hat{{\mathcal{O}}}}\right]_{\rm ren}$ with $\hat{{\mathcal{O}}}=\int_{\Sigma_p}{\mathcal{O}}$ and showing that codimension-one defects IR-factorize into a Cardy-state projector ${|{\mathcal{B}}\rangle\langle {\mathcal{B}}|}$ on the bulk rigged Hilbert space. The factorization is established through a careful regularization of the unbounded operator ${\hat{{\mathcal{O}}}}$ within a Gelfand triple, followed by weak limits in spectral parameters and renormalization, yielding an IR description in terms of conformal boundary conditions; symmetry constraints via fusion categories further refine the allowed factorization channels, yielding idempotent defect fusion. The authors demonstrate this mechanism by solving pinning flows in 2d diagonal minimal models (including Ising and tricritical Ising) and in the 3d ${\rm O}(N)$ CFT, obtaining explicit IR decompositions into Cardy boundaries and familiar boundary universality classes (ordinary, special, extraordinary-log). These results unify defect RG flows with conformal boundary data, constrain long-distance defect dynamics, and offer a tractable route to predicting interface physics in strongly coupled CFTs, with potential extensions to holography and QCD-like conformal windows.

Abstract

We introduce generalized pinning fields in conformal field theory that model a large class of critical impurities at large distance, enriching the familiar universality classes. We provide a rigorous definition of such defects as certain unbounded operators on the Hilbert space and prove that when inserted on codimension-one surfaces they factorize the spacetime into two halves. The factorization channels are further constrained by symmetries in the bulk. As a corollary, we solve such critical impurities in the 2d minimal models and establish the factorization phenomena previously observed for localized mass deformations in the 3d ${\rm O}(N)$ model.

Figures (4)

• Figure 1: IR factorization of the pinning flow via the strip flow by resolution. Here it is assumed that strip flow of the CFT ${\mathcal{T}}$ ends in a gapped phase (orange region) described by TQFT ${\mathcal{T}}_{\rm gap}$ with Cardy branes $|\nu\rangle$.
• Figure 2: The fusion graphs for rank $r$ fusion categories ${\mathcal{W}}_r$ and ${\mathcal{R}}_r$. The shaded node is the generator of the graph. The $n$-th node corresponds to either ${\mathcal{L}}_{1,2n-1}$ or ${\mathcal{L}}_{2n-1,1}$ (see discussion around \ref{['smallrankWR']}).
• Figure 3: The fusion graphs for the rank $r{-}1$ module category of ${\mathcal{R}}_r$. This module category can be represented in terms of the other topological defects in the minimal model, with the $n$-th node corresponding to either ${\mathcal{L}}_{1,2n}$ or ${\mathcal{L}}_{2n,1}$.
• Figure 4: The first diagram defines the generalized pinning flows from ${\mathcal{N}}$ with operator ${\mathcal{O}}$ in the ${\mathbb Z}_2$ twisted sector. The last two diagrams arise from topological moves in the Ising CFT relating different pinning flows.