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arXiv:2504.06203 · Physical Review Letters

Factorizing Defects from Generalized Pinning Fields

Fedor K. Popov, Yifan Wang

Published Nov 12, 2025• 7 citations
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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 O(N) model.

DOI
10.1103/rh7j-yg11

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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 O(N) model.
The factorization proposal claims that the codimension one “pinning defect,” on which a local relevant operator is integrated, factorizes the space into two halves in general conformal field theories in the infrared limit. In this paper, we study a two-dimensional long-range Ising model at criticality with a line defect or an interface, which physically corresponds to changing the local temperature on it. We show that in the perturbative regime, it is not factorizing even in the infrared limit. An intuitive explanation of the nonfactorization is that the long-range Ising model is equivalent to a local conformal field theory in higher dimensions. In this picture, the space is still connected through the “extra dimension” across the defect line.
We consider defect conformal field theories (DCFTs) at finite temperature, where a ppp-dimensional conformal defect wraps the thermal circle. Extending the previous study of line defects to defects of general dimensions, we determine the general forms of the thermal one-point functions of bulk scalars, conserved currents and the stress tensor by imposing the residual symmetry and the conservation laws. The low temperature expansion of the thermal one-point functions is shown to be reproduced by the bulk-defect operator expansion, allowing us to read off the thermal one-point coefficients of defect operators. We examine the general results in four classes of examples with free bulk theories: the trivial defect, free scalar and fermion theories with a boundary, the free scalar theory with a localized ϕϕϕ deformation in d=2 p+2d=2\,p+2d=2p+2 dimensions and the free O(N)\mathrm{O}(N)O(N) model with a localized ϕ2ϕ^2ϕ2 deformation in d=p+2−εd=p+2-εd=p+2−ε dimensions. In the last example with ε=1ε= 1ε=1, where the defect becomes an interface, we find that the thermal one-point functions coincide with those of the free scalar theory with the Dirichlet boundary condition, in accordance with the conjectured factorization of an interface CFT into two decoupled boundary CFTs.
Motivated by the three-dimensional topological field theory/two-dimensional conformal field theory (CFT) correspondence, we study a broad class of one-dimensional quantum mechanical models, known as anyonic chains, which can give rise to an enormously rich (and largely unexplored) space of two-dimensional critical theories in the thermodynamic limit. One remarkable feature of these systems is the appearance of non-local microscopic “topological symmetries” that descend to topological defects of the resulting CFTs. We derive various model-independent properties of these theories and of this topological symmetry/topological defect correspondence. For example, by studying precursors of certain twist and defect fields directly in the anyonic chains, we argue that (under mild assumptions) the two-dimensional CFTs correspond to particular modular invariants with respect to their maximal chiral algebras and that the topological defects descending from topological symmetries commute with these maximal chiral algebras. Using this map, we apply properties of defect Hilbert spaces to show how topological symmetries give a handle on the set of allowed relevant deformations of these theories. Throughout, we give a unified perspective that treats the constraints from discrete symmetries on the same footing as the constraints from topological ones.
A bstract The study of quantum impurities has long been a central and inspiring theme in quantum many-body physics. Localized impurities are modeled by line defects in quantum field theory. We describe a line defect in Liouville CFT realized as a “localized cosmological constant”: a non-topological line insertion into the Liouville path integral that is tractable at both weak and strong defect coupling. At weak coupling, we analyze the defect perturbatively and characterize it through its correlations with local operators, energy and information transport, the Casimir energies associated with fusion, and corrections to the open string channel spectrum. We also study the effect of a cuspidal deformation of the defect locus on these observables and describe novel monotonicity properties as the cusp angle is varied. These results derived using perturbation theory are more generally applicable to pinning defects constructed from scalar primary operators in compact 2 d CFTs. At strong coupling, in a semiclassical limit, the defect admits a geometric interpretation in terms of a discontinuity in the extrinsic curvature of the 1 d defect locus embedded in 2 d hyperbolic geometries. The observables characterizing the defect in this regime are computed by gluing hyperbolic surfaces across the defect, and are compared with the corresponding weak coupling results. The correlations across the defect, both at weak and strong coupling, can also be realized by an effective “decohered FZZT interface” constructed by diagonal gluing of two copies of the fixed-length FZZT boundary state. These line defects also have interesting interpretations in other models, in terms of end-of-the-world branes in Jackiw-Teitelboim gravity, dust shells in AdS 3 gravity, and interfaces with a proliferation of non-abelian Wilson loops in 4 d N \mathcal{N} N N = 2 gauge theories.
Abstract In this paper we study the constraints imposed by conformal invariance on extended objects a.k.a. defects in a conformal field theory. We identify a particularly nice class of defects that is closed under conformal transformations. Correlation function of the defect with a bulk local operator is fixed by conformal invariance up to an overall constant. This gives rise to the notion of defect expansion, where the defect itself is expanded in terms of local operators. This expansion generalizes the idea of the boundary state. We will show how one can fix the correlation function of two defects from the knowledge of the defect expansion. The defect correlator admits a number of conformal cross-ratios depending on their dimensionality. We find the differential equation obeyed by the conformal block and solve them in certain special cases.
Abstract We investigate non-local conserved charges in perturbed two-dimensional conformal field theories (CFT) from the point of view of the 3d symmetry TFT (SymTFT) of the unperturbed theory. In the SymTFT we state a simple commutation condition which results in a pair of compatible bulk and defect perturbations, such that the perturbed line defects are conserved in the perturbed CFT. In other words, the perturbed defects are rigidly translation invariant, and such defects form a monoidal category which extends the topological symmetries. As examples we study the A-type Virasoro minimal models M ( p , q ) . Our formalism provides one-parameter families of commuting non-local conserved charges for perturbations by a primary bulk field with Kac label (1, 2), (1, 3), or (1, 5), which are the standard integrable perturbations of minimal models. We find solutions to the commutation condition also for other bulk perturbations, such as (1, 7), and we contrast this with the existence of local conserved charges. There has been recent interest in the possibility that in certain cases perturbations by fields such as (1, 7) can be integrable, and our construction provides a new way in which integrability can be found without the need for local conserved charges.