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arXiv:2504.05277 · Journal of Physics A Mathematical and Theoretical

Non-local charges from perturbed defects via SymTFT in 2d CFT

Federico Ambrosino, Ingo Runkel B, G.M.T. Watts

Published Sep 24, 2025• 3 citations
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Abstract

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.

DOI
10.1088/1751-8121/ae0b10

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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.
The modern way to understand symmetries of a quantum field theory is via its topological defects in various dimensions. In this contribution to the proceedings we focus on line defects in 2d QFT and we point out that topological defects naturally embed into a larger class, namely translation invariant defects. The latter still allow for non-singular fusion and one obtains a monoidal category of translation invariant defects which contains that of topological defects as a full subcategory. We give a simple perturbative description of translation invariant defects in a perturbed conformal field theory via chiral three-dimensional topological field theory. We show in the example of the Ising CFT and the Lee-Yang CFT that even if no topological defects survive the deformation, some translation invariant defects still do.
We describe a general way of constructing integrable defect theories as perturbations of conformal field theory by local defect operators. The method relies on folding the system onto a boundary field theory of twice the central charge. The classification of integrable defect theories obtained in this way parallels that of integrable bulk theories which are a perturbation of the tensor product of two conformal field theories. These include local defect perturbations of all c < 1 minimal models, as well as of the coset theories based on SO(2n), obtained in this way. We discuss in detail the former case of all the Virasoro minimal models. In the Ising case our construction corresponds to having a spin field as a defect operator; in the folded formulation this is mapped onto an orbifolding of the boundary sine-Gordon theory at β2/8π = 18, or a version of the anisotropic Kondo model.
A bstract We embark on a systematic study of continuous non-invertible symmetries, focusing on 1+1d CFTs. We describe a generalized version of Noether’s theorem, where continuous non-invertible symmetries are associated to non-local conserved currents: point-like operators attached to extended topological defects. The generalized Noether’s theorem unifies several constructions of continuous non-invertible symmetries in the literature, and allows us to exhibit many more examples in diverse theories of interest. We first review known examples which are non-intrinsic (i.e., invertible up to gauging), and then describe new examples in Wess-Zumino-Witten models and products of minimal models. For some of these new examples, we show that these continuous non-invertible symmetries are intrinsic if we demand that a certain global symmetry is preserved. The continuous non-invertible symmetries in products of minimal models also allow us to construct new examples of defect conformal manifolds in a single copy of a minimal model. Finally, we comment on continuous non-invertible symmetries in higher dimensions.
A bstract Defects in conformal field theories are interesting objects to study from both formal and applied points of view. In this paper, we construct conformal defects in free scalar field CFTs in diverse dimensions. After discussing the possible quadratic defects, we explore interacting setups. These are realized by coupling the bulk free scalar to lower-dimensional theories, including the unitary family of minimal models M(m,m+1) \mathcal{M}\left(m,m+1\right) M(m,m+1) M m m + 1 . Another example involves coupling to a two-dimensional free scalar field, from which we construct surface defects for the bulk dimensions three and five. Additionally, we consider monodromy defects associated with a global U(1) flavour symmetry. In these theories, we study both self-defect interactions and couplings to Minimal Models, finding new IR defect fixed points. For all our examples, we provide results for correlation functions, such as those involving the bulk stress tensor and the displacement operator, and for the defect central charges.
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.
Topological defects in two-dimensional conformal field theories (CFTs) are canonical examples of generalized symmetries in quantum field theories. They not only encode key features of the underlying theory, but their study has also enhanced our understanding of renormalization group (RG) flows in theories that contain them. Lattice models exist which discretize the states, energy-momentum tensor, and primary fields of many 2D CFTs, raising the natural question of whether topological defects can be similarly discretized. In this thesis, we demonstrate that in many cases all topological defects of minimal model CFTs can be discretized, enabling analytical and ab-initio numerical computations of their properties. In some of these constructions, we make essential use of the integrable structure of the underlying lattice models. The lattice realization of these operators also permits the study of defect RG flows between CFTs with different topological defects inserted. These topological defects and defect RG flows are realized on the lattice not only for Hermitian but also for non-Hermitian lattice Hamiltonians. Studying topological defects in minimal model CFTs on the lattice has also opened the door to a deeper understanding of topological defects in CFTs where they remain poorly understood.