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.
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.
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.
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.
We study perturbations of topological defects by a relevant holomorphic defect field in two-dimensional conformal field theory. This produces translation invariant defects and these necessarily flow to topological defects in the infrared. Translation invariant defects form a monoidal category which extends the fusion category of topological defects. We argue that the renormalisation group flow defines a semiring homomorphism from the semiring of translation invariant defects to an "asymptotic semiring" which we construct from topological defects. The semiring homomorphism preserves functional relations, and we illustrate in the example of Virasoro minimal models that the resulting relations in the asymptotic semiring constrain the defect flow endpoints. The companion paper [1] contains a detailed study of such defect flows in several minimal models, both numerically and analytically.
Abstract In this paper and its sequel, we construct topologically invariant defects in two-dimensional classical lattice models and quantum spin chains. We show how defect lines commute with the transfer matrix/Hamiltonian when they obey the defect commutation relations , cousins of the Yang–Baxter equation. These relations and their solutions can be extended to allow defect lines to branch and fuse, again with properties depending only on topology. In this part I, we focus on the simplest example, the Ising model. We define lattice spin-flip and duality defects and their branching, and prove they are topological. One useful consequence is a simple implementation of Kramers–Wannier duality on the torus and higher genus surfaces by using the fusion of duality defects. We use these topological defects to do simple calculations that yield exact properties of the conformal field theory describing the continuum limit. For example, the shift in momentum quantization with duality-twisted boundary conditions yields the conformal spin 1/16 of the chiral spin field. Even more strikingly, we derive the modular transformation matrices explicitly and exactly.
A bstract We consider topological defect lines (TDLs) in two-dimensional conformal field theories. Generalizing and encompassing both global symmetries and Verlinde lines, TDLs together with their attached defect operators provide models of fusion categories without braiding. We study the crossing relations of TDLs, discuss their relation to the ’t Hooft anomaly, and use them to constrain renormalization group flows to either conformal critical points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-degenerate gapped state. For various massive flows, we determine the infrared TQFTs completely from the consideration of TDLs together with modular invariance.
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.