A bstract In this work, we study the realization of non-invertible duality symmetries along the toroidal branch of the c = 2 conformal manifold. A systematic procedure to construct symmetry defects is implemented to show that all Rational Conformal Field Theories along this branch enjoy duality symmetries. Furthermore, we delve into an in-depth analysis of two representative cases of multicritical theories, where the toroidal branch meets various orbifold branches. For these particular examples, the categorical data and the defect Hilbert spaces associated with the duality symmetries are obtained by resorting to modular covariance. Finally, we study the interplay between these novel symmetries and the various exactly marginal and relevant deformations, including some representative examples of Renormalization Group flows where the infrared is constrained by the non-invertible symmetries and their anomalies.
A bstract In this work, we study the realization of non-invertible duality symmetries along the toroidal branch of the c = 2 conformal manifold. A systematic procedure to construct symmetry defects is implemented to show that all Rational Conformal Field Theories along this branch enjoy duality symmetries. Furthermore, we delve into an in-depth analysis of two representative cases of multicritical theories, where the toroidal branch meets various orbifold branches. For these particular examples, the categorical data and the defect Hilbert spaces associated with the duality symmetries are obtained by resorting to modular covariance. Finally, we study the interplay between these novel symmetries and the various exactly marginal and relevant deformations, including some representative examples of Renormalization Group flows where the infrared is constrained by the non-invertible symmetries and their anomalies.
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.
A bstract We discuss two-dimensional conformal field theories (CFTs) which are invariant under gauging a non-invertible global symmetry. At every point on the orbifold branch of c = 1 CFTs, it is known that the theory is self-dual under gauging a ℤ 2 × ℤ 2 symmetry, and has Rep( H 8 ) and Rep( D 8 ) fusion category symmetries as a result. We find that gauging the entire Rep( H 8 ) fusion category symmetry maps the orbifold theory at radius R to that at radius 2/ R . At R = 2 2 , which corresponds to two decoupled Ising CFTs (Ising 2 in short), the theory is self-dual under gauging the Rep( H 8 ) symmetry. This implies the existence of a topological defect line in the Ising 2 CFT obtained from half-space gauging of the Rep( H 8 ) symmetry, which commutes with the c = 1 Virasoro algebra but does not preserve the fully extended chiral algebra. We bootstrap its action on the c = 1 Virasoro primary operators, and find that there are no relevant or marginal operators preserving it. Mathematically, the new topological line combines with the Rep( H 8 ) symmetry to form a bigger fusion category which is a ℤ 2 -extension of Rep( H 8 ). We solve the pentagon equations including the additional topological line and find 8 solutions, where two of them are realized in the Ising 2 CFT. Finally, we show that the torus partition functions of the Monster 2 CFT and Ising × Monster CFT are also invariant under gauging the Rep( H 8 ) symmetry.
We develop a systematic construction of renormalization group (RG) domain walls between two-dimensional rational conformal field theories. Using phantom currents, we consider an auxiliary chiral extension linked to the folded ultraviolet--infrared theory by a topological interface. The construction of RG walls then reduces to identifying suitable automorphism-twisted Cardy states in the auxiliary theory. We find two such types of automorphisms: outer automorphisms exchanging identical current algebra factors, as in Gaiotto's construction, and continuous inner automorphisms generated by the zero modes of spin-one phantom currents. In the latter case, the RG domain walls generically admit a continuous defect conformal manifold. We further show that, when an RG flow can be orbifolded to another one, the orbifold may lift defect conformal moduli of the RG brane constructed using inner automorphisms. The two types of RG branes can then be mapped to each other via a topological interface defined by the orbifold. We demonstrate the two types of RG brane construction via various examples of flows between non-unitary minimal models, from parafermions to minimal models, and between generalized parafermions. Many physical observables of the RG walls are computed non-perturbatively, including boundary entropies and spectra, and UV--IR operator mixings. We also find interesting nontrivial boundary monodromy: a closed cycle in the defect conformal moduli space can carry an RG domain wall into other distinct non-trivial conformal interfaces.
Two-dimensional Conformal Field Theories (CFTs) allow us to describe quantum field theories at critical points and in general systems without a characteristic scale. It is possible to implement such theories on topologically non-trivial surfaces. If we formulate a two-dimensional CFT on a toroidal surface, a new symmetry emerges known as modular invariance. This symmetry corresponds to the set of transformations that act without changing the toroidal surface, consequently, the partition function of the theory must remain invariant under these transformations. This invariance is used as a constraint by the modular bootstrap method to obtain information about the spectrum without using the Lagrangian and operator content of the theory. By imposing that the theory is consistent with modular symmetry, it is possible to determine the limits of the spectrum beyond which modular invariance can no longer be satisfied, thus constraining the space of admissible two-dimensional conformal field theories. To study CFTs that present discrete global symmetries, it is essential to introduce Topological Defect Lines (TDLs). These are one-dimensional, topologically invariant operators that implement the action of the symmetry group on the Hilbert space. The presence of TDLs modifies the topology of the system and introduces new partition functions associated with different symmetry sectors. Under modular transformations, the TDLs mix among themselves, and consequently, the corresponding partition functions also mix, thus promoting the modular constraint to a vector-valued condition. This formalism makes it possible to implement discrete symmetries also considering the possible presence of anomalies. The objective of this thesis is therefore to understand how discrete symmetries and anomalies affect the spectrum of a CFT. Based on the modular bootstrap framework developed by Lin and Shao for the Z2 symmetry, the analysis was first analytically extended to the Abelian case Z3. This was useful in extending the discussion to a more complex case, namely the non-Abelian symmetry S3. This extension constitutes the original theoretical contribution of the thesis, providing the explicit formulation of modular bootstrap equations for CFTs with non-Abelian discrete symmetry. The analytical modular constraints were reformulated numerically, bridging the modular bootstrap framework with semidefinite programming in SDPB. This analysis paves the way for a numerical study based on convex optimization techniques to explore the constraints arising from modular bootstrap equations.
In this Letter we continue the investigation of RG flows between Virasoro minimal models of two-dimensional conformal field theories that are protected by noninvertible symmetries. RG flows leaving unbroken a subcategory of noninvertible symmetries are associated with anomaly matching conditions that we employ systematically to map the space of flows between minimal models beyond the Z_{2}-symmetric proposed recently in the literature. We introduce a family of nonlinear integral equations that appear to encode the exact finite-size, ground-state energies of these flows, including nonintegrable cases, such as the recently proposed M(kq+I,q)→M(kq-I,q). Our family of NLIEs encompasses and generalizes the integrable flows known in the literature: ϕ_{(1,3)}, ϕ_{(1,5)}, ϕ_{(1,2)} and ϕ_{(2,1)}. This work uncovers a new interplay between exact solvability and noninvertible symmetries. Furthermore, our nonperturbative description provides a nontrivial test for all the flows conjectured by anomaly matching conditions, but so far not observed by other means.
RG flows of minimal W-algebra CFTs via non-invertible symmetries
In this letter we study renormalization group (RG) flows between 2d conformal field theories enjoying extended higher-spin W-symmetry. We propose a new class of RG flows between the diagonal minimal models of WN-algebra that take the form WN(p,q)→WN(p,kp−q). These are obtained by matching the anomalies of the non-invertible symmetry Rep[SU(N)p−N] (and its discrete quotients) that is preserved by special relevant primary fields. This large non-invertible symmetry includes the familiar ZN symmetry of the minimal models. Our new flows furnish a significant generalization of the ones recently found in the case of Virasoro algebra, and include all previously known RG flows of WN. They have the remarkable property of being uniform in the rank N of the W-algebra.