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arXiv:2306.16555 · Advances in Theoretical and Mathematical Physics

Topological defect lines in bosonized Parafermionic CFTs

Haghighat Babak, Youran Sun

Published Jan 1, 2024• 2 citations
arXivHTMLPDF

Abstract

Topological defect lines (TDLs) are extended line operators which act on the Hilbert space of two-dimensional CFTs and satisfy non-trivial fusion algebras when forming junctions. Among the most interesting fusion algebras are the so-called Tambara-Yamagami (TY) fusion categories which are realized in (bosonized) Parafermionic CFTs. The corresponding TY[$\mathbb{Z}_k$]-categories have been explicitly realized for the cases $k=2$, $k=3$, and $k=4$ together with the action of the defect lines on the Hilbert space of the corresponding CFTs. For each of the cases, different methods have been used in the previous literature. In the current paper, we present a unified framework for finding the TDLs in bosonized Parafermionic CFTs. Our approach relies on combining several previously used methods and the definition of an extended $S$ matrix. We apply the method to the cases $k=2$ to $k=5$ to extract corresponding TDL fusion algebras.

DOI
10.4310/atmp.241031012317

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Topological defect lines (TDLs) are extended line operators which act on the Hilbert space of two-dimensional CFTs and satisfy non-trivial fusion algebras when forming junctions. Among the most interesting fusion algebras are the so-called Tambara-Yamagami (TY) fusion categories which are realized in (bosonized) Parafermionic CFTs. The corresponding TY[Zk\mathbb{Z}_kZk​]-categories have been explicitly realized for the cases k=2k=2k=2, k=3k=3k=3, and k=4k=4k=4 together with the action of the defect lines on the Hilbert space of the corresponding CFTs. For each of the cases, different methods have been used in the previous literature. In the current paper, we present a unified framework for finding the TDLs in bosonized Parafermionic CFTs. Our approach relies on combining several previously used methods and the definition of an extended SSS matrix. We apply the method to the cases k=2k=2k=2 to k=5k=5k=5 to extract corresponding TDL fusion algebras.
We consider topological defect lines (TDLs) in two-dimensional fermionic conformal field theories (CFTs). Besides inheriting all the properties of TDLs in bosonic CFTs, TDLs in fermionic CFTs could host fermionic defect operators at their endpoints and junctions. Furthermore, there is a new type of TDLs, called q-type TDLs, that have no analog in bosonic CFTs. Their distinguishing feature is an extra one-dimensional Majorana fermion living on the worldline of the TDLs. The properties of TDLs in fermionic CFTs are captured in the mathematical language of the super fusion category. We propose a classification of the rank-2 super fusion categories generalizing the \mathbb{Z}_8 ℤ 8 classification for the anomalies of \mathbb{Z}_2 ℤ 2 symmetries. We explicitly solve the F-moves for all the nontrivial categories, and derive the corresponding spin selection rules that constrain the spectrum of the defect operators. We find the full set of TDLs in the standard fermionic minimal models and a partial set of TDLs in the two exceptional models, which give CFT realizations to the rank-2 super fusion categories. Finally, we discuss a constraint on the renormalization group flow that preserves a q-type TDL.
We discuss in this paper the lattice discretizations of all topological defect lines (TDLs) for diagonal, minimal CFTs, using integrable restricted solid-on-solid (RSOS) models. For these CFTs, the TDLs can be labeled by the Kac labels. In the case of (1,s)(1,s)(1,s) TDLs, lines that are exactly topological on the lattice can be obtained using the centralizer of the underlying Temperley-Lieb algebra, all the other lines become topological in the continuum limit only. Our general construction relies on insertions of rows/columns of faces with modified spectral parameters, and can therefore be studied using integrability techniques. We determine the regions of spectral parameters realizing the different (r,s)(r,s)(r,s) TDLs, and in particular calculate analytically all the associated eigenvalues (and degeneracy factors). We also show how fusion of TDLs can be obtained from fusion hierarchies in the algebraic approach to the Bethe-ansatz. All our results are checked numerically in detail for several minimal CFTs.
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.
We study generalized discrete symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. In particular, we describe 't Hooft anomalies and classify gapped phases stabilized by these symmetries, including new 1+1D topological phases. The algebra of these operators is not a group but rather is described by their fusion ring and crossing relations, captured algebraically as a fusion category. Such data defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this fusion category acting as a global symmetry defines a boundary condition. This is akin to gauging a discrete global symmetry at the boundary of Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category symmetry in these boundary conditions and separate the symmetry-preserving phases from the symmetry-breaking ones. For Tambara-Yamagami categories and their generalizations, which are associated with Kramers-Wannier-like self-dualities under orbifolding, we develop gauge theoretic techniques which simplify the analysis. We include some examples of CFTs with fusion category symmetry derived from Kramers-Wannier-like dualities as an appetizer for the Part II companion paper.
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.
A bstract Topological defect lines (TDLs) in two-dimensional conformal field theories (CFTs) are standard examples of generalized symmetries in quantum field theory. Integrable lattice incarnations of these TDLs, such as those provided by spin/anyonic chains, provide a crucial playground to investigate their properties, both analytically and numerically. Here, a family of parameter-dependent integrable lattice models is presented, which realize different TDLs in a given CFT as the parameter is varied. These models are based on the general quantum-inverse scattering construction, and involve inhomogeneities of the spectral parameter. Both defect hamiltonians and (defect) line operators are obtained in closed form. By varying the inhomogeneities, renormalization group flows between different TDLs (such as the Verlinde lines associated with the Virasoro primaries (1 , s ) and ( r, 1) in diagonal minimal CFTs) are then studied using different aspects of the Bethe-ansatz as well as ab-initio numerical techniques. Relationships with the anisotropic Kondo model as well as its non-Hermitian version are briefly discussed.