Abstract
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.
Highlights
Two-dimensional conformal field theories (CFTs) are usually defined in terms of the data of local “bulk” point-like operators, namely the spectrum of Virasoro primaries and their structure constants, subject to the associativity of operator product expansion (OPE) and modular invariance
We consider UV CFTs realizing twisted siblings of the Rep(S3) fusion category, and rule out the possibility of flowing to IR topological quantum field theories (TQFTs) with unique vacuum. It is a priori not obvious whether a fusion category of topological defect lines (TDLs) can always be realized by some TQFT, as the latter requires the construction of defect operators and is subject to modular invariance, neither of which is directly captured by the fusion category structure
Minimal model perturbed by φ2,1 to flow to a TQFT in the IR that admits the TDLs X√ and Y
Summary
Two-dimensional conformal field theories (CFTs) are usually defined in terms of the data of local “bulk” point-like operators, namely the spectrum of Virasoro primaries and their structure constants, subject to the associativity of operator product expansion (OPE) and modular invariance. In all known examples of global symmetries in a CFT, the corresponding invertible lines are subject to a strong locality property, namely, that it can end on defect operators, which obey an extended set of OPEs.. The general TDLs of interest in this paper are models of a more general mathematical structure known as fusion category [41, 42] (at least when there are finitely many simple lines), which still requires the crossing relations to obey the pentagon identity, but does not require braiding. We consider UV CFTs realizing twisted siblings of the Rep(S3) fusion category, and rule out the possibility of flowing to IR TQFTs with unique vacuum It is a priori not obvious whether a fusion category of TDLs can always be realized by some TQFT, as the latter requires the construction of defect operators and is subject to modular invariance, neither of which is directly captured by the fusion category structure. Some further details of the H-junction crossing kernels and explicit solutions to the pentagon identities are given in appendices A and C
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