Topological defect lines and renormalization group flows in two dimensions

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.

Anomalies and bounds on charged operators

“…An invariant way to characterize a global symmetry and its 't Hooft anomaly is by the associated (invertible) topological defect lines L [11][12][13][14][15][16][17][18][19][20][21][22][23]. See [24,25] for modern applications of topological defect lines to renormalization group flows and gauging. 4 These are extended objects in quantum field theory whose contraction of a loop around a local operator φ(x) implements the symmetry transformation.…”

“…4 In this paper, we focus on invertible topological defect lines, which are associated to global symmetries. There are also non-invertible ("non-symmetry") topological defect lines that have interesting consequences on the dynamics of quantum field theory (QFT) under renormalization group (RG) flows [25].…”

“…Note that even though we start with a bosonic CFT, there are anyonic or fermionic operators living at the end of the Z 2 line, depending on whether the Z 2 is anomalous or not. In the non-anomalous case, this is analogous to the emergent fermionic excitations of lattice spin models [28], and we will review a general CFT derivation in [25]. This spin selection rule in turn constrains the light operator spectrum in the Hilbert space of local operators H via modular transformations, as we will discuss below.…”

Open topological defects and boundary RG flows

“…In the simplest situation the original insertion remains intact and we can say that the defect commutes with this bulk insertion. As shown in [11], if we have some defects that commute with a bulk relevant operator then there are interesting consequences for the bulk renormalisation group (RG) flows triggered by this operator. The fusion algebra of such commuting defects between themselves must be robust under the fusion and this places constraints on the end points of the flows (triggered by the same operator with positive or negative coupling) particularly when the flows are massive and the end points may be described by non-trivial topological theories.…”

3d one-form mixed anomaly and entanglement entropy

“…We check our proposal for known examples. 1-form symmetry in 3d bulk topological field theory becomes 0-form symmetry in 2d boundary theory, because 1-dimensional symmetry lines become the topological defect lines of 0-form symmetry in 2d [30,31]. This can be seen from the 2d counterpart of 3d Chern-Simons theory, the Wess-Zumino-Witten models.…”