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JHEP01(2019)026

Published for SISSA by Springer

Received: July 12, 2018

Revised: December 11, 2018

Accepted: December 23, 2018

Published: January 3, 2019

Topological defect lines and renormalization group

ﬂows in two dimensions

Chi-Ming Chang,

Ying-Hsuan Lin,

Shu-Heng Shao,

Yifan Wang

and Xi Yin

Center for Quantum Mathematics and Physics (QMAP), University of California,

Davis, CA 95616, U.S.A.

Walter Burke Institute for Theoretical Physics, California Institute of Technology,

Pasadena, CA 91125, U.S.A.

School of Natural Sciences, Institute for Advanced Study,

Princeton, NJ 08540, U.S.A.

Joseph Henry Laboratories, Princeton University,

Princeton, NJ 08544, U.S.A.

Jeﬀerson Physical Laboratory, Harvard University,

Cambridge, MA 02138 U.S.A.

E-mail: wychang@ucdavis.edu, yhlin@caltech.edu, shuhengshao@gmail.com,

yifanw@princeton.edu, xiyin@fas.harvard.edu

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.

Keywords: Anomalies in Field and String Theories, Conformal Field Theory, Global

Symmetries

ArXiv ePrint: 1802.04445

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP01(2019)026

JHEP01(2019)026

but typically obey strong notions of locality that do not obviously follow from those of the

bulk local operators.

A basic example is a global symmetry element g, which by deﬁnition is a linear trans-

formation on the bulk local operators that preserve their OPEs: the action of g on a bulk

local operator may be viewed as the contraction of a loop of a topological defect line (TDL)

on the bulk local operator [13, 32–34]. A TDL that corresponds to a global symmetry will

be referred to as an invertible line in this paper, for the reason that such a line is associ-

ated to a symmetry action and therefore its inverse must exist.

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.

In the case of a continuous global symmetry such as U(1), Noether’s theorem

states that there must be an associated conserved spin-one current j

. The contour inte-

gral of this conserved current e

iθ

then deﬁnes a family of invertible TDLs labeled by

θ ∈ S

, where the topological property follows from the conservation equation. In the case

of discrete global symmetry such as Z

, the existence of the associated invertible TDLs

with the above-mentioned properties can be thought of as a discrete version of Noether’s

theorem. It has nontrivial implications on the action of g on the bulk local operators that

do not obviously follow from the standard axioms on the bulk local operators.

Interestingly, there are TDLs that do not correspond to any global symmetries, and

they are ubiquitous in 2d CFTs [2, 7, 13, 19, 25]. This is possible because the general

TDLs need not obey group-like fusion relations, but instead form a (semi)ring (generally

non-commutative) under fusion. Though invariant under isotopy transformations (by def-

inition), a general TDL cannot simply be reconnected within the same homological class;

rather, it obeys nontrivial crossing relations under the splitting/joining operation, and

consequently, the action of a general TDL on bulk local operators by contraction need not

preserve the OPE as a global symmetry action would.

A special class of TDLs (not necessarily invertible) are known as Verlinde lines [2,

7, 35, 36]. They exist in rational CFTs deﬁned by diagonal modular invariants. The

fusion ring generated by the Verlinde lines in an RCFT is formally identical to that of

the representations of the chiral vertex algebra (although the physical interpretation of the

fusion of Verlinde lines is entirely diﬀerent from the OPEs). In particular, such a fusion ring

is commutative and admits braiding, which is not the case for the most general system of

TDLs. The fusion of general TDLs may not be commutative (as is the case for nonabelian

global symmetry), and even when they are commutative, they may not admit braiding.

The structure of fusion and crossing relations of the Verlinde lines is captured by what

is known as a modular tensor category [37–40], which requires the crossing relations to

See [28–31] for the lattice realization of some topological line defects and their constraints on in-

frared phases.

Here the inverse L

−1

of a line L is deﬁned such that their fusion relation is LL

−1

= L

−1

L = 1. For a

more general TDL L (such as the N line in the critical Ising model), its inverse might not exist.

In the absence of an ’t Hooft anomaly, such defect operators are related to orbifold twisted sector states

upon a symmetry-invariant projection, but the existence of the defect operator Hilbert space is more general

and applies to global symmetries with an ’t Hooft anomaly as well.

– 2 –

JHEP01(2019)026

obey the pentagon identity, and braiding relations to further obey the hexagon identity.

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. In a sense, fusion categories modeled by TDLs unify and generalize

the notions of both nonabelian symmetry groups and modular tensor category (modeled

by Verlinde lines). See [27, 43] for physicists’ expositions on this subject, and [27] in

particular for the relation to gauging and the ’t Hooft anomaly. The goal of this paper

is to explore the possible types of TDLs realized in unitarity, compact 2d CFTs, and

their implications on renormalization group (RG) ﬂows by a generalization of the ’t Hooft

anomaly matching [44, 45].

We will begin by describing a set of physically motivated deﬁning properties of TDLs

in section 2, and discuss their relations to the notion of fusion category in section 3. In the

context of global symmetry groups corresponding to group-like categories, we will discuss

’t Hooft anomalies, orbifolds, and discrete torsion [46, 47] in relation to invertible TDLs in

section 4.

In section 5, we discuss TDLs in rational CFTs that are generally not invertible.

In particular, we review Verlinde lines in section 5.1, and describe the explicit crossing

relations of Verlinde lines in diagonal Virasoro minimal models. Next, in section 5.2, we

will discuss examples of TDLs in rational CFTs that are neither Verlinde lines nor invertible

lines. The ﬁrst example is a set of TDLs in the three-state Potts model that preserves the

Virasoro algebra but not the W

algebra, found in [7]. The second example is given by the

topological Wilson lines in WZW and coset models, generalizing the construction of [48].

The third example is a set of TDLs in the non-diagonal SU(2)

WZW model of E

type (or

in the (A

, E

) minimal model), which realizes the so-called

fusion category [49, 50].

This fusion category consists of just three simple lines, has commutative fusion relation,

and yet does not admit braiding [49].

The primary interest of this paper is to explore the constraints of TDLs on the dynamics

of QFTs when the fusion and crossing relations of TDLs are known. A consequence of these

relations of TDLs is the restriction on the spin content of defect operators at the end of

the TDLs. Typically, only speciﬁc fractional spins are allowed for the defect operators at

the end of a given type of TDLs. This is the subject of section 6.

When certain TDLs are preserved along an RG ﬂow, say the ones that commute with

the relevant deformation of the UV CFT, these TDLs will survive in the IR. The IR TDLs

obey the same fusion and crossing relations as in the UV. Physically, this follows from

the topological property of the TDL as there is no intrinsic scale associated to it. More

rigorously, the consistent solutions of the fusion and crossing relations are discrete and

therefore cannot be deformed continuously under RG ﬂows. This is known as the Ocneanu

rigidity in category theory [42]. This basic observation has interesting implications on an

RG ﬂow to a massive phase, where the IR dynamics is described by a topological quantum

ﬁeld theory (TQFT) [51]. The TDLs of the TQFT inherited from the UV CFT will

constrain and often allow us to completely determine the fully extended TQFT [52–55],

i.e. with all lines and defect operators included. In particular, we show that if a TDL L with

– 3 –

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