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JHEP02(2020)150

Published for SISSA by Springer

Received: December 7, 2019

Revised: January 9, 2020

Accepted: January 30, 2020

Published: February 25, 2020

A mathematical theory of gapless edges of 2d

topological orders. Part I

Liang Kong

and Hao Zheng

a,b

Shenzhen Institute for Quantum Science and Engineering, and Department of Physics,

Southern University of Science and Technology,

Shenzhen 518055, China

Department of Mathematics, Peking University,

Beijing 100871, China

E-mail: kongl@sustech.edu.cn, zhengh@sustech.edu.cn

Abstract: This is the ﬁrst part of a two-part work on a uniﬁed mathematical theory of

gapped and gapless edges of 2d topological orders. We analyze all the possible observables

on the 1+1D world sheet of a chiral gapless edge of a 2d topological order, and show that

these observables form an enriched unitary fusion category, the Drinfeld center of which is

precisely the unitary modular tensor category associated to the bulk. This mathematical

description of a chiral gapless edge automatically includes that of a gapped edge (i.e. a

unitary fusion category) as a special case. Therefore, we obtain a uniﬁed mathematical

description and a classiﬁcation of both gapped and chiral gapless edges of a given 2d topo-

logical order. In the process of our analysis, we encounter an interesting and reoccurring

phenomenon: spatial fusion anomaly, which leads us to propose the Principle of Univer-

sality at RG ﬁxed points for all quantum ﬁeld theories. Our theory also implies that all

chiral gapless edges can be obtained from a so-called topological Wick rotations. This fact

leads us to propose, at the end of this work, a surprising correspondence between gapped

and gapless phases in all dimensions.

Keywords: Anyons, Conformal Field Theory, Topological Field Theories, Topological

States of Matter

ArXiv ePrint: 1905.04924

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP02(2020)150

JHEP02(2020)150

Some results of this two-part work were announced in [79] without providing the details.

In Part I, we focus on chiral gapless edges.

Throughout this work, we use “nd” to mean the spatial dimension and “n`1D” to

mean the spacetime dimension, and by a “2d topological order”, we mean an anomaly-free

2d topological order without symmetry. We use “Theorem” to represent a mathematical

(rigorous) result and “Theorem

” to summarize important physical (unrigorous) results.

Topological orders are the universal classes of gapped local Hamiltonian lattice models

at zero temperature. Since the system is gapped, correlation functions decay exponentially.

In the long wave length limit, the only local observables are topological excitations. It was

known that a 2d topological order is determined by its (particle-like) topological excitations

uniquely up to E

states. Mathematically, the fusion-braiding properties of the topolog-

ical excitations in a 2d topological order can be described by a unitary modular tensor

category (UMTC) (see for example [27, 28] and a review [62], appendix E). Therefore, a

2d topological order is described mathematically by a pair pC, cq, where C is the UMTC

of topological excitations and c is the chiral central charge. If a topological order pC, 0q

admits a gapped edge, it is called a non-chiral topological order. The mathematical theory

of gapped edges is known. More precisely, a gapped edge can be mathematically described

by a unitary fusion category (UFC) M such that its Drinfeld center ZpMq coincides with

C [37, 64, 69]. The fact that the bulk phase is determined by an edge as its Drinfeld center

is called the boundary-bulk relation.

When a 2d topological order is chiral, it has topologically protected gapless

edges [45, 89, 105, 106] (see reviews [93, 106, 107] and references therein). Observables

on a gapless edge is signiﬁcantly richer than those on a gapped edge because gapless edge

modes are described by a 1+1D conformal ﬁeld theory (CFT) [7, 90], the mathematical

structures of which are much richer than that of a UFC [47, 88, 90, 100]. It seems that a

categorical description of a gapless edge as simple as that of a gapped edge is impossible.

Nevertheless, in the last 30 years, the mathematical theory of boundary-bulk (or open-

closed) CFT’s has been successfully developed at least in three diﬀerent approaches:

1. the conformal-net approach (see [59, 88, 97, 98] and references therein),

2. the 2+1D-TQFT approach (see [23, 25, 33, 35] and references therein),

3. the vertex-operator-algebra approach (see [49, 54, 55, 67] and references therein).

These mathematical developments have revealed a universal phenomenon: the mathemat-

ical structures of a boundary-bulk CFT can be split into two parts.

1. One part consists of a chiral algebra V (or a conformal net), also called a vertex

operator algebra (VOA) in mathematics (see for example [86]), such that the category

Mod

of V -modules is a modular tensor category [51].

2. The other parts are purely categorical structures, including certain algebras in Mod

and an algebra in the Drinfeld center ZpMod

q of Mod

(see Theorem 4.7 and 4.21).

– 2 –

JHEP02(2020)150

This suggests that it might be possible to describe a chiral gapless edge of a 2d topological

order pC, cq by a pair pV, X

q, where V is a VOA and X

is a purely categorical structure that

can be constructed from Mod

, C and perhaps some additional categorical data. The main

goal of this paper is to show that this is indeed possible. More precisely, the main result of

this work, summarized in Theorem

6.7, says that X

is an Mod

-enriched unitary fusion

category (Deﬁnition 6.4) satisfying some additional properties.

We explain the layout of this paper. In section 2, we brieﬂy review the mathematical

theory of 2d topological orders and that of gapped edges. We emphasize previously over-

looked details, such as the so-called spatial fusion anomalies, so that it makes our study of

chiral gapless edge looks like a natural continuation. In section 3, we carefully describe all

possible observables (at a RG ﬁxed point) on the 1+1D world sheet of a chiral gapless edge

of a 2d topological order pC, cq. In particular, we show that the observables on the 1+1D

world sheet of a chiral gapless edge include a family of topological edge excitations, 0+1D

boundary CFT’s [12–14], 0D domain walls between boundary CFT’s [31] and two kinds

of fusions among domain walls. These boundary CFT’s and domain walls are required to

preserve a chiral symmetry given by a VOA V such that Mod

is assumed to be a UMTC.

This symmetry condition allows us to describe all boundary CFT’s, domain walls and their

fusions as objects or morphisms in Mod

. As a consequence, all observables organize them-

selves into a single categorical structure X

called an Mod

-enriched monoidal category.

Therefore, a chiral gapless edge can be described by a pair pV, X

In order to further explore the additional hidden structures in X

(in section 6), we need

nearly all important mathematical results of rational CFT’s in last 30 years. These results

are unknown to most working physicists especially to those in the ﬁeld of condensed matter

physics. A brieﬂy review of these results is necessary. In section 4, we review the mathemat-

ical theory of boundary-bulk rational CFT’s in the VOA approach [49, 51, 57, 67, 86, 112].

This mathematical theory is not only a rigorous version of the physical theory but also a

reformulation in terms of new and eﬃcient categorical language, which, together with a

classiﬁcation result, play a crucial role in this work. In particular, in section 4.1, we recall

a Segal-type mathematical deﬁnition of a boundary-bulk CFT. In section 4.2, we recall the

classiﬁcation theory of boundary-bulk rational CFT’s [25, 33, 67, 71, 88, 97, 98]. In sec-

tion 4.3, we explain the notion of an internal hom. In section 4.4, we discuss issues related

to unitary CFT’s and show that boundary CFT’s and domain walls among them can all

be constructed from internal homs, and summarize all useful results in Theorem

4.21.

After the preparation in section 4, we are able to give a natural and explicit con-

struction of chiral gapless edges in section 5.1. It turns out that the enriched monoidal

categories appearing there are only special cases of the so-called canonical construction (see

Theorem 5.3) [91]. In section 5.2, we will construct more general gapless edges by fusing

the gapless edges constructed in section 5.1 with gapped domain walls, or equivalently,

by the so-called topological Wick rotations. We will leave a loophole of our reasoning in

section 5.2 and ﬁx it in section 6.3. Interestingly, these mathematical constructions au-

tomatically include all the gapped edges as special cases. Therefore, we obtain a uniﬁed

mathematical theory of both gapped and chiral gapless edges.

– 3 –

JHEP02(2020)150

In section 6, we continue our exploration of the additional hidden structures in X

. In

particular, in section 6.1, using the classiﬁcation theory of boundary-bulk rational CFT’s

summarized in Theorem

4.21, we reveal the hidden relations between Mod

and the

underlying category X of X

, and show that X

is an Mod

-enriched unitary fusion category

(see Deﬁnition 6.4). In section 6.2, using the boundary-bulk relation [75] and the results

in [78], we obtain a precise and complete mathematical description and a classiﬁcation

theory of chiral gapless edges of 2d topological orders. This is the main result of this

paper, and is summarized in Theorem

6.7. In the process of deriving the classiﬁcation

result, we will also discuss and propose a deﬁnition of a phase transition between two

gapless phases via topological Wick rotations. In section 6.3, motivated by a recurring

phenomenon in this work, we propose a very general principle:

Principle of Universality at RG ﬁxed points. A physical theory at a RG

ﬁxed point always satisﬁes a proper universal property in the mathematical

sense.

We use it to ﬁx the last loophole of our reasoning introduced in section 5.2. This principle

also provides a mathematical formulation of the spatial fusion anomaly.

In section 7, we provide some outlooks for the study of gapless boundaries of higher

dimensional topological orders. In particular, we will propose a surprising correspondence

between gapped phases and gapless phases in all dimensions. In appendix, we provide the

mathematical deﬁnitions of some categorical notions in the enriched settings.

It is worthwhile to point out what is new in this paper. The main result Theorem

6.7

was ﬁrst announced in [79] without providing any proofs. This paper contain many missing

details and a complete proof. All physical (mathematically unrigorous) arguments used

in the proof are explicitly spelled out. They are the No-Go Theorem (see section 3.3),

Naturality Principle in physics (see section 6.1), the generalization of the mathematically

rigorous Theorem 4.10 to unitary cases (see Theorem

4.21), our deﬁnition of a purely

edge phase transition (see section 6.2) and Principle of Universality at RG ﬁxed points

(see section 6.3). All the rest steps in the proof are mathematically rigorous. In this

work, we also introduce a few brand new physical concepts for the ﬁrst time, including

spatial fusion anomaly, topological Wick rotation, a model-independent deﬁnition of a

phase transition between gapless edges (see section 6.2), Principle of Universality and

Gapped-Gapless Correspondence (see section 7).

In Part II [80], we will develop a mathematical theory of non-chiral gapless edges and

0d defects on a gapless edge. We will also give explicit calculations of various dimensional

reduction processes and a complete mathematical description of boundary-bulk relation

including both gapped and gapless edges. It is also worthwhile to mention that our theory

of gapless edge provides a mathematical description of the critical points of topological

phase transitions on the edges of 2d topological orders. An example was explained in [16].

– 4 –

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