A mathematical theory of gapless edges of 2d topological orders. Part I

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 topological 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 Universality 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.


Introduction
Topological phases of matter have attracted a lot of attentions in recent years among physicists because they go beyond Landau's paradigm of phases and phase transitions (see a recent review [109] and references therein). In this work and part II [80], we develop a unified mathematical theory of the gapped and gapless edges of 2d topological orders.

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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 ph " 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 8 states. Mathematically, the fusion-braiding properties of the topological 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 openclosed) CFT's has been successfully developed at least in three different approaches:

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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 7 q, where V is a VOA and X 7 is a purely categorical structure that can be constructed from Mod V , 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 ph 6.7, says that X 7 is an Mod V -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 overlooked 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][13][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 V 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 V . As a consequence, all observables organize themselves into a single categorical structure X 7 called an Mod V -enriched monoidal category. Therefore, a chiral gapless edge can be described by a pair pV, X 7 q.
In order to further explore the additional hidden structures in X 7 (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 mathematical 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 section 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 ph 4.21.
After the preparation in section 4, we are able to give a natural and explicit construction 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 automatically include all the gapped edges as special cases. Therefore, we obtain a unified mathematical theory of both gapped and chiral gapless edges.

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In section 6, we continue our exploration of the additional hidden structures in X 7 . In particular, in section 6.1, using the classification theory of boundary-bulk rational CFT's summarized in Theorem ph 4.21, we reveal the hidden relations between Mod V and the underlying category X of X 7 , and show that X 7 is an Mod V -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 ph 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 ph 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 ph 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].

JHEP02(2020)150 2 Gapped defects of 2d topological orders
In this section, we review some of the basic facts of 2d topological orders, including the categorical description of its particle-like topological excitations, that of gapped edges and the boundary-bulk relation.

Particle-like topological excitations
An nd topological order is called anomaly free if it can be realized by an nd local Hamiltonian lattice model, and is called anomalous if otherwise [73].
Before the discovery of fractional quantum Hall effect, a 2d gapped local Hamiltonian lattice model at zero temperature was viewed as "trivial" because all the correlation functions decay exponentially. It seems that there is no "local observables" in the long wave length limit. It was realized latter, however, the ground state degeneracy (GSD) of such 2d systems on surfaces with non-trivial topology, such as spheres with holes, torus, higher genus surfaces (with edges), etc., are non-trivial and different for different systems [94,95,101]. It means that there are different kinds of "trivialness" of such "trivial" systems, which are, therefore, non-trivial.
Note that the GSD is a global observable [3]. But the notion of a phase or a topological order is a local concept that is defined on an open disk (in the infinite system size limit). It was realized later that the local observables that are responsible for the non-trivial global observable GSD are the particle-like topological excitations (or anyons).
It is well known that all particle-like topological excitations form a mathematical structure called a unitary modular tensor category (UMTC) (see a review [62], appendix E). The physical meanings of some (not all) ingredients of a UMTC C are precisely the observables in the long wave length limit, and are explained below.
1. An object in C represents a particle-like topological excitation (or an anyon). The tensor unit 1 C represents the trivial topological excitation. A simple object represents a simple anyon, and a non-simple object, i.e. a direct sum of simple objects (e.g. ' i x i for simple x i ), represents a composite anyon.
2. For any two anyons x and y, a morphism from x to y is an instanton, which is a localized defect on time axis (e.g. at t " t 0 ), with two boundary conditions x (for t ă t 0 ) and y (for t ą t 0 ). If x and y are simple, then the space of instantons from the boundary condition x to y is given by hom C px, yq " δ x,y C. For two composite anyons ' n x n and ' m y m , where x n , y m are simple, we have hom C p' n x n , ' m y m q " ' n,m δ xn,ym C.
3. The fusion of instantons along the time axis, called a temporal fusion, defines a composition of morphisms hom C py, zqˆhom C px, yq Ñ hom C px, zq pf, gq Þ Ñ f˝g (2.1)

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Our convention is that the time coordinate of f is later than that of g. There is a distinguished morphism 1 x P hom C px, xq, which is called the identity morphism, represents the trivial instanton. It is trivial in the sense that f˝1 x " f and 1 y˝f " f for f P hom C px, yq.
4. The tensor product functor b : CˆC Ñ C defined by px, yq Þ Ñ x b y describes the fusion between two anyons in spatial dimensions, called a spatial fusion. The trivial anyon is denoted by 1 C . We have 5. The spatial fusion b of anyons also induces a spatial fusion of instantons. The naive guess of the space of instantons after the fusion is hom C px, yq b C hom C px 1 , y 1 q. But this naive guess is wrong because an anyon, viewed as a 0d topological order, is anomalous. The correct one is hom C px b y, x 1 b y 1 q. In general, there is a map (induced from the universal property of hom C px b y, for x, x 1 , y, y 1 P C. Its failure of being an isomorphism is called spatial fusion anomalies. This is a recurring phenomenon in this work, and will be discussed in section 6.3. It is also worthwhile to point out that the spatial fusion anomaly vanishes when x " y " x 1 " y 1 " 1 C . This result has a non-trivial analogue for chiral gapless edges (see Remark 6.3).
6. The creation and annihilation of a particle-antiparticle pair are described by the duality morphisms where x˚is the anti-particle of x, satisfying some natural properties. Mathematically, this amounts to the rigidity of a UMTC.

7.
Since the associated Hamiltonian model is gapped, one can adiabatically move one anyon around another one. This adiabatic movement defines a braiding isomorphism c x,y : x b y Ñ y b x for x, y P C. The composed morphism x b y cx,y Ý Ý Ñ y b x cy,x Ý Ý Ñ x b y is called a double braiding, which amounts to adiabatically moving anyon x around y (say clockwise) in a full circle.
The coherence conditions satisfied by the above data are all physically obvious. The nondegeneracy condition of the double braiding of a UMTC is stated as follows: p‹q For a given simple anyon x, if its double braiding with all anyons y (including x itself) is trivial, i.e. px b y cx,y The condition p‹q is an anomaly-free condition, which says that all anyons in a topological order should be able to detect themselves via double braidings. If a system of anyons does not satisfy the condition p‹q, it must represent an anomalous 2d topological order, which must be a gapped boundary of a non-trivial 3+1D topological order [73].

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UFC's: L N M UMTC's: C " ZpLq " ZpMq " ZpNq Figure 1. This picture depicts a 2d topological order pC, 0q, where C is a UMTC, with three gapped edges given by UFC's L, M, N. The 2d bulk is oriented as the usual R 2 with the normal direction pointing out of the paper in readers' direction. The arrows indicate the induced orientation on the edge.
Remark 2.1. The space of instantons hom C px, yq has another physical meaning. It is also the space of ground states of the topological order defined on a 2-sphere together with two anyons x˚, y. Two physical meanings of the same hom-space is a manifestation of the state-field correspondence of a TQFT or a topological order.
For each UMTC C, one can define a topological central charge c top C , which is defined modulo 8. We use C to denote the same monoidal category as C but equipped with the braidings given by the anti-braidings in C. C is also a UMTC. It was known that the Drinfeld center ZpCq of C is given by C b C [92], where b is the Deligne tensor product. The simplest UMTC is the category of finite dimensional Hilbert spaces, denoted by H.
In the long wave length limit, are there any other physical observables in a 2d topological order? It turns out that there is a special state called the E 8 -state (see for example [63]), which is a non-trivial topological order but has no non-trivial anyon. It was known that its 1d edge is gapless and is given by the E 8 chiral conformal field theory of central charge c " 8. It was generally accepted that a 2d topological order is completely determined by its anyons up to the E 8 -states (i.e. E bN 8 ) (see a review [108]). We summarize this well-known result below.
A 2d (anomaly-free) topological order can be described mathematically by a pair pC, cq, where C is a UMTC and c is the chiral central charge such that c top C " c pmod 8q.
The pair pH, 0q describes the trivial 2d topological order.

1d gapped edges
A 2d topological order pC, 0q admitting a gapped edge is called a non-chiral 2d topological order. As depicted in figure 1, the bulk phase pC, 0q might have several different gapped edges, each of which represents a 1d anomalous topological order. Such an anomalous 1d topological order can be described mathematically by a unitary fusion category (UFC) L, in which JHEP02(2020)150 2. for topological edge excitations x, y P L, the space of morphisms hom L px, yq is the space of instantons (or localized defects on time axis) with boundary conditions x and y; 3. the composition of two morphisms describes the fusion of two instantons along the time axis; 4. the fusion product b in L represents the fusion of two such edge excitations in the spatial dimension according to the orientation of the edge. 1 L denote the trivial topological edge excitation, i.e.
5. This fusion also induces a fusion of two instantons in the spatial dimension. Again, the naive guess hom L px, yq b C hom L px 1 , y 1 q for the space of instantons after the fusion is wrong. The correct one is hom L px b x 1 , y b y 1 q. In general, there is a map for x, x 1 , y, y 1 P L. This map is not an isomorphism in general. Its failure of being an isomorphism is an indication of spatial fusion anomaly. This phenomenon is a recurring theme of this work. We will further explore this phenomenon as a special case of a more general principle in section 6.3.
Note that there is no braiding because these topological excitations are restricted on the edge, thus cannot be braided. The relation between the boundary phase and the bulk phase, or simply boundary-bulk relation, was known [64,69].
Boundary-bulk relation. The 2d bulk phase is uniquely determined by the anomalous 1d topological order on its boundary (or edge), and the UMTC C describing the 2d bulk phase is given by the Drinfeld center of the UFC L (or M, N) describing a 1d edge (as depicted in figure 1), i.e. C » ZpLq » ZpMq » ZpNq.
Mathematically, different gapped edges L, M, N that share the same bulk or Drinfeld center if and only if they are Morita equivalent as UFC's [22,92].
Importantly, although a UFC L determines an anomalous 1d topological order, it does not fix a gapped edge of a 2d topological order completely. The same UFC might be realized as different edges of the same bulk. For example, the Z 2 2d topological order has two different gapped edges [5], both of which realize the same 1d anomalous topological order defined by the UFC ReppZ 2 q.
The additional physical data that is needed to uniquely determine the edge is the information of how anyons in the bulk can be fused onto the edge. This information is given by a monoidal functor L : C Ñ L which factors through the forgetful functor f : ZpLq Ñ L, i.e. there exists a braided monoidal functor L 1 : C Ñ ZpLq such that the JHEP02(2020)150 is commutative. Such a functor L is called a central functor [19].

What happens to gapless edges?
We have seen that we have a beautiful mathematical theory of gapped edges of 2d topological orders. Ironically, experimentally discovered 2d topological orders, such as those discovered in quantum Hall effects, all have topologically protected gapless edges. The natural question is whether there is a similar mathematical theory for gapless edges. As depicted in figure 2, this question contains at least the following three parts: 1. What is the mathematical description of a gapless edge of pC, cq?
2. Does the boundary-bulk relation, i.e. bulk = the center of the edge, still hold?
3. What is the mathematical description of a 0d gapless domain wall between two gapless 1d edges?
In [75], using a very formal argument, we have shown that the bulk topological order (in any dimensions) should be given by the center of its boundary, i.e.
regardless whether the boundary is gapped or gapless ( [75], Remark 5.7), and regardless what the mathematical description of a gapless boundary is.

Remark 2.3.
A reader might be puzzled by the above statement. How could one know the notion of the center of certain algebraic object before we have a complete mathematical definition of the algebraic object. This is because the notion of center is defined by its universal property, which only depends on the notion of a morphism between two such algebraic objects. In [75], the notion of a morphism between two anomalous nd topological orders was introduced without knowing how to define an nd topological order.
This result ( [75], Remark 5.7) convinced us that there should be a unified mathematical theory of gapped and gapless edges of 2d topological orders. We will answer the first question for chiral gapless edges in this work, and answers the second and third questions in [80].
Remark 2.4. The result of ( [75], Remark 5.7) and the success of this work have non-trivial and exciting predictions for the study of higher dimensional gapped and gapless phases. We will briefly discuss them in section 7.

Chiral gapless edge I
In this section, we will try to find all observables living on the 1+1D world sheet of a chiral gapless edge of a 2+1D topological order, which is fixed to be pC, cq.

Vertex operator algebras
Suppose that the 2d bulk phase pC, cq is realized on an open 2-disk as depicted in figure 3.
We choose a complex coordinate z on the 1+1D world sheet of the chiral gapless edge (i.e. the 2D boundary of the solid cylinder) such that the time axis is the real axis. We assume that gapless edge modes are completely chiral. It is known that the chiral edge modes are states in a chiral conformal field theory with the central charge c (see [45,89,105,106], reviews in [93,106,107] and references therein). This chiral CFT is defined on the 1+1D world sheet of the edge. We denote this chiral CFT by U . By the state-field correspondence in a 2D CFT, there is a bijective map Y from U to the space of chiral fields, still denoted by U : The notation φpzq is commonly used in physics literature and Y pφ, zq is commonly used in mathematics literature. We prefer to use the mathematical notation Y pφ, zq sometimes in this work because the first notion can be quite ambiguous whenever we discuss intertwining operators. We should view these chiral fields as "local observables" on the world sheet. Since these chiral fields can live on the entire 1+1D world sheet, they cannot be multivalued. Therefore, these chiral fields, say φpzq, contains only integer powers of the complex variable z, i.e.
φpzq " where φ n P EndpU q. Any two chiral fields in U can have the so-called operator product expansion (OPE) as shown below: This OPE is commutative, i.e. φpz 1 qψpz 2 q " ψpz 2 qφpz 1 q. It provides U with an algebraic structure, which is called a chiral algebra in physics, and is rigorously defined in mathematics under a different name: a vertex operator algebra (VOA) (see for example [86] JHEP02(2020)150 Figure 3. This picture depicts a 2d topological order pC, cq on a 2-disk, together with a 1d gapless edge, propagating in time. A chiral field φpzq P U is depicted on the 1+1D world sheet.
for a review). In this work, we use the following working definition of a VOA, which is sometimes called a CFT-type VOA. We avoid to use formal variables that are commonly used in mathematics literature.
Definition 3.1. A vertex operator algebra (VOA) U consists of the following data: • a C-linear vector space U graded by non-negative integers (called conformal weights), i.e. U " ' nPN U pnq , • a distinguished element 1 P U p0q called vacuum state, • a distinguished element ω U P U p2q called Virasoro element, satisfying the following axioms: 1. dim U pnq ă 8 and U p0q " C1; 2. if u P U pkq , then u n maps U pmq into U pm`k´n´1q ; 3. Y p1, zq " id U and u´11 " u for u P U ; 4. for u, v, w P U and w 1 P U 1 :" ' n Up nq , the following three expressions xw 1 , Y pu, z 1 qY pv, z 2 qwy, xw 1 , Y pv, z 2 qY pu, z 1 qwy, xw 1 , Y pY pu, z 1´z2 qv, z 2 qwy converge absolutely in three different domains |z 1 | ą |z 2 | ą 0, |z 2 | ą |z 1 | ą 0, |z 2 | ą |z 1´z2 | ą 0, respectively, to the same rational function f pz 1 , z 2 q (with poles only at z 1 , z 2 " 0, 8 and z 1 " z 2 ); 5. we have Y pω U , zq " ř nPZ Lpnqz´n´2 such that Lpnq, n P Z generate the famous Virasoro algebra, i.e. rLpmq, Lpnqs " pm´nqLpm`nq`c 12 pm 3´m qδ m`n,0 where the number c is called the central charge; JHEP02(2020)150 Figure 4. This picture depicts a 2d topological order pC, cq on a 2-disk, together with a 1d gapless edge, propagating in time. At t " 0, we move a bulk particle a to the edge. The blue world line is supported on x.
6. for u P U pnq , we have Lp0qu " nu; Remark 3.2. Y pω U , zq is called the energy momentum tensor in physics. It generates a sub-VOA xω U y, which is the smallest sub-VOA of U . The number c is the same central charge as the one in the 2d bulk phase pC, cq.

Open-string vertex operator algebras
Is this VOA U the only observables on the 1+1D world sheet of the edge? Consider the situation depicted in figure 4. At t " 0, a topological excitation a in the bulk is moved to the edge. It becomes a topological edge excitation x. As a typical quantum quenching scenario, this movement suddenly changes the microscopic physics at the location of x. As a consequence, the RG flow will drive the world line supported on x to a fixed point theory, which is denoted by A x . In particular, A x should contain all the physical observables on the world line at the RG fixed point. It is clear that when a " 1 C , x is the trivial topological edge excitations, denoted by 1. Therefore, we must have A 1 " U . More general topological edge excitations can be created by inserting impurities from outside. In other words, there are potentially more topological edge excitations than those obtained from moving bulk excitations to the edge. At the RG fixed point, the conformal symmetry is restored. The observables in A x are again chiral fields that can have OPE along the world line. But the chiral fields in A x are potentially different from those in U . By our choice of complex coordinate, the world line (or the time axis) is the real axis. There is no multi-value issue here because the chiral fields are restricted on the real axis. As a consequence, the chiral fields on the world line can have non-integer powers, i.e.
φprq " ÿ nPR φ n r´n´1, Moreover, two chiral fields on the world line can also have OPE as follows: pψ k φqpr 2 q pr 1´r2 q k`1`p ψ k´1 φqpr 2 q pr 1´r2 q k`¨¨¨, for some k P R.
More precisely, OPE among all these chiral fields defines an algebraic structure on A x called an open-string vertex operator algebra (OSVOA) [54]. We recall this notion below.
3. An open-string vertex operator algebra (OSVOA) A of central charge c consisting of the following data: • two distinguished elements 1 P A p0q and ω A P A p2q , satisfying the following conditions: 1. dim A pnq ă 8 and A pnq " 0 for n ! 0; 2. Y p1, rq " id A and u´11 " u for u P A; 3. for u 1 ,¨¨¨, u n , w P A and w 1 P A 1 " ' n Ap nq , the series: where P k : A A pkq is the projection operator, converges absolutely when r 1 ą¨¨¨ą r n ą 0 and can be extended to a potentially multi-valued complex analytic function in pCˆq n with only possible singularities at z i " z j for i, j " 1,¨¨¨, n and i ‰ j; 4. the following two expressions xw 1 , Y A pu 1 , r 1 qY A pu 2 , r 2 qwy, xw 1 , Y A pY A pu, r 1´r2 qv, r 2 qwy converge absolutely in two different domains r 1 ą r 2 ą 0 and r 2 ą r 1´r2 ą 0, respectively, and equal on the intersection of the above two domains; 5. we have Y A pω A , rq " ř nPZ Lpnqr´n´2 such that tLpnqu nPZ generate the Virasoro algebra of central charge c; 6. for u P A pnq , we have Lp0qu " nu; rq. An OSVOA is a non-commutative generalization of the notion of a VOA. In particular, a VOA is automatically an OSVOA. An OSVOA contains a smallest subalgebra xω A y Ă A generated by ω A . Actually, xω A y is a VOA. More generally, the following subspace of an OSVOA A: C 0 pAq " tu P ' nPZ A pnq |Y A pu, rq " ÿ nPZ u n r´n´1, Y A pv, rqu " e rLp´1q Y pu,´rqv, @v P Au, (3.1) defines a subalgebra of A called the meromorphic center of A. Moreover, it is a VOA. The defining property of C 0 pAq is equivalent to the following condition:

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p‚q For u P C 0 pAq, v, w P A and w 1 P A 1 , there exists a (possibly multi-valued) analytic function on such that it is single valued in z 1 and equals to xw 1 , Y A pu, z 1 qY A pv, r 2 qwy, xw 1 , Y A pv, r 2 qY A pv, z 1 qwy, xw 1 , Y A pY A pu, z 1´r2 qv, r 2 qwy in the domains |z 1 | ą r 2 ą 0, r 2 ą |z 1 | ą 0, r 2 ą |z 1´r2 | ą 0, respectively; We introduce a formal vertex operator: where x is a formal variable.

Theorem 3.4 ([54]
). The condition (‚) is equivalent to the statement that the formal vertex operator Y f A p´, xq´is an intertwining operator of C 0 pAq.

Boundary CFT's and domain walls
More general situations can occur on the same world line as depicted in figure 5 (a). More precisely, one can insert an impurity at t " t 1 ą 0 on the world line so that the topological edge excitation is changed from x to y. As a consequence, the OSVOA living on the tt ą t 1 u-part of the world line is given by A y . The domain wall between A x and A y also contains chiral fields (sometimes called defect fields in CFT's). This domain wall is similar to the instantons on a gapped edge (recall section 2.2). We denote the space of such defect fields by M x,y . Then it is clear that the space of defect fields localized around x is M 1,x . Moreover, we should have M x,x " A x and M 1,1 " A 1 " U . This space M x,y should also be an R-graded vector space, i.e. M x,y " ' nPR pM x,y q pnq , such that dimpM x,y q pnq ă 8. Similar to the fusion of instantons along the time axis in the gapped edge case, defect fields in different domain walls can also have OPE along the world line. This type of OPE defines a vertex operator: This OPE should also be associative. More precisely, it satisfies the following condition.
JHEP02(2020)150 It is important to note that we have not yet discussed the issue if these M x,y or A x are really different from U . At least, it is clear that there should be some relations among U , A x and M x,y . For example, A x acts on M x,y from one side and A y acts from the other side, and U acts on M x,y from above (or below) the time axis, and U can be mapped into A x , etc. In order to formulate them precisely, we need first to show that these A x are more than just OSVOA's. They are actually boundary CFT's [12][13][14] that share the same bulk 1+1D CFT. Indeed, consider a dimensional reduction process, which starts from figure 5 (a), and gradually squeeze the solid cylinder, and ends up in the 1+1D world line depicted in figure 5 (b). The 0+1D boundary of this 1+1D world sheet is precisely given by the world line supported on x. Observables on this world line remain the same during this process. It is clear that both chiral and anti-chiral fields live on the resulting 1+1D world sheet. It was shown in ( [11], section 2.1) that they form a modular invariant bulk CFT A bulk . This fact was emphasized in [84,99] as a consequence of the following "no-go theorem": Any 1+1D CFT's realized by 1d lattice Hamiltonian models should be modular invariant. Therefore, the OSVOA's A x and A y must be the boundary CFT's sharing the same modular invariant bulk CFT A bulk , and M x,y is a domain wall between them and is also compatible with the same bulk. Mathematically, however, it is not a clear statement because there are different mathematical definitions of a boundary-bulk CFT. Here we prefer to make this statement mathematically precise. We propose a stronger "no-go theorem".
No-go Theorem. A 1+1D boundary-bulk CFT realized by a 1d lattice Hamiltonian model with boundaries should satisfy all the axioms in the mathematical definition of a boundary-bulk CFT given in Definition 4.1 or 4.4.
In particular, the boundary-bulk CFT should satisfy all the consistence conditions, including the famous modular invariant condition [7,90] and Cardy condition [13,14].

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Remark 3.5. Using boundary CFT's to study 0d defects or impurities in other condensed matter systems has a long history [1,2]. But we are not aware of any earlier works mentioning the appearance of the boundary CFT's on the gapless edges of 2d topological orders. That a chiral vertex operator living in M 1,x was known in 90's (see [110,111]).
Remark 3.6. Once we established the precise mathematical description of a chiral gapless edge, we can compute this dimensional reduction process and identify precisely which modular invariant bulk CFT is obtained at the end of this process [79] (see more details in [80]).
In summary, observables on the 1+1D world sheet of the edge of a 2d topological order form a "not-yet-categorical" structure: 1. objects are topological edge excitations: x, y,¨¨¨; 2. the space of morphisms are M x,y (boundary CFT's and domain walls); 3. there is a map given by the associative OPE Y pz,y,xq : We can not yet claim it is a categorical structure not only because the OPE is not yet a composition map but also because the "identity morphisms" are missing. We will discuss this important missing data in the next subsection.

Chiral symmetries
The no-go theorem demands A x to be a boundary CFT, which requires many additional structures beyond that of an OSVOA. One of them is called the boundary condition of a boundary CFT. It says that, as a boundary CFT, A x should satisfy a conformal invariant boundary condition (as the minimal requirement) or a V -invariant boundary condition in general, where V is called the chiral symmetry. We explain this in detail in this subsection.
Consider moving chiral fields in U into those in A x along a path γ (e.g. γ 1 , γ 2 in figure 6). This process defines a linear map ι γ : U Ñ A x , which is clearly independent of the homotopy type of the path. Therefore, as shown in figure 6, there are essentially two independent ways of mapping U into A x along the paths γ 1 , γ 2 , respectively, i.e. ι γ 1 , ι γ 2 : U Ñ A x . It is clear that the vacuum state and the OPE of the chiral fields must be preserved in these processes. Therefore, ι γ 1 , ι γ 2 are two OSVOA homomorphisms. The minimal requirement for A x to give a consistent boundary CFT is that ι γ should satisfy the following condition: • Conformal-invariant boundary condition [12,66]: Let V be a sub-VOA of U , i.e. xω U y Ă V Ă U . In most general situation, we expect the following condition to be true.  Figure 6. This picture depicts how the chiral fields in U can be mapped to A x .
We denote that path independent map by ι In other words, chiral fields in V can move transparently on the entirely 1+1D world (except those 0D walls M x,y for x ‰ y). If the V -invariant boundary condition holds for all A x , then, for a sub-VOA V 1 of V , the V 1 -invariant boundary condition holds automatically. If we require the set of topological edge excitations to be fixed, then there will be the maximal This VOA V will be called the chiral symmetry of the edge. From now on, we take V to be the chiral symmetry of the edge.
Remark 3.7. If we demands V 1 Ĺ V to be the chiral symmetry, then the set of topological edge excitations can be enlarged accordingly. We will discuss this issue in section 6.2.
Remark 3.8. It can happen that the chiral symmetry is maximal, i.e V " U , as we will show in section 5.1. We will also show in section 5.2 that xω U y Ĺ V Ĺ U in general.
By the V -invariant boundary condition, M x,y are all V -modules in the usual sense [29]. We denote the category of V -modules by Mod V . We obtain the following property of Y Ax : • for φ P V , v, w P A x and w 1 P A 1 x , the operator products converge absolutely in the domains |z| ą t ą 0, t ą |z| ą 0, |z| ą |t´z| ą 0, respectively, and are analytic continuation of each other along any path in the zplane (for example γ 3˝γi for i " 1, 2 as depicted in figure 6).
By the results in section 3.2, this implies that V lies in the meromorphic center of A x . By Theorem 3.4, Y Ax is an intertwining operator of V . This analysis can be generalized to more general vertex operators introduced in (3.2), where Y pz,y,xq defines the OPE between defect fields in M y,z and those in M x,y . Similarly, by the V -invariant boundary condition, we obtain the following property of Y pz,y,xq : JHEP02(2020)150 • for φ P V , v P M y,z , w P M x,y and w 1 P M 1 x,z , the operator products xw 1 , φpzqY pz,y,xq pv, tqwy, xw 1 , Y pz,y,xq pv, tqφpzqwy, xw 1 , Y pz,y,xq pφpzqv, t´zqv, zqwy converge absolutely in the domains t ą |z| ą 0, |z| ą t ą 0, |z| ą |t´z| ą 0, respectively, and are analytic continuation of each other along any paths in the zplane.
By Theorem 3.4 again, Y pz,y,xq is an intertwining operator of V [29]. Note that Until the end of this section, we make the following assumption as a natural requirement in physics: the chiral symmetry V is a unitary rational VOA such that the category of unitary V -modules, still denoted by Mod V , is a UMTC.
There is a list of conditions on the VOA V that guarantees Mod V to be a modular tensor category by Huang's theorem [51,52]. In general, it is not clear if Mod V is a UMTC when V is unitary. See [42][43][44] for discussion of the relation between the unitarity of a VOA V and that of Mod V .
According to the tensor category theory of rational VOA developed by Huang and Lepowsky [48,57,58], the intertwining operator Y pz,y,xq of V is equivalent to a morphism Note that we have chosen the convention that the left factor of the relative tensor product´b V´h as a higher time coordinate. This convention is compatible with our OPE (recall (3.2)) and our convention in eq. (2.1).
The associativity of OPE of intertwining operators of V (recall the paragraph below eq. (3.2)) is commutative [51]. When w " x " y " z, this commutative diagram together with the unit property V b V A x » A x imply that the triple pA x , e, ι x q defines an algebra in Mod V (see Definition 4.5) [54], Theorem 4.3. Similarly, one can see that M x,y is an A y -A xbimodule, which also induces the same V -action on M x,y because the V -action on M x,y is path independent. This leads to the following commutative diagrams: In summary, all the observables on the 1+1D world sheet of a chiral gapless edge of a 2d topological order pC, cq can be described by a pair pV, X 7 q, where X 7 is a categorical structure:

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• objects in X 7 are topological edge excitations: x, y, z,¨¨¨; • for each pair px, yq of objects in X 7 , there is a space of morphisms: hom X 7 px, yq :" M x,y , which is an object in Mod V ; • there is a map id satisfying the commutative diagrams in (3.3) and (3.4). By Definition A.1, the categorical structure X 7 is nothing but a category enriched in Mod V , or an Mod V -enriched category [60].
There is a canonical C-linear category X associated to X 7 defined as follows: • the objects are the objects in X 7 (i.e. topological edge excitations); • the space of morphisms hom X px, yq :" hom Mod V p1 Mod V , M x,y q; • the identity morphism is given by id • the composition morphism˝: hom X py, zq b hom X px, yq Ñ hom X px, zq is defined by the following composed morphism: Note that id x is indeed the identity morphism in the usual sense, i.e. id y˝f " f " f˝id x for f P hom X px, yq, because of the commutative diagrams in (3.4). Mathematically, this category X is called the underlying category of X 7 . Physically, X is nothing but the category of topological edge excitations. We will call the UMTC Mod V the background category of X 7 .

X 7 is an enriched monoidal category
Two topological edge excitations x 1 and x can be fused along the edge (in the spatial dimension) to give a new edge excitation x 1 b x as depicted in figure 7. This spatial fusion automatically induces the spatial fusion of all observables on two world lines. If M x 1 ,y 1 and M x,y can both be viewed as defects in the spatial dimension, we should expect this spatial fusion to be given by M x 1 ,y 1 b U M x,y , where M x 1 ,y 1 and M x,y are viewed as U -U -bimodules and the relative tensor product b U is well-defined in Mod V . However, it turns out to be wrong in general. The spatial fusion of observables (or instantons) in temporal dimension are very different from those in spatial dimensions. Recall that this is the recurring phenomena also happening on the gapped edges and when we move a bulk particle to the edge. The spatial fusion of A x 1 and A x is again a typical quantum quenching scenario. When we move x 1 closer to x at t " 0, the Hamiltonian around x changes suddenly. Renormalization flow will drive the world line theory supported on x 1 b x to a new fixed point theory, which is given by is not only physically universal but also universal in a mathematical sense, which automatically demands such a morphism (see Remark 3.10 below). By generalizing the above argument to M x 1 ,y 1 and M x,y , we expect that there is a natural morphism, for x, y, x 1 , y 1 P X 7 , which is not an isomorphism in general. Its failure of being an isomorphism is an indication of spatial fusion anomaly.
Remark 3.10. The universal property of A x and M x,y will be explained in details in section 4.3 and section 6.3. This is a recurring theme that will be summarized as a principle in section 6.3 for theories at RG fixed points. It is worthwhile to pointing out that, in some important situations, spatial fusion anomalies vanish (see section 5.1 and Remark 6.3).
Composing the morphism (3.6) with the canonical morphism M x 1 , The morphism b should satisfy some natural properties: 1. The chiral symmetry condition should be preserved under the spatial fusion. In other words, V should be mapped into M xby,xby canonically (as ι xby : V Ñ M xby,xby ) and independent of the paths we choose. In particular, this implies that the following is commutative.
2. Consider the situation depicted in figure 7. If we fuse M x,y , M y,z , M x 1 ,y 1 , M y 1 ,z 1 horizontally and vertically and let it flows to the fixed point theory M x 1 x,z 1 z . For convenience, we abbreviated x 1 b x to x 1 x. This process should be independent of which fusion (horizontal or vertical) we do first. This implies that the following commutative diagram (called the braided interchange property [91]): Our braiding convention is explained in Remark 3.11 below.
Mathematically, these two properties simply says that the spatial fusion b defines an enriched functor X 7ˆX7 Ñ X 7 .
This spatial fusion is clearly associative and unital with respect to the tensor unit given by the trivial edge excitation 1. As a consequence, this spatial fusion upgrades X 7 to an Mod V -enriched monoidal category (see Definition A.8) [78,91].
Remark 3.11. Our convention of the braidings in Mod V is that when M x 1 ,y 1 is moving around M y,z along a path sitting on the left side of the world line supported on x, i.e. along the world line supported on x 1 in figure 7, then M x 1 ,y 1 will stay on the top during the braiding. More precisely, in this case, the initial (resp. final) time coordinate of M x 1 ,y 1 is higher (resp. lower) than that of M y,z , the adiabatic move gives the braiding We will use this convention throughout this work. We summarize the main result of this section as a physical theorem.

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Theorem 3.13. The complete set of observables on a chiral gapless edge of a 2d topological order forms a pair pV, X 7 q, where • V is the chiral symmetry, which is a unitary rational VOA such that Mod V is a UMTC; • X 7 is an Mod V -enriched monoidal category, whose objects are topological edge excitations x, y, z,¨¨¨, and whose morphisms M x,y are boundary CFT's (if x » y) or domain walls consisting of boundary condition changing operators (if x fi y).
As a consequence, the underlying category X, i.e. the category of topological edge excitations, is a C-linear monoidal category.

Boundary-bulk CFT's
Further study of X 7 needs the classification theory of rational CFT's (see Theorem ph 4.21 and Remark 4.26). In this section, we review the mathematical theory of rational CFT's based on the representation theory of VOA's.

Definitions of boundary-bulk CFT's
The theory of conformal field theory (CFT) was developed by physicists at the end of 80's (see [7,90] and [26] for a lengthy review). It took, however, about 20 years for mathematicians to develop a successful mathematical foundation of CFT's. This foundation is not merely a rigorization of the existing physical theory in its old formalism, but a new mathematical theory written in a beautiful new language with powerful new tools, and it leads to many new results far beyond those in [26]. These new results play a crucial role in this work, and are quite unfamiliar to most condensed matter physicists. Therefore, we briefly outline here where these results come from. See [68] for a more detailed review.
When physicists Belavin, Polyakov and Zamolodchikov published the first systematic study of 2D CFT's in 1984 [7], mathematicians Frenkel, Lepowsky and Meurman published their independent discovery of similar mathematical structures also in 1984 [32]. In 1987, the preprint of Segal's mathematical definition of a 2D CFT [100] came out and have made a big impact to mathematics community since then. We will give a sketchy presentation of an open-closed (or boundary-bulk) generalization of Segal's definition [46,49,68].
We define Bord cpx op´cl to be the category, in which (1) the objects are finite ordered set with two colors: "c" (for closed strings) and "o" (for open strings), e.g. to 1 , c 2 , o 3 , c 4 , c 5 u; (2) the spaces of morphisms are those of the conformal equivalence classes of open-closed complex bordisms, i.e. the moduli spaces of Riemann surfaces with boundary components being either completely parametrized or not parametrized but having none or some parametrized line segments, e.g.

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where the label "in{out" is associated to domain/codomain of the morphism and is determined by the relation between the orientation of the boundary component induced from that of the surface and that induced from the parametrization.
The category Bord cpx op´cl has the structure of a symmetric monoidal category with the tensor product defined by the disjoint union. Moreover, it has a˚-structure defined by flipping the orientation of the Riemann surfaces. This flipping exchanges the domains with the codomains. Let H 8 be the category of Hilbert spaces. It also has a˚-structure hom H 8 px, yq   Since the spaces of morphisms in Bord cpx op´cl are moduli spaces naturally equipped with complex structures, "real-analytic" means that the functor, restricting on the spaces of morphisms, gives real-analytic functions. If the functor is complex analytic, then such CFT's are called holomorphic CFT's.

Remark 4.3.
A CFT functor F maps tc 1 u, i.e. the set containing a single "c"-colored element, to H bulk , and maps to 1 u to H bdy . The monoidalness of F means that This definition of CFT includes all consistence conditions, such as the modular invariant condition and the famous Cardy condition, etc. Unfortunately, this definition is not directly workable because chiral fields φpzq in a CFT are associated to insertion at a point instead of a boundary component (or a line segment). It suggests to consider Riemann surfaces with parametrized punctures in the interior (closed strings stretched to infinity) or on the boundaries (open strings stretched to infinity) [47,68,104]. We will denote this surface-with-puncture variation of Bord cpx op´cl by Bord cpx,8 op´cl . Actually, Bord cpx,8 op´cl is not a category because the composition of morphisms are only partially defined, thus is called a partial category. Similarly, we replace H 8 by a partial category of graded vector spaces (with a subtle definition of morphisms and their compositions), denoted by GV [47]. It is also possible to endow each of Bord cpx, 8 op´cl and GV with a˚-structures [42]. Then we obtain a working definition of a CFT.

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To distinguish it from the first definition, we denote F ptc 1 uq by V bulk and F pto 1 uq by V bdy .

Classification theory of boundary-bulk CFT's
If we restrict a CFT functor F to the partial subcategory consisting of only bordisms of genus zero (Riemann spheres) from tc 1 ,¨¨¨, c n u 8 n"0 to tc 1 u and assume F is complex analytic, it was proved by Huang that this restricted F endows V bulk with the structure of a vertex operator algebra (VOA) [47]. Therefore, a VOA is a substructure of a CFT. In general, a VOA is not modular invariant except for holomorphic VOA's (such as the Monster Moonshine VOA [32]). The idea to find a modular invariant bulk (or closed) CFT V bulk is to realize it as certain extension of V b C V , where V is a VOA and V is the same as the VOA V in the formal variable but contains only the anti-chiral fields φpzq " ř n φ nz´n´1 . Therefore, we need study the representation theory of VOA. Building on many earlier works by Huang and Lepowsky on tensor category theory for VOA's [48,57,58], Zhu's influential work on modular invariance [112] and Huang's proof of Verlinde formula [51], Huang proved that the category Mod V of V -modules for V satisfying certain rationality conditions is a modular tensor category [52]. As a consequence, a bulk CFT V bulk is a commutative algebra in the category of V b C V -modules [65], or equivalently, in the category Mod V bMod V , which is also the Drinfeld center ZpMod V q of Mod V . Similarly, the boundary CFT V bdy is a chiral extension of a VOA.
Before we state the classification result, we need recall some basic notions in tensor categories.
Definition 4.5. An algebra in a monoidal category A is a triple A " pA, m, ηq where A is an object of A, m (the multiplication) is a morphism A b A Ñ A and η (the unit) is a morphism 1 A Ñ A such that An algebra A in a braided monoidal category is called commutative if m A˝cA,A " m A .
Similarly, one can define a coalgebra A " pA, ∆, q, where ∆ : A Ñ A b A and : A Ñ 1 A obey the following coassociativity and counit conditions: Definition 4.6. A Frobenius algebra A " pA, m, η, ∆, q is both an algebra and a coalgebra such that the coproduct ∆ is an A-A-bimodules map, i.e.
We will use the following graphical representation:

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Let B be a modular tensor category with the tensor product b and a tensor unit 1 B . The duality maps are expressed graphically (read from bottom to top) as follows: (4.1) A Frobenius algebra A in a modular tensor category B is called special if m˝∆ " λ 1 1 A and ˝η " λ 2 dim A for some λ 1 λ 2 " 1. It is called symmetric if it satisfies the following identity:  Let B " Mod V for a rational VOA V such that it is a modular tensor category [51]. We have ZpBq " B b B [92]. We denote the right adjoint functor to the tensor product functor b : B b B Ñ B by b R , which maps Frobenius algebras in B to Frobenius algebras in B b B [71]. We denote the finite set of isomorphism classes of simple objects in B by IrrpBq. For A P ZpBq, we also choose a basis tb pibj;αq A u of hom C pA, i b jq and its dual basis tb A pibj;βq u of hom C pi b j, Aq for i, j P IrrpBq, i.e. b pibj;αq A˝b A pibj;βq " δ αβ 1 ibj , and use the following graphical notations For an algebra A in B, its full center ZpAq in ZpBq will be defined in Definition 4.20. If A is Frobenius algebra, ZpAq can be defined by the subalgebra of b R pAq given by the image of the morphism in eq. (4.4) for A bdy " A [25,71]. Now we are ready to state a classification result for boundary-bulk CFT's. 3. ι : A bulk Ñ b R pA bdy q is an algebra homomorphism factoring through ZpA bdy q ãÑ b R pA bdy q JHEP02(2020)150 satisfying the modular invariant condition, for all i, j P IrrpMod V q, and the Cardy condition where ι˚is the right dual of ι. The conjecture is that this triple is sufficient to give a boundary-bulk CFT defined in Definition 4.4 [87].
Remark 4.8. The complete structure of a boundary-bulk CFT, including spectrums, correlation functions, OPE, partition functions, etc., is rather complicated. The drastic simplification achieved in Theorem 4.7 might looks surprising to physicists. This miracle happens because Mod V has not only the structure of a modular tensor category but also a much richer structure called "vertex tensor category" [52,57], which makes the reduction possible.
Definition 4.9. An algebra A in a fusion category is called separable if the multiplication morphism m splits as an A-A-bimodule map. A separable algebra A is called simple if A is a simple A-A-bimodule. A commutative separable algebra A in a braided fusion category A is called connected if dim hom A p1 A , Aq " 1. A connected commutative separable algebra is also called a condensable algebra [69]. A condensable algebra in a modular tensor category A is called a Lagrangian algebra if pdim Aq 2 " dim A.
Using the results in [33], it is easy to show that a condensable algebra in a modular tensor category can be automatically upgraded to a commutative simple special symmetric Frobenius algebra (CSSSFA), which is unique up to a scalar factor in the definition of the counit (or equivalently, the comultiplication). • the bulk CFT A bulk is a Lagrangian algebra in ZpBq, which can be upgraded to a CSSSFA (unique up to a scalar factor for the counit) [71]; • boundary CFT's that share the same bulk A bulk are simple special symmetric Frobenius algebras (SSSFA) A bdy in B such that A bulk » ZpA bdy q as algebras, where ZpA bdy q is the full center of A bdy [25,33,71];

M N
A bulk " ZpA 1 q " ZpA 2 q " ZpA 3 q P ZpBq Figure 8. This picture depicts a 1+1D world sheet, on which lives boundary CFT's A 1 , A 2 , A 3 P B and a bulk CFT A bulk P ZpBq given by the full center of the boundary CFT's. The domain wall M is an invertible A 1 -A 2 -bimodule and N is an invertible A 2 -A 3 -bimodule.
• a domain wall between two different boundary CFT's A 1 and A 2 is given by a canonical invertible A 1 -A 2 -bimodule in Mod V [18,31].
Two boundary CFT's are Morita equivalent if and only if their full centers are isomorphic as algebras [70]. These results are illustrated in figure 8. and the unit map defined by the canonical embedding 1 B b 1 B ãÑ Zp1 B q (see for example [30,71]). Note that Zp1 B q gives the famous charge conjugate modular invariant partition function; • the domain wall between two boundary CFT's A " y b y˚and B " x b x˚is given by the invertible A-B-bimodule y b x˚ [31].

Internal homs
It turns out that the object y b x˚, appeared in Example 4.11, is the simplest example of a universal construction called the internal hom. Since the notion of an internal hom plays a crucial role in this work, we review its definition.

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Definition 4.12. Let A be a monoidal category. A left A-module M is a category equipped with an action functor d : AˆM Ñ M, pa, xq Þ Ñ a d x and natural isomorphisms: satisfying the following commutative diagrams: Example 4.13. A itself is a left A-module with the action functor defined by the tensor product functor b : AˆA Ñ A.
Definition 4.14 (Internal hom by adjunction). The internal hom rx, ys A , or rx, ys for simplicity, is an object in A uniquely determined (up to unique isomorphism) by a family of isomorphisms: which are natural in all variables a, x, y.
If f : a d x Ñ y is mapped to g : a Ñ rx, ys under the isomorphism (4.8), then f and g are called the mates of each other. Taking a " rx, ys, we denote the mate of the identity map 1 rx,ys : rx, ys Ñ rx, ys by ev : rx, ys d x Ñ y.
Example 4.15. When A is a UFC and M " A, we have rx, ys » ybx˚and ev : rx, ysdx Ñ y is precisely given by 1 y b v x : y b x˚b x Ñ y.
We can equivalently define the notion of an internal hom by its universal property. which is universal among all such pairs. It means that for any such a pair pQ, Q b x f Ý Ñ yq there exists a unique morphism f : Q Ñ rx, ys rendering the following diagram commutative: Remark 4.17. The standard mathematical symbol D! represents the phrase "there exists a unique". It is an important and enjoyable exercise to show that this universal property determines rx, ys up to unique isomorphism. Intuitively, the internal hom rx, ys is in some sense the "maximal" one among all possible pQ, f q.

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There are two canonical morphisms defined by their mates shown as follows: ry, zs b rx, ys Ñ rx, zs ÐÑ ry, zs b rx, ys b x ev˝p1bevq ÝÝÝÝÝÝÑ z (4.10) When x " y " z, these morphisms endow rx, xs with the structure of an algebra.   In particular, rA,´s : A A Ñ A is the precisely the forgetful functor.
The notion of full center can also be defined as an internal hom [17]. Let A be a fusion category and ZpAq be its Drinfeld center. Let A be a simple separable algebra in A. Then the category A A|A of A-A-bimodules in A is a fusion category [96]. There is an action functor ZpAqˆA A|A Ñ A A|A defined by pz, xq Ñ z b x.
Definition 4.20. The full center ZpAq of A is the internal hom rA, As ZpAq .

Unitary boundary-bulk CFT's
We have discussed the classification result of boundary-bulk rational CFT's. What about unitary rational CFT's? The representation theory of a unitary VOA has not been seriously studied until recently [42][43][44]. As far as we know, the classification theory of unitary rational CFT's derived directly from Definition 4.1 or 4.4, is not yet available. There is, however, another formulation of a unitary rational CFT based on the representation theory of rational conformal nets, and its classification theory was known [88,97,98]. It is essentially the same as that of boundary-bulk CFT's given in Theorem 4.7 but with a rational VOA replaced by a rational conformal net, whose module category is a UMTC, and the two types of Frobenius algebras replaced by two types of :-Frobenius algebras (see [72] for a review). A :-Frobenius algebra in a UMTC is a Frobenius algebra satisfying the conditions ∆ " m : and " η : . For example, the Frobenius algebra constructed in (4.5) is a :-SSSFA. We will assume that this classification also works for the unitary boundarybulk CFT's defined in Definition 4.1 or Definition 4.4. We make the classification result more precise below.
From now on, we assume that V is a unitary rational VOA of central charge c such that the category of unitary V -modules (see [42]), still denoted by Mod V , is a UMTC.
If B is a UMTC and M is an indecomposable unitary left B-module, then the category Fun B pM, Mq of B-module functors from M to M can be upgraded to a UFC [38]. Therefore,

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for non-zero x P M, the category B rx,xs|rx,xs » Fun B pM, Mq can be upgraded to a UFC. In this case, the simple separable algebra rx, xs is automatically a Frobenius algebra with ∆ :" m : and :" η : . Moreover, it is a :-SSSFA because m : is automatically a bimodule map by the unitarity of B rx,xs|rx,xs .
Conversely, all :-SSSFA's in B can be realized by internal homs. Indeed, for a :-SSSFA A, the category B A of right A-modules is an indecomposable unitary left B-module. If x is a right A-module, then x˚is automatically a left A-module. For x, y P B A , we have [96] rx, ys " px b A y˚q˚» y b A x˚.
In particular, A » rA, As as Frobenius algebras.
If B is a :-SSSFA in B that is Morita equivalent to A, we can also realize B as an internal hom in the following way. Suppose that an invertible B-A-bimodule x defines the Morita equivalence. Since´b B x : B B Ñ B A defines an equivalence between the two left B-modules and maps B to x, we obtain B » rB, Bs » rx, xs as :-SSSFA's. Combining the above discussion with the assumption that unitary boundary-bulk CFT's based on VOA's is equivalent to those based on conformal nets [88,97,98], we obtain the following physical "theorem".
Theorem ph 4.21. Let B " Mod V be a UMTC. For a given unitary bulk CFT A bulk in ZpBq with a unique vacuum, we have the following assertions.
• The category of boundary conditions of A bulk is given by an indecomposable unitary left B-module M which is canonically associated to A bulk . We have M » B A for a unitary boundary CFT A, i.e. a :-SSSFA in B such that ZpAq » A bulk as algebras.
An object x P M is called a boundary condition.
• A bulk " rid M , id M s ZpBq is a Lagrangian algebra in ZpBq.
• For x P M, the unitary boundary CFT associated to the boundary condition x is given by rx, xs B , which is a :-SSSFA in B such that Zprx, xs B q » A bulk ; • For x, y P M, the domain wall between the two boundary CFT's rx, xs B and ry, ys B is precisely given by the invertible bimodule rx, ys B .
Remark 4. 23. Strictly speaking, we should require not only the bulk CFT to have a unique vacuum, but each boundary CFT also to have a unique vacuum. Usually, a QFT (defined on S n or R n ) with multiple vacuums is not stable. It will flow to a stable one (with a unique vacuum) under perturbation. This requirement amounts to consider only simple boundary conditions in M and boundary CFT's rx, xs B for simple x P M, or equivalently, haploid :-SSSFA's A (i.e. dim hom B p1 B , Aq " 1). For this reason, we sometimes discuss only simple boundary conditions and simple topological edge excitations in many physical discussions.  In particular, in this case, B " H, the only Lagrangian algebra in H is C. The category of boundary conditions associated to the algebra C is H, which is the unique indecomposable semisimple H-module. For x, y P H, the associated boundary TQFT's are matrix algebras rx, xs " x b C x˚" hom H px, xq and ry, ys, respectively, and the domain wall between them is rx, ys " y b x˚" hom H px, yq. Note that the full center of rx, xs is just the usual center Zprx, xsq " C.
Remark 4.26. Theorem ph 4.21 plays a crucial in this work. We want physics readers to keep in mind that this radical simplification of wall-boundary-bulk CFT's to internal homs does not lose any physical information. More precisely, a single internal hom rx, ys B or rid M , id M s ZpBq (for B " Mod V ) automatically includes the information of the spectrum, chiral (or non-chiral) fields, correlation functions, OPE, structure constants and partition functions on torus or any other higher genus surfaces, etc. The key point of this miracle is stated in Remark 4.8.

Constructions of gapless edges
In section 5.1, we give an explicit construction of the so-called canonical gapless edge of a 2d chiral topological order, and show that it is a special case of the so-called canonical construction. In section 5.2, we construct more gapless edges by fusing canonical gapless edges with gapped domain walls.

A natural construction of chiral gapless edge of pB, cq
We are ready to give an explicit construction of a chiral gapless edge for a 2d chiral topological order defined by a pair pB, cq, where B is a UMTC realized by a unitary JHEP02(2020)150 A y " ry, ys M x,y " rx, ys U " V a Figure 9. This picture depicts a 2d topological order pB, cq, together with the canonical gapless edge pV, B Bq, in which homB B px, yq :" rx, ys " y b x˚P B for x, y P B.
rational VOA V , i.e. B " Mod V and c is the central charge of V . We denote the tensor product in B by b " b V In this case, we give a very natural construction of a chiral gapless edge of the 2d topological order pB, cq. We describe its ingredients explicitly as follows (also illustrated in figure 9): • Topological edge excitations are all obtained from moving topological bulk excitations to the edge. They are labeled by objects in B.
• U " M 1 B ,1 B " r1 B , 1 B s " 1 B " V , namely, all the boundary CFT's A x and walls between them M x,y are required to preserve the largest chiral symmetry V . As a consequence, M x,y are objects in B.
• Defect fields in M x,y can be fused with those in M y,z to give defect fields in M x,z . This amounts to a morphism ry, zs b rx, ys Ñ rx, zs in B, which is defined as follows: In this case, rx, ys is automatically an invertible ry, ys-rx, xs-bimodule.
• The spatial fusion of topological edge excitations M x 1 ,y 1 b M x,y Ñ M x 1 bx,y 1 by (depicted in figure 7) is a morphism in B defined as follows: If it indeed exists, it is not known how many there are. In a special case, when c " 0, the only unitary VOA of central charge zero is V " C [40]. It means that, in this case, a 2d topological order pB, 0q can have gapped edges (V " C) but does not have any non-trivial chiral gapless edges. It may have non-chiral gapless edges [16,80].
These observables on the above chiral gapless edge of pB, cq can be summarized by a pair pV, B 7 q, where B 7 is a B-enriched monoidal category defined as follows: • an object in B 7 is a topological edge excitation, i.e. an object in B; • hom B 7 px, yq " rx, ys " y b x˚; • the identity morphism id x : 1 B Ñ rx, xs " x b x˚is defined by u x : 1 B Ñ x b x˚; • the composition morphism e : ry, zs b rx, ys Ñ rx, zs is defined by eq. (5.1); • the morphism b : rx 1 , y 1 s b rx, ys Ñ rx 1 b x, y 1 b ys is defined by eq. (5.2). [91] that this categorical structure B 7 is indeed a B-enriched monoidal category. Moreover, it is just a special case of the so-called canonical construction. [91]). Let D be a braided monoidal category and Y a monoidal category equipped with a braided oplax monoidal functor F Y : D Ñ ZpYq such that F Y p1q " 1. It endows Y with structure of an action functor d : DˆY Ñ Y defined by pc, yq Þ Ñ c d y :" F Y pcq b y. We assume that internal hom rx, ys in D exists for all x, y P Y. Then we obtain from the triple pD, Y, F Y q a D-enriched monoidal category, denoted by D Y, as follows:

Theorem 5.3 (Canonical Construction
1. objects in D Y are precisely the objects in Y; 2. for x, y P Y, homD Y px, yq :" rx, ys in D; 3. the identity morphism id x : 1 D Ñ rx, xs is the morphism in D given by the mate of the canonical isomorphism 1 D d x » x in Y; 4. the composition morphism˝: ry, zs b rx, ys Ñ rx, zs is the mate of the following composed morphism: JHEP02(2020)150 5. b : rx 1 , y 1 s b rx, ys Ñ rx 1 b x, y 1 b ys is the mate of the following composed morphism: The underlying category of D Y is Y, and the background category of D Y is D.
Using the notation of the canonical construction, we see that B 7 " B B. From now on, we denote pV, B 7 q by the pair pV, B Bq. As we will show later, only physically relevant F Y are braided monoidal functors (NOT oplax). Hence, among all possible Y (viewed as a finite left B-module), Y " B is the canonical choice. For this reason, from now on, we will refer to pV, B Bq as the canonical chiral gapless edge of pB, cq.
Remark 5.4. Strictly speaking, for a given 2d topological order pB, cq, there is no canonical chiral gapless edge even if we assume the existence of a VOA V such that B » Mod V because such VOA's are often not unique (see [10] and [20,21,81]). Once V is fixed, however, the categorical data B B is indeed canonical with respect to B.
We would like to provide a graphical notation for this canonical gapless edge as shown below. In the second picture, we try to provide different "physical meanings" to the two "B"s in B B.
• The background category "B", as a UMTC, can be interpreted as a 2d topological order. Therefore, we use this "B" to label a "fictional 2d phase" in the time direction (depicted as the vertical plane). Note that the fictional vertical plane also remind us the 1+1D world sheet of this chiral gapless edge.
• The underlying category "B", viewed as a UFC (by forgetting its braiding), is the category of topological edge excitations. It can be interpreted as a "fictional gapped domain wall" between the fictional 2d phase in the time direction and the physical 2d phase pB, cq in the spatial dimensions. Notice that B is indeed a legitimate gapped domain wall (actually the trivial one) between two 2d phases pB, cq and pB, cq.
Note that the whole fictional vertical plane, including the fictional "wall" labeled by B, should be viewed as a single gapless edge pV, B Bq. This graphical notation of canonical gapless edge in (5.3) has an immediate advantage. It seems to suggest that this gapless JHEP02(2020)150 edge can be obtained, as illustrated below, by starting from a horizontal plane in the spatial dimensions then folding the left half of the plane to a vertical spacetime plane. Of course, such a folding process is physically impossible. A picture might be closer to the physical reality is the following one. Imagine that we cut out the left half of a 2d topological order pB, cq defined on R 2 by brutal force. It takes the system in the neighborhood of the edge away from a RG fixed point. As time goes by, the edge will undergo a self-healing process by flowing to a new RG fixed point. One of the possible RG fixed point is the canonical chiral gapless edge pV, B Bq. We will show later that, in general, there are other RG fixed points corresponding to different chiral gapless edges. What makes this picture so amazing is its holographical nature. That is, the information lost through the brutal force cutting in the spatial dimensions can be completely restored in the temporal dimension! Therefore, we believe that this fictional folding process can not be just an ad hoc bookkeeping trick. It should have some yet-to-be-clarified deep physical meanings, and will be proved to be very useful later. Therefore, we would like to refer to this fictional folding trick as a "topological Wick rotation".
Remark 5.5. It is generally accepted that a chiral gapless edge of a 2d topological order is given by a so-called "chiral CFT", the precise meaning of which has never been clarified. Our result in this subsection gives a precise meaning to a "chiral CFT" as a pair pV, B Bq. Note that the spaces of conformal blocks do not live on the 1+1D world sheet of the gapless edge directly. It can, however, be recovered by the underlying category of B B.
Remark 5.6. The unitary boundary-bulk CFT obtained by applying the dimensional reduction depicted in figure 5 on the canonical gapless edge is precisely the Cardy case presented in Example 4.11.
It turns out that the canonical construction of enriched monoidal categories also naturally includes the mathematical description of a gapped edge of a 2d topological order, i.e. a UFC, as a special case. Indeed, for an ordinary UFC M, there is a unitary braided monoidal functor H ãÑ ZpMq defined by C Þ Ñ 1 ZpMq . The UFC M can be viewed as the H-enriched monoidal category obtain from the triple pH, M, H ãÑ ZpMqq via the canonical construction, i.e. M " H M. Moreover, H can be viewed as the UMTC Mod V for V " C, which should be viewed as the trivial unitary rational VOA of central charge c " 0. As a consequence, a gapped edge M of a 2d topological order is described by the pair pC, H Mq, where the underlying category is again the category of topological edge excitations M and the background category is H. In this case, rx, xs H " hom M px, xq, as a direct sum of matrix algebras, should be viewed as a boundary TQFT of the trivial bulk TQFT given by its full center Zprx, xs H q " C.

General gapless edges
Let B " Mod V . Let pC, cq be a 2d topological order that is Witt equivalent to pB, cq. In other words, the 2d topological orders pB, cq and pC, cq can be connected by a gapped domain wall, which is described by Ý Ñ ZpMq is that if the orientation of the wall is the same (resp. the opposite) as the induced orientation of a bulk, then this bulk phase acts on the wall from right (resp. left). We will use this convention throughout this work.
We will postpone the proof of (5.6) until section 6.3. Let us take (5.6) for granted for now. We list a few basic ingredients of the gapless edge pV, B Mq.

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1. Topological edge excitations x, y, z,¨¨¨are objects in M, and the trivial topological edge excitation 1 is given by the tensor unit 1 M of M.
2. The VOA U " M 1,1 that can live on the entire 1+1D world sheet is given by r1 M , 1 M s B , which a condensable algebra in B [19]. According to [53], U is indeed a VOA extension of V . More generally, we have M x,y " rx, ys B for x, y P M.
From the explicit construction of the gapless edge pV, B Mq, we can see some general features of gapless edges.
• In general, V Ĺ U . For example, when pC, cq is obtained by condensing a non-trivial condensable algebra U in pB, cq (i.e. dim U ą 1), we have C " B 0 U , where B 0 U is the category of local U -modules in B [19,69]. This condensation also produces a gapped domain wall between pB, cq and pC, cq described by the UFC B U of right U -modules in B. In this case, M " B U , and U " r1 M , 1 M s B ‰ V is a non-trivial VOA extension of V . By results in [53], we have B 0 U " Mod U . Therefore, C fi Mod V in this case.
• In general, Mod V fi C fi Mod U . For example, when B " Mod V is obtained by condensing a non-trivial condensable algebra A in C, i.e. B " C 0 A , the condensation produces a gapped domain wall M " C A . In this case, the central functor B Ñ M is the canonical embedding C 0 Recall that the category of boundary conditions of the unitary bulk CFT ZpU q is given by the category B U of right U -modules in B (Theorem ph 4.21). It is illuminating to study the relation between the category B U of boundary conditions and that of topological edge excitations M on the gapless edge pV, B Mq. Namely, the category of boundary conditions B U is a fusion subcategory of that of topological edge excitations. We illustrate these functors in the following diagram: where left (resp. right) adjoints form a commutative diagram. The functor L :

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• In general, we have B U Ĺ M. For example, let A be a non-trivial condensable algebra A P C. When Mod V " B " C 0 A and M " C A , on the gapless edge pV, B Mq, we have U " A " V and B U Ĺ M. In this case, M splits as a direct sum of indecomposable unitary left B-modules, i.e.
For two simple objects x P B U and y P M i for i " 1,¨¨¨, N , we have Rpyq " 0 and M x,y " 0 because hom B pb, M x,y q » hom M pb d x, yq " 0 for all b P B. But M 1,yby˚‰ 0. Moreover, the boundary CFT M y,y " ry, ys ‰ 0 because 1 B ãÑ ry, ys is non-zero. In general, the boundary CFT's M y,y and M x,x are not compatible in the sense that they have different bulk CFT's. By Theorem 4.21, each unitary left B-modules is associated to a unique bulk CFT. The bulk CFT's associated to B U and M i are different in general. In other words, the dimensional reduction process depicted in figure 5 might produce different bulk CFT's in general. Also note that both L and R L are monoidal functors, but L R and R are not monoidal in general.

Chiral gapless edges II
In this section, we continue the analysis started in section 3 on the structures and properties of observables on the 1+1D world sheet of a chiral gapless edge. This analysis leads us to a classification theory of all chiral gapless edges of any 2d topological orders.

X 7 is an enriched unitary fusion category
At the end of section 3, we conclude that the observables on the 1+1D world sheet of a chiral gapless edge of the 2d topological order pC, cq can be summarized by a pair pV, X 7 q, where V is the chiral symmetry and X 7 is an Mod V -enriched monoidal category. The objects in X 7 are topological edge excitations, and the hom spaces hom X 7 px, yq :" M x,y are boundary CFT's (if x » y) or domain walls between boundary CFT's (if x fi y). In order to gain a better understanding of the hidden structure of X 7 , it is crucial to understand the mysterious relation between the objects in X 7 and the morphisms in X 7 . In terms of the underlying category X of X 7 , it amounts to ask for the relation between X and B " Mod V .
Consider the 1+1D world sheets depicted in figure 10 (a), which is essentially the same as figure 5 (b). The boundary CFT's A x 0 , A y 0 , A z 0 are compatible with U in the sense that they share the same bulk CFT ZpU q. As we have shown at the end of section 5, it is possible that there are x P X such that M 1,x " 0, i.e. there is no boundary condition changing operator that can change the boundary condition associated to 1 to that associated to x. As a consequence, the whole enriched category X 7 splits into a direct sum of connected components: where 1 P X 7 0 and M x i ,x j " 0 for x i P X 7 i , x j P X 7 j , i, j " 0,¨¨¨, N and i ‰ j. Within each connected component X 7 i , all boundary CFT's and domain walls M x i ,y i (for x i , y i P JHEP02(2020)150 Figure 10. These two pictures depicts two possible 1+1D world sheets obtained by applying dimensional reduction to figure 5 (a). The associated two sets of boundary CFT's tA 1 , A x0 , A y0 u and tA xi , A yi , A zi u are potentially incompatible and M x0,xi " 0 for x 0 P X 0 , x i P X i .
X i ) are necessarily compatible. Figure 10 (b) depicts three compatible boundary CFT's A x i , A y i , A z i for x i , y i , z i P X 7 i and their domain walls and the associated bulk CFT. By the Naturality Principle in physics, if A x i appears as a boundary CFT on the 1+1D world sheet of the gapless edge pV, X 7 q, then all boundary CFT's and their domain walls that are compatible with A x i should appear on this world sheet. 1 As a consequence, by Theorem 4.21 and Remark 4.24, each connected component X 7 i of X 7 , is precisely the enriched category B X i via the canonical construction (see Example A.3), where X i is an indecomposable unitary left B-module and, at the same time, the category of boundary conditions associated to the bulk CFT ZpA x i q. Therefore, we obtain as B-enriched categories and X » X 0 ' X 1 '¨¨¨' X N as unitary left B-modules. We denote the left B-action on X by d : BˆX Ñ X. In other words, X 7 » B X as B-enriched categories with rx, ys B » M x,y for x, y P X. We want to show that it is also an equivalence of enriched monoidal categories. Remark 6.1. When V " C, B " H and the edge pC, H Xq is gapped. In this case, H has a unique indecomposable unitary left H-module H. Therefore, X » H '¨¨¨' H as unitary left H-modules, i.e. X i " H. In this case, M x i ,x j " 0 for x i P X i , x j P X j and i ‰ j. For x i P X i " H, the matrix algebra rx i , x i s H " hom X px i , x i q is actually a 0+1D boundary TQFT, which has a trivial bulk TQFT given by its center Zprx i , x i sq " C. The ordinary center is also the full center in this case.

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Morrison-Penneys' classification of B-enriched monoidal categories [91] indicates that X 7 is necessarily obtained from a triple pB, X, F X q via the canonical construction, where F X : B Ñ ZpXq is a braided oplax monoidal functor such that the composed functor is left adjoint to a functor M 1,´: X Ñ B, which will be defined later, i.e.
In our case, since X is already a unitary left B-module with rx, ys B " M x,y , the result turns out to be even stronger: F X is a unitary braided monoidal functor (NOT oplax), or equivalently, L is a central functor (recall Diagram (2.5)). To see this, let us go through some parts of Morrison-Penneys' proof that are adapted and simplified to suit our case.
1. There is a well-defined C-linear functor M 1,´: X Ñ B defined by y Þ Ñ M 1,y and It is clear that M 1,´p X i q " 0 for i ą 0.
2. Let L : B Ñ X be the left adjoint functor of M 1,´. The category of topological edge excitations X is monoidal. The natural physical requirement is that each topological edge excitation should have its anti-particle. This requires X to be rigid thus a UFC. We want to show that L is a central functor.
3. By definition (see [91], section 2.7), X 7 is rigid if and only if X is rigid.
Thus Lpaq b x » a d x for a P B and x, y P X. Then Lpaq » a d 1. This further implies the following isomorphisms: Therefore, L is monoidal.

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5. The monoidal functor L : B Ñ X is a central functor. This amounts to show that each Lpaq can be endowed with the structure of a half-braiding [19]. Indeed, we have hom X pLpaq b x, yq » hom B pa, M x,y q » hom B pa, M 1,x˚by q » hom X pLpaq, x˚b yq Therefore, we obtain a half-braiding isomorphism Lpaq b x » x b Lpaq as desired by Yoneda Lemma. Further details of the proof could be found in [91]. The physical meaning of above proof is not very explicit. In the rest of this subsection, we would like to explain some of its hidden structures in more physical terms and some physical consequences.
1. Since 1 P X 0 and A 1 " U , by Theorem 4.21, we must have X 0 » B U as left Bmodules. We want to construct a canonical equivalence X 0 » B U explicitly. Note that, for x 0 P X 0 , there is a natural right U -action on M 1,x 0 (see figure 11): which endows M 1,x 0 with the structure of a right U -module. Therefore, the functor M 1,´: X Ñ B factors through the forgetful functor f : B U Ñ B, i.e. M 1,´" f˝R, where R : X Ñ B U is defined by x Þ Ñ M 1,x . We want to show that the functor R : X 0 Ñ B U is the canonical equivalence we are looking for.
2. It is clear that RpX i q " 0 for i ą 0. By Theorem 4.21, each boundary condition X P B U can be realized as the domain wall rU, Xs B " X between the boundary CFT's U and rX, Xs B because rX,´s B : B U Ñ B is precisely the forgetful functor (recall Example 4.19). Therefore, M 1,x 0 for x 0 P X 0 recovers all boundary conditions in B U , or equivalently, R : X 0 Ñ B U is essentially surjective. Moreover, for x P X 0 , the domain wall Rpxq " M 1,x as an invertible domain wall uniquely determines the boundary CFT M x,x , which has to be rRpxq, Rpxqs B by Theorem 4.21. Then, for x, y P X 0 , we must have M x,y » rRpxq, Rpyqs B and It is implies that R is fully faithful. Therefore, R : X 0 » Ý Ñ B is an equivalence of finite unitary categories.

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3. Let R L be the left adjoint functors of R. The left adjoint functor of f is´b V U . we obtain the following diagram (similar to the diagram (5.9)): where the right and left adjoint functors form two commutative diagrams, respectively. Since RpX i q " 0 for i ą 0, R L factors as B U Ñ X 0 ãÑ X.
4. We explain that B U is also a UFC. Mathematically, for y P X 0 , the right U -module M 1,y may have more than one possible left U -actions. But there is only one that is physically meaningful. It is determined 3 by the adiabatic move along the path γ 3 depicted in figure 11. According to our braiding convention (recall Remark 3.11), this left U -action is defined by where c´,´is the braiding in B. This left U -action allows us to fuse M 1,y 1 with M 1,y horizontally (in figure 11) to give M 1,y 1 b U M 1,y . As a consequence, B U is a UFC with the tensor product b U and the tensor unit U . Moreover, the functor´b V U is monoidal.
5. Since L : B Ñ X is a central functor, by ( [19], Lemma 3.5), the functor R L defineds a monoidal equivalence from B U to its image in X. Since the left B-module structure on X is induced from the monoidal functor L, the monoidal functor R L is also a left B-module functor. Since X 0 » B U as left B U -modules, R L : B U Ñ X 0 must be an equivalence of left B U -modules. Therefore, R : X 0 Ñ B U is an equivalence of left B U -modules and a monoidal equivalence. In particular, we have a canonical isomorphism: M 1,y 1 by » M 1,y 1 b U M 1,y , for y, y 1 P X 0 .  Figure 11. This picture depicts a special situation on the 1+1D world sheet of the chiral gapless edge pV, X 7 q for x, y, x 1 , y 1 P X 0 . It shows that the horizontal U -actions on M 1,y 1 , M x,1 , M 1,y , M x,1 are determined by the vertical U -actions via the braidings long the fours paths, respectively.
can be split into M 1,y and M x,1 without altering the physics (depicted in the right half of figure 11). It is also clear that the dual formula of (6.3): also holds. As a consequence, the spatial fusion of M x 1 ,y 1 and M x,y can be achieved by fusing M 1,y 1 , M x 1 ,1 , M 1,y , M x,1 as shown in figure 11. The braided interchange property (recall Diagram 3.8) implies that the morphism (3.6) must be an isomorphism: This formula does not hold for general x, x 1 , y, y 1 P X. This also explain why (5.2) is an isomorphism because X " X 0 for the canonical chiral gapless edge. When V " 0 (i.e. edge is gapped), X 0 " H and (6.4) simply says that the tensor product of two matrix algebras is again a matrix algebra. More formula of this type will be given and studied in [76].
The results in this subsection motivates us to introduce the following definition. Then the results in this subsection can be summarized as follows: • A chiral gapless edge pV, X 7 q of a 2d topological order pC, cq is precisely given by pV, B Xq, where B " Mod V is a UMTC and B X is a B-enriched unitary fusion category. JHEP02(2020)150

Classification theory of chiral gapless edges
The last constraint on the mathematical description of a chiral gapless edge is that it must satisfy the so-called boundary-bulk relation, which says that the center of a gapped/gapless edge should coincide with the UMTC describing the bulk phase [75]. Motivated by the physical meaning of the boundary-bulk relation for 2d topological orders, we introduced the notion of the center of an enriched monoidal category in [78]. We will not give the precise definition of this notion in this work. Instead, we will do that in Part II [80].
Theorem 6.5 ( [78]). For a B-enriched fusion category B X obtained from a triple pB, X, F X q, its center Zp B Xq is given by the centralizer of the image of B in ZpXq, denoted by F X pBq 1 | ZpXq .
Therefore, the requirement of boundary-bulk relation, i.e. Zp B Xq » C, is equivalent to the condition: F X pBq 1 | ZpXq » C as UMTC's. Since B is a UMTC, the functor F X is necessarily a braided monoidal embedding. Therefore, we obtain B b C » ZpXq as UMTC's [19]. Similar to gapped edges, to uniquely determine the edge, we need specify how the topological excitations in the bulk are mapped to those on the edge [69]. This is given by a central functor C Ñ X (recall Diagram (2.5)). Together with F X , we obtain a unitary braided monoidal equivalence: which should be viewed as a defining data of the edge. This completes our analysis.
Remark 6.6. The existence of a braided monoidal equivalence φ X in (6.5) should be viewed as an anomaly-free condition. Namely, the bulk pC, cq should resolve all the anomalies of the edge pV, B Xq viewed as an anomalous 1d phase.
How do we know whether we have found all the physical requirements of B X? We have indeed found all of them because the pairs pV, B Xq are precisely those chiral gapless edges constructed in section 5.2 via the physical process of fusing canonical gapless edges with gapped domain walls (with a loophole to be fixed in section 6.3). In other words, topological Wick rotations realize all chiral gapless edges.
We have found the precise mathematical description of the observables on the 1+1D world sheet of a chiral gapless edge as a pair pV, B Xq. We need to answer a few physical questions before we claims that this mathematical description actually classify all chiral gapless edges.
1. What do we mean by a chiral gapless edge physically? In reality, if a chiral gapless edge of a 2d topological order is realized in lab, it might not have any topological edge excitations on it at all before we introduce topological defects (or impurities) onto the edge. In this context, by a chiral gapless edge described by pV, B Xq, we mean the maximal way of inserting topological defects onto the edge without breaking the chiral symmetry V .

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2. How do we know different pairs describes different edges? We provide three answers. First, two different edges can be obtained by fusing canonical edges with two different gapped domain walls along two potentially different 2d topological orders. Therefore, a transition between two chiral gapless edges can be understood as a gap-closing topological phase transition between two 2d topological orders, together with a gapclosing transition between two gapped domain walls. Secondly, for two different B X and B 1 X 1 , the categories of topological edge excitations are different; mathematically, B X and B 1 X 1 , as two E 1 -algebras, can be viewed as different topological invariants defined on open 1-disks. They lead to different global topological invariant by integration or factorization homology [3]. Changing of topological invariants is associated with a phase transition in general. Thirdly, for a given chiral gapless edge pV, B Xq, when we introduce more defects that break the chiral symmetry V to a smaller one (see Remark 6.11), we believe that this process of breaking of chiral symmetry should cause a purely edge phase transition.
3. How to define a phase transition between two 1+1D gapless phase? A phase transition between two gapped phases are defined by closing the gap. But a phase transition between two gapless phases is not so clear from this perspective. As far as we know, there is no model-independent definition of such a phase transition. Interestingly, the physical intuition of chiral gapless edges actually provides us three possible model-independent definitions of a phase transition between two potentially anomalous 1+1D gapless phases.
(a) We can define such a phase transition as a process of changing or breaking local quantum symmetries (i.e. chiral symmetries in this case).
(b) Since all chiral gapless edges can be obtained by fusing the canonical chiral gapless edges with gapped domain walls along two 2d bulk phases, we can define the phase transition between two chiral gapless edges by a gap-closing topological phase transition between two bulk phases and that between two gapped domain walls.
(c) We can define such a 1+1D phase transition by a 2d topological phase transition via a topological Wick rotation. More precisely, a 2d topological phase transition from a 2d topological order pB, cq to a new one pD, cq can be achieved by first introducing some islands of pD, cq-phases into the 2d topological order pB, cq as depicted in the first picture in figure 12. When these islands proliferated, a topological phase transition from pB, cq to pD, cq occurs. By Wick rotating this process, we obtain a description (or a definition) of a purely edge 1+1D phase transition as depicted in the second picture in figure 12. This definition automatically makes sense in any dimensions. We will return to this point in section 7.
As a consequence of these definitions, any chiral gapless edge pV 1 , B 1 X 1 q can be obtained from another one pV, B Xq via a purely edge phase transition.

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topological Wick rotation Ý ÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝÝ Ñ Figure 12. These two pictures depict a physical description of pure edge phase transition via a topological Wick rotation.
We conclude that the mathematical description of observables on a 1+1D world sheet of a chiral gapless edge as a pair pV, B Xq provides a classification of all chiral gapless edges. We summarize this result as a physical theorem.
Theorem ph 6.7. Gapped and chiral gapless edges of a 2d topological order pC, cq are precisely described and classified by pairs pV, B Xq, where • V is a unitary rational VOA of central charge c such that B :" Mod V is a UMTC; • X is a UFC equipped with a unitary braided monoidal equivalence φ X : BbC Ñ ZpXq; • B X is the B-enriched UFC obtained via the canonical construction from the triple pB, X, F X q, where F X is the unitary braided monoidal functor defined as follows: Equivalently, for the convenience of numerical computation, • all gapped and chiral gapless edges of pC, cq are classified by pairs pV, Aq, where A is a Lagrangian algebra in Mod V b C.
When V " C, the edge is gapped. In this case, Mod V " H. Remark 6.8. Unstable edges naturally occur if we fuse a chiral gapless edge with a gapped domain wall. If we allow unstable edges in our mathematical description, we can simply replace the condition that X is UFC by a weaker condition that X is a unitary multi-fusion category (see also Remark 6.15).
Remark 6.9. There is a nice classification of gapless edges for abelian 2d topological orders given in [9]. It will be very interesting to compare their results with ours. Constructing new gapless edges via anyon condensations of a non-abelian bulk phase pC, cq was considered in some special cases in [6]. Our constructions of gapless edges are more general than those obtained by condensing the bulk [19,69]. supmq nˆs upnq m Ă supmnq 1 , spp2mq nˆs pp2nq m Ă sop4mnq 1 , sopmq nˆs opnq m Ă sopmnq 1 , sopmq 4ˆs up2q m Ă spp2mq 1 ,¨¨¨.

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Then A can be viewed as a condensable algebra in B " Mod V [19,53] and we have Mod A » B 0 A . Therefore, the two topological orders pB, cq and pMod A , cq can be connected by a gapped domain wall given by B A . By topological Wick rotations, we obtain • a chiral gapless edge of pB, cq defined by pA, Mod A pB A qq, in which M 1,1 " A; • a chiral gapless edge of pMod A , cq defined by pV, B pB rev A qq, in which M 1,1 " A.
Remark 6.11. Recall Remark 3.7, for a given chiral gapless edge, the chiral symmetry is chosen to be the maximal one that is transparent on the entire 1+1D world sheet of the edge. If we choose a smaller one, i.e. V 1 Ĺ V , then the category of boundary conditions for the same bulk CFT ZpU q will be enlarged to pMod V 1 q U . Note that The category of topological edge excitations will be enlarged to pMod It is not yet possible to list explicitly all chiral gapless edges for a given chiral 2d topological order because, for a fixed UMTC B, how many unitary rational VOA's V satisfy Mod V » B as UMTC's is still an open question. It was conjectured that every UMTC C (without fixing c) can be realized by the category of unitary modules over a unitary VOA (see for example [39,102]). It is not true if we also fix c. More precisely, for a pair pC, cq, in general, it is not possible to find a VOA V of central charge c such that C » Mod V . For example, when C is non-chiral and C fi H, such V does not exist because the only unitary VOA of central charge 0 is the trivial one V " C. In other words, the non-chiral 2d topological order pC, 0q does not have any chiral gapless edges. It has only gapped and non-chiral gappable gapless edges (recall Remark 5.2). It is, however, possible to find V of central charges of 8Z such that C » Mod V [39]. It is interesting to note that our classification result actually supports a different conjecture: Conjecture 6.12. For a UMTC C and a central charge c such that c top C " c pmod 8q, there is at least one unitary VOA V of central charge c such that Mod V is a UMTC Witt equivalent to C. Remark 6.13. Interestingly, if there are only finitely many chiral gapless edges of a given bulk phase pC, cq, an assumption which is not totally unreasonable, then our theory suggests that there are only finitely many unitary rational VOA's with central charge c such that their module categories are UMTC's that are Witt equivalent to C. Remark 6.14. A domain wall between two 2d topological orders can be realized as an edge by the folding trick. Therefore, the classification theory of the gapped and chiral gapless edges also provides that of the gapped and chiral gapless domain walls.

Universality at RG fixed points
The reasoning behind our classification theory of chiral gapless edges given in section 6.2 will not be complete unless we fill the one last loophole. Note that we have not yet provided any explanation of the mysterious formula (5.6). We will do that in this subsection. Instead of explaining the mysterious formula (5.6), we would like to explain a more general formula, which describes the fusion of two chiral gapless domain walls (recall Remark 6.14). We illustrate two chiral gapless walls before the fusion in figure 13 (a) and after the fusion in figure 13 (b). More precisely, A, B, C, D, E are UMTC's, and M and N are UFC's equipped with unitary braided monoidal equivalences φ M : 4 The vertical direction is the direction of time. Two vertical planes depict the 1+1D world sheets (or fictional bulk phases) of two chiral gapless domain walls pV A , A Mq and pV B , B Nq. Note that V A and V B have central charge c 2 and c 3 , respectively. They precisely make up the difference of the chiral central charges of the two sides of the wall. Now we propose a formula for the spatial fusion of these two walls:

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Note that this formula covers the formula (5.6) as a special case.
Remark 6.15. Note that M b D N is in general a multi-fusion category even if both M are N are fusion ones. In this case, it describes an unstable domain wall, which can flow to a stable one under RG flow [3]. Since the multi-fusion categories naturally appear in such a fusion process, it is also natural to use unitary multi-fusion categories to describe the category of topological wall excitations.
Before we explain the formula (6.6), we first recall that M x,y " rx, ys B . By Definition 4.16, M x,y is universal in the mathematical sense. More precisely, rx, ys B , as a space of boundary condition changing operators, is equipped with a map ev : rx, ys B d x Ñ y, which JHEP02(2020)150 specifies how the boundary condition changing is done. For any space of boundary condition changing operators, i.e. pQ, f : Q d x Ñ yq, where Q is the space and f specifies how the boundary condition changing is done, there exists a unique morphism f : Q Ñ rx, ys B exhibiting the following diagram commutative. We believe that it is just a special case of a more general principle for physics at a RG fixed point. More precisely, we propose 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.
Another example of the Principle of Universality is the boundary-bulk relation, i.e. bulk = the center of a boundary [75]. More precisely, the unique gapped n`1d bulk BpXq of a (gapped or gapless) nd boundary X satisfies the following universal property: where BpXq is viewed as an nd topological order by forgetting additional structures, b is the stacking operation of two phases, and both m and g are morphisms between potentially anomalous nd (gapped or gapless) phases (introduced in [75]) and define a unital BpXqaction and a unital Q-action on X, respectively. When X is a gapped boundary of a 2d topological order, the 2d bulk BpXq is precisely given by the Drinfeld center ZpXq of X, which is viewed as a unitary fusion category. Mathematically, in this case, b is the Deligne tensor product; m is just the tensor product functor b of X, and is unital and monoidal; and the unital action g is also monoidal; and (6.8) is nothing but the mathematical universal property of the Drinfeld center.
More examples will appear below. Note that the condition "at RG fixed point" is crucial. This principle can not be true for general QFT's or many-body condensed matter systems that are not at RG fixed point. However, for general QFT's, we believe that observables (such as those in the bulks, boundaries, walls or other defects, instantons, etc.) should factor through the universal ones. Now we return to the physical situation depicted in figure 13. A typical example of physical observables on the 1+1D world sheet of each gapless wall before the fusion are rx, ys A for x, y P M and rp, qs B for p, q P N. If we treat two vertical planes as two real bulk phases in space, then, when we fuse them, the observables rx, ys A and rp, qs B should also fuse and give rx, ys A b rp, qs B P A b B (see figure 13 (b)). This naive picture contradicts to both the formula (6.6) and the Principle of Universality.

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It is again a typical quantum quenching scenario. When we fuse two walls in the spatial dimensions, only thing we should expect is that the categories of wall excitations M and N are fused together to give M b D N. Indeed, the topological excitations in D acts on those in M from right according to pm, dq Þ Ñ m d d and on those in N from left according to pd, nq Þ Ñ d d n. Assume that this fusion produces a functor F from M b N to the not-yet-known category of topological excitations in the final wall. We should expect that F pm d d, nq » F pm, d d nq for m P M, n P N. Such F is called a D-balanced functor. The relative tensor product b D : MˆN Ñ M b D N, defined by pm, nq Þ Ñ m b D n, is an example of such D-balanced functors, i.e. pm d dq b D n » m b D pd d nq, and is universal among all D-balanced functors. That is, for any D-balanced functor G : is commutative. By the Principle of Universality, we conclude that the category of topological excitations on the wall after the fusion is given by M b D N. It is worthwhile to remind the readers that this is precisely how two gapped walls are fused [37,64,69]. As a consequence, by ( [77], Theorem 3.3.6) and the boundary-bulk relation (recall Theorem 6.5), the background category of the wall after the fusion has to be A b B.
Assume that this fusion happens at t " 0. Then two wall excitations x P M and p P N are fused into a new wall excitation xp :" x b D p P M b D N. According to the discussion in section 3.2, observables living on the world line support on it will be changed to the boundary CFT M xp,xp , which is precisely given by rxp, xps AbB . Similarly, topological excitations y P M and q P N is fused to yq :" y b D q. Therefore, the domain wall between the two boundary conditions xp and yq should be given by M xp,yq " rxp, yqs AbB .
In general, the spatial fusion of instantons rx, ys A b rp, qs B is not invariant under RG flow, 5 which will drive it to a new RG fixed point given by rxp, yqs AbB . Indeed, rx, ys A b rp, qs B is a space of boundary condition changing operators because it is equipped with a natural way to do the "boundary condition changing": prx, ys A b rp, qs B q d px b D pq :" prx, ys A d xq b D prp, qs B d pq Therefore, there exists a canonical morphism g : rx, ys A brp, qs B Ñ rxp, yqs by the universal property of the internal hom (6.7). When g is not an isomorphism, it means that additional operators such as excitations tunneling between two 1d walls becoming local as the walls getting close, and it is a manifestation of the spatial fusion anomaly and, at the same time, a mathematical description of the RG flow. The appearance of spatial fusion anomaly is due to the fact that both domain walls are anomalous 1d phases which allows additional operators to become local as two 1d walls getting close. Indeed, when both of the 1d phases JHEP02(2020)150 are anomaly-free, what live on two world sheets are modular invariant bulk CFT's. In this case, g is indeed an isomorphism, or equivalently, spatial fusion anomaly vanishes.
In summary, we have provided an explanation of the formula (6.6) in terms of RG flow and Principle of Universality. Therefore, the reasoning behind our main result Theorem ph 6.7 is complete.

Conclusions and outlooks
In this work, for a 2d topological order pC, cq, we have found a complete and precise mathematical description of all observables on the 1+1D world sheet of a 1d chiral gapless edge. This description also provides a mathematical classification of all chiral gapless edges as summaried in Theorem ph 6.7. As a consequence, all chiral gapless edges can be obtained from topological Wick rotations.
The significance of this work is manifold. As we will show in [80], the mathematical theory of chiral gapless edges immediately implies a theory of non-chiral gapless edges and a generalization of the old boundary-bulk relation ( [77], Theorem 3.3.7) to include gapless edges. Besides these immediate generalizations, there are more surprising and exciting implications of this work.
1. It provides a new and systematic way of describing and classifying all gapless edges of 2d symmetry protected/enriched topological orders.
2. It provides a new and systematic way of studying all 1+1D phase transitions among gapped or gapless 1d phases.
More importantly, this work also suggests that we can study higher dimensional gapless boundaries similarly because many of the physical arguments used in this works, such as dimensional reductions, topological Wick rotations and Principle of Universality at RG fixed points, should work automatically in any dimensions.
1. As we mentioned in section 2.3, the main results of [75] says that a topological order in any dimension is determined by one of its boundaries by taking center regardless if this boundary is gapped or gapless (see [75], Remark 5.7). As a consequence, if an n`1d topological order has a precise mathematical description based on higher categories (see [73,74]), so does its nd gapless boundary. This prediction is already highly non-trivial and rather surprising. In particular, this work can be viewed as a consequence of this prediction.
2. This work also suggests that we should be able to construct gapless boundaries of higher dimensional topological orders via the topological Wick rotations together with additional information of local quantum symmetries such as the chiral symmetry in this work. Note that the topological Wick rotation depicted in (5.7) automatically makes sense in all dimensions. This provides a new way to construct and describe potentially anomalous gapless phases in higher dimensions by certain enriched higher categories. Although the topological Wick rotation alone can not determine local quantum symmetries, it does provide a severe constraint on the possible JHEP02(2020)150 local quantum symmetries. Moreover, many results can be derived without knowing local quantum symmetries. If this surprising speculation is indeed correct, it will be a very exciting progress for a systematic study of gapless phases in higher dimensions. Moreover, this work also suggests that a phase transition between two n`1D gapless phases can be defined by a gap-closing n`2D topological phase transitions via topological Wick rotations as illustrated in figure 12, which automatically makes sense in any dimensions.
We summarize above observations by a correspondence between gapped and gapless phases. It will be called Gapped-Gapless Correspondence, and will serve as a guiding principle for our future studies.
Gapped-Gapless Correspondence: All nd gapless boundaries (or potentially anomalous nd gapless phases), including higher codimensional domain walls, of an n`1d topological order can be obtained from topological Wick rotations together with additional information of local quantum symmetries. Moreover, a phase transition between two gapless boundaries (without altering the bulk topological order) can be defined by a gap-closing topological phase transition via a topological Wick rotation (see figure 12).
The typical structures of local quantum symmetries for QFT's in higher dimension will be briefly discussed in ( [80], Remark 3.1).

(A.3)
Remark A.2. We distinguish the notation id x and 1 x , where the former one is the identity morphism in an enriched category, and 1 x is the identity morphism in an ordinary category.
Note that an ordinary category is a category enriched in the category of sets. We will call an element f P hom A p1, hom C 7 px, yqq a morphism from x to y, denoted by f : x Ñ y. A morphism f : x Ñ y is called an isomorphism (or invertible) if there is a morphism g : y Ñ x such that g˝f " id x and f˝g " id y . Now we define the underlying category of C 7 , denoted by C. The category C consists of the same objects as those in C 7 and • hom C px, yq :" hom A p1, hom C 7 px, yqq for x, y P ObpC 7 q; • the identity morphism 1 x P hom C px, xq is just id x : 1 Ñ hom C 7 px, xq; • the composition map hom C py, zqˆhom C px, yqÝ Ñ hom C px, zq is defined by the following composed map hom A p1, hom C 7 py, zqqˆhom A p1, hom C 7 px, yqq b Ý Ñ hom A p1, hom C 7 py, zq b hom C 7 px, yqq Ñ hom A p1, hom C 7 px, yqq.
Example A.3. (Canonical Construction) Let M be a left A-module category which has internal homs in A, i.e. the functor´d x : A Ñ M admits a right adjoint rx,´s : M Ñ A for every x P M. Then M can be promoted to an A-enriched category M 7 which has the same objects as M and hom M 7 px, yq " rx, ys. The composition˝is given by the canonical morphism ry, zs b rx, ys Ñ rx, zs and id x is given by the canonical morphism 1 Ñ rx, xs. We also denote this enriched category by A M. Note that its underlying category is just M because hom M px, yq » hom A p1, rx, ysq.
Example A.4. If A is rigid, then it can be promoted to an A-enriched category A 7 . In this case, hom A 7 px, yq " y b x˚where x˚is the left dual of x, hom A 7 p1, 1q " 1, : pz b y˚q b py b x˚q Ñ z b x˚is induced by the counit map v y : y˚b y Ñ 1, and id x is given by the unit map u x : 1 Ñ x b x˚.

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Definition A.5. An enriched functor F : C 7 Ñ D 7 between A-enriched categories consists of a map F : ObpC 7 q Ñ ObpD 7 q and a morphism F : hom C 7 px, yq Ñ hom D 7 pF pxq, F pyqq for every pair x, y P C 7 such that the following diagrams commute for x, y, z P C 7 : 1 idx y y id F pxq 9 9 hom C 7 px, xq F G G hom D 7 pF pxq, F pxqq, hom C 7 py, zq b hom C 7 px, yq˝G G F bF hom C 7 px, zq F hom D 7 pF pyq, F pzqq b hom D 7 pF pxq, F pyqq˝G G hom D 7 pF pxq, F pzqq. Definition A.7. An enriched natural transformation ξ : F Ñ G between two enriched functors F, G : C 7 Ñ D 7 consists of a morphism ξ x : F pxq Ñ Gpxq for every x P C such that the following diagram commutes for x, y P C 7 : x hom D 7 pF pxq, F pyqq ξy˝´G G hom D 7 pF pxq, Gpyqq. (A.5) An enriched natural transformation ξ is called an enriched natural isomorphism if each ξ x is an isomorphism. Now we assume A is a braided monoidal category equipped with braiding c x,y : xby Ñ y b x for x, y P A. Let C 7 , D 7 be A-enriched categories. The Cartesian product C 7ˆD7 is an A-enriched category defined as follows: • ObpC 7ˆD7 q " ObpC 7 qˆObpD 7 q; • hom C 7ˆD7 ppx, yq, px 1 , y 1 qq " hom C 7 px, x 1 q b hom D 7 py, y 1 q; • the composition : hom C 7ˆD7 ppx 1 , y 1 q, px 2 , y 2 qqbhom C 7ˆD7 ppx, yq, px 1 , y 1 qq Ñ hom C 7ˆD7 ppx, yq, px 2 , y 2 qq is given by hom C 7 px 1 , x 2 q b hom D 7 py 1 , y 2 q b hom C 7 px, x 1 q b hom D 7 py, y 1 q 1bc´1b1 Ý ÝÝÝÝÝ Ñ hom C 7 px 1 , x 2 q b hom C 7 px, x 1 q b hom D 7 py 1 , y 2 q b hom D 7 py, y 1 q (A.6) bÝ ÝÝÝÝ Ñ hom C 7 px, x 2 q b hom D 7 py, y 2 q.
Remark A.9. An enriched monoidal category is strict if λ, ρ, α are the identity natural transformations ( [91], Definition 2.1). In the strict case, an A-enriched monoidal category defined here is equivalent to "an A-enriched monoidal category" defined in [91].
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