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Nuclear Physics B 886 (2014) 436-482

www. elsevier. com/locate/nuclphysb

Anyon condensation and tensor categories

Liang Konga,b'*

a Institute for Advanced Study (Science Hall), Tsinghua University, Beijing 100084, China b Department of Mathematics and Statistics, University of New Hampshire, Durham, NH 03824, USA

Received 10 January 2014; received in revised form 6 June 2014; accepted 2 July 2014

Available online 9 July 2014

Editor: Hubert Saleur

Abstract

Instead of studying anyon condensation in various concrete models, we take a bootstrap approach by considering an abstract situation, in which an anyon condensation happens in a 2-d topological phase with anyonic excitations given by a modular tensor category C; and the anyons in the condensed phase are given by another modular tensor category D. By a bootstrap analysis, we derive a relation between anyons in D-phase and anyons in C-phase from natural physical requirements. It turns out that the vacuum (or the tensor unit) a in D-phase is necessary to be a connected commutative separable algebra in C, and the category D is equivalent to the category of local A-modules as modular tensor categories. This condensation also produces a gapped domain wall with wall excitations given by the category of a-modules in C. A more general situation is also studied in this paper. We will also show how to determine such algebra a from the initial and final data. Multi-condensations and 1-d condensations will also be briefly discussed. Examples will be given in the toric code model, Kitaev quantum double models, Levin-Wen types of lattice models and some chiral topological phases.

© 2014 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

1. Introduction

Anyon condensation is important subject to study in the field of topological order because it tells us how a topological phase is transformed into another topological phase. In 2002, Bais,

* Correspondence to: Institute for Advanced Study (Science Hall), Tsinghua University, Beijing 100084, China. E-mail address: kong.fan.liang@gmail.com.

http://dx.doi.org/10.1016Zj.nuclphysb.2014.07.003

0550-3213/© 2014 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3.

Schroers, Slingerland initiated a systematic study of anyon condensation based on the idea of Hopf symmetry breaking [12,13]. The theory was further developed by Bais and Slingerland in an influential work [10], and was followed and further developed by many researchers (see for example [17,11,16,14,9,56,1] and references therein). In spite of many works in this direction, the fundamental mathematical structure that controls the anyon condensation has not been identified in its full generality. Recently, Kapustin and Saulina [44], followed by Levin [56] and Barkeshli, Jian and Qi [1], successfully identified the notion of Lagrangian subgroup [25] with gapped boundaries. But these studies are restricted to the gapped boundaries in abelian Chern-Simons models and Kitaev quantum double models. A general condensation theory is still not available.

On the other hand, it has been known to physicists for a long time that a system of anyons can be described by a (unitary) modular tensor category (see Appendix A.1 for its definition). Then it is clear that anyon condensation should be related to some mathematical structures in modular tensor categories. Mathematicians know how to obtain new (unitary) modular tensor categories from a given (unitary) modular tensor category C since the seminal works of Bockenhauer, Evans and Kawahigashi [4-6] in 1999-2001 and that of Kirillov Jr. and Ostrik [53] in 2001 (see also [34]). They proved that given a connected commutative separable algebra A in C, the category Cl0c of local A-modules is a (unitary) modular tensor category. All of these mathematical notions will be defined in Appendix A. What this mathematical result suggests to us is obvious: an anyon condensation is determined by a connected commutative separable algebra A in the un-condensed phase C. In the spring of 2009, Alexei Kitaev told me this connection between anyon condensation and connected commutative separable algebra in a modular tensor category. He also provided a brief physical proof based on many-body wave functions [50]. This connection was announced independently by Alexei Davydov in an international workshop in Sydney in 2011 [21]. In that talk, he stated explicitly this connection and examples were also provided there, but he did not provide any explanation why this connection is physically reasonable. This connection was also known to Fuchs, Schweigert and Valentino and was briefly mentioned in Section 4 in their work [36]. Anyon condensation was also studied in [15] in the framework of Kitaev quantum double models. But the general theory was not given there.

The condensed matter physics community has not fully embraced this connection yet. This delay is perhaps partially due to the abstractness of the language used by mathematicians. But perhaps more important reason is that it is unclear why these mathematical structures emerge naturally in physics. Through the influence of the works [44,51,15,36], recently, many physicists start to notice a possible link to mathematical literatures and expressed the willingness to understand these abstract structures in more physical way. The main goal of this paper is to provide a detailed explanation of how each ingredient of the complete mathematical structures emerge naturally from concrete and natural physical requirements in anyon condensation. We hope that this analysis can convince physicists that the tensor-categorical language, although abstract, is a powerful and efficient language for anyon condensation.

In general, an anyon condensation is a very complicated physical process. The mathematical structure associated to it cannot be very trivial. Instead, it is very rich. This mathematical structure captures the universal structure in anyon condensation which is model independent. Working with examples sometimes does not shed enough lights on the underlining universal structure because a concrete example usually contains too many structures that might mislead us. Ideally, we would like to take no more assumptions than what we absolutely need. This suggests us to take a bootstrap approach towards anyon condensation. Namely, instead of studying concrete models directly, we start from an abstract setting in which only two abstract sets of data, i.e. two systems of anyons before and after the condensation, are given (see Fig. 1). We want to work

Fig. 1. The set-up of bootstrap analysis: We consider an anyon condensation in a topological phase described by a modular tensor category c. The condensed phase is described by another modular tensor category d, and the gapped excitations on the domain wall is described by a spherical fusion category £.

out all the necessary relations between these two sets of data C and D. We will show in details in Section 2-4 how these relations in terms of abstract tensor-categorical structures emerge from natural physical requirements. The final result is summarized below (see Appendix A for the definitions of various mathematical notions).

Main results of bootstrap analysis (Theorembs 4.7). If a system of anyons, described by a (unitary) modular tensor category D, is obtained from a condensation in another system of anyons given by a (unitary) modular tensor category C, together with a gapped domain wall with wall excitations given by a (unitary) spherical fusion category £ (see Fig. 1), then we must have

1. The vacuum of D-phase is given by (or condensed from) a connected commutative separable algebra A in C and D ~ Clo as (unitary) modular tensor categories, where QlAf is the category of local A-modules in C.

2. Let CA be the category of right A-modules in C. Then we have £ ~ CA as (unitary) spherical fusion categories.

3. Anyons in the D-phase can move onto the wall according to the monoidal functor (also called a bulk-to-wall map) - <g> A : C ^ CA defined by C ^ C < A for all C e C.

4. Anyons in the D-phase can move onto the wall according to the embedding Clo ^ CA, then move out to the D-side freely.

Remark 1.1. In order to summarize and highlight the results obtained from our bootstrap analysis, we will borrow some mathematical terminologies to state these bootstrap results as theorembs, lemmabs, propositionbs and corollarybs, where we use an extra superscript bs to distinguish them from real mathematical results. For example, the above main result is summarized in Theorembs 4.7.

Actually, we will carry out our bootstrap study with a weaker assumption on the domain wall. More precisely, we assume that the vacuum B of £ comes from C but is not necessarily given by the same algebra A. We will analyze the excitations on the domain wall (see Fig. 1) and their relations with two bulk phases. This bootstrap approach has been carried out in Sections 2-4. In the end, we obtain that the vacuum B of £ must be a (not necessarily commutative) connected separable algebra in C and £ ~ CB|B, where CB¡B is the category of B-bimodules in C. The complete

bootstrap result is summarized in Theorembs 4.2 (see also Fig. 3). The results in Theorembs 4.7 is a special case of Theorembs 4.2 when B = A. Our bootstrap analysis also shows that B can be viewed as a connected separable algebra in CA. This suggests that this general situation can be viewed as a two-step condensation, in which the first condensation of A is followed by the second 1-d condensation of B in the 1-d phase CA on the wall. We will briefly discuss them in Sections 4 and 8.

Once this connection is fully established. In Section 5, We will use existing mathematical theory, in particular the seminal work by Davydov, Muger, Nikshych and Ostrik [23], to study the structures of multi-condensations and answer some questions in anyon condensation, such as the existence and uniqueness of condensation algebras A and B from given physically detectable data. Similar to many bootstrap studies, bootstrap results might include solutions that are not entirely physical. Indeed, we will show that the precise information of the algebra B is not entirely physical, even though the second 1-d condensation is physical. Namely, there can be two non-isomorphic candidates for the algebra B that cannot be distinguished from any physical detectable data. Instead, only the Morita equivalence class of B is physically detectable. This nuance is partially due to the artificial generality of our bootstrap analysis. But more importantly, it is perhaps due to the fact that macroscopic physics is insensitive to the small perturbation of microscopic physics. Indeed, one can lift this unphysicalness microscopically (but not macro-scopically) by considering extended string-net models defined on a unitary tensor category A equipped with a fiber functor [2]. The boundary (or wall) lattice in such a model depends on a choice of A-module category M together with a fiber functor «: M ^ Hilb which can select an algebra B from its Morita class.

Physicists are invited to pay more attention on the special case B = A. Such a condensation is called a one-step condensation. This case is completely physical. Namely, the structures contain in Theorembs 4.7 are all physical. More complicated multi-step condensations can all be decomposed into one-step condensations. We will study the mathematical structures underlining multi-step condensations in Section 7. In particular, in Section 7.2, we will show that any two topological phases connected by a gapped wall can be obtained from a two-step condensation in a single phase. In Section 6, we will provide many examples of one-step condensation from the toric code model, Kitaev quantum double models, Levin-Wen type of lattice models and condensations in chiral topological phases.

Remark 1.2. For most physical applications, we need the assumption of unitarity [49]. Since our theory works pretty well in the non-unitary cases, we only assume the modular tensor category without unitarity in the main body of this paper. We will put all results in the unitary cases in Remarks.

After the appearance of the 3rd version of this paper on arXiv, I was informed by Sander Bais that this work has some overlaps with Sebas Eliens' thesis [26], in which a commutative algebra object as Bose condensates was discussed (see [26, Section 6.2]). See also their recent paper [27] joint with Romers.

The basic mathematical structures used in this work have already appeared in the seminal works [4-6] by Bockenhauer, Evans and Kawahigashi in 1999-2001. Moreover, they worked in the unitary setting, which is the most relevant case in physics. But they used the language of the subfactor theory instead of the tensor-categorical language. The dictionary provided by Kawahigashi (Table 1) is helpful.

This paper contains no new mathematical result.

Table 1

A dictionary between the tensor-categorical language used in this work and the subfactor language used in the works of Bockenhauer, Evans and Kawahigashi.

Tensor-categorical language

Subfactor language

Connected comm. separable algebra A Category ca of A-modules Category of local A-modules The bulk-to-wall map Boundary-bulk duality

Local Q-system «-Induced system Ambichiral system «^-Induction

Quantum double of a-induced system

The layout of the paper is: in Section 2, we carry out this bootstrap analysis for the condensed phase D; in Section 3, we carry out a bootstrap analysis for the domain wall between the C-phase and the D-phase; in Section 4, we will analyze the relation between wall excitations and two bulk excitations, thus complete our bootstrap analysis; in Section 5, we will discuss how to use initial data and final data to determine the condensation, i.e. finding (A, B); in Section 6, we will provide examples; in Section 7, we will discuss multi-condensations and Witt equivalence; at last in Section 8, we will give a remark to 1-d condensations, some conclusions and outlooks; Appendix A contains the definitions of all tensor-categorical notions appeared in this work.

2. Bootstrap analysis I: the condensed phase D

Let us start with a 2-d topological phase containing a system of anyonic excitations which are described by a modular tensor category C (see Definition A.12), which is equipped with a tensor product (or < for simplicity), a tensor unit 1g (or 1 for simplicity), an associator ax,Y,z : (X < Y) < Z —> X < (Y < Z), a braiding cx,y : X < Y —> Y < X and a twist 9x : X —> X for all X, Y, Z e C. This is our initial data. Notice that we have intentionally ignored unit isomorphisms and duality maps from the data because the role they play in our presentation is implicit. We assume that an anyon condensation happens in a region inside of a 2-d phase C as depicted in Fig. 1, and the anyons in the condensed phase are given by another modular tensor category D, which is equipped with a tensor product a tensor unit 1l, an associator aL,M,N : (L M) N —> L (M N), a braiding cLN : M N —> N M and a twist OjL : M —> M for all L, M, N e D. This is our final data. Note that we do not require C and D to be non-chiral (i.e. a monoidal center of a fusion category). The goal of this section is to explore the relation between these two sets of data.

Before we start our bootstrap analysis, we need first clarify the physical meaning of direct sum in a modular tensor category C. In general, an object X in C is a Z>0-linear combination (or direct sum) of simple objects, i.e.

X = i © j © k © i © •••.

This Z>0-linear combination should be viewed as a categorification of the superposition of states in quantum mechanics.1 Then, X can be viewed as a categorical wave function. The vacuum

1 In quantum mechanics, the superposition of two states ¡1) and ¡2) is given by ¡1) + ¡2) (up to a factor). But a topological excitation (or an anyon), according to [51], cannot be described by a single quantum state in general. Instead, it is described by a space of states, which is invariant under the action of a local operator algebra Q, i.e. a Q-module. Then the superposition of states must be replaced by the direct sum of spaces of states. We will refer to such a space of states as a categorified quantum state or wave function. For example, in the Ising topological phase, we have a < a = 1 © ty (see Eq. (42)), which means that the fusion product of two a-anyons is a superposition of 1 and ty, or equivalently, it can split into either 1 or ty.

1e can be viewed as a categorical ground-state (or vacuum) wave function. The category C can be viewed as the categorical Hilbert space especially when the objects of C are given by the representations of some symmetry (quantum) group.2

For convenience, we will call such X a composite anyon, and call a simple object in C a simple anyon to distinguish them. We will also use the term X-anyons. For example, three X-anyons means X ® X ® X.

Now we are ready to start our bootstrap analysis. We will start from a few basic physical building blocks of the relation between C and D.

1. Any composite anyon in D is made of (or a Z>0-linear combination of) simple anyons in C. The condensation process does not affect the ingredients of such a composite anyon. More precisely, any condensation, no matter if it is anyonic or not, is a process of selecting a subspace of the original large Hilbert space H. The ground state or other states in the condensed phase are those states in H that survives the condensation. Therefore, all objects in D are automatically objects in C, the condensation simply induces the identity condensation map M M for all M in D. In particular, the categorical vacuum wave function (or the tensor unit) 1d should be viewed as an object A in C, i.e. 1® = A. In general, A is a composite anyon in C unless the condensation is trivial. The object A should be viewed as a categorical ground-state wave function in the condensed phase. In general, objects in D do not cover all objects in C except in the case of trivial condensation. Namely, not all composite anyons in C survive the condensation.

2. Since all anyons in D are made of simple anyons in C, all the possible fusion-splitting channels in the condensed D-phase must come from those in the uncondensed C-phase. The information of these channels is given by hom spaces. Therefore, we must have an embedding:

homD (M, N) — homC(M, N).

Namely, horn® (M, N) should be viewed as a subspace of home (M, N). In other words, D can be viewed as a subcategory of C.

Remark 2.1. All physical observables are encoded in the hom spaces. Very often, physicists like to understand a morphism f : X — Y by the canonically associated linear maps: / : home (i, X) — home(i, Y) defined by g — f o g for all simple objects i e C. These two points of view are equivalent. In this work, we will treat f as a physical observable and use it directly instead of /*, and call f as a morphism or a map.

Remark 2.2. Since D is a subcategory of C, if a simple object in C survives the "condensation", it must still be simple in D. It was known, however, in physics that a simple object in C can split after "condensing" to the boundary. This superficial contradiction is actually a confusion in language. These two "condensations" are referring to two different ways to compare two different categories. We will explain this point in Remarks 5.5 and 5.6.

3. The vacuum 1e in C-phase should condense into the vacuum 1® in D. Mathematically, this means that there exists a morphism iA : 1e — A in C.

2 This is always possible for some weak Hopf algebras (but in general not in a unique way) by a generalized version of Tannaka-Krein duality [39] (see also [4,51] for a graphic construction).

4. The difference between these two phases lies mainly in the way they fuse anyons. Therefore, we would like to know the difference and relation between M ® N and M N for any pair of anyons M, N e D. The condensation process should be able to produce M N from M ® N. Therefore, we expect that there is an onto map (or a quotient map), called condensation map,

Pm,n : M ® N ^ M N (1)

in C. Moreover, we require that M N lies in M ® N in a canonical way (automatic in the unitary cases). By that we mean, there is a canonical morphism

eMN : M N ^ M ® N (2)

such that pm,n ◦ &mn = idM®DN.

Since 1D = A, we must have A A = A and A M = M = M A. We denote the

map pa,a : A ® A ^ A = A A by /A and eA,A by eA.

Since C is semisimple, /A and eA define a decomposition of A ® A:

A ® A = A © X (3)

where X can be chosen to be the cokernel of eA. By the mathematical definition of direct sum, it amounts to the existence of maps eX, rX, together with eA, /A, as shown in the following diagram:

ea rx A-->- A ® A^-

/¿a ex

satisfying /A o eA = idA, rx o ex = idx, and

/A o ex = 0, rx o eA = 0, eA o /a + ex o rx = idA®A- (4)

Remark 2.3. If C is unitary (see Definition A.6), then we can choose pMyN and eMyN to be a part of orthonormal basis such that eMiN = p*M N.

These are the basic data associated to an anyon condensation. We will explore the properties of these data below.

1. Associativity of /A. If we condense three A-anyons3 in the bulk of C-phase, this process is independent of which pair of A condense first. This independence leads to the following

3 In physics, a condensation involves a large number of particles. It does not make any sense to say "condense three anyons". A condensation in an anyon system is triggered by interaction among anyons. This interaction (e.g. adding a pair-wise interaction 1 — pm N to the Hamiltonian) makes the subspace M N of M ® N more favorable in energy. We believe that a condensation in a region R in the bulk can be realized by turning on the interaction in many small disjoint disks, each of which contains only a small number of anyons, and gradually enlarging the disk area such that the entire region R is covered by the disks. By "condensing three anyons", we mean turning on the interaction in a small disk containing only 3 anyons and projecting the local Hilbert space associated to the small disk onto the subspace of energy favorable states. A real condensation is a combination of such projections in a large quantity (in the thermodynamics limit). We use the terminology "condense three anyons" here just for convenience. We will use it and similar terms in many places later. On the other hand, to tell a complete story of anyon condensation, one would like to really write down a Hamiltonian system that can realize a given phase transition. It is an important problem in physics (see [14]), but beyond the scope of this paper. We hope that the gap between a complete physical theory of anyon condensation and the bootstrap study in this work can be filled in the near future.

commutative diagram:

(A ® A) ® A —

(A A) ® A

(A A) A-

&a.a.a

■>A ® (A ® A)

A ® (A A)

= A (A A)

which means that ia : A ® A u A is an associative multiplication. 2. Unit properties. The identity condensation map idA : A u A should be stable under a perturbation of the vacuum 1g in C. This leads to the following commutative diagrams:

1C ® A -^U- A ® A

A ® A-

A ® 1c

where the first diagram says that if we start with an A-anyon, then "create" a vacuum nearby, then condense it into A, then condense this A further with the second A into the new vacuum A, this process is physically not distinguishable with doing nothing (or the identity condensation map). The meaning of the second commutative diagram in (6) is similar.

3. Commutativity. The condensation of two vacuums A ® A is independent of whether we move one A-particle around the other A-particle along an arbitrary path before or after the condensation. This leads to the following commutative diagram:

A ® A-

A = A A-

ca,a=ida

-> A ® A

■>■ A A = A

Remark 2.4. The commutative diagrams (5), (6) and (7) are nothing but the defining properties of a commutative C-algebra for the triple (A. ia. iA) (recall Definition A.13).

4. The stability of the vacuum A in A ® A under the A-action.4 The vacuum A, which lies in A ® A, i.e. eA : A u A ® A, should be stable under the screening of a cloud of vacuum (see Fig. 2). It implies that the A-action on A ® A cannot create any splitting channels from A to X. Otherwise the vacuum A in an A-cloud can decay, which is physically unnatural. More precisely, this means that the following two composed maps

A ® A —U A ® A ® A U A ® A X (8)

A ® A U A ® A ® A A ® A X (9)

4 This stability is different from the usual stability of the vacuum under the small perturbations of Hamiltonian.

Fig. 2. Stability of the vacuum A in A ® A under the action of A: The condensed vacuum A in D can be viewed as a canonical building block of A ® A. This information is encoded in the map ea : A ^ A ® A. When A ® A is shrouded by (or simply a part of) an A-cloud (a tensor product A®" for large n), A in A ® A should be stable under the action of A on A ® A from both sides. In other words, splitting A into X under such action is forbidden. Otherwise, A is not stable. Therefore, we obtain that the map eA must be stable under the A-action on A ® A from both sides. Mathematically, this condition says that two composed maps (8) and (9) are zero maps.

must be zero maps. We will use the conditions (8) and (9) to show that the algebra A is separable (see Definition A.16) in the next paragraph. Physicists can skip it. Notice that A ® A is naturally an A-bimodule. By the associativity, the map /A is automatically an A-bimodule map. Using this fact, together with (8) and (9) being zero maps, it is easy to show that the map eA is an A-bimodule map. Moreover, using (4), it is easy to show that the following composed map:

A ® X -—-A ® A ® A U A ® A X defines a left A-module structure on X. Similarly, the following composed map:

X ® A U A ® A ® A U A ® A X

defines a right A-module structure on X. These two module structures are compatible so that X is an A-bimodule. Using the fact that eA and /A are A-bimodule maps, it is easy to show that both eX and rX are A-bimodule maps. In other words, the decomposition (3) is also a decomposition of A-bimodules.

In mathematics, such an algebra (A, /A, iA) is called separable (see Definition A.16). An important property of an separable algebra in C is that both the category CA of A-modules in C and the category CA|A of A-bimodules in C are semisimple (see for example [53]).

5. The algebra (A, /A, iA) is connected, i.e. home(1e, A) ~ C (see also Definition A.16): As we will show later that all objects M in D are A-modules and morphisms are A-module maps. Therefore,

C ~ homD(A, A) = homA(A, A) ~ home(1e,A) (10)

Above bootstrap results can be summarized as follows:

Lemmabs 2.5. D is a subcategory of C. The vacuum A = 1d of the D-phase is a connected commutative separable algebra in C.

To simplify our terminology, we introduce the following notion.

Definition 2.6. An algebra in a modular tensor category is called condensable if it is a connected commutative separable algebra.

Example 2.7. In the toric code model, two examples of condensable algebra in the monoidal center Z(RepZ2) of the fusion category RepZ2 are 1 © e and 1 © m. More examples of condensable algebra in other models will be given in Section 6.

Since dimhomg (A, 1) = 1, we can choose a map eA : A ^ such that eA ◦ iA = dim A ■ idA. It is known that the pairing A ® A > A is non-degenerate. This implies [32,

Lemma 3.7] that A has a unique Frobenius algebra structure (see Definition A.21). Moreover, by [32, Corollary 3.10], this Frobenius algebra is automatically symmetric; by [32, Lemma 3.11], it is also normalized-special (see Definition A.21). As a consequence, the coproduct Aa satisfies lA o Aa = idA and Aa o ia is a projector on A ® A. Moreover, Aa is an A-bimodule map because of the defining property of a Frobenius algebra. In other words, Aa give a splitting of the A-bimodule map iA : A ® A ^ A. Using results in [32], one can prove that

eA = Aa. (11)

Therefore, we have shown that a condensable algebra gives a simple normalized-special commutative symmetric Frobenius algebra in C [32]. Conversely, the latter algebra can reproduce the original condensable algebra. In other words, these two concepts are equivalent.

Remark 2.8. We are trying to keep the mathematical structure to the minimum for physics oriented readers. That is why we choose to let Frobenius algebra structure emerge automatically. Alternatively, one can argue directly that eA : A ^ A ® A gives a coassociative comultiplication because the vacuum A lies in A ® A ® A in a canonical way. Moreover, the separability of eA is equivalent to the defining property (Eq. (65)) of a Frobenius algebra. Since A is connected, i.e. dimhomg (A, 1) = 1, for any f e homg (A, 1), the map (1 ® f) o eA is an A-module map and thus must be c ■ idA for some scalar c e C because A is a simple A-module (proved later). Then we can choose the counit e : A ^ 1 so that the counit condition hold. Again we obtain a structure of a normalized-special commutative symmetric Frobenius algebra on A.

Remark 2.9. When C is unitary, we have eA = i*A. Then the coassociativity follows from the associativity automatically. Choose the counit eA := t*A. Then the counit property is automatic. By [32], we have eA o iA = dim A ■ idA automatically.

The condensation also must preserve the twist (a generalized notion of spin). This leads to the following conditions:

A—A M-^M

idA (12)

' - D "

for all M e D. Since A is the vacuum, = idA. We must require that 0A = idA. In physical language, it means that A must be a boson. This condition turns out to be a redundant condition

because a commutative Frobenius algebra A is symmetric if and only if 0A = idA [34, Proposition 2.25]. Therefore, we obtain

Corollary 2.10. A condensable algebra A in C is automatically a boson, i.e. 0A = idA.

The second diagram in (12) simply means 0D = 0M . Before we discuss its meaning, we would like to first explore the properties of the condensation maps \xM := pA,M : A ® M ^ A ®D M = M for all M e D and eA M : A ®D M A ® M.

1. The pair (M, mM) is a left A-module:

(a) Associativity. As before, if we condense two A-anyons and an M-anyon, the process should not depends on which two of them condense first. This leads to the following commutative diagram:

A ® (A ® M) —

A ® (A ®d M)

A ®D (A ®D M)--

<*a,a,m

->(A ® A) ® M Ma1

(A ®d A) ® M

■ (A ®D A) ®D M

(b) Unit property. Due to the similar physical reason behind the unit property of A, we have

1C ® M

■A ® M

Above two commutativity diagrams (13) and (14) are the defining properties of a left A-module for the pair (M, ^M) (see Definition A.14).

2. Similarly, M equipped with a right A-action pM,A : M ® A ^ M is a right A-module.

3. (M, /j,m) is a local A-module: condensation process is irrelevant to how you arrange the initial configuration of an A-anyon and an M-anyon. More precisely, it means that if you start with an arbitrary initial position of these two anyons, then move one around the other along a path, then condense them, it is equivalent to first condense them, then move them around the same path. Mathematically, it means that the condensation respects the braiding. Thus, we obtain the following commutative diagram:

A ® M CA,M> M ® A^-AUA ® M

where A ®D M = M = M ®D A and c^ A = c^ M = idM [47, Proposition XIII.1.2]. Therefore, we obtain

Mm ◦ CM,a ◦ CA,m = Mm-

Such an A-module M is called local (see Definition A.15). This commutative diagram also means that the left A-module structure determines the right A-module structure in a unique way via braiding, i.e.

Pm.a = Im ◦ Cm. a = Im ◦ ca]m-

For this reason, we will also denote pM.A by im .

4. Stability of a condensed anyon M in A ® M and M ® A under the A-action. Similar to the stability of the vacuum A, we require that the condensed particle M in A ® M, i.e. eA.M : M c—> A ® M is stable under the screening of a cloud of the vacuum A. It implies that left A-action on A ® M cannot create any splitting channels from the subobject M to other complementary subobjects in A ® M. Similar to that of the stability of the vacuum A, we obtain that eA^M is a left A-module map. By the locality of M, the map eM^A : M — M ® A is automatically a right A-module map and an A-bimodule map.

5. Compatibility among eA, eA M and eM.A. Consider two physical processes described by the two paths in the first of the following two diagrams:

M-® M

A ® a ® A ® M

-M ® A

M ® A^-^M ® A ® A

Notice that the physical processes described by the composed map A ® M

-a— a ® M can be viewed as something virtually happening all the time. Of course,

one can embed A into more A-anyons (or an A-cloud) and then fuse them with M until the last A. It is a natural physics requirement that eA,M must be stable under such virtual processes. Therefore, we conclude that the first diagram in (16) is commutative. Similarly, we can convince ourselves that the commutativity of the second diagram in (16) is also a physical requirement.

Using the Frobenius property of Aa = eA and the identities: im o eAM = idM and im o eM A = idM, and their graphic expressions (see Appendix A.2), we obtain the following identities:

eA,M =

V eM>A=V

Remark 2.11. When C is unitary, then identities (17), together with eA = i*a and eA = i*A, implies that eA,M = I*M and eM,A = P'M,A.

6. Morphisms in D are A-module homomorphisms. The morphisms in D determine the fusionsplitting channels in the phase D. These fusion-splitting process must come from those fusion-splitting process (or morphisms) in C and survived the condensation process. In particular, it means that such a morphism must remain intact after the screening by a cloud of the vacuum A. In other words, we have the following commutative diagrams:

A ® M-

A M = M-

-^ A ® N

->-N = A N

for all f e hom® (M, N). By the fact that both eA,M and ¡xN are left A-module maps, it is clear that the commutativity of the diagram (18) is equivalent to the condition that f is an A-module map, i.e. hom® (M, N) = homA(M, N). Mathematically, it means that the

embedding D ^ ClAc is fully faithfully. This fact implies, in particular, the identity (10). We would also like to point out that the upper path in diagram (18) defines a screening map ScA : homg (M, N) ^ home/„c (M, N) given by5

Sca : f ^ ßM ◦ (if ) ◦ eA,M-

An A-module map is automatically an A-A-bimodule map. This screening map ScA is very natural from physical point of view because a fusion-splitting channel in C-phase screened by a cloud of the vacuum A is automatically a fusion-splitting channel in D-phase. Using (17) and the locality of M as A-module, the screening map defined in (19) can be equivalently defined graphically as follows:

SC A(g) =

Using the normalized-specialness of the Frobenius algebra A, it is also easy to see that the screening map ScA is a projector, i.e. ScA o ScA = ScA, and surjective. We adapt the superficially new definition, which appeared in [53,34], not only because it looks pictorially more like a screening of M by a cloud of the vacuum A, but also because the new definition has other applications which does not work for the definition (19). For example, if M is a non-local left A-module and g = idM : M ^ M, the screening operator defined in (20) is actually a projection onto the largest local sub-A-module of M [53,34]. An important example of morphisms in D is 0® = 0M (recall the second diagram in (12)). Actually, for a left A-module M, the condition that 0M e homA(M, M) is equivalent to the condition that M is a local A-module [34, Proposition 3.17].

Remark 2.12. When C is unitary, by Remark 2.9, it is easy to see that ScA o * = * o ScA. This

implies that ClA is a *-category (see Definition A.6).

5 In the special case M = N, assuming that A is commutative symmetric special Frobenius, this screening map was given as the Qm-operator defined in Eq. (3.35) in [34] (see also (20)). But we cannot use the Qm-operator directly here because we want to apply the result to prove Eq. (10), which was further used to prove that A is a special symmetric commutative Frobenius algebra.

7. ®m = ®A (see Definition A.22 for the definition of ®A). Notice first that the condensation cannot distinguish the following two condensations: (N ® A) ®m M and N ® (A ®m M). Namely, we must have (N ® A) ®m M ~ N ® (A ®m M). The rest argument is a little mathematical. Notice that the canonical map fN,M : N ®A M ^ N ®m M must be an epi-morphism because pN,M is an epimorphism. It is enough to show that the kernel of fN,M is zero. Since N can always be realized as a submodule of N ® A (recall (17)), it is enough to prove that the map fN®A,M : (N ® A) ®A M ^ (N ® A) ®m M is an isomorphism. On the one hand, fN®A,M is an onto map. On the other hand, the domain is isomorphic to the codomain as objects:

(N ® A) ®AM ~ N ® (A ®A M) ~ N ® (A ®M M) ~ (N ® A) ®M M.

Therefore, fN®A,M can only be an isomorphism. By the universal properties of ®A, these isomorphisms fN,M defines an natural isomorphism between two tensor product functors f :®A -=> ®m. Hence, we can take ®m = ®A. Moreover, for f : M ^ M' and g : N ^ N', it is easy to show that

f ®Ag = PM' ,N (f ® g) ◦ eM,N. (21)

The associator a£ M N : L ® A (M ®A N) ^ (L ®A M) ®A N is uniquely determined by m,n and the universal property of ®A. Therefore, we must have M N = at M N■ More precisely, it can be expressed as follows:

aL,M,N = PL®aM,N ◦ (PL,M ® idN) ◦ aL,M,N ◦ eL, M®AN ◦ (idL ®A eM,N). (22)

Remark 2.13. When C is unitary, it is easy to show that (f ®A g)* = f * ®A g* and (aA M N)* = (aA M N). The unitarity of the unit morphisms in (62) is trivially true here. In other words, CAc is a monoidal *-category (see Definition A.8).

What we have shown so far is that D must be a full sub-tensor category of the tensor category eAc of local A-modules in C. Moreover, there is a natural braiding in CAC [5,53], defined by descending the braiding cM, N : M ® N ^ N ® M to a braiding cM N : M ®A N ^ N ®A M via the universal property of ®A:

M ® N^^N ® M

M ®AN N ®AM

On the other hand, above diagram is still commutative if we replace cM N in above diagram by

M N for the exact the same reason as those discussed above the diagram (15). By the universal property of ®A, such cM N is unique. Therefore, we must have cM N = cM N. We can express cAM N more explicit as follows:

cM,N = cM,N = Pn,m ◦ cm,n ◦ eM,N. (23)

Above bootstrap results can be summarized as follows.

Lemmabs 2.14. D is a full braided monoidal subcategory of Cl0c.

Remark 2.15. When C is unitary, Eq. (23) implies that (cM N)* = (cM N)-1. In other words, Gl0_c is a braided monoidal *-category (see Definition A.8), and D is a braided monoidal ★-subcategory of CAc.

The category CAc is also rigid (see Definition A.3). The duality maps can be naturally defined. For example, if M e Cl^c, then the right dual Mv in C is automatically a local A-module. Moreover, the birth (or coevaluation) map bM : A u M ® A Mv is given by

A -A A ® M ® Mv -M-U M ® Mv --M M ®a M and the death (or evaluation) map dA : Mv ®AM u A is given by

Mv ®a M (eMV-M)l\ mv® M ® A Mv® M ® A ® A Mv® M ® A M-U A

lbmk a ^nt ^ .w Mm 1 ,, „ ijv pm,m ,, _ ^

where Aa = eA and eMv M is defined in (2) and it splits pMv M. The duality maps in D must coincide with the duality maps in CAc because ®d = ®A. The quantum dimensions in D can be easily obtained from those in C as follows:

dim® M = dimg M/dime A,

and dim(CAc) = dig) [53,34].

Although we have not completed our bootstrap analysis, as we will show later from our bootstrap study of domain wall, there is no additional relation between C and D that can tell us which objects in CAc shall be excluded in D except the condition that D is modular. In general, if a local A-module is excluded from D, there must be a principle, such as a symmetry constraint, to tell us why such exclusion happens. Since there is no such symmetry constraint in sight except the requirement of the modularity of D, we conclude that D must be a maximum modular tensor subcategory in cAc.

On the other hand, we recall an important mathematical theorem proved in [5, Theorem 4.2] (in unitary setting) and [53, Theorem 4.5] (see also [34, Proposition 3.21]).

Theorem 2.16. If A is a condensable algebra in a modular tensor category C, then the category Cl0c of local A-modules in C is also modular.

Remark 2.17. If C is a unitary modular tensor category, since 0® = 0M, we have (0®)* = 0*M = 0—1 = (0MD)-1. In other words, CAc is a ribbon *-category, hence, a unitary modular tensor category [5, Theorem 4.2].

In particular, let I, y be two simple objects in CAc, the i-matrix in CAc is given by [5], [53, Eq. (4.3)] (see also [34, Eq. (3.56)]):

»iY = ^T^ (24)

dim A'

Therefore, we draw our conclusion:

Theorembs 2.18. If a system of anyons, described by a (unitary) modular tensor category D, is obtained via a condensation from another system of anyons given by a (unitary) modular tensor category C, then there is a condensable algebra A in C such that D ~ Cl°c as (unitary) modular tensor categories and A is the vacuum in D.

We have thus completed our bootstrap analysis on the condensed phase D.

3. Bootstrap analysis II: domain wall

If the domain wall between the C-phase and the D-phase is gapped, it gives a 1-dimensional topological phase. The wall excitations can fuse but not braid with each other. As a result, they form a unitary tensor category E. Moreover, we require that a pair of particle and its antipar-ticle can be annihilated or created from the vacuum, and the vacuum degeneracy is trivial,6 i.e. homE (1E, 1 e) ~C. Therefore, E must be a unitary fusion category, which has a unique spherical structure [49,28]. For discussion in non-unitary cases, we assume sphericalness. Unitarity will be discussed in remarks. Our bootstrap analysis for the domain wall is entirely similar to that for the condensed phase D. So we will be brief here.

All objects in E should come from objects in C. These objects partially survive the condensation but are confined so that they can only live on the domain wall. It is reasonable to view E as a condensed phase except that the condensed particles can only survive on a 1-d wall. Once we take this point of view, many basic building blocks in E can be analyzed similar to those in D.

1. E is a subcategory of C. If X e E, we must have the identity condensation map idX : X ^ X. The vacuum 1E in E can be viewed as an object B in C. From the bootstrap point of view, it seems unnatural to take B = A as a priori. We would prefer to start our bootstrap study with minimum assumptions. We will see later that bootstrap study will tell us some relation between A and B.

2. We must have an embedding: homg (M, N) homg (M, N).

3. The vacuum 1g should condense into B. Namely, there is a morphism iB : 1g ^ B in C. On the other side, the vacuum 1d = A in D-phase should also fuse into the vacuum on the wall when we move the vacuum A close to the wall. Therefore, we have an morphism iBA : A ^ B

in C. A

4. For any X, Y e E, There should be a condensation map: pi Y : X ® Y ^ X ®E Y and a canonical embedding efY : X ®E Y ^ X ® Y such that

PX, Y ° eX , Y = idX®gY-

Since C is semisimple, we can have a decomposition: X ® Y = X ®E Y © U for some U e C.

UIE UIE

In other words, we have eX Y : U

^ X ® Y and y : X ® Y ^ U such that

U |E U |E E U |E n U |E E n

rx,Y ° eX,Y = idU, Px,y ° eX,Y = 0 , rx 'Y ° eX,Y = 0 , (25)

E E , U |E U |E

eX,Y ° Px,y + eX'Y ° rX'Y = idX®Y- (26)

In particular, we define ßB := p| B and eB := e| B.

6 If this condition is not satisfied, the associated 1-d topological phase is not stable [68].

Remark 3.1. If C is unitary, we can choose ef Y = (pf Y)*.

These are the basic data associated to the quasiparticles on the wall. We will explore their properties below.

1. Associativity of ¡i.B. Consider the process of condensing three B-anyons, this process is independent of which pair of B condenses first. This leads to a the same commutative diagram as (5) but with A replaced by B.

2. Unit properties. This is also similar to the proof of the unit properties of A. This leads to the same diagrams as (6) but with all A replaced by B.

3. Stability of the vacuum B in B ® B under the B -action. By the same argument of the stability of the vacuum A, we obtain the stability of the vacuum B which implies that both maps in (8) and (9) with A replaced by B are zero maps. As a consequence, eB is a B-B-bimodule map. Therefore, B is separable, and the category CB of B-modules and the category CB|B of B -B -bimodules are both semisimple.

4. Connectivity of B. A disconnected separable algebra decomposes into direct sum of connected separable algebras. If B is disconnected, the category CB|B is a multifusion category. As we will show later that E can be embedded into CB|B fully faithfully and E is a fusion category with a simple unit B. Therefore, B must be connected, i.e. dimhomg (1g, B) — 1.

Above bootstrap results can be summarized as follows.

Lemmabs 3.2. The vacuum 1g on the wall can be viewed as a connected separable algebra B in C.

Similarly to the arguments above Remark 2.8, in which the commutativity is not used, B has a unique simple normalized-special symmetric Frobenius algebra structure with the comultipli-cation AB := eB : B ^ B ® B.

Now we would like to explore the properties of /X := pf X : B ® X ^ X = B ®f X and nX := pf B : X ® B ^ X = X ®f B for all X e E.

1. The pair (X, /X) gives a left B-module: the proof is entirely similar to that of left A-module.

2. Similarly, X equipped with a right B-action /X : X ® B ^ X is a right B-module.

3. The triple (X, /X , /X) is a B-B-bimodule. This follows from the following commutative diagram:

(B ® X) ® B -

(B ®E X) ® B (B ®E X) ®E B -

->-B ® (X ® B)

B ® (X ®E B)

f¿x®£b

B ®E (X ®E B) ,

the physical meaning of which is obvious. 4. Stability of X in B ® X and X ® B under the B-action. Notice first that, by the associativity, it is automatically true that the map /X is a left B-module map and /X a right B -module

map for all X e E. Similar to the previously discussed stabilities, we obtain that ef X is a left B-module map and ef B a right B-module map.

5. Compatibility among eB, ef X and ef B. Similar to the compatibility among eA, eAiM and eM, A, we can show that the same diagrams (16) but with all A replaced by B and all M by X are commutative due to the same physical requirements. As a consequence, using the Frobenius properties of AB, we obtain the following identities:

6. Morphisms in E are B-bimodule maps: A morphism f : X ^ Y in E should be stable under the screening of the vacuum B from both sides. In other words, we should have the following two commutative diagrams:

B ® X-

B ®E X = X-

B ® Y

+ Y = B(

X ® B-

Y ® B

X®E B = X-

■ Y = Y® E B

Since ef X, /¿L are left B-module maps and ef B, ¡xR are right B-module maps, we obtain that f is a B -bimodule map if and only if diagrams in (28) and (29) are commutative. Namely, we have homE(X, Y) = homB|B(X, Y), where homB|B(X, Y) denotes the set of B-B -bimodule maps from X to Y.

Similar to the D-phase case, using (27), we obtain a screening map: ScB : home (X, Y) ^ homB|B(X, Y) defined by, for g e home(X, Y),

Using the normalized-specialness of the Frobenius algebra B, it is easy to see that ScB is a projector, i.e. ScB o ScB — ScB. 7. — ®B: the category CB|B of B-B-bimodules is a tensor category with tensor product ®B. We have B ®B X — X — B ®E X. Moreover, the condensation cannot distinguish

the following condensations: (X ® B) ®E Y and X ® (B ®E Y). By the same argument for ®D = ®A, we obtain that ®E = ®B.

What we have shown is that E can be fully-faithfully embedded in CB|B as a spherical fusion subcategory. As the tensor unit of E, the algebra B must be a simple B-B -bimodule. Since an separable algebra decomposes into connected subalgebras, therefore we must have dimhome(1e, B) = 1. Again, by the same argument above Theorem 2.16, we conclude that E should be a maximum spherical fusion subcategory in CBb. Using [35, Lemma 4.1], one can show that CB|B is actually spherical. Therefore, we must have:

Theorembs 3.3. E ~ CB|B as spherical fusion categories.

Remark 3.4. If C is unitary, we choose efY = (pf Y)* and eB = i*B. Then B is automatically a normalized-special symmetric Frobenius algebra. Similar to algebra A, we can show that ScB commutes with *. We obtain that CB|B is a *-category. It is a routine to check that CB|B is a unitary fusion category, which has a unique spherical structure [49,28].

We have completed the analysis on the internal properties of wall excitations. 4. Bootstrap analysis III: final results

In this section, we will complete our bootstrap analysis. Only thing that remains to be studied is the interrelation between wall excitations and bulk excitations from two sides. The final conclusion of our bootstrap analysis is summarized in Theorembs 4.2.

Let us continue our bootstrap analysis.

1. The bulk-to-wall map L : C ^ E. It is quite clear from the physical intuition that as an anyon in C move close to the wall from left, it is facing a vacuum B-cloud on the wall. Therefore, this bulk-to-wall map is given by the functor: - ® B : C ^ CB|B, i.e. C ^ C ® B for all C e C, the left B-module structure on C ® B is defined by

B ® C ® B^-^C ® B ® B —>C ® B (31)

where we use a braiding convention: what lives on the left side of the wall, e.g. an anyon C in the C-bulk, should stay on the top during the braiding. This convention is systemically used below. The right B -module structure is the obvious one. The bulk-to-wall map is also called a±-induction in mathematical literature [3,4].

2. L is monoidal (see Definition A.1). This condition is a physical requirement that was illustrated schematically in the two diagrams in Eq. (3.2) in [36]. Since L = - ® B, one can show that it is automatically monoidal. More explicitly, we have an isomorphism

L(U) ®£ L(V) = (U ® B) ®b(V ® B) -> (U ® V) ® B = L(U ® V)

for all U, V e C. It is also easy to see that the identity 1e ® B = B is just the preserving-the-unit condition of a monoidal functor. These isomorphisms automatically satisfy all coherence conditions in the definition of a monoidal functor.

3. L is central (see Definition A.29). This condition is a physical requirement which was illustrated schematically in the two diagrams in Eq. (3.4) in [36]. We briefly recall the argument

below. When an anyon U in C-phase move to the E-wall closely enough, it can be viewed as a particle on the wall. It is easy to see that it can braid with a wall excitation in a unique way. Namely, it can exchange positions with a wall excitation X as long as the path of U is in C-bulk and the path of X is restricted on the wall. This braiding is only a half-braiding. It implies that L is a central functor (see Definition A.29). Namely, it can be obtained by

forget .-.-„

a composition C u Z(E)-> E of functors. This physical requirement is automatically

satisfied by L = - ® B. Because there is a natural half braiding given by

-i c,x

X ®B (C ® B) ~ X ® C —U C ® X ~ (C ® B)

for C e C, X e E, satisfying all the coherence conditions of a central functor. Notice that the braiding convention in (32) is chosen according to the convention in (31).

4. L is dominant (see Definition A.30). This means that all wall excitations should contain in the image of L as subobjects. Mathematically, it is automatic because ef B : X u X ® B is an embedding of B-bimodule. In other words, X ® B contains X as subobjects for all X e Cb\B .

5. One data that has not been studied so far is the morphism iBA : A u B. Our physical intuition immediately suggests that the following diagrams:

is commutative. It is also natural that iBA o iA = iB. Therefore, iBA : A u B is an algebra homomorphism. Since A is simple, it implies that iBA is an embedding. As a consequence, the wall excitations CB\B can be embedded into the category CA\A.

6. B is an algebra over A (see Definition A.23). Consider a D-vacuum A and an E-vacuum B, then let A fuse into the wall, then fuse with B from either left or right side. Physically, two possible paths give the same fusion process A ® B u B. This leads to the following conditions:

A ® B—^-B ® B-^^B

B ® A-

B ® B

Such algebra B is called an algebra over A. It is equivalent to say that B is an algebra in the category CA of right A-modules. 7. The bulk-to-wall map R : D u E. The anyon in D moving close to the wall from right is facing a cloud of vacuum B in E. Therefore, this bulk-to-wall map must be given by B ®A - : CloAc u CB\B. Note that there is a natural B-bimodule structure on B ®A M, for

M e CloAc. The left B-module structure is obvious. The right B-module structure is defined by: A

eb,m 1

B®M®B

B ®AM ® B

B ® M B ®A M

U B ® B ® M

£ = C^is-wall

D = e^-phase

a bulk excitation

IXI a wall excitation

C-phase

a bulk excitation

Fig. 3. The final result of bootstrap analysis: the vacuum 1d in d-phase is given by a condensable algebra A in c; the vacuum 1g is given by a connected separable algebra B in c; there is an algebraic homomorphism A : A ^ B. Moreover, c ~ c'ac as modular tensor categories and e ~ cb|b as spherical fusion categories. The two bulk-to-wall maps are given in the picture, where C e c, X e e and M e d.

where the braiding convention is chosen according to (31). Using the commutative diagram (34) and the fact that A is a normalized-special symmetric Frobenius algebra, it is easy to check that this defines a B -bimodule.

8. R is monoidal. It is similar to L [36]. Mathematically, the bulk-to-wall map: B ®A - : D ^ E = CB|B is automatically monoidal.

9. R is central. It is similar to L [36]. Mathematically, the functor B - is automatically central because there is a naturally defined isomorphism:

«£ i X ®B (B ®A M) Z,Bt9-M> X ® (B ®A M) X ® B ® M

Cx,b®m n ^ ^ „ pb®am,x°(pb,m 1)

^ B ® M ® X--> (B ®A M) ®B X.

satisfying all the coherence conditions. It is relatively easier to see this result if we view B as an algebra in CA (see for example Section 3.4 in [23]). 10. R is rarely dominant. When B = A, R = A ®A - is an embedding. When CAc = Vect, R is the trivial embedding Vect ^ E.

Remark 4.1. When C is unitary, both functors - ® B and B ®A - preserve adjoints.

We have thus completed our bootstrap analysis. We summarize our bootstrap results in the following theorembs (see also Fig. 3):

Theorembs 4.2. If a system of anyons, described by a (unitary) modular tensor category D, is obtained via condensation from another system of anyons given by a (unitary) modular tensor category C, together with a gapped domain wall with wall excitations given by a (unitary) spherical fusion category E, then we must have

1. The vacuum in D is a condensable algebra A in C and D ~ CAc as (unitary) modular tensor categories;

2. The vacuum in E is a connected separable algebra B in C and E ~ CB|B as (unitary) spherical fusion categories;

3. There is an algebraic homomorphism iA : A ^ B in C such that B is an algebra over A;

4. The bulk-to-wall map from C-side is given by the monoidal functor

-®B : C ^ CB|B, defined by C ^ C ® B, VC e C; (36)

5. The bulk-to-wall map from D-side is given by the monoidal functor

B ®a -:Cloc ^ Cb|b, definedby M ^ B ®aM, VM e CloAc. (37)

In the rest of this section, we will consider a simplified situation. We would like to motivate this situation by first discussing two simple mathematical results.

Lemma 4.3. The canonical embedding CB|B CA|A induced from the algebra homomorphism iBA : A ^ B can be factorized through CA, i.e.

CBBC-CA|A

where the embedding F^ is chosen according to the convention (39).

Proof. Since A is commutative, a right A-module is automatically a left A-module by applying braiding. There are two different ways to do this. But for a wall excitation viewed as right A-module, only one braiding choice is physically meaningful because A can move out of wall only to the D-side of the wall. So the left A-module structure on a wall excitation is determined by its right A-module structure via a braiding according to moving out of the wall to the D side then acting from left, and vise versa. Let CA be the category of right A-modules. We choose a convention that the left A-module structure /X on X is defined by its right module structure /X as follows:

/X = /X ◦ cx,A. (39)

Notice that this choice of braiding is compatible with the convention fixed in (31). Then CA is also a tensor category with tensor product ®A. Therefore, we obtain a natural embedding F$ : CBb ^ CA such that diagram (38) is commutative. □

Lemma 4.4. CA is actually a (unitary) spherical fusion category. Moreover, B is a connected separable algebra in CA.

Proof. The first statement follows from the proof of [35, Lemma 4.1]. The unitarity can easily checked by the results in Remarks 2.13, 2.15. The second statement is easy to check. □

What do above two lemmas tell us? Consider a system of quasi-particles confined to a 1-d domain wall. It can be described by a spherical fusion category E'. If we run a bootstrap analysis on a possible condensation for this 1-d system of quasi-particles, we will find that the vacuum of the condensed 1-d phase is given by a connected separable algebra in E'. Therefore, our physical

interpretation of Lemmas 4.3 and 4.4 is that a CB|B-phase on the wall can be obtained by first condensing A in C and obtaining a gapped CA-wall, then condensing B in the 1-d phase CA. This becomes rather clear if we consider the case A = 1g and B = 1g. In this case, the condensation happens only on a line with the excitations given by the spherical fusion category CB|B. Both sides of the line remain intact.

Remark 4.5. More general 1-d condensation can happen. For example, if an arbitrary C-phase and an arbitrary D-phase is bounded by an 1-d domain wall with excitations given by a spherical fusion category E. It is possible to have a condensation happens in a line segment on the wall. If we run a similar bootstrap analysis on this 1-d system, we can show that the 1-d condensed phase is given by EB'\B', where B' is a connected separable algebra in E, and what lies between the E-phase and the condensed EB|B -phase is a 0-d defect given by objects in the category EB' of (left or right) B-modules in E. We will come back to this point in Section 8.

Our bootstrap analysis cannot rule out the possibility that there are more than one condensations between the initial data and the final data. To simplify the situation, we introduce the following notion:

Definition 4.6. An anyon condensation is called an one-step condensation if it is determined by a single condensable algebra in the original phase.

In a one-step condensation, all anyons in the condensed phase can automatically cumulate on the wall. In particular, the vacuum A of D is also the vacuum of the wall. In addition to these particles, those A-bimodules such that the left action and the right action are related by (39) can also live on the wall. Therefore, particles on the wall are given by the category CA of A-modules. In this case, anyons in D-phase can first move into the wall according to the natural embedding R : Cloc ^ CA, then move out of the wall to the D-side freely. Mathematically, it is just the fact that Ry o R = id®, where Ry is the right adjoint of R. In this case, our bootstrap analysis gives the following conclusion.

Theorembs 4.7. If a system of anyons, described by a (unitary) modular tensor category D, is obtained via a one-step condensation from another system of anyons given by a (unitary) modular tensor category C, together with a gapped domain wall with wall excitations given by a (unitary) spherical fusion category E, then we must have:

1. The vacuum in D is a condensable algebra A in C and D ~ CAc as modular tensor categories;

2. E ~ CA as (unitary) spherical fusion categories;

3. Anyons in the C-phase can move onto the wall according to the monoidal functor

-®A : C ^ CA, defined by C ^ C ® A, VC e C; (40)

4. Anyons in the D-phase can move onto the wall according to the embedding CloA ^ CA, then move out to the D-side freely.

By the so-called folding trick, E is a boundary theory for a doubled system C D, where D is the same monoidal category D but with the braiding given by the antibraiding in D. Let Z(E) be the monoidal center of E (see Definition A.26 and Remark A.27). By Muger's theorem [60] (see

also Theorem A.28), Z(E) is a modular tensor category. The so-called boundary-bulk duality says that C K D ~ Z(E) as modular tensor categories. This result was first obtained by Kiteav [50], and a proof in the framework of Levin-Wen types of lattice models was given in [51], and a model-independent proof was given in [36, Section 3]. Therefore, our bootstrap results must pass the test of boundary-bulk duality. It is guaranteed by the following important mathematical result [6, Corollary 4.8], [23, Corollary 3.30].

Theorem 4.8. Let A be a condensable algebra in a modular tensor category C, then we have C K CA ~ Z(CA).

Moreover, since B is a connected separable algebra in CA, it is known that CB|B, which can be naturally identified with (CA)B|B, is Morita equivalent to CA [64] (see Definition A.31) and we have Z(CB|B) ~ Z(CA) as modular tensor categories [59,29]. Therefore, the results given in

Theorem 4.2 also pass this test, i.e. C K C'^ ~ Z(Cb|b). 5. Determine the condensation from physical data

In the previous sections, we have seen that a one-step condensation is described by a single condensable algebra A in the initial phase C. The condensed phase is given by the modular tensor category CAoc and the domain wall is given by the spherical fusion category CA. We even know how anyons in both bulks are fused into the domain wall (see Theorem 4.2). This is a quite satisfying picture.

However, many important questions still remains. For example, if we only have the abstract data of the initial phase C, the condensed phase D and E-domain wall, how can we determine the condensable algebra A and the algebra B ? Are such algebras A and B unique? If not, is that possible to add more physically detectable information so that we can determine A and B uniquely? We would like to answer these questions in this section.

5.1. Gapped boundaries

We would like to first consider a special case in which D = Vect (or Hilb if we assume unitarity). Namely, we have a gapped boundary given by E.

Let us first look at a simple example: the toric code model. In this case, the bulk excitations are given by the modular tensor category Z(RepZ2), which is the monoidal center of the unitary fusion category RepZ2 of representations of the Z2-group. It contains four simple anyons 1, e, m, e. There are two different types of boundary: the smooth boundary and the rough boundary [8,51]. The boundary excitations in both cases are given by the same unitary fusion category RepZ2. The difference between these two types of boundary lies in how bulk anyons condense when they approach the boundary. In the smooth boundary case, m -particles are condensed but e-particles are confined on the boundary; in the rough boundary case, m-particles are confined on the boundary but e particles are condensed. As we will show in Section 6.1, in the smooth boundary case, the associated condensable algebra is 1 © m; in the rough boundary case, the associated condensable algebra is 1 © e. In other words, the condensation can be uniquely fixed by specifying the complete information of the bulk-to-boundary map, which is a monoidal functor F : Z(RepZ2) ^ RepZ2. If bulk-to-boundary map is not given as initial data, then the possible condensation in general is not unique. This phenomenon carries on to the most general cases.

In a general case, we have a bulk phase given by a modular tensor category C with a gapped boundary phase given by E. Even if the bulk-to-boundary map is not given, we must have a braided monoidal equivalence C — Z(E) according to [36]. Therefore, there exists a monoidal

forget

functor given by L : C — Z(E)-> E. Let Lv be the right adjoint of L (see Definition A.2).

By [29,23], Lv(1E) has a natural structure of a condensable algebra in C. Moreover, it is a Lagrangian algebra (see DefinitionA.18). By TheoremA.19, we have Z(Cl[V^ )) — Vect. Moreover, since Lv(X) is naturally a right Lv(1E)-module, we obtain a functor Lv : E — CLv(1g), which was proved to be a monoidal equivalence [29]. Therefore, we can certainly realize an E -boundary via a one-step condensation but not necessarily in a unique way.

In general, for given C and E, it is possible to have more than one bulk-to-boundary maps. Sometimes, this phenomenon can be explained by the existence of non-trivial braided automorphisms of C. Indeed, if j : C — C is a braided equivalence, then L o j : C — E gives a potentially different monoidal functor. Then j-1(Lv(1E)) is also a Lagrangian algebra. The condensation of jj(Lv(1 E)) gives exactly the same boundary excitations, i.e.

Cj(Lv (1£)) — CLv(1g) — E.

In the case of toric code model, the bulk excitation Z(RepZ2) has a Z2 automorphism group. The non-trivial automorphism in this Z2 group is called electric-magnetic duality (see for example [2]) which exchanges an e-particle with an m-particle. Therefore, any one of two bulk-to-boundary maps in the toric code model (discussed before) can be obtained from the other by applying the electric-magnetic duality.

The bulk-to-boundary map L : C — E is a physically detectable data [36]. Once it is given, then Lv(1E) gives a Lagrangian algebra in C. The gapped boundary E can be obtained from C by a one-step condensation of the algebra Lv(1E). Indeed, we have the following commutative diagrams:

f°rget C C -®Lv(1£) C

CLv(1g) \- CLv(1g)

where the second diagram is obtained by taking left adjoints from in the first diagram. The second commutative diagram in (41) simply says that not only the boundary excitations E coincide with CLv(1g), but also their associated bulk-to-boundary maps (recall (40)) coincide. This says in particular that Lv(1E) is unique up to isomorphisms. We summarize these results below.

Theorembs 5.1. Given a topological bulk phase C with a gapped boundary phase E, together with a given bulk-to-boundary map: a monoidal functor L : C — E, there is a unique Lagrangian algebra Lv(1E) realizing these topological data by a one-step condensation. More precisely, we have E — CLv(1g) and L coincides with the functor - ® Lv(1E) : C — CLv(1g). In other words, gapped boundaries of a C-bulk are one-to-one corresponding to Lagrangian algebras in C.

Example 5.2. Consider the Ising topological order with anyon 1, f, a and the fusion rules:

a ® a = 1 © f, a ® e = a, f ® f = 1. (42)

We use Ising to denote the corresponding unitary modular tensor category. By double folding the Ising topological phase along a line, we obtain a double layered system Ising Kl Ising with a

gapped boundary, boundary excitations on which are given by the unitary fusion category Ising. The bulk-to-wall map is given by the usual fusion product functor Ising Kl Ising Ij~®> Ising. We have

Lv(1ising) = (1 H 1) © (f H f) © (a H a), (43)

and (Ising H Ising)Lv(iIsing) ~ Ising as fusion categories. The algebraic structure on Lv(1Ising) is guaranteed by abstract nonsenses [29,55,23]. But an explicit construction in terms of chosen bases of hom spaces is available in literature (see for example [60, Proposition 4.1], [34, Lemma 6.19], [55, Proposition 2.25]).

Remark 5.3. In the Abelian Chern-Simons theory based on the modular tensor category C(G, q), where G is a finite abelian group and q a non-degenerate quadratic form, there is a one-to-one correspondence between Lagrangian algebras in the category C(G, q) and Lagrangian subgroups of G [36, Theorem 5.5]. So in this case, we recover the result in [44,56,1].

Remark 5.4. By the folding trick, a domain wall between a C-phase and a D-phase can be viewed as a boundary of a C K D-phase. Therefore, the 1-d phases on a gapped domain wall between a C-phase and a D-phase are classified by Lagrangian algebras in C K D. In the case C = D, by [23, Proposition 4.8], such domain walls are equivalently classified by indecomposable semisimple C-modules.

Remark 5.5. It is known in physics literature that a simple anyon in C can split into two particles on the boundary (see for example [10,14]). For example, in the case of an Ising K Ising-bulk with an Ising-boundary, the simple anyon a K a in the bulk is mapped to a ® a = 1 © f (see (42)) on the boundary. This phenomenon has no contradiction to the fact that E can be viewed as a subcategory of C (recall Remark 2.2). To compare C with E, we need first map one to the other. But there are many ways to do this. To view E as a subcategory of C, we use the forgetful functor in the first diagram in (41); to see that a simple anyon in C can split into two particles on the boundary, we use the bulk-to-boundary functor - ® Lv(1E) in the second diagram in (41). Note that these two functors are adjoints of each other. For example, in the topological order C = Ising K Ising, the condensation of Lv (1Ismg) in (43) produces the trivial phase D = Hilb and a gapped boundary Ising. The simple anyon a K a in C maps to the gapped boundary via the functor - ® Lv(1E) and becomes

(a K a)® A = ((a ® a) K 1) ® A = ((1 © f) K 1) ® A = A © {(f K 1) ® A),

which is decomposable on the boundary, or equivalently, via the functor L and becomes L(a K a) = a ® a = 1 © f.

If we allow a two-step condensation to realize the same data: C —> E ^ Vect. Then the uniqueness of A and B is not guaranteed. More precisely, as we will show later, A can still be fixed uniquely as Lv(1E). This follows as a special case of Eq. (50) when D = Vect. However, this data is not enough to fix B. As you can see from (41), any L : C ^ E is equivalent to the standard functor - ® Lv(1E) : C ^ CLv(1g). Except in the case B = A, B is noncommutative and cannot be Lv(1E) in general. Actually, Lv(1E) is only the left center C/(B) of B. We will define this notion now. Let Y be a B-bimodule. The following map:

defines an idempotent from Y to Y [32]. We define the left center of Y, denoted by Cl(Y), to be the image of this map. In particular, for Y = B, we have Cl(B) = Im PB. Choose a split iY : Cl(Y) ^ Y, rY : Y ^ Cl(Y) such that iY o rY = PY and rY o iY = idCl(Y). It is easy to show that the following two maps: for f e homB\B(C 0 B, Y) and g e homC(C, Cl(Y)),

f ^ rY o f o (idC 0 ib),

g ^ HY o (lY 0 idB) o (g 0 idB)

are well-defined and inverse to each other. Moreover, they define an natural isomorphism between the following two hom spaces:

homB|B(C 0 B, Y) — hom^C, C(Y)).

Then it is clear from Definition A.2 of the right adjoint functor that we have (- ® B)y = Cl (-). Since - ® B is dominant, by [23, Lemma 3.5], we have CBb ~ GCl(B) as fusion categories and the following diagram:

-®Ci(B)

CCi(B)

Cb|b is commutative.

Therefore the bulk-to-boundary map L : C ^ E only determine the left center of B. Since there is no additional physically detectable data available to us. We can conclude that the algebra B is not entirely physical. Only its left center, which is nothing but A in this case, is physically detectable. Actually, in this case, the left center Cl(B) coincides with the so-called full center of B [31,54]. It is defined by Ci(Fv(B)), where F : Z(Cb\b) ^ Cb\b is the forgetful functor, as an object in Z(CB\B) ~ C Kl Vect = C. The notion of full center is equivalent to the Morita equivalent class of B (see Definition A.17) [54, Theorem 3.24]. Therefore, only the Morita class of B is physical. Indeed, by definition, two separable algebras B1 and B2 are Morita equivalent if CBl ~ CB2, which further implies CBl\Bl ~ CB2\B2. We won't be able to distinguish them by macroscopic physics.

5.2. One-step condensations

We would like to answer the questions raised at the beginning of this section for the data C —> E ^ D for general D. If the condensation is one-step, by our bootstrap analysis, we have E ~ CA Cl0c — D for some algebra A in C, and the functor L : C ^ CA — E is given by

forget

- ® A. Notice that the right adjoint of - ® A is nothing but the forgetful functor CA-> C.

This means that A can be recovered from (— 0 A)v (1g).

Therefore, if we start from an abstract data C E ^ D, one can immediately recover A as Lv(1E). By [23, Lemma 3.5], Lv(1E) has a canonical structure of a condensable algebra. In this case, by [23, Lemma 3.5], we obtain that E — CLv(1g) as monoidal categories. Moreover, we have the following commutative diagram (recall (41)):

-®LV(1g )

CLV(1g)

Therefore, the bulk-to-wall map L can be identified with the standard bulk-to-wall map - ® LV(1g). The category D is just C^OV(1g). Since we have already assume that the condensation is one-step, i.e. B = LV(1g), the information E ^ D is redundant. Actually, we will show in the next subsection that LV(1g) can also be recovered in a different way by using the data

E - CLV(1g).

rp) _ ploc

D = eLV (1g)

5.3. General condensations

In general, given the abstract data

where L is central and dominant and R is central. To determine A is more complicated than the boundary case. We will do that below.

The data (47) gives a functor L K R : C K D — E which is also central [36]. Then we obtain the following commutative diagram:

C H D G

where G is the canonical braided monoidal equivalence induced from the central structure of L K R. Our strategy is to identify A as an subalgebra of (L K R)v(1E) by using the one-to-one correspondence between the condensable subalgebras of (L K R)v(1E) and the indecomposable fusion subcategories of E [23, Theorem 4.10]. Let R(D) be the image of D in E under the functor R. Consider the relative center ZR(D)(E) (recall Definition A.26), the forgetful functor

forget

E can be factorized as follows:

Z(E) -

■zr(d)(e)

where FR(D) is the forgetful functor which is monoidal. Let FR^) be the right adjoint functor of Fr(D) and 1zr(d)(E) be the tensor unit in Zr(D)(E). By [23, Lemma 3.5], F^(D)(1zr(d)(£})

is a condensable algebra in Z(E). C can be embedded in C K D via C — C K 1d for all C e C. By [23, Theorem 4.10, Remark 4.12], there is a unique C-algebra A such that

A K 1D — G-1(Frr(B)(1zr(d)(£))). (50)

This defines A uniquely. Note that the algebra A K 1d is a subalgebra of the Lagrangian algebra (L K R)v(1E). It was proved in [23, Theorem 4.10, Remark 4.12] that D — Cl°c. The condensation of A produces a domain wall with excitations given by CA. In the case D = Vect, it is easy to see that ZR(D)(E) = E and G-1 o F^d) = Lv. Therefore, we have

A = Lv(1e) in this case. In a general one-step condensation, we have E = Ca and D = Cl°c, and A K 1Cioc — G-1(F\oc(1Z loc(eA))) is simply a fact that was proved in [23].

a ca ca

Similar to the boundary case, we cannot fix the algebra B from the data (47). By the discussion in the boundary case, we have shown that Lv(1E) = Cl(B). By [23, Lemma 3.5], we obtain that L coincides with the functor - 0 Lv(1E) : C ^ CLv(1e). Hence, the left center Cl(B) of B encodes the entire information of the functor L. Using both L and R, we obtain the full center of B

Z(B) := Cl((L K R)v(B)) = (L K R)v(B) e C K D,

where the equality is due to the fact that (L K R)v(B) is commutative. This is the best we can get because we have used all physically detectable data. Notice that the left center of B is uniquely determined by the full center of B via the canonical functor f : C — C K Vect ^ C K D because Lv = fv o (L K R)v. Therefore, it is equivalent to say that B is physically determined only up to its full center. By [54, Theorem 3.24], the full centers of such algebras are one-to-one corresponding to the Morita equivalence classes of such algebras. In other words, only the Morita equivalence class of B is physical.

Remark 5.6. In physics literature (see for example [10,14]), a simple anyon in the un-condensed phase C can also be decomposable in the condensed phase D = ClAc. It is a phenomenon similar to the one discussed in Remark 5.5. To compare these two phases, we need find functors between C and D. There is a forgetful functor F : Cl^c ^ C, which can be realized as the composition of the following two functors:

F : Co -> Ca —-^ C.

Its adjoint Fv : C ^ ClAc is given by Fv = Rv o L. In other words, both functor F and Fv are the wall-tunneling maps between the C-phase and the D-phase. Note that even though there is no wall in the original setup, the condensation choose the wall automatically. For a simple anyon i in C, Fv(i) is not simple in D in general; for a simple anyon M in C, F(M) is not simple in C in general. When physicists discuss the phenomenon of the splitting (in D-phase) of a simple anyon in C-phase, they applied the functor Fv implicitly. For example, in the topological order Ising K Ising, take the condensable algebra A = (1 K1) © (f K f), which is a subalgebra of (43), then

L : a K a ^ (a K a) 0 A = (a K a) © (a K a).

Note that a K a is a simple local A-module, thus survives in D = Cl^c. Then Fv(a K a) = (a K a) © (a K a) is not simple. Note that the splitting of a K a in D-phase was studied in [14], and the two summands in Fv(a Ka) were denoted by (1, 1 )0 and (2, |)i, respectively, in [14].

6. Examples

In this section, we give some examples of one-step condensations in non-chiral and chiral topological phases. Recently, the gapped boundaries and domain walls have been studied intensively from various perspectives (see for example [15,51,36,56,1,67,46,37]).

6.1. Toric code model

In the toric code model [48], the bulk excitations are given by the modular tensor category Z(RepZ2), which is the monoidal center of the fusion category RepZ2. It contains four simple anyons 1, e, m, e. There are two different types of boundary: a smooth boundary and a rough boundary [8]. They are characterized by two RepZ2-modules [51]. By [51], the smooth boundary, viewed as a Levin-Wen type of lattice model, is defined by a boundary lattice associated to the RepZ2-module RepZ2 (see Definition A.24); for the rough boundary, the boundary lattice is defined by the RepZ2 -module Vect. The boundary excitations in these two cases are given, respectively, by FunRepZ2 (RepZ2, RepZ2) and FunRepZ2 (Vect, Vect), which are equivalent as fusion categories, i.e.

FunRepZ2 (RepZ2, RepZ2) — RepZ2 — FunRepZ2 (Vect, Vect).

The difference of these two boundaries can be detected by how bulk anyons approach the boundary.

In the case of smooth boundary, it was shown in [8] that when an m-anyon moves from the bulk to the boundary it simply disappeared. In this case, the condensation algebra A1, or the categorified ground wave function of the condensed phase, is given by

A1 := 1 © m.

The boundary is given by Z(RepZ2 )A1 which is monoidally equivalent to RepZ2. According to Theorem 4.7, the bulk-to-boundary map is given by the monoidal functor - ® Ai: Z(RepZ2) — Z(RepZ2 )A1. Indeed, under this functor, we have

1 — 1 ® (1 © m) = 1 © m, m — m ® (1 © m) = 1 © m, e — e ® (1 © m) = e © e, e — e ® (1 © m) = e © e.

Clearly, m is mapped to the vacuum of the boundary theory. Even though the object e © e is not simple in RepZ2, it is the only simple right A1-module other than A1. Notice that e © e as an A1-module is not local. So 1 © m is the only simple object in Z(RepZ2A, i.e. Z(RepZ2)Ac — Vect. 1 © m and e © e are the simple excitations on the smooth boundary. Their fusion products:

(1 © m) ®a1 (e © e) = (e © e) ®A1 (1 © m) = (e © e), (e © e) ®A1 (e © e) = 1 © m

coincide with those in RepZ2. Moreover, Z(RepZ2)A1 — RepZ2 as fusion categories.

The case of rough boundary is entirely similar. In this case, the condensation algebra A is given by A2 := 1 © e, which is Lagrangian, i.e. Z(RepZ2 Y^ = Vect. The boundary excitations are given by the fusion category Z(RepZ2)A2 which is monoidally equivalent to RepZ2. The bulk-to-boundary map is given by the monoidal functor -®A2 : Z(RepZ2) — Z(RepZ2)A2, in which

e — e ® (1 © e) = 1 © e, m — m ® (1 © e) = m © e.

The boundary excitations and bulk-to-boundary maps associated to these two different types of boundaries are related by interchanging e with m.

a 6-wall JVC 6 M

: C-lattice

Fig. 4. A bulk-to-boundary map in Levin-Wen type of lattice models: the bulk is defined by a c-lattice and the boundary by an m-lattice. A bulk anyon, given by a c-c-bimodule functor f: c — c, can be viewed as an excitation on a trivial c-domain wall. As it is moving close to the m-boundary, the trivial c-wall fuse with the m-boundary and becomes a c kg m-boundary. Then it is intuitively clear that this bulk-to-boundary map Lm is given by (51).

6.2. Levin-Wen types of lattice models

The toric code model with boundaries is just one of a large family of Levin-Wen type of lattice models constructed in [58,51]. In these models, a bulk lattice is defined by a spherical fusion category C and the boundary lattice is defined by a C-module M (or gM if we want to make the C-action explicit) (see Definition A.24). In this case, the boundary excitations are given by the category CM := Fune(M, M) of C-module functors from M to M (see Definition A.25). The vacuum on the boundary is just the identity functor idM : M — M. The bulk excitations are given by the monoidal center Z(C) of C. The category Z(C) is defined as the category Fung|g(C, C) and was proved to be a modular tensor category [60]. In such a model, the bulk-to-boundary map LM : Z(C) — CM is completely determined by moving a bulk excitation closer to the boundary (see Fig. 4). Mathematically, it is given by a monoidal functor:

LM : (C C) — (M — C Ke M F^eidM) C Ke M — M). (51)

Given such a functor LM, we can determine an condensable algebra in C in the following way. Let LM be the right adjoint of the LM. For M = C and an object X e C, the object Lg(X) in Z(C), when it is viewed as an object in C by applying forgetful functor, is given by Lg(X) = 0; i ® X ® iv (together with a well-defined half braiding [19]), where the direct sum runs over all simple objects i in C. In particular, the condensable algebra is given by:

Lg(1e) = 0 i ® iv.

When C is modular tensor category, Z(C) — C K C, then we can also write:

Lg(1e) = 0 i K iv.

We want to remark that above algebra is also the famous charge-conjugate construction of modular invariant closed conformal field theory [32]. More generally, any condensable algebra A is Lagrangian if and only if it is modular invariant [55, Theorem 3.4]. This fact might suggest something deeper in physics.

C-lattice ■

For general M, A = LM(idM) is a condensable algebra in Z(C). Moreover, LM(idM) is also Lagrangian, i.e.

dim(LM(idm))2 = dim Z(C).

As a consequence, we have CAc — Vect [23]. Moreover, the functor Lv induces a monoidal equivalence: Lv : CM — CA defined by F ^ Lv(F) [29]. Namely, any quasiparticle F in CM can be realized in CA thus also in Z(C) as Lv(F). This implies the following commutative diagram:

forget

LM(idM)

which, by taking left adjoint, further implies the following commutative diagrams:

-®LM(idM)

■CLv(idM)

The above diagram simply says that not only the boundary excitations CM coincide with the boundary excitations CLv(idM) obtained from the condensation of Lv(idM), their associated bulk-to-boundary maps also coincide. Therefore, we conclude that the same topological phase determined by an gM-boundary lattice model can be obtained by condensing the algebra LM(idM) in a Z(C)-bulk. It turns out that there is a one-to-one correspondence between Lagrangian algebras in Z(C) and indecomposable semisimple C-modules [23, Proposition 4.8]. Therefore, there is a unique condensable algebra LM(idM) in Z(C) reproduce the gapped boundary phase defined by the gM-boundary lattice model.

When C is a modular tensor category, we can make the Lagrangian algebras LM(idM) more explicit in the following way. We can choose a simple object M in M. Then the internal hom A = [M, M], defined by the following natural isomorphism

homM (X 0 M,M) — homC (X, [M, M]),

have a natural structure of a connected separable C-algebra [62], which can also be endowed with a natural simple symmetric special Frobenius algebra structure. Moreover, M — CA as categories [62]. Then LM(idM) — Z(A) where Z(A) is the full center of A. If we denote irreducible A-modules in C as MX for X e J where J is a finite set, then we can express LM(idM) more explicitly as follows [54]:

LM(idM) = 0 Mx 0aM:

with a properly defined half braiding. According to Section 5.1, the choice of A is not entirely physical, but its full center Z(A) or its Morita class is physical. But as we pointed out in Introduction, this unphysical choice can be lifted microscopically to a physical one by lifting these lattice models to extended string-net models [2]. In these cases, the boundary lattice is defined by a C-module M together with a fiber functor M ^ Vect (see [2, Section 9]). This additional

data can select an algebra in C from its Morita class. However, such microscopic fine structures do not play a role in macroscopic physics.

Remark 6.1. These algebras LM(idM) — Z(A) and A, together with an algebraic homomor-phism Z(A) — A, form a so-called Cardy algebra [52], a notion which classifies open-closed rational conformal field theories (see also [32,31,55]). This connection between anyon condensation and rational CFT is not accidental and was clarified in [45] and in a mathematically rigorous way in [36, Section 6].

6.3. Kitaev quantum-double models

Kitaev quantum double models [48] cover a subset of non-chiral topological phases defined by Levin-Wen models. In this case, a complete classification of any condensations is known [20]. We will discuss this classification in this subsection.

Let G be a finite group with unit e. The bulk phase of a Kitaev quantum-double model is given by the unitary modular tensor category Z(RepG). If a gapped boundary theory is given by the unitary fusion category RepG, the bulk-to-boundary map is given by the forgetful functor F : Z(RepG) — RepG. Then the associated condensation is given by a Lagrangian algebra F v(C) where C is the trivial representation of G and the tensor unit of RepG. In this case, Fv(C) is given by the group algebra C[G].

It is known that the set of Lagrangian algebra in Z(RepG) is one-to-one corresponding to the set of indecomposable semisimple module category over RepG [23]. This set can be characterized by a pair (H, m), where H is a subgroup and m e H2(H, Cx) [63].

There are more condensable algebras in Z(RepG). They have been classified by Davydov in [20]. Because of its importance in the physical applications, we would like to spell out this classification explicitly.

An explicit description of the category Z(RepG) is given in [20, Proposition 3.1.1]. Its objects is a pair (X, pX), where X is a G-graded vector spaces, i.e. X = 0geG Xg, and pX : G x X — X is a compatible G-action, which means for f, g e G (fg)(v) = f(g(v)), e(v) = v for all v e X and f(Xg) = Xfgf-1. The tensor product of (X, pX) and (Y, pY) is just usual tensor product of G-graded vector spaces with the G-action pX®Y defined by g(x ® y) = g(x) ® g(y) for x e X, y e Y. The tensor unit is C which is viewed as a G-grade vector space supported only on the unit e and equipped with a trivial G-action. The braiding is given by

cx,y(x ® y) = f(y) ® x, x e Xf, y e Y, f e G. The dual object Xv = 0geG(X'v)g is given by

(Xv)g = (Xg-1 )v = hom(Xf-1, C)

with action g(l)(x) = l(g-1(x)) for l e hom(Xf-1, C), x e Xgf-1g-1. The twist is given by SX(x) = f-1 (x) for x e Xf. The quantum dimension dimX is just the usual vector space dimension.

By [20, Theorem 3.5.1], a condensable algebra A = A(H, F, y, e) is determined by a subgroup H c G, a normal subgroup F in H, a cocycle y e Z2(F, Cx) and e : H x F — Cx satisfying the following conditions:

egh(f ) = eg{hfh-1)eh(f), Vg,h e H, f e F,

y(f,g)eh(fg) = £h(f)£h(g)y(hff-1,hgh-1)e(f), Vh e H, f,g e F,

Y(f,g) = f(g)Y(fgf-1,f), Vf,g e F. (54)

This algebra A = A(H, F, y, e) as a vector space is spanned by agff, g e G, f e F, modulo the relations

agh,f = eh(f)ag,hfh-1, Vh e H, (55)

together with a G-grading ag,f e Agfg-1 and a G-action h(ag,f) = ahg,f. The multiplication is given by

ag,fag',f' = sg,g' y{f,f 0 ag,ff'.

By [20, Theorem 3.5.3], the algebra A(H, F, y, e) is Lagrangian if and only if F = H. In this case, e is uniquely determined by y in (54). Such algebra is determined by a pair (H, y ) (see also [63]).

Among all of these algebras, a special class is very simple. Let F be the trivial group. Both y and e are trivial. In this case, by (55), agh,1 = ag, 1. Therefore, the algebra is spanned by the coset G/H. Moreover, the G-action on A = A[H ], given by f(ag,i) = afg,1,Vf, g e G, is an algebraic automorphism, i.e. f(ab) = f(a)f(b) for a, b e A. This algebra A[H] is nothing but the function algebra on the coset G/H. In this case, the condensed phase Z(RepG)AcH] defined by A[H] is nothing but the unitary modular tensor category of Z(RepH) [20], which can be realized by a quantum double model associated to the group H. Therefore, this condensation can be viewed as a symmetric broken process from gauge group G to H (see [13,10] for the idea of Hopf symmetry broken).

6.4. Condensations in chiral topological phases

If a topological phase, given by a modular tensor category C, is chiral, it means that it cannot have a gapped boundary, or equivalently, C is not a monoidal center of any fusion category. Therefore, there is no Lagrangian algebra in C. But C can still have non-Lagrangian condensable algebras.

Many examples of condensable algebras in chiral topological orders are given by conformal embedding of rational vertex operator algebras (VOA) [57]. Let U and V be two rational vertex operator algebras. The rationality means, in particular, that the category RepU of U-modules and the category RepV of V-modules are modular tensor categories [40]. If U — V as a sub-VOA (preserve the Virasoro element), then V is a finite extension of U and can be viewed as an algebra in RepU. Moreover, V is a condensable algebra in RepU and (Rep^Vf — RepV (see also [42, Theorem 4.3, Remark 4.4]). In other words, a topological phase associated to the modular tensor category RepV can be obtained by condensing the condensable algebra V in the topological phase associated to the modular tensor category RepU. This V as an algebra in RepU is rarely Lagrangian.

For example, let Vg k denotes the VOA associated to affine Lie algebra g at level k. A few well-known conformal embedding are:

VTt2,4 ^ Vsl3,1, Vsl2,10 ^ VSp4,1, Vil2,6 0C Vsl2,6 V?09,1, Vsum,n 0C Vsun,m Vsumn,1, Vsom,n 0C Vson,m Vsomn,1.

Examples of conformal embedding can be found in many places (see for example [23, Appendix]).

7. Multi-condensations and Witt equivalence

An anyon condensation transforms one topological phase to another one. It provides a powerful tool to study the moduli space of all topological phases.

7.1. Multi-condensations

Definition 7.1. (See [23].) A modular tensor category is completely anisotropic if the only condensable algebra A e C is A = 1e.

Therefore, if a topological phase described by a completely anisotropic modular tensor category, then it cannot be condensed further. We will call such topological phase completely anisotropic.

Example 7.2. A few examples of completely anisotropic modular tensor categories: (1) Fibonacci categories [22]: simple objects are 1 and x with fusion rule x ® x = 1 © x. (2) Tensor powers of Fibonacci categories [22]. (3) Ising model: simple objects are 1, f, a with fusion rules given in (42). It was proved in [33] that only two simple special symmetric Frobenius algebra are 1 and 1 © f — a ® a v. But it is easy to see that the algebra a ® a v is not commutative and not a boson. Therefore, the only condensable algebra is the trivial algebra 1.

In general, a modular tensor category C might contain a lot of condensable algebras. Let A be a condensable algebra in C. A commutative algebra B over A (recall Definition A.23) is naturally a commutative algebra in CAc. We have CB — (CA)B.

By [34, Lemma 4.3], [20, Proposition 2.3.3], a commutative algebra over A is separable (connected) if and only the corresponding algebra in CAoc is separable (connected). Therefore, condensing a condensable algebra B over A in C-phase can be obtained by a two-step condensation: first condensing A, then condensing B in the condensed phase CAoc. In particular, we have

ploc ^ /ploc\loc CB — (CA )B ■

A maximum condensable algebra A in C will create a completely anisotropic topological phase Cloc. This completely anisotropic topological phase is trivial only if the original phase C is non-chiral, i.e. a bulk with a gapped boundary. The condensations in chiral topological phases are discussed in Section 6.4.

7.2. Witt equivalence

Two phases C and D are called Witt equivalent if they can be connected by a gapped domain wall. This is a well-defined equivalence relation, which was first introduced in mathematics [23] to classify rational conformal field theory. It was proved to be relevant to the problem of connecting two topological phases by a gapped domain wall in [36, Section 4] (also known to Kitaev [50]). Mathematically, two modular tensor categories C and D are Witt equivalence if there is a spherical fusion category C such that

C K D — Z(E). (56)

By taking quotient of this relation from all modular tensor categories, we obtain a group, called Witt group [23]. This Witt group is an infinite group. The unit element is given by the Witt class [Vect] determined by the category Vect of finite-dimensional vector spaces. The multiplication

of the group is given by the Deligne tensor product K, which, in physics, amounts to putting one anyon system on the top of another anyon system, i.e. a double-layer system. The inverse is given by [C]-1 = [C]. In particular, a topological phase C can have a gapped boundary or non-chiral if and only if [C] = [Vect] in Witt classes (see [36, Section 3] for a proof). An interesting result proved in [23, Theorem 5.13] is that in each Witt class, there is a unique (up to braided equivalence) completely anisotropic modular tensor category. A further study on Witt equivalence was carried out in [24].

Remark 7.3. Theorem 5.13 in [23] is stated for non-degenerate braided fusion category. But it is also true for (unitary) modular tensor category which can be viewed as a non-degenerate braided fusion category with a (unitary) spherical structure. The proof of Theorem 5.13 can be easily adapted to the (unitary) modular case because, for a maximal connected commutative separable algebra A, the category ClAoc is automatically (unitary) spherical.

If two topological phases C and D are Witt equivalent, in general, you cannot obtain D by condensing anyons in C. But you can obtain both phases via a two-step condensation from a single phase.

Indeed, by [23, Corollary 5.9] (see also [36, Section 4]), C and D are Witt equivalent if and only if there are spherical fusion categories C1 and C2 such that C K Z(C1) — D K Z(C2) as braided tensor categories. These two categories C1 and C2 can be determined as follows. If (56) is true, multiplying both sides by D, we obtain

C K (D K D) — D K Z(E).

Since D is modular, we have (D K D) — Z(D) as braided tensor categories. Therefore, given the data in (56), we can choose two categories C1 = D and C2 = E such that there is a braided monoidal equivalence G : C K Z(D) — D K Z(E). One can start from a topological phase given by A := D K Z(E), then condense two condensable algebras A1 and A2 in A:

A1 := G(1e K FD (1d)), A2 := 1d K Fg (1£),

where FD and Fg are the right adjoint functors of the forgetful functors Fd : Z(D) — D and FE : Z(E) — E, respectively. After the condensations, we obtain two phases [23, Proposition 5.15]:

C — Al^ and D — AA2c.

As a consequence, we have shown that any two Witt equivalent topological phases can be obtained from a single phase via a two-step condensation.

The gapped domain wall in above condensation can be determined as follows. Assume that we obtain a very thick wall between the C-phase and the D-phase after the two-step condensation. According to our bootstrap study of one-step condensations, the excitations on the left side of the thick wall must be given by the spherical fusion category AA1; those on the right side of the wall must be given by AA2; in the middle of the thick wall is the original phase A. Therefore, viewed from far away, this thick wall becomes a 1-d wall with excitations given by AA1 Ka AA2. The gapped domain walls between a C-phase and a D-phase are not unique of course. They are classified by Lagrangian algebras in C K D.

GBi|Bi-wall

XI a codimension-2 defect

,52(8,4-

C-phase

es2|B2-wall D = Cj|c-phase

Fig. 5. 1-d condensation: a 1-d condensation on a c^ ^-wall can create a new 1-d phase B2, where B2 is a connected separable algebra in cB1 b (or equivalently an algebra over B1 in c), together with a codimension-2 defect which is given by an object X in the category cB1IB2. Two 1-d bulk-to-wall maps are given by — ® X and X ®-.

8. Conclusions and outlooks

We have established a general theory for anyon condensation and showed that a one-step condensation is determined by a condensable algebra in the initial modular tensor category.

We have also briefly mentioned 1-d condensation happened on a domain wall or a boundary. If we run a bootstrap analysis for 1-d condensation, we will obtain a rather complete picture (see Fig. 5). In this case, a 1-d condensation on a CB1|B1-wall can create on this wall a new 1-d phase CB2|B2, where B2 is a connected separable algebra in CB1|B1. It also creates between these two 1-d phases a codimension 2 defect, which is given by an object X in the category CB1|B2 of B1-B2-bimodules. The two 1-d bulk-to-wall maps are given by the functors - ® X and X , where X is a B1-B2-bimodule. These structures can also be summarized by the following commutative (up to natural isomorphisms) diagram:

Above diagram was first appeared in the study of Levin-Wen type of lattice models enriched by defects in [51]. According to Section 5.1, the choice of B1 and B2 is not entirely physical, but their Morita classes are physical. When B1 = B2 = B, a codimensional 2 defect is just a wall excitation in the original CB1 |B1 -phase. It can also be viewed as some kind of 0-dimensional condensation. Codimensional 3 defects are instantons. A brief discussion of such defects was given in [51].

The bootstrap approach taken in this work does not tell a complete story of anyon condensation. Ideally, one would like to start from concrete lattice models and turn on the interaction between anyons in a given region (see [14]) and see how the phase transition really happens and how it matches with the bootstrap results given in this work. To introduce interactions among anyons is amount to selecting a proper subspace (as energy favorable) of a multi-anyon

CB1|B1

CB2|B2

space [30]. The condensation map pM,N : M ® N ^ M ®d N introduced in the work provides a crucial information for possible constructions. We hope to address this issue in the future.

The mathematical theory of anyon condensation described in this work can also be applied to the condensations in a symmetry enriched topological orders [43] with small variations. In this case, we must work with algebras in G-crossed braided tensor categories [38]. We will give more details elsewhere.

Acknowledgements

I thank Alexei Kitaev for sharing his unpublished works with me and Alexei Davydov for sending me the slides of his talk. I thank Ling-Yan Hung, Chao-Ming Jian and Yi-Zhuang You for motivating me to write up this paper. Their comments on the first version of this paper lead to clarification in Remarks 2.2 and 5.5, 5.6. I want to thank Michael Müger and Dmitri Nikshych for clarifying the notion of unitary category, and thank Jürgen Fuchs and Christoph Schweigert for clarifying the connection to their works and for many suggestions for improvement. I want to thank Sander Bais and Joost Slingerland for clarifying their contributions to this subject, and Yasuyuki Kawahigashi for clarifying the connection to the subfactor theory. I thank Xiao-Liang Qi, Xiao-Gang Wen, Zhong Wang, Yong-shi Wu for helpful discussion. I would like to thank the referee for many important suggestions for improvement. The author is supported by Basic Research Young Scholars Program, Initiative Scientific Research Program at Tsinghua University, and NSFC under Grant No. 11071134.

Appendix A

For the convenience of physics readers, we include in this appendix the mathematical definitions of various tensor-categorical notions appeared in this work. We will not spell out explicitly the coherence conditions used in some of these notions because they are usually lengthy and mysterious to the first time readers. For more details, readers should consult with reviews of this subject (see for example [7,18,61,65,66]).

A.1. Modular tensor categories

In this subsection, we review the definition of spherical fusion category and that of modular tensor category. A beautiful introduction to the later notion from the point of view of anyons can be found in Appendix E in [49].

A monoidal category (or tensor category) is a category equipped with a tensor product ® and a tensor unit 1 (or vacuum in physical language). The tensor product ® is associative with the associativity isomorphisms:

are required to satisfy the triangle relations. A braiding is a family of isomorphisms cXj : X ® Y —> Y ® X, satisfying the hexagon relations.

ax,Y,z : X ® (Y ® Z) —> (X ® Y) ® Z WX,Y,Z e C, which are required to satisfy the pentagon relations. The unit isomorphisms:

1 ® X XX X ® 1

Definition A.1. A monoidal functor F : C ^ D between two monoidal categories C and D is a functor such that there are isomorphisms F(X ® Y) -=> F(X) ® F(Y) (preserving the tensor products) and F(1) -=> 1 (preserving the unit) satisfying some coherence properties. If both C and D are braided, F is called braided monoidal if the following diagram:

F(X ® Y) —F(X) ® F(Y)

F(cx,y)

cf(x) , f(y)

F(Y ® X)—^ F(Y) ® F(X) is commutative for all X, Y e C.

Definition A.2. A right adjoint of a functor F : C ^ D between two categories is a functor Fv : D ^ C such that there are natural isomorphisms:

homD (F(X), Y) ~ homC (X,Fv(Y)), VX e C , Y e D.

A C-linear category means that all hom spaces home (A, B) for A , B e C are vector spaces over C. C is semisimple if every object in C is a direct sum of simple objects. C is called finite if there are only finite number of inequivalent simple objects. We denote the set of equivalence classes of simple objects in C by I, elements in I by i, j, k, l e I. We have \I| < to. A simple unit means the unit 1 is in I.

In a finite semisimple C-linear category, it is possible to translate the associativity and unit isomorphisms to some very concrete data. The isomorphism (57) can be recovered from the following isomorphisms:

home ((i ® j) ® k,l) home (i ® (j ® k),l) In terms of the chosen basis, F can be expressed by what is called fusion matrices in physics.

Definition A.3. A tensor category C is called rigid if each U e C has a left dual vU and a right dual Uv, together with the following duality maps:

r» = dU : Uv® U ^ 1, A diU : U ®vU ^ 1, uv u u vu

u uv vu u ~

y^j = bU : 1 ^ U ® Uv, = bU : 1 ^ vU ® U, (59)

where letter "b" stands for "birth" and "d" for "death", such that all the following conditions: vu u u uv

idvy, = idU' f\J = idU' C^J = iduV

VC7 U U Uv

are satisfied. C is called sovereign if VU = Uv for all U e C.

Definition A.4. A multi-fusion category is a finite semisimple C-linear rigid tensor category C with finite-dimensional hom spaces. If the tensor unit in C is simple, C is called a fusion category.

Let C be a rigid tensor category and U e C is an object. We naturally have (vU)v = U and v(Uv) = U. If a e homC (U, Uvv), we define a left trace

TrL (a) : 1 b—U ® Uv — Uvv ® Uv -U— 1. If a e homg (U, vvU), we define a right trace:

Tr* (a): 1 vU ® U — vU ®vvU 1.

Definition A.5. A pivotal structure on a rigid tensor category C is an isomorphism a : ide — vv, i.e. a collection of isomorphisms aU : U — Uvv natural in U and satisfying aU®V = aU ® aV. In this case, we have vvU — U — Uvv. C is called spherical if TrL(aU) = TrR(aU) for all U e C. We set Tr = TrL/R in this case.

If C is spherical, we define quantum dimension for U e C by dimU := Tr(aU).

Definition A.6. A *-category C is a C-linear category equipped with a functor * : C ^ Cop which acts trivially on the objects and is antilinear and involutive on morphisms, i.e. * : Hom(A, B) ^ Hom(B, A) is defined so that

(g ◦ f)* = f *◦ (f)*= f *, f ** = f (60)

for f : U ^ V, g : V ^ W, h : X ^ Y, X e Cx. A *-category is called unitary if * satisfies the positive condition: f o f * = 0 implies f = 0.

Remark A.7. That * preserves the identity maps follows from (60). More precisely, for X e C, we have idX = (idX o idX)* = idX o idX = idX.

A functor F : C — D between two *-categories is required to be adjoint preserving, i.e. F(f*) = F(f)*.

Definition A.8. A monoidal *-category C is a monoidal category such that * is compatible with the monoidal structures, i.e.

(g ® h)* = g*® h*, Wg : V ^ W, h : X ^ Y, (61)

aX,Y,Z = aXJ,Z, lX = lXl, rX= rX1' (62)

A braided monoidal *-category requires that * is compatible with the braiding, i.e. cX y — cX y for all X, Y. , ,

In a monoidal *-category C, if an object X has a right dual (Uv, bU, dU), it automatically has a left dual which is given by (U^, d'U, bU). Similarly, a left dual automatically gives a right dual. For this reason, we will adopt a symmetric notation for duality, we denote both the right and left duals as U, i.e. U := Uv = vU, and we set bU = dU and dU = b'*j. A unitary fusion category has a unique pivotal structure which is spherical [48,28].

Proposition A.9. (See [28].) For a unitary fusion category, the quantum dimensions of objects are real and positive.

Definition A.10. A ribbon category is a rigid braided tensor category C with a twist 0U ■ U —> U such that the following conditions holds:

01 = idi, 0uv = (0u)v, 0U®v = cv,u ◦ cu,v o (0u ® 0y), where (0u)v ■= (du ® idUv) o (idUv ® 0u ® idUv) o (idUv ® bu).

If the category C is ribbon, we can identify vU = Uv, i.e. C is sovereign.

Definition A.11. A ribbon *-category C is a braided monoidal *-category which is ribbon and 0 x = 0 x1 for all objects X. C is unitary ribbon if * is also positive.

Definition A.12. A (unitary) modular tensor category is a C-linear semisimple finite (unitary) ribbon category such that the matrix [si,j] defined by

Sij =j , (63)

is non-degenerate.

We have si,j = sj,i and so,i = dim i. The dimension of C is defined by

dim C = (dim i)2

A.2. Algebras in a modular tensor category

Let C be a braided tensor category.

Definition A.13. An algebra in C (or a C-algebra) is a triple (A, ¡x, i), where A is an object in C, m is a morphism A ® A ^ A and i : 1 ^ A satisfying the following conditions:

X o (x ® idA) ◦ aA,A,A = X ◦ (idA ® x), X o (i ® idA) = idA = X o (idA ® i)■

The algebra A is called commutative if x = X ◦ cA,A.

We denote the ingredients of an algebra graphically as follows:

X = i = I

Definition A.14. A left module over an algebra A = (A, ¡x, i) is a pair (M, xm), where M is an object in C and xM ■ A ® M ^ M such that

xM o (x ® idM) o aA,A,M = Xm o (idA ® xm)

and /M o (iA ® idM) = idM. The definition of a right A-module (M, ¿M) is similar. An A-B -bimodule is a triple (M, /M, IM) such that (M, /¿M) is a left A-module and (M, /M) is a right B -module such that

iM ◦ {iM ® id^ ◦ aA,M,B = iM ◦ (idA ® iM).

We denote the module structure graphically as follows:

Definition A.15. For a commutative algebra A, a module (M, /M) is called local if /M = /M o

CM,A o CA,M.

Definition A.16. A C-algebra (A, ß, i) is called separable if ß : A ® A ^ A splits as a morphism of A-bimodule. Namely, there is an A-bimodule map e : A ^ A ® A such that ß o e = idA. A separable algebra is called connected if dim home (1, A) = 1. A commutative separable algebra is also called étale algebra in [23].

If a C-algebra A is separable, the category CA of A-module and the category CA|A of A-bimodules are both semisimple. In this paper, a connected separable commutative C-algebra is also called a condensable algebra for simplicity.

Definition A.17. Two C-algebras A and B are called Morita equivalent if CA = CB. Or equiv-alently, there are an A-B -bimodule M and a B-A-bimodule N such that M ®B N ~ A and N ®AM ~ B as bimodules.

Let C be a modular tensor category.

Definition A.18. (See [25].) A connected commutative separable algebra A is called Lagrangian if (dim A)2 = dim C.

Theorem A.19. (See for example [23].) For a Lagrangian algebra A in C, the category CAc of local A-modules is trivial, i.e. CAC — Vect.

Definition A.20. A coalgebra is a triple (A, A, e) where A : A ^ A ® A and e : A ^ 1 obey the co-associativity:

A o (A ® idA) — aAA,A ◦ A o (idA ® A) and the counit condition (e ® idA) o A — idA — (idA ® e) o A.

We will use the following graphical representation for the morphisms of a coalgebra, A A

V • -!

Definition A.21. A Frobenius algebra A = (A, x, i, A, t) is an algebra and a coalgebra such that the coproduct is an intertwiner of A-bimodules, i.e.

(idA ® x) o (A ® idA) — A o x — (x ® idA) ◦ (idA ® A). Let now C be a sovereign tensor category. A Frobenius algebra in C is symmetric iff Av Av

A Frobenius algebra is called normalized-special if

X o A — idA and e o i — dim(A) idi.

Definition A.22. Let A be a normalized-special Frobenius algebra in a modular tensor category C and let M be a right A-module and N be a left A-module. The tensor product M ® A N can be defined by the image of the following idempotent (or projector):

P®A =

Moreover, there exists morphisms eA : M ®A N ^ M ® N and rA : M ® N ^ M ®A N such that rA o eA — idM®an and eA o rA — P®a .

Definition A.23. Let A be an algebra in C. An algebra B is called an algebra over A if there is an algebra homomorphism f : A ^ B such that the following diagram commutes:

A® B—^B ® B^^B

B ® A-

B ® B

B is an algebra over A is equivalent to the statement that B is an algebra in CA.

A.3. Module categories and centers

Definition A.24. A left module category over a tensor category C (or a left C-module) is a category M equipped with a C-action: a functor ® : C x M ^ M such that there are isomorphisms:

X ® (Y ® M) —> (X ® Y)® M

for X, Y e C and M e M and 1 ® M —> M, satisfying some obvious coherence conditions [62]. A C-module M is called semisimple if every object in M is a direct sum of simple objects. M is called indecomposable if it cannot be written as a direct sum of two C-modules.

The definition of a right C-module is similar. For two tensor categories C and D, a C-D-bimodule is a category M equipped with a left C-module and a right D-module structure, such that there are isomorphisms X ® (M ® Y) —=> (X ® M) ® Y satisfying some natural coherence conditions.

Definition A.25. A C module functor F : M ^ N is a functor from M ^ N together with an isomorphism F(X ® M) —=> X ® F(M) satisfying some coherence conditions [62].

Definition A.26. Let A be a fusion category and M an A-bimodule. We define the center Za (M) of M by the category of A-bimodule functors, i.e. ZA(M) := FunA|A(M, M). In the case that A c B as a fusion subcategory, ZA(B) is also called the relative center of B. If A = B, Za(A) is called monoidal center of A, also denoted by Z(A).

Remark A.27. An object in Z(A) is a pair (Z, z), where Z is an object in A and z is a family of isomorphisms {zX : Z ® X —=> X ® Z}XeA, satisfying some consistency conditions. There is a forgetful functor Z(A) ^ A defined by (Z, z) ^ Z. This functor is monoidal. When A is modular, Z(A) = A K A, where A is the same tensor category as A but with the braiding given by the anti-braiding of A. In this case, the forgetful functor coincides with the usual tensor product a K b ^ a ® b for a, b e A.

An important result of Muger [60] says:

Theorem A.28. If A is a spherical fusion category, then the monoidal center Z(A) is a modular tensor category.

Definition A.29. A functor F : A ^ B from a braided tensor category A to a (not necessarily braided) tensor category B, is called central, or equipped with a structure of a central functor, if there are natural isomorphisms B ® F(A) —=> F(A) ® B satisfying some coherence conditions such that F can be lifted to a braided monoidal functor to the center Z(B) of B. More precisely, there is a functor F : A ^ Z(B) such that the following diagram:

A—^Z(B)

forget

is commutative.

Definition A.30. A functor F : A ^ B is called dominant if, for any B e B, there are A e A such that homB (B, F(A)) — 0.

Definition A.31. Two fusion categories A and B are called Morita equivalent if there is an indecomposable semisimple A-module M such that B ~ FunA(M, M)®op as fusion categories, where Fun^(M, M)®°p is the category of A-module functors from M to M but with the tensor product ®op defined by f ®op g — g ® f — g o f.

It was proved in [29] that two fusion categories A and B are Morita equivalent if and only if Z(A) ~ Z(B) as braided fusion categories.

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