A (Dummy's) Guide to Working with Gapped Boundaries via (Fermion) Condensation

We study gapped boundaries characterized by"fermionic condensates"in 2+1 d topological order. Mathematically, each of these condensates can be described by a super commutative Frobenius algebra. We systematically obtain the species of excitations at the gapped boundary/ junctions, and study their endomorphisms (ability to trap a Majorana fermion) and fusion rules, and generalized the defect Verlinde formula to a twisted version. We illustrate these results with explicit examples. We also connect these results with topological defects in super modular invariant CFTs. To render our discussion self-contained, we provide a pedagogical review of relevant mathematical results, so that physicists without prior experience in tensor category should be able to pick them up and apply them readily


Introduction
There are many works that study gapped boundaries in 2+1 d bosonic topological orders that are characterized by anyon condensation [ The most signatory set of physical properties of a gapped boundary includes the collection of topological defects it supports, their quantum dimensions and fusion properties which controls ground state degeneracies of the system in an open manifold. As alluded to above, these gapped boundaries can be understood in terms of (super) Frobenius algebra. The physics of the gapped boundaries should be encoded in the mathematics, which in principle could be extracted systematically. This is indeed the case particularly for bosonic gapped boundaries -except that the techniques are dispersed in the physics and mathematics literature, the latter of which is often shrouded in a language completely foreign to physicists, and that the formal principles laid out may not be readily converted into a practical computation. A practical way of computing fusion rules of defects localized at junctions between different gapped boundaries have been elucidated in [SH19]. In the case of fermion condensation which receives attention only more recently [ALW19, BGK17, LKW16, WW17, BGK17], a systematic study including non-Abelian fermion condensation and junctions remain largely an open problem.
We propose that a super-commutative separable special Frobenius algebra in the bulk topological order is responsible for characterizing its fermionic boundary conditions -to our knowledge this is the first systematic use of the "super-commutative" version of the Frobenius algebra to describe the fermionic boundaries and through which to work out their properties.
We elucidate properties of defects in a gapped boundary or junctions characterized by fermion condensations. This includes obtaining the full collection of topological defects, identifying their endomorphisms, and also computing their fusion rules, that can be summarized by a (super) defect Verlinde formula. We extended the results of [ALW19, BGK17, LKW16, WW17, BGK17] to include non-Abelian condensates, and also the study of junctions between (fermionic) gapped boundaries. We also develop new ways to compute the half-linking numbers.
To understand these results, it is most convenient if the reader is familiar with the computational tools available in braided tensor categories and their algebra objects. Relevant mathematical results are mostly scattered in many different places, which maybe a major hindrance 1 Let us emphasize here that gapped interfaces are also characterized by anyon condensation. However, every gapped interface can be understood in terms of a gapped boundary by the folding trick. The main difference between a gapped interface and a gapped boundary is that across the interface there are still non-trivial bulk excitations, while the phase is trivial across a gapped boundary. The discussion here, to avoid clutter, addresses directly the gapped boundaries. The discussion however can be easily turned around into a discussion of gapped interface.
to entering the subject. We therefore collect the most relevant tools to make the paper selfcontained. We give up some mathematical rigor to make the language more readily accessible to working physicists with minimal prior experience in tensor category theories, with explicit examples illustrating various computations. A formal and proper introduction to the subject can be found in numerous places in the literature. Of particular use are [FRS02,FRS04,JO01], and references therein. The paper is organized as follows. In section 2 we first give a brief review of (super) braided tensor categories. We review also algebra objects in a category, and their representations. These results are then extended to include super commutative algebra. The computation of half-linking number and the fusion rules of modules and bi-modules, in addition to their endomorphisms, are discussed.
In section 3, we illustrate the results obtained in section 2 in explicit examples, namely the toric code model and the D(S 3 ) quantum double, where we explicitly obtain the Frobenius algebra and the bi-modules that describe fermionic boundaries. We also demonstrate how their fusion rules are computed.
In section 4, we describe the connection of the current results to supersymmetric CFT's and their topological defects. We also discuss the twisted version of the Verlinde formula that produces the difference in fermion parity even and odd channels in the fusion of primaries.
In section 5, we conclude with some miscellaneous facts about fermion condensation, and various open problems to be addressed in the future.
There are several lengthy computations that we have relegated into the appendix. In particular, the computation of the 6j symbol describing associativity of fusion of the boundary excitations, are explained and illustrated in detail there. We also include more sophisticated examples of fermion condensates in the SU (2) 10 and D(D 4 ), which involve condensates of multiple fermions. In particular, in the latter example some of these (super) modular invariants do not appear to correspond to a condensate that preserve fermion parity. Whether these examples have physical meanings should be explored in greater details in the future. We present in appendix B the counting of Majorana modes localized at junction using the Abelian Chern-Simons description of the toric code order, and compute entanglement entropy on a cylinder with different fermionic/bosonic boundary conditions. Some topological data of the D(S 3 ) quantum double is reviewed in appendix C.

A Physicist's skeletal manual to tensor category and gapped boundaries -review and generalizations to fermionic boundaries
Tensor category covers a huge class of mathematical structures. To the author, the framework has a structure not unlike an onion where extra structures can be included layer by layer, adding to the complexity of the situation.
As far as 2+1 d topological order is concerned, the categories that are of interest are (braided, or in fact modular) fusion categories.
In the following, we will collect the most important results that would actually be used in the rest of the paper. Rather than listing all the algebraic equations in one full swoop as if all the properties are supposed to appear together from the beginning, we are presenting these results in a way to emphasize that many of the properties are in fact independent. Each add-on property is an extra mathematical structure to the construct, and each such addition has to be made consistent with all the other qualifiers already included, very often leading to extra consistency constraints, which is the origin of the many algebraic equations characterizing a certain tensor category.

The basics of Braided fusion tensor category
Let us summarize the most basic concepts below. 2

Simple Objects
The most basic structure is the collection of objects. Simple objects describe different species of point excitations in a 2+1 d topological order. Physically interesting theories are semi-simple categories, where every object can be decomposed as direct sums of simple objects, which form a basis of elementary particles.
The multiplicity m ai should be non-negative integers.

Morphism
Morphisms are maps between objects. Morphisms reveal structures of the objects. In a bosonic theory, simple objects, describing point particles having no internal structure has a 1-dimensional "endomorphism" space. i.e. there is a unique map mapping a simple object to itself (endomorphism). That map is just the identity map. Graphically it is often represented as a straight line. Between two simple objects a and b there is no map between them, unless a = b. For example if we have where c i are the simple objects in C, We can construct a basis for these morphisms from the composite object a to c i . This is illustrated in figure morphism, where the basis index α runs from 1 to m ai . States in a Hilbert space are constructed from basis of morphisms. Particularly when we attempt to count the number of states in a topological order with a given number of anyons, the Hilbert space is basically the space of morphisms that map the collection of anyons to appropriate objects -e.g. to the trivial anyons if the system is in a closed manifold with no boundaries. These "maps of collection of anyons to another anyon" are part of an extra mathematical structure -namely fusion, that is discussed below. One important technical issue is a phase ambiguity in constructing a basis for morphisms. Rescaling a given basis in (2.4) by ζ α (a, c i ) defines an equally good basis.

Fusion
Anyons obey a commutative and associative fusion algebra: where N c ab = N c ba is a non-negative integer specifying the number of different ways in which anyons a and b can fuse to c. A special object 1 called the trivial object(vacuum) fuses trivially with all other objects: 1 ⊗ a = a. 3 The building block of the states of anyonic Hilbert space is the fusion basis represented diagrammatically by a vertex: where µ = 1, . . . , N c ab . The number d i is the quantum dimension of anyon i, which will be briefly reviewed later in this section. The collection of all fusion trees with the same input/output legs spans a subspace of the Hilbert space, namely the fusion space V ab c . Note that in the construction of explicit basis of these linear maps V ab c , or equivalently the definition of the states (2.7), is ambiguous up to a phase ξ ab c , which is the same kind of rescaling of morphism basis as we have seen in (2.5). Larger fusion bases are constructed from the building blocks by taking tensor product of the building blocks in an appropriate order.
Associativity of anyon fusion is given by (a ⊗ b) ⊗ c = a ⊗ (b ⊗ c). It follows that the corresponding fusion space V abc d has two sets of basis with respect to the fusion order, and the basis transformation in this fusion space is captured by the F symbols 4 as shown in (2.8).
The F -symbols are not independent, they're related by the coherent condition known as the pentagon equation, as shown in figure 1. Pentagon equations are sufficient to solve for all F -symbols in an anyonic model.
Note that because of the phase ambiguity mentioned above the F -symbols are not invariant under these rescalings. It allows one to fix some of the components of F .

Quantum dimension
A category is called pivotal if every simple object a has a dual a * . Given any simple object a, the dual of a is a simple object a * satisfying a ⊗ a * = 1 + · · · , (2.9) where 1 is the trivial object(vacuum). Diagrammatically, any line labeled by a * is equivalent to a line labeled by a but with the direction reversed. The pivotal structure is essential in the definition of quantum dimension. The quantum dimension of a simple object is defined as the quantum trace (the pivotal trace) of an identity operator.
The diagram in (2.10) is direction-irrelevant, so we can freely replace a by a * and therefore d a = d a * . The quantum dimension of the trivial object 1 is d 1 = 1 in any anyonic model. Quantum dimension is conserved under anyon fusion, following the fusion (2.6) we have The notion of quantum dimension can be generalized to an arbitrary object in the category C, in particular dim(A) = a m a d a , for any object A = a m a a ∈ C. (2.12) The dimension of category C is defined as where D = a d 2 a is called the total quantum dimension of C.
Braiding and Twist (spin) Using the fusion tree basis, the braiding exchange operator can be represented by R symbols shown in figure In the special case where b = a * and c = 1, the R symbol is reduced to the spin of an anyon. R-symbols and F -symbols are not independent. In order for braiding to be compatible with fusion, it is found that some coherent condition must be satisfied by the F -symbols and Rsymbols, which may be expressed diagrammatically as the following Hexagon equation. Given the solution of F -symbols, hexagon equations are sufficient to solve for all R-symbols. A note on Super-fusion category A useful place for the discussion of super-fusion category can be found in [GWW15,ALW19]. The above discussion applies to a generic fusion category. In the presence of fermion condensation which we will be interested below, the resultant gapped boundary would carry extra structure connected to the Z 2 fermion parity. To accommodate that structure, we need to upgrade the notion of a fusion category to a superfusion category. There are many definitions of super-categories. At the level of the objects, it may involve a decomposition of the objects to a direct sum of even and odd parity objects 5 .
However, this definition is quite restrictive. In the gapped boundaries that are considered, and more so when it comes to defects localized between boundaries, such a decomposition is not very clear. Therefore, we will adopt the discussion in [GWW15,ALW19] and not discuss such a decomposition (2.18).
Then the most distinctive characteristic of a super-fusion category is its morphisms. First, the allowed endomorphism space of simple objects would be enlarged. Simple objects could potentially carry a "fermion parity odd" map to itself, in addition to the usual identity map which carries even fermion parity. i.e. In this case, dim [Hom(a, a) = 2]. Pictorially, the two "basis maps" to itself are represented as in figure below. Simple objects having a two dimensional endomorphism are referred to as q-type objects in the literature [ALW19]. 5 See for example [BE17] and also some of the references that appear in [ALW19] Second, fusion, being morphisms from C ⊗ C → C, could also acquire both parity even and odd channels. They are illustrated in (2.20). Odd channels are often represented with an extra red dot on the vertex. There is an ambiguity in the definition of the fusion coefficients. Consider the following fusion process: The fusion is an element in Hom(a ⊗ b, c). If c is a q-type object that has a two dimensional endomorphism space, the fusion map could be concatenated by a non-trivial endomorphism in c and remains an element in Hom(a ⊗ b, c). In other words, while defining ∆ ab , we have implicitly made a choice in discarding possible endomorphism in c. This does not appear natural. Therefore, it is proposed in [ALW19] to enlarge the fusion space to Under this definition, it would mean that the fermionic "dots" can be freely moved from the vertex to the connecting anyon lines if they are q-type objects [ALW19].
Since fusion spaces are Z 2 graded, and that the old channels essentially carry a Majorana mode which leads to sign changes under swapping of labels [GWW15,ALW19], the pentagon equation has to be upgraded to keep track of these labellings and signs. This leads to the super-pentagon equation illustrated in (2.23). Note that the operation K refers to the exchange of the labels of the vertices. Depending on their fermion parity, (i.e. if both vertices carry odd parity), there is a sign change there.

Gapped boundaries and (super)-Frobenius Algebra in tensor categories
It is well known that each bosonic gapped boundariy of a non-chiral bosonic topological order C in 2+1 dimensions is characterized by a (commutative separable symmetric) Frobenius algebra in C [Kon14,JO01]. 6 Physically the algebra encodes the condensation of a collection of bosons at the boundary. While there are already ample hints elsewhere such as [ALW19, WW17, BGK17], it has not been formulated completely explicitly. Here, we propose that a fermionic gapped boundary is also encoded in a separable symmetric Frobenius algebra, except where "commutativity" is relaxed to "super-commutativity", to accommodate condensation of fermionic anyons. i.e. To reiterate, the relevant mathematical structure is a super-commutative separable symmetric Frobenius algebra.
Each of these labels will be discussed below.
We collect some necessary facts about Frobenius algebra and anyon condensation below. In this section, we rely heavily on [JO01], and particularly [FRS02,FRS04], which have developed many useful tools and proved numerous identities related to Frobenius algebra and their modules, that assist us significantly in our quest -the reason being that a super-commutative Frobenius algebra is a Frobenius algebra after all. Applications of these tools to understand super Frobenius algebra and their q-type modules/bi-modules are some of the main goals of the current paper.

Algebra and co-algebra
An algebra in the category C is a collection A of simple objects. This collection is expressible as [JO01,FRS02] A This collection of anyons when equipped with the appropriate set of structures that we will discuss below, would be identified with the set of anyons that condense at the gapped boundary. The collection, as an algebra, is equipped with a product µ, which maps A × A → A. We have already constructed all the basis of maps (or homomorphisms) from C × C → C in the previous section. Therefore this product µ must be expressible in terms of the basis constructed out of the simple objects. This is illustrated in (2.25). where ζ labels the fusion channel i ⊗ j → k in the bulk.
The multiplicity W i1 determines the dimension of the maps (homomorphisms) from A to the simple object c i . We therefore introduce a label α as we did in figure morphism. Defining the product µ is equivalent to solving for the coefficients defining the linear combination of basis maps -note that they are subjected to the same phase ambiguity as discussed in (2.5).
A unit ι A is a morphism from the vacuum 1 to A. This morphism is in fact an embedding ι A : 1 → A, so in any mathematical expression the unit can be simply understood as the vacuum object 1 despite its nature of morphism. A more accurate yet pedagogical understanding of the unit is to think of it as "taking out the vacuum 1 from A". Or alternatively, when we do arithmetic, any number x = 1.x. We can freely multiply any number by unity.
The vacuum fuses trivially with all objects in the category, translating this back to a morphism we see immediately that the unit has to satisfy the morphism equality µ•(ι A ⊗id A ) = id A . Here id A is the identity map in A. This equality is illustrated in (2.27).
Each algebra in a category has a unique unit, which means the vacuum appears exactly once in the algebra A.
A co-algebra A is a collection of simple objects A equipped with a co-product ∆, which maps A → A × A. While this operation may look mysterious, we have a very familiar example in physics. Consider for example two electrons with spins S 1 and S 2 respectively. The action of spatial rotations on these spins are effected through the total angular momentum operator S =Ŝ 1 +Ŝ 2 . This is an example where we are "splitting" an SU (2) group element into the products of two SU (2) group elements, expressed as a sum of operators in the corresponding Lie algebra. Like in the case of the product µ, the map ∆ can be expressed in terms of the basis maps constructed in C. This is illustrated in the following.
A bi-algebra is a collection of objects A that is equipped with a product and a co-product at the same time. By now, it should be familiar that every time an extra structure is introduced, we have to determine how the new structure and all the previous structures already in place should fit together. In a bi-algebra the product µ and the coproduct ∆ satisfy the following relation: A counit ε A is the same morphism as the unit, but with the direction reversed, namely a morphism from A to the vacuum 1. Like the unit, the counit satisfies a similar morphism equality: which is a direction reversed version of (2.27).
A Frobenius algebra is a bi-algebra that is trivially associative. Its properties are conveniently summarised in the following pictures. We have also included the conditions for the algebra being "separable" and symmetric.
A Frobenius algebra A is called separable if there exists a map e : A → A ⊗ A such that µ • e = id A . A Frobenius algebra A is called special(a.k.a. strongly-separable) if the product µ is the inverse of the coproduct ∆ and the unit ι A is the inverse of the counit ε A , namely 7 The separability of A allows a well-defined notion of simple objects in the representation category Rep A, and of non-simple objects as direct sums of simple objects.
A Frobenius algebra A is called symmetric if the product µ and the counit ε A satisfy This condition is sometimes called normalized special.
A commutative algebra is one where µ • R A,A = µ. This is illustrated in the following figure. The collection of anyons A can condense physically if they are mutually local, and that they are bosonic. It turns out that the above condition is sufficient to imply both.
We have explained all the necessary qualifiers of the algebra object that describes a condensate. The condensate should ultimately behave like the vacuum, or trivial anyon in the condensed phase, where intuitively it could be freely created or annihilated without causing any changes to the states. These mathematical structures introduced above are therefore physical requirements of the condensed anyons.
To accommodate also fermionic anyons condensing, one has to relax the commutative condition to "super-commutativity".
A super algebra is one which is graded by the Z 2 (fermion parity) symmetry. Therefore, the algebra A acquires a decomposition [CKM17] A = A 0 ⊕ A 1 . (2.33) The fermion parity σ(i) of an anyon i belonging to A p is (−1) p , for p ∈ 0, 1.
This decomposition allows us to define super-commutativity, which is given by 8 where R c i ,c j is the half-braid of the modular tensor category C. The mathematical definition has a simple physical interpretation -this is precisely to bind the fermionic anyon with a "free fermion" so that the pair together behaves like a boson and condenses. The idea of pairing is also discussed in [ALW19]. Here we made it explicit that this is the physical realization of super-commutativity in the mathematics literature.
The dimension D A of the algebra is defined as A (super)-Lagrangian algebra is a (super)-commutative Frobenius algebra satisfying . This is sufficient and necessary condition for the bosonic algebra to recover a modular invariant, and apparently a sufficient condition to recover a super-modular category. As we will demonstrate in the example of D(D 4 ) in the appendix D.2, there are examples where a supermodular invariant with positive integer coefficients that does not admit an interpretation as a fermion condensate.

Defects via construction of modules
Having introduced the concept of an algebra in a category C which plays the role of our condensate at the gapped boundary, we would like to obtain the collection of allowed defects, or excitations, at the boundary.
When an anyon in the bulk phase approaches the gapped boundary, it would generally become an excitation at the boundary. However, since a bunch of anyons are "condensed" at the boundary, which can be freely created and annihilated there, it means that the fusion product between a bulk anyon and a condensed anyon might no longer be distinguishable at the boundary. In other words, the bulk anyons form "multiplets" under fusion with the condensed anyons. The corresponding mathematical jargon would be that the boundary excitations are modules (or representations) of the condensate algebra A. We note that for a (super)-commutative Frobenius algebra, these left/right modules form a fusion category. In physical terms -excitations at the gapped boundary have well defined fusion rules.
In this subsection, we would summarise how to recover these modules.

Left (Right) Modules
Each module M of A in C is also a collection of anyons in C. i.e. (2.37) Again, there are maps from M → C, expressed in the figure (2.37), as well as its dual map, from C → M As a "representation" of the algebra A, they admit an action by A onto it. i.e. There is a linear map ρ M A : A × M → M .
Since these anyons have non-trivial mutual braiding, we should specify whether A is acting on the left or on the right of M at the gapped boundary. Here we will assume that the action is on the left, making M a "left-module". In the case of commutative and super-commutative algebra A, the right action can be generated from the left action, simply by composing the product with an R-crossing [FRS02]. In the following, unless otherwise specified, we will first explicitly discuss left actions, which would automatically apply to right actions.
Again, these (left or right) actions are linear maps which can be expressed in terms of the basis of morphisms C × C → C (fusion) we have constructed in the previous section. The map ρ M A can thus be explicitly expressed in terms of these basis, as illustrated in (2.39). As a representation of the algebra A, ρ M A must satisfy (2.46). This is nothing but a generalization of our familiar property of a group representation, in which where ρ M (g) ab is the representation matrix of the group element g ∈ G. Besides, it is well known that the irreducible representations of a group satisfy an orthogonality relation Here {α} are labels of basis of ρ M when expanded explicitly as maps in C. It will be made explicit in the following.
To extract the coefficients λ we can use the orthogonality relation (2.42) which then gives (2.44) [FRS02].
Note that the basis abstractly denoted {α} shows up above in the basis map projecting the modules to anyons in C i.e. α, β. This identity is very useful. We note that (super)commutativity of the Frobenius algebra A allows us to work simply with left modules. It also ensures that the resultant collection of boundary excitations (modules) form a (super) fusion category (i.e. the structure of fusion is well defined.) [JO01,FRS02] . To a physicist -it means it makes sense to look for edge excitations which is always expressible as a linear combination of some basic excitations (the simple/irreducible representations) of the algebra, and that these excitations have well defined fusion rules.
The 6j-symbols responsible for associativity of the fusion of the boundary excitations can be computed systematically, as soon as the precise multiplication µ of the condensate algebra A and the left/right action of the algebra on its modules are solved. Since this is relatively tedious and lengthy, we would relegate the computation to the appendix.
In practice, W c i M is crucial data to work out ρ M A using equation (2.46). One important handle towards solving for W is via the inspection of induced modules.
Modules can be "induced" by fusion with the condensate A. The product µ would automatically supply the correct structure to produce a left (right) action satifying (2.46) described above. This is illustrated in (2.45) [JO01,FRS02].
These induced representations following from A⊗c i , denoted Ind A (c i ) are generally reducible. They can be expressed in terms of the simple (irreducible) modules as [FRS02] These parameters λ can be solved using the identity illustrated in (2.44). This identity allows very efficient computation of the modules -particularly when the induced module is itself simple. It is possible to generate all the simple modules through constructing induced modules. The deduction of the W -matrix is greatly faciliated by the identities relating quantum dimensions discussed in 2.3.2 below.

Endomorphisms
As we have emphasized in multiple instances, one novel ingredient in a fermionic gapped system (describable by a super-fusion category) is that some of the modules have non-trivial endomorphism -i.e. the q-type objects we have referred to earlier. We need to identify which of the modules we obtained correspond to q-type objects. This is discussed in [ALW19] in the context of Abelian fermion condensation. There, it is observed that anyons that are "fixed points" under fusion with the condensing fermion are q-type objects when considered as modules (or boundary excitations) of the condensate algebra A. i.e. fixed point anyons satisfy where ψ ∈ A, and d ψ = 1.
In the case of non-Abelian condensation and where the defects are localized at the junctions, the above condition (2.48) is not well defined. In these cases, endomorphisms of the modules can be deduced by applying identities discussed in [FRS02] and also solving for the modules explicitly. These methods are to be reviewed and extended in the next two subsections. The identities applicable to junction defects will be discussed separately in section 2.4. A necessary signature of non-trivial endomorphism is that the same module acquires two independent left (right) action. The situation can easily be confused with the case when a single anyon is splitted into two boundary excitations (i.e. two simple modules). This situation has been discussed for example in [ERB14]. These two situations are distinguished precisely using identities relating quantum dimensions in the parent and the condensed theory that we will discuss below.

Quantum dimension and endomorphism-some useful identities
This section explains a novel application of various useful identities proved in [FRS02,FRS04]. These identitis connect the quantum dimensions of the defect and that of the anyons composing it . Since they are very useful and powerful, we would like to reproduce some of them here. To simplify notations, let us follow [FRS02,FRS04] and denote dim [Hom(a, b)] = a, b . (2.49) A module M as a collection of anyons in C has a quantum dimension in C given by where now we can write The subscript C serves to remind us that this is counting the dimension of homeomorphisms (or maps) from the point of view of C.
The dimension of a module M as an object in the representation category of A can be defined by the quantum trace in A. This is illustrated in the following.
The shorthand M, M A denotes the dimension of endomorphism of M as a "simple" object in the representation category of A. i.e. This is equal to 2 for a q-type representation, and 1 otherwise. This is a generalization of the result in [JO01,FRS02], allowing for the left (right) action coming in two independent copies for q-type excitations.
There is also a very useful theorem The Reciprocity theorem. It states that In words, it says the dimension of space of maps between M and c i in C is the same as that of maps between M and the induced module of c i when they are treated as objects in the representation category of A, or in other words, as boundary excitations. The is proved in [FRS02].
Two useful relations follow from the above theorem. They are given by It is a generalization of Corollary 4.14 in [FRS02] allowing for q-type objects. The generalization follows from the fact that a q-type object carries two independent left action which should be implicitly summed over in x. Since the two sets of left action belong to the same module M x for M x a q-type object, we replace the sum by the dimension of the endomorphism of M x . Physically, this has a very simple interpretation, which applies to both fermionic and bosonic condensation. It can be re-written as This means that quantum dimension is "conserved" as a bulk anyon is "decomposed" into boundary excitations in (2.55).
Moreover, (2.55) together with (2.37) imply that Equation (2.53, 2.56, 2.58) are powerful handles to determining W , and also the endomorphism -specifically to distinguish it from the situation in which an anyon "splits" at the boundary, actually participating in two distinct representations. Specifically, when two independent solutions of an irreducible representation can be solved involving the same collection of anyons, the dimension of the endomorphism must be consistent with a quantum dimension of the resultant excitation that is greater than 1. This will be illustrated in the example section with explicit solutions.

Fermion parity and spin structures
We note that there are two other new ingredients in working with fermion condensation. Here, we extended the techniques in [FRS02,CCW16] to accommodate these new ingredients.
Firstly, one would expect to work out σ M c i , which is the fermion parity assignment to the anyon c i in the representation M . In a non-Abelian theory, it is possible that c i participates in multiple modules. Rather than assigning a fermion parity to individual anyons, it is more appropriate to determine fermion parity of the "condensation channel". i.e. Among the W ix different ways a bulk anyon is mapped to the boundary excitation M x , some of the maps have even parity and others, odd parity.
One can work out the fermion parity of these condensation channels systematically starting from the parity assignment of the condensate algebra A. This follows from a twist of the relation (2.58).Ã where Ω cx gives the difference between the number of even and odd participation channels for c in x, andÃ i.e. there is a minus sign for every fermion in the condensate.
Since Ω ix is the difference between the number of even and odd "condensation" channels of c i in x, one can see that for x a q-type excitation, Ω ix = 0. This has an impact on the derivation of the "twisted Verlinde formula" to be discussed below. In the case where all W i1 < 2, Ω i1 can be directly treated as the fermion parity σ i of the anyon c i in defining the super commutative algebra (2.34). For cases where some W i1 ≥ 2, on first sight we might have to assign multiple parities to the same anyon participating in the algebra A. However, we suspect this could never happen -i.e. a fermionic anyon could never enter a super Frobenius algebra A twice having W i1 > 2.
We note that since (2.59) shows up quadratically on the r.h.s., there is a sign ambiguity for Ω cx for x = 0. Practically in the examples we work with, we make a specific choice. We are not aware of a canonical choice at present.  Secondly, in the presence of fermions which are sensitive to spin-structures, there are anyons that are responsible for Ramond (or anti-periodic) boundary conditions for the free fermions -or in other words, they fit with the Ramond type spin-structure when they are inserted in a closed manifold with non-contractible cycle. This is illustrated in figure 2. Under a fermion condensation, the boundary excitations can either be responsible for the Neveu-Schwarz (NS) type spin structure or Ramond (R) type in a non-trivial cycle. This can be checked by checking the monodromy matrix: where ψ ∈ A has fermionic self-statistics, with θ ψ = −1, and both c i,j belong to the same boundary excitation, with W ix and W jx non-vanishing for some x, and that their Ω ix and Ω jx comes with opposite signs. The boundary excitation x is R type if the monodromy matrix defined above evaluates to -1 for all i, j in x, and NS type in the case of +1. This generalizes the discussion in [WW17, ALW19, LKW16] to accommodate non-Abelian fermionic condensates.
We note however, that in a condensate involving bosonic anyons not generated by fusion of two fermions, the confined anyons do not necessarily have a well defined spin structure -since they are non-local wrt the bosonic components of the new vacuum made up of the condensed anyons.

Fusion rules
Physically, when we observe a cluster of excitations from sufficiently far away i.e. at a distance large compared to the separation between them, then the cluster would effectively appear as some point excitation. The fusion maps in the bulk is part of the data that defines the topological theory. The fusion between boundary excitations however are "derived" properties that can be worked out from the choice of the condensed anyons A and the bulk fusion rules.
Mathematically, we have contended that boundary excitations are representations (or modules) of the condensed algebra A. Therefore, the physical concept of fusion simply correspond to fusion of representations. Diagrammatically, when we have a pair of representations, we should be able to define a new left (and right) action on the combined system. Directly analogous to the situation in combining spins where we takeŜ =Ŝ 1 +Ŝ 2 -as already explained when we introduce the co-product -the new left action on the combined system should require the use of the co-product. This is illustrated diagrammatically in the middle figure in (2.64) for fusion of left modules. An extra intermediate A line connecting M 1 and M 2 is introduced. As illustrated in (2.64), it implements automatically (2.63) The intermediate A-line indeed acts a projector. i.e. If one puts in two parallel A-lines between the modules M 1 , M 2 as in (2.63), using the fact that the algebra A is Frobenius and separable (2.31) and that M 1,2 both satisfy (2.40), one can show that it can be reduced to having only one line. This would be left as a simple exercise for the readers.
To summarise, the fusion map of modules of A is defined such that one mods out the relation (2.63). This fusion map is denoted ⊗ A , and practically implemented using the projector involving the A-line introduced in (2.64).
Note however, the module resulting from the fusion is not irreducible. It is important information to recover the decomposition of the fusion product in terms of irreducible representations. This has been considered in [FRS02]. The decomposition coefficients can be computed using (2.44).
To recover only the fusion coefficients however, we can make use of the identity [JO01] (which has been generalized here to accommodate for non-trivial endomorphisms where n z xy are the fusion coefficients counting the total number of fusion channels mapping M x × M y to M z in the boundary, as defined in (2.22) for a super-fusion category.
As reviewed already in the previous section, the fusion channels in the condensed phase describable by a super fusion category also come with even or odd fermion parities. The above equation should thus be refined. We found a twisted version of the above relation, using also (2.59) The twisted fusion coefficientñ z xy is the difference between the number of even and odd fusion channels taking x ⊗ y to z. Here, M x , M x δ A denotes the difference between the number of even and odd endomorphisms of M x . For example, a q-type object with one even and one odd endomorphic maps satisfies M x , M x δ A = 0. As discussed previously, the dimension of endomorphism space for simple objects is either 1 (non q-type) or 2 (q-type) in a super-fusion category. Also, Ω ix vanishes for x a q-type object. Therefore, the sum over x in (2.67) might as well be restricted to non-q-type excitations, to givẽ Now this totally parallels (2.65).

(twisted) Defect Verlinde formula
In [SH19], we described a formula relating the fusion coefficients of the boundary excitations with the "half-link" between the boundary excitations and the condensed anyons. Here we would like to generalize it to the case accommodating fermion condensation, and also to express the "half-link" in terms of a trace of the different linear maps whose basis we have constructed explicitly in the previous section.
First, let us obtain the pair of twisted defect Verlinde formula for a given gapped boundary characterized by A.
This can be derived using (2.65), which takes an identical form in fermionic condensates as in bosonic ones discussed in [SH19] The matrix V is invertible -the first index i runs over only c i ∈ A and the index x enumerates the boundary excitations. As we argued in [HW15b,SH19], the number of anyons in A is always equal to the number of boundary excitations, so that V is a square matrix.
As observed in [SH19], the matrix V is related to the "half-linking" number as follows: Here, we would also like to express the half-linking number in terms of the basic defining properties of the condensate algebra A and the modules, in the incarnation of a quantum trace that is illustrated in (2.71).
We observe that the normalization constant takes the following form In a super fusion category, fusion channels can acquire even or odd fermion parities. The defect Verlinde formula given above relates the total number of fusion channels with the halflinking numbers.
There is an independent equation that relates the difference between the number of even and odd fusion channels to a "twisted" half-linking number. This can be derived using (2.67) using very similar techniques as the derivation of (2.69). We first define the matrix This is the analogue of the V matrix defined in (2.69). As already noted earlier when Ω jx was defined, Ω jx = 0 for x a q-type excitation. Therefore, in the matrix v ix , x only runs over the gapped excitations that are not q-type. The other index j now runs over the anyons c j belonging to the gapped excitation x f responsible for generating the fermion parity. i.e. There is a special gapped boundary excitation x f such that the monodromy with the condensate produces a +1 on all the bosonic condensed anyons, and a -1 on all the fermionic ones. -This will be further discussed in section 4 below, where this special excitation can be readily worked out by a simple modular transformation in the bulk using (4.14) . Surprisingly, v is also a square matrix -there is always an equal number of anyons in the special boundary excitation x f as there are non-q-type boundary excitations! i.e. Once restricting x to non-q-type objects, equation (2.68) and (2.65) take the same form.
Thus we obtain the following twisted Verlinde formula simply by replacing V by v in (2.69), which givesñ The sum over i runs over anyons participating in x f . We note that this is not imposed by hand -but simply follows from the property of v. This is one of the main results of this paper.

Defects at junctions and bimodules
In the previous section, we have focussed on excitations in a given gapped boundary where A condensed. Here, we would like to extend the discussion to excitations localized at the junction of two different gapped boundaries characterized by condensate algebra A and B should correspond to irreducible left-right bi-modules of A and B. Each bi-module is again a collection of anyons in C. The left and right action of A and B respectively should commute. This is illustrated in (2.75).
It is shown that the bimodules together also form a semi-simple fusion category [FRS02]. Exactly analogous to the case of left (right) modules, one can generate induced modules from any anyon c i ∈ C, by sandwiching c i by A and B on the left and right of c i respectively. i.e Repeating (2.45) with a copy of B on the right as well.
The induced bimodule obtained are reducible (not simple), and thus can be decomposed in terms of simple ones. By inspecting the fusion A ⊗ c i ⊗ B that generates the induced bimodule, it is possible to isolate all the independent simple (irreducible) modules, and recover the W-matrix. Without the simple conservation formula for quantum dimensions as in (2.56) and also the analogue of (2.65), it is not apparent if there is a simple formula for the W-matrix. Moreover, as in the case of left (right) modules, one has to work out the endomorphism of a given module.
An identity particularly useful for the purpose is the following [FRS04] : where here we are using ∼ = loosely to mean the two sides are isomorphic. This also implies As we will see in the example of D(S 3 ) in section 3.2.2, the formula assists us in determining non-trivial endomorphisms in a junction excitation.
Exactly as in the case of left (right) modules, one can work out the endomorphism of a given module.
There is a new complication in the presence of fermionic condensates. As we have discussed, free fermions have been introduced into the system to enrich the theory into a spin-TQFT, and the condensate algebra a super-Frobenius algebra. In a spin TQFT, it is possible to introduce localized Majorana modes. Therefore, every excitation at the junctions would become Z >0 graded -the non-negative integer "grading" keeping track of the number of Majorana modes that have been added to the spot. This has been observed in [BJQ13a,BJQ13b] where gapped boundaries of Abelian spin TQFT were discussed. For each extra Majorana mode that is added, the quantum dimension of the defect would be raised by a factor of √ 2. We note that when we add a pair of Majorana mode to the same spot, they could pair up as a Dirac fermion mode and be gapped out by a local Hamiltonian. Therefore, the grading is not topologically robust, and could be reduced to a Z 2 structure.
In the current paper where we focus on bosonic bulk topological orders, it is observed that fusion of defects between bosonic junctions with defects at bosonic-fermionic junctions could generate different flavours (or grading) of the excitations.

Fusion rules and the Defect Verlinde formula
Fusion of bi-modules (or excitations localized at junctions) follows a similar playbook as the fusion of the modules. For A, B, C ⊂ C, the fusion map is given by (2.78) Practically, ⊗ B , which we have already discussed while defining the fusion map for left (right) modules, can be implemented by inserting a B line. This is illustrated in (2.79).
Again, we can decompose the resultant A|C bimodule in terms of irreducible (or simple) A|C modules. This can be done by using equation (2.42, 2.44) again. We note that these equations work equally well for a bimodule -we simply need to view the bimodule as the left-module of the algebra A ⊗ C rev -where the super-script rev refers to folding C. Practically however since the left and right action of A and C respectively commute, one is basically including in (2.42) an extra C loop also on the right.
In actual applications, one is often only interested in working out the fusion coefficients. As such, we might hope to adopt a strategy similar to (2.65). However, for bi-modules, we do not know of a simple analogue. For Abelian bulk topological order however, we can work it out as follows. For simplicity, we will assume that A ∩ B = 1. i.e. The trivial anyon in C is the only anyon in the intersection between the two condensates. In this case, every induced bimodule Ind A|B (c i ) is also simple. The strategy to work out the fusion coefficients is that the fusion operation ⊗ A can be implemented by modding out redundant copies of the condensates as the induced bimodules fuse. i.e.
We have included extra super-scripts over the W-matrix to distinguish the data of different condensates. The above is clearly a A|C bimodule which is then decomposed into simple bimodules.
• Extra trapped Majorana modes at junctions The above rules apply to bosonic condensates A, B, C. There is a caveat when it comes to algebra involving fermionic condensate as well. As already mentioned in the previous subsection, junctions between boundaries can host Majorana zero modes in the presence of free fermions, so that every simple bimodules that we work out based on seeking representations of the algebra A, B comes in an infinite number of versions -differing by the number of extra Majorana zero modes that they host. When considering fusion of junctions, different versions are often being generated, even if we start with a canonical choice obtained via the induced modules A ⊗ c i ⊗ B.
To accommodate this complication, we note the following. The fermionic anyons that condensed had in fact formed Cooper pairs with free fermions introduced at the gapped boundary. Therefore it is more proper to write the condensate algebra as i.e. A free fermion is denoted ψ 0 which pairs up with the condensed fermion.
An extra majorana mode trapped at a junction would correspond to an extra copy of ⊗(1 ⊕ ψ 0 ) introduced there. This is an appropriate way of keeping track of these Majorana modes, since they can absorb or release a Dirac fermion, and so behaves as a genuine fermion condensate at a point -and whose only module would be the "condensate" itself : χ ≡ 1 ⊕ ψ 0 . When we fuse two such modes (which are modules of χ) we should get On the rhs, it is understood that the fermion is no longer localized at a 0-dimensional junction. This then correctly recovers the fusion rule of Majorana modes -that produces the direct sum of the trivial state and a single fermion state. When considering general fusion of junctions with possible extra Majorana modes, it is key to keep track of copies of χ. We will illustrate this technicality in the examples section 3.1.2 and 3.2.2, which is crucial towards keeping track of the quantum dimensions of junction excitations.
If C = A, the resultant gapped boundary excitations should be further reduced from an A|A bimodule to a left (right) module (recall that the left and right modules can be generated from each other since we are considering (super)-commutative Frobenius algebra).

Then we have
where we have made use of (2.58, 2.37) in the last equality.
In [SH19], we obtained a defect Verlinde formula describing the fusion of bi-modules. In the presence of both bosonic and fermionic condensates, one can write down a defect Verlinde formula too. Given the extra complication of Majorana fermion modes, we will have to fix the ambiguity in the defect Verlinde formula too. The half-linking number across a junction expressed as a quantum trace is illustrated in (2.71). The junction excitation involved here is the "canonical" choice obtainable from the induced modules, and our defect Verlinde formula would describe the fusion of these canonical junction excitations.
The defect Verlinde formula that describes the canonical fusion coefficients as defined above takes exactly the same form as in [SH19]. For completeness, we reproduce it here and the inverse of M is taken by treating it as a matrix with indices α, β, while the inverse of γ is taken wrt the indices {c α,β , z} i.e. the number of z indices is equal to c α,β . We have taken extra pains to include subscripts for the α, β to indicate the precise condensate these condensation channels are related to. It should be clear that x lives at the junction between A and B, and y between B and C, and finally z between A and C.
Similarly to (2.71), for half-linking numbers considering the fusion of bi-modules we propose (2.86) with normalization constant given by

Note: the M3J and M6J symbols and VLCs
To assist our readers in the sea of literature, here we would like to comment on the relationship between the condensate algebra and some of the linear maps introduced elsewhere.

M-symbols
The notion of M-symbols were introduced in [CCW16]. The idea is that the gapped boundary is an interface where bulk anyons could end on it. As one consider multiple bulk anyons ending on the boundary, it is possible to consider changing the order of fusion of the bulk anyons before they end on the boundary. These different processes should be related by linear maps, which is given by the M3J and M6J symbols. This is illustrated in figure 2.88. As expected, the M-symbols are directly related to the defining data of the condensate algebra A and its modules (up to appropriate normalizations). This is illustrated in (2.89) .
It is not very convenient to solve for M using the above relation. It is often easier to solve for M directly based on its consistency conditions that is the analogue of the pentagon equation. Therefore, we generalize the consistency condition for purely bosonic condensate in [CCW16] to accommodate fermionic condensates, where the M6J symbols would satisfy a twisted form of such a consistency identity. The major difference is to recognize that the junction at which a bulk anyon enters the gapped boundary is precisely described by the condensation maps that we have defined in (2.38) and (2.26). They come in fermion parity even and odd versions in the presence of a fermionic condensate, and one has to keep track of the ordering of these junctions, similar to the derivation of the super-pentagon identity.
Similarly to bosonic condensation the M6J symbol for fermion condensation carries several groups of indices. First, there are labels of bulk anyons a, b, c. Second, there are boundary excitations x, y, z they condense to and third, condensation channel labels µ, ν, λ. In addition  to those, it has an extra index to label the fusion channels of the boundary excitations. Then we introduce s a x (µ), s xy w (ω) to denote the parity of the condensation channel and fusion channel respectively, 0 for even and 1 for odd. To avoid dependence of fermionic wave functions on the ordering of odd channels , we introduce one "Majorana number" θ x on each vertex x of condensation and also each fusion vertex of the boundary excitations, which are denoted by red dots in the above diagram. These Majorana numbers satisfy In [zheng-cheng gu, zhenghan wang and xiao-gang wen, 1010.1517] they introduce 6j symbols that carry the Majorana numbers along. They are defined as Similarly, one could define a new M-tensor carrying the Majorana numbers: VLC's were introduced in [ERB14]. They are linear maps that map fusion basis in the bulk theory to the boundary fusion basis. This is illustrated in (2.95).
Note that the notation introduced there is applicable for all W ix ∈ {0, 1}.
These VLC's can be separated into three classes.
First, there are the "vacuum vertex" where three vacuum lines in the boundary theory meet. These vertices are precisely defining the product and co-product of the condensed algebra A.
Then, there are vertices where the boundary vacuum line meets a boundary excitation, leaving it "invariant". These vertices are precisely defining the left (right) action of the algebra on the module corresponding to the boundary excitation.
Finally, there are three boundary excitations meeting at a vertex, defining fusion maps in the condensed theory. Fusion of modules are defined in (2.64). These can be decomposed in terms of irreducible (simple) modules using (2.44) which we have discussed. These decomposition coefficients can be related to the VLC's defining the fusion map as illustrated in (2.96) by mapping them to the parent theory, and compare fusion basis by basis. This connection with [ERB14] is valid when we restrict to the situation where W ix ∈ {0, 1}, and so the channel labels do not feature here. 9 Note that the right most diagram in (2.96) is nothing but the expansion of the following diagram (2.97) in basis form.
A lesson learned here is that the defining property of the condensed theory is basically the product of the algebra A, and its left (right) modules, from which everything else derives. The precise mathematical formulation also allows extension to cases where W ix > 1 rather seamlessly.

Illustrating with examples
Having developed the formal computational tools based on (super)-Frobenius algebra and their modules, here we would like to illustrate these tools in explicit examples, and in the process, understanding interesting features of boundaries.

Beginner's level -the Toric code
The toric code is the paradigmatic example of a bosonic topological order in 2+1 dimensions. We are going to see that most of the important physics of gapped boundaries and junctions can already be understood here.
It has four excitations, {1, e, m, f }. As it is well known, there are three kinds of gapped boundaries for the toric code topological order in 2+1 dimensions. Among these boundaries, two of them are conventional ones obtained from condensing bosons. Specifically one is termed the electric boundary where the electric charges condense (i.e. A e = 1 ⊕ e), and the other, termed the magnetic boundary where the magnetic charges condense (i.e. A m = 1 ⊕ m).
Here we would like to discuss in detail the third type of gapped boundary following from condensing the e − m bound state which is a fermion. This has been mentioned before in [BGK17]. We will also study junctions between these boundaries.

The fermion condensate
The fermionic Frobenius algebra is given by The boundary excitations are characterized by We can summarize this data in terms of the W and Ω matrices : The fusion rules can be obtained using (2.65).
⊗ 0 One should also check that X f is a non-q-type object with trivial endomorphism. Here, one can check that X f is responsible for generating the fermion parity. The number of objects it contains as a module of A f is equals 2. This is the same as the total number of non-q-type defects in the gapped boundary -which is also 2 (where the "trivial defect" has to be included.) This is a confirmation of the claim made after (2.73).
The algebra admits the following product µ satisfying the Frobenius equation The 6j symbols of the condensed phase can be read off following the discussion in the previous section.
They are given by F 000 0;00 = F x00

The Bosonic-Fermionic junctions
As alluded to in the previous subsection, the toric code model admits two bosonic gapped boundaries that correspond to the electric A e and magnetic A m condensates. i.e.
For completeness, let us recall also that in each of these bosonic boundaries, there is one nontrivial excitation. Let us denote the one in the electric boundary by X e and that in the magnetic boundary by x m . They are given by One can readily check using (2.65) that they satisfy a Z 2 fusion rules.
We would like to consider junctions between these bosonic boundaries with the fermionic boundary introduced in the previous sub section.
First, we consider the e − f junction. Results of the m − f would follow in a completely analogous manner. By considering the "induced" bimodule A e ⊗c i ⊗A f , we find that there is only one excitation X ef localized at the e − f junction. i.e. The four different anyons c i ∈ {1, e, m, f } in the toric code model would generate exactly the same bimodule. i.e.
Since this is an Abelian model, one can work out the fusions readily using (2.80). The fusion rules are given by Where X e and X f are non-trivial excitations of e and f boundaries respectively. From these fusion rules we can conclude that the quantum dimension of this excitation is √ 2.
We note that (3.7) takes the same form as that in the fusion of defects localized at the e − m junction. There, one could also readily check that Therefore it is known that X em also has quantum dimension Recall in section 2.4.1 that there is generically a subtlety regarding Majorana modes, and that in the computation above one should make the replacement This does not affect the conclusion in (3.7), or any of the fusion rules in a single gapped boundary -all it does is to tag an odd fusion channel by an explicit factor of ψ 0 .
It however makes a crucial difference below as we are going to see.
Consider the fusion of excitations in different types of junctions. Specifically, this is illustrated in figure 3. We might expect the following fusion rule where # should be some positive integer. But one could readily see that this is not possible if quantum dimensions are conserved in the process of fusion -which it should -to ensure that the counting of ground state degeneracy a robust topological number. This is because using the methods above, we claimed that all the defects should have quantum dimension √ 2, so that # could not possibly be an integer. Now we reconsider (3.10) by introducing the free fermions ψ 0 . The fusion described in (3.10) can now be computed as follows: This shows that we obtain at the e − m junction X em and also a Majorana mode (1 ⊕ ψ 0 ) -here this has to be interpreted as such since it is localized at the e − m junction! This may appear somewhat mysterious. To elucidate the physics, we demonstrate it using two different methods.
First, let us study lattice model of the toric code and also explicit constructions of its gapped boundaries. It is convenient to describe these boundaries using the Wen-Plaquette version of the toric code topological order [Wen03], as had been thoroughly discussed in [YZK13].  [Wen03]. Note that in the former, the spin-degrees of freedom lives on the links, whereas in the latter, they live on the vertices. Therefore, where the black lines intersect the blue lattice lives a spin 1/2 degree of freedom. The plaquettes are divided into two sets, the Z and X plaquettes. The Hamiltonian acts in a way depending on this division, as reviewed briefly in (3.12).
is reproduced here where σ s are Pauli matrices. By acting a σ z operator on a vertex, a pair of e can be created in two adjacent X plaquettes. Similarly by acting a σ x on a vertex, a pair of m will be created in two adjacent Z plaquettes. The fermion gapped boundary appears as a "smooth" boundary in the blue lattice. A "smooth" fermionic boundary on the Wen plaquette model was discussed in [YZK13]. To visualize the boundary modes, it is most convenient to fermionize the boundary spin degrees of freedom, and turn it into a set of Majorana modes {c i }, one at each boundary vertex i. As a check, a fermion string operator can be applied at the boundary as shown in the figure, showing that individual f anyon can be created or destroyed there -justifying the claim that f condenses at the boundary [YZK13,BGK17].
The boundary is gapless if translation invariance is preserved [YZK13]. There are multiple ways to turn it gapped. One way, discussed in [BGK17], is to introduce an extra set of Majorana modes {γ i }, one at each vertex at the boundary. Another possibility is simply to give up translation invariance, and introduce a boundary Hamiltonian that pairs neighbouring Majorana modes. For our purpose, this suffices to illustrate the fusion rules of junctions discussed above. The same results can also be understood from the perspective of Abelian Chern-Simons theory.
This will be relegated to the appendix.

The bimodules and computing the half-linking number
In the previous sections, we have obtained some "coarse-grained" data regarding the bimodules. Here we would like to provide details of some "fine-grained" data of these boundaries -namely the actual Frobenius algebra characterizing the boundary, and the left/right action of the bimodules, to illustrate the general principles laid out in earlier sections.
For concreteness, let us focus on the e − f junction. To begin with, we need to solve for two Frobenius algebra A e and A f . Using the conditions discussed in section 2.2 and also the 6j-symbols of the toric code topological order, we obtain (3.13, 3.14). Then we would like to obtain the unique simple bimodule X ef already discussed in (3.1.2). Note that here we have used the freedom to rescale discussed in (2.5) and introduce ζ Ae and ζ A f f to set all the coefficients in (3.13, 3.14) to 1. Then we would like to obtain the unique simple bimodule X ef already discussed in (3.1.2). A bi-module is separately a left module of A e and right module of A f . Therefore the left-right actions must separately satisfy (2.40) and its right-action counter part. But as a bi-module, it must satisfy commutativity between the left and right action as illustrated in (2.75) too. These results in the bimodule illustrated in (3.15).
We still have enough phase rescaling freedom here (ζ and ζ X ef f ) to set all the coefficients to unity. Substituting the algebra and the bimodules into (2.86), we obtain The normalization N ef is given by which recovers the fusion rules (3.7), confirming (2.87).

Intermediate level -D(S 3 )
The quantum double model D(S 3 ) is the paradigmatic example of non-Abelian topological orders that illustrate non-trivial features that could arise.
For completeness, we include the topological data of the bulk theory in the appendix, which sets the notations of the anyons that we will use below. The bosonic gapped boundaries of D(S 3 ) have been studied in many places [CCW16,CCW17a,SH19]. These condensates and the junctions between them are also summarized in the appendix.
In addition to the well known bosonic gapped boundaries, there is also one fermionic gapped boundary. This is already noted in [WW17]. Let us study it in somewhat more detail below.

The fermionic boundary
The condensate is given by This condensation is closely related to the fermionic boundary of the toric code. In this case, one can readily work out the W matrix and Ω matrix using the methods in section 2.3.4. The (3.15) fusion rules between defects are also readily obtainable. We will slightly delay the presentation of these results, by taking a somewhat longer route.
As discussed in [WW17], for a fermionic condensate that preserves fermion parity, one could consider splitting the condensation into two steps -first condensing the bosons in A f , which should form a closed Frobenius sub-algebra, before condensing the fermion, which would be reduced to an Abelian anyon in the intermediate condensed phase. Applying this logic here, it would imply that one could first consider condensing For completeness, let us present the Frobenius algebra α AC in (3.20). Similarly to the case of the Toric code, here we have chosen the phase ambiguities (ζ 1 A and ζ 1 C ) such that all the coefficients including A to be 1. The virtue of the sequential condensation is that it allows one to work out A f in (3.18) as a Lagrangian algebra of D(S 3 ) by treating it as condensation of simple modules of α AC . Un-packaging it into fusion basis in D(S 3 ) is simple.
One could work out the intermediate phase where C is condensed. The methods discussed in section 2.3.4 continues to apply, even though α AC is not a Lagrangian algebra that defines a gapped boundary. In this case, one finds that the condensed phase is described by a fusion category that contains the toric code category as a sub-category. It has been noted that the toric code order remains "deconfined" such that the braiding structure is preserved, in addition to sectors identified as "confined defects" that are non-local wrt to the condensate, and thus whose braided structure is lost [JO01].
Let us summarize the properties of the intermediate phase in the table below: One could also work out the precise left actions of the algebra α AC on these modules. They are presented in (3.22). One observes that there are multiple solutions in each given module in (3.22). In this case however, they all correspond to a phase redundancy following from the choice of phase for the fusion basis discussed in (2.5). In other words, they do not lead to independent modules. This should be contrasted with a q-type object that we will study below, where a single module gives rise to two truly independent solutions.
The fermionic gapped boundary is then generated by condensing f , exactly as in the toric code case. What is new here is that X and Y are non-Abelian defects, and they display interesting properties, particularly when we begin considering junctions between gapped boundaries.
The fusion rules involving X and Y are summarized in the table below: Finally, as we condense f , it is not hard to see that X and Y together form a module, while the toric code sub-category behaves in exactly the same way described in section 3.1.1.
Let us summarize the overall W and Ω matrix: The fusion rules between these defects are given by These fusion rules satisfy the (twisted) defect Verlinde formula. It confirms that all the defects have trivial endomorphism. Now here, we again confirm the claim made after (2.73), that the number of objects N f in the defect responsible for generating the fermion parity -in this case it is X f = B ⊕ C ⊕ D containing 3 objects i.e. N f = 3 -equals the total number of non-q-type defects in the gapped phase -i.e. {1, X f , Z}!

A bosonic-fermionic junction -Take 2
It is particularly interesting to revisit the bosonic-fermionic junction corresponding to juxtaposing the magnetic boundary and the fermionic boundary in the toric code. There is new physics precisely because of the presence of non-Abelian confined defects.
The magnetic boundary described in terms of a Frobenius algebra in D(S 3 ) can be summarized by the following W-matrix: Their fusion rules are identical to (3.25) by replacing X f → X m and Z → Z m . One observes that the above table is equivalent to (3.24) upon exchanging D and E. There is this curious situation that Z m as a defect in the magnetic boundary contains the same list of anyons as the Z defect in the fermionic boundary. Now we would like to work out the junction defects. Following the playbook in section 2.4, one could identify two different bimodules by inspecting all the induced modules one by one. i.e. We inspect A m ⊗ c i ⊗ A f , ∀c i ∈ D(S 3 ). The two junction modules are summarized as follows: One can see from the table (3.21), that X mf is the same X mf we have discussed previously in (3.7). i.e. The fusion of X mf is given by The fusion of Z mf is slightly trickier. To understand it, it is necessary to see that it is actually a q-type object with non-trivial endomorphism. This is where studying the intermediate phase where A ⊕ C has already been condensed simplifies the problem significantly.
We can make use of the identity (2.77), applying it on the intermediate phase where α AC has condensed. We notice that (3.29) The identity (2.77) then implies , Ind Am|A f (X)) = Hom(X, 2(X ⊕ Y )) (3.30) The space of maps from X to 2(X ⊕ Y ) has to be 2 dimensional, for X, Y simple objects. Therefore the endomorphism space of Ind Am|A f (X) = Z is also 2 dimensional. We note that one might entertain the possibility that Z mf is not a simple object (irreducible representation) -that could also support a non-trivial space of endomorphism. However, the dimension should take the form of x n 2 x M x , M x , where x runs through all the irreducible representations contained in Z mf and n x is the multiplicity of M x appearing in Z and so n x ∈ Z >0 . Again for simple objects M x , M x can only be 1 for a non-q-type object, or 2 for a q-type object. Clearly, Z mf , Z mf Am|A f = 2 is only compatible with Z mf being a simple q-type object.
There is another manifestation of the non-trivial endomorphism. Consider solving for the left-right action of A m and A f on Z mf using the methods described in (2.40) and (2.75). One should be able to obtain 2 independent solutions, despite the fact that Z mf remains simple. These would form basis of the two generators of the endomorphism maps! For illustration purpose, we solve for them explicitly in the next subsection.
Finally, we are ready to recover the fusion rules of the junctions. Again to be careful with Majorana modes, we should upgrade the Frobenius algebra to include the free fermion explicitly, exactly as in (2.81, 3.9).
Using notations of the intermediate phase, we obtain the following fusion rules (3.32) Note that we have made the reduction from a A m |A m bimodule to a left (right) A m module by implicitly modding out by a factor of A m .
A warning here has to be flagged: the extra factor of (1 ⊕ ψ 0 ) is not a Majorana mode. We have actually confronted this situation in (2.82), Recall that Majorana modes are localized at a point. Here, it is roaming free along the entire magnetic condensate boundary. It is only making explicit that there are two fusion channels, one with even and the other odd fermion parity. Had we kept ψ 0 explicit in (2.65) the odd channels would be tagged by a copy of ψ 0 too. Now we see that ψ 0 is an important book-keeping device -if localized at a junction it accounts for quantum dimensions of √ 2. On the other hand if it roams free in the 1 dimensional boundary or in the bulk, they account for factors of 2 . Their introduction allows one to keep track of quantum dimensions of defects clearly -which are naturally conserved under fusion. Recall that Z mf is a q-type object, and in cases as such, it is expected that its fusion maps always carry an even number of even and odd channels, since they could be converted between each other by composing with an odd endomorphism. This has been briefly discussed in the introduction of super-category in section 2.
The quantum dimension of Z mf is thus given by again with the ambiguity of adding Majorana modes at the junction on top of this "canonical" basis. This is rather amusing, since Z mf , Z m and Z all contain exactly the same list of anyons The computation of the half-linking numbers and a check of the defect Verlinde formula will be discussed below.

Half-linking numbers and the defect Verlinde formula -Take 2
In the previous subsection we studied the bi-modules are argued that Z mf should carry nontrivial endomorphism. One manifestation of the non-trivial endomorphism is the emergence of two independent solutions when one solves for the left-right action of the algebras on the module after fixing all the phase ambiguities. To illustrate this point, we solve for the left-right actions explicitly below. We note that the Frobenius algebra A m = 1⊕m is identical to (3.13), replacing e by m.
We can make use of the explicit form of the algebra and modules and compute the gamma We found that

(super)-modular invariants and twisted characters
Bosonic gapped boundaries in a topological order are in 1-1 correspondence with modular invariants. For topological orders corresponding to the representation category C of the tensor product of a chiral algebra and an anti-chiral algebra , each of these bosonic gapped boundaries correspond to a modular invariant CFT. The In the case of fermionic gapped boundaries, each of them certainly defines a "super"-modular invariant i.e. a modular invariant under S and T 2 transformation on a torus [Lev13]. The reverse is not true however, as we will describe interesting examples in the appendix. It is argued that invariance under T 2 and S is the appropriate generalization of the concept of modular invariance for spin CFT [Lev13]. 10 Meanwhile, the (super) modular invariant essentially defines a Hilbert space H A .
and V i are the representations of the chiral algebra that defines the topological order introduced at the beginning of the section.
The excitations at the gapped boundary correspond to topological defects in the (super) modular invariant CFT.
The defect operator takes the following form where |c i is a short hand for the primary and also descendents in H i . As a topological defect, the descendents are summed over in a way where the levels match in the bra and ket [PZ01]. The indices α, β ∈ {1, · · · , W A i1 }. The coefficients γ x i α,β correspond precisely to half-linking numbers between the condensed anyon c i and the boundary excitation x in the topological theory.
Taking the trace on a torus produces the twisted character χ X (−1/τ, −1/τ ), where χ i is an abuse of notation corresponding to χ i (τ )χī(τ ) that follows from tracing the holomorphic and anti-holomorphic parts in H i . It is customary to denotẽ q = e 2iπτ ,τ = −1/τ, q = e 2iπτ . (4.5) The rhs of (4.4) can be rewritten using its S modular transformation property to yield where the S matrix here correspond precisely to that of the bulk phase. When the condensation multiplicities (i.e. elements of the W matrix) are either 0 or 1, one can readily show, using the identities (2.69, 2.70), that (4.6) reduces to the following: We note that here j runs also over sectors outside of A. While we do not have a direct proof of this result for general W i1 > 1 , from physical considerations -that the edge excitation admits a decomposition as bulk anyons -the result (4.7) should remain true.
Parts of these have been discussed in [LSH19,SLH19] where A defines a bosonic modular invariant. These results readily apply to super modular invariants, with the fusion algebra of the topological defects again given by the defect Verlinde formula (2.69, 2.70).
The novel structure that comes with a super modular invariant is the presence of fermion parity. In the following, we will discuss also twisted characters with R-type boundary conditions in the time direction in the following.

Topological defects in the CFT and the fermion parity defect
It is well known that in a CFT carrying global symmetries, one can define characters twisted by a generator g of the global symmetry group G in the time direction.
χ g X (q,q) = tr(gXq L 0 −c/24qL 0 −c/24 ). (4.8) As a global symmetry g would commute with L 0 andL 0 . In the context of the fermion parity symmetry, g = (−1) F , where F is the fermion parity operator. We thus define By considering the fermion parity of the different sectors appearing in (4.7), one concludes that (4.10) Moreover, in a spin CFT, we should keep track of the spin structure in the spatial direction that follows from the insertion of X. For X corresponding to an N S (R)-type object, it generates a N S (R) type spin structure in the spatial cycle. We note that since A is a Lagrangian algebra, among the A-modules only A is NS type. As expected-none of the topological line operators are local wrt to A. An important fact is that under an S transformation, the spin structures of the two cycles are expected to swap. i.e.
where s, t ∈ {N S, R} denote the spin structures along the time and spatial cycle, and S s,t denotes the S transformation matrix that would swap the spin structures, so that we only sum over defects Y with spin structure s. The "defect S-matrix" is related to the W matrix and the bulk S matrix. It is given by: where we have introduced the shorthand W 1 ≡ Ω and W 0 ≡ W . Equation (4.11) has a handy application. As discussed in (2.3.5), there is a special defect x f that generates an R-type spin structure. One can work out the the components W ix f readily if we have Ω i1 by applying (4.11) -noting that the trivial defect is the only N S sector defect here: This finally gives Generically, given any other symmetries g, and corresponding defects generating g-twisted boundary conditions, the W matrix of the latter can be extracted using analogues of (4.14). For example, using this method, we identify analogues of RR, NSR, RNS, defects in a condensed theory involving multiple non-Abelian fermion condensation. These example concerning SU (2) 10 and D(D 4 ) will be relegated to the appendix.

Revisiting the (twisted)-Verlinde formula
The discussion above inspires a re-visit of the Verlinde formula for a spin CFT. The structure of the S-matrix of characters in a spin -CFT being decomposed into different sectors, namely {S N S,N S , S N S,R , S R,N S , S R,R }, have long been discussed in the literature.
The supersymmetric CFT literature has given a Verlinde formula (see for example [AANN95,EH94]), although that does not distinguish even and odd fusion channels. On the other hand, in a spin CFT that is graded by fermion parity, one should distinguish parity even and odd fusion channels. There is a separate identity isolating the difference between the even and odd channels, in the form of the twisted Verlinde formula. The derivation is based on consideration of dimensional reduction of the 3d spin TQFT to a 2d spin TQFT in [ALW19]. We supplied an alternative derivation in the context of (non Abelian) fermion condensation in (2.74). Here, we will obtain a third derivation that depends solely on properties of the decomposition of the S matrix recalled above.
An extra key observation is that the fusion of two excitations whose worldlines cut across a common twist lines would fuse in a twisted way. The wave-function satisfies i.e.
Now we can also evaluate the wavefunction via The expression Y.Z corresponds to the aggregate spin structure after fusing two objects with spin structures Y and Z respectively. We note that they form a Z 2 group structure, with R playing the role of a Z 2 generator satisfying R.R = N S.
The second equality above is obtained by considering (4.17). It is a "spin-structure enriched" version of a well-known identity.
Combining with (4.15, 4.16), we obtain where we have again introduced the short-hand notation n N S ≡ n, n R ≡ñ. (4.19) Let us emphasize that the S X,Y matrix we are working with is not in a unitary basis. Let us take the C 2 theory as an example which follows from the Ising theory (with three sectors 0, ψ, σ ) with the fermion ψ condensed, this theory only has 1 NS sector and 1 R sector. i.e. (4.20) The corresponding characters in the C 2 theory can be written as In this basis, the S X,Y -matrix is given by These are some general considerations that a priori are not connected to anyon condensation.

Conclusion
The main aim of the current paper is to study in detail gapped boundaries of 2+1 d topological orders characterized by an anyon condensate that contains fermions. The physics of these gapped boundaries include the different species of excitations, their fusion rules, and also the properties of junctions when two different gapped boundaries meet. In the case of bosonic condensates, these issues have been studied in detail by many, which we have mentioned in the introduction. It is realized that the underlying mathematical structure characterizing each boundary condensate is a commutative Frobenius algebra, and that the boundary excitations and junction excitations are modules and bi-modules of these Frobenius algebra, respectively. In the current paper, we have generalized these considerations to cover gapped boundaries following from fermionic anyon condensation. The mathematical generalization is to replace t "commutativity" by "super-commutativity". This has been discussed to some extent in [ALW19] for simple current condensations. Here we generalize it to accommodate arbitrary fermionic anyon condensation. Moreover, we also extended the discussion to include junction excitations. Particularly, we developed systematic ways to read off the endomorphism of a (bi)-modulewhich describes whether the corresponding defect could host a Majorana mode. Along the way, we clarify and generalize the defect Verlinde formula discussed in [SH19] to fermionic boundaries, as well as providing a systematic recipe to compute the half-linking numbers central to the formula.
We also discussed the connection between these defects in a super-condensate and line operators in a super modular invariant CFTs -as in the bosonic case, each fermionic condensate defines a "super" modular invariant, and these gapped boundary excitations are topological line operators.
There are some miscellaneous facts that we have omitted in the main text, but which maybe of interest.
Fermion condensation that preserves fermion parity at the end can be reduced to an Abelian fermion condensation [WW17]. Consider a fermionic gapped boundary of a bosonic bulk topological order. If we adopt the strategy of a sequential condensation that condenses the bosons in the condensate first, the intermediate phase has only three different possible choices. Namely, the toric code order (c=0), the Ising order (c=1/2) , and the 3-fermion order (D 4 , 1) (c=4) [RSW07]. These are the only bosonic modular tensor categories with at least one fermionic simple current and that they have quantum dimension 2 -which would become fully confined by condensing just one more fermion. This simple fermion is responsible for carrying the odd fermion parity, and it is necessarily a simple current in the intermediate phase, as demonstrated in [WW17]. This fact gives a simple check that decides if a bulk order has a fermionic gapped boundary -i.e. by staring at the topological central charge which is preserved under anyon condensation.
Another curious fact is the apparent scarcity of fermionic gapped boundaries, particularly those beyond simple fermion condensation. In the entire SU (2) k series of modular tensor categories, only SU (2) 10 contains a Lagrangian super-Frobenius algebra. We could not find any in SU (3) k , by following the principle of sequential condensation. These conclusions are made by adopting the philosophy of "sequential" condensation. We first look up modular invariants of these models (SU (2) k series adopts an ADE classification, and SU (3) k has been classified for example in [Gan94].) and among them look for candidates where one can condense a further simple fermionic current.
To conclude, we note that there are various questions that are still pending. We looked into super modular invariants in D(D 4 ), and there, we found examples where the super-modular invariants do not appear to correspond to a super-commutative Frobenius algebra as we have defined it in the main text. It rather appeared as a non-commutative algebra where anyons in the condensate can be non-local wrt each other. It would also imply that the product of the tentative algebra must break fermion parity -the product of total parity even anyons gives a parity odd anyon. One could perhaps discard them as simply being unphysical. However, the mutual non-locality among the condensed anyons was only at worst a minus sign. It is a curiosity whether there might after all be an interpretation to these super modular invariants.
The generalization to fermionic condensates suggest that it is possible to further extend the idea of anyon condensation to include anyons of arbitrary spin -the new ingredient is to couple to the condensing anyon an appropriate gauge field -the counter part of spin structures that would render the condensation consistent. It is believed that such condensates might be related to gapless boundary conditions. Finally, it is realized recently that these gapped boundaries are examples of spontaneous breaking of a categorical symmetry[TW19, JW19, KLW + 20a]. It is important to understand if there are other implications of the categorical symmetry, or how to systematically reverse the process of condensation (i.e. the analogue of gauging), and how these ideas could be generalized to higher dimensions.

A Computing F-symbols of the gapped boundary/interface
In the main text, we have reviewed that the representations of the super Frobenius algebra forms a super-fusion category. We have discussed in detail how to recover the fusion rules of the resultant super-fusion category. Since fusion remains associative, there is a corresponding set of F symbols for the representation category, which can be obtained using data of the parent phase. This has also be discussed in detail in the case of a bosonic condensates in [ERB14], but only mentioned briefly in [ALW19] in the case of a fermionic condensate. Here we would like to present the details in a simple example for illustration purpose. The tricky part is to fix various sign conventions and keeping track of spin structures in constructing fusion basis in the representation category. In the following, we will make the process completely explicit in a simple but non-trivial example.

A.1 Illustration via SU (2) 6
The basic topological data of SU (2) 6 is shown in table 2. The anyon 6 in su(2) 6 is a simple current fermion with quantum dimension d 6 = 1. The monodromy [BGH + 17] of each anyon with this fermion defines a Z 2 grading on su(2) 6 , thus dividing the bulk anyons into Neveu-Schwartz sector (anyons 0, 2, 4, 6) and Ramond sector (anyons 1, 3, 5). The fusion of sector a with the fermion is simply given by a ⊗ 6 = 6 − a. After condensing the fermion 6, anyon a and 6 − a are identified as a single object X a = {a + (6 − a)} in the gapped interface. The spin structure of an interface object is directly inherited from the parent, since a and 6 − a have the same spin structure in this case. The list of interface objects is given in table 3. Note that the anyon 3 becomes a q-type object X 3 in the interface, because 3 is invariant under the action of the fermion 6 ⊗ 3 = 3.  Table 2: su(2) 6 topological sectors, quantum dimensions, and topological spins. The label 2j gives the SU (2) spins of the corresponding sector.
The folling notation is used to denote the interface excitations after condensing the simple current fermion 6 in su(2) 6 . The phase λ ∈ U (1) which will appear later is defined as the fermion line with two end points as shown in figure 8. For su (2) 6 the fermion f is the sector 6. The λ can not be canceled from the solution of super Pentagon equations by a gauge transformation. Indeed, it can be shown that λ is purely imaginary [ALW19] and therefore λ = ±i. We use the following convention to solve the super Pentagon equation in child phase. If the fusion channel X ⊗ Y → Z is even, then we identify the channel X ⊗ Y → Z with the parent fusion channelX ⊗Ỹ →Z whereX,Ỹ ,Z are bulk representatives of X, Y, Z respectively. If the fusion channel X ⊗ Y → Z is odd, then we identify this odd fusion channel with an even fusion channel where a fermion line segment ends at theZ leg, the interval between the vertex and the ending point is changed to fZ. We choose 0, 1, 2, 3 as the bulk representatives of X 0 , X 1 , X 2 , X 3 respectively, therefore we have a "vertex lifting rule" with respect to the above convention: • For any even channel vertex, replace leg label X i by bulk anyon i. As an example we look at the even channel in X 2 ⊗ X 2 → C 1|1 X 2 , this vertex is replaced by 2 ⊗ 2 → 2 vertex in the bulk.
• For any odd channel vertex, remove the oddness by attaching a fermion line and move downward according to figure 9. Here we list all the odd channel lifting rules: To calculate all the F -symbols in the child phase, we first lift all the vertices to bulk vertices according to figure 9, then we do F -moves and R-moves in the bulk, and finally we translate the bulk vertices back to child theory using figure 9 again. Consider the following child phase F -move, the legs α, β, γ, δ, ξ, η ∈ {X 0 , X 1 , X 2 , X 3 } label an interface excitation, and the vertices µ, ν, µ , ν ∈ Z 2 label an even/odd channel. Without loss of generality, we study the case where µ = ν = 0, namely both vertices in the LHS of A.9 are even. We provide a step-by-step calculation of the F -symbols in this case, calculation in other cases can be carried out with similar steps, and are therefore omitted in this paper. To perform an interface F -move, vertices are first lift to the bulk, followed by a bulk F -move. When η ∈ {4, 5, 6}, the bulk fusion tree can not be transformed to an interface fusion tree at first sight, because neither the first or the second rule in figure 9 can be directly applied to the two vertices. However, the following diagram reveals the relation between the bulk and interface fusion trees.
where in the first step we apply the vertex lifting rule in figure 9, in the second and third step we perform a bulk F -move, and in the last step we replace the fermion lines by λ according to figure 8. For simplicity we sometimes label X 6−η by 6 − η, this notation is unambiguous. It follows that Now we have found all the interface F -symbols in the case µ = ν = 0.
With similar steps we can solve for all the F -symbols in the interface, the detailed calculation is omitted. A part of the solutions are shown in figure 10, and the complete list can be found in the attached file. The action of a generic Abelian Chern Simons theory is characterized by a K matrix , · · · , N }. It is well known that the Chern-Simons theory with K matrix is the low energy effective field theory of the Toric code topological order. Excitations of the topological order is characterzed by their charge vectors l i ∈ Z, i ∈ {1, · · · , N } [WW15]. The fermionic excitation f , is characterized by the class of charge vectors To form a gapped boundary, where f is condensed, we have to pair up the fermionic excitation with a free fermion. This can be achieved by introducing extra fields by extending the K matrix as follows The extended block in the K-matrix carries odd integers in the diagonal components. It is well known that the resultant Chern-Simons theory is a "spin-TQFT" [BM05], that is sensitive to the spin-structure of the manifold. Indeed physically, it corresponds to introducing free fermions, whose charge vectors are given by the classes of vectors as follows: One could easily check that the exchange statistics of two charge vectors l 1,2 ∈ {l ψ } obtained by πl 1 K −1 l 2 gives π.
In the presence of boundaries, one could obtain a boundary action from e.g. gauge fixing a i 0 = 0 and solving for the flatness constraint that follows from the eom of a i 0 which gives Substituting into the action (B.1) produces a 1+1 d effective action on the bosons φ i . For completeness, it takes the following form where it is discussed in many places [Wen95] that V ij determines the velocity of the edge modes that is not determined universally by the topological bulk, but it is controlled by the specific materials that realize the theory. Regardless of the precise values of V ij , as long as the matrix is not singular, the φ i and φ j can be related to free left and right moving modes in the edge. In the theory defined by the K-matrix (B.4) i.e. There are two pairs of free (anti)-chiral bosons in this edge theory.
Now we would like to consider a disk geometry as in figure 11, and we would like to understand the kind of excitations that are localized at the junction. One way to go about this problem is to consider regulating the junction by considering a small segment ending at the two gapped boundaries. This is also illustrated in figure 11.
One then quantizes the edge theory on the little segment that ends on the gapped boundary.
It is already noted in [SLH19, DFLN08] that a segment ending on a gapped boundary satisfies boundary conditions depending on the charge vectors of anyons condensing at the boundary [SLH19,DFLN08]. To be precise, the boundary conditions at a given boundary is where x 0 is the position of the end point of the segment, L A is a collection of charge vectors that condense at the gapped boundary, with mutual statistics l i .K −1 .l j = 0 for all l i,j ∈ L A .
The charge vectors that define the condensate for the electric, magnetic and fermionic gapped boundaries are given respectively by [Lev13,WW15,KS11] (B.10) We note that the choice of these charge vectors are subjected to some ambiguities. i.e. a, b, c, d can take either ±1 and they would still describe the condensation of the same topological charges at the boundary. This lead to freedom in determining boundary conditions in (B.9). As we are going to see -this ambiguity is precisely the freedom to insert Majorana modes at a junction. Now we put three distinct boundaries on a disk. As mentioned above, we analyse each junction by considering the open segment ending on two different adjacent boundaries. This is illustrated in figure 11. Let x be the parameter along the strings and takes value between 1 and l. Then the boundary conditions for three strings are given by Figure 11: Three different boundaries separated by three junctions on a disk. The bulk phase is the toric code order.
Where the upper index {Y, B, G} indicating the strings with color yellow, blue and green. It is also convenient to present these boundary conditions by "unfolding" the gluing conditions connecting a left and a right moving mode above and combine them into a "closed string" with either periodic or anti-periodic boundary conditions.
1 (l) Φ Here we are focusing on the oscillator parts of the expansion. In the case of a periodic boundary condition, the sum over n runs over integers. On the other hand, for anti-periodic boundary conditions, n ∈ Z + 1/2. These half-integer modes are related to the majorana mode (and its Virasoro descendent). The bottom line is therefore, that by making different choice of a, b, c, d in (B.10), one could change the boundary conditions of the bosons describing the junctions altering their moding, which is equivalent to adding or deleting Majorana modes at the given junction. However, we can only have altogether an even number of chiral bosons with antiperiodic boundary conditions as we take different combinations of these variables a, b, c, d = ±1. This is consistent with the fact that any physically realizable situation has an even number of Majorana modes.
In principle we can have more junctions on a disk and it is straight forward to generalize the result above to those cases. This reproduces the observation made in section 3.1.2 on the lattice model.

B.1 Entanglement of a strip region in a cylinder
In the previous subsection, we reviewed the Chern-Simons formulation of these gapped boundaries. In this section we take up the formulation to compute the entanglement of a strip region in a cylinder with e − f boundaries. These are direct generaliztions of the computations in [LSH19, SLH19].
We will first consider a strip region on a cylinder. The top and bottom boundaries are each characterized by some condensate, or Lagrangian algebra. This is illustrated in figure 12. l 1 r 1 l 2 r 2 Figure 12: Entanglement entropy of a strip R on a cylinder, cutting through the top and bottom boundaries. Each of these boundaries are gapped, and is characterized by some condensate.

B.1.1 Anti-periodic boundary condition
First we consider the Lagrangian algebras L e and L f characterizing the top and bottom boundaries respectively. For concreteness, we first consider taking a = c = d = −1. The boundary conditions for edge modes living at the entanglement cuts are thus given by Again it is convenient to combine the left and right moving modes and express it in terms of a chiral boson with anti-periodic boundary condition.
Hence the mode expansion is Exactly as in [SLH19], one can construct the Ishibashi state describing the gluing across the entanglement cut: where the Hamiltonian H i = L i 0 +L i 0 − 1/12 is inserted as a UV regularization with the cutoff scale . The normalization constant of this state is where η(q) θ 4 (q) and the θ 2 (q) below are the Dedekind η-function and Jacobi θ-functions respectively. Then the reduced density matrix ρ is obtained by tracing out the chiral or anti-chiral part of the density matrix N |0 0|. Therefore the trace of the n-th-power of the reduced density matrix is given by Hence the entanglement entropy of this state is given by Here the − ln √ 2 is the contributoin of the two ground states, indicating the trapped Majorana mode in the junction of e − f boundary.

B.1.2 Periodic boundary condition
Then we consider the same Lagrangian algebra with a = −c = −d = 1. The only difference from the previous calculation is that here we have periodic boundary condition and non trivial zero modes.
So we have the following mode exopansion The Ishibashi state across the entanglement cut is given by Following the same method, the entanglement entropy is S = 2 · πl 6 (B.29) As shown before the junction between e − f boundaries can be chosen to trap an unpaired Majorana mode or not. The ground states on a cylinder with different boundary conditions at the top and bottom could be understood as a blown up of the junction. Such an ambiguity in the trapped Majorana mode can be revealed by studying the entanglement entropy.

B.2 Entanglement of a cylindrical region in a cylinder
In this section we consider the entanglement of a cylindrical region embedded in the cylinder. This is illustrated in figure 13. Recall in [LSH19] that in caseas as such, the gluing condition defining the Ishibashi states is determined by the allowed anyon lines crossing the cut. In the configuration of the entanglement we have chosen, the anyon line allowed has to be a common condensed anyon shared by the top and bottom boundary. The only such anyon is the trivial anyon, and therefore, the Ishiashi state contains only the trivial sector. The entanglement entropy can be similarly computed, which is given by Here ln 2 is again the quantum dimension of the bulk toric code order, and there is a factor of two, following from independent contributions from the two entanglement cuts.
C Some topological data of D(S 3 ) and its gapped boundaries Their fusion rules are given in Table 4 The S-matrix of D(S 3 ) is given by There're four types of gapped boundaries for the D(S 3 ) bulk, labeled by the four subgroups of S 3 , or equivalently, by the four Lagrangian algebras A 1,2,3,4 shown in Table 5. The boundary excitations at one particular boundary form a fusion category, as represented by the diagonal Table 5: Summary of the distinct boundaries labeled by four different condensates A 1,2,3,4 , and the quantum dimension of defects/excitations localized between them. This is reproduced from [CCW17b]. The diagonal cells give the fusion category describing the boundary excitations of each type of boundary.
cells of Table 5. The off-diagonal cells give the number and quantum dimension of distinct defects located between two different boundaries.

D Advanced level -multiple non-Abelian fermionic condensation
In the discussion in the main text, we have focused on situations where the condensate only contains at most one fermionic anyon. We consider here more complicated examples where the condensate could contain multiple fermions. One reason for considering these examples is that (modular invariant) supersymmetric CFT with multiple super-charges can also be understood in terms of anyon condensates -with each super charge related to a species of condensed fermion. In these supersymmetric CFT's, one could allow each fermionic sector to carry independent spin structures. One would have naively expect that this structure should carry through. It doeswhen the condensed fermions are all simple currents with quantum dimension equals 1. We study multiple fermion condensations in SU (2) 10 , and also in D(D 4 ). We found that not all the cases have a well defined independent spin structures. The independent spin structures may appear "deformed" -as in the case of SU (2) 10 . It is known that Lagrangian algebra in the modular tensor category is in 1-1 correspondence with modular invariants. In D(D 4 ), we find that for every super Frobenius algebra we studied it corresponds to a "super" modular invariant (i.e. combinations of anyons that are invariant under S and T 2 ). However, the converse is no longer true. Some super modular invariants do not appear to correspond to super Frobenius algebra. There are also super modular invariants that appear to describe condensates that break fermion parity. It is not clear whether such modular invariants or strange condensates are physically relevant or realizable. Junctions between boundaries following from these strange condensates are also mysterious, if physical at all.
For completeness, we will give the list of these strange super modular invariants below.

D.1 SU (2) 10
The distinct topological sectors in the SU (2) 10 topological order is reviewed in the  In earlier works [ALW19], the theory is part of the series SU (2) 4k+2 for all integers k > 0 which contains a fermionic simple current (i.e with SU (2) spin 2j = 4k + 2 that can be condensed. In the case of SU (2) 10 , the fermionic simple current is 2j = 10.
The condensate of interest to us however, is A = 0 ⊕ 4 ⊕ 6 ⊕ 10, which has also been discussed in [WW17]. There, it is noted that this forms a Lagrangian algebra, and therefore all the topological sectors except A itself is confined, giving rise to a gapped boundary. Here, we look into the confined sectors more closely. Using the methods discussed in the main text, it is possible to work out all the modules of A. These results are summarized in the table below. Since 2j = 4 and 2j = 10 are both fermionic, one might wonder whether one can define independent spin structures.
The way we determine spin structures is based on the trick described in (4.14). We assign the following fermion parity to the condensates 2j 0 4 6 10 (σ 2j 4 , σ 2j 10 ) (1,1) (-1,1) (1,-1) (-1,-1) We obtain Here, we have substituted χ x 4 ,10 = 1 if x is in the R-sector wrt 4 and 10 respectively, and 0 in the corresponding NS-sector. While taking the (NS,NS) sector simply recovers the W matrix of A, and the (R,R) sector above produces the defect 3⊕7, the (N S, R) , and the (N S, N S)and (R, N S) produces a linear combinations of defects! In particular, the (R,NS) defect is a combination of A and 2 ⊕ 4 ⊕ 6 ⊕ 8, while (NS,R) a combination of 3 ⊕ 7 and 1 ⊕ 3 ⊕ 2 × 5 ⊕ 7 ⊕ 9. This explains why the spin-structure labels of these two defects are "in quotes", since they appear non-standard.
This appears to suggest that the defects 2 ⊕ 4 ⊕ 6 ⊕ 8 and 1 ⊕ 3 ⊕ 2 × 5 ⊕ 7 ⊕ 9 generate some symmetry transformation that does not simply form a Z 2 group. This should be an example of the algebraic symmetry discussed recently in [KLW + 20b].
The super modular invariants and super Lagrangian algebras thus found actually contain usual bosonic condensates such as A = abcekk. To this end we introduce the notion of proper fermion condensation, by proper we mean there's at least one fermion in the condensate.

Double Toric Code T C T C
The above super Lagrangian algebras and super modular invariants can be understood from sequential condensation of D(D 4 ). First we perform the simple boson condensation A = a + b in D(D 4 ), where b is another boson with quantum dimension 1. It turns out that the unconfined child theory of this boson condensation is the double toric code T C T C. There's a gapped interface between the bulk D(D 4 ) phase and child T C T C phase. Under the bulk-to-boundary condensation functor F : i → i ⊗ A, the bulk anyons are mapped to interface excitations as shown in the table(8). In the first part of the table, we see that the 8 simple current bosons in the bulk become 4 simple current bosons in the interface, these 4 bosons are unconfined and can enter the child phase. In the third part of the table, we see that the interface excitations "ip", "ls", "mt" and "ov" are confined in the sense of bosonic anyon condensation, because each has a lift to bulk anyons with different spin.
Another abelian topological order with 10 simple current bosons and 6 simple current fermions is the double Toric Code T C T C. In the 10 bosons, 4 are different from others and are known as the diagonal bosons, namely 11, eē, mm and ff .
Indeed, the S-matrix of the child phase can be calculated by means of [ERB14] and identified with the S-matrix of the double Toric Code T C T C. We omit the calculation and only show the object identification of the child phase and T C T C. First, the 4 special bosons in the two theories are identified: ab = 11, df = eē, ce = mm and gh = ff . Second, the remaining 6 bosons and 6 fermions in the two theories are identified through fusion rules. The full dictionary is listed in the last column of table(8). However, this identification is not unique due to the topological symmetries in T C T C. Similar analysis and identification with T C T C can be carried out in three other boson condensation A = a + c, A = a + d and A = a + h. The dictionaries are shown respectively in table(9), table(10) and table(11).

Classification of fermion condensation in D(D4)
The above found super modular invariants and super Lagrangian algebras of D(D 4 ) are the results of sequential condensation, and can be understood from anyon condensation in the child phase T C T C. With our knowledge of T C T C, the proper fermion condensation in D(D 4 ) can be regrouped according to fermion condensation in T C T C. In each case we can do the sequential condensation, namely we first arrive at stage T C T C and try condense the fermions in T C T C. Since T C T C is abelian, fermion condensation in T C T C is much easier than in the original D(D 4 ). i i ⊗ A T C T C particle a, c ac 11 b, e be mm d, g dg eē  Table 10: Identify child theory of A = a + d condensation with T C T C.
Fermion condensation in T C T C is classified according to the number of fermions in the condensate, and are divided into the following three collections.

b+b+b+f
There's only one fermion in the condensate in this case. For example, we can condense A = 11 + eē + mm + ff in T C T C according to table(8). In terms of the D(D 4 ) anyons this condensate of T C T C can be rewritten as A = ab + j + + k + + u + , and hence equivalent to a direct condensation of A = a + b + j + k + u in D(D 4 ). There're 12 such condensates, namely { abjku, abjnr, abknq, acikt, acimr, ackmp, adijs, adilq, adjlp, ahlmu, ahlnt, ahmns }.

b+f+f+f
There're three fermions in the condensate in this case. There're 4 such condensates, when rewritten using D(D 4 ) labels, they are { abqru, acprt, adpqs, ahstu }. i i ⊗ A T C T C particle a, h ah 11 b, g bg mm c, f cf eē d, d de ff l l + + l − 1ē + e1 m m + + m − 1m + m1 n n + + n − em + mē s s + + s − mf + fm t t + + t − ef + fē u u + + u − 1f + f1 Table 11: Identify child theory of A = a + h condensation with T C T C.

Super modular invariants in T C T C
We use a similar algorithm to generate all super modular invariants of gapped boundary in T C T C.
The super modular invariants with at least one fermion are: 11 + eē + mf + fm 11 + ef + mm + fē 11 + 1f + f1 + ff 11 + 1ē + f1 + fē ↔ 11 + e1 + 1f + ef 11 + 1f + m1 + mf ↔ 11 + f1 + 1m + fm 11 + ef + mē + fm ↔ 11 + fē + em + mf It is found that all super modular invariants of T C T C with fermion present are of the form b+b+f+f. As the second step of sequential condensation in D(D 4 ), it is weird that the above listed b+b+b+f and b+f+f+f cases are NOT super modular invariants in T C T C, although they're super modular invariants in the parent theory D(D 4 ).
Strange as it may seem, we observe that a special linear combination of super modular invariants of form b+b+b+f (or b+f+f+f ) is indeed a super modular invariant of T C T C. For example χ = χ a + χ b + χ j + χ k + χ u is a super modular invariant of D(D 4 ), after condensing A = a + b these anyons become a, b → 11, j → 1ē, k → 1m, u → 1f as shown in table 8. It can be easily checked that χ 1 = χ 11 + χ 1ē + χ 1m + χ 1f is not a super modular invariant of T C T C. However, due to the topological symmetry in T C T C there's another identification a, b → 11, j → e1, k → m1, u → f1. Under this identification the super modular invariant χ descends to χ 2 = χ 11 + χ e1 + χ m1 + χ f1 . Although neither χ 1 nor χ 2 is a super modular invariant in the child phase T C T C, the arithmetic average of the two if a super modular invariant in T C T C: 1 2 (χ 1 + χ 2 ) = χ 11 + 1 2 (χ 1ē + χ e1 ) + 1 2 (χ 1m + χ m1 ) + 1 2 (χ 1f + χ f1 ) is invariant under S-transformation.
For the above reason, we have in total 16 weird condensates in D(D 4 ), which have form b+b+b+f or b+f+f+f in the child pahse T C T C from the point of view of sequential condensation, shown in the following table.  If we take these super modular invariants as fermion condensations, the resulting phases are all trivial with fermion parity broken. However there is still a Z 2 -symmetry generated by a combination of fermion and boson or two fermions in the condensate. All these examples have similiar structures so in the following table we select one example to illustrate.

Condensate Modules
Z 2 parity odd (1) Z 2 parity odd (2) {a, b, k, n, q} {l, o, s, v} In the above table we list the anyons with odd parity for two Z 2 symmetries in the 3 rd and 4 th columun. These Z 2 symmetries are generated by the combination of {k, q} and {n, q} respectively. The fusion rule of these modules is given by