Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases
GGapped boundaries and string-like excitations in(3+1)d gauge models of topological phases
Alex Bullivant, (cid:50)
Clement Delcamp (cid:68) , (cid:55) (cid:50) Department of Pure Mathematics, University of Leeds, Leeds, LS2 9JT, UK (cid:68)
Max-Planck-Institut f¨ur QuantenoptikHans-Kopfermann-Str. 1, 85748 Garching, Germany (cid:55)
Munich Center for Quantum Science and Technology (MCQST)Schellingstr. 4, D-80799 M¨unchen [email protected] , [email protected] We study lattice Hamiltonian realisations of (3+1)d Dijkgraaf-Witten theory with gapped bound-aries. In addition to the bulk loop-like excitations, the Hamiltonian yields bulk string-like excitationsthat terminate at point-like gapped boundary excitations. Using a tube algebra approach, we classifysuch excitations and derive the corresponding representation theory. Via a dimensional reductionargument, we relate this tube algebra to that describing (2+1)d boundary point-like excitations atinterfaces between two gapped boundaries. Such point-like excitations are well known to be encodedinto a bicategory of module categories over the input fusion category. Exploiting this correspondence,we define a bicategory that encodes the string-like excitations ending at gapped boundaries, showingthat it is a sub-bicategory of the centre of the input bicategory of group-graded 2-vector spaces. Inthe process, we explain how gapped boundaries in (3+1)d can be labelled by so-called pseudo-algebraobjects over this input bicategory. a r X i v : . [ c ond - m a t . s t r- e l ] J un ontents j -symbols 315.5 Canonical basis for (2+1)d boundary excited states 34 Vec α G Vec α G Vec α G Vec α G -module categories 506.7 Bicategory of boundary excitations in (2+1)d gauge models 516.8 Pseudo-algebra objects and gapped boundaries in (3+1)d gauge models 546.9 Bicategory of boundary excitations in (3+1)d gauge models 58 A.1 Proof of the orthogonality relations (5.12) 64A.2 Proof of the invariance property (5.28) 64A.3 Proof of the defining relation of the 6 j -symbols 66A.4 Proof of the pentagon identity 68 ∼ i ∼ Canonical basis for boundary excitations in (2+1)d 71
B.1 Proof of the canonical algebra product (5.50) 71B.2 Ground state projector on the annulus 71B.3 Proof of the diagonalisation property (5.69) 72 ∼ ∼ ECTION 1
Introduction
A prominent class of gapped quantum phases of matter are given by so-called topological phasesof matter . Such phases can be defined as equivalence classes of gapped quantum models whose low-energy effective descriptions realise topological quantum field theories (TQFTs) [1]. In (2+1)d, sphericalfusion categories can be used to define a state-sum TQFT known as the
Turaev-Viro-Barrett-Westbury
TQFT [2, 3]. Given such data, one can define an exactly solvable Hamiltonian model on a closedmanifold, in a canonical manner, that describes non-chiral topological phases in (2+1)d [4–6]. Suchmodels support topological excitations referred to as anyons , which display exotic braiding and fusionstatistics. Topological excitations are typically described via the so-called
Drinfel’d center of theinput spherical fusion category [7]. For any spherical fusion category, the center construction definesa modular tensor category, which is widely accepted as being the right classification tool for anyonsin (2+1)d [8, 9].Given an open manifold, it is often possible to extend the lattice Hamiltonian to the boundary,while preserving the gap. Equivalence classes of such extensions define the notion of gapped bound-aries , which realise anomalous TQFTs. These are found to be described by indecomposable modulecategories over the input spherical category. Furthermore, boundary Hamiltonians yield point-likeexcitations that can be classified through the language of module category functors [10]. Domain wallsbetween distinct topological phases can be considered in a similar fashion. By iterating the procedure,it is possible to further extend such models to interfaces between different gapped boundaries. Thecorresponding zero-dimensional Hamiltonians yield point-like excitations in their own right. Thesedifferent settings have received a lot of attention in recent years within the topological order commu-nity [10–19], partly due to their application to the field of topological quantum computation [13, 20].Mathematically, these fit in the wider topic of defect
TQFTs [21–28].Despite tremendous progress in our understanding of (2+1)d topological models, a lot of questionsremain open regarding generalizations to higher dimensions. It is expected that topological modelsin (3+1)d should take as input a spherical fusion bicategory. Although the precise definition of suchnotion remains partly elusive, a compelling definition has been recently put forward by Douglas et al.in [29]. In this manuscript, the authors show that their definition encompasses a large class of four-dimensional state-sum invariants. Ultimately, we would like to derive properties of (3+1)d topologicalmodels within this general higher category theoretical framework, which is admittedly tantalizing butdifficult. In order to make progress in this direction, we decide to focus on so-called gauge modelsof topological phases , i.e. models that have a lattice gauge theory interpretation [8, 30–32]. Thesemodels are interesting for diverse reasons. Technically, they are particularly manageable allowing tocarry out computations in full detail, and they are easily definable in any dimensions. Physically, theyhappen to be extremely relevant in (3+1)d as they seem to encapsulate a large class of Bosonic modelsdisplaying topological order [33–36].In (2+1)d, topological gauge models are obtained by choosing as input the category of G -gradedvector spaces, with G a finite group and monoidal structure twisted by a cohomology class in H ( G, U(1)).The corresponding state-sum invariant is referred to as the Dijkgraaf-Witten invariant [37]. In thiscontext, (bulk) anyonic excitations are described in terms of the so-called twisted quantum double ofthe group, whose irreducible representations provide the simple objects of the Drinfel’d centre of thecategory of G -graded vector spaces [38, 39]. Gapped boundaries are found to be labelled by a simpleset of data, namely a subgroup of the input group and a 2-cochain that is compatible with the input3-cocycle [11, 40], and their excitations have been considered for instance in [13, 17, 28, 41, 42]. ∼ ∼ ore generally, given a closed ( d +1)-manifold, the input data of Dijkgraaf-Witten theory is afinite group G and a cohomology class [ ω ] ∈ H d +1 ( G, U(1)). It is always possible to define a latticeHamiltonian realization of the theory on a d -dimensional hypersurface Σ, such that the ground statesubspace of the model is provided by the image of the partition function assigned to the cobordismΣ × [0 , twisted quantumtriple of the group, which is a natural extension of the twisted quantum double. Although a generaltheory of gapped boundaries in (3+1)d is still lacking, examples have already been proposed in thecase of topological gauge models [12]. These are labelled by a set of data akin to (2+1)d, namely asubgroup of the input group and a 3-cochain compatible with the input 4-cocycle. The main objectiveof our manuscript is to study excitations for such gapped boundaries in (3+1)d.In order to reveal the algebraic structure underlying the bulk excitations in arbitrary spatialdimension, several strategies exist. Our focus is on the so-called tube algebra approach [5, 32, 41, 43–47], which is a generalization of Ocneanu’s tube algebra [48, 49]. In general, the ‘tube’ refers tothe manifold ∂ Σ × [0 , ∂ Σ is the boundary left by removing a regular neighbourhood ofthe excitation in question, and the ‘algebra’ to an algebraic extension of the gluing operation ( ∂ Σ × [0 , ∪ ∂ Σ ( ∂ Σ × [0 , (cid:39) ( ∂ Σ × [0 , S × [0 ,
1] and T × [0 , ∂ Σ [32]. This strategy has been extensively applied to general two-dimensionalmodels, and more recently to gauge and higher gauge models in three dimensions [32, 47].The tube algebra approach can be adapted in order to study excitations on defects and gappedboundaries, and has been employed in some specific cases in [10, 41, 42, 50]. In this context, thetube possesses two kinds of boundary: a physical gapped boundary that corresponds to the one ofthe spatial manifold, and a boundary obtained by removing a local neighbourhood of an excitationincident on the boundary of the spatial manifold. Although, the method is very general and could beused to study any pattern of boundary excitations in (3+1)d, we shall focus on a specific configuration,namely bulk string-like excitations that terminate at point-like gapped boundary excitations. Thereare several motivations to consider these specific excitations. The first one is that, due to the topologyof the problem, we can relate the corresponding tube algebra to the one relevant to the study ofpoint-like excitations at the zero-dimensional interface of two gapped boundaries in (2+1)d. Thisis a generalization of what happens in the bulk, where upon dimensional reduction, bulk loop-likeexcitations can be treated as point-like anyons [51, 52]. In [32], this mechanism was made precisein terms of so-called lifted models, where we showed that higher-dimensional tube algebras could berecast in terms of lower-dimensional analogues using the language of loop groupoids. We generalizethese techniques in this manuscript by introducing the notion of relative groupoid algebras , which weuse to unify both the (2+1)d and (3+1)d tube algebras.Although this correspondence between two seemingly very different types of excitations is interest-ing per se, it turns out to be a precious technical tool. Indeed, since it allows us to recast the (3+1)dtube algebra as a (2+1)d one, we can use the (2+1)d scenario, which is easier to visualise and intuit,as a guideline for the more complex (3+1)d case. Using this framework, we derive the irreduciblerepresentations of the (3+1)d tube algebra, which classify the elementary string-like excitations whoseendpoints lie on gapped boundaries. We further define a notion of tensor product that encodes theconcatenation of these excitations, and compute the Clebsch-Gordan series compatible with this tensorproduct. Moreover, we find the 6j-symbols that ensure the quasi-coassociativity of this tensor product. ∼ ∼ ll these mathematical notions can then be put to use in order to define canonical bases of groundstates or excited states in the presence of gapped boundaries.The second reason we decide to focus on such string-like excitations terminating at boundariespertains to category theory. The same way the relevant category theoretical data to describe gaugemodels in (2+1)d is the category of G -graded vector spaces, the one relevant to describe (3+1)d gaugemodels is the bicategory of G -graded 2-vector spaces . In a recent work [53], Kong et al. applied thegeneralised centre construction to this bicategory and demonstrated that the result was given by thebicategory of module categories over the multi-fusion category of loop-groupoid-graded vector spaces.This is a categorification of the well-know result that the centre of the category of group-graded vectorspaces can be described as the category of modules for the loop-groupoid algebra [54]. The latterrelation can be appreciated from the point of view of the tube algebra approach, which we use toargue that the centre of the bicategory of G -graded 2-vector spaces describes string-like excitationswhose endpoints terminate on boundaries of the spatial manifold.In order to prove this statement, we construct explicitly the bicategory of module categoriesover the multi-fusion category of groupoid-graded vector spaces. To do so, we rely on the familiarcorrespondence between indecomposable module categories and category of module over algebra objects [55–57]. When applied to the group treated as a one-object groupoid, this provides a description for(2+1)d point-like excitations at the interface between two gapped boundaries. When applied to theloop-groupoid of the group, we demonstrate that it describes string-like excitations terminating onspatial boundaries, which string-like excitations ending at gapped boundary point-like excitations is asubclass of. Organisation of the paper
In sec. 2 we review the construction of the lattice Hamiltonian realization of Dijkgraf-Witten theoryin any spatial dimension. We then describe an extension of the Hamiltonian model to introducegapped boundary conditions. In the subsequent discussion, we apply the tube algebra approach topoint-like excitations at the interface of two one-dimensional gapped boundaries in sec. 3. In sec. 4,we consider string-like bulk excitations that terminate at point-like boundary excitations and applythe tube algebra approach to this scenario. We also introduce in this section the notion of relativegroupoid algebra that unifies the (2+1)d and (3+1)d computations. The representation theory ofthe tube algebras is presented in full detail in sec. 5. Finally, the category theoretical structurescapturing the properties of boundary excitations in (2+1)d and (3+1)d are developed in sec. 6. Thecorrespondence with the centre construction of the bicategory of group-graded 2-vector spaces is alsoestablished in this section. ∼ ∼ ECTION 2
Dijkgraaf-Witten Hamiltonian Model
In this section, we first review the definition of the Dijkgraaf-Witten theory and the construction of itsHamiltonian realisation. We then generalise the construction to include gapped boundaries.
The input for the ( d +1)-dimensional Dijkgraaf-Witten theory is given by a pair ( G, [ ω ]) where G isa finite group and [ ω ] ∈ H d +1 ( G, U(1)) is a ( d +1)-cohomology class. Given a closed manifold, thistheory can be conveniently expressed as a sigma model with target space the classifying space BG of the group G . In order to extend the definition of the partition function to open manifolds, it isnecessary to endow the manifold with a triangulation, in which case the partition function is obtainedby summing over G -labellings of the 1-simplices that satisfy compatibility constraints. Ultimately, weare interested in lattice Hamiltonian realisations of such theory, for which we need the expression ofthe partition function that the Dijkgraaf-Witten theory assigns to a special class of open manifoldsreferred to as pinched interval cobordisms . We shall directly define the partition function for thisspecial class of manifolds. Details regarding more basic aspects of this theory can be found in [32, 37].Let Ξ be a compact, oriented d -manifold with a possibly non-empty boundary. We define the pinched interval cobordism Ξ × p I over Ξ as the quotient manifoldΞ × p I ≡ Ξ × I / ∼ , (2.1)where I ≡ [0 ,
1] denotes the unit interval, and the equivalence relation ∼ is such that ( x, i ) ∼ ( x, i (cid:48) ),for all ( x, i ) , ( x, i (cid:48) ) ∈ ∂ Ξ × I . By definition, we have ∂ (Ξ × p I ) = Ξ ∪ ∂ Ξ Ξ and Ξ ∩ Ξ = ∂ Ξ, where Ξ isthe manifold Ξ with reversed orientation. In contrast, the boundary of the interval cobordism Ξ × I over Ξ reads ∂ (Ξ × I ) = Ξ ∪ Ξ ∪ ( ∂ Ξ × I ). To illustrate this distinction, we can consider the followingsimple examples: [0 , × p [0 ,
1] = , [0 , × [0 ,
1] = . Naturally, if ∂ Ξ = ∅ , then we have the identification Ξ × p I = Ξ × I .In order to define the Dijkgraaf-Witten partition function, we shall further require our pinchedinterval (spacetime) manifold be equipped with a choice of triangulation , i.e. a ∆-complex whose geo-metric realisation is homeomorphic to the manifold. We shall further assume that every triangulationhas a chosen total ordering of its 0-simplices (vertices), referred to as a branching structure . A choiceof branching structure for a triangulation naturally encodes the structure of a directed graph on thecorresponding one-skeleton. By convention, we choose the 1-simplices (edges) to be directed from thelowest ordered vertex to the highest ordered vertex. Given a compact, oriented d -manifold Ξ, we notatea triangulation of the pinched interval cobordism Ξ × p I by (cid:52) (cid:48) Ξ (cid:52) , such that ∂ ( (cid:52) (cid:48) Ξ (cid:52) ) = Ξ (cid:52) ∪ ∂ Ξ (cid:52)(cid:48) Ξ (cid:52) (cid:48) ,where Ξ (cid:52) and Ξ (cid:52) (cid:48) denote two possibly different triangulations of Ξ. Let us remark that by definition,we have ∂ (Ξ (cid:52) ) = ∂ (Ξ (cid:52) (cid:48) ).Let Ξ × p I be a ( d +1)-dimensional pinched interval cobordism endowed with a triangulation (cid:52) (cid:48) Ξ (cid:52) .We define a G -colouring of (cid:52) (cid:48) Ξ (cid:52) as an assignment of group elements g v i v j ∈ G to every oriented 1-simplex ( v i v j ) ⊂ (cid:52) (cid:48) Ξ (cid:52) , with v i < v j , such that for every 2-simplex ( v i v j v k ) ⊂ (cid:52) (cid:48) Ξ (cid:52) , with v i < Here U(1) denotes the circle group as a G -module with action (cid:46) : G × U(1) → U(1) given by g (cid:46) u = u for all g ∈ G and u ∈ U(1). ∼ ∼ j < v k , the condition g v i v j g v j v k = g v i v k is satisfied. The set of G -colourings on (cid:52) (cid:48) Ξ (cid:52) is notated byCol( (cid:52) (cid:48) Ξ (cid:52) , G ). Given a G -colouring g ∈ Col( (cid:52) (cid:48) Ξ (cid:52) , G ) and an n -simplex (cid:52) ( n ) = ( v v . . . v n ) ⊂ (cid:52) (cid:48) Ξ (cid:52) ,we denote by g [ v v . . . v n ] ≡ ( g v v , . . . , g v n − v n ) ∈ G n , the n group elements specifying the restrictionof g to a G -colouring of ( v v . . . v n ). Using this notation, we further define the evaluation of a( d +1)-cocycle ω ∈ Z d +1 ( G, U(1)) on a G -colouring g ∈ Col( (cid:52) (cid:48) Ξ (cid:52) , G ) restricted to a ( d +1)-simplex( v . . . v d +1 ) ⊂ (cid:52) (cid:48) Ξ (cid:52) as ω ( g [ v . . . v d +1 ]) ≡ ω ( g v v , . . . , g v d v d +1 ) . Equipped with the above, let us now define the partition function that the ( d +1)-dimensional Dijkgraaf-Witten theory assigns to a given pinched interval cobordism. Letting Ξ be a compact, oriented d -manifold and (cid:52) (cid:48) Ξ (cid:52) a triangulation of Ξ × p I , the partition function defines a linear operator Z Gω [ (cid:52) (cid:48) Ξ (cid:52) ] : H Gω [Ξ (cid:52) ] → H Gω [Ξ (cid:52) (cid:48) ] , where the Hilbert spaces H Gω [Ξ (cid:52) ] and H Gω [Ξ (cid:52) (cid:48) ] are defined according to H Gω [Ξ ∗ ] ≡ (cid:79) (cid:52) (1) ⊂ Ξ ∗ C [ G ] . (2.2)In the equation above, the tensor product is over all 1-simplices (cid:52) (1) in the corresponding triangulation,and C [ G ] denotes the Hilbert space spanned by {| g (cid:105)} ∀ g ∈ G with inner product (cid:104) g | h (cid:105) = δ g,h , ∀ g, h ∈ G .Explicitly, the linear operator Z Gω [ (cid:52) (cid:48) Ξ (cid:52) ] reads Z Gω [ (cid:52) (cid:48) Ξ (cid:52) ] ≡ | G | (cid:52)(cid:48) Ξ (cid:52) ) (cid:88) g ∈ Col( (cid:52)(cid:48) Ξ (cid:52) ,G ) (cid:89) (cid:52) ( d +1) ⊂ (cid:52)(cid:48) Ξ (cid:52) ω ( g [ (cid:52) ( d +1) ]) (cid:15) ( (cid:52) ( d +1) ) (cid:79) (cid:52) (1) ⊂ Ξ (cid:52)(cid:48) | g [ (cid:52) (1) ] (cid:105) (cid:79) (cid:52) (1) ⊂ Ξ (cid:52) (cid:104) g [ (cid:52) (1) ] | , where (cid:52) (cid:48) Ξ (cid:52) ) := | (cid:52) (cid:48) Ξ (cid:52) (0) | − | ∂ (cid:52) (cid:48) Ξ (cid:52) (0) | − | ∂ Ξ (0) (cid:52) | and (cid:15) ( (cid:52) ( d +1) ) ∈ ± d +1)-simplex (cid:52) ( d +1) ⊂ (cid:52) (cid:48) Ξ (cid:52) .Before concluding this section, let us describe some of the salient features of the partition functionabove. Firstly, given a pinched interval cobordism Ξ × p I and two choices of triangulation (cid:52) (cid:48) Ξ (cid:52) and (cid:52) (cid:48) ˜Ξ (cid:52) such that ∂ ( (cid:52) (cid:48) Ξ (cid:52) ) = ∂ ( (cid:52) (cid:48) ˜Ξ (cid:52) ), we find the operators Z Gω [ (cid:52) (cid:48) Ξ (cid:52) ] = Z Gω [ (cid:52) (cid:48) ˜Ξ (cid:52) ] to be equal.This property follows directly from the ( d +1)-cocycle condition satisfied by ω , i.e. d ( d +1) ω = 1. Thisimplies that the operator Z Gω is boundary relative triangulation independent , i.e. it remains invariantunder retriangulation of the interior int( (cid:52) (cid:48) Ξ (cid:52) ) := (cid:52) (cid:48) Ξ (cid:52) \ ∂ ( (cid:52) (cid:48) Ξ (cid:52) ) of (cid:52) (cid:48) Ξ (cid:52) but does depend on achoice of boundary triangulation. Using this boundary relative triangulaton independence, we findthe crucial relation Z Gω [ (cid:52) (cid:48)(cid:48) Ξ (cid:52) (cid:48) ] Z Gω [ (cid:52) (cid:48) Ξ (cid:52) ] = Z Gω [ (cid:52) (cid:48)(cid:48) Ξ (cid:52) ] . Secondly, given a d -manifold Σ equipped with a triangulation Σ (cid:52) and Ξ (cid:52) a subcomplex of int(Σ (cid:52) ),there is a natural action of Z Gω [ (cid:52) (cid:48) Ξ (cid:52) ] on H Gω [Σ (cid:52) ] such that Z Gω [ (cid:52) (cid:48) Ξ (cid:52) ] : H Gω [Σ (cid:52) ] → H Gω [Σ (cid:52) (cid:48) ]where Σ (cid:52) (cid:48) is a triangulation of Σ induced from Σ (cid:52) by replacing the subcomplex Ξ (cid:52) ⊂ int(Σ (cid:52) ) withΞ (cid:52) (cid:48) , while keeping the remaining triangulation the same. On the subspace V Gω [Σ (cid:52) ] := Im Z Gω [ (cid:52) (cid:48) Ξ (cid:52) ] ⊂ H Gω [Ξ (cid:52) ] , (2.3) ∼ ∼ he operator Z Gω [ (cid:52) (cid:48) Ξ (cid:52) ] further defines a unitary isomorphism Z Gω [ (cid:52) (cid:48) Ξ (cid:52) ] : V Gω [Σ (cid:52) ] ∼ −→ V Gω [Σ (cid:52) (cid:48) ] . (2.4)This follows directly from the boundary relative triangulation independence of Z Gω as well as theHermicity condition Z Gω [ (cid:52) (cid:48) Ξ (cid:52) ] † = Z Gω [ (cid:52) Ξ (cid:52) (cid:48) ] . (2.5) Let us now construct an exactly solvable model that is the lattice Hamiltonian realisation of Dijkgraaf-Witten theory in d spatial dimensions [30–32]. The input of the model is a pair ( G, ω ) where G isa finite group and ω a normalised representative of a cohomology class in H d +1 ( G, U(1)). Given anoriented (possibly open) d -manifold Σ representing the spatial manifold of the theory, and a choice oftriangulation Σ (cid:52) , the microscopic Hilbert space of the model is given by H Gω [Σ (cid:52) ] ≡ (cid:79) (cid:52) (1) ⊂ Σ (cid:52) C [ G ] , as in (2.2). A natural choice of basis for H Gω [Σ (cid:52) ] is given by an assignment of g v i v j ∈ G for eachoriented edge ( v i v j ) ⊂ Σ (cid:52) defined by the vertices v i < v j . Henceforth, we shall refer to such states as graph-states .The bulk Hamiltonian is obtained as a sum of mutually commuting projectors that come in twofamilies. Firstly, to every 2-simplex ( v v v ) ⊂ int(Σ (cid:52) ) of the interior of Σ (cid:52) , we assign an operator B ( v v v ) that is defined via the following action on a graph-state | g (cid:105) ∈ H Gω [Σ (cid:52) ]: B ( v v v ) : | g (cid:105) (cid:55)→ δ g v v g v v , g v v | g (cid:105) . This definition can be extended linearly to an operator on any state | ψ (cid:105) ∈ H Gω [Σ (cid:52) ]. Secondly, to every0-simplex ( v ) ⊂ int(Σ (cid:52) ), we assign an operator A ( v ) which acts on a local neighbourhood of ( v )defined as the subcomplex Ξ v := cl ◦ st( v ) ⊂ Σ (cid:52) . Here st( − ) and cl( − ) are the star and the closure operations, respectively, so that Ξ v corresponds to the smallest subcomplex of Σ (cid:52) that include allthe simplices of which ( v ) is a subsimplex. The definition of A ( v ) requires the triangulated pinchedinterval cobordism Ξ v Ξ Ξ v defined as Ξ v Ξ Ξ v := ( v (cid:48) ) (cid:116) j cl ◦ st( v ) , where − (cid:116) j − denotes the join operation. Given two simplices (cid:52) ( n ) ≡ ( v v . . . v n ) and (cid:52) ( n (cid:48) ) ≡ ( v n +1 v n +2 . . . v n + n (cid:48) +1 ), the join operation creates the new simplex (cid:52) ( n ) (cid:116) j (cid:52) ( n (cid:48) ) ≡ ( v v . . . v n + n (cid:48) +1 ).In the definition above, ( v (cid:48) ) refers to an auxiliary vertex such that v < v (cid:48) < v , and which follows theordering of ( v ) with respect to the other vertices in Σ (cid:52) . For the sake of concreteness, we illustratethese various definitions with the following two-dimensional example:Σ (cid:52) = and (0 (cid:48) ) (cid:116) j cl ◦ st(0) = (0 (cid:48) ) (cid:116) j 0 = (cid:48) . ∼ ∼ inally, given a state | ψ (cid:105) ∈ H Gω [Σ (cid:52) ], the action of the operator A ( v ) is defined via A ( v ) : | ψ (cid:105) (cid:55)→ Z Gω [( v (cid:48) ) (cid:116) j cl ◦ st( v )] | ψ (cid:105) . (2.6)For instance, in (3+1)d the action of the operator A (4) on a vertex (4) shared by four 3-simplicesexplicitly reads A (4) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:43) = Z Gπ (cid:34) (cid:48) (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:43) = 1 | G | (cid:88) k ∈ G π ( g g , g , g , g (cid:48) ) π ( g , g , g g , g (cid:48) ) π ( g , g , g , g (cid:48) ) π ( g , g g , g , g (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:48) (cid:43) , where π ∈ Z ( G, U(1)). The lattice Hamiltonian is finally obtained as H Gω [Σ (cid:52) ] bulk = − (cid:88) (cid:52) (2) ⊂ int(Σ (cid:52) ) B (cid:52) (2) − (cid:88) (cid:52) (0) ⊂ int(Σ (cid:52) ) A (cid:52) (0) , (2.7)where the sums run over all the 2-simplices and 0-simplices in the interior of Σ (cid:52) , respectively. Itfollows from the definitions and the boundary relative triangulation independence that the operators { A (cid:52) (0) , B (cid:52) (2) } ∀ (cid:52) (0) , (cid:52) (2) ⊂ int(Σ (cid:52) ) satisfy the algebra A ( v i ) A ( v i ) = A ( v i ) , A ( v i ) A ( v j ) = A ( v j ) A ( v i ) , B ( v j v k v l ) B ( v j v k v j ) = B ( v j v k v l ) , B ( v j v k v l ) B ( v (cid:48) j v (cid:48) k v (cid:48) l ) = B ( v j (cid:48) v k (cid:48) v l (cid:48) ) B ( v j v k v l ) , A ( v i ) B ( v j v k v l ) = B ( v j v k v l ) A ( v i ) , for all ( v i ) , ( v i (cid:48) ) , ( v j v k v l ) , ( v (cid:48) j v (cid:48) k v (cid:48) l ) ⊂ Σ (cid:52) . All the operators are mutually commuting projectors andthe Hamiltonian is exactly solvable. It follows that the ground state projector P bulkΣ (cid:52) simply reads P bulkΣ (cid:52) := (cid:89) (cid:52) (0) ⊂ int(Σ (cid:52) ) A (cid:52) (0) (cid:89) (cid:52) (2) ⊂ int(Σ (cid:52) ) B (cid:52) (2) . (2.8)Notice that the ordering in the product is superfluous by the commutativity of the operators. Fur-thermore it follows from inspection that P bulkΣ (cid:52) = Z Gω [ (cid:52) Σ (cid:52) ] , (2.9)such that the ground state subspace of H Gω [Σ (cid:52) ] bulk is given byIm P bulkΣ (cid:52) = Im Z Gω [ (cid:52) Σ (cid:52) ] ≡ V Gω [Σ (cid:52) ] , (2.10)with the last equality following from (2.3). This is the space spanned by linear superpositions | ψ (cid:105) of graph-states fulfilling the stabiliser constraints A (cid:52) (0) | ψ (cid:105) = | ψ (cid:105) and B (cid:52) (2) | ψ (cid:105) = | ψ (cid:105) at every (cid:52) (0) , (cid:52) (2) ⊂ int(Σ (cid:52) ). ∼ ∼ et us conclude this construction by making two observations. The first one is that we showed in(2.4) how given two triangulations Σ (cid:52) and Σ (cid:52) (cid:48) of Σ such that ∂ Σ (cid:52) = ∂ Σ (cid:48)(cid:52) , the subspaces V Gω [Σ (cid:52) ]and V Gω [Σ (cid:52) (cid:48) ] were unitarily isomorphic. This signifies that it is always possible to perform localchanges of the triangulation in the interior of Σ while remaining in the same gapped phase. This willturns out to be very useful when performing explicit computations. In particular, we shall often applyunitary isomorphisms obtained from pinched interval cobordisms describing so-called Pachner moves .The second observation is that the Hamiltonian operators do not mix ground states with differingboundary G -colourings, so that there exists a natural decomposition of the Hilbert space as V Gω [Σ (cid:52) ] = (cid:77) a ∈ Col( ∂ Σ (cid:52) ,G ) V Gω [Σ (cid:52) ] a (2.11)where V Gω [Σ (cid:52) ] a ⊆ V Gω [Σ (cid:52) ] denotes the subspace of states identified by the boundary colouring a ∈ Col( ∂ Σ (cid:52) , G ). More details regarding the construction up to that point can be found in [32]. Given an open d -dimensional surface Σ endowed with a triangulation Σ (cid:52) , we reviewed above howto define an exactly solvable model as the Hamiltonian realisation of Dijkgraaf-Witten theory whoseinput data is a finite group G and normalised ( d +1)-cocycle in H d +1 ( G, U(1)). The lattice Hamiltonian H Gω [Σ (cid:52) ] bulk was obtained as a sum of mutually commuting projectors that act on the interior of Σ (cid:52) .We would like to extend this Hamiltonian to ∂ Σ (cid:52) while preserving the gap of the system, givingrise to the notion of gapped boundaries. In order to do so, we shall first define a generalisation ofthe partition function introduced in sec. 2.1 for spacetime ( d +1)-manifolds presenting two types ofboundaries.Let us begin by introducing the notion of relative pinched interval cobordisms . Let Ξ be a compact,oriented, d -manifold with non-empty boundary and Ω ⊆ ∂ Ξ a choice of ( d − × Ωp I over Ξ with respect to Ω is definedas the quotient manifold Ξ × Ωp I ≡ Ξ × I / ∼ Ω , (2.12)where ∼ Ω is defined such that ( x, i ) ∼ Ω ( x, i (cid:48) ), for all ( x, i ) , ( x, i (cid:48) ) ∈ ( ∂ Ξ \ int(Ω)) × I . By definition, wehave ∂ (Ξ × Ωp I ) = Ξ ∪ Ω (Ω × p I ) ∪ ∂ Ξ Ξ and Ξ ∩ Ξ = ∂ Ξ \ int(Ω). To illustrate this definition we considerthe following simple examples:[0 , × p [0 ,
1] = , [0 , × Ωp [0 ,
1] = , with Ω ≡ ⊂ { , } = ∂ I . Henceforth, we shall utilise the convention that Ξ × Ωp I defines a cobordismΞ × Ωp I : Ξ → Ξ , (2.13)and refer to Ω × p I ⊂ ∂ (Ξ × Ωp I ) as a time-like boundary . A triangulation of Ξ × Ωp I can be constructedas follows: Let Ξ (cid:52) , Ξ (cid:52) (cid:48) be a pair of triangulations of Ξ such that Ω (cid:52) ⊂ ∂ Ξ (cid:52) and Ω (cid:52) (cid:48) ⊂ ∂ Ξ (cid:52) (cid:48) definetwo possibly different triangulations of Ω satisfying ∂ Ξ (cid:52) \ int(Ω (cid:52) ) = ∂ Ξ (cid:52) (cid:48) \ int(Ω (cid:52) (cid:48) ) . (2.14) ∼ ∼ onsidering a triangulation (cid:52) (cid:48) Ω (cid:52) of the time-like boundary Ω × p I , we define (cid:52) (cid:48) Ξ Ω (cid:52) as the triangula-tion of the relative pinched interval cobordism Ξ × Ωp I whose boundary reads Ξ (cid:52) ∪ Ω (cid:52) (cid:52) (cid:48) Ω (cid:52) ∪ ∂ Ξ (cid:52)(cid:48) Ξ (cid:52) (cid:48) .Given a triangulation (cid:52) (cid:48) Ξ Ω (cid:52) of Ξ × Ωp I , let us now define a generalisation of the ( d +1)-dimensionalDijkgraaf-Witten theory with input data ( G, ω ) such that the corresponding partition function eval-uated on (cid:52) (cid:48) Ξ Ω (cid:52) remains invariant under triangulation changes of both the interior of (cid:52) (cid:48) Ξ Ω (cid:52) andthe interior of the time-like boundary (cid:52) (cid:48) Ω (cid:52) . Let Ω = (cid:116) i Ω i be a decomposition of Ω into con-nected components Ω i , each with triangulations Ω (cid:52) ,i ⊂ ∂ Ξ (cid:52) and Ω (cid:52) (cid:48) ,i ⊂ ∂ Ξ (cid:52) (cid:48) . The generalisedtheory associates to each connected component Ω i a pair ( A i , φ i ), where A i ⊂ G is a subgroup and φ i ∈ C d ( A i , U(1)) a normalised group d -cochain such that d ( d ) φ i = ω − | A i . We refer to the data( A i , φ i ) as a choice of gapped boundary condition . We define a ( G, { A i } )-colouring g of (cid:52) (cid:48) Ξ Ω (cid:52) asa G -colouring such that g [ (cid:52) (cid:48) Ω (cid:52) ,i ] ∈ Col( (cid:52) (cid:48) Ω (cid:52) ,i , A i ). The set of ( G, { A i } )-colourings on (cid:52) (cid:48) Ξ Ω (cid:52) isdenoted by Col( (cid:52) (cid:48) Ξ Ω (cid:52) , G, { A i } ). Equipped with such choices, we define the generalised partitionfunction as follows: Z G, { A i } ω, { φ i } [ (cid:52) (cid:48) Ξ Ω (cid:52) ] = 1 | G | (cid:52)(cid:48) Ξ Ω (cid:52) ) (cid:81) i | A i | (cid:52)(cid:48) Ω (cid:52) ,i ) (2.15) (cid:88) g ∈ Col( (cid:52)(cid:48) Ξ Ω (cid:52) ,G, { A i } ) (cid:89) i (cid:18) (cid:89) (cid:52) ( d ) ⊂ (cid:52)(cid:48) Ω (cid:52) ,i φ i ( g [ (cid:52) ( d ) ]) (cid:15) ( (cid:52) ( d ) ) (cid:19) (cid:89) (cid:52) ( d +1) ⊂ (cid:52)(cid:48) Ξ Ω (cid:52) ω ( g [ (cid:52) ( d +1) ]) (cid:15) ( (cid:52) ( d +1) ) (cid:79) (cid:52) (1) ⊂ Ξ (cid:52)(cid:48) | g [ (cid:52) (1) ] (cid:105) (cid:79) (cid:52) (1) ⊂ Ξ (cid:52) (cid:104) g [ (cid:52) (1) ] | , where (cid:52) (cid:48) Ξ Ω (cid:52) ) := | int( (cid:52) (cid:48) Ξ (cid:52) ) (0) | + | int(Ξ (cid:52) (cid:48) ) (0) | + | int(Ξ (cid:52) ) (0) | and (cid:52) (cid:48) Ω (cid:52) ,i ) := | int( (cid:52) (cid:48) Ω (cid:52) ,i ) (0) | + | int(Ω (cid:52) (cid:48) ,i ) (0) | + | int(Ω (cid:52) ,i ) (0) | .As stated previously, the partition function remains invariant under retriangulation of the interiorof (cid:52) (cid:48) Ω (cid:52) as well as the interior of (cid:52) (cid:48) Ξ Ω (cid:52) . In this manner, the partition function Z G, { A i } ω, { φ i } [ (cid:52) Ξ Ω (cid:52) ]defines a projection operator and we associate to the triangulation Ξ (cid:52) the following Hilbert space:: V G, { A i } ω, { φ i } [Ξ (cid:52) ] := Im Z G, { A i } ω, { φ i } [ (cid:52) Ξ Ω (cid:52) ] . (2.16)Furthermore, akin to equations (2.4) and (2.5), the triangulation invariance properties of the partitionfunction together with the Hermicitiy condition Z G, { A i } ω, { φ i } [ (cid:52) (cid:48) Ξ Ω (cid:52) ] † = Z G, { A i } ω, { φ i } [ (cid:52) Ξ Ω (cid:52) (cid:48) ] (2.17)demonstrate that the operator Z G, { A i } ω, { φ i } [ (cid:52) (cid:48) Ξ Ω (cid:52) ] : V G, { A i } ω, { φ i } [Ξ (cid:52) ] ∼ −→ V G, { A i } ω, { φ i } [Ξ (cid:52) (cid:48) ] (2.18)defines a unitary isomorphism of Hilbert spaces. In sec. 2.2, we described the Hamiltonian realisation H Gω [Σ (cid:52) ] bulk of the Dijkgraaf-Witten theory in d spatial dimensions in the presence of open boundary conditions. Utilising the partition function(2.15) introduced in the previous section, we shall now define an extension of the Hamiltonian modelto include gapped boundary conditions [11, 12]. In sec. 6, we shall revisit gapped boundary conditions from a category theoretical point of view. ∼ ∼ et us consider an oriented d -manifold Σ with non-empty boundary and a choice of triangulationΣ (cid:52) . The input of the model is a pair ( G, ω ) and a choice of gapped boundary conditions { ( A i , φ i ) } for each connected component ∂ Σ (cid:52) ,i ⊂ Σ (cid:52) , where A i ⊂ G is a subgroup and φ i ∈ C d ( A i , U(1)) is anormalised group d -cochain satisfying the condition d ( d ) φ i = ω − | A i . In the interior of Σ (cid:52) , the (bulk)Hamiltonian was defined in eq. 2.7. Given such a choice of gapped boundary conditions, let us nowdefine an operator that acts on a local neighbourhood of a boundary vertex ( v ) ⊂ ∂ Σ (cid:52) ,i . Mimickingthe definition of the bulk vertex operator, we consider the subcomplex Ξ v := cl ◦ st( v ), whichcorresponds to the smallest subcomplex that includes all the simplices of which ( v ) is a subsimplex.We next define the triangulated relative pinched interval cobordism over Ξ v with respect to Ω :=cl ◦ st( v ) ∩ ∂ Σ (cid:52) ,i Ξ v Ξ Ξ v := ( v (cid:48) ) (cid:116) j cl ◦ st( v ) , (2.19)whose boundary is given by ∂ ( Ξ v Ξ Ξ v ) = Ξ v ∪ Ω ( Ω v Ω Ω v ) ∪ ∂ Ξ v Ξ v (2.20)where Ω v := ( v (cid:48) ) (cid:116) j Ω. Given this relative pinched interval cobordism, we define the action of theoperator A A i ,φ i ( v ) on a state | ψ (cid:105) ∈ H G,A i ω,φ i [Σ (cid:52) ] via A A i ,φ i ( v ) : | ψ (cid:105) (cid:55)→ Z G,A i ω,φ i [ Ξ v Ξ Ξ v ] | ψ (cid:105) . (2.21)The gapped boundary Hamiltonian is finally defined as H G, { A i } ω, { φ i } [Σ (cid:52) ] = H Gω [Σ (cid:52) ] bulk + (cid:88) ∂ Σ (cid:52) ,i ⊂ ∂ Σ (cid:52) H G,A i ω,φ i [ ∂ Σ (cid:52) ,i ] bdry , (2.22)where H G,A i ω,φ i [ ∂ Σ (cid:52) ,i ] bdry := − (cid:88) (cid:52) (0) ⊂ ∂ Σ (cid:52) ,i A A i ,φ i (cid:52) (0) . (2.23)From the triangulation invariance properties of the partition function Z G, { A i } ω, { φ i } follows that the Hamil-tonian is a sum of mutually commuting projection operators, and as such it is still exactly solv-able. Furthermore, analogously to the bulk Hamiltonian, we can identify the ground-state subspace V G, { A i } ω, { φ i } [Σ (cid:52) ] with Im Z G, { A i } ω, { φ i } [ (cid:52) Σ ∂ Σ (cid:52) ] ≡ V G, { A i } ω, { φ i } [Σ (cid:52) ] , (2.24)and verify that the unitary isomorphism Z G, { A i } ω, { φ i } [ (cid:52) (cid:48) Σ ∂ Σ (cid:52) ] : V G, { A i } ω, { φ i } [Σ (cid:52) ] ∼ −→ V G, { A i } ω, { φ i } [Σ (cid:52) (cid:48) ] (2.25)commutes with the Hamiltonian. This last statement implies that we can always replace a giventriangulated subcomplex Ω (cid:52) ⊂ ∂ Σ (cid:52) by Ω (cid:52) (cid:48) while remaining in the ground state sector.Note finally that in the subsequent discussion, we shall also refer to gapped interfaces betweenseveral gapped boundaries. However, we will not require an explicit form of the Hamiltonian for suchinterfaces, and as such we omit here the explicit definition. Despite such an omission, the correspondingHamiltonian can be explicitly defined in close analogy with the construction of the gapped boundaryHamiltonian presented in this section. ∼ ∼ n order to illustrate the definition and some properties of the gapped boundary Hamiltonian, let usnow specialize to two dimensions (see also [11]). We consider a two-dimensional surface Σ endowedwith a triangulation Σ (cid:52) and a single connected boundary component ∂ Σ (cid:52) . The input data for thebulk Hamiltonian is a finite group G and a normalised group 3-cocycle α . Furthermore, we define on ∂ Σ (cid:52) a gapped boundary whose input data is a pair ( A, φ ), where A ⊂ G is a subgroup and φ a group2-cochain satisfying d (2) φ = α − | A which is explicitly expressed via α − ( a, a (cid:48) , a (cid:48)(cid:48) ) ! = d (2) φ ( a, a (cid:48) , a (cid:48)(cid:48) ) = φ ( a (cid:48) , a (cid:48)(cid:48) ) φ ( a, a (cid:48) a (cid:48)(cid:48) ) φ ( aa (cid:48) , a (cid:48)(cid:48) ) φ ( a, a (cid:48) ) , (2.26)for every a, a (cid:48) , a (cid:48)(cid:48) ∈ A ⊂ G . We consider the following situation: where the dashed area represents the bulk of the manifold, whereas the coloured line stands forthe gapped boundary. The black lines represent the 1-simplices on the interior Σ (cid:52) that are includedin cl ◦ st (1) . We first want to write down the action of the boundary operator at the vertex (1) ongraph-states of the formSpan C (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) g (cid:20) (cid:21)(cid:29)(cid:27) ∀ g ∈ Col(cl ◦ st(1) ,G,A ) ≡ Span C (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) ag a (cid:48)− gga a (cid:48) (cid:29)(cid:27) ∀ g ∈ G ∀ a,a (cid:48) ∈ A . (2.27)The boundary vertex operator A A,φ (1) boils down to evaluating the partition function (2.15) on therelative pinched interval cobordism (023) × (02)p I defined by , (2.28)such that < < ˜1 < < and the orange edges represent the time-like boundary. Explicitly, theaction of this boundary vertex operator reads A A,φ (1) (cid:12)(cid:12)(cid:12)(cid:12) ag a (cid:48)− gga a (cid:48) (cid:29) = 1 | A | (cid:88) ˜ a ∈ A α ( a, ˜ a, ˜ a − g ) φ (˜ a, ˜ a − a (cid:48) ) α (˜ a, ˜ a − a (cid:48) , a (cid:48)− g ) φ ( a, ˜ a ) (cid:12)(cid:12)(cid:12)(cid:12) ag a (cid:48)− g ˜ a − ga ˜ a ˜ a − a (cid:48) (cid:29) . (2.29)Let us now compute a triangulation changing boundary operator on a graph state (2.27). Morespecifically, let us construct the isomorphism that replaces the boundary subcomplex (01) ∪ (12) by asingle 1-simplex (02) . The corresponding operator is conveniently obtained by evaluating the partition ∼ ∼ unction (2.15) on the relative pinched interval cobordism , (2.30)with time-like boundary (012) , implementing the isomorphism (cid:12)(cid:12)(cid:12)(cid:12) ag a (cid:48)− gga a (cid:48) (cid:29) (cid:39) | A | α ( a, a (cid:48) , a (cid:48)− g ) φ ( a, a (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) ag a (cid:48)− gaa (cid:48) (cid:29) . (2.31)We can now confirm that this triangulation changing operator does commute with the Hamilto-nian operator. This follows from the cocycle relations d (2) φ ( a, ˜ a, ˜ a − a (cid:48) ) = α − ( a, ˜ a, ˜ a − a (cid:48) ) and d (3) α ( a, ˜ a, ˜ a − a (cid:48) , a (cid:48)− g ) = 1. SECTION 3
Tube algebra for gapped boundary excitations in (2+1)d
In this section, we apply the tube algebra approach in order to derive the algebraic structure underlyingthe boundary point-like excitations in two spatial dimensions.
Let us consider an open two-dimensional surface Σ. Its boundary ∂ Σ is referred to as the physicalboundary of the system. In the previous section, we explained how to construct the lattice Hamiltonianrealisation of Dijkgraaf-Witten theory on a triangulation of Σ. We further detailed how this modelcould be extended to the physical boundary of Σ in such way as to remain gapped. Bulk excitationsof this model were studied in detail in general dimensions in [32]. In addition to bulk excitations, thelattice Hamiltonian yields point-like boundary excitations that are excitations obtained by violatingsome of the stabiliser constraints on the boundary. We are interested in the classification and thestatistics of such gapped boundary excitations. More specifically, we consider the situation where twodifferent one-dimensional gapped boundaries meet at a zero-dimensional interface, and are interestedin the point-like excitations living at such interface. This situation can be locally depicted as follows: A φ B ψ . (3.1)Given that the input data for the bulk theory is a pair ( G, α ), where α is a normalized representativeof a cohomology class in H ( G, U(1)), the thick coloured lines stand for two gapped boundariescharacterized by the boundary conditions A φ ≡ ( A, φ ) and B ψ ≡ ( B, ψ ), respectively, while theblack dot illustrates the binary interface between them. The boundary conditions A φ and B ψ , whichwere defined in the previous section, are such that A, B ⊂ G , d (2) φ = α − | A and d (2) ψ = α − | B .We denote the lattice Hamiltonian for this specific choice of boundary conditions by H G,A,Bα,φ,ψ [Σ], andits associated ground state subspace by V G,A,Bα,φ,ψ [Σ]. In the following discussion, we will suppose thatthe Hamiltonian is further extended to the interface, but we do not require the explicit form of the ∼ ∼ orresponding operator. Note that although we restrict our attention to gapped boundaries, ourexposition could be easily generalised to accommodate domain walls , which can be thought of asshared gapped boundaries between two (possibly different) topological phases.By definition, given a point-like excitation at the interface of two one-dimensional gapped bound-aries, there is a local neighbourhood of Σ for which the energy density is higher than the one of theground state. Removing such a local neighbourhood leaves a new boundary component, referred toas the excitation boundary , that is incident on the physical boundary ∂ Σ of the manifold. We denotethe resulting manifold by Σ o and the excitation boundary by ∂ Σ o | ex . . We illustrate this configurationas follows: → , (3.2)where the dashed area represents the region whose energy density is higher than the one of theground state. The black line represents the excitation boundary, whose topology is the one of the unitinterval I ≡ [0 , o with a triangulation, we are interested in the lattice Hamiltonian H G,A,Bα,φ,ψ [Σ o (cid:52) \ ∂ Σ o (cid:52)| ex . ] obtained by removing all the operators whose supports are on ∂ Σ o (cid:52)| ex . . In a wayreminiscent to the bulk Hamiltonian in sec. 2.2, this Hamiltonian displays open boundary conditions such that the corresponding ground state subspace can be decomposed over them. Properties of thepoint-like excitations can then be encoded into the boundary conditions, so that a classification ofthe boundary conditions induces a classification of the corresponding point-like excitations. In otherwords, ground states in V G,A,Bα,φ,ψ [Σ o (cid:52) ], which are characterised by a given excitation boundary colouring,define specific excitations with respect to ground states in the Hilbert space V G,A,Bα,φ,ψ [Σ (cid:52) ]. In general,any such excitation is a superposition of elementary point-like excitations . In order to find these point-like elementary boundary excitations, we apply the tube algebra approach, whose general constructioncan be found in [32].Let us consider the manifold ∂ Σ o | ex . × I . Naturally, it has the topology of a 2-cell but we wouldlike to emphasize the fact that it has two kinds of boundary components, namely a pair of physicalboundary components and a pair of excitation boundary components. More precisely, it is the systemobtained by removing from the two-disk D local neighbourhoods at the interface of two differentphysical boundaries: → (cid:39) , (3.3)where the nomenclature is the same as before. A crucial, yet trivial, fact is that we can always glue acopy of ∂ Σ o | ex . × I to Σ o along ∂ Σ o | ex . without modifying its topology, i.e. ∪ → (cid:39) . As explained in more detail in [32], given a triangulation of Σ o and making use of the triangulationchanging unitary isomorphisms, this simple gluing operation induces a symmetry map on the groundstate subspace, whose simple modules classify the boundary conditions on ∂ Σ o | ex . and as such thecorresponding point-like boundary excitations. In order to compute these simple modules, we further ∼ ∼ emark that it is always possible to apply a diffeomorphism so that a local neighbourhood of ∂ Σ o | ex . is of the form ∂ Σ o | ex . × I so that the corresponding ground state subspaces are isomorphic. The effectof such diffeomorphism is to localise the action of the symmetry map so that it only involves degreesof freedom living within ∂ Σ o | ex . × I . Consequently, it is enough to consider the symmetry map thatcorresponds to the gluing of two copies of the manifold ∂ Σ o | ex . × I , i.e.( ∂ Σ o | ex . × I ) ∪ ∂ Σ o | ex . ( ∂ Σ o | ex . × I ) (cid:39) ∂ Σ o | ex . × I . (3.4)We pictorially summarize these operations below: ∪ (cid:39) ∪ reduces −−−−→ to ∪ → (cid:39) . Given a triangulation of ∂ Σ o | ex . × I , this symmetry map in turn endows the associated ground statesubspace with a finite-dimensional algebraic structure referred to as the tube algebra . Irreduciblerepresentations of the tube algebra label the simple modules of the original symmetry map, classifyingboundary conditions on ∂ Σ o | ex . , and thus the corresponding point-like boundary excitations. Let us now derive the tube algebra for the configuration described above so as to determine theelementary boundary excitations at the interface of two one-dimensional gapped boundaries. First, weneed to specify the ground state subspace on ∂ Σ o | ex . × I by picking a triangulation. Crucially, the choiceof triangulation does not matter. Indeed, given a triangulation of the excitation boundary, changingthe discretisation of the physical boundary or the bulk of ∂ Σ o | ex . × I yields an isomorphic ground statesubspace, which would in turn induce an isomorphic tube algebra. Furthermore, a different choiceof triangulation for the excitation boundary would yield a Morita equivalent tube algebra, which bydefinition has the same simple modules as the original algebra. As such, we should make the simplestchoice of triangulation possible. We choose to discretise the excitation boundary by a single 1-simplexand ∂ Σ o | ex . × I as a triangulated 2-cell. The resulting triangulated manifold is denoted by T [ I (cid:52) ] andthe corresponding ground state subspace explicitly reads V G,A,Bα,φ,ψ [ T [ I ]] := Span C (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) g (cid:20) (cid:48) (cid:48) (cid:21)(cid:29)(cid:27) ∀ g ∈ Col( T [ I ] ,G,A,B ) ≡ Span C (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) g a − gbba (cid:48) (cid:48) (cid:29)(cid:27) ∀ g ∈ G ∀ ( a,b ) ∈ A × B ≡ Span C (cid:8)(cid:12)(cid:12) g a −→ b (cid:11)(cid:9) ∀ g ∈ G ∀ ( a,b ) ∈ A × B , (3.5)where some labellings are left implicit since they can be deduced from the flatness constraints, i.e.the stabiliser constraints with respect to the B (cid:52) (2) -operators. The tube algebra can be computedusing the following algorithm: Recall that the tube algebra is an extension of the gluing operation Note that we rotated the drawings by 90 ◦ for convenience. We refer the reader to [32] for a general and more detailed definition of the tube algebra. ∼ ∼ [ I ] ∪ I T [ I ] (cid:39) T [ I ] to the ground state subspace V G,A,Bα,φ,ψ [ T [ I ]]. Using the relation (2.11), we obtain thefollowing decomposition of the Hilbert space V G,A,Bα,φ,ψ [ T [ I ]]: V G,A,Bα,φ,ψ [ T [ I ]] = (cid:77) g ∈ Col( I ×{ } ,G ) g ∈ Col( I ×{ } ,G ) V G,A,Bα,φ,ψ [ T [ I ]] g ,g . The gluing itself is then performed via an injective map
GLU defined according to
GLU : V G,A,Bα,φ,ψ [ T [ I ]] ⊗ V G,A,Bα,φ,ψ [ T [ I ]] → (cid:77) g ,g (cid:48) ∈ Col( I ×{ } ,G ) g ,g (cid:48) ∈ Col( I ×{ } ,G ) V G,A,Bα,φ,ψ [ T [ I ]] g ,g ⊗ V G,A,Bα,φ,ψ [ T [ I ]] g (cid:48) ,g (cid:48) , which acts on states | ψ g ,g (cid:105) ∈ V G,A,Bα,φ,ψ [ T [ I ]] g ,g and | ψ (cid:48) g (cid:48) ,g (cid:48) (cid:105) ∈ V G,A,Bα,φ,ψ [ T [ I ]] g (cid:48) ,g (cid:48) via identification ofthe boundary conditions along the gluing interface, i.e. GLU : | ψ g ,g (cid:105) ⊗ | ψ (cid:48) g (cid:48) ,g (cid:48) (cid:105) (cid:55)→ δ g ,g (cid:48) | ψ g ,g (cid:105) ⊗ | ψ (cid:48) g ,g (cid:48) (cid:105) . This map can be linearly extended to states displaying mixed grading. Importantly, the image ofthis map typically differs from the ground state subspace V G,A,Bα,φ,ψ [ T [ I ] ∪ I T [ I ]] since all the stabiliserconstraints might not be satisfied along the gluing interface. This can be resolved by applying theHamiltonian projection operator P T [ I ] ∪ I T [ I ] with respect to the full Hamiltonian H G,A,Bα,φ,ψ [ T [ I ] ∪ I T [ I ]],which was defined in sec. 2.4. Finally, we can apply a triangulation changing isomorphism in orderto obtain a final state in V G,A,Bα,φ,ψ [ T [ I ]]. Putting everything together, this defines a (cid:63) -product, whichtogether with V G,A,Bα,φ,ψ [ T [ I ]] defines the tube algebra: (cid:63) : V G,A,Bα,φ,ψ [ T [ I ]] ⊗ V G,A,Bα,φ,ψ [ T [ I ]] GLU −−→ H
G,A,Bα,φ,ψ [ T [ I ] ∪ I T [ I ]] P T [ I ] ∪ I T [ I ] −−−−−−→ V G,A,Bα,φ,ψ [ T [ I ] ∪ I T [ I ]] ∼ −→ V G,A,Bα,φ,ψ [ T [ I ]] . Given two basis states of V G,A,Bα,φ,ψ [ T [ I ]] as defined in (3.5), let us now compute explicitly this (cid:63) -product.Firstly, the G -colourings along the gluing interface are identified via the map GLU , i.e.
GLU (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) g a − gbba (cid:48) (cid:48) (cid:29) ⊗ (cid:12)(cid:12)(cid:12)(cid:12) g (cid:48) a (cid:48)− g (cid:48) b (cid:48) b (cid:48) a (cid:48) (cid:48) (cid:48) (cid:29)(cid:19) = δ g (cid:48) ,a − gb (cid:12)(cid:12)(cid:12)(cid:12) g ( aa (cid:48) ) − gbb (cid:48) b b (cid:48) a a (cid:48) (cid:48) (cid:48) (cid:48) (cid:29) . Secondly, we apply the Hamiltonian projector P T [ I ] ∪ I T [ I ] in order to enforce the gauge invariance at thephysical boundary vertices that are along the gluing interface. This operator is obtained by evaluatingthe partition function (2.15) on the relative pinched interval cobordism (cid:48) (cid:48) (cid:48) ˜1 (cid:48) , (3.6) ∼ ∼ nd its action explicitly reads P T [ I ] ∪ I T [ I ] (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) g ( aa (cid:48) ) − gbb (cid:48) b b (cid:48) a a (cid:48) (cid:48) (cid:48) (cid:48) (cid:29)(cid:19) = 1 | A || B | (cid:88) (˜ a, ˜ b ) ∈ A × B ϑ ABg ( a, ˜ a | b, ˜ b ) ϑ ABa − gb (˜ a, ˜ a − a (cid:48) | ˜ b, ˜ b − b (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) g ( aa (cid:48) ) − g (cid:48) bb (cid:48) b ˜ b ˜ b − b (cid:48) a ˜ a ˜ a − a (cid:48) (cid:48) (cid:48) (cid:48) (cid:29) , where we introduced the cocycle data ϑ ABg ( a, a (cid:48) | b, b (cid:48) ) := ψ ( b, b (cid:48) ) φ ( a, a (cid:48) ) α ( a, a (cid:48) , a (cid:48)− a − gbb (cid:48) ) α ( g, b, b (cid:48) ) α ( a, a − gb, b (cid:48) ) . (3.7)It follows from α − | A = d (2) φ and α − | B = d (2) ψ , as well as the cocycle conditions d (3) α ( a, a (cid:48) , a (cid:48)(cid:48) , a (cid:48)(cid:48)− a (cid:48)− a − gbb (cid:48) b (cid:48)(cid:48) ) = 1 d (3) α ( a, a − gb, b (cid:48) , b (cid:48)(cid:48) ) = 1 d (3) α ( a, a (cid:48) , a (cid:48)− a − gbb (cid:48) , b (cid:48)(cid:48) ) = 1 d (3) α ( g, b, b (cid:48) , b (cid:48)(cid:48) ) = 1that ϑ AB satisfies d (2) ϑ ABg ( a, a (cid:48) , a (cid:48)(cid:48) | b, b (cid:48) , b (cid:48)(cid:48) ) := ϑ ABa − gb ( a (cid:48) , a (cid:48)(cid:48) | b (cid:48) , b (cid:48)(cid:48) ) ϑ ABg ( a, a (cid:48) a (cid:48)(cid:48) | b, b (cid:48) b (cid:48)(cid:48) ) ϑ ABg ( aa (cid:48) , a (cid:48)(cid:48) | bb (cid:48) , b (cid:48)(cid:48) ) ϑ ABg ( a, a (cid:48) | b, b (cid:48) ) = 1 , (3.8)which in particular implies the following property ϑ ABa − gb ( a − , a | b − , b ) = ϑ ABg ( a, a − | b, b − ) . (3.9)Furthermore, given that α , φ and ψ are normalized cocycles, we have the normalisation conditions: ϑ ABg ( A , a (cid:48) | B , b (cid:48) ) = ϑ ABg ( a, A | b, B ) = 1 = ϑ ABg ( A , a (cid:48) | b, B ) = ϑ ABg ( a, A | B , b (cid:48) ) . (3.10)Going back to the tube algebra, it remains to apply a triangulation changing isomorphism in order torecover the initial triangulation. This can be done by evaluating the partition function for the pinchedinterval cobordism (012) + × I endowed with the triangulation depicted below:(012) + × I := (cid:48) (cid:48) (cid:48) ≡ (00 (cid:48) (cid:48) (cid:48) ) + ∪ (011 (cid:48) (cid:48) ) − ∪ (0122 (cid:48) ) + . (3.11)The corresponding operator implements the isomorphism (cid:12)(cid:12)(cid:12)(cid:12) g ( aa (cid:48) ) − g (cid:48) bb (cid:48) b ˜ b ˜ b − b (cid:48) a ˜ a ˜ a − a (cid:48) (cid:48) (cid:48) (cid:48) (cid:29) (cid:39) | A | | B | ϑ ABg ( a ˜ a, ˜ a − a (cid:48) | b ˜ b, ˜ b − b (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) g ( aa (cid:48) ) − gbb (cid:48) bb (cid:48) aa (cid:48) (cid:48) (cid:48) (cid:29) . ∼ ∼ utting everything together, we obtain (cid:12)(cid:12)(cid:12)(cid:12) g a − gbba (cid:48) (cid:48) (cid:29) (cid:63) (cid:12)(cid:12)(cid:12)(cid:12) g (cid:48) a (cid:48)− g (cid:48) b (cid:48) b (cid:48) a (cid:48) (cid:48) (cid:48) (cid:29) = δ g (cid:48) ,a − gb | A | | B | ϑ ABg ( a, a (cid:48) | b, b (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12) g ( aa (cid:48) ) − gbb (cid:48) bb (cid:48) aa (cid:48) (cid:48) (cid:48) (cid:29) , where we used the cocycle relation d (2) ϑ ABg ( a, ˜ a, ˜ a − a (cid:48) | b, ˜ b, ˜ b − b (cid:48) ) = 1. Using the more symbolicnotation introduced in (3.5), the (cid:63) -product reads (cid:12)(cid:12) g a −→ b (cid:11) (cid:63) (cid:12)(cid:12) g (cid:48) a (cid:48) −−→ b (cid:48) (cid:11) = δ g (cid:48) ,a − gb | A | | B | ϑ ABg ( a, a (cid:48) | b, b (cid:48) ) (cid:12)(cid:12) g aa (cid:48) −−−→ bb (cid:48) (cid:11) . (3.12) Before concluding this section about boundary point-like excitations in (2+1)d, we are going to showthat the tube algebra derived above can be recast as a twisted groupoid algebra [54]. Although this mightseem a little bit artificial at the moment, this will turn out to be very useful in the subsequent sections.Indeed, we will show that in the language of groupoid algebras, both the tube algebras in (2+1)d and in(3+1)d can be unified allowing for a simultaneous study of the corresponding representation theories.Let us first review some basic category theoretical definitions. More details can be found forexample in [56, 58]. Given a category C , the set of objects and the set of morphisms between objectsare denoted by Ob( C ) and Hom( C ), respectively. Given two objects X, Y ∈ Ob( C ), the set of morphismsfrom X to Y is written Hom C ( X, Y ) (cid:51) f : X → Y , such that X = s( f ) and Y = t( f ) are the source and target objects of f , respectively. Composition rule of morphisms is defined according to X f −−→ Y f (cid:48) −−−→ Z = X ff (cid:48) −−−−→ Z .
Furthermore, for every object X ∈ Ob( C ), the corresponding identity morphisms is denoted by id X ∈ Hom C ( X, X ). Finally, we notate the set of n composable morphims in C by C n comp := { ( f , . . . , f n ) ∈ Hom( C ) n | t( f i ) = s( f i +1 ) , ∀ i ∈ , . . . , n − } . Let us now specialize to groupoids : Definition . A (finite) groupoid G is a category whose object and morphism setsare finite and all morphisms are invertible, i.e. for each morphism g ∈ Hom G ( X, Y ) , there existsa morphism g − ∈ Hom G ( Y, X ) such that gg − = id X and g − g = id Y . Every finite group provides a finite one-object groupoid refers to as the delooping of the group:
Example . Let G be a finite group. The delooping of G is the one-object groupoid G with Ob( G ) = { • } and morphism set Hom G ( • , • ) = G with the composition rulebeing provided by the group multiplication in G . Henceforth, we shall identify any group G and its delooping G , denoting both by G . Generalizing thenotion of group cohomology in an obvious way, we obtain the notion of groupoid cohomology: Definition . Let G be a finite groupoid and M a G -module. Giventhe set of n composable morphisms G n comp in G , we define an n -cochain on G as a map ω n : Analogously to group cohomology, groupoid cohomology of a groupoid is implicitly defined as the simplicial coho-mology of its classifying space. ∼ ∼ n comp → M . On the space C n ( G , M ) of n -cochains, the coboundary operator d ( n ) : C n ( G , M ) → C n +1 ( G , M ) is defined via d ( n ) ω n ( g , . . . , g n +1 ) (3.13):= g (cid:46) ω n ( g , . . . , g n +1 ) ω n ( g , . . . g n ) ( − n +1 n (cid:89) i =1 ω n ( g , . . . , g i − , g i g i +1 , g i +2 , . . . , g n +1 ) ( − i . The n -th cohomology group of groupoid cocycles is then defined as usual by H n ( G , M ) := Ker d ( n ) Im d ( n − ≡ Z n ( G , M ) B n ( G , M ) . (3.14)Throughout this manuscript, we shall always consider cohomology groups of the form H n ( G , U(1)),where U(1) is taken to be the G -module with the trivial groupoid action. Naturally, the cohomology ofa group coincides with the groupoid cohomology of its delooping. Furthermore, we shall often require,without loss of generality, that cocycles are normalised : Definition . Given a groupoid n -cocycle [ ω n ] ∈ H n ( G , U(1)) , we call ω n ∈ [ ω n ] a normalised representative if ω n ( g , . . . , g n ) = 1 , whenever any of the arguments is anidentity morphism. In particular there always exists a normalised representative of each n-cocycleequivalence class [ ω n ] ∈ H n ( G , U(1)) . Utilising the technology of groupoid cohomology, we can now introduce twisted groupoid algebras ,generalising the theory of twisted group algebras [54]:
Definition . Given a finite groupoid G and a normalised 2-cocycle ϑ ∈ Z ( G , U(1)) , the twisted groupoid algebra C [ G ] ϑ is the algebra defined over the vector space Span C {| g (cid:105) | ∀ g ∈ Hom( G ) } (3.15) with algebra product | g (cid:105) (cid:63) | g (cid:48) (cid:105) := δ t( g ) , s( g (cid:48) ) ϑ ( g , g (cid:48) ) | gg (cid:48) (cid:105) . (3.16) The requirement that ϑ is a 2-cocycle ensures that C [ G ] ϑ is an associative algebra. Putting everything together, let us now recast the (2+1)d tube algebra as a twisted groupoid algebra.Let G AB be the (finite) groupoid whose objects are given by group elements in G , and whose morphismsread g a −→ b a − gb ≡ g a −→ b , where ( a, b ) ∈ A × B with the composition given by the multiplication in G : g a −→ b a − gb a (cid:48) −−→ b (cid:48) a (cid:48)− a − gbb (cid:48) = g aa (cid:48) −−−→ bb (cid:48) a (cid:48)− a − gbb (cid:48) . (3.17)Utilising this definition, we can conveniently redefine ϑ AB as a normalised groupoid 2-cocycle in H ( G AB , U(1)), in such a way that the tube algebra defined earlier is isomorphic to the groupoidalgebra C [ G AB ] ϑ AB ≡ C [ G AB ] αφψ of G AB twisted by ϑ AB . Notice that the normalization conditions (3.10) do not state that the cocycle is equal to one whenever any of theentry is one, but instead whenever any of the morphism in the corresponding groupoid is the identity. It is thereforecompatible with the definition given earlier. ∼ ∼ ECTION 4
Tube algebra for gapped boundary excitations in (3+1)d
In this section, we apply the tube algebra approach to study boundary excitations in (3+1)d. Althoughthe excitation content of the model is rich in (3+1)d, we focus on a special configuration, which turnsout to be related to that considered in the previous section via a dimensional reduction argument.
The strategy we presented in sec. 3 applies identically in three dimensions. Given a pattern of two-dimensional gapped boundaries, excitations can be classified by considering boundary conditions ofthe manifold obtained by removing local neighbourhoods of these excitations. Given that the inputdata for the bulk theory is a pair (
G, π ), where is π a normalized representative of a cohomologyclass in H ( G, U(1)), we are interested in the situation where two two-dimensional gapped boundariescharacterized by the boundary conditions A λ ≡ ( A, λ ) and B µ ≡ ( B, µ ) meet at a one-dimensionalinterface. The boundary conditions are such that
A, B ⊂ G , d (3) λ = π − | A and d (3) µ = π − | B . Wedenote the Hamiltonian defined according to (2.7) for these boundary conditions as H G,A,Bπ,λ,µ [Σ].Given this situation, several types of excitations could be studied. For instance, we could investi-gate point-like boundary excitations at the one-dimensional interface. Instead, we consider a pair ofpoint-like boundary excitations, one for each gapped boundary, that are linked by a bulk string-likeexcitation. In other words, we consider a bulk string-like excitation that terminates at the two gappedboundaries. This situation can be depicted as follows: A λ B µ → , (4.1)where the dark volume represents a local neighbourhood of the string-like excitation, and thus theregion whose energy density is higher than that of the ground state. Removing this local neighbour-hood leaves an excitation boundary ∂ Σ o | ex . that has the topology of cylinder. Classifiying boundaryconditions on such cylinder corresponds to classifying the string-like excitations.Let us consider the manifold ∂ Σ o | ex . × I . This manifold has the topology of a hollow cylinder,which has two kinds of boundary components, namely a pair of physical boundary components anda pair of excitation boundary components. Given the 3-ball endowed with two gapped boundaries,the same manifold can be obtained by removing local neighbourhoods of the interface and of a stringterminating at the two gapped boundaries: (cid:39) . ∼ ∼ y construction, this manifold can be glued to the original system along the excitation boundary ∂ Σ o | ex . without affecting its topology. It follows from the discussion in sec. 3 that there is a tube algebraassociated with the gluing of two copies of this tube-like manifold, whose irreducible representationsclassify this special type of string-like excitations. Let us derive the tube algebra for the special configuration described above. As before, we first need tospecify the ground state subspace on ∂ Σ o | ex . × I by choosing a discretisation. We choose to discretise ∂ Σ o | ex . × I as a triangulated cube with two opposite faces identified. The resulting triangulatedmanifold is denoted by T [ S × I ] and the corresponding ground state subspace explicitly reads V G,A,Bπ,λ,µ [ T [ S × I ]] := Span C (cid:40)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ga b b (cid:48) (cid:48)
10 ˜0 (cid:48) ˜1 (cid:48) ˜1˜0 (cid:43)(cid:41) ∀ g ∈ G | g = a − gb ∀ a ,a ∈ A ∀ b ,b ∈ B (4.2) ≡ Span C (cid:8)(cid:12)(cid:12) ( g, a , b ) a −−→ b (cid:11)(cid:9) ∀ g ∈ G | g = a − gb ∀ a ,a ∈ A ∀ b ,b ∈ B , (4.3)where we make the identifications (0) ≡ (˜0) , (0 (cid:48) ) ≡ (˜0 (cid:48) ) , (1) ≡ (˜1) , (1 (cid:48) ) ≡ (˜1 (cid:48) ) , (00 (cid:48) ) ≡ (˜0˜0 (cid:48) ) , (01) ≡ (˜0˜1) , (0 (cid:48) (cid:48) ) ≡ (˜0 (cid:48) ˜1 (cid:48) ) and (11 (cid:48) ) ≡ (˜1˜1 (cid:48) ) . As before, some labellings are left implicit since they can be deducedfrom the flatness constraints. Let us now compute the (cid:63) -product for two such states adapting in theobvious way the definition of the previous section. Firstly, colourings along the gluing interface areidentified via the map GLU , i.e.
GLU (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ga b b (cid:48) (cid:48)
10 ˜0 (cid:48) ˜1 (cid:48) ˜1˜0 (cid:43) ⊗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a (cid:48) g (cid:48) a (cid:48) b (cid:48) b (cid:48) (cid:48) (cid:48)
21 ˜1 (cid:48) ˜2 (cid:48) ˜2˜1 (cid:43)(cid:33) = δ g (cid:48) ,a − gb δ a (cid:48) ,a a δ b (cid:48) ,b b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a (cid:48) b (cid:48) a ga b b (cid:48) (cid:48)
10 ˜0 (cid:48) ˜1 (cid:48) ˜1˜0 2 (cid:48) (cid:48) ˜2 (cid:43) , where we introduced the notation x y := y − xy . Secondly, we apply the Hamiltonian projector P T [ S × I ] ∪ S × I T [ S × I ] in order to enforce the twisted gauge invariance at the physical boundary ver-tices along the gluing interface. This operator can be expressed by evaluating the partition function ∼ ∼ P T [ S × I ] ∪ S × I T [ S × I ] (cid:32)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a (cid:48) b (cid:48) a ga b b (cid:48) (cid:48)
10 ˜0 (cid:48) ˜1 (cid:48) ˜1˜0 2 (cid:48) (cid:48) ˜2 (cid:43)(cid:33) (4.4)= 1 | A || B | (cid:88) (˜ a, ˜ b ) ∈ A × B (cid:37) ABg,a ,b ( a , ˜ a | b , ˜ b ) (cid:37) ABa − gb ,a a ,b b (˜ a, ˜ a − a (cid:48) | ˜ b, ˜ b − b (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ a − a (cid:48) ˜ b − b (cid:48) a ˜ aga b b ˜ b (cid:48) (cid:48)
10 ˜0 (cid:48) ˜1 (cid:48) ˜1˜0 2 (cid:48) (cid:48) ˜2 (cid:43) , where we introduced the cocycle data (cid:37) ABg,a ,b ( a , a (cid:48) | b , b (cid:48) ) := T b ( µ )( b , b (cid:48) ) T a ( λ )( a , a (cid:48) ) T a ( π )( a , a (cid:48) , a (cid:48)− a − gb b (cid:48) ) T a ( π )( g, b , b (cid:48) ) T a ( π )( a , a − gb , b (cid:48) ) (4.5)in terms of the cocycle data T ( λ ), T ( µ ) and T ( π ) that are itself defined according to T x ( α )( y , y ) := α ( x, y , y ) α ( y , y , x y y ) α ( y , x y , y ) , T x ( π )( y , y , y ) := π ( y , x y , y , y ) π ( y , y , y , x y y y ) π ( x, y , y , y ) π ( y , y , x y y , y ) , for any group elements x, y , y , y ∈ H in a finite group H and group cochains α ∈ C ( H, U(1)), π ∈ C ( H, U(1)). Defining d (2) T x ( α )( y , y , y ) := T x y ( α )( y , y ) T x ( α )( y , y y ) T x ( α )( y y , y ) T x ( α )( y , y ) , (4.6) d (3) T x ( π )( y , y , y , y ) := T x y ( π )( y , y , y ) T x ( α )( y , y y , y ) T x ( π )( y , y , y ) T x ( π )( y y , y , y ) T x ( α )( y , y , y y ) , (4.7)it follows from the cocycle conditions d (4) π = 1, d (3) λ = π − | A and d (3) µ = π − | B that d (3) T ( π ) = 1, d (2) T ( λ ) = T ( π ) − | A and d (2) T ( µ ) = T ( π ) − | B . Utilising the cocycle conditions d (3) T a ( π )( a , a (cid:48) , a (cid:48)(cid:48) , a (cid:48)(cid:48)− a (cid:48)− a − gb b (cid:48) b (cid:48)(cid:48) ) = 1 d (3) T a ( π )( a , a − gb , b (cid:48) , b (cid:48)(cid:48) ) = 1 d (3) T a ( π )( a , a (cid:48) , a (cid:48)− a − gb b (cid:48) , b (cid:48)(cid:48) ) = 1 d (3) T a ( π )( g, b , b (cid:48) , b (cid:48)(cid:48) ) = 1 , we finally obtain that (cid:37) AB satisfies d (2) (cid:37) ABg,a ,b ( a , a (cid:48) , a (cid:48)(cid:48) | b , b (cid:48) , b (cid:48)(cid:48) ) := (cid:37) ABa − gb ,a a ,b b ( a (cid:48) , a (cid:48)(cid:48) | b (cid:48) , b (cid:48)(cid:48) ) (cid:37) ABg,a ,b ( a , a (cid:48) a (cid:48)(cid:48) | b , b (cid:48) b (cid:48)(cid:48) ) (cid:37) ABg,a ,b ( a a (cid:48) , a (cid:48)(cid:48) | b b (cid:48) , b (cid:48)(cid:48) ) (cid:37) ABg,a ,b ( a , a (cid:48) | b , b (cid:48) ) = 1 . (4.8)Going back to the tube algebra, it remains to apply a triangulation changing isomorphism in order torecover the initial triangulation, and thus a state in V G,A,Bπ,λ,µ [ T [ S × I ]]. This is done by evaluating the ∼ ∼ artition function for the pinched interval cobordism (012) + × S × I endowed with the triangulationdefined as (012) + × S × I = (0122 (cid:48) ˜2 (cid:48) ) + ∪ (012˜2˜2 (cid:48) ) − ∪ (01˜1˜2˜2 (cid:48) ) + ∪ (0˜0˜1˜2˜2 (cid:48) ) − ∪ (011 (cid:48) (cid:48) ˜2 (cid:48) ) − ∪ (011 (cid:48) ˜1 (cid:48) ˜2 (cid:48) ) + ∪ (01˜1˜1 (cid:48) ˜2 (cid:48) ) − ∪ (0˜0˜1˜1 (cid:48) ˜2 (cid:48) ) + ∪ (00 (cid:48) (cid:48) (cid:48) ˜2 (cid:48) ) + ∪ (00 (cid:48) (cid:48) ˜1 (cid:48) ˜2 (cid:48) ) − ∪ (00 (cid:48) ˜0 (cid:48) ˜1 (cid:48) ˜2 (cid:48) ) + ∪ (0˜0˜0 (cid:48) ˜1 (cid:48) ˜2 (cid:48) ) − . (4.9)The corresponding operator implements the isomorphism (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˜ a − a (cid:48) ˜ b − b (cid:48) a ˜ aga b b ˜ b (cid:48) (cid:48)
10 ˜0 (cid:48) ˜1 (cid:48) ˜1˜0 2 (cid:48) (cid:48) ˜2 (cid:43) (cid:39) | A | | B | (cid:37) ABg,a ,b ( a ˜ a, ˜ a − a (cid:48) | b ˜ b, ˜ b − b (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a a (cid:48) ga b b b (cid:48) (cid:48) (cid:48)
20 ˜0 (cid:48) ˜2 (cid:48) ˜2˜0 (cid:43) . Putting everything together, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a ga b b (cid:48) (cid:48)
10 ˜0 (cid:48) ˜1 (cid:48) ˜1˜0 (cid:43) (cid:63) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a (cid:48) g (cid:48) a (cid:48) b (cid:48) b (cid:48) (cid:48) (cid:48)
21 ˜1 (cid:48) ˜2 (cid:48) ˜2˜1 (cid:43) (4.10)= δ g (cid:48) ,a − gb δ a (cid:48) ,a a δ b (cid:48) ,b b | A | | B | (cid:37) ABg,a ,b ( a , a (cid:48) | b , b (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a a (cid:48) ga b b b (cid:48) (cid:48) (cid:48)
20 ˜0 (cid:48) ˜2 (cid:48) ˜2˜0 (cid:43) , where we used the cocycle relation d (2) (cid:37) ABg,a ,b ( a , ˜ a, ˜ a − a (cid:48) | b , ˜ b, ˜ b − b (cid:48) ). Using the more symbolicnotation introduced in (4.2), we obtained (cid:12)(cid:12) ( g, a , b ) a −−→ b (cid:11) (cid:63) (cid:12)(cid:12) ( g (cid:48) , a (cid:48) , b (cid:48) ) a (cid:48) −−→ b (cid:48) (cid:11) = δ g (cid:48) ,a − gb δ a (cid:48) ,a a δ b (cid:48) ,b b | A | | B | (cid:37) ABg,a ,b ( a , a (cid:48) | b , b (cid:48) ) (cid:12)(cid:12) ( g, a , b ) a a (cid:48) −−−−→ b b (cid:48) (cid:11) . Similarly to its (2+1)d analogue, the tube algebra found above can be recast as a twisted groupoidalgebra. Interestingly, due to the topology of the problem, we shall notice how in this language the(3+1)d tube algebra can be recast in terms of the (2+1)d one, unifying both computations. This isreminiscent of the notion of lifted models and lifted tube algebras developed in [32] in the context ofbulk excitations.An important ingredient of our construction is the notion of loop groupoid : ∼ ∼ efinition . Given a finite groupoid G , the loop groupoid Λ G is the groupoidwith object set { g ∈ End G ( X ) | ∀ X ∈ Ob( G ) } and morphisms of the form h : g → h − gh , for every g ∈ End G ( X ) and h ∈ Hom G ( X, Y ) . Composition in Λ G is inherited from the one in G . Specialising to the case where the finite groupoid is taken to be the delooping of a finite group G , weobtain that Λ G is the groupoid with object set Ob(Λ G ) = G and morphism set Hom(Λ G ) = { g a −→ a − ga | ∀ g, a ∈ G } . Composition is given by multiplication in G such that g a −→ a − ga a (cid:48) −−→ ( aa (cid:48) ) − gaa (cid:48) = g aa (cid:48) −−→ ( aa (cid:48) ) − gaa (cid:48) , for all g, a, a (cid:48) ∈ G . Using this terminology, we can check that the cocycle data T ( π ), T ( λ ) and T ( µ )defined in (4.5) actually correspond to loop groupoid cocycles in Z (Λ G, U(1)), Z (Λ A, U(1)) and Z (Λ B, U(1)), respectively. More generally, for any group G , we have a map T : Z • ( G, U(1)) → Z •− (Λ G, U(1)) referred to as the S -transgression map. More details regarding this map can befound in [32, 54, 59]. We further require the notion of relative groupoid : Definition . Given a groupoid G , and a pair of subgroupoids A , B ⊆ G ,the relative groupoid G AB is the groupoid with object set Ob( G AB ) := { g ∈ Hom( G ) | s( g ) ∈ Ob( A ) , t( g ) ∈ Ob( B ) } and morphism set provided by g a −→ b a − gb ≡ g a −→ b , (4.11) for all g ∈ Ob( G AB ) , a ∈ Hom A (s( g ) , − ) and b ∈ Hom B (t( g ) , − ) . Composition is defined by g a −→ b a − gb a (cid:48) −−→ b (cid:48) a (cid:48) − a − gbb (cid:48) = g aa (cid:48) −−−→ bb (cid:48) a (cid:48) − a − gbb (cid:48) , (4.12) for all composable pairs ( a , a (cid:48) ) ∈ A and ( b , b (cid:48) ) ∈ B . It follows immediately from the definition above that the groupoid G AB , whose twisted groupoidalgebra is isomorphic to the (2+1)d tube algebra, actually corresponds to the relative groupoid definedfor the delooping of the groups. We are almost ready to define the (3+1)d tube algebra in thislanguage. The last item we require is a notion of normalised cocycle for relative groupoid. To this endwe introduce ( G , α )-subgroupoids: Definition . Given a finite groupoid G and a normalised 3-cocycle α ∈ Z ( G , U(1)) , we calla pair ( A , φ ) a ( G , α ) -subgroupoid when A ⊆ G is a subgroupoid of G and φ ∈ C ( A , U(1)) is a2-cochain satisfying the condition d (2) φ ( a , a (cid:48) , a (cid:48)(cid:48) ) = α − ( a , a (cid:48) , a (cid:48)(cid:48) ) |A for all composable ( a , a (cid:48) , a (cid:48)(cid:48) ) ∈A . For any pair of ( G , α )-subgroupoids ( A , φ ) and ( B , ψ ), we construct a normalised 2-cocycle ϑ AB ∈ Z ( G AB , U(1)) for the relative groupoid G AB via: ϑ AB ( g a −→ b , a − gb a (cid:48) −−→ b (cid:48) ) := ψ ( b , b (cid:48) ) φ ( a , a (cid:48) ) α ( a , a (cid:48) , a (cid:48)− a − gbb (cid:48) ) α ( g , b , b (cid:48) ) α ( a , a − gb , b (cid:48) ) (4.13) ≡ ϑ AB g ( a , a (cid:48) | b , b (cid:48) ) (4.14)for all composable morphisms g a −→ b , a − gb a (cid:48) −−→ b (cid:48) ∈ G AB , (4.15) ∼ ∼ here we are using the shorthand notation introduced in (4.11). It follows from α − |A = d (2) ψ and α − |B = d (2) φ , as well as the cocycle conditions d (3) α ( a , a (cid:48) , a (cid:48)(cid:48) , a (cid:48)(cid:48)− a (cid:48)− a − gbb (cid:48) b (cid:48)(cid:48) ) = 1 d (3) α ( a , a − gb , b (cid:48) , b (cid:48)(cid:48) ) = 1 d (3) α ( a , a (cid:48) , a (cid:48)− a − gbb (cid:48) , b (cid:48)(cid:48) ) = 1 d (3) α ( g , b , b (cid:48) , b (cid:48)(cid:48) ) = 1that ϑ AB satisfies the 2-cocycle relation d (2) ϑ AB g ( a , a (cid:48) , a (cid:48)(cid:48) | b , b (cid:48) , b (cid:48)(cid:48) ) := ϑ AB a − gb ( a (cid:48) , a (cid:48)(cid:48) | b (cid:48) , b (cid:48)(cid:48) ) ϑ AB g ( a , a (cid:48) a (cid:48)(cid:48) | b , b (cid:48) b (cid:48)(cid:48) ) ϑ AB g ( aa (cid:48) , a (cid:48)(cid:48) | bb (cid:48) , b (cid:48)(cid:48) ) ϑ AB g ( a , a (cid:48) | b , b (cid:48) ) = 1 . (4.16)Unsurprisingly, this equation mimics (3.8). Furthermore, given that α is a normalized cocycle, wehave the normalisation conditions: ϑ AB g (id s( a (cid:48) ) , a (cid:48) | id s( b (cid:48) ) , b (cid:48) ) = ϑ AB g ( a , id t( a ) | b , id t( b ) ) = 1 ϑ AB g (id s( a (cid:48) ) , a (cid:48) | b , id t( b ) ) = ϑ AB g ( a , id t( a ) | id s( b (cid:48) ) , b (cid:48) ) = 1 , which further imply ϑ AB a − gb ( a − , a | b − , b ) = ϑ AB g ( a , a − | b , b − ) . (4.17)Let G be a finite group and π ∈ Z ( G, U(1)). We consider two subgroups
A, B ⊂ G and λ ∈ C ( A, U(1)), µ ∈ C ( B, U(1)) such that d (3) λ = π − | A and d (3) µ = π − | B . It follows from thecomputations in sec. 4 that (Λ A, T ( λ )) and (Λ B, T ( µ )) are (Λ G, T ( π ))-subgroupoids. We define ϑ Λ A Λ B by applying the formula (4.13) for α ≡ T ( π ), φ ≡ T ( λ ) and ψ ≡ T ( µ ). Putting everything together, weobtain the twisted relative groupoid algebra C [Λ G Λ A Λ B ] ϑ Λ A Λ B . We can show that this twisted relativegroupoid algebra is isomorphic to the (3+1)d tube algebra by identifying( g, a , b ) a −−→ b ≡ g a −−→ b , (4.18)such that a g −→ b ≡ g ∈ Ob(Λ G Λ A Λ B ), a a −→ a a ≡ a ∈ Hom Λ A (s( g ) , − ) and b b −→ b b ≡ b ∈ Hom Λ B (t( g ) , − ), as well as ϑ Λ A Λ B ≡ (cid:37) AB , which was defined in (4.5).Thereafter, we make use of the shorthand notations Λ( G AB ) ≡ Λ G Λ A Λ B and C [Λ( G AB )] ϑ Λ( AB ) ≡ C [Λ( G AB )] αφψ ≡ C [Λ G Λ A Λ B ] ϑ Λ A Λ B to refer to this relative groupoid algebra. We purposefully choose anotation very similar to describe the (2+1)d and (3+1)d tube algebras in order to emphasize the factthat the framework presented in this section unifies both. As a matter of fact, we can obtain the (2+1)dalgebra from the (3+1)d one by restricting the loop groupoid Λ G to morphisms whose source and targetobjects are the identity in G and by replacing the loop groupoid 3-cocycle α ≡ T ( π ) ∈ Z (Λ G, U(1)),where π ∈ Z ( G, U(1)), by a group 3-cocycle α ∈ Z ( G, U(1)). In virtue of this last remark, we maynow focus on the algebra relevant to the (3+1)d boundary excitations, namely C [Λ( G AB )] αφψ , anddeduce the results for the (2+1)d scenario as a limiting case.We conclude this section with a remark regarding the notation. Since the morphisms a ∈ Hom Λ A (s( g ) , − ) and b ∈ Hom Λ B (t( g ) , − ) in (4.18) are specified by a choice of group variables inthe finite groups A and B , respectively, we shall often loosely identify both in the following for nota-tional convenience. ∼ ∼ ECTION 5
Representation theory and elementary boundary excitations
In this section, we derive the irreducible representations of the algebra C [Λ( G AB )] αφψ , and elucidatetheir physical interpretation as a classifier for the elementary boundary excitations in (3+1)d. Asmentioned earlier, due to the topology of the problem, and the common description as relative groupoidalgebras, this study can be straightforwardly applied to describe elementary boundary excitations in(2+1)d. Given a finite group G , two subgroups A, B ⊂ G and cocycle data π ∈ Z ( G, U(1)), λ ∈ C ( A, U(1)), µ ∈ C ( B, U(1)) satisfying d (4) π = 1, d (3) λ = π − | A , d (3) µ = π − | B , respectively, we define α ≡ T ( π ) ∈ Z (Λ G, U(1)), φ ≡ T ( λ ) ∈ C (Λ A, U(1)) and ψ ≡ T ( µ ) ∈ C (Λ B, U(1)). We explained abovethat the simple modules of the groupoid algebra C [Λ( G AB )] αφψ ≡ C [Λ( G AB )] ϑ Λ( AB ) classify elementarystring-like excitations terminating at gapped boundaries. Let us now derive these simple modules.We shall find that they are labelled by a pair ( O , R ), where O is an equivalence class of boundarycolourings with respect to the action of the tube algebra, and R is a projective group representationthat decomposes the symmetry action of the tube algebra on a given boundary colouring.We begin by first decomposing the algebra C [Λ( G AB )] αφψ into a direct sum of subalgebras. To thisend, we notice that the tube algebra defines an action on the set of boundary colourings yielding anequivalence relation on Ob(Λ( G AB )) given by g ∼ g (cid:48) , if ∃ g a −→ b ∈ Hom(Λ( G AB )) such that g (cid:48) = t (cid:0) g a −→ b (cid:1) . The subsets of Ob(Λ( G AB )), i.e. boundary colourings of the tube, that are in the same equivalenceclass form a partition of Ob(Λ( G AB )) into disjoint sets. Let us denote by O AB , O (cid:48) AB ⊆ Ob(Λ( G AB ))two such equivalence classes. Considering two basis elements of the form (cid:12)(cid:12) g a −→ b (cid:11) , (cid:12)(cid:12) g (cid:48) a (cid:48) −−→ b (cid:48) (cid:11) (5.1)such that g ∈ O AB and g (cid:48) ∈ O (cid:48) AB , it follows from the definition of the algebra that the product ofthese two states necessarily vanishes. Consequently, each equivalence class of Ob(Λ( G AB )) defines asubalgebra ( C [Λ( G AB )] αφψ ) O AB ⊂ C [Λ( G AB )] αφψ whose defining vector space isSpan C (cid:8)(cid:12)(cid:12) g a −→ b (cid:11)(cid:9) ∀ g a −→ b ∈ Hom(Λ( G AB ))s . t . g ∈O AB . (5.2)Since orbits O AB form a partition of Ob(Λ( G AB )), we have the following decomposition C [Λ( G AB )] αφψ = (cid:77) O AB ⊂ G ( C [Λ( G AB )] αφψ ) O AB . (5.3)Given an equivalence class O AB , we notate its elements by { o i } i =1 ,..., |O AB | and call o the representative element of O AB . We further consider the set { p i , q i } i =1 ,..., |O AB | ⊆ Hom(Λ A ) × Hom(Λ B ) defined bya choice of morphism o i p i −→ q i o ∈ Hom(Λ( G AB )) , ∀ o i ∈ O AB ∼ ∼ nd the requirement ( p , q ) = (id s( o ) , id t( o ) ). The stabiliser group of O AB is then defined as Z O AB := { ( a , b ) ∈ Hom(Λ A ) × Hom(Λ B ) | o = a − o b } . (5.4)Remark that the orbit-stabiliser theorem implies | Z O AB | · |O AB | = | A || B | . Finally, we construct thetwisted group algebra C [ Z O AB ] as the algebra with defining vector spaceSpan C (cid:8)(cid:12)(cid:12) a −→ b (cid:11)(cid:9) ∀ ( a , b ) ∈ Z O AB (5.5)and product rule (cid:12)(cid:12) a −→ b (cid:11) (cid:63) (cid:12)(cid:12) a (cid:48) −−→ b (cid:48) (cid:11) = ϑ Λ( AB ) o ( a , a (cid:48) | b , b (cid:48) ) (cid:12)(cid:12) aa (cid:48) −−−→ bb (cid:48) (cid:11) . (5.6)Given that α is normalized, it follows from definition (4.13) that ϑ Λ( AB ) o is a representative normalisedgroup 2-cocycle in H ( Z O AB , U(1)). For each simple unitary ϑ Λ( AB ) o -projective representation ( D R , V R )of Z O AB , we can define a simple representation of the relative groupoid algebra C [Λ( G AB )] αφψ via ahomomorphism D O AB ,R : C [Λ( G AB )] αφψ → End( V O AB ,R ) where V O AB ,R := Span C {| o i , v m (cid:105)} ∀ i =1 ,..., |O AB |∀ m =1 ,..., dim( V R ) . (5.7)For i, j ∈ { , . . . , |O AB |} , m, n ∈ { , . . . , dim( V R ) } the matrix elements are defined to be D O AB ,R [ im ][ jn ] (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) = δ g , o i δ a − gb , o j ϑ Λ( AB ) o ( p − i , a | q − i , b ) ϑ Λ( AB ) o ( p − i ap j , p − j | q − i bq j , q − j ) D Rmn (cid:0)(cid:12)(cid:12) p − i ap j −−−−−→ q − i bq j (cid:11)(cid:1) (5.8)such that | o i , v m (cid:105) (cid:46) D O AB ,R (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) = |O AB | (cid:88) i,j =1 dim( V R ) (cid:88) m,n =1 D O AB ,R [ im ][ jn ] (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) | o j , v n (cid:105) . (5.9)Henceforth, we make use of the shorthand notation ρ AB ≡ ( O AB , R ), I ≡ [ im ], J ≡ [ jn ] and d ρ AB ≡ d O AB ,R = |O AB | · dim( V R ). It follows immediately from the definition and the linearity of the ϑ Λ( AB ) o -projective representations of Z O AB that these matrices define an algebra homomorphism, i.e. (cid:88) K D ρ AB IK (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ AB KJ (cid:0)(cid:12)(cid:12) g (cid:48) a (cid:48) −−→ b (cid:48) (cid:11)(cid:1) = δ g (cid:48) , a − gb ϑ Λ( AB ) g ( a , a (cid:48) | b , b (cid:48) ) D ρ AB IJ (cid:0)(cid:12)(cid:12) g aa (cid:48) −−−→ bb (cid:48) (cid:11)(cid:1) . (5.10)Furthermore, the matrix elements satisfy the conjugation relation D ρ AB IJ (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) = 1 ϑ Λ( AB ) g ( a , a − | b , b − ) D ρ AB JI (cid:0)(cid:12)(cid:12) a − gb a − −−−→ b − (cid:11)(cid:1) , (5.11)which follows from the unitarity of the projective representation D R of the stabilizer subgroup Z O AB ,inducing a unitary representation of C [Λ( G AB )] αφψ . This endows C [Λ( G AB )] αφψ with the structure ofa *-algebra which in turn implies its semi-simplicity due to finiteness. Finally, the representationsmatrices satisfy the following orthogonality and completeness conditions1 | A || B | (cid:88) g a −→ b ∈ Λ( G AB ) D ρ AB IJ (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ (cid:48) AB I (cid:48) J (cid:48) (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) = δ ρ AB ,ρ (cid:48) AB d ρ AB δ I,I (cid:48) δ J,J (cid:48) (5.12)1 | A || B | (cid:88) ρ AB (cid:88) I,J d ρ AB D ρ AB IJ (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ AB IJ (cid:0)(cid:12)(cid:12) g (cid:48) a (cid:48) −−→ b (cid:48) (cid:11)(cid:1) = δ g , g (cid:48) δ a , a (cid:48) δ b , b (cid:48) . (5.13)A proof of the orthogonality relation can be found in app. A.1, the completeness following from similararguments. ∼ ∼ .2 Comultiplication map and concatenation of string-like excitations The simple modules of the relative groupoid algebra C [Λ( G AB )] αφψ classify string-like bulk excitationsterminating at gapped boundaries labelled by A λ and B µ , such that φ ≡ T ( λ ) and ψ ≡ T ( µ ). Let usnow delve deeper into the exploration of the properties of this algebra, in relation to the concatenationof the corresponding excitations. We consider the following system of three gapped boundaries andstring-like excitations terminating at these gapped boundaries: A λ B µ C ν . (5.14)The two string-like excitations depicted above are characterized by the relative groupoid algebras C [Λ( G AB )] αφψ and C [Λ( G BC )] αψϕ , respectively, where ϕ ≡ T ( ν ). We will show that these string-like excitations can be concatenated, and the result of this concatenation is a string-like excitationterminating at the gapped boundaries labelled by A λ and C ν . More specifically, we will demonstratethat a pair of modules for the relative groupoid algebras C [Λ( G AB )] αφψ and C [Λ( G BC )] αψϕ can becomposed to form a module for the relative groupoid algebra C [Λ( G AC )] αφϕ .Let us consider a pair of elementary string-like excitations with internal Hilbert spaces V ρ AB and V ρ BC , respectively. In the absence of external constraints, the corresponding join Hilbert spaceis provided by the tensor product V ρ AB ⊗ V ρ BC . It remains to understand how the tube algebraacts on this join Hilbert space. We introduce an algebra homomorphism ∆ B : C [Λ( G AC )] αφϕ → C [Λ( G AB )] αφψ ⊗ C [Λ( G BC )] αψϕ defined by∆ B (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) := 1 | B | (cid:88) g ∈ Ob(Λ( G AB )) g ∈ Ob(Λ( G BC )) g g = gb ∈ Hom Λ B (t( g ) , s( g )) ζ Λ( ABC ) a , b , c ( g , g ) (cid:12)(cid:12) g a −→ b (cid:11) ⊗ (cid:12)(cid:12) g b −→ c (cid:11) (5.15)where ζ Λ( ABC ) a , b , c ( g , g ) := α ( g , g , c ) α ( a , a − g b , b − g c ) α ( g , b , b − g c ) . (5.16)As mentioned earlier, when no confusion is possible, we shall loosely identify b ∈ Hom Λ B (t( g ) , s( g ))and the group variable b ∈ B it evaluates to in order to make the notation lighter. By analogy withthe theory of Hopf algebras, we refer in the following to ∆ B as the B - comultiplication map of thetwisted groupoid algebra C [Λ( G AC )] αφϕ . It follows from the cocycle conditions d (3) α ( a , a (cid:48) , a (cid:48)− a − g bb (cid:48) , b (cid:48)− b − g cc (cid:48) ) = 1 d (3) α ( a , a − g b , b − g c , c (cid:48) ) = 1 d (3) α ( a , a − b , b (cid:48) , b (cid:48)− b − g cc (cid:48) ) = 1 d (3) α ( g , b , b − g c , c (cid:48) ) = 1 d (3) α ( g , b , b (cid:48) , b (cid:48)− b − g cc (cid:48) ) = 1 d (3) α ( g , g , c , c (cid:48) ) = 1 Because of the geometry of the operation under consideration, we refrain from referring to this process as the‘fusion’ of the corresponding string-like excitations. That being said, in (2+1)d, the same map defines the usual fusionof point-like excitations. ∼ ∼ hat ζ Λ( ABC ) a , b , c satisfies the relation ϑ Λ( AB ) g ( a , a (cid:48) | b , b (cid:48) ) ϑ Λ( BC ) g ( b , b (cid:48) | c , c (cid:48) ) ϑ Λ( AC ) g g ( a , a (cid:48) | c , c (cid:48) ) = ζ Λ( ABC ) aa (cid:48) , bb (cid:48) , cc (cid:48) ( g , g ) ζ Λ( ABC ) a , b , c ( g , g ) ζ Λ( ABC ) a (cid:48) , b (cid:48) , c (cid:48) ( a − g b , b − g c ) (5.17)ensuring that the map ∆ B is an algebra homomorphism, i.e.∆ B (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) ◦ ∆ B (cid:0)(cid:12)(cid:12) g (cid:48) a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) = ∆ B (cid:0)(cid:12)(cid:12) g a −→ c (cid:11) ◦ (cid:12)(cid:12) g (cid:48) a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) . (5.18)Putting everything together, given the relative groupoid algebras C [Λ( G AB )] αφψ , C [Λ( G BC )] αψϕ and apair of representations ( D ρ AB , V ρ AB ) and ( D ρ BC , V ρ BC ), the comultiplication ∆ B allows us to definethe tensor product representation (( D ρ AB ⊗ D ρ BC ) ◦ ∆ B , V ρ AB ⊗ V ρ BC ), where( D ρ AB ⊗ D ρ BC ) ◦ ∆ B : C [Λ( G AB )] αφψ ⊗ C [Λ( G BC )] αψϕ → End( V ρ AB ⊗ V ρ BC ) (5.19)such that( D ρ AB ⊗ D ρ BC ) (cid:0) ∆ B (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1)(cid:1) = 1 | B | (cid:88) g ∈ Ob(Λ( G AB )) g ∈ Ob(Λ( G BC )) g g = gb ∈ B ζ Λ( ABC ) a , b , c ( g , g ) D ρ AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) ⊗ D ρ BC (cid:0)(cid:12)(cid:12) g b −→ c (cid:11)(cid:1) , where we loosely identified b ∈ Hom Λ B (t( g ) , s( g )) and the corresponding group variable for nota-tional convenience. In the following, it will be often useful to write the so-called truncated tensorproduct ⊗ B of representation matrices defined as D ρ AB ⊗ B D ρ BC := ( D ρ AB ⊗ D ρ BC ) ◦ ∆ B . (5.20)Using the semisimplicity of relative groupoid algebras, the tensor product representations defined aboveare generically not simple and as such admit a decomposition into direct sum of simple representations,i.e. D ρ AB ⊗ B D ρ BC ∼ = (cid:77) ρ AC N ρ AC ρ AB ,ρ BC D ρ AC , (5.21)where the number N ρ AC ρ AB ,ρ BC ∈ Z +0 is referred to as the multiplicity of the simple C [Λ( G AC )] αφϕ rep-resentation ( D ρ AC , V ρ AC ) appearing in the tensor product of the representations ( D ρ AB , V ρ AB ) and( D ρ BC , V ρ BC ). Henceforth, we assume multiplicity-freeness of the multifusion category of representa-tions, i.e. N ρ AC ρ AB ,ρ BC ∈ { , } in order to simplify the notations. Note however that it is straightforwardto lift this assumption. Using the orthogonality relations of the irreducible representations, we find auseful expression to compute explicitly this number, namely N ρ AC ρ AB ,ρ BC = 1 | A || C | (cid:88) g a −→ c ∈ Λ( G AC ) tr (cid:104) ( D ρ AB ⊗ B D ρ BC ) (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) D ρ AC (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) (cid:105) . (5.22)Note finally that given the algebras C [Λ( G AA )] αφφ and C [Λ( G BB )] αψψ , the regular modules C [Λ( G AA )] αφφ C [Λ( G BB )] αψψ satisfy the unit module properties C [Λ( G AA )] αφφ ⊗ A ρ AB ∼ = ρ AB ∼ = ρ AB ⊗ B C [Λ( G BB )] αψψ (5.23) The regular module of an algebra is defined as the algebra viewed as a module over itself. ∼ ∼ s C [Λ( G AB )] αφψ modules.As explained above, thanks to our formulation in terms of relative groupoid algebras, we can easilyextract all the relevant structures for the (2+1)d algebra as a limiting case. This is done in the nextsection, where we define a canonical basis of excited states. In this scenario, the comultiplication mapyields the fusion of the corresponding point-like excitations. In preparation for the later discussion, let us study further the properties of the comultiplicationmap introduced earlier. Since the comultiplication map ∆ B is an algebra homomorphism, there existintertwining unitary maps U ρ AB ,ρ BC : (cid:77) ρ AC V ρ AC → V ρ AB ⊗ B V ρ BC , (5.24)where the sum is over labels ρ AC such that D ρ AC ∈ D ρ AB ⊗ B D ρ BC , that satisfy the defining relation( D ρ AB I AB J AB ⊗ B D ρ BC I BC J BC ) (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) = (cid:88) ρ AC I AC ,J AC U ρ AB ,ρ BC [ I AB I BC ][ ρ AC I AC ] D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) U ρ AB ,ρ BC [ J AB J BC ][ ρ AC J AC ] . Henceforth, we will denote the matrix elements of this unitary map as (cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105) := U ρ AB ,ρ BC [ I AB I BC ][ ρ AC I AC ] , and refer to them as Clebsch-Gordan coefficients. Using the orthogonality of the representation ma-trices, we obtain the equivalent defining relation d ρ AC | A || C | (cid:88) g a −→ c ∈ Λ( G AC ) ( D ρ AB I AB J AB ⊗ B D ρ BC I BC J BC ) (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) = (cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105)(cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105) . The unitarity of U ρ AB ,ρ BC imposes the following orthogonality and completeness relations: (cid:88) I AB ,I BC (cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105)(cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ (cid:48) AC I (cid:48) AC (cid:105) = δ I AC ,I (cid:48) AC δ ρ AC ,ρ (cid:48) AC (5.25) (cid:88) ρ AC ,I AC (cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105)(cid:104) ρ AB I (cid:48) AB ρ BC I (cid:48) BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105) = δ I AB ,I (cid:48) AB δ I BC ,I (cid:48) BC . (5.26)Furthermore, the Clebsch-Gordan coefficients satisfy the following crucial property (cid:88) g ∈ Hom(s( a ) , s( c )) (cid:88) { J } ( D ρ AB I AB J AB ⊗ B D ρ BC I BC J BC ) (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1)(cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105) = (cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105) (5.27)referred to as the gauge invariance of the coefficients. This property can be checked as follows: Firstly,utilise the unitarity of the intertwining maps to rewrite the defining equation as the intertwiningproperty (cid:88) J AB ,J BC ( D ρ AB I AB J AB ⊗ B D ρ BC I BC J BC ) (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1)(cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105) = (cid:88) I AC (cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105) D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) . ∼ ∼ econdly, multiply this equation on both side by D ρ AC K AC J AC (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) and use the identity (cid:88) g ∈ Hom(s( a ) , s( c )) (cid:88) J AC D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) D ρ AC K AC J AC (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) = (cid:88) g ∈ Hom(s( a ) , s( c )) (cid:88) J AC ϑ Λ( AC ) g ( a , a − | c , c − ) D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) D ρ AC J AC K AC (cid:0)(cid:12)(cid:12) a − gc a − −−−→ c − (cid:11)(cid:1) = (cid:88) g ∈ Hom(s( a ) , s( c )) D ρ AC I AC K AC (cid:0)(cid:12)(cid:12) g A −−−→ C (cid:11)(cid:1) = δ I AC ,K AC , where we used (5.11). Note that we use the notation A to refer to the morphism in Hom(s( a ) , − )characterized by the group variable A ∈ A , and similarly for C . Summing over J AC = 1 , . . . , d ρ AC finally yields the gauge invariance. This invariance of the Clebsch-Gordan coefficients further implies (cid:88) { J } D ρ AB I AB J AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ BC I BC J BC (cid:0)(cid:12)(cid:12) g b (cid:48) −−→ c (cid:11)(cid:1)(cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105) D ρ AC J AC I AC (cid:0)(cid:12)(cid:12) g a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) (5.28)= 1 | B | (cid:88) ˜ b ∈ B ϑ Λ( AB ) g ( a , ˜ a | b , ˜ b ) ϑ Λ( BC ) g ( b (cid:48) , ˜ b | c , ˜ c ) ζ Λ( ABC )˜ a , ˜ b , ˜ c ( a − g b , b (cid:48)− g c ) ϑ Λ( AC ) g (˜ a , ˜ a − a (cid:48) | ˜ c , ˜ c − c (cid:48) ) δ g , a − g bb (cid:48)− g c × (cid:88) { K } D ρ AB I AB K AB (cid:0)(cid:12)(cid:12) g a ˜ a −−→ b ˜ b (cid:11)(cid:1) D ρ BC I BC K BC (cid:0)(cid:12)(cid:12) g b (cid:48) ˜ b −−−→ c ˜ c (cid:11)(cid:1)(cid:104) ρ AB K AB ρ BC K BC (cid:12)(cid:12)(cid:12) ρ AC K AC (cid:105) D ρ AC K AC I AC (cid:0)(cid:12)(cid:12) ˜ a − g ˜ c ˜ a − a (cid:48) −−−−→ ˜ c − c (cid:48) (cid:11)(cid:1) , which is true for all composable morphisms a , ˜ a in Λ A and c , ˜ c in Λ C . A proof of this identity can befound in app. A.2. It is straightforward to check that this last relation induces another one, namely (cid:88) { J } D ρ AB J AB I AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) g b (cid:48) −−→ c (cid:11)(cid:1)(cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105) D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) g a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) (5.29)= 1 | B | (cid:88) ˜ b ∈ B ϑ Λ( AB ) g (˜ a , ˜ a − a | ˜ b , ˜ b − b ) ϑ Λ( BC ) g (˜ b , ˜ b − b (cid:48) | ˜ c , ˜ c − c ) ζ Λ( ABC )˜ a , ˜ b , ˜ c ( g , g ) ϑ Λ( AC ) g ( a (cid:48) , ˜ a | c (cid:48) , ˜ c ) − × (cid:88) { K } δ a (cid:48)− g c (cid:48) , g g D ρ AB K AB I AB (cid:0)(cid:12)(cid:12) ˜ a − g ˜ b ˜ a − a −−−−→ ˜ b − b (cid:11)(cid:1) D ρ BC K BC I BC (cid:0)(cid:12)(cid:12) ˜ b − g ˜ c ˜ b − b (cid:48) −−−−→ ˜ c − c (cid:48) (cid:11)(cid:1) × (cid:104) ρ AB K AB ρ BC K BC (cid:12)(cid:12)(cid:12) ρ AC K AC (cid:105) D ρ AC I AC K AC (cid:0)(cid:12)(cid:12) g a (cid:48) ˜ a −−−→ c (cid:48) ˜ c (cid:11)(cid:1) . j -symbols Given two relative groupoid algebras C [Λ( G AB )] αφψ , C [Λ( G BC )] αψϕ and a pair of representations definedby ( D ρ AB , V ρ AB ), ( D ρ BC , V ρ BC ), we constructed earlier the tensor product representation (( D ρ AB ⊗D ρ BC ) ◦ ∆ B , V ρ AB ⊗ V ρ BC ) of C [Λ( G AC )] αφϕ . Let us now consider the quasi-invertible algebra elementΦ ABCD ∈ C [Λ( G AB )] αφψ ⊗ C [Λ( G BC )] αψϕ ⊗ C [Λ( G CD )] αϕχ defined asΦ ABCD := (cid:88) g ∈ Ob(Λ( G AB )) g ∈ Ob(Λ( G BC )) g ∈ Ob(Λ( G CD )) α − ( g , g , g ) (cid:12)(cid:12) g A −−−→ B (cid:11) ⊗ (cid:12)(cid:12) g B −−−→ C (cid:11) ⊗ (cid:12)(cid:12) g C −−−→ D (cid:11) , (5.30)such that g , g and g are composable morphisms in Λ G . The cocycle conditions d (3) α ( a , a − g b , b − g c , c − g d ) = 1 d (3) α ( g , g , c , c − g d ) = 1 d (3) α ( g , b , b − g c , c − g d ) = 1 d (3) α ( g , g , g , d ) = 1 ∼ ∼ mply the identity ζ Λ( BCD ) b , c , d ( g , g ) ζ Λ( ABD ) a , b , d ( g , g g ) ζ Λ( ACD ) a , c , d ( g g , g ) ζ Λ( ABC ) a , b , c ( g , g ) = α ( g , g , g ) α ( a − g b , b − g c , c − g d ) , (5.31)which in turn ensures that the comultiplication is quasi-coassociative , i.e(∆ B ⊗ id)∆ C (cid:0)(cid:12)(cid:12) g a −→ d (cid:11)(cid:1) = Φ ABCD (cid:63) (cid:2) (id ⊗ ∆ C )∆ B (cid:0)(cid:12)(cid:12) g a −→ d (cid:11)(cid:1)(cid:3) (cid:63) Φ − ABCD , ∀ (cid:12)(cid:12) g a −→ d (cid:11) ∈ C [Λ( G AB )] αφχ . (5.32)This signifies that the truncated tensor product of representations ( D ρ AB ⊗ B D ρ BC ) ⊗ C D ρ CD and D ρ AB ⊗ B ( D ρ BC ⊗ C D ρ CD ) defined as( D ρ AB ⊗ B D ρ BC ) ⊗ C D ρ CD := ( D ρ AB ⊗ D ρ BC ⊗ D ρ CD ) ◦ (∆ B ⊗ id)∆ C (5.33) D ρ AB ⊗ B ( D ρ BC ⊗ C D ρ CD ) := ( D ρ AB ⊗ D ρ BC ⊗ D ρ CD ) ◦ (id ⊗ ∆ C )∆ B (5.34)must be isomorphic as C [Λ( G AD )] αφχ -modules. More specifically, it follows immediately from thequasi-coassociativity condition that the mapsΦ ρ AB ,ρ BC ,ρ CD := ( D ρ AB ⊗ D ρ AB ⊗ D ρ AB )(Φ ABCD ) ∈ End( V ρ AB ⊗ V ρ BC ⊗ V ρ CD ) (5.35)define intertwiners between the tensor product of representations above such thatΦ ρ AB ,ρ BC ,ρ CD [ D ρ AB ⊗ B ( D ρ BC ⊗ C D ρ CD )] = [( D ρ AB ⊗ B D ρ BC ) ⊗ C D ρ CD ]Φ ρ AB ,ρ BC ,ρ CD . (5.36)Let us consider two vector spaces V ρ AB and V ρ BC . These are spanned by vectors | ρ AB I AB (cid:105) and | ρ BC I BC (cid:105) , respectively, such that the corresponding groupoid algebras act on these basis vectors fromthe right. We define the truncated tensor product of two such vectors as | ρ AB I AB (cid:105) ⊗ B | ρ BC I BC (cid:105) := (cid:0) | ρ AB I AB (cid:105) ⊗ | ρ BC I BC (cid:105) (cid:1) (cid:46) ∆ B ( AC ) , (5.37)which span the vector space V ρ AB ⊗ B V ρ BC ⊂ V ρ AB ⊗ V ρ BC . More specifically. we have | ρ AB I AB (cid:105) ⊗ B | ρ BC I BC (cid:105) = (cid:88) ρ AC I AC | ρ AB ⊗ B ρ BC ; ρ AC , I AC (cid:105) (cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105) , (5.38)where we define | ρ AB ⊗ B ρ BC , ρ AC I AC (cid:105) := (cid:88) I AB ,I BC (cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105)(cid:0) | ρ AB I AB (cid:105) ⊗ | ρ BC I BC (cid:105) (cid:1) . (5.39)Noting that | ρ AB ⊗ B ρ BC , ρ AC I AC (cid:105) ( D ρ AB ⊗ B D ρ BC ) (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) = | ρ AB ⊗ B ρ BC , ρ AC I AC (cid:105)D ρ AC (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) , (5.40)we realize that Span C {| ρ AB ⊗ B ρ BC , ρ AC I AC (cid:105)} ∀ I AC ∼ = V ρ AC as C [Λ( G AC )] αφϕ representations throughthe map | ρ AB ⊗ B ρ BC , ρ AC I AC (cid:105) (cid:55)→ | ρ AC I AC (cid:105) . Similarly, we can define the following truncated tensorproduct of vectors (cid:0) | ρ AB I AB (cid:105) ⊗ B | ρ BC I BC (cid:105) (cid:1) ⊗ C | ρ CD I CD (cid:105) := (cid:0) | ρ AB I AB (cid:105) ⊗ | ρ BC I BC (cid:105) ⊗ | ρ CD I CD (cid:105) (cid:1) (cid:46) [(∆ B ⊗ id)∆ C ]( AD ) | ρ AB I AB (cid:105) ⊗ B (cid:0) | ρ BC I BC (cid:105) ⊗ C | ρ CD I CD (cid:105) (cid:1) := (cid:0) | ρ AB I AB (cid:105) ⊗ | ρ BC I BC (cid:105) ⊗ | ρ CD I CD (cid:105) (cid:1) (cid:46) [(id ⊗ ∆ C )∆ B ]( AD ) , ∼ ∼ hich define basis vectors in ( V ρ AB ⊗ B V ρ BC ) ⊗ C V ρ BC and V ρ AB ⊗ B ( V ρ BC ⊗ C V ρ BC ), respectively. Wethen find that Φ ABCD induces the following isomorphism:( V ρ AB ⊗ B V ρ BC ) ⊗ C V ρ BC ∼ = V ρ AB ⊗ B ( V ρ BC ⊗ C V ρ BC ) . (5.41)Vectors (cid:0) | ρ AB I AB (cid:105) ⊗ B | ρ BC I BC (cid:105) (cid:1) ⊗ C | ρ CD I CD (cid:105) are typically not linearly independent, however a basisfor the vector space ( V ρ AB ⊗ B V ρ BC ) ⊗ C V ρ BC is provided by the vectors (cid:88) { I } (cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105)(cid:104) ρ AC I AC ρ CD I CD (cid:12)(cid:12)(cid:12) ρ AD K AD (cid:105) | ρ AB I AB (cid:105) ⊗ | ρ BC I BC (cid:105) ⊗ | ρ CD I CD (cid:105) . (5.42)We obtain that Φ ABCD acts on such basis vectors as (cid:88) ρ AC (cid:88) { I } (cid:110) ρ AB ρ AD ρ BC ρ AC ρ CD ρ BD (cid:111)(cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105)(cid:104) ρ AC I AC ρ CD I CD (cid:12)(cid:12)(cid:12) ρ AD K AD (cid:105) | ρ AB I AB (cid:105) ⊗ | ρ BC I BC (cid:105) ⊗ | ρ CD I CD (cid:105) (cid:46) Φ ABCD = (cid:88) { I } (cid:104) ρ AB I AB ρ BD I BD (cid:12)(cid:12)(cid:12) ρ AD K AD (cid:105)(cid:104) ρ BC I BC ρ CD I CD (cid:12)(cid:12)(cid:12) ρ BD I BD (cid:105) | ρ AB I AB (cid:105) ⊗ | ρ BC I BC (cid:105) ⊗ | ρ CD I CD (cid:105) (5.43)such that the so-called 6 j -symbols are defined as (cid:110) ρ AB ρ AD ρ BC ρ AC ρ CD ρ BD (cid:111) := 1 d ρ AD (cid:88) { I } α ( o i AB , o i BC , o i CD ) (cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105)(cid:104) ρ AC I AC ρ CD I CD (cid:12)(cid:12)(cid:12) ρ AD I AC (cid:105)(cid:104) ρ AB I AB ρ BD I BD (cid:12)(cid:12)(cid:12) ρ AD I AD (cid:105)(cid:104) ρ BC I BC ρ CD I CD (cid:12)(cid:12)(cid:12) ρ BD I BD (cid:105) , where the notation is the one of definition (5.8) of the representation matrices. This establishes theisomorphism (5.41). A detailed proof of the defining relation (5.43) can be found in app. A.3.Furthermore, given the vector space (( V ρ AB ⊗ B V ρ BC ) ⊗ C V ρ CD ) ⊗ D V ρ DE , we find that[(id ⊗ id ⊗ ∆ D χ )(Φ ABCE )] (cid:63) [(∆ B ⊗ id ⊗ id)(Φ ACDE )]and ( AB ⊗ Φ BCDE ) (cid:63) [(id ⊗ ∆ C ⊗ id)(Φ ABDE )] (cid:63) (Φ ABCD ⊗ DE ) (5.44)induce the same isomorphism. This is referred to as the so-called pentagon identity and ensures theself-consistency of the quasi-coassociativity. A proof of the pentagon identity can be found in app. A.4.In a similar vein, it can be shown that the regular C [Λ( G BB )] αψψ -module satisfies the so-called triangle identity such that the following diagram commutes( ρ AB ⊗ B C [Λ( G BB )] αψψ ) ⊗ B ρ BC ρ AB ⊗ B ( C [Λ( G BB )] αψψ ⊗ B ρ BC ) ρ AB ⊗ B ρ BC Φ ABBC ∼ = ∼ = (5.45)as C [Λ( G AB )] αφϕ -modules for all C [Λ( G AB )] αφψ -modules ρ AB and C [Λ( G BC )] αψϕ -modules ρ BC . ∼ ∼ .5 Canonical basis for (2+1)d boundary excited states So far we have been dealing with the groupoid algebra C [Λ( G AB )] αφψ , which is isomorphic to the (3+1)dtube algebra derived in (4). We have defined its simple modules, which classify elementary string-likeexcitations terminating at gapped boundaries, and introduced a comultiplication map that defines anotion of concatenation for these string-like excitations. Furthermore, we constructed the Clebsch-Gordan series and 6 j -symbols associated with this comultiplication map. As mentioned earlier, wehave been using the language of relative groupoid algebras, since it unifies both the tube algebrasin (2+1)d and in (3+1)d. More specifically, we explained earlier how to obtain the (2+1)d algebrafrom the (3+1)d one by restricting the object in Λ( G AB ) to group variables in G and by replacingthe loop groupoid 3-cocycle α ≡ T ( π ) ∈ Z (Λ G, U(1)), where π ∈ Z ( G, U(1)), by a group 3-cocycle α ∈ Z ( G, U(1)). We shall now use this mechanism to adapt all the notions derived so far to thestudy of elementary point-like excitations at the interface between two gapped boundaries in (2+1)d.Thanks to our formulation, the notations remain almost identical. Concretely, it simply amountsto replacing g ∈ Ob(Λ(G AB )) by g ∈ G , and ( a , b ) ∈ Λ A × Λ B by ( a, b ) ∈ A × B , and to picking α in H ( G, U(1)), the other cocycle data descending from it. Note that replacing ( a , b ) by ( a, b ) ismerely formal as we have often identified the morphisms a and b with the group variables they arecharacterized by for notational convenience.Using the definition of the representation matrices together with the Clebsch-Gordan series, weshall now illustrate the mathematical structures introduced earlier by defining a complete and or-thonormal basis of excited states for any pattern of elementary point-like excitations in (2+1)d. Thesame basis can also be used to define ground state subspaces in the absence of excitations. Naturally,the same construction could be carried out in (3+1)d since we have derived all the relevant notionsin this case, which encompasses the (2+1)d one. However, we choose to focus in (2+1)d where it iseasier to visualise the construction.First, let us derive the canonical basis for a pair of dual elementary point-like excitations livingat the interfaces of two gapped boundaries labelled by the data ( A, φ ) and (
B, ψ ). This correspondsto the situation depicted in (3.3) so that we are merely looking for a canonical basis for the vectorspace C [ G AB ] αφψ . For each simple module labelled by ρ AB , this basis is defined by the set of elements | ρ AB IJ (cid:105) ∈ C [ G AB ] αφψ , with I, J ∈ { , . . . , d ρ AB } , such that | ρ AB IJ (cid:105) = (cid:16) d ρ AB | A || B | (cid:17) (cid:88) g ∈ G ( a,b ) ∈ A × B D ρ AB IJ (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) (cid:12)(cid:12) g a −→ b (cid:11) . (5.46)This transformation defines an isomorphism such that the inverse is provided by the formula (cid:12)(cid:12) g a −→ b (cid:11) = (cid:16) | A || B | (cid:17) (cid:88) ρ d ρ AB (cid:88) I,J D ρ AB IJ (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) | ρ AB IJ (cid:105) . (5.47)The latter formula expresses the fact that a given state describing such point-like boundary excitationscan be written as a sum of states describing elementary excitations. It follows immediately from theorthonormality (5.12) of the representation matrices that this basis is orthonormal: (cid:10) ρ (cid:48) AB I (cid:48) J (cid:48) (cid:12)(cid:12) ρ AB IJ (cid:11) = d ρ AB d ρ (cid:48) AB | A || B | (cid:88) g,g (cid:48) ∈ G ( a,b ) , ( a (cid:48) ,b (cid:48) ) ∈ A × B D ρ (cid:48) AB I (cid:48) J (cid:48) (cid:0)(cid:12)(cid:12) g (cid:48) a (cid:48) −−→ b (cid:48) (cid:11) D ρ AB IJ (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) (cid:10) g (cid:48) a (cid:48) −−→ b (cid:48) (cid:12)(cid:12) g a −→ b (cid:11) = δ ρ (cid:48) AB ,ρ AB δ I (cid:48) ,I δ J (cid:48) ,J (5.48) ∼ ∼ nd complete: (cid:88) ρ AB ,I,J (cid:10) ρ AB IJ (cid:12)(cid:12) ρ AB IJ (cid:11) = (cid:88) ρ AB ,I,J d ρ AB | A || B | (cid:88) g ∈ G ( a,b ) ∈ A × B D ρ AB IJ (cid:0)(cid:12)(cid:12) g a −→ b (cid:11) D ρ AB IJ (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) = (cid:88) g ∈ G ( a,b ) ∈ A × B | G | · | A | · | B | = (cid:12)(cid:12) C [ G AB ] αφψ (cid:12)(cid:12) . (5.49)Crucially, the canonical basis diagonalizes the (cid:63) -product (see proof in app. B.1): | ρ AB IJ (cid:105) (cid:63) | ρ (cid:48) AB I (cid:48) J (cid:48) (cid:105) = | A | | B | δ ρ AB ,ρ (cid:48) AB δ J,I (cid:48) d ρ AB | ρIJ (cid:48) (cid:105) . (5.50)As a useful corollary, we have that (cid:12)(cid:12) g a −→ b (cid:11) (cid:63) | ρ AB IJ (cid:105) = (cid:88) I (cid:48) D ρ AB II (cid:48) (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) | ρI (cid:48) J (cid:105) (5.51) | ρ AB IJ (cid:105) (cid:63) (cid:12)(cid:12) g a −→ b (cid:11) = (cid:88) I (cid:48) D ρ AB J (cid:48) J (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) | ρIJ (cid:48) (cid:105) . (5.52)Let Z C [ G AB ] αφψ be the centre of C [ G AB ] αφψ consisting of all elements | ψ (cid:105) ∈ C [ G AB ] αφψ that satisfy | ψ (cid:105) (cid:63) (cid:12)(cid:12) g a −→ b (cid:11) = (cid:12)(cid:12) g a −→ b (cid:11) (cid:63) | ψ (cid:105) , ∀ (cid:12)(cid:12) g a −→ b (cid:11) ∈ C [ G AB ] αφψ . (5.53)Let us consider the states | ρ AB (cid:105) := 1 d ρ AB (cid:88) I | ρ AB II (cid:105) . (5.54)It follows immediately from corollaries (5.51) and (5.52) that these states are central, i.e. | ρ AB (cid:105) (cid:63) (cid:12)(cid:12) g a −→ b (cid:11) = (cid:12)(cid:12) g a −→ b (cid:11) (cid:63) | ρ AB (cid:105) , ∀ (cid:12)(cid:12) g a −→ b (cid:11) ∈ C [ G AB ] αφψ , (5.55)from which we can easily deduce that | ρ AB (cid:105) form a complete and orthonormal basis for the centre: Z C [ G AB ] αφψ = Span C (cid:8) | ρ AB (cid:105) (cid:9) ∀ ρ AB . (5.56)We now would like to show that this centre describes the ground state subspace of our model for the annulus O depicted below: . (5.57)A triangulation O (cid:52) for O can be inferred from T [ I ] defined in (3.5) by imposing the identifications (0) ≡ (1) , (0 (cid:48) ) ≡ (1 (cid:48) ) and (00 (cid:48) ) ≡ (11 (cid:48) ) . It further follows that we can identify the space of colouredgraph-states on O (cid:52) as the subspace of coloured graph-states on T [ I ] that satisfy g = a − gb . The groundstate subspace can be finally obtained by enforcing the twisted gauge invariance at the two vertices ∼ ∼ ia the Hamiltonian projector P O (cid:52) . This operator can be easily deduced from the one appearing inthe definition of the (2+1)d open tube algebra: P O (cid:52) = 1 | A || B | (cid:88) g ∈ G ( a,b ) ∈ A × B (cid:88) (˜ a, ˜ b ) ∈ A × B δ g,a − gb ϑ ABg ( a, ˜ a | b, ˜ b ) ϑ ABg (˜ a, ˜ a − a ˜ a | ˜ b, ˜ b − b ˜ b ) (cid:12)(cid:12) ˜ a − g ˜ b ˜ a − a ˜ a −−−−−→ ˜ b − b ˜ b (cid:11)(cid:10) g a −→ b (cid:12)(cid:12) . (5.58)Crucially, this operator can be identically expressed in terms of algebra elements in C [ G AB ] αφψ asfollows (cf. proof in app. B.2) P O (cid:52) = 1 | A || B | (cid:88) g ∈ G ( a,b ) ∈ A × B (cid:88) ˜ g ∈ G (˜ a, ˜ b ) ∈ A × B (cid:16)(cid:12)(cid:12) ˜ g ˜ a −→ ˜ b (cid:11) − (cid:63) (cid:12)(cid:12) g a −→ b (cid:11) (cid:63) (cid:12)(cid:12) ˜ g ˜ a −→ ˜ b (cid:11)(cid:17)(cid:10) g a −→ b (cid:12)(cid:12) , (5.59)where (cid:12)(cid:12) ˜ g ˜ a −→ ˜ b (cid:11) − = 1 ϑ AB ˜ g (˜ a, ˜ a − | ˜ b, ˜ b − ) (cid:12)(cid:12) ˜ a − ˜ g ˜ b ˜ a − −−−→ ˜ b − (cid:11) . (5.60)Note furthermore that we can express the identity algebra element in C [ G AB ] αφψ as (cid:12)(cid:12) AB (cid:11) = (cid:88) ˜ g ∈ G (˜ a, ˜ b ) ∈ A × B (cid:12)(cid:12) ˜ g ˜ a −→ ˜ b (cid:11) − (cid:63) (cid:12)(cid:12) ˜ g ˜ a −→ ˜ b (cid:11) (5.61)such that (cid:12)(cid:12) AB (cid:11) (cid:63) (cid:12)(cid:12) g a −→ b (cid:11) = (cid:12)(cid:12) g a −→ b (cid:11) = (cid:12)(cid:12) g a −→ b (cid:11) (cid:63) (cid:12)(cid:12) AB (cid:11) , ∀ (cid:12)(cid:12) g a −→ b (cid:11) ∈ C [ G AB ] αφψ . (5.62)It implies that the image of the Hamiltonian projector P O (cid:52) is spanned by states | ψ (cid:105) ∈ C [ G AB ] αφψ satisfying | ψ (cid:105) (cid:63) (cid:12)(cid:12) g a −→ b (cid:11) = (cid:12)(cid:12) g a −→ b (cid:11) (cid:63) | ψ (cid:105) , ∀ (cid:12)(cid:12) g a −→ b (cid:11) ∈ C [ G AB ] αφψ , (5.63)which is precisely the definition of the centre of | ψ (cid:105) ∈ C [ G AB ] αφψ . We deduce that the ground statesubspace on O (cid:52) is spanned by the states | ρ AB (cid:105) : V G,A,Bα,φ,ψ [ O (cid:52) ] = Im P O (cid:52) = Z C [ G AB ] αφψ = Span C (cid:8) | ρ AB (cid:105) (cid:9) ∀ ρ AB . (5.64)As an immediate consequence of this statement is the fact that the ground state degeneracy of theannulus equals the number of elementary boundary point-like excitations at the interface of two gappedboundaries. This mimics the well-know result that the number of bulk point-like excitations equalsthe ground state degeneracy on the torus.Let us pursue our construction by defining the canonical basis associated with the following configu-ration: A φ B ψ C ϕ → (cid:39) , (5.65)i.e. the two-disk D from which local neighbourhoods at the interface of the three gapped boundarieshave been removed. This manifold is referred to as the thrice-punctured two-disk and is denoted ∼ ∼ y Y . We choose a triangulation Y (cid:52) for this manifold and consider the following space of colouredgraph-states:Span C (cid:40)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) g (cid:34) (cid:48) (cid:48) (cid:48) (cid:48)(cid:48) (cid:48)(cid:48) (cid:48)(cid:48) (cid:35)(cid:43)(cid:41) ∀ g ∈ Col( Y (cid:52) ,G,A,B,C ) ≡ Span C (cid:40)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a (cid:48) g g c (cid:48)− a − g b b (cid:48) − g c a a (cid:48) c (cid:48) cb b (cid:48) (cid:48) (cid:48) (cid:48) (cid:48)(cid:48) (cid:48)(cid:48) (cid:48)(cid:48) (cid:43)(cid:41) ∀ g ,g ∈ G ∀ a,a (cid:48) ∈ A ∀ b,b (cid:48) ∈ B ∀ c,c (cid:48) ∈ C ≡ | g , a, b, g , b (cid:48) , c, a (cid:48) , c (cid:48) (cid:105) Y (cid:52) . We are interested in the ground state subspace V G,A,B,Cα,φ,ψ,ϕ [ Y (cid:52) ] on this manifold. In order to obtain thisHilbert space, we need to apply the Hamiltonian projector P Y (cid:52) simultaneously at all three physicalboundary vertices. This operator is obtained by evaluating the partition function (2.15) on the relativepinched interval cobordism (cid:48)(cid:48) (cid:48) (cid:48)(cid:48) ˜1 (cid:48) ˜1 (cid:48)(cid:48) (cid:48)(cid:48) (cid:48) ˜1 3 (cid:48) (5.66)and its action explicitly reads P Y (cid:52) (cid:0) | g , a, b, g , b (cid:48) , c, g , a (cid:48) , c (cid:48) (cid:105) Y (cid:52) (cid:1) (5.67)= 1 | A || B || C | (cid:88) ˜ a ∈ A ˜ b ∈ B ˜ c ∈ C ϑ ACa (cid:48) g g c (cid:48)− ( a (cid:48) , ˜ a | c (cid:48) , ˜ c ) ϑ ABg (˜ a, ˜ a − a | ˜ b, ˜ b − b ) ϑ BCg (˜ b, ˜ b − b (cid:48) | ˜ c, ˜ c − c ) ζ ABC ˜ a, ˜ b, ˜ c ( g , g ) × | ˜ a − g ˜ b, ˜ a − a, ˜ b − b, ˜ b − g ˜ c, ˜ b − b (cid:48) , ˜ c − c, a (cid:48) ˜ a, c (cid:48) ˜ c (cid:105) Y (cid:52) . (5.68)Let us now define the following basis states | ρ AB I AB , ρ BC I BC , ρ AC I AC (cid:105) Y (cid:52) := (cid:88) { g ∈ G } (cid:88) a,a (cid:48) ∈ Ab,b (cid:48) ∈ Bc,c (cid:48) ∈ C (cid:88) { J } D ρ AB J AB I AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) g b (cid:48) −−→ c (cid:11)(cid:1)(cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105) D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) a (cid:48) g g c (cid:48)− a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) × | g , a, b, g , b (cid:48) , c, a (cid:48) , c (cid:48) (cid:105) Y (cid:52) . We can show using the invariance property (5.29) of the Clebsch-Gordan coefficients that these basisstates diagonalise the action of the Hamiltonian projector, i.e. for every { ρ x I x } x = AB,BC,AC we have P Y (cid:52) (cid:0) | ρ AB I AB , ρ BC I BC , ρ AC I AC (cid:105) Y (cid:52) (cid:1) = | ρ AB I AB , ρ BC I BC , ρ AC I AC (cid:105) Y (cid:52) . (5.69)A proof of this crucial relation can be found in app. B.3. We refer to these states as the canonical basisstates for Y (cid:52) . It follows from the orthogonality and the completeness of the representation matricesas well as the Clebsch-Gordan series, that this basis is orthogonal and complete. ∼ ∼ t is now possible to use the canonical basis states we have derived so far in order to defineexcited states associated with more complicated boundary patterns. For instance, the case of D withfour different gapped boundaries can be treated easily by noticing that the manifold resulting fromremoving local neighbourhoods at every interface can be realised as the gluing of two copies of Y (cid:52) .Similarly, canonical basis states for this manifold are obtained via the (cid:63) -product by contracting twostates of Y (cid:52) along one magnetic index. Interestingly, two different bases can be defined following thisscheme, but they are equivalent. This is ensured by the quasi-coassociativity, and more specificallythe isomorphism (5.41). As a matter of fact, the two bases can be explicitly related to each other viathe 6 j -symbols as defined in (5.43), which was the motivation for introducing them. More generally,any number of gapped boundaries can be treated in a similar fashion by gluing several copies Y (cid:52) according to a fusion binary tree . Thanks to the quasi-coassociativity, the choice of tree is not relevantas the corresponding bases are all equivalent. SECTION 6
Gapped boundaries and higher algebras
In this section, we describe a higher categorical construction capturing the salient features of thegapped boundary excitations considered in the previous sections. We begin by reviewing the definitionsof monoidal categories and bicategories before introducing the theory of module categories. For moredetails on such constructions, see for example [29, 56, 59, 60]. Building upon such notions, we thendemonstrate the relation between gapped boundary excitations and bicategories of module categories.In particular we review that the bicategory
MOD ( Vec αG ) provides a convenient description of gappedboundary excitations in (2+1)d Dijkgraaf-Witten theory [10], and show that MOD ( Vec T ( π )Λ G ) describesstring-like bulk excitations terminating at the boundary in (3+1)d Dijkgraaf-Witten theory. We begin this section by first introducing higher category theory . In order to motivate the ethos ofhigher category theory, it is illuminating to first consider the notion of categorification . Generally,categorification refers to a collection of techniques in which statements about sets are translated intostatements about categories. Let us consider a simple example. Given a pair of sets
X, Y and atriple of functions f, g, h : X → Y , it is natural to pose relations between such functions in terms ofequations. For instance, we may have f = g and g = h as functions from X to Y , from which we caninfer the relation f = h by transitivity. In this setting, categorification is the process whereby each set X is replaced by a category C X , and each function f : X → Y is sent to a functor F f : C X → C Y . Usingthe additional structure proper to categories, we have a choice about the way we lift the equations f = g and g = h . We could either require the corresponding functors to be equal, i.e. F f = F g and F g = F h implying F f = F h , or alternatively, we could instead require only the existence of naturalisomorphisms, i.e. η fg : F f ∼ −→ F g and η gh : F g ∼ −→ F h . In the latter case, we use equations on thenatural transformations in order to prescribe a natural transformation η fh = η fg ◦ η gh : F f → F h replacing transitivity.Building upon the idea of categorification, let us now introduce bicategories , which will form themodel of higher category theory utilised in the following discussion. Given a (small) category C , recallthat we denote by Hom C ( X, Y ) the set of (1-)morphisms (hom-set) between the objects
X, Y ∈ Ob( C ).Roughly speaking, a bicategory is obtained by applying the categorification mechanism spelt out aboveto such sets of morphisms. More specifically, we replace Hom C ( X, Y ) with a category that we denoteby
Hom C ( X, Y ). The composition function ◦ : Hom C ( X, Y ) × Hom C ( Y, X ) → Hom C ( X, Z ) is then ∼ ∼ eplaced with a composition bifunctor ⊗ : Hom C ( X, Y ) × Hom C ( Y, Z ) → Hom C ( X, Z ). Moreover,equations between morphisms are replaced with natural transformations between functors togetherwith equations defined for such natural transformations. With this idea in mind, we now define ournotion of bicategory:
Definition
Bicategory ) . A bicategory B i consists of: • A set of objects
Ob( B i ) . • For each pair of objects
X, Y ∈ Ob( B i ) , a category Hom B i ( X, Y ) , whose objects and morphismsare referred to as 1- and 2-morphisms, respectively. Given a 1-morphism f ∈ Hom B i ( X, Y ) , X =: s( f ) and Y =: t( f ) are referred to as the ‘source’ and the ‘target’ objects of f , respectively.The composition of 2-morphisms in Hom B i ( X, Y ) is designated as the ‘vertical’ composition. • For each triple of objects
X, Y, Z ∈ Ob( B i ) , a binary functor ⊗ : Hom B i ( X, Y ) × Hom B i ( Y, Z ) → Hom B i ( X, Z ) designated as the ‘horizontal’ composition. • For each object X ∈ Ob( B i ) , a 1-morphism X ∈ Ob(
Hom B i ( X, X )) , and for each morphism f : X → Y , a pair of natural isomorphism (cid:96) f : X ⊗ f → f and r Y : f ⊗ Y → f called the‘left’ and ‘right’ unitors, respectively. • For each triple of composable 1-morphisms f, g, h , a natural isomorphism α f,g,h : ( f ⊗ g ) ⊗ h → f ⊗ ( g ⊗ h ) called the 1-associator.This data is subject to coherence relations encoded in the commutativity of the diagrams (( f ⊗ g ) ⊗ h ) ⊗ k ( f ⊗ ( g ⊗ h )) ⊗ k ( f ⊗ g ) ⊗ ( h ⊗ k ) f ⊗ (( g ⊗ h ) ⊗ k ) f ⊗ ( g ⊗ ( h ⊗ k )) α f ⊗ g , h , k α f,g,h ⊗ k α f , g , h ⊗ i d k α f,g ⊗ h,k id f ⊗ α g,h,k and ( f ⊗ t( f ) ) ⊗ g f ⊗ ( t( f ) ⊗ g ) f ⊗ g α f, t( f ) ,g id f ⊗ (cid:96) g r f ⊗ id g (6.1) for all composable 1-morphisms f, g, h, k , referred to as the pentagon and the triangle relations,respectively. As in conventional category theory, it is customary to depict relations in a bicategory using diagram-matic calculus. Unlike the directed graph structure utilised in category theory, the diagrammaticpresentation of bicategories is given in terms of so-called pasting diagrams of the form
X Y fgF (6.2) ∼ ∼ here X, Y ∈ Ob( B i ) are objects, f, g ∈ Ob(
Hom B i ( X, Y )) are 1-morphisms and F ∈ Hom
Hom B i ( X,Y ) ( f, g )is a 2-morphism. In this notation, horizontal and vertical compositions are depicted as X Y Z fgF f (cid:48) g (cid:48) F (cid:48) = X Z f ⊗ f (cid:48) g ⊗ g (cid:48) F ⊗ F (cid:48) and X Y ff (cid:48)(cid:48) FG = X Y ff (cid:48)(cid:48) F G , respectively. Explicit examples of bicategories will be provided in sec. 6.5 and 6.6 Vec α G We shall now apply the idea of categorification to groupoid algebras, yielding a notion of ‘highergroupoid algebra’. First, let us review the relation between monoids and categories. A monoid isdefined by a set X equipped with a function · : X × X → X called the product, and a distinguishedelement ∈ X called the unit, satisfying the relations · x = x = x · , ∀ x ∈ X . Alternatively,a monoid can be defined as a (small) category C with a single object • and Hom( C ) = Hom C ( • , • )such that the composition function ◦ : Hom C ( • , • ) × Hom C ( • , • ) → Hom C ( • , • ) provides the monoidproduct on Hom C ( • , • ), and the identity morphism id • provides the corresponding monoid unit. Usingthis presentation of a monoid as a one-object category, we recover upon categorification the notion of monoidal category as a one-object bicategory: given a bicategory B i with a single object Ob( B i ) = { • } ,the category of homorphisms Hom B i ( • , • ) defines a monoidal category equipped with a tensor productstructure provided by the bifunctor ⊗ : Hom B i ( • , • ) × Hom B i ( • , • ) → Hom B ( • , • ). In particular, the1-associator in B i induces the (0-)associator in the monoidal category Hom B i ( • , • ).Akin to the categorification of a monoid to a monoidal category, one can consider a categorificationof an algebra over a field. Instead of presenting the general case, we shall restrict ourselves to thecategorification of groupoid algebras. Recall that given a finite groupoid G , the (complex) groupoidalgebra C [ G ] is the algebra defined over the vector space Span C {| g (cid:105) | ∀ g ∈ Hom( G ) } with algebraproduct | g (cid:105) (cid:63) | g (cid:48) (cid:105) := δ t( g ) , s( g (cid:48) ) | gg (cid:48) (cid:105) . One natural categorification of C [ G ] is given by replacing thecomplex field with the (symmetric) monoidal category Vec of finite dimensional complex-vector spaces,which yields the monoidal category of groupoid-graded vector spaces:
Definition
Category of G -graded vector spaces ) . Let G be a finite groupoid. A G -gradedvector space is a vector space of the form V = (cid:76) g ∈ Hom( G ) V g . We call a G -graded vector space V ‘homogeneous’ of degree g ∈ Hom( G ) if V g (cid:48) is the zero vector space for all g (cid:48) (cid:54) = g . The monoidalcategory Vec G is then defined as the category whose objects are G -graded complex-vector spaces,and morphisms are grading preserving linear maps. The tensor product is defined on homogeneouscomponents V g and W g (cid:48) according to V g ⊗ W g (cid:48) = (cid:40) ( V ⊗ W ) gg (cid:48) if t( g ) = s( g (cid:48) )0 otherwise (6.3) with unit object = (cid:76) g ∈ Hom( G ) δ id g . There are | Hom( G ) | simple objects denoted by C g , ∀ g ∈ Hom( G ) . Every object is isomorphic to a direct sum of simple objects, making Vec G semi-simple.Finally, the associator is given by the canonical map id C gg (cid:48) g (cid:48)(cid:48) : ( U g ⊗ V g (cid:48) ) ⊗ W g (cid:48)(cid:48) ∼ −→ U g ⊗ ( V g (cid:48) ⊗ W g (cid:48)(cid:48) ) . (6.4) ∼ ∼ ote that by choosing the groupoid to be the delooping of a finite group, we recover the more familiarfact that the category of G -graded vector spaces is a categorification of the notion of group algebra.Analogously to the twisting of a groupoid algebra by a groupoid 2-cocycle, we can twist the associatorof Vec G by a normalised groupoid 3-cocycle α ∈ Z ( G , U(1)) so as to define the monoidal category
Vec α G , whereby the associator on simple objects is provided by α C g , C g (cid:48) , C g (cid:48)(cid:48) = α ( g , g (cid:48) , g (cid:48)(cid:48) ) · id C gg (cid:48) g (cid:48)(cid:48) : ( C g ⊗ C g (cid:48) ) ⊗ C g (cid:48)(cid:48) ∼ −→ C g ⊗ ( C g (cid:48) ⊗ C g (cid:48)(cid:48) ) . (6.5)The monoidal category Vec α G has the additional property of being a multi-fusion category : Definition
Multifusion category ) . A category C is called multi-fusion if C is a finite semi-simple, C -linear, abelian, rigid monoidal category such that tensor product ⊗ : C × C → C isbilinear on morphisms. If additionally
Hom C ( , ) ∼ = C then we call C a fusion category. We shall not expand on this definition here, but instead refer the reader to the chapter 4 of [56].Conceptually, the observation that
Vec α G is a multi-fusion category plays a similar role to semi-simplicityin the theory of algebras. Recall that given a semi-simple algebra A , every module is isomorphic toa direct sum of simple modules. These simple modules can be found via the notion of primitiveorthogonal idempotents . An idempotent in an algebra A is an element e ∈ A such that e · e = e , anda pair of idempotents e, e (cid:48) ∈ A are orthogonal if e · e (cid:48) = δ e,e (cid:48) e . Such an idempotent is called primitiveif it cannot be written as sum of non-trivial idempotents. Specifying a complete set of primitiveorthogonal idempotents { e , . . . , e n } for A , we can define a simple right A -module M i = e i · A , foreach i ∈ , . . . , n . In the following, we will review the notion of module category over a multi-fusioncategory, categorifying the notion of module over a semi-simple algebra. In this setting the analogueof idempotent will be given by so called separable algebra objects . In this part, we introduce the notions of module category over multi-fusion category C , and modulecategory functors following closely [56]. These happen to be relevant notions to describe gappedboudaries and their excitations [10]. However, as we explain below, we use in practice an equivalentdescription in terms of separable algebra objects. First, let us define a module category: Definition C -Module category) . Given a multi-fusion category
C ≡ ( C , ⊗ , , (cid:96), r, α ) , a (left) C -module category is defined by a triple ( M , (cid:12) , ˙ α ) consisting of a category M , an action bifunctor (cid:12) : C × M → M and a natural isomorphism ˙ α X,Y,M : ( X ⊗ Y ) (cid:12) M ∼ −→ X (cid:12) ( Y (cid:12) M ) , ∀ X, Y ∈ Ob( C ) and M ∈ Ob( M ) , (6.6) referred to as the module associator, such that the diagram (( X ⊗ Y ) ⊗ Z ) (cid:12) M ( X ⊗ ( Y ⊗ Z )) (cid:12) M ( X ⊗ Y ) (cid:12) ( Z (cid:12) M ) X (cid:12) (( Y ⊗ Z ) (cid:12) M ) X (cid:12) ( Y (cid:12) ( Z (cid:12) M )) ˙ α X ⊗ Y , Z , M ˙ α X,Y,Z (cid:12) M α X , Y , Z ⊗ i d M ˙ α X,Y ⊗ Z,M id X ⊗ ˙ α Y,Z,M (6.7) ∼ ∼ ommutes for every X, Y, Z ∈ Ob( C ) and M ∈ Ob( M ) . Additionally there is a unit isomorphism (cid:96) M : (cid:12) M ∼ −→ M , where is the tensor unit of C , such that the following diagram commutes: ( X ⊗ ) (cid:12) M X ⊗ ( (cid:12) M ) X (cid:12) M ˙ α X, ,M id X ⊗ (cid:96) M r X ⊗ id M , (6.8) for all X ∈ Ob( C ) , M ∈ Ob( M ) . Every module category can be decomposed into so-called indecomposable module categories [55]:
Definition . A C -module category M is said to be ‘inde-composable’ when M is not equivalent to a direct sum of non-zero C -module categories. Indecomposable module categories will turn out to be the relevant data to label gapped boundaries.To describe excitations, we further require the notion module category functors:
Definition . Given a multi-fusion category C and a pair ( M , M ) of C -module categories with module associators ˙ α and ¨ α , respectively, a C -module functor is a pair ( F, s ) where F : M → M is a functor, and s is natural isomorphism given by s X,M : F ( X (cid:12) M ) → X (cid:12) F ( M ) , ∀ X ∈ Ob( C ) and M ∈ Ob( M ) , (6.9) such that the diagram F ( X (cid:12) ( Y (cid:12) M )) F (( X ⊗ Y ) (cid:12) M ) ( X ⊗ Y ) (cid:12) F ( M ) X (cid:12) F ( Y (cid:12) M ) X (cid:12) ( Y (cid:12) F ( M )) F ( ˙ α X,Y,M ) s X,Y ⊗ M id X (cid:12) s Y,M s X ⊗ Y,M ¨ α X,Y,F ( M ) (6.10) commutes for every X, Y ∈ Ob( C ) and M ∈ Ob( M ) . We are almost ready to define a bicategory, the remaining ingredient is a notion of morphism formodule functors:
Definition . Given a multi-fusion category C and two C -module functors ( F, s ) and ( F (cid:48) , s (cid:48) ) , a morphism of module functors between F and F ’ is a naturaltransformation η : F → F (cid:48) such that the diagram F ( X (cid:12) M ) X (cid:12) F ( M ) F (cid:48) ( X (cid:12) M ) X (cid:12) F (cid:48) ( M ) s X,M id X (cid:12) η M η X (cid:12) M s (cid:48) X,M (6.11) commutes for every X ∈ Ob( C ) and M ∈ Ob( M ) . Putting everything together, we obtain the following definition of a bicategory of module categories ∼ ∼ efinition . Given a multi-fusion category C , we denote by MOD ( C ) the bicategory with objects, C -module categories, 1-morphisms, C -module functors, and2-morphisms, C -module natural transformations. The remainder of this section is dedicated to providing a more practical formulation of this bicategoryusing the fact that for a multi-fusion category C , every indecomposable C -module category is equivalentto the category of module objects for a separable algebra object in C [56]. Using this latter formulation,we shall then explain how the bicategory of module categories is indeed the relevant notion to describegapped boundaries and their excitations in gauge models of topological phases. Vec α G Let us now present the notion of algebra objects in the multi-fusion category
Vec α G thought as acategorification of the groupoid algebra over G . In the subsequent discussion, we will build uponthis notion in order to define module categories over higher groupoid algebras as a categorification ofmodules over semi-simple algebras. Definition . Given a multi-fusion category
C ≡ ( C , ⊗ , , (cid:96), r, α ) , an (asso-ciative) algebra object in C is defined by a triple ( A, m, u ) consisting of an object A as well asmorphisms m : A ⊗ A → A and u : → A in C referred to as multiplication and unit, respectively,such that the diagrams below commute: • A ssociativity: ( A ⊗ A ) ⊗ A A ⊗ ( A ⊗ A ) A ⊗ AA ⊗ A A α id A ⊗ mmm ⊗ id A m , (6.12) • U nit: A A ⊗ A ⊗ A ⊗ A A ⊗ A A r − id A ⊗ u m(cid:96) − u ⊗ id A m i d A , (6.13) where α, (cid:96), r refer to the associator, left unitor and right unitor for the monoidal structure of C ,respectively. Given the above definition, an important observation is that algebra objects in the fusion category
Vec correspond to associative, unital, finite-dimensional, complex algebras. Let us now consider algebraobjects in
Vec α G . For each ( G , α )-subgroupoid ( A , φ ), as defined in sec. 4.3, we construct an algebraobject A φ ≡ ( (cid:76) a ∈ Hom( A ) C a , m, u ) with multiplication and unit defined according to m : A φ ⊗ A φ → A φ : a ⊗ a (cid:48) (cid:55)→ δ t( a ) , s( a (cid:48) ) φ ( a , a (cid:48) ) aa (cid:48) and u ( Vec α G ) := (cid:88) X ∈ Ob( A φ ) id X , respectively. In particular, we remark that the algebra object A φ in Vec α G corresponds to a generali-sation of a twisted groupoid algebra over A , where the twisting by a 2-cocycle is instead given by the ∼ ∼ -cochain φ . Since φ is not a groupoid 2-cocycle, algebra objects are not associative as conventionalalgebras, but instead are only associative within Vec α G due to the condition d (2) φ = α − |A . We leaveit to the reader to check that every algebra object in Vec α G is in one-to-one correspondence with a( G , α )-subgroupoid and Vec α G algebra objects.Given an algebra object A in a multi-fusion category C , we are interested in modules over A referred to as A -module objects : Definition . Let C be a multi-fusion category and A ≡ ( A, m, u ) analgebra object in C . A right module object over A (or right A -module) consists of a pair ( M, p ) ,with M ∈ Ob( C ) and p : M ⊗ A → M ∈ Hom( C ) such that the diagrams below commute: • C ompatibility: ( M ⊗ A ) ⊗ A M ⊗ AM ⊗ ( A ⊗ A ) M ⊗ A M p ⊗ id A pα id M ⊗ m p , (6.14) • U nit: M MM ⊗ M ⊗ A id M pr − u . (6.15)Homorphisms between modules over a given algebra object are then defined in an obvious way: Definition . Given an algebra object A in a multifusioncategory C , let ( M , p ) and ( M , p ) be two right A -modules. An A -homomorphism between these A -modules is a morphism f ∈ Hom C ( M , M ) such that the diagram M ⊗ A M ⊗ AM M f ⊗ id A p p f (6.16) commutes. It follows from the definition above that A -homomorphisms between a pair of A -module objects( M , p ) and ( M , p ) in C define a subspace of Hom C ( M , M ), which is notated via Hom A ( M , M )in the following. Moreover, composing A -homomorphisms yields another A -homomorphism so thatwe can define a category of A -modules as follows: Definition . Given a multi-fusion category C and an algebraobject A = ( A, m, u ) , we define the category Mod C ( A ) as the category with objects A -moduleobjects in C and morphisms A -module homomorphisms. In a similar vein, we can define a left A -module objects and left A -module homomorphisms. We leaveit to the reader to derive the corresponding axioms. Combining both left and right modules over analgebra object yields the notion bimodule object : ∼ ∼ efinition . Let C be a multi-fusion category and ( A, B ) a pair of algebraobjects in C . We define an ( A, B ) -bimodule object in C as a triple ( M, p, q ) such that ( M, p ) is aright B -module object, ( M, q ) is a left A -module object and the diagram ( A ⊗ M ) ⊗ B M ⊗ BA ⊗ ( M ⊗ B ) A ⊗ M M q ⊗ id B pα id A ⊗ p q . (6.17) commutes. Noticing that the monoidal identity of any multi-fusion category C naturally defines an algebra object,we can identify the ( , A )-bimodule ( M, (cid:96) M , p ), for a given algebra object A , with the right A -module( M, p ), and similarly the ( A, )-bimodule ( M, r M , q ) with the left A -module ( M, q ). Definition . Let ( M , p , q ) and ( M , p , q ) be a pairof ( A, B ) -bimodule objects in a multi-fusion category C . An ( A, B ) -homomorphism between these ( A, B ) -bimodules is a morphism f ∈ Hom C ( M , M ) such that f : ( M , p ) → ( M , p ) is a right B -module homomorphism, f : ( M , q ) → ( M , q ) is a left A -module homomorphism, and thefollowing diagram commutes: ( A ⊗ M ) ⊗ B M ⊗ B ( A ⊗ M ) ⊗ B M ⊗ BA ⊗ ( M ⊗ B ) A ⊗ M M A ⊗ ( M ⊗ B ) A ⊗ M M q ⊗ id B p α id A ⊗ p q q ⊗ id B p α id A ⊗ p q ( i d A ⊗ f ) ⊗ i d B i d A ⊗ ( f ⊗ i d B ) id A ⊗ f f ⊗ i d B f . It follows from the definition that (
A, B )-homomorphisms between a pair of (
A, B )-bimodule ob-ject ( M , p , q ) and ( M , p , q ) in C define a subspace of Hom C ( M , M ), which will be denoted byHom A,B ( M , M ) in the following. Moreover, composing two ( A, B )-homomorphisms yields another(
A, B )-homomorphism so that we can define the following category of (
A, B )-bimodules:
Definition . Given a multi-fusion category C and a pairof algebra objects A and B , we define the category Bimod C ( A, B ) as the category with objects ( A, B ) -bimodules and morphisms ( A, B ) -bimodule homomorphisms. Let us now go back to our example of interest, namely the higher groupoid algebras
Vec α G , and describethe corresponding bimodule objects. We consider a pair ( A , φ ), ( B , ψ ) of ( G , α )-subgroupoids, and thecorresponding algebra objects A φ ≡ ( (cid:76) a ∈ Hom( A ) C a , m A , u A ), and B ψ ≡ ( (cid:76) b ∈ Hom( B ) C b , m B , u B ).Let ( M, p, q ) be an ( A φ , B ψ )-bimodule in Vec α G such that M = (cid:76) g ∈ Hom( G ) M g , p : M ⊗ B ψ → M and ∼ ∼ : A φ ⊗ M → M . Let us consider the Vec α G morphism pq ≡ q ◦ (id A ⊗ p ) such that pq : A φ ⊗ ( M ⊗ B ψ ) → M : C a ⊗ ( M g ⊗ C b ) (cid:55)→ δ t( a ) , s( g ) δ s( b ) , t( g ) [ M g (cid:46) pq ( M g , C a , C b )] ∈ M agb where pq ( M g , C a , C b ) : M g → M agb is a linear map which includes a G -grading shift. In virtue of thecompatibility conditions satisfied by p and q , the diagram A φ ⊗ (( A φ ⊗ ( M ⊗ B ψ )) ⊗ B ψ ) A φ ⊗ ( M ⊗ B ψ ) A φ ⊗ ( M ⊗ B ψ ) M id A ⊗ ( pq ⊗ id B ) pq A m B pq , (6.18)commutes, where A m B decomposes as A m B : A φ ⊗ (( A φ ⊗ ( M ⊗ B ψ )) ⊗ B ψ ) id A ⊗ α A ,M ⊗B , B −−−−−−−−−−→ A φ ⊗ ( A φ ⊗ (( M ⊗ B ψ ) ⊗ B ψ )) id A ⊗ (id A ⊗ α M, B , B ) −−−−−−−−−−−−→ A φ ⊗ ( A φ ⊗ ( M ⊗ ( B ψ ⊗ B ψ ))) id A ⊗ (id A ⊗ (id M ⊗ m B )) −−−−−−−−−−−−−−−→ A φ ⊗ ( A φ ⊗ ( M ⊗ B ψ )) α − A ,A,M ⊗B −−−−−−−→ ( A φ ⊗ A φ ) ⊗ ( M ⊗ B ψ ) m A ⊗ id M ⊗B −−−−−−−−→ A φ ⊗ ( M ⊗ B ψ ) . (6.19)Furthermore, it acts on non-zero basis vectors ( a , a (cid:48) , b , b (cid:48) ) ∈ C a × C a (cid:48) × C b × C b (cid:48) and v g ∈ M g as A m B : a (cid:48) ⊗ (( a ⊗ ( v g ⊗ b )) ⊗ b (cid:48) ) (cid:55)→ δ t( a (cid:48) ) , s( a ) δ t( b ) , s( b (cid:48) ) (cid:36) AB g ( a , a (cid:48) | b , b (cid:48) ) [ a (cid:48) a ⊗ ( v g ⊗ bb (cid:48) )] (6.20)for any set of a (cid:48) , a , g , b , b (cid:48) composable morphisms in G , where we introduced the cocycle data (cid:36) AB g ( a , a (cid:48) | b , b (cid:48) ) := α ( a , gb , b (cid:48) ) α ( g , b , b (cid:48) ) α ( a (cid:48) , a , gbb (cid:48) ) φ ( a (cid:48) , a ) ψ ( b , b (cid:48) ) . (6.21)Writing pq : A φ ⊗ ( M ⊗ B ψ ) → M : a ⊗ ( v g ⊗ b ) (cid:55)→ v g (cid:46) pq ( v g , a , b ) ∈ M agb , it follows from equation (6.20) that pq ( v g , a , b ) ∈ End( M ) satisfies the algebra pq ( v g , a , b ) (cid:46) pq ( v g (cid:48) , a (cid:48) , b (cid:48) ) = δ t( a (cid:48) ) , s( a ) δ t( b ) , s( b (cid:48) ) δ g (cid:48) , agb (cid:36) AB g ( a , a (cid:48) | b , b (cid:48) ) pq ( v g , a (cid:48) a , bb (cid:48) ) (6.22)for all g , g (cid:48) ∈ Hom G (Ob( A ) , Ob( B ))), a ∈ Hom A ( − , s( g )), a (cid:48) ∈ Hom A ( − , s( g (cid:48) )), b ∈ Hom B (t( g ) , − )and b (cid:48) ∈ Hom B (t( g (cid:48) ) , − ). Such data can be concisely described by introducing the groupoid (cid:101) G AB withobject set Hom G (Ob( A ) , Ob( B )) and morphism set given by g −−−→ a , b agb ≡ g −−−→ a , b , (6.23)for all g ∈ Ob( (cid:101) G AB ), a ∈ Hom A ( − , s( g )) and b ∈ Hom B (t( g ) , − ). Composition is defined by g −−−→ a , b agb −−−−→ a (cid:48) , b (cid:48) a (cid:48) agbb (cid:48) = g −−−−−→ a (cid:48) a , bb (cid:48) a (cid:48) agbb (cid:48) , (6.24) ∼ ∼ or all composable pairs ( a (cid:48) , a ) ∈ A . and ( b , b (cid:48) ) ∈ B . . Noting that [ (cid:36) AB ] ∈ H ( (cid:101) G AB , U(1))defines a (cid:101) G AB pq can then be described via a weak functor F pq M : (cid:101) G AB → Vec : g ∈ Ob( (cid:101) G AB ) (cid:55)→ M g ⊂ M : g −−−→ a , b ∈ Hom( (cid:101) G AB ) (cid:55)→ pq ( v g , a , b ) : M g → M agb , such that every isomorphism pq ( v g , a , b ) satisfies the composition relation (6.22). Using the equivalencebetween representations and modules of algebraic structures, we can thus view the pair ( M, F pq M ) asa module over the twisted groupoid algebra C [ (cid:101) G AB ] (cid:36) AB . Considering the diagram A φ ⊗ ( M ⊗ B ψ ) A φ ⊗ ( M ⊗ B ψ ) M M A ⊗ ( f ⊗ id B ) pq pq f , (6.25)for a pair of ( A φ , B ψ )-bimodules ( M , pq ( − )) and ( M , pq ( − )), we conclude that an ( A φ , B ψ )-bimodulehomomorphism is defined via a natural transformation f : pq → pq , or equivalently, as an inter-twiner for representations of C [ (cid:101) G AB ] (cid:36) AB . Putting everything together, we obtain the equivalence Bimod
Vec α G ( A φ , B ψ ) (cid:39) Mod ( C [ (cid:101) G AB ] (cid:36) AB ). Vec α G Pursuing our construction, we shall now introduce a special class of algebra objects known as separablealgebra objects . We will then construct a bicategory whose objects are separable objects, and mor-phisms are bimodule objects between them. First, let us define what it means for an algebra objectto be separable:
Definition . Let C be a multi-fusion category and A ≡ ( A, m, u ) an algebra object in C . The algebra object A is said to be ‘separable’ if the multiplication map m : A ⊗ A → A admits a ‘section’ map ∆ : A → A ⊗ A such that A ∆ −→ A ⊗ A m −→ A = A id A −−→ A , as an ( A, A ) -bimodule homomorphism. Let us now define a binary functor. Let
A, B, C be three separable algebra objects in a multi-fusioncategory C , M AB ≡ ( M AB , q A , p B ) ≡ ( M AB , pq M AB ) an ( A, B )-bimodule, and M BC ≡ ( M BC , q B , p C ) ≡ ( M BC , pq M BC ) a ( B, C )-bimodule. Using this data, we want to construct an (
A, C )-bimodule, whichwe shall notate via ( M AB ⊗ B M BC , pq M AB ⊗ B pq M BC ). First, let us define the morphism pq M AB ⊗ M BC : ∼ ∼ ⊗ (( M AB ⊗ M BC ) ⊗ C ) → M AB ⊗ M BC that decomposes as A ⊗ (( M AB ⊗ M BC ) ⊗ C ) id A ⊗ α MAB,MBC,C −−−−−−−−−−−−→ A ⊗ ( M AB ⊗ ( M BC ⊗ C )) (6.26) α − A,MAB,MBC ⊗ C −−−−−−−−−−−→ ( A ⊗ M AB ) ⊗ ( M BC ⊗ C ) ( (cid:96) − A ⊗ MAB ) ⊗ id MBC ⊗ C −−−−−−−−−−−−−−→ (( A ⊗ M AB ) ⊗ ) ⊗ ( M BC ⊗ C ) (id A ⊗ MAB ⊗ u B ) ⊗ id MBC ⊗ C −−−−−−−−−−−−−−−−−−→ (( A ⊗ M AB ) ⊗ B ) ⊗ ( M BC ⊗ C ) (id A ⊗ MAB ⊗ ∆ B ) ⊗ id MBC ⊗ C −−−−−−−−−−−−−−−−−−→ (( A ⊗ M AB ) ⊗ ( B ⊗ B )) ⊗ ( M BC ⊗ C ) α − A ⊗ MAB,B,B ⊗ id MBC ⊗ C −−−−−−−−−−−−−−−−→ ((( A ⊗ M AB ) ⊗ B ) ⊗ B ) ⊗ ( M BC ⊗ C ) ( α A,MAB,B ⊗ id B ) ⊗ id MBC ⊗ C −−−−−−−−−−−−−−−−−−−→ (( A ⊗ ( M AB ⊗ B )) ⊗ B ) ⊗ ( M BC ⊗ C ) α A ⊗ ( MAB ⊗ B ) ,B,MBC ⊗ C −−−−−−−−−−−−−−−−→ ( A ⊗ ( M AB ⊗ B )) ⊗ ( B ⊗ ( M BC ⊗ C )) pq MAB ⊗ pq MBC −−−−−−−−−−→ M AB ⊗ M BC . Using this morphism, let us further define the endomorphism e M AB ⊗ M BC : M AB ⊗ M BC → M AB ⊗ M BC that decomposes as M AB ⊗ M BC r − MAB ⊗ MBC −−−−−−−−→ ( M AB ⊗ M BC ) ⊗ (cid:96) − MAB ⊗ MBC ) ⊗ −−−−−−−−−−−→ ⊗ (( M AB ⊗ M BC ) ⊗ ) (6.27) u A ⊗ (id MAB ⊗ MBC ⊗ u C ) −−−−−−−−−−−−−−−−→ A ⊗ (( M AB ⊗ M BC ) ⊗ C ) pq MAB ⊗ MBC −−−−−−−−→ M AB ⊗ M BC . By the requirement that ∆ : B → B ⊗ B is a ( B, B )-bimodule section to the (
B, B )-bimodulehomomorphism m : B ⊗ B → B , together with the compatibility conditions spelt out above and thenaturalness of the associator α , we can show that e M AB ⊗ M BC is an idempotent endomorphism in C , i.e. e M AB ⊗ M BC ◦ e M AB ⊗ M BC = e M AB ⊗ M BC . The requirement that the multi-fusion category C is abelianensures that every idempotent is a split idempotent : Definition . An idempotent a e −→ a is called split when there exists anobject b and morphisms a s −→ b , b r −→ a such that b r ◦ s −−→ b = b id b −−→ b and a s ◦ r −−→ a = a e −→ a . We define the object M AB ⊗ B M BC ∈ C as a choice of splitting object for the idempotent e M AB ⊗ M BC such that M AB ⊗ M BC s MAB,MBC −−−−−−−→ M AB ⊗ B M BC and M AB ⊗ B M BC r MAB,MBC −−−−−−−→ M AB ⊗ M BC , where s M AB ,M BC ◦ r M AB ,M BC = e M AB ⊗ M BC and r M AB ,M BC ◦ s M AB ,M BC = id M AB ⊗ M BC . Crucially, a choice ofsplitting object is unique up to isomorphism, and independent of a choice of section up to isomorphism.Using this data, let us further define the following morphism: pq M AB ⊗ B pq M BC := r M AB ,M BC ◦ ( pq M AB ⊗ M BC ) ◦ s M AB ,M BC . (6.28)Putting everything together, we obtain that ( M AB ⊗ B M BC , pq M AB ⊗ B pq M BC ) defines an ( A, C )-bimodule in C . So we have obtained a way to define an ( A, C )-bimodule out of an (
A, B )- and a(
B, C )-bimodule given three separable algebra objects
A, B, C . This can expressed in terms of thebifunctor ⊗ B : Bimod C ( A, B ) × Bimod C ( B, C ) → Bimod C ( A, C ) , (6.29)where objects M AB ∈ Ob(
Bimod C ( A, B )) and M BC ∈ Ob(
Bimod C ( B, C )) are mapped via ⊗ B : M AB × M BC (cid:55)→ M AB ⊗ B M BC , (6.30) ∼ ∼ nd bimodule homomorphisms f AB ∈ Hom(
Bimod C ( A, B )), f BC ∈ Hom(
Bimod C ( B, C )) are sent to ⊗ B : f AB × f BC (cid:55)→ f AB ⊗ B f BC := s M AB ,M BC ◦ ( f AB ⊗ f BC ) ◦ r M AB ,M BC . (6.31)In order to obtain a bicategory, we are left to define a left unitor, a right unitor and an associator.Considering A and B as ( A, A )- and (
B, B )-bimodules, respectively, one can verify that for any (
A, B )-bimodule M AB A ⊗ A M AB ∼ = M AB ∼ = M AB ⊗ B B , (6.32)as (
A, B )-bimodule in C . This property demonstrates that an algebra A seen as an ( A, A )-bimoduledefines a notion a unit morphism for an algebra object A . The corresponding left unitor isomorphism,which is an ( A, B )-bimodule, is defined via the maps A ⊗ A M AB r A,MAB −−−−−→ A ⊗ M AB q A −−→ M AB and M AB (cid:96) − MAB −−−−→ ⊗ M AB ∆ ⊗ id MAB −−−−−−−→ ( A ⊗ A ) ⊗ M AB α
A,A,MAB −−−−−−−→ A ⊗ ( A ⊗ M AB ) id A ⊗ q A −−−−−→ A ⊗ M AB s A,MAB −−−−−→ A ⊗ A M AB , which can be shown to satisfy the triangle relations. The right unitor can be defined in a similarfashion. Finally, for any quadruple of algebra objects A, B, C, D and (
A, B )-bimodule M AB , ( B, C )-bimodule M BC and ( C, D )-bimodule M CD , the morphism( M AB ⊗ B M BC ) ⊗ C M CD r MAB ⊗ MBC,MCD −−−−−−−−−−−→ ( M AB ⊗ B M BC ) ⊗ M CD r MAB,MBC ⊗ id MCD −−−−−−−−−−−−−→ ( M AB ⊗ M BC ) ⊗ M CD α
MAB,MBC,MBC −−−−−−−−−−−→ M AB ⊗ ( M BC ⊗ M CD ) id MAB ⊗ s MBC,MCD −−−−−−−−−−−−−→ M AB ⊗ ( M BC ⊗ C M CD ) id MAB ⊗ s MBC,MCD −−−−−−−−−−−−−→ M AB ⊗ B ( M BC ⊗ C M CD )defines an isomorphism of ( A, D )-bimodules in C satisfying the pentagon relation. Putting everythingtogether, we obtain the following bicategory: Definition . Given a multi-fusion C , we no-tate via sAlg ( C ) the bicategory with objects, separable algebras objects in C , and hom-category Hom sAlg ( C ) ( A, B ) :=
Bimod C ( A, B ) for all separable algebra objects A, B in C . The compositionbifunctor is provided by ⊗ B : Bimod C ( A, B ) × Bimod C ( B, C ) → Bimod C ( A, C ) as defined in thissection. Let us now apply the definition above to the multi-fusion category
Vec α G . First of all, every algebraobject in Vec α G can be shown to be separable. Indeed, given an algebra object A φ in Vec α G , a choice ofsection ∆ : A φ → A φ ⊗ A φ is provided by the following map on basis elements:∆ : a (cid:55)→ | Hom A (s( a ) , − ) | (cid:88) a , a ∈ Hom( A ) a a = a φ ( a , a ) a ⊗ a . (6.33)Algebra objects equipped with the section defined above form the objects of the bicategory sAlg ( Vec α G ).Let A φ , B ψ , C ϕ be three objects in sAlg ( Vec α G ), we consider the 1-morphisms M AB ≡ ( M AB , pq M AB ) ∈∼ ∼ b( Bimod
Vec α G ( A φ , B ψ )) and M BC ≡ ( M BC , pq M BC ) ∈ Ob(
Bimod
Vec α G ( B ψ , C ϕ )). Following (6.26), themap pq M AB ⊗ M BC acts on basis elements of C a ⊗ (([ M AB ] g ⊗ [ M BC ] g ) ⊗ C c ) as pq M AB ⊗ M BC : a ⊗ (( v AB g ⊗ v BC g ) ⊗ c ) (cid:55)→ | Hom( B ) | (cid:88) b ∈ Hom( B ) α ( g , g , c ) α ( a , g , b ) α ( ag b , b − , g c ) ψ ( b , b − ) α ( a , g , g c ) α ( ag , b , b − ) v AB g (cid:46) pq ( v AB g , a , b ) ⊗ v BC g (cid:46) pq ( v BC g , b − , c ) . Applying the formula above to a = id s( g ) and c = id t( g ) , we obtain that the map s M AB ,M BC : M AB ⊗ M BC → M AB ⊗ B M BC acts on basis elements as s M AB ,M BC : v AB g ⊗ v BC g (cid:55)→ | Hom( B ) | (cid:88) b ∈ Hom( B ) ψ ( b , b − ) α ( g b , b − , g ) α ( g , b , b − ) v AB g (cid:46) pq ( v AB g , id s( g ) , b ) ⊗ v BC g (cid:46) pq ( v BC g , b − , id t( g ) ) , whereas r M AB ,M BC : M AB ⊗ B M BC → M AB ⊗ M BC is given by the inclusion. We can finally check thatthe binary functor simplifies such that pq M AB ⊗ B pq N BC = pq M AB ⊗ M BC . (6.34)Left unitor, right unitor and associator can now be readily obtained. Finally, let us remark thatthe above bifunctor can be conveniently rephrased as a comultiplication map (cid:101) ∆ B : C [ (cid:101) G AC ] (cid:36) AC → C [ (cid:101) G AB ] (cid:36) AB ⊗ C [ (cid:101) G BC ] (cid:36) BC defined by (cid:101) ∆ B (cid:0)(cid:12)(cid:12) g −−→ a , c (cid:11)(cid:1) := 1 | Hom( B ) | (cid:88) g ∈ Ob( (cid:101) G AB ) g ∈ Ob( (cid:101) G BC ) g g = gb ∈ Hom B (t( g ) , t( g )) α ( g , g , c ) α ( a , g , b ) α ( ag b , b − , g c ) ψ ( b , b − ) α ( g , g , g g ) α ( gg , g , g − ) (cid:12)(cid:12) g −−−→ a , b (cid:11) ⊗ | g −−−−→ b − , c (cid:11) . (6.35) Vec α G -module categories We are now ready to describe the bicategory
MOD ( Vec α G ) by spelling out equivalence with the bicate-gory sAlg ( C ) described above. In the following, we will describe how this is the relevant structure todescribe boundary excitations in gauge models of topological phases.Letting A φ be a (separable) algebra object in Vec α G , the category Mod
Vec α G ( A φ ) of right A φ -modulesis a left module category for Vec α G . Let us spell out this correspondence. The module functor (cid:12) : Vec α G × Mod
Vec α G ( A φ ) → Mod
Vec α G ( A φ ) (6.36)is defined on objects V ∈ Ob(
Vec α G ) and ( M A , p A ) ∈ Ob(
Mod
Vec α G ( A φ ) by (cid:12) : V × M A (cid:55)→ V ⊗ M A , (6.37)where V ⊗ M A ∈ Ob(
Mod
Vec α G ( A φ )) is the A φ -module with action defined by the following compositionof morphisms in Vec α G :( V ⊗ M A ) ⊗ A φ α V,M A , A −−−−−−→ V ⊗ ( M A ⊗ A φ ) id V ⊗ p A −−−−−→ V ⊗ M A . (6.38) ∼ ∼ he functor takes morphisms to their tensor product over the field C . The module associator ˙ α V,W,M reduces to the associator in
Hom α G such that for V, W ∈ Ob(
Vec α G ) one has( V ⊗ W ) ⊗ M A α V,W,M A −−−−−−→ V ⊗ ( W ⊗ M A ) . (6.39)A Vec α G -module category Mod
Vec α G ( A φ ) is then indecomposable if and only if the algebra object A φ isnot isomorphic a direct sum of two non-trivial algebra objects [56].Let us now describe Vec α G -module functors. Let A φ and B ψ be any pair of algebra objects in Vec α G , with Mod
Vec α G ( A φ ) and Mod
Vec α G ( B ψ ) the corresponding category of module objects. To each( A φ , B ψ )-bimodule object M AB , we can define a Vec α G -module functor − ⊗ A M AB : Mod
Vec α G ( A φ ) → Mod
Vec α G ( B ψ ) , (6.40)which acts on objects M A ∈ Ob(
Mod
Vec α G ( A φ )) via the map − ⊗ A M AB : M A (cid:55)→ M A ⊗ A M AB , (6.41)and sends morphisms f ∈ Hom(
Mod
Vec α G ( A φ )) to f ⊗ id M AB . The natural isomorphism s : ( Vec α G ⊗ Mod
Vec α G ( A φ )) ⊗ A Bimod ( A φ , B ψ ) → Vec α G ⊗ ( Mod
Vec α G ( A φ ) ⊗ A Bimod ( A φ , B ψ )) (6.42)is given on objects V ∈ Ob(
Vec α G ) and M A ∈ Mod
Vec α G ( A φ ) via the associator α in Vec α G such that: s V,M A : ( V ⊗ M A ) ⊗ A M AB r V ⊗ M A ,M AB −−−−−−−−→ ( V ⊗ M A ) ⊗ M AB α V,M A ,M AB −−−−−−−−→ V ⊗ ( M A ⊗ M AB ) id V ⊗ s M A ,M AB −−−−−−−−−−→ V ⊗ ( M A ⊗ A M AB ) . (6.43)In a similar vein, morphisms of Vec α G -module functors are induced by natural transformations betweenbimodules. Together, this yields the desired equivalence: Proposition . There exists an equivalence of bicategories sAlg ( Vec α G ) and MOD ( Vec α G ) bysending separable algebra objects in Vec α G to their category of (right) modules in Vec α G , bimod-ule objects M AB ∈ Hom sAlg ( Vec α G ) ( A φ , B ψ ) are sent to the Vec α G -module functor − ⊗ A M AB : Mod
Vec α G ( A ) → Mod
Vec α G ( B ) and bimodule natural transformations are sent to morphisms of Vec α G -module functors. Using the technology developed in this section, we are now ready to describe gapped boundaries andtheir excitations in (2+1)d gauge models of topological phases within the language of bicategories.More specifically, we shall define a bicategory
Bdry αG whose objects are given by gapped boundary con-ditions, 1-morphisms provide gapped boundary excitations, and 2-morphisms define fusion processesof gapped boundary excitations. We shall then demonstrate that Bdry αG is equivalent, as a bicategory,to MOD ( Vec αG ).Let us begin with a brief review of the results obtained in the first part of this manuscript withinthe tube algebra approach. Hamiltonian realisations of (2+1)d Dijkgraf-Witten theory are definedin terms of pairs ( G, α ), where G is a finite group and α is a normalised 3-cocycle in H ( G, U(1)).In sec. 2, it was argued that gapped boundaries can be indexed by pairs (
A, φ ), where A ⊂ G is a ∼ ∼ ubgroup of G and φ ∈ C ( A, U(1)) is a 2-cochain satisfying the condition d (2) φ = α − | A . In sec. 3, weshowed that boundary excitations at the interface of two one-dimensional gapped boundaries labelledby ( A, φ ) and (
B, ψ ), respectively, were classified via representations of the boundary tube algebrathat is isomorphic to the twisted groupoid algebra C [ G AB ] αφψ .We now collect the previous results into a bicategory Bdry αG . The objects of Bdry αG are given bythe set of all gapped boundary conditions { ( A, φ ) } . For each pair ( A, φ ), (
B, ψ ) of gapped boundaryconditions, we assign the hom-category
Hom
Bdry αG (( A, φ ) , ( B, ψ )) :=
Mod ( C [ G AB ] αφψ ) , (6.44)where Mod ( C [ G AB ] αφψ ) denotes the category of C [ G AB ] αφψ -modules and intertwiners. In this way, the1-morphisms ρ AB ∈ Ob(
Hom
Bdry αG (( A, φ ) , ( B, ψ ))) correspond to boundary excitations incident at theinterface between gapped boudaries labelled by (
A, φ ) and (
B, ψ ). The composition bifunctor ⊗ : Mod ( C [ G AB ] αφψ ) × Mod ( C [ G BC ] αψϕ ) → Mod ( C [ G AC ] αφϕ ) (6.45)is defined on 1-morphisms ρ AB ∈ Ob(
Mod ( C [ G AB ] αφψ )) and ρ BC ∈ Ob(
Mod ( C [ G BC ] αψϕ )) via ⊗ : ρ AB × ρ BC (cid:55)→ ρ AB ⊗ B ρ BC := ( ρ AB ⊗ ρ BC ) (cid:46) ∆ B ( AC ) , (6.46)as described in sec. 5.2, and on 2-morphisms f AB : ρ AB → ρ (cid:48) AB ∈ Hom(
Mod ( C [ G AB ] αφψ )), f BC : ρ BC → ρ (cid:48) BC ∈ Hom(
Mod ( C [ G BC ] αψϕ )) via ⊗ : f AB × f BC (cid:55)→ ( f AB ⊗ B f BC : ρ AB ⊗ B ρ BC → ρ (cid:48) AB ⊗ B ρ (cid:48) BC ) , (6.47)where the morphism on the r.h.s decomposes as f AB ⊗ B f BC : ρ AB ⊗ B ρ BC (cid:44) −→ ρ AB ⊗ ρ BC f AB ⊗ f BC −−−−−−→ ρ (cid:48) AB ⊗ ρ (cid:48) BC → ρ (cid:48) AB ⊗ B ρ (cid:48) BC . (6.48)In the sequence of linear maps above, the first arrow notates the injection of ρ AB ⊗ B ρ BC into ρ AB ⊗ ρ BC ,and the last arrow notates the projection map ρ (cid:48) AB ⊗ ρ (cid:48) BC (cid:55)→ ( ρ (cid:48) AB ⊗ ρ (cid:48) BC ) (cid:46) ∆ B ( AC ) = ρ (cid:48) AB ⊗ B ρ (cid:48) BC . (6.49)Furthermore, a 2-morphism of the form ζ : ρ AB ⊗ B ρ BC → ρ AC ∈ Hom(
Mod ( C [ G AC ] αφϕ )) is an inter-twiner interpreted as describing the process of fusing a pair of boundary excitations at the interfacesof gapped boundaries labelled by ( A, φ ), (
B, ψ ) and (
B, ψ ), (
C, ϕ ), respectively: A φ B ψ C ϕ . (6.50)The identity morphism associated with the object ( A, φ ) is given by the regular module C [ G AA ] αφφ ∈ Ob(
Mod ( C [ G AA ] αφφ )) with left and right unitors the intertwiner isomorphisms (cid:96) : C [ G AA ] αφφ ⊗ A ρ AB ∼ −→ ρ AB , r : ρ AB ⊗ B C [ G BB ] αψψ ∼ −→ ρ AB , (6.51)as described in sec. 5.2. Finally, the 1-associator for a triple of 1-morphisms ρ AB ∈ Ob(
Mod ( C [ G AB ] αφψ )), ρ BC ∈ Ob(
Mod ( C [ G BC ] αψϕ )), ρ CD ∈ Ob(
Mod ( C [ G CD ] αϕχ )) is given by the intertwiner isomorphism inHom( Mod ( C [ G AD ] αφχ ))Φ ρ AB ρ BC ρ CD : ( ρ AB ⊗ B ρ BC ) ⊗ C ρ CD → ρ AB ⊗ B ( ρ BC ⊗ C ρ CD ) , (6.52) ∼ ∼ s described explicitly in sec. 5.2. It follows from the results of the first part of this manuscript thatsuch data satisfy the pentagon and triangle relations ensuring we do obtain a bicategory.So we have recast our results obtained in the first part of this manuscript in terms of the boundary tubealgebra and its representation theory as the bicategory
Bdry αG . We shall now establish the followingequivalence of bicategories: Bdry αG (cid:39) MOD ( Vec αG ) . (6.53)More precisely, we shall establish the equivalence of the bicategories Bdry αG (cid:39) sAlg ( Vec αG ), from whichwe can induce the equivalence above through prop. 6.1, by noting equivalence of bicategories is tran-sitive . First, we need to introduce a notion of homomorphism between bicategories: Definition . Given a pair of bicategories B i and B i (cid:48) ,a strict homomorphism F : B i → B i (cid:48) of bicategories consists of • a function F : Ob( B i ) → Ob( B i (cid:48) ) , • a family of functors F XY : Hom B i ( X, Y ) → Hom B i (cid:48) ( F ( X ) , F ( Y )) referred to as hom-functors ,for each pair of objects X, Y ∈ Ob( B i ) ,such that F X,Y ( f ) ⊗ F Y,Z ( g ) = F X,Z ( f ⊗ g ) B i (cid:48) F ( X ) = F X,X ( B i X ) F ( α B i f,g,h ) = α B i (cid:48) F X,Y ( f ) , F Y,Z ( g ) , F Z,W ( h ) F ( r B i X ) = r B i (cid:48) F ( X ) , F ( (cid:96) B i X ) = (cid:96) B i (cid:48) F ( X ) , for all objects W, X, Y, Z ∈ Ob( B i ) and morphisms f ∈ Ob(
Hom B i ( X, Y )) , g ∈ Ob(
Hom B i ( Y, Z )) , h ∈ Ob(
Hom B i ( Z, W )) . Recall that a functor between categories defines an equivalence if and only if it is full , faithful andessentially surjective . In a similar vein, a sufficient condition for a strict homomorphism of bicategories F to define an equivalence of bicategories is that the map is surjective on objects, and the functors F X,Y for all
X, Y ∈ Ob( B i ) define equivalences of the categories Hom B i ( X, Y ) (cid:39) Hom B i (cid:48) ( F ( X ) , F ( Y )).Using this sufficient condition, let us now establish the equivalence of bicategories F : Bdry αG (cid:39) −→ sAlg ( Vec αG ). We begin by defining the function F : Ob( Bdry αG ) → Ob( sAlg ( Vec αG )). It is given bysending each boundary condition ( A, φ ) to the corresponding separable algebra object A φ in Vec αG .From the previous discussion, we know that both boundary conditions and separable algebra objectsare indexed by subgroups of G and 2-cochains satisfying the compatibility conditions with α . It followsthat the function F is a bijection, and thus surjective. The hom-functors are required to define thefollowing equivalence of categories: Hom
Bdry αG (( A, φ ) , ( B, ψ )) :=
Mod ( C [ G AB ] αφψ ) (cid:39) Mod ( C [ (cid:101) G AB ] (cid:36) AB ) =: Hom sAlg ( Vec αG ) ( A φ , B ψ ) , where the groupoid (cid:101) G AB and its 2-cocycle (cid:36) AB is obtained by applying the definition at the end ofsec. 6.4 to the delooping of G . In order to establish this equivalence, it suffices to demonstrate theisomorphism of twisted groupoid algebras C [ (cid:101) G AB ] (cid:36) AB (cid:39) C [ G AB ] αφψ ≡ C [ G AB ] ϑ AB , for all boundary Recall that the derivations in sec. 5.2, and more generally in sec. 5, were carried out explicitly for the boundarytube algebra in (3+1)d. However, we explained that the (2+1)d boundary tube algebra, which is the one relevant here,is obtained as a limiting case. ∼ ∼ onditions ( A, φ ) and (
B, ψ ). The equivalence
Mod ( C [ G AB ] αφψ ) (cid:39) Mod ( C [ ˜ G AB ] (cid:36) AB ) of their modulecategories then follows by pre-composition. Noting from the definition that both groupoids have thesame dimension, the isomorphism is provided by the following map on basis elements: (cid:12)(cid:12) g −−−→ a,b (cid:11) (cid:55)→ φ ( a − , a ) α ( a − , a, gb ) (cid:12)(cid:12) g a − −−−→ b (cid:11) , ∀ (cid:12)(cid:12) g −−−→ a,b (cid:11) ∈ C [ (cid:101) G AB ] (cid:36) AB . (6.54)Furthermore, one can check that such isomorphism is compatible with the respective comultiplicationmaps through the following commuting diagram C [ G AC ] αφϕ C [ G AB ] αφψ ⊗ C [ G BC ] αψϕ C [ (cid:101) G AC ] (cid:36) AC C [ (cid:101) G AB ] (cid:36) AB ⊗ C [ (cid:101) G BC ] (cid:36) BC ∆ B (cid:39)(cid:39) (cid:101) ∆ B . (6.55)Commutativity is ensured by the relation φ ( a − , a ) α ( a − , a, g b ) ψ ( b, b − ) α ( b, b − , g c ) α ( g , g , c ) α ( a, g , b ) α ( ag b, b − , g c ) ψ ( b, b − ) α ( a, g , g c ) α ( ag , b, b − )= φ ( a − , a ) α ( a − , a, g g c ) α ( g , g , c ) α ( a − , ag b, b − g c ) α ( g , b, b − g c ) , which follows from the cocycle relation d (3) α ( a − , a, g b, b − g c ) = 1 , d (3) α ( a − , a, g g c ) = 1 , d (3) α ( ag , b, b − , g c ) = 1 . Since the composition functors in both bicategories are induced from the respective comultiplicationmaps, it can be verified that such hom-functors satisfy the conditions of a strict homomorphism ofbicategories, hence establishing the required equivalence of bicategories.
In the previous discussion, we argued that, given a lattice Hamiltonian realisation of (2+1)d Dijkgraaf-Witten theory with input data (
G, α ), gapped boundary conditions are in bijection with algebra objectsin the fusion category
Vec αG . We shall now outline the analogue of this statement for lattice Hamiltonianrealisations of (3+1)d Dijkgraaf-Witten theory.Given a fixed input data ( G, π ), where G is a finite group and π is normalized group 4-cocyclein H ( G, U(1)), it has been argued that the relevant category theoretical structure is provided bythe monoidal bicategory πG of G -graded 2-vector spaces [29, 32, 33, 53, 61, 62]. Let us begin bydescribing the salient features of the monoidal bicategory as a categorification of Vec . Thereexist several definitions of this bicategory, see e.g. [63–65], in the following we shall consider as the bicategory of finite dimensional, semi-simple
Vec -module categories,
Vec -module functors and
Vec -module functor homomorphisms. As customary, objects of will be referred to as . There is a single simple object provided by the
Vec -module category
Vec , which implies thatfor all objects X ∈ Ob( ), there exists a
Vec -module equivalence X (cid:39) (cid:1) i Vec . The monoidalstructure of is defined on objects via the weak 2-functor (cid:2) : × → : X × Y (cid:55)→ X (cid:2) Y , ∼ ∼ or all X, Y ∈ Ob( ), where (cid:2) denotes the
Deligne tensor product of abelian categories [66]. Inparticular, for a pair of 2-vector spaces X and Y , the Deligne tensor product yields the category X (cid:2) Y ,whose set of objects is Ob( X (cid:2) Y ) := Ob( X ) × Ob( Y ) and set of morphisms given by Hom( X (cid:2) Y ) :=Hom( X ) ⊗ C Hom( Y ). The composition in Hom( X (cid:2) Y ) is induced from the ones in Hom( X ) andHom( Y ), accordingly. This monoidal structure is equipped with a pseudo-natural adjoint equivalenceof Vec -module categories ( X (cid:2) Y ) (cid:2) Z α X,Y,Z −−−−→ X (cid:2) ( Y (cid:2) Z ) , (6.56)together with a Vec -module functor isomorphism π known as the pentagonator :(( X (cid:2) Y ) (cid:2) Z ) (cid:2) W ( X (cid:2) ( Y (cid:2) Z )) (cid:2) W ( X (cid:2) Y ) (cid:2) ( Z (cid:2) W ) X (cid:2) (( Y (cid:2) Z ) (cid:2) W ) X (cid:2) ( Y (cid:2) (( Z (cid:2) W )) α X (cid:2) Y , Z , W α αX,Y,Z (cid:2) W α X , Y , Z (cid:2) i d W α X,Y (cid:2)
Z,W id X (cid:2) α Y,Z,W π X , Y , Z , W . (6.57)Both α and π can be shown to evaluate to the identity 1- and 2-morphisms, respectively. Note that the pseudo-naturality of α specifies that for any triple of 2-vector spaces X, Y, Z and
Vec -module functors f X : X → X (cid:48) , f Y : Y → Y (cid:48) and f Z : Z → Z (cid:48) there exists a 2-isomorphism( X (cid:2) Y ) (cid:2) Z X (cid:2) ( Y (cid:2) Z )( X (cid:48) (cid:2) Y (cid:48) ) (cid:2) Z (cid:48) X (cid:48) (cid:2) ( Y (cid:48) (cid:2) Z (cid:48) ) α X,Y,Z f X (cid:2) ( f Y (cid:2) f Z )( f X (cid:2) f Y ) (cid:2) f Z α X,Y,Z (cid:39) . (6.58)Henceforth, we shall not draw arrows for such 2-isomorphisms but instead notate the 2-cell with the (cid:39) symbol.Akin to a monoidal category, the monoidal bicategory admits a monoidal unit ∈ Ob( ),which is equipped with the
Vec -module category pseudo-natural adjoint equivalences X (cid:2) r X −−→ X and (cid:2) X (cid:96) X −−→ X , (6.59)for all X ∈ Ob( ), together with
Vec -module functor isomorphisms τ , τ , τ referred to as trian-gulators :( (cid:2) X ) (cid:2) Y (cid:2) ( X (cid:2) Y ) X (cid:2) Y α ,X,Y (cid:96) X (cid:2) Y (cid:96) X (cid:2) id Y τ , ( X (cid:2) ) (cid:2) Y X (cid:2) ( (cid:2) Y ) X (cid:2) Y α X, ,Y id X (cid:2) (cid:96) Y r X (cid:2) id Y τ , (6.60)( X (cid:2) Y ) (cid:2) X (cid:2) ( Y (cid:2) ) X (cid:2) Y α X,Y, id X (cid:2) r Y r X (cid:2) Y τ . (6.61) Although we use a similar notation, the associator of the monoidal structure is not to be confused with the 1-associator natural isomorphism of the underlying bicategory. ∼ ∼ hese isomorphisms can be all be shown to evaluate to the identity 1- and 2-morphisms, respectively.More generally, for an arbitrary monoidal bicategory, such data is subject to a series of coherence datawhich we shall not provide here, instead pointing the reader to e.g. [60, 63, 67].Having described the most notable features of , we now describe the monoidal bicategory πG , which is obtained following a process analogous to the lift of Vec to Vec αG . Let G be a finitegroup and π a normalised group 4-cocycle in H ( G, U(1)). A G -graded 2-vector space is a 2-vectorspace of the form X = (cid:1) g ∈ G X g . We call a G -graded 2-vector space homogeneous of degree g ∈ G if X = X g . The monoidal bicategory πG is then defined as the bicategory whose objects aregiven by G -graded 2-vector spaces, 1-morphisms are G -grading preserving Vec -module functors, and2-morphisms are
Vec -module functor homomorphisms. The simple objects of πG are given by thecategories Vec g , for all g ∈ G , and every object is equivalent to a direct sum of simple objects. Themonoidal structure of πG is given on homogeneous components via the weak 2-functor (cid:2) : Vec g × Vec g (cid:48) → Vec gg (cid:48) , (6.62)for all g, g (cid:48) ∈ G . Since π is a normalised representative of [ π ] ∈ H ( G, U(1)), the adjoint equivalences(
Vec g (cid:2) Vec g (cid:48) ) (cid:2) Vec g (cid:48)(cid:48) α Vec g, Vec g (cid:48) , Vec g (cid:48)(cid:48) −−−−−−−−−−→ Vec g (cid:2) ( Vec g (cid:48) (cid:2) Vec g (cid:48)(cid:48) ) , (6.63) Vec g (cid:2) Vec G r Vec g −−−→ Vec g , , Vec G (cid:2) Vec g (cid:96)
Vec g −−−→ Vec g (6.64)are the identity 1-morphisms, the triangulators τ , τ , τ are the identity 2-morphisms, whereas thepentagonator 2-isomorphism is given by π Vec g , Vec g (cid:48) , Vec g (cid:48)(cid:48) , Vec g (cid:48)(cid:48)(cid:48) := π ( g, g (cid:48) , g (cid:48)(cid:48) , g (cid:48)(cid:48)(cid:48) ) · id Vec gg (cid:48) g (cid:48)(cid:48) g (cid:48)(cid:48)(cid:48) for all g, g (cid:48) , g (cid:48)(cid:48) , g (cid:48)(cid:48)(cid:48) ∈ G . It is straightforward to verify that the requirement that π is a 4-cocycle ensures thecoherence relations for the pentagonator are satisfied.Having defined the monoidal bicategory πG , we shall now argue that gapped boundary condi-tions in (3+1)d gauge models of topological phases correspond to pseudo-algebra objects [68] in πG ,categorifying the relation between algebra objects in Vec αG and gapped boundaries in (2+1)d gaugemodels: Definition . Let B i ≡ ( B i , ⊗ , , α, r, (cid:96), π, τ , τ , τ ) be a monoidalbicategory. A pseudo-algebra object in B i is a sextuple ( A, m, u, ς m , ς r , ς (cid:96) ) consisting of an object A ∈ Ob( B i ) , a pair of 1-morphisms m : A ⊗ A → A , u : → A , and a triple of 2-isomorphisms ς m , ς r , ς (cid:96) defined according to ( A ⊗ A ) ⊗ A A ⊗ ( A ⊗ A ) A ⊗ AA ⊗ A A α id A ⊗ mmm ⊗ id A mς m ,A A ⊗ A ⊗ A A r − id A ⊗ u m id A ς r , A ⊗ A A ⊗ A A (cid:96) − u ⊗ id A m id A ς (cid:96) , ∼ ∼ nd subject to the following coherence relations: (( A ⊗ A ) ⊗ A ) ⊗ A ( A ⊗ A ) ⊗ A A ⊗ A ( A ⊗ ( A ⊗ A )) ⊗ A ( A ⊗ A ) ⊗ A AA ⊗ (( A ⊗ A ) ⊗ A ) A ⊗ ( A ⊗ A ) A ⊗ ( A ⊗ ( A ⊗ A )) A ⊗ ( A ⊗ A ) A ⊗ A ( m ⊗ id A ) ⊗ id A m ⊗ id A mα A,A,A ⊗ id A α A,A ⊗ A,A id A ⊗ α A,A,A (id A ⊗ id A ) ⊗ m id A ⊗ m m ( A ⊗ m ) ⊗ id A id A ⊗ ( m ⊗ id A ) α A,A,A m ⊗ i d A i d A ⊗ m (cid:39) ς m ς m ⊗ i d A i d A ⊗ ς m is equal to (( A ⊗ A ) ⊗ A ) ⊗ A ( A ⊗ A ) ⊗ A A ⊗ A ( A ⊗ ( A ⊗ A )) ⊗ A A ⊗ ( A ⊗ A )( A ⊗ A ) ⊗ ( A ⊗ A ) A ⊗ A AA ⊗ (( A ⊗ A ) ⊗ A ) ( A ⊗ A ) ⊗ AA ⊗ ( A ⊗ ( A ⊗ A )) A ⊗ ( A ⊗ A ) A ⊗ A ( m ⊗ id A ) ⊗ id A m ⊗ id A mα A,A,A ⊗ id A α A,A ⊗ A,A id A ⊗ α A,A,A id A ⊗ m m id A ⊗ ( m ⊗ id A ) α A ⊗ A , A , A α A , A , A ⊗ A m ⊗ ( i d A ⊗ i d A ) i d A ⊗ m ( i d A ⊗ i d A ) ⊗ m m ⊗ i d A mα A,A,A α A,A,A (cid:39) π (cid:39)(cid:39) ς m ς m and ( A ⊗ ) ⊗ A ( A ⊗ A ) ⊗ A A ⊗ AA ⊗ A AA ⊗ ( ⊗ A ) A ⊗ ( A ⊗ A ) A ⊗ A r − ⊗ id A ( A ⊗ u ) ⊗ id A m ⊗ id A m id A ⊗ (cid:96) − id A ⊗ ( u ⊗ id A ) id A ⊗ m m i d A ⊗ i d A m i d A ⊗ i d A ς r ⊗ i d A (cid:39)(cid:39) i d A ⊗ ς − (cid:96) ∼ ∼ s equal to ( A ⊗ ) ⊗ A ( A ⊗ A ) ⊗ A A ⊗ AA ⊗ A AA ⊗ ( ⊗ A ) A ⊗ ( A ⊗ A ) A ⊗ A r − ⊗ i d A (id A ⊗ u ) ⊗ id A m ⊗ id A m i d A ⊗ (cid:96) − id A ⊗ ( u ⊗ id A ) id A ⊗ m m α A, ,A α A,A,A τ (cid:39) ς m . Given the above definition, a first observation is that a pseudo-algebra object in corresponds toa finite-dimensional, semi-simple monoidal category. This relies in particular on the fact that semi-simple abelian categories always have a unique structure of semi-simple
Vec -module category [69].Let us now apply this definition to πG . For each pair ( A, λ ), where A ⊂ G is a subgroup and λ ∈ C ( A, U(1)) is a 3-cochain satisfying the condition d (3) λ = π − | A , we construct a pseudo-algebraobject Vec
A,λ ≡ ( (cid:1) a ∈ A Vec a , m, u, ς m , ς r , ς (cid:96) ) such that: the multiplication m : Vec
A,λ (cid:2)
Vec
A,λ → Vec
A,λ is given on homogeneous components via the functor m Vec a , Vec a (cid:48) : Vec a (cid:2) Vec a (cid:48) (cid:55)→ Vec aa (cid:48) forall a, a (cid:48) ∈ A , the unit map u is defined in an obvious way, the 2-isomorphisms ς r and ς (cid:96) are trivial,and the 2-isomorphism ς m : α Vec
A,λ , Vec
A,λ , Vec
A,λ ◦ (id Vec
A,λ ◦ m (cid:2) m ) ⇒ ( m (cid:2) id Vec
A,λ ) ◦ m (6.65)defines an associator for the product map m that is determined by λ . This associator acts on homoge-nous components labelled by a, a (cid:48) , a (cid:48)(cid:48) ∈ A as λ a,a (cid:48) ,a (cid:48)(cid:48) : α Vec a , Vec a (cid:48) , Vec a (cid:48)(cid:48) ◦ (id Vec a (cid:2) m Vec a (cid:48) , Vec a (cid:48)(cid:48) ) ◦ m Vec a , Vec a (cid:48) a (cid:48)(cid:48) ⇒ ( m Vec a , Vec a (cid:48) (cid:2) id Vec a (cid:48)(cid:48) ) ◦ m Vec aa (cid:48) , Vec a (cid:48)(cid:48) . The condition d (3) λ = π − | A demonstrates that Vec
A,λ is not a monoidal category in the conventionalsense since the associator λ fails to satisfy the pentagon equation (6.1). Instead, the associator satisfiesthe following equation on homogeneous components labelled by a, a (cid:48) , a (cid:48)(cid:48) , a (cid:48)(cid:48)(cid:48) ∈ A :( λ a,a (cid:48) ,a (cid:48)(cid:48) (cid:2) id Vec a (cid:48)(cid:48)(cid:48) ) ◦ λ a,a (cid:48) a (cid:48)(cid:48) ,a (cid:48)(cid:48)(cid:48) ◦ (id Vec a (cid:2) λ a (cid:48) ,a (cid:48)(cid:48) ,a (cid:48)(cid:48)(cid:48) ) ◦ π a,a (cid:48) ,a (cid:48)(cid:48) ,a (cid:48)(cid:48)(cid:48) = λ aa (cid:48) ,a (cid:48)(cid:48) ,a (cid:48)(cid:48)(cid:48) ◦ λ a,a (cid:48) ,a (cid:48)(cid:48) a (cid:48)(cid:48)(cid:48) . (6.66)In this way, we see that Vec
A,λ defines a monoidal category which is associative inside 2
Vec πG but notas a conventional monoidal category. This result provides a categorification of the observation thatan algebra object A φ in Vec αG defines a twisted groupoid algebra, which is associative inside Vec αG butnot as a conventional algebra. Mimicking the analysis carried out in sec. 6.7, we shall now introduce a category theoretical formula-tion of gapped boundaries in (3+1)d gauge models and string-like excitations terminating at gappedboundaries, which we studied from a tube algebra point of view in sec. 4. In particular, we shall definea bicategory πG that is analogous to Bdry αG . We shall then relate this construction to the workof Kong et al. in [53] arguing that πG forms a full sub-bicategory of Z ( πG ), i.e. the centre of πG .Let us begin with a brief review of the results obtained in the first part of this manuscript withinthe tube algebra approach. Hamiltonian realisations of (3 + 1) d Dijkgraaf-Witten theory are defined in ∼ ∼ erms of pairs ( G, π ), where G is a finite group and π a normalised 4-cocycle in H ( G, U(1)). In sec. 2.4,it was argued that gapped boundaries can be indexed by pairs (
A, λ ), where A ⊂ G is a subgroup of G and λ ∈ C ( A, U(1)) is a 3-cochain satisfying the condition d (3) λ = π − | A . In the previous section,we explained that such data is in bijection with pseudo-algebra objects Vec
A,λ in πG . Moreover,we showed in sec. 4 within the tube algebra approach that given a pair of two-dimensional gappedboundaries labelled by ( A, λ ) and (
B, µ ), respectively, string-like excitations threading through thebulk from the former boundary to the latter were defined as modules of the twisted relative groupoidalgebra C [Λ( G AB )] T ( π ) T ( λ ) T ( µ ) , where Λ( G AB ) ≡ Λ G Λ A Λ B and T : Z • ( G, U(1)) → Z •− (Λ G, U(1)). Viathe introduction of a comultiplication map, we further described the concatenation of such string-likeexcitations in sec. 5.Let us now collect these results into a bicategory πG , in a way akin to the definition of Bdry αG .The objects of πG are given by pairs (Λ A, T ( λ )) for every gapped boundary condition labelled by( A, λ ). Given a pair of objects (Λ A, T ( λ )), (Λ B, T ( µ )), we define the hom-category Hom πG (cid:0) (Λ A, T ( λ )) , (Λ B, T ( µ )) (cid:1) := Mod (cid:0) C [Λ( G AB )] T ( π ) T ( λ ) T ( µ ) (cid:1) , (6.67)where Mod ( C [Λ( G AB )] T ( π ) T ( λ ) T ( µ ) ) denotes the category of C [Λ( G AB )] T ( π ) T ( λ ) T ( µ ) -modules and intertwiners.The composition functors, associator and unitors are given analogously to the construction of Bdry αG .From this definition, we interpret the objects (Λ A, T ( λ )) of πG as defining boundary conditions forthe end-points of a string-like excitation that terminates on a gapped boundary labelled by ( A, λ ). The1-morphisms ρ AB ∈ Ob(
Hom π G (( A, λ ) , ( B, µ ))) then correspond to string-like excitations terminat-ing at gapped boundaries labelled by (
A, λ ) and (
B, µ ), respectively. The bifunctor on 1-morphismsprovides a notion of concatenation for a pair of string-like excitations that share a boundary end-point,as described in sec. 5.2. The 2-morphisms correspond to intertwiners, so that a 2-morphism of theform ζ : ρ AB ⊗ B ρ BC → ρ AC can be interpreted as implementing the renormalization of a pair of con-catenated string-like excitations. Identity 1-morphisms and unitors are defined analoguously to Bdry αG .Similarly, the 1-associator for a triple of 1-morphisms ρ AB , ρ BC , ρ CD in the appropriate hom-categoriesis given by the intertwiner isomorphism Φ ρ AB ρ BC ρ CD : ( ρ AB ⊗ B ρ BC ) ⊗ C ρ CD → ρ AB ⊗ B ( ρ BC ⊗ C ρ CD ),as described explicitly in sec. 5.2.It is well-known that, given a lattice Hamiltonian realisation of (2+1)d Dijkgraaf-Witten theory withinput data ( G, α ), algebraic properties of the (bulk) anyonic excitations can be encoded into the cen-tre Z ( Vec αG ) of the fusion category Vec αG , this centre being in particular a braided monoidal category.The objects of Z ( Vec αG ) are interpreted as the elementary excitations of the model, or anyons, andthe morphisms implement space-time processes of such anyons. The monoidal structure describesthe fusion and splitting processes of the excitations, whereas the braiding structure encodes theirexchange statistics. Recently, Kong et al. studied in [53] the analogue of this result in (3+1)d.The relevant category theoretical structure in (3+1)d being the monoidal bicategory πG , theycomputed the braided monoidal bicategory Z ( πG ) obtained as the categorified centre of πG ,arguing that such a bicategory should describe string-like excitations and their statistics in (3+1)dgauge models. More specifically, they demonstrated that as a bicategory Z ( πG ) is equivalent to thebicategory MOD ( Vec T ( π )Λ G ). Using this equivalence, they suggested that objects of Z ( πG ) could beinterpreted as string-like topological excitations, 1-morphisms as particle-like topological excitations,and 2-morphisms as instantons. Relating this bicategory to the boundary tube algebra in (3+1)d, weshall argue that objects of Z ( πG ) should be interpreted as particle excitations at the endpoints of Recall that Λ G refers to the loop groupoid of the group G treated as a one-object groupoid (see sec. 4). ∼ ∼ string-like excitation terminating at boundary components of the spatial manifold, the 1-morphismsas string-like topological excitations, and 2-morphisms as implementating the renormalisation of con-catenated string-like excitations.In order to establish the interpretation spelt out above, we begin by showing that πG isequivalent as a bicategory to a full sub-bicategory ( Vec T ( π )Λ G ) of MOD ( Vec T ( π )Λ G ). Our argumentmirrors the equivalence of bicategories Bdry αG (cid:39) MOD ( Vec αG ) established in sec. 6.7. Utilising prop. 6.1,we know that, up to equivalence, all Vec T ( π )Λ G -module categories can be expressed as the category ofmodule objects for an algebra object in Vec T ( π )Λ G . Moreover, we established in sec. 6.4 that all suchalgebra objects were indexed by (Λ G, T ( π ))-subgroupoids, as defined in sec. 4.3. Given the data( A, λ ) of gapped boundary condition in (3+1)d, we explained in sec. 4 that the loop groupoid Λ A together with the groupoid 2-cochain T ( λ ) defines such a (Λ G, T ( π ))-subgroupoid. Henceforth, weshall refer to groupoids of this form as (Λ G, T ( π ))-subgroupoids. In this vein, we define the bicat-egory ( Vec T ( π )Λ G ) as the full sub-bicategory of MOD ( Vec T ( π )Λ G ) whose objects are Vec T ( π )Λ G -modulecategories induced from (Λ G, T ( π ))-subgroupoids, and hom-categories are the corresponding ones in MOD ( Vec T ( π )Λ G ). Similarly, we define ( Vec T ( π )Λ G ) as the full sub-bicategory of sAlg ( Vec T ( π )Λ G ), whoseobjects are algebra objects in Vec T ( π )Λ G of the form Λ A T ( λ ) , and hom-categories are the correspondingcategories of bimodule objects in Vec T ( π )Λ G . Mimicking our proof of the equivalence Bdry αG (cid:39) sAlg ( Vec αG ),we can show the equivalence between πG and ( Vec T ( π )Λ G ). This equivalence relies in particularon the isomorphism C [Λ( G AB )] T ( π ) T ( λ ) T ( µ ) ≡ C [Λ G Λ A Λ B ] ϑ Λ A Λ B (cid:39) C [ (cid:103) Λ G Λ A Λ B ] (cid:36) Λ A Λ B of twisted relativegroupoid algebras, which is realised by an obvious generalisation of (6.54). Utilising the proof ofprop. 6.1, it follows that ∂ sAlg ( Vec αG ) (cid:39) ∂ MOD ( Vec T ( π )Λ G ), hence establishing the equivalence πG (cid:39) ( Vec T ( π )Λ G ) . (6.68)Let us now explain how we can generalise our approach so as to obtain the bicategory MOD ( Vec T ( π )Λ G ),which we recall was shown to be equivalent to Z ( πG ). When considering the boundary tube algebrafor the (3+1)d gauge models in sec. 4, we could have allowed for a larger spectrum of boundarycolourings beyond the ones inherited from the gapped boundary conditions. More specifically, wecould have considered G -colourings that are provided by morphisms in any (Λ G, T ( π ))-subgroupoid( X , φ ) such that d (2) φ = T ( π ) |− X . Given a pair of (Λ G, T ( π ))-subgroupoids ( X , φ ) and ( Y , ψ ), wecould have then considered G -coloured graph-states of the form (cid:12)(cid:12) g x −→ y (cid:11) ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x gx y y (cid:48) (cid:48)
10 ˜0 (cid:48) ˜1 (cid:48) ˜1˜0 (cid:43) ≡ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) gx x − gy y y y x x y x (cid:43) (6.69)where we borrowed the notation from sec. 4 and x = x x −→ x x ∈ Hom( X ) , y = y y −→ y y ∈ Hom( Y ) , g x −→ y , ∈ Hom(Λ G X Y ) , such that Λ G X Y denotes the relative groupoid over Λ G defined by X and Y . In this setting, thereexists a natural multiplication of such boundary tubes defining an algebra isomorphic to the twisted ∼ ∼ roupoid algebra C [Λ G X Y ] T ( π ) φψ . Letting String πG denote the bicategory defined in the same manner as πG with objects all (Λ G, T ( π ))-subgroupoids and hom-categories Hom
String πG (( X , φ ) , ( Y , ψ )) := Mod ( C [Λ G X Y ] T ( π ) φψ ) , (6.70)we obtain the following equivalence of bicategories: String πG (cid:39) MOD ( Vec T ( π )Λ G ) . (6.71)Utilising this equivalence of bicategories, together with the physical interpretation inherited from thetube algebra approach, we interpret the Vec T ( π )Λ G -module category Mod
Vec T ( π )Λ G ( X φ ) for a (Λ G, T ( π ))-subgroupoid ( X , φ ) as the [70], for a point-like particle pinned to the boundary ofthe spatial manifold that appears at the end-point of a string-like (bulk) excitation. As before, 1-morphisms are naturally interpreted as bulk string-like excitations.The motivation for calling Vec T ( π )Λ G -module categories Mod
Vec T ( π )Λ G ( X φ ) 2-Hilbert spaces is as follows.In finite-dimensional quantum mechanics, given a finite set X of classical field configurations, the cor-responding Hilbert space H [ X ] is given by the free vector space of functions f : X → C . Categorifyingthe set of classical field configurations to a groupoid G , whose objects correspond to classical fieldconfigurations and morphisms, the symmetries of the field configurations. The category [ G , Vec ] β of(weak) functors F : G →
Vec for [ β ] ∈ H ( G , U(1)) provides a natural categorification of H [ X ] whichcan be shown to correspond to a finite 2-vector space (see sec. 6.8). The category [ G , Vec ] β can then beshown to admit a categorification of the inner-product of finite Hilbert spaces given by the hom-functor (cid:104)− , −(cid:105) : ([ G , Vec ] β ) op (cid:2) [ G , Vec ] β → Vec . (6.72)Recalling that Mod
Vec T ( π )Λ G ( X φ ) is defined by a category of weak functors from a groupoid to Vec , theterm 2-Hilbert space seems most appropriate. We interpret the objects (also called 2-vectors) of
Mod
Vec T ( π )Λ G ( X φ ) as describing the Hilbert space of a boundary excitation.We conclude this section by showing that, in general, objects in ( Vec T ( π )Λ G ) are not indecomposable as Vec T ( π )Λ G -module categories. For convenience, we shall focus on the limiting case where the group G is abelian , but our analysis can be extended to the non-abelian scenario. Analogously to indecomposablemodules over an algebra, an indecomposable module category is a module category which is notequivalent to the direct sum of non-zero module categories. Using the equivalence between Vec T ( π )Λ G -module categories and the categories of module objects for a separable algebra object in Vec T ( π )Λ G , wehave that a Vec T ( π )Λ G -module category is indecomposable if only if the corresponding algebra object isnot Morita equivalent to a direct sum of non-zero algebra objects. Given a (3+1)d gauge model withinput data ( G, π ), and a choice of gapped boundary condition (
A, λ ), an algebra object Λ A T ( λ ) in Vec T ( π )Λ G naturally decomposes as a direct sum viaΛ A T ( λ ) = (cid:77) a ∈ A (Λ A a ) T a ( λ ) , (6.73)where Λ A a denotes the groupoid with unique object a ∈ A and set of morphisms { a a (cid:48) −→ a } ∀ a (cid:48) ∈ A . The2-cochain T a ( λ ) ∈ C (Λ A a , U(1)) is then given by the restriction of T ( λ ) ∈ C (Λ A, U(1)) to Λ A a .This decomposition yields Mod
Vec T ( π )Λ G (Λ A T ( λ ) ) (cid:39) (cid:77) a ∈ A Mod
Vec T ( π )Λ G ((Λ A a ) T a ( λ ) ) (6.74) ∼ ∼ s Vec T ( π )Λ G -module categories, so that the category of module objects is not indecomposable as amodule category unless A = G is the trivial subgroup of G . Generically, for possibly non-abelian G an indecomposable Vec T ( π )Λ G -module category can be specified by a triple ( O , H, φ ) , where O denotesa conjugacy class of G , H is a subgroup of the centralizer Z o ⊆ G for a representative o ∈ O , and φ ∈ C ( H, U(1)) is 2-cochain satisfying d (2) φ = T ( π ) | H [53]. The corresponding algebra object is thengiven by ( H o ) φ o , where H o denotes the groupoid with unique object o ∈ O and hom-set { h : o → o } ∀ h ∈ H with composition given by multiplication in H , and the 2-cochain φ o ∈ C ( H o , U(1)) isdefined by the relation φ o ( h : o → o , h (cid:48) : o → o ) := φ ( h, h (cid:48) ) for all h, h (cid:48) ∈ H . SECTION 7
Discussion
Gapped boundaries of topological models have been under scrutiny in the past years. Focusing onlattice Hamiltonian realisations of Dijkgraaf-Witten theory, a.k.a gauge models of topological phases,we studied gapped boundaries and their excitations in (2+1)d and (3+1)d. More specifically, the goalof this paper was two-fold: Apply the tube algebra approach to classify gapped boundary excitationsand, using these results, elucidate the higher-category theoretical formalism relevant to describe gappedboundaries in (3+1)d.As explained in detail in [32], local operators of lattice Hamiltonian realisations of Dijkgraaf-Wittentheory can be conveniently expressed in terms of the partition function of the theory applied to so-called pinched interval cobordisms. We introduced a generalisation of the Dijkgraaf-Witten partitionfunction, from which the gapped boundary Hamiltonian operators could be defined in analogy with thebulk Hamiltonian operators using the language of relative pinched interval cobordisms. Given gappedboundaries labelled by subgroups of the input group and cochains compatible with the input cocycle,we applied the tube algebra approach in order to reveal the algebraic structure underlying two types ofexcitations: ( i ) Point-like excitations at the interface of two gapped boundaries in (2+1)d, where the‘tube’ has the topology of I × I , and ( ii ) string-like (bulk) excitations terminating at point-like gappedboundary excitations, where the ‘tube’ has the topology of ( S × I ) × I . Crucially, both tube algebrascan be related via a lifting (or dimensional reduction) argument, and as such can be studied in parallel.This statement was formalised using the notion of relative groupoid algebra. When applied to theinput group treated as a one-object groupoid, this notion yields the (2+1)d tube algebra, whereasit yields the (3+1)d tube algebra when applied to the loop groupoid of the group. We subsequentlystudied the representation theory of the (3+1)d tube algebra in full detail, which encompasses the(2+1)d one as a limiting case, deriving the irreducible representations as well as the correspondingrecoupling theory.In the second part of this manuscript, we reformulated the previous statements in category theo-retical terms. In (2+1)d, the relevant notion to describe gapped boundaries and their excitations is thebicategory MOD ( Vec αG ) of module categories over the category Vec αG of group-graded vector spaces.In practice, a module category can be obtained as a category of modules over an algebra object inthe input category. The bicategory of module categories above can then be shown to be equivalentto a bicategory of separable algebra objects, such that objects correspond to the gapped boundaryconditions and morphisms to representations of a groupoid algebra isomorphic to the (2+1)d tubealgebra. The identification with the tube algebra allowed us to elucidate the physical interpretationof the category theoretical notions at play. Mimicking this (2+1)d construction, we further defineda bicategory that encodes the string-like excitations terminating at point-like excitations on gappedboundaries and found that is was equivalent to a sub-bicategory of the bicategory MOD ( Vec T ( π )Λ G ) of ∼ ∼ odules categories over the category Vec T ( π )Λ G of loop-groupoid-graded vector spaces. Comparing withthe work of Kong et al. [53], MOD ( Vec T ( π )Λ G ) is equivalent to the higher categorical centre Z ( πG ) ofthe category πG of G -graded 2-vector spaces, which is the input category of (3+1)d gauge models.In virtue of the physical interpretation inherited from the tube algebra approach, we thus suggestedthat Z ( πG ) describes bulk string-like excitations whose end-points terminate at point-like particlesthat are pinned to the boundary of the spatial manifold. This is the higher-dimensional analogueof the well-known statement that bulk point-like excitations in (2+1)d are described by the centre Z ( Vec αG ) of the input category.The distinction between the gapped boundary string-like excitations we focused on, and the moregeneral ones encoded in the centre Z ( πG ) can be appreciated from an extended TQFT point ofview. We should think of Z ( πG ) as describing the object the extended 4-3-2-1 Dijkgraaf-WittenTQFT assigns to the circle. It follows from our analysis that such extended TQFT is more generalthan what gapped boundary conditions provide. Working out the details of this more general scenariowill be the purpose of another paper.The study carried out in this manuscript can be generalized in several ways. First of all, we couldstudy gapped domains walls instead of gapped boundaries and consider string-like excitations thatterminate at gapped domains walls point-like excitations. In (2+1)d, the so-called folding trick can beused in order to map a gapped domain wall configuration to a gapped boundary one. It would certainlybe interesting to consider how this generalizes in higher dimensions. Once this more general scenariois well-understood, we could then apply our results to so-called fracton models, which were recentlysuggested in [71–73] to have an interpretation in terms of defect TQFTs. A related question wouldbe to study invertible domain walls such as duality defects and derive the underlying mathematicalstructure in category theoretical terms.Another follow-up work pertains to the relation between the string-like excitations as describedby Z ( πG ) and the loop-like excitations of the model. In a recent paper [32], the authors showedthat loop-like excitations and their statistics were captured by the category of modules over the so-called twisted quantum triple algebra . This algebra can be expressed as the twisted groupoid algebra C [Λ G ] T ( π ) of the loop groupoid of the loop groupoid of G . In comparison, recall that the twistedquantum double is isomorphic to C [Λ G ] T ( α ) in this language. This groupoid algebra was shownby the authors to be isomorphic to the tube algebra associated with the manifold T × I , a localneighbourhood of a loop-like object being a solid torus. Intuitively, we may expect loop-like excitationsto descend from the string-like ones via a tracing mechanism. This can be formalized using the notionof categorical trace , building upon the fact that it maps a module category over Vec G to a module over C [Λ G ] [59, 74]. Another way to establish the connection between string-like and loop-like excitationsconsists in first realising that, as braided monoidal categories, we have the equivalences Z ( Vec T ( π )Λ G ) (cid:39) Mod ( C [Λ G ] T ( π ) ) and Z ( Vec T ( π )Λ G ) (cid:39) Dim ( MOD ( Vec T ( π )Λ G )), where Dim denotes the dimension of abicategory [59, 75] obtained via an appropriate categorification of the dimension of a vector space.The details of this correspondence will be presented in a forthcoming paper.
Acknowledgments
CD would like to thank David Aasen and Dominic Williamson for very useful discussions on closelyrelated topics. CD is funded by the European Research Council (ERC) under the European UnionsHorizon 2020 research and innovation programme through the ERC Starting Grant WASCOSYS(No. 636201) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) underGermanys Excellence Strategy EXC-2111 390814868. AB is funded by the EPSRC doctoral prizefellowship scheme. ∼ ∼ PPENDIX A
Representation theory of the relative groupoid algebra
In this appendix, we collect the proofs of several important results of the representation theory of therelative groupoid algebra C [Λ( G AB )] αφψ . A.1 Proof of the orthogonality relations (5.12)Let us confirm that the representation matrices as defined in (5.8) satisfy the orthogonality relation(5.12):1 | A || B | (cid:88) g a −→ b ∈ Λ( G AB ) D ρ AB IJ (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ (cid:48) AB I (cid:48) J (cid:48) (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) = 1 | A || B | (cid:88) g a −→ b ∈ Λ( G AB ) δ g , o i δ a − gb , o j ϑ Λ( AB ) o ( p − i ap j , p − j | q − i bq j , q − j ) ϑ Λ( AB ) o ( p − i , a | q − i , b ) D Rmn (cid:0)(cid:12)(cid:12) p − i ap j −−−−−→ q − i bq j (cid:11)(cid:1) × δ g , o (cid:48) i δ a − gb , o (cid:48) j ϑ Λ( AB ) o ( p (cid:48)− i , a | q (cid:48)− i , b ) ϑ Λ( AB ) o ( p (cid:48)− i ap (cid:48) j , p (cid:48)− j | q (cid:48)− i bq (cid:48) j , q (cid:48)− j ) D R (cid:48) m (cid:48) n (cid:48) (cid:0)(cid:12)(cid:12) p (cid:48)− i ap (cid:48) j −−−−−−→ q (cid:48)− i bq (cid:48) j (cid:11)(cid:1) = 1 | A || B | (cid:88) o i a −→ b ∈ Hom(Λ( G AB )) δ O AB , O (cid:48) AB δ i,i (cid:48) δ j,j (cid:48) δ a − o i b , o j D Rmn (cid:0)(cid:12)(cid:12) p − i ap j −−−−−→ q − i bq j (cid:11)(cid:1) D R (cid:48) m (cid:48) n (cid:48) (cid:0)(cid:12)(cid:12) p − i ap j −−−−−→ q − i bq j (cid:11)(cid:1) = 1 | Z O AB | (cid:88) ( a , b ) ∈ Z O AB δ O AB , O (cid:48) AB δ i,i (cid:48) δ j,j (cid:48) D Rmn (cid:0)(cid:12)(cid:12) a −→ b (cid:11)(cid:1) D R (cid:48) m (cid:48) n (cid:48) (cid:0)(cid:12)(cid:12) a −→ b (cid:11)(cid:1) = δ ρ AB ,ρ (cid:48) AB δ I,I (cid:48) δ J,J (cid:48) d ρ AB , where we first expanded the representation matrices according to definition (5.8) and then used theorthogonality of the irreducible representation in Z O AB together with the relation | Z O AB | · |O AB | = | A || B | . A.2 Proof of the invariance property (5.28)Let us prove the invariance property (5.28), which we reproduce below for convenience (cid:88) { J } D ρ AB I AB J AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ BC I BC J BC (cid:0)(cid:12)(cid:12) g b (cid:48) −−→ c (cid:11)(cid:1)(cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105) D ρ AC J AC I AC (cid:0)(cid:12)(cid:12) g a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) (A.1)= 1 | B | (cid:88) ˜ b ∈ B ϑ Λ( AB ) g ( a , ˜ a | b , ˜ b ) ϑ Λ( BC ) g ( b (cid:48) , ˜ b | c , ˜ c ) ζ Λ( ABC )˜ a , ˜ b , ˜ c ( a − g b , b (cid:48)− g c ) ϑ Λ( AC ) g (˜ a , ˜ a − a (cid:48) | ˜ c , ˜ c − c (cid:48) ) δ g , a − g bb (cid:48)− g c × (cid:88) { K } D ρ AB I AB K AB (cid:0)(cid:12)(cid:12) g a ˜ a −−→ b ˜ b (cid:11)(cid:1) D ρ BC I BC K BC (cid:0)(cid:12)(cid:12) g b (cid:48) ˜ b −−−→ c ˜ c (cid:11)(cid:1)(cid:104) ρ AB K AB ρ BC K BC (cid:12)(cid:12)(cid:12) ρ AC K AC (cid:105) D ρ AC K AC I AC (cid:0)(cid:12)(cid:12) ˜ a − g ˜ c ˜ a − a (cid:48) −−−−→ ˜ c − c (cid:48) (cid:11)(cid:1) , ∼ ∼ et us consider the left-hand side of (A.1). In virtue of the gauge invariance (5.27) of the Clebsch-Gordan coefficients, this is equal tol . h . s(A.1) = (cid:88) g ∈ Hom(s(˜ a ) , s(˜ c )) (cid:88) { J,K } D ρ AB I AB J AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ BC I BC J BC (cid:0)(cid:12)(cid:12) g b (cid:48) −−→ c (cid:11)(cid:1) D ρ AC J AC I AC (cid:0)(cid:12)(cid:12) g a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) × ( D ρ AB J AB K AB ⊗ B D ρ BC J BC K BC ) (cid:0)(cid:12)(cid:12) g ˜ a −→ ˜ c (cid:11)(cid:1) D ρ AC J AC K AC (cid:0)(cid:12)(cid:12) g ˜ a −→ ˜ c (cid:11)(cid:1)(cid:104) ρ AB K AB ρ BC K BC (cid:12)(cid:12)(cid:12) ρ AC K AC (cid:105) = 1 | B | (cid:88) g (cid:48) ∈ Ob(Λ( G AB )) g (cid:48) ∈ Ob(Λ( G BC ))˜ b ∈ B (cid:88) { J,K } D ρ AB I AB J AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ BC I BC J BC (cid:0)(cid:12)(cid:12) g b (cid:48) −−→ c (cid:11)(cid:1) D ρ AC J AC I AC (cid:0)(cid:12)(cid:12) g a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) × D ρ AB J AB K AB (cid:0)(cid:12)(cid:12) g (cid:48) a −→ ˜ b (cid:11)(cid:1) D ρ BC J BC K BC (cid:0)(cid:12)(cid:12) g (cid:48) b −→ ˜ c (cid:11)(cid:1) D ρ AC J AC K AC (cid:0)(cid:12)(cid:12) g (cid:48) g (cid:48) a −→ ˜ c (cid:11)(cid:1) × ζ Λ( ABC )˜ a , ˜ b , ˜ c ( g (cid:48) , g (cid:48) ) (cid:104) ρ AB K AB ρ BC K BC (cid:12)(cid:12)(cid:12) ρ AC K AC (cid:105) , where we applied the definitions of the truncated tensor product ⊗ B and the comultiplication map∆ B . Using D ρ AC J AC K AC (cid:0)(cid:12)(cid:12) g (cid:48) g (cid:48) a −→ ˜ c (cid:11)(cid:1) = 1 ϑ Λ( AC ) g (cid:48) g (cid:48) (˜ a , ˜ a − | ˜ c , ˜ c − ) D ρ AC K AC J AC (cid:0)(cid:12)(cid:12) ˜ a − g (cid:48) g (cid:48) ˜ c ˜ a − −−−→ ˜ c − (cid:11)(cid:1) , together with the fact that the representation matrices define algebra homomorphisms yieldsl . h . s(A.1) = 1 | B | (cid:88) g (cid:48) , g (cid:48) ˜ b ∈ B (cid:88) { K } ϑ Λ( AB ) g ( a , ˜ a | b , ˜ b ) ϑ Λ( BC ) g ( b (cid:48) , ˜ b | c , ˜ c ) ϑ Λ( AC )˜ a − g (cid:48) g (cid:48) ˜ c (˜ a − , a (cid:48) | ˜ c − , c (cid:48) ) ϑ Λ( AC ) g (cid:48) g (cid:48) (˜ a , ˜ a − | ˜ c , ˜ c − ) × D ρ AB I AB K AB (cid:0)(cid:12)(cid:12) g a ˜ a −−→ b ˜ b (cid:11)(cid:1) D ρ BC I BC K BC (cid:0)(cid:12)(cid:12) g b (cid:48) ˜ b −−−→ c ˜ c (cid:11)(cid:1) D ρ AC K AC I AC (cid:0)(cid:12)(cid:12) ˜ a − g (cid:48) g (cid:48) ˜ c ˜ a − a (cid:48) −−−−→ ˜ c − c (cid:48) (cid:11)(cid:1) × δ g (cid:48) , a − g b δ g (cid:48) , b (cid:48)− g c δ g , g (cid:48) g (cid:48) ζ Λ( ABC )˜ a , ˜ b , ˜ c ( g (cid:48) , g (cid:48) ) (cid:104) ρ AB K AB ρ BC K BC (cid:12)(cid:12)(cid:12) ρ AC K AC (cid:105) = 1 | B | (cid:88) ˜ b ∈ B (cid:88) { K } ϑ Λ( AB ) g ( a , ˜ a | b , ˜ b ) ϑ Λ( BC ) g ( b (cid:48) , ˜ b | c , ˜ c ) ϑ Λ( AC )˜ a − g ˜ c (˜ a − , a (cid:48) | ˜ c − , c (cid:48) ) ϑ Λ( AC ) g (˜ a , ˜ a − | ˜ c , ˜ c − ) × D ρ AB I AB K AB (cid:0)(cid:12)(cid:12) g a ˜ a −−→ b ˜ b (cid:11)(cid:1) D ρ BC I BC K BC (cid:0)(cid:12)(cid:12) g b (cid:48) ˜ b −−−→ c ˜ c (cid:11)(cid:1) D ρ AC K AC I AC (cid:0)(cid:12)(cid:12) ˜ a − g ˜ c ˜ a − a (cid:48) −−−−→ ˜ c − c (cid:48) (cid:11)(cid:1) × δ g , a − g bb (cid:48)− g c ζ Λ( ABC )˜ a , ˜ b , ˜ c ( a − g b , b (cid:48)− g c ) (cid:104) ρ AB K AB ρ BC K BC (cid:12)(cid:12)(cid:12) ρ AC K AC (cid:105) . Finally, using d (2) ϑ Λ( AC ) g (˜ a , ˜ a − , a (cid:48) | ˜ c , ˜ c − , c (cid:48) ) = 1, we obtainl . h . s(A.1) = 1 | B | (cid:88) ˜ b ∈ B (cid:88) { K } ϑ Λ( AB ) g ( a , ˜ a | b , ˜ b ) ϑ Λ( BC ) g ( b (cid:48) , ˜ b | c , ˜ c ) ζ Λ( ABC )˜ a , ˜ b , ˜ c ( a − g b , b (cid:48)− g c ) ϑ Λ( AC ) g (˜ a , ˜ a − a (cid:48) | ˜ c , ˜ c − c (cid:48) ) × D ρ AB I AB K AB (cid:0)(cid:12)(cid:12) g a ˜ a −−→ b ˜ b (cid:11)(cid:1) D ρ BC I BC K BC (cid:0)(cid:12)(cid:12) g b (cid:48) ˜ b −−−→ c ˜ c (cid:11)(cid:1) D ρ AC K AC I AC (cid:0)(cid:12)(cid:12) ˜ a − g ˜ c ˜ a − a (cid:48) −−−−→ ˜ c − c (cid:48) (cid:11)(cid:1) × δ g , a − g bb (cid:48)− g c (cid:104) ρ AB K AB ρ BC K BC (cid:12)(cid:12)(cid:12) ρ AC K AC (cid:105) , which is the right-hand side of (A.1), as expected. Note that the above is true for every morphism˜ a , ˜ c . ∼ ∼ .3 Proof of the defining relation of the j -symbols In this appendix, we confirm the definition of the 6 j -symbols (cid:110) ρ AB ρ AD ρ BC ρ AC ρ CD ρ BD (cid:111) := 1 d ρ AD (cid:88) { I } α ( o i AB , o i BC , o i CD ) (cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105)(cid:104) ρ AC I AC ρ CD I CD (cid:12)(cid:12)(cid:12) ρ AD I AC (cid:105)(cid:104) ρ AB I AB ρ BD I BD (cid:12)(cid:12)(cid:12) ρ AD I AD (cid:105)(cid:104) ρ BC I BC ρ CD I CD (cid:12)(cid:12)(cid:12) ρ BD I BD (cid:105) , such that they satisfy the relation (cid:88) ρ AC (cid:88) { I } (cid:110) ρ AB ρ AD ρ BC ρ AC ρ CD ρ BD (cid:111)(cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105)(cid:104) ρ AC I AC ρ CD I CD (cid:12)(cid:12)(cid:12) ρ AD K AD (cid:105) | ρ AB I AB (cid:105) ⊗ | ρ BC I BC (cid:105) ⊗ | ρ CD I CD (cid:105) (cid:46) Φ ABCD = (cid:88) { I } (cid:104) ρ AB I AB ρ BD I BD (cid:12)(cid:12)(cid:12) ρ AD K AD (cid:105)(cid:104) ρ BC I BC ρ CD I CD (cid:12)(cid:12)(cid:12) ρ BD I BD (cid:105) | ρ AB I AB (cid:105) ⊗ | ρ BC I BC (cid:105) ⊗ | ρ CD I CD (cid:105) . (A.2)Inserting the definition of the 6 j -symbols into equation (A.2) and writing down explicitly the actionof Φ ABCD using (5.8), we find that the left-hand side is equal tol . h . s(A.2) = 1 d ρ AD (cid:88) ρ AC (cid:88) { I,J } α ( o j AB , o j BC , o j CD ) α ( o i AB , o i BC , o i CD ) × (cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105)(cid:104) ρ AC J AC ρ CD J CD (cid:12)(cid:12)(cid:12) ρ AD J AD (cid:105)(cid:104) ρ AB J AB ρ BD J BD (cid:12)(cid:12)(cid:12) ρ AD J AD (cid:105)(cid:104) ρ BC J BC ρ CD J CD (cid:12)(cid:12)(cid:12) ρ BD J BD (cid:105) × (cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105)(cid:104) ρ AC I AC ρ CD I CD (cid:12)(cid:12)(cid:12) ρ AD K AD (cid:105) | ρ AB I AB (cid:105) ⊗ | ρ BC I BC (cid:105) ⊗ | ρ CD I CD (cid:105) The defining relation of the Clebsch-Gordan coefficients yields (cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105)(cid:104) ρ AC J AC ρ CD J CD (cid:12)(cid:12)(cid:12) ρ AD J AD (cid:105)(cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105)(cid:104) ρ AC I AC ρ CD I CD (cid:12)(cid:12)(cid:12) ρ AD K AD (cid:105) = d ρ AC d ρ AD | A | | C || D | (cid:88) g a −→ c ∈ Λ( G AC ) g (cid:48) a (cid:48) −→ d ∈ Λ( G AD ) ( D ρ AB J AB I AB ⊗ B D ρ BC J BC I BC ) (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) D ρ AC J AC I AC (cid:0)(cid:12)(cid:12) g a −→ c (cid:11)(cid:1) × ( D ρ AC J AC I AC ⊗ C D ρ CD J CD I CD ) (cid:0)(cid:12)(cid:12) g (cid:48) a (cid:48) −−→ d (cid:11)(cid:1) D ρ AD J AD K AD (cid:0)(cid:12)(cid:12) g (cid:48) a (cid:48) −−→ d (cid:11)(cid:1) = d ρ AC d ρ AD | A | | B || C | | D | (cid:88) g , g , g (cid:48) , g (cid:48) ( a , c ) ∈ A × C ( a (cid:48) , d ) ∈ A × D ( b , c (cid:48) ) ∈ B × C D ρ AB J AB I AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) g b −→ c (cid:11)(cid:1) D ρ AC J AC I AC (cid:0)(cid:12)(cid:12) g g a −→ c (cid:11)(cid:1) × D ρ AC J AC I AC (cid:0)(cid:12)(cid:12) g (cid:48) a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) D ρ CD J CD I CD (cid:0)(cid:12)(cid:12) g (cid:48) c (cid:48) −−→ d (cid:11)(cid:1) D ρ AD J AD K AD (cid:0)(cid:12)(cid:12) g (cid:48) g (cid:48) a (cid:48) −−→ d (cid:11)(cid:1) × ζ Λ( ABC ) a , b , c ( g , g ) ζ Λ( ACD ) a (cid:48) , c (cid:48) , d ( g (cid:48) , g (cid:48) ) , where the second sum is over g ∈ Ob(Λ( G AB )), g ∈ Ob(Λ( G BC )), g (cid:48) ∈ Ob(Λ( G AC )), g (cid:48) ∈ Ob(Λ( G CD )) and the corresponding morphisms, which we loosely identify with the group variablesthey are characterized by. Furthermore, we have that1 | A || C | (cid:88) ρ AC I AC ,J AC d ρ AC D ρ AC J AC I AC (cid:0) | g (cid:48) a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) D ρ AC J AC I AC (cid:0)(cid:12)(cid:12) g g a −→ c (cid:11)(cid:1) (A.3)= δ a − g g c , a (cid:48)− g (cid:48) c (cid:48) | A || C | (cid:88) ρ AC d ρ AC tr (cid:2) D ρ AC (cid:0) | g (cid:48) a (cid:48) a − −−−−→ c (cid:48) c − (cid:11)(cid:1)(cid:3) = δ g g , g (cid:48) δ a , a (cid:48) δ c , c (cid:48) , (A.4) ∼ ∼ here we made use of the orthogonality relation (5.12) so that (cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105)(cid:104) ρ AC J AC ρ CD J CD (cid:12)(cid:12)(cid:12) ρ AD J AD (cid:105)(cid:104) ρ AB I AB ρ BC I BC (cid:12)(cid:12)(cid:12) ρ AC I AC (cid:105)(cid:104) ρ AC I AC ρ CD I CD (cid:12)(cid:12)(cid:12) ρ AD K AD (cid:105) = d ρ AD | A || B || C || D | (cid:88) g , g , g (cid:48) ( a , c ) ∈ A × C d ∈ D b ∈ B D ρ AB J AB I AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) g b −→ c (cid:11)(cid:1) × D ρ CD J CD I CD (cid:0)(cid:12)(cid:12) g (cid:48) c −→ d (cid:11)(cid:1) D ρ AD J AD K AD (cid:0)(cid:12)(cid:12) g g g (cid:48) a −→ d (cid:11)(cid:1) × ζ Λ( ABC ) a , b , c ( g , g ) ζ Λ( ACD ) a , c , d ( g g , g (cid:48) ) . Putting everything together so far, we obtainl . h . s(A.2) = 1 | A || B || C || D | (cid:88) g , g , g (cid:48) ( a , c ) ∈ A × C ( b , d ) ∈ B × D (cid:88) { I,J } α ( o j AB , o j BC , o j CD ) α ( o i AB , o i BC , o i CD ) ζ Λ( ABC ) a , b , c ( g , g ) ζ Λ( ACD ) a , c , d ( g g , g (cid:48) ) × D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) g b −→ c (cid:11)(cid:1) D ρ CD J CD I CD (cid:0)(cid:12)(cid:12) g (cid:48) c −→ d (cid:11)(cid:1)(cid:104) ρ BC J BC ρ CD J CD (cid:12)(cid:12)(cid:12) ρ BD J BD (cid:105) × D ρ AB J AB I AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ AD J AD K AD (cid:0)(cid:12)(cid:12) g g g (cid:48) a −→ d (cid:11)(cid:1)(cid:104) ρ AB J AB ρ BD J BD (cid:12)(cid:12)(cid:12) ρ AD J AD (cid:105) × | ρ AB I AB (cid:105) ⊗ | ρ BC I BC (cid:105) ⊗ | ρ CD I CD (cid:105) . In virtue of the definition of the representation matrices, we observe that we must have o i AB = a − g b , o i BC = b − g c , o i CD = c − g (cid:48) d , o j AB = g , o j BC = g and o j CD = g (cid:48) in order for the whole expressionnot to vanish. Applying the quasi-coassociativity condition ζ Λ( BCD ) b , c , d ( g , g (cid:48) ) ζ Λ( ABD ) a , b , d ( g , g g (cid:48) ) ζ Λ( ACD ) a , c , d ( g g , g (cid:48) ) ζ Λ( ABC ) a , b , c ( g , g ) = α ( g , g , g (cid:48) ) α ( a − g b , b − g c , c − g (cid:48) d ) , (A.5)we obtainl . h . s(A.2) = 1 | A || B || C || D | (cid:88) g , g , g (cid:48) ( a , c ) ∈ A × C ( b , d ) ∈ B × D (cid:88) { I,J } ζ Λ( BCD ) b , c , d ( g , g (cid:48) ) ζ Λ( ABD ) a , b , d ( g , g g (cid:48) ) × D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) g b −→ c (cid:11)(cid:1) D ρ CD J CD I CD (cid:0)(cid:12)(cid:12) g (cid:48) c −→ d (cid:11)(cid:1)(cid:104) ρ BC J BC ρ CD J CD (cid:12)(cid:12)(cid:12) ρ BD J BD (cid:105) × D ρ AB J AB I AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ AD J AD K AD (cid:0)(cid:12)(cid:12) g g g (cid:48) a −→ d (cid:11)(cid:1)(cid:104) ρ AB J AB ρ BD J BD (cid:12)(cid:12)(cid:12) ρ AD J AD (cid:105) × | ρ AB I AB (cid:105) ⊗ | ρ BC I BC (cid:105) ⊗ | ρ CD I CD (cid:105) . Let us now insert the resolution of the identity δ J BD ,J BD = (cid:88) h , h (cid:48) ∈ Ob(Λ( G BD )) (cid:88) I BD D ρ BD J BD I BD (cid:0)(cid:12)(cid:12) h (cid:48) b −→ d (cid:11)(cid:1) D ρ BD J BD I BD (cid:0)(cid:12)(cid:12) h b −→ d (cid:11)(cid:1) , (A.6)where h and h (cid:48) are implicitly identified via the algebra product. As a special case of (5.28), we have (cid:88) { J } (cid:88) c ∈ C D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) g b −→ c (cid:11)(cid:1) D ρ CD J CD I CD (cid:0)(cid:12)(cid:12) g (cid:48) c −→ d (cid:11)(cid:1) D ρ BD J BD I BD (cid:0)(cid:12)(cid:12) h b −→ d (cid:11)(cid:1)(cid:104) ρ BC J BC ρ CD J CD (cid:12)(cid:12)(cid:12) ρ BD J BD (cid:105) = (cid:88) { J } (cid:88) c ∈ C D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) g b −→ c (cid:11)(cid:1) D ρ CD J CD I CD (cid:0)(cid:12)(cid:12) g (cid:48) c −→ d (cid:11)(cid:1) D ρ BD J BD I BD (cid:0)(cid:12)(cid:12) g g (cid:48) b −→ d (cid:11)(cid:1) (cid:104) ρ BC J BC ρ CD J CD (cid:12)(cid:12)(cid:12) ρ BD J BD (cid:105) δ h , g g (cid:48) . ∼ ∼ e can finally use the gauge invariance of the Clebsch-Gordan coefficients1 | B | (cid:88) { J } (cid:88) h (cid:48) , g ζ Λ( ABD ) a , b , d ( g , h (cid:48) ) D ρ AB J AB I AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ BD J BD I BD (cid:0)(cid:12)(cid:12) h (cid:48) b −→ d (cid:11)(cid:1) D ρ AD J AD K AD (cid:0)(cid:12)(cid:12) g h (cid:48) a −→ d (cid:11)(cid:1) (cid:104) ρ AB J AB ρ BD J BD (cid:12)(cid:12)(cid:12) ρ AD J AD (cid:105) = (cid:104) ρ AB I AB ρ BD I BD (cid:12)(cid:12)(cid:12) ρ AD K AD (cid:105) and1 | C | (cid:88) { J } (cid:88) g , g (cid:48) ζ Λ( BCD ) b , c , d ( g , g (cid:48) ) D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) g b −→ c (cid:11)(cid:1) D ρ CD J CD I CD (cid:0)(cid:12)(cid:12) g (cid:48) c −→ d (cid:11)(cid:1) D ρ BD J BD I BD (cid:0)(cid:12)(cid:12) g g (cid:48) b −→ d (cid:11)(cid:1) (cid:104) ρ BC J BC ρ CD J CD (cid:12)(cid:12)(cid:12) ρ BD J BD (cid:105) = (cid:104) ρ BC I BC ρ CD I CD (cid:12)(cid:12)(cid:12) ρ BD I BD (cid:105) , so as to yield (A.2) as expected. A.4 Proof of the pentagon identity
As explained in the main text, the pentagon identity is the statement that the algebra elements[(id ⊗ id ⊗ ∆ D )(Φ ABCE )] (cid:63) [(∆ B ⊗ id ⊗ id)(Φ ACDE )]and ( AB ⊗ Φ BCDE ) (cid:63) [(id ⊗ ∆ C ⊗ id)(Φ ABDE )] (cid:63) (Φ ABCD ⊗ DE )induce the same isomorphism on the four-particle vector space (( V ρ AB ⊗ B V ρ BC ) ⊗ C V ρ CD ) ⊗ D V ρ DE .In light of the definition of the truncated tensor product of vector spaces, this can be demonstratedexplicitly by showing the equality:( AB ⊗ Φ BCDE ) (cid:63) [(id ⊗ ∆ C ⊗ id)(Φ ABDE )] (cid:63) (Φ ABCD ⊗ DE ) (cid:63) ((( AB ) C ) D ) E = [(id ⊗ id ⊗ ∆ D )(Φ ABCE )] (cid:63) [(∆ B ⊗ id ⊗ id)(Φ ACDE )] (cid:63) ((( AB ) C ) D ) E , (A.7)where we defined ((( AB ) C ) D ) E := [(∆ B ⊗ id) ◦ (∆ C ⊗ id) ◦ ∆ D ]( AE ) . Writing down explicitly the definition of the comultiplication maps, we have[(id ⊗ ∆ C ⊗ id)(Φ ABDE )] = 1 | C | (cid:88) { g } c ∈ C ζ Λ( BCD ) B ,c, D ( g , g ) α ( g , g g , g ) (cid:12)(cid:12) g A −−−→ B (cid:11) ⊗ (cid:12)(cid:12) g B −−−→ c (cid:11) ⊗ (cid:12)(cid:12) g c −−−→ D (cid:11) ⊗ (cid:12)(cid:12) g D −−−→ E (cid:11) , [(id ⊗ id ⊗ ∆ D )(Φ ABCE )] = 1 | D | (cid:88) { g } d ∈ D ζ Λ( CDE ) C ,d, E ( g , g ) α ( g , g , g g ) (cid:12)(cid:12) g A −−−→ B (cid:11) ⊗ (cid:12)(cid:12) g B −−−→ C (cid:11) ⊗ (cid:12)(cid:12) g C −−−→ d (cid:11) ⊗ (cid:12)(cid:12) g d −−−→ E (cid:11) , [(∆ B ⊗ id ⊗ id)(Φ ACDE )] = 1 | B | (cid:88) { g } b ∈ B ζ Λ( ABC ) A ,b, C ( g , g ) α ( g g , g , g ) (cid:12)(cid:12) g A −−−→ b (cid:11) ⊗ (cid:12)(cid:12) g b −−−→ C (cid:11) ⊗ (cid:12)(cid:12) g C −−−→ D (cid:11) ⊗ (cid:12)(cid:12) g D −−−→ E (cid:11) , ∼ ∼ nd ((( AB ) C ) D ) E = 1 | B || C || D | (cid:88) { g } ( b , c , d ) ∈ B × C × D ζ Λ( ADE ) A , d , E ( g g g , g ) ζ Λ( ACD ) A , c , d ( g g , g ) ζ Λ( ABC ) A , b , c ( g , g ) × (cid:12)(cid:12) g A −−−→ b (cid:11) ⊗ (cid:12)(cid:12) g b −→ c (cid:11) ⊗ (cid:12)(cid:12) g c −→ d (cid:11) ⊗ (cid:12)(cid:12) g d −−−→ E (cid:11) . Applying the definition of the algebra product, we then obtain[(id ⊗ id ⊗ ∆ D )(Φ ABCE )] (cid:63) [(∆ B ⊗ id ⊗ id)(Φ ACDE )]= 1 | B || D | (cid:88) { g } ( b , d ) ∈ B × D ζ Λ( CDE ) C , d , E ( g , g ) ζ Λ( ABC ) A , b , C ( g , g ) α ( g , g , g g ) α ( g g , g d , d − g ) × (cid:12)(cid:12) g A −−−→ b (cid:11) ⊗ (cid:12)(cid:12) g b −−−→ C (cid:11) ⊗ (cid:12)(cid:12) g C −−−→ d (cid:11) ⊗ (cid:12)(cid:12) g d −−−→ E (cid:11) and ( AB ⊗ Φ BCDE ) (cid:63) [(id ⊗ ∆ C ⊗ id)(Φ ABDE )] (cid:63) (Φ ABCD ⊗ DE )= 1 | C | (cid:88) { g } c ∈ C ζ Λ( BCD ) B , c , D ( g , g ) α ( g , g , g ) α ( g , g g , g ) α ( g , g c , c − g ) × (cid:12)(cid:12) g A −−−→ B (cid:11) ⊗ (cid:12)(cid:12) g B −−−→ c (cid:11) ⊗ (cid:12)(cid:12) g c −−−→ D (cid:11) ⊗ (cid:12)(cid:12) g D −−−→ E (cid:11) . It remains to multiply both expression from the right by ((( AB ) C ) D ) E . First, we compute the right-andside of (A.7):r . h . s(A.7) = 1 | B | | C || D | (cid:88) { g } b , b (cid:48) , c , d , d (cid:48) ζ Λ( CDE ) C , d , E ( g , g ) ζ Λ( ABC ) A ,b, C ( g , g ) α ( g , g , g g ) α ( g g , g d , d − g ) × ζ Λ( ADE ) A , d (cid:48) , E ( g g g d , d − g ) ζ Λ( ACD ) A , c , d (cid:48) ( g g , g d ) ζ Λ( ABC ) A , b (cid:48) , c ( g b , b − g ) × ϑ Λ( AB ) g ( A , A | b , b (cid:48) ) ϑ Λ( BC ) g ( b , b (cid:48) | C , c ) ϑ Λ( CD ) g ( C , c | d , d (cid:48) ) × ϑ Λ( DE ) g ( d , d (cid:48) | E , E ) (cid:12)(cid:12) g A −−−→ b (cid:11) ⊗ (cid:12)(cid:12) g b −−−→ C (cid:11) ⊗ (cid:12)(cid:12) g C −−−→ d (cid:11) ⊗ (cid:12)(cid:12) g d −−−→ E (cid:11) . Using the cocycle relations ϑ Λ( AB ) g ( A , A | b , b (cid:48) ) ϑ Λ( BC ) g ( b , b (cid:48) | C , c ) ϑ Λ( AC ) g g ( A , A | C , c ) = ζ Λ( ABC ) A , bb (cid:48) , c ( g , g ) ζ Λ( ABC ) A , b , C ( g , g ) ζ Λ( ABC ) A , b (cid:48) , c ( g b , b − g c )and ϑ Λ( CD ) g ( C , c | d , d (cid:48) ) ϑ Λ( DE ) g ( d , d (cid:48) | E , E ) ϑ Λ( CE ) g g ( C , c | E , E ) = ζ Λ( CDE ) c , dd (cid:48) , E ( g , g ) ζ Λ( CDE ) C , d , E ( g , g ) ζ Λ( CDE ) c , d (cid:48) , E ( g d , d − g )as well as the quasi-coassociativity conditions ζ Λ( CDE ) c , d (cid:48) , E ( g d , d − g ) ζ Λ( BCE ) B , c , E ( g g , g g ) ζ Λ( BDE ) B , d , E ( g g g d , d − g ) ζ Λ( BCD ) B , c , d (cid:48) ( g g , g d ) = α ( g g , g d , d − g ) α ( g g c , c − g dd (cid:48) , d (cid:48)− d − g ) ∼ ∼ nd ζ Λ( CDE ) c , dd (cid:48) , E ( g , g ) ζ Λ( BCE ) B , c , E ( g g , g g ) ζ Λ( BDE ) B , dd (cid:48) , E ( g g g , g ) ζ Λ( BCD ) B , c , dd (cid:48) ( g g , g ) = α ( g g , g , g ) α ( g g c , c − g dd (cid:48) , d (cid:48)− d − g )yields r . h . s(A.7) = 1 | B || C || D | (cid:88) { g } b,c,d ζ Λ( CDE ) C , d , E ( g g g , g ) ζ Λ( BCD ) B , c , d ( g g , g ) ζ Λ( ABC ) A , b , c ( g , g ) α ( g , g , g g ) α ( g g , g , g ) × (cid:12)(cid:12) g A −−−→ b (cid:11) ⊗ (cid:12)(cid:12) g b −→ c (cid:11) ⊗ (cid:12)(cid:12) g c −→ d (cid:11) ⊗ (cid:12)(cid:12) g d −−−→ E (cid:11) . Let us repeat the same procedure in order to compute the left-hand side of (A.7):l . h . s(A.7) = 1 | B || C | | D | (cid:88) { g } b , c , c (cid:48) , d ζ Λ( BCD ) B , c , D ( g , g ) α ( g , g , g ) α ( g , g g , g ) α ( g , g c , c − g ) × ζ Λ( ADE ) A , d , E ( g g g , g ) ζ Λ( ACD ) A , c (cid:48) , d ( g g c , c − g ) ζ Λ( ABC ) A , b , c (cid:48) ( g , g c ) × ϑ Λ( AB ) g ( A , A | B , b ) ϑ Λ( BC ) g ( B , b | c , c (cid:48) ) ϑ Λ( CD ) g ( c , c (cid:48) | D , d ) × ϑ Λ( DE ) g ( D , d | E , E ) (cid:12)(cid:12) g A −−−→ B (cid:11) ⊗ (cid:12)(cid:12) g B −−−→ c (cid:11) ⊗ (cid:12)(cid:12) g c −−−→ D (cid:11) ⊗ (cid:12)(cid:12) g D −−−→ E (cid:11) . Using the cocycle relation ϑ Λ( BC ) g ( B , b | c , c (cid:48) ) ϑ Λ( CD ) g ( c , c (cid:48) | D , d ) ϑ Λ( BD ) g g ( B , b | D , d ) = ζ Λ( BCD ) b , cc (cid:48) ,d ( g , g ) ζ Λ( BCD ) B , c , D ( g , g ) ζ Λ( BCD ) b , c (cid:48) , d ( g c , c − g )as well as the quasi-coassociativity conditions ζ Λ( BCD ) b , c (cid:48) , d ( g c , c − g ) ζ Λ( ABD ) A , b , d ( g , g g ) ζ Λ( ACD ) A , c (cid:48) , d ( g g c , c − g ) ζ Λ( ABC ) A , b , c (cid:48) ( g , g c ) = α ( g , g c , c − g ) α ( g b , b − g cc (cid:48) , c (cid:48)− c − g d )and ζ Λ( BCD ) b , cc (cid:48) , d ( g , g ) ζ Λ( ABD ) A , b , d ( g , g g ) ζ Λ( ACD ) A , cc (cid:48) , d ( g g , g ) ζ Λ( ABC ) A , b , cc (cid:48) ( g , g ) = α ( g , g , g ) α ( g b , b − g cc (cid:48) , c (cid:48)− c − g d )yields l . h . s(A.7) = 1 | B || C || D | (cid:88) { g } b , c ,d ζ Λ( ADE ) A , d , E ( g g g , g ) ζ Λ( ACD ) A , c , d ( g g , g ) ζ Λ( ABC ) A , b , c ( g , g ) α ( g , g , g ) α ( g , g g , g ) α ( g , g , g ) × (cid:12)(cid:12) g A −−−→ b (cid:11) ⊗ (cid:12)(cid:12) g b −→ c (cid:11) ⊗ (cid:12)(cid:12) g c −→ d (cid:11) ⊗ (cid:12)(cid:12) g d −−−→ E (cid:11) . The equality between l . h . s(A.7) and r . h . s(A.7) finally follows from the groupoid 3-cocycle condition d (3) α = 1, hence the pentagon identity. ∼ ∼ PPENDIX B
Canonical basis for boundary excitations in (2+1)d
In this appendix, we collect the proofs of some properties crucial to the definition of the canonical basispresented in sec. 5.5.
B.1 Proof of the canonical algebra product (5.50)Using transformations (5.46) and (5.47), as well as the definition of the (cid:63) -product, we have | ρ AB IJ (cid:105) (cid:63) | ρ (cid:48) AB I (cid:48) J (cid:48) (cid:105) = ( d ρ AB d ρ (cid:48) AB ) | A || B | (cid:88) g,g (cid:48) ∈ G ( a,b ) , ( a (cid:48) ,b (cid:48) ) ∈ A × B D ρ AB IJ (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ (cid:48) AB I (cid:48) J (cid:48) (cid:0)(cid:12)(cid:12) g (cid:48) a (cid:48) −−→ b (cid:48) (cid:11)(cid:1) (cid:12)(cid:12) g a −→ b (cid:11) (cid:63) (cid:12)(cid:12) g (cid:48) a (cid:48) −−→ b (cid:48) (cid:11) = ( d ρ AB d ρ (cid:48) AB ) | A || B | (cid:88) g,g (cid:48) ∈ G ( a,b ) , ( a (cid:48) ,b (cid:48) ) ∈ A × B D ρ AB IJ (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ (cid:48) AB I (cid:48) J (cid:48) (cid:0)(cid:12)(cid:12) g (cid:48) a (cid:48) −−→ b (cid:48) (cid:11)(cid:1) δ g (cid:48) ,a − gb ϑ ABg ( a, a (cid:48) | b, b (cid:48) ) (cid:12)(cid:12) g aa (cid:48) −−−→ ab (cid:48) (cid:11) = ( d ρ AB d ρ (cid:48) AB ) | A || B | (cid:88) g,g (cid:48) ∈ G ( a,b ) , ( a (cid:48) ,b (cid:48) ) ∈ A × B D ρ AB IJ (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ (cid:48) AB I (cid:48) J (cid:48) (cid:0)(cid:12)(cid:12) g (cid:48) a (cid:48) −−→ b (cid:48) (cid:11)(cid:1) δ g (cid:48) ,a − xb ϑ g ( a, a (cid:48) | b, b (cid:48) ) × (cid:16) | A || B | (cid:17) (cid:88) ρ (cid:48)(cid:48) AB d ρ (cid:48)(cid:48) AB (cid:88) I (cid:48)(cid:48) ,J (cid:48)(cid:48) D ρ (cid:48)(cid:48) AB I (cid:48)(cid:48) J (cid:48)(cid:48) (cid:0)(cid:12)(cid:12) g aa (cid:48) −−−→ bb (cid:48) (cid:11)(cid:1) | ρ (cid:48)(cid:48) AB I (cid:48)(cid:48) J (cid:48)(cid:48) (cid:105) . But by linearity of the representation matrices, we have δ g (cid:48) ,a − gb ϑ ABg ( a, a (cid:48) | b, b (cid:48) ) D ρ (cid:48)(cid:48) AB I (cid:48)(cid:48) J (cid:48)(cid:48) (cid:0)(cid:12)(cid:12) g aa (cid:48) −−−→ bb (cid:48) (cid:11)(cid:1) = (cid:88) K D ρ (cid:48)(cid:48) AB I (cid:48)(cid:48) K (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ (cid:48)(cid:48) AB KJ (cid:48)(cid:48) (cid:0)(cid:12)(cid:12) g (cid:48) a (cid:48) −−→ b (cid:48) (cid:11)(cid:1) . (B.1)Orthogonality of the representation matrices finally yields the desired expression | ρ AB IJ (cid:105) (cid:63) | ρ (cid:48) AB I (cid:48) J (cid:48) (cid:105) = | A | | B | δ ρ AB ,ρ (cid:48) AB δ J,I (cid:48) d ρ AB | ρ AB IJ (cid:48) (cid:105) . (B.2) B.2 Ground state projector on the annulus
Let us evaluate the quantity1 | A || B | (cid:88) g ∈ G ( a,b ) ∈ A × B (cid:88) ˜ g ∈ G (˜ a, ˜ b ) ∈ A × B (cid:16)(cid:12)(cid:12) ˜ g ˜ a −→ ˜ b (cid:11) − (cid:63) (cid:12)(cid:12) g a −→ b (cid:11) (cid:63) (cid:12)(cid:12) ˜ g ˜ a −→ ˜ b (cid:11)(cid:17)(cid:10) g a −→ b (cid:12)(cid:12) , (B.3)and confirm that it is equal to P O (cid:52) as defined in (5.58). By direct computation, we have (cid:12)(cid:12) g a −→ b (cid:11) (cid:63) (cid:12)(cid:12) ˜ g ˜ a −→ ˜ b (cid:11) = δ ˜ g,a − gb ϑ ABg ( a, ˜ a | b, ˜ b ) (cid:12)(cid:12) g a ˜ a −−→ b ˜ b (cid:11) (B.4)and (cid:12)(cid:12) ˜ g ˜ a −→ ˜ b (cid:11) − (cid:63) (cid:12)(cid:12) g a ˜ a −−→ b ˜ b (cid:11) = δ ˜ g,g ϑ AB ˜ a − g ˜ b (˜ a − , a ˜ a | ˜ b − , b ˜ b ) ϑ AB ˜ g (˜ a, ˜ a − | ˜ b, ˜ b − ) (cid:12)(cid:12) ˜ a − g ˜ b ˜ a − a ˜ a −−−−−→ ˜ b − b ˜ b (cid:11) (B.5) ∼ ∼ o that (cid:12)(cid:12) ˜ g ˜ a −→ ˜ b (cid:11) − (cid:63) (cid:12)(cid:12) g a −→ b (cid:11) (cid:63) (cid:12)(cid:12) ˜ g ˜ a −→ ˜ b (cid:11) = δ ˜ g,g δ ˜ g,a − gb ϑ ABg ( a, ˜ a | b ˜ b ) ϑ AB ˜ a − g ˜ b (˜ a − , a ˜ a | ˜ b − , b ˜ b ) ϑ AB ˜ g (˜ a, ˜ a − | ˜ b, ˜ b − ) (cid:12)(cid:12) ˜ a − g ˜ b ˜ a − a ˜ a −−−−−→ ˜ b − b ˜ b (cid:11) . (B.6)Using the groupoid cocycle condition d (2) ϑ ABg (˜ a, ˜ a − , a ˜ a | ˜ b, ˜ b − , b ˜ b ) and performing the summationsfinally yield the desired result. B.3 Proof of the diagonalisation property (5.69)Given the action of the Hamiltonian projector (5.67) on Y (cid:52) , we show that the basis states defined as | ρ AB I AB , ρ BC I BC , ρ AC I AC (cid:105) Y (cid:52) := (cid:88) g ,g ∈ Ga,a (cid:48) ∈ Ab,b (cid:48) ∈ Bc,c (cid:48) ∈ C (cid:88) { J } D ρ AB J AB I AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) g b (cid:48) −−→ c (cid:11)(cid:1)(cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105) D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) a (cid:48) g g c (cid:48)− a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) × | g , a, b, g , b (cid:48) , c, a (cid:48) , c (cid:48) (cid:105) Y (cid:52) satisfy the relation P Y (cid:52) (cid:0) | ρ AB I AB , ρ BC I BC , ρ AC I AC (cid:105) Y (cid:52) (cid:1) = | ρ AB I AB , ρ BC I BC , ρ AC I AC (cid:105) Y (cid:52) . (B.7)By direct computation, we have P Y (cid:52) (cid:0) | ρ AB I AB , ρ BC I BC , ρ AC I AC (cid:105) Y (cid:52) (cid:1) = (cid:88) { g ∈ G } a,a (cid:48) ∈ Ab,b (cid:48) ∈ Bc,c (cid:48) ∈ C (cid:88) { J } D ρ AB J AB I AB (cid:0)(cid:12)(cid:12) g a −→ b (cid:11)(cid:1) D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) g b (cid:48) −−→ c (cid:11)(cid:1)(cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105) D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) a (cid:48) g g c (cid:48)− a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) × | A || B || C | (cid:88) ˜ a ∈ A ˜ b ∈ B ˜ c ∈ C ϑ ACa (cid:48) g g c (cid:48)− ( a (cid:48) , ˜ a | c (cid:48) , ˜ c ) ϑ ABg (˜ a, ˜ a − a | ˜ b, ˜ b − b ) ϑ BCg (˜ b, ˜ b − b (cid:48) | ˜ c, ˜ c − c ) ζ ABC ˜ a, ˜ b, ˜ c ( g , g ) × | ˜ a − g ˜ b, ˜ a − a, ˜ b − b, ˜ b − g ˜ c, ˜ b − b (cid:48) , ˜ c − c, a (cid:48) ˜ a, c (cid:48) ˜ c (cid:105) Y (cid:52) . Using the invariance property (5.29) of the Clebsch-Gordan series, we can rewrite the previous quantityas P Y (cid:52) (cid:0) | ρ AB I AB , ρ BC I BC , ρ AC I AC (cid:105) Y (cid:52) (cid:1) = 1 | A || B | | C | (cid:88) { g ∈ G } ˜ a,a,a (cid:48) ∈ A ˜ b, ˜ b (cid:48) ,b,b (cid:48) ∈ B ˜ c,c,c (cid:48) ∈ C (cid:88) { J } D ρ AB J AB I AB (cid:0)(cid:12)(cid:12) ˜ a − g ˜ b (cid:48) ˜ a − a −−−−→ ˜ b (cid:48)− b (cid:11)(cid:1) D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) ˜ b (cid:48)− g ˜ c ˜ b (cid:48)− b (cid:48) −−−−→ ˜ c − c (cid:11)(cid:1) × (cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105) D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) a (cid:48) g g c (cid:48)− a (cid:48) ˜ a −−−→ c (cid:48) ˜ c (cid:11)(cid:1) × ϑ ABg (˜ a, ˜ a − a | ˜ b (cid:48) , ˜ b (cid:48)− b ) ϑ BCg (˜ b (cid:48) , ˜ b (cid:48)− b (cid:48) | ˜ c, ˜ c − c ) ζ ABC ˜ a, ˜ b (cid:48) , ˜ c ( g , g ) ϑ ABg (˜ a, ˜ a − a | ˜ b, ˜ b − b ) ϑ BCg (˜ b, ˜ b − b (cid:48) | ˜ c, ˜ c − c ) ζ ABC ˜ a, ˜ b, ˜ c ( g , g ) × | ˜ a − g ˜ b, ˜ a − a, ˜ b − b, ˜ b − g ˜ c, ˜ b − b (cid:48) , ˜ c − c, a (cid:48) ˜ a, c (cid:48) ˜ c (cid:105) Y (cid:52) . Let us now use the fact that ζ ABC ˜ a, ˜ b (cid:48) , ˜ c ( g , g ) ζ ABC ˜ a, ˜ b, ˜ c ( g , g ) = ζ ABC A , ˜ b − ˜ b (cid:48) , C (˜ a − g ˜ b, ˜ b − g ˜ c ) ϑ ABg (˜ a, A | ˜ b, ˜ b − ˜ b (cid:48) ) ϑ BCg (˜ b, ˜ b − ˜ b (cid:48) | ˜ c, C ) ∼ ∼ s well as the groupoid cocycle conditions d (2) ϑ ABg (˜ a, A , ˜ a − a | ˜ b, ˜ b − ˜ b (cid:48) , ˜ b (cid:48)− b ) = 1 and d (2) ϑ BCg (˜ b, ˜ b − ˜ b (cid:48) , ˜ b (cid:48)− b | ˜ c, C , ˜ c − c )in order to rewrite ϑ ABg (˜ a, ˜ a − a | ˜ b (cid:48) , ˜ b (cid:48)− b ) ϑ BCg (˜ b (cid:48) , ˜ b (cid:48)− b (cid:48) | ˜ c, ˜ c − c ) ζ ABC ˜ a, ˜ b (cid:48) , ˜ c ( g , g ) ϑ ABg (˜ a, ˜ a − a | ˜ b, ˜ b − b ) ϑ BCg (˜ b, ˜ b − b (cid:48) | ˜ c, ˜ c − c ) ζ ABC ˜ a, ˜ b, ˜ c ( g , g )= ϑ AB ˜ a − g ˜ b ( A , ˜ a − a | ˜ b − ˜ b (cid:48) , ˜ b (cid:48)− b ) ϑ BC ˜ b − g ˜ c (˜ b − ˜ b (cid:48) , ˜ b (cid:48)− b (cid:48) | C , ˜ c − c ) ζ ABC A , ˜ b − ˜ b (cid:48) , C (˜ a − g ˜ b, ˜ b − g ˜ c ) . Performing a simple relabelling of summation variables, we then obtain P Y (cid:52) (cid:0) | ρ AB I AB , ρ BC I BC , ρ AC I AC (cid:105) Y (cid:52) (cid:1) = 1 | A || B | | C | (cid:88) { g ∈ G } ˜ a,a,a (cid:48) ∈ A ˜ b, ˜ b (cid:48) ,b,b (cid:48) ∈ B ˜ c,c,c (cid:48) ∈ C (cid:88) { J } D ρ AB J AB I AB (cid:0)(cid:12)(cid:12) g ˜ b − ˜ b (cid:48) a −−−−→ ˜ b (cid:48)− b (cid:11)(cid:1) D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) ˜ b (cid:48)− ˜ bg b (cid:48)− b (cid:48) −−−−→ c (cid:11)(cid:1) × (cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105) D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) g a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) × ϑ ABg ( A , a | ˜ b − ˜ b (cid:48) , ˜ b (cid:48)− b ) ϑ BCg (˜ b − ˜ b (cid:48) , ˜ b (cid:48)− b (cid:48) | C , c ) ζ ABC A , ˜ b − ˜ b (cid:48) , C ( g , g ) × | g , a, ˜ b − b, g , ˜ b − b (cid:48) , c, a (cid:48) , c (cid:48) (cid:105) Y (cid:52) . Moreover, let us notice that (5.29) induces1 | B | (cid:88) ˜ b (cid:48) (cid:88) { J } D ρ AB J AB I AB (cid:0)(cid:12)(cid:12) g ˜ b − ˜ b (cid:48) a −−−−→ ˜ b (cid:48)− b (cid:11)(cid:1) D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) ˜ b (cid:48)− ˜ bg b (cid:48)− b (cid:48) −−−−→ c (cid:11)(cid:1) × (cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105) D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) a (cid:48) g g c (cid:48)− a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) × ϑ ABg ( A , a | ˜ b − ˜ b (cid:48) , ˜ b (cid:48)− b ) ϑ BCg (˜ b − ˜ b (cid:48) , ˜ b (cid:48)− b (cid:48) | C , c ) ζ ABC A , ˜ b − ˜ b (cid:48) , C ( g , g )= (cid:88) { J } D ρ AB J AB I AB (cid:0)(cid:12)(cid:12) g a −−−−→ ˜ b − b (cid:11)(cid:1) D ρ BC J BC I BC (cid:0)(cid:12)(cid:12) g b − b (cid:48) −−−−→ c (cid:11)(cid:1)(cid:104) ρ AB J AB ρ BC J BC (cid:12)(cid:12)(cid:12) ρ AC J AC (cid:105) D ρ AC I AC J AC (cid:0)(cid:12)(cid:12) a (cid:48) g g c (cid:48)− a (cid:48) −−→ c (cid:48) (cid:11)(cid:1) . A final relabelling of summation variables yields the desired result.
References [1] M. Atiyah,
Topological quantum field theories , Inst. Hautes Etudes Sci. Publ. Math. (1989) 175–186.[2] V. G. Turaev and O. Y. Viro, State sum invariants of 3 manifolds and quantum 6j symbols , Topology (1992) 865–902.[3] J. W. Barrett and B. W. Westbury, Invariants of piecewise linear three manifolds , Trans. Am. Math.Soc. (1996) 3997–4022, [ hep-th/9311155 ].[4] M. A. Levin and X.-G. Wen,
String net condensation: A Physical mechanism for topological phases , Phys. Rev.
B71 (2005) 045110, [ cond-mat/0404617 ].[5] R. Koenig, G. Kuperberg and B. W. Reichardt,
Quantum computation with Turaev-Viro codes , Annalsof Physics (Dec., 2010) 2707–2749, [ ].[6] A. K. Jr,
String-net model of turaev-viro invariants , . ∼ ∼
7] S. Majid,
Foundations of quantum group theory . Cambridge university press, 2000.[8] A. Yu. Kitaev,
Fault tolerant quantum computation by anyons , Annals Phys. (2003) 2–30,[ quant-ph/9707021 ].[9] A. Kitaev,
Anyons in an exactly solved model and beyond , Annals Phys. (2006) 2–111.[10] A. Kitaev and L. Kong,
Models for gapped boundaries and domain walls , Communications inMathematical Physics (2012) 351–373.[11] A. Bullivant, Y. Hu and Y. Wan,
Twisted quantum double model of topological order with boundaries , Phys. Rev.
B96 (2017) 165138, [ ].[12] H. Wang, Y. Li, Y. Hu and Y. Wan,
Gapped boundary theory of the twisted gauge theory model ofthree-dimensional topological orders , Journal of High Energy Physics (2018) 114.[13] I. Cong, M. Cheng and Z. Wang,
Topological quantum computation with gapped boundaries , arXivpreprint arXiv:1609.02037 (2016) .[14] M. Barkeshli, P. Bonderson, M. Cheng and Z. Wang, Symmetry, Defects, and Gauging of TopologicalPhases , .[15] B. Yoshida, Gapped boundaries, group cohomology and fault-tolerant logical gates , Annals Phys. (2017) 387–413, [ ].[16] S. Beigi, P. W. Shor and D. Whalen,
The Quantum Double Model with Boundary: Condensations andSymmetries , Communications in Mathematical Physics (Sept., 2011) 663–694, [ ].[17] H. Bombin and M. A. Martin-Delgado,
Family of non-abelian kitaev models on a lattice: Topologicalcondensation and confinement , Phys. Rev. B (Sep, 2008) 115421.[18] S. B. Bravyi and A. Yu. Kitaev, Quantum codes on a lattice with boundary , quant-ph/9811052 .[19] H. Bombin, Topological order with a twist: Ising anyons from an abelian model , Phys. Rev. Lett. (Jul, 2010) 030403.[20] M. Barkeshli, C.-M. Jian and X.-L. Qi,
Twist defects and projective non-abelian braiding statistics , Phys. Rev. B (Jan, 2013) 045130.[21] S. Morrison and K. Walker, Higher categories, colimits, and the blob complex , Proceedings of theNational Academy of Sciences (2011) 8139–8145.[22] N. Carqueville,
Lecture notes on 2-dimensional defect TQFT , 2016, .[23] N. Carqueville, I. Runkel and G. Schaumann,
Line and surface defects in Reshetikhin-Turaev TQFT , .[24] N. Carqueville, C. Meusburger and G. Schaumann, , Adv. Math. (2020) 107024, [ ].[25] N. Carqueville, I. Runkel and G. Schaumann,
Orbifolds of n-dimensional defect TQFTs , Geom. Topol. (2019) 781–864, [ ].[26] N. Carqueville, I. Runkel and G. Schaumann, Orbifolds of Reshetikhin-Turaev TQFTs , .[27] J. Fuchs, J. Priel, C. Schweigert and A. Valentino, On the Brauer Groups of Symmetries of AbelianDijkgraafWitten Theories , Commun. Math. Phys. (2015) 385–405, [ ].[28] J. Fuchs, C. Schweigert and A. Valentino,
A geometric approach to boundaries and surface defects inDijkgraaf-Witten theories , Commun. Math. Phys. (2014) 981–1015, [ ].[29] C. L. Douglas and D. J. Reutter,
Fusion 2-categories and a state-sum invariant for 4-manifolds , . ∼ ∼
30] Y. Hu, Y. Wan and Y.-S. Wu,
Twisted quantum double model of topological phases in two dimensions , Phys. Rev.
B87 (2013) 125114, [ ].[31] Y. Wan, J. C. Wang and H. He,
Twisted Gauge Theory Model of Topological Phases in ThreeDimensions , Phys. Rev.
B92 (2015) 045101, [ ].[32] A. Bullivant and C. Delcamp,
Tube algebras, excitations statistics and compactification in gauge modelsof topological phases , JHEP (2019) 216, [ ].[33] T. Lan, L. Kong and X.-G. Wen, A classification of 3+1D bosonic topological orders (I): the case whenpoint-like excitations are all bosons , ArXiv e-prints (Apr., 2017) , [ ].[34] C. Zhu, T. Lan and X.-G. Wen,
Topological non-linear σ -model, higher gauge theory, and a realizationof all 3+ 1D topological orders for boson systems , arXiv preprint arXiv:1808.09394 (2018) .[35] R. Thorngren, TQFT, Symmetry Breaking, and Finite Gauge Theory in 3+1D , .[36] T. Johnson-Freyd, On the classification of topological orders , .[37] Dijkgraaf, Robbert and Witten, Edward, Topological gauge theories and group cohomology , Communications in Mathematical Physics (Apr, 1990) 393–429.[38] V. G. Drinfeld,
Quasi Hopf algebras , Alg. Anal. (1989) 114–148.[39] R. Dijkgraaf, V. Pasquier and P. Roche,
Quasi hopf algebras, group cohomology and orbifold models , Nuclear Physics B Proceedings Supplements (Jan., 1991) 60–72.[40] S. Beigi, P. W. Shor and D. Whalen, The quantum double model with boundary: Condensations andsymmetries , Communications in Mathematical Physics (Jun, 2011) 663694.[41] T. Lan and X.-G. Wen,
Topological quasiparticles and the holographic bulk-edge relation in (2+1)-dimensional string-net models , Phys. Rev.
B90 (2014) 115119, [ ].[42] J. C. Bridgeman and D. Barter,
Computing data for Levin-Wen with defects , .[43] C. Delcamp, Excitation basis for (3+1)d topological phases , JHEP (2017) 128, [ ].[44] N. Bultinck, M. Mari¨en, D. J. Williamson, M. B. S¸ahino˘glu, J. Haegeman and F. Verstraete, Anyonsand matrix product operator algebras , Annals of physics (2017) 183–233.[45] D. Aasen, E. Lake and K. Walker,
Fermion condensation and super pivotal categories , .[46] C. Delcamp, B. Dittrich and A. Riello, Fusion basis for lattice gauge theory and loop quantum gravity , JHEP (2017) 061, [ ].[47] A. Bullivant and C. Delcamp, Excitations in strict 2-group higher gauge models of topological phases , JHEP (2020) 107, [ ].[48] A. Ocneanu, Chirality for operator algebras , Subfactors (Kyuzeso, 1993) (1994) 39–63.[49] A. Ocneanu,
Operator algebras, topology and subgroups of quantum symmetry–construction of subgroupsof quantum groups , in
Taniguchi Conference on Mathematics Nara , vol. 98, pp. 235–263, 2001.[50] D. J. Williamson, N. Bultinck and F. Verstraete,
Symmetry-enriched topological order in tensornetworks: Defects, gauging and anyon condensation , .[51] C. Wang and M. Levin, Braiding statistics of loop excitations in three dimensions , Phys. Rev. Lett. (2014) 080403, [ ].[52] J. Wang and X.-G. Wen,
Non-Abelian string and particle braiding in topological order: Modular SL(3, Z )representation and (3+1) -dimensional twisted gauge theory , Phys. Rev.
B91 (2015) 035134,[ ]. ∼ ∼
53] L. Kong, Y. Tian and S. Zhou,
The center of monoidal 2-categories in 3+1D Dijkgraaf-Witten theory , Adv. Math. (2020) 106928, [ ].[54] S. Willerton,
The twisted drinfeld double of a finite group via gerbes and finite groupoids , Algebraic &Geometric Topology (2008) 1419–1457.[55] V. Ostrik, Module categories, weak Hopf algebras and modular invariants , arXiv Mathematics e-prints (Nov., 2001) math/0111139, [ math/0111139 ].[56] P. Etingof, S. Gelaki, D. Nikshych and V. Ostrik, Tensor categories , vol. 205. American MathematicalSoc., 2016.[57] V. Ostrik,
Module categories over the drinfeld double of a finite group , math/0202130 .[58] S. Mac Lane, Category theory for the working mathematician , .[59] B. Bartlett,
On unitary 2-representations of finite groups and topological quantum field theory , arXivpreprint arXiv:0901.3975 (2009) .[60] C. J. Schommer-Pries, The classification of two-dimensional extended topological field theories , 2011.[61] D. V. Else and C. Nayak,
Cheshire charge in (3+1)-dimensional topological phases , Phys. Rev. B (Jul, 2017) 045136.[62] C. Delcamp and A. Tiwari, From gauge to higher gauge models of topological phases , JHEP (2018)049, [ ].[63] M. M. Kapranov and V. A. Voevodsky, , in Proc.Symp. Pure Math , vol. 56, pp. 177–260, 1994.[64] J. C. Baez and A. S. Crans,
Higher-Dimensional Algebra VI: Lie 2-Algebras , Theor. Appl. Categor. (2004) 492–528, [ math/0307263 ].[65] J. Lurie, On the Classification of Topological Field Theories , .[66] P. Deligne, Cat´egories tannakiennes , in
The Grothendieck Festschrift , pp. 111–195. Springer, 2007.[67] N. Gurski,
Loop spaces, and coherence for monoidal and braided monoidal bicategories , .[68] S. Lack, A coherent approach to pseudomonads , Advances in Mathematics (2000) 179 – 202.[69] M. Neuchl,
Representation theory of Hopf categories , Ph.D. thesis, University of Munich, 1997.[70] J. C. Baez,
Higher-dimensional algebra ii. 2-hilbert spaces , Advances in Mathematics (1997)125–189.[71] X.-G. Wen,
A systematic construction of gapped non-liquid states , .[72] D. Aasen, D. Bulmash, A. Prem, K. Slagle and D. J. Williamson, Topological defect networks forfractons of all types , 2020.[73] J. Wang,
Non-Liquid Cellular States , .[74] N. Ganter and M. Kapranov, Representation and character theory in 2-categories , math/0602510 .[75] J. C. Baez and J. Dolan, Higher dimensional algebra and topological quantum field theory , J. Math.Phys. (1995) 6073–6105, [ q-alg/9503002 ]. ∼76