On lattice models of gapped phases with fusion category symmetries

We construct topological quantum field theories (TQFTs) and commuting projector Hamiltonians for any 1+1d gapped phases with non-anomalous fusion category symmetries, i.e. finite symmetries that admit SPT phases. The construction is based on two-dimensional state sum TQFT whose input datum is an $H$-simple left $H$-comodule algebra, where $H$ is a finite dimensional semisimple Hopf algebra. We show that the actions of fusion category symmetries $\mathcal{C}$ on the boundary conditions of these state sum TQFTs are represented by module categories over $\mathcal{C}$. This agrees with the classification of gapped phases with symmetry $\mathcal{C}$. We also find that the commuting projector Hamiltonians for these state sum TQFTs have fusion category symmetries at the level of the lattice models and hence provide lattice realizations of gapped phases with fusion category symmetries. As an application, we discuss the edge modes of SPT phases based on these commuting projector Hamiltonians. Finally, we mention that we can extend the construction of topological field theories to the case of anomalous fusion category symmetries by replacing a semisimple Hopf algebra with a semisimple pseudo-unitary connected weak Hopf algebra.


Introduction and summary
Symmetries of physical systems are characterized by the algebraic relations of topological defects. For instance, ordinary group symmetries are associated with invertible topological defects with codimension one. When the codimensions of invertible topological defects are greater than one, the corresponding symmetries are called higher form symmetries [1]. We can generalize these symmetries by relaxing the invertibility of topological defects. Symmetries associated with such non-invertible topological defects are called non-invertible symmetries, which are studied recently in various contexts . The algebraic structures of non-invertible symmetries are in general captured by higher categories [40][41][42][43][44]. In particular, non-invertible symmetries associated with finitely many topological defect lines in 1+1 dimensions are described by unitary fusion categories [13]. These symmetries are called fusion category symmetries [15] and are investigated extensively .
Fusion category symmetries are ubiquitous in two-dimensional conformal field theories (CFTs). A basic example is the symmetry of the Ising CFT [27,45,46]: the Ising CFT has a fusion category symmetry generated by the non-invertible Kramers-Wannier duality defect and the invertible Z 2 spin-flip defect. 1 More generally, any diagonal RCFTs have fusion category symmetries gener-ated by the Verlinde lines [48]. Fusion category symmetries are also studied in other CFTs such as CFTs with central charge c = 1 [20,23,49] and RCFTs that are not necessarily diagonal [50][51][52][53][54][55][56]. 2 We can also consider fusion category symmetries in topological quantum field theories (TQFTs). In particular, it is shown in [15,19] that unitary TQFTs with fusion category symmetry C are classified by semisimple module categories over C. This result will be heavily used in the rest of this paper. This classification reveals that fusion category symmetries do not always admit SPT phases, i.e. symmetric gapped phases with unique ground states. If fusion category symmetries do not have SPT phases, they are said to be anomalous [15], and otherwise non-anomalous.
Fusion category symmetries exist on the lattice as well. Remarkably, 2d statistical mechanical models with general fusion category symmetries are constructed recently in [27,28]. There are also examples of 1+1d lattice models known as anyonic chains [29][30][31]. These models might cover all the gapped phases with fusion category symmetries. However, to the best of my knowledge, systematic construction of 1+1d TQFTs and corresponding gapped Hamiltonians with fusion category symmetries is still lacking.
In this paper, we explicitly construct TQFTs and commuting projector Hamiltonians for any 1+1d gapped phases with arbitrary non-anomalous fusion category symmetries. For this purpose, we first show that a TQFT with fusion category symmetry, which is formulated axiomatically in [13], is obtained from another TQFT with different symmetry by a procedure that we call pullback. This is a natural generalization of the pullback of an SPT phase with finite group symmetry by a group homomorphism [57]. Specifically, we can pull back topological defects of a TQFT with symmetry C by a tensor functor F : C → C to obtain a TQFT with symmetry C. This corresponds to the fact that given a C -module category M and a tensor functor F : C → C , we can endow M with a C-module category structure. By using this technique, we can construct any TQFTs with non-anomalous fusion category symmetries. 3 To see this, we first recall that non-anomalous symmetries are described by fusion categories that admit fiber functors [15,19,22]. Such fusion categories are equivalent to the representation categories Rep(H) of finite dimensional semisimple Hopf algebras H. Therefore, TQFTs with non-anomalous fusion category symmetries are classified by semisimple module categories over Rep(H). Among these module categories, we are only interested in indecomposable ones because any semisimple module category can be decomposed into a direct sum of indecomposable module categories. Every indecomposable semisimple module category over Rep(H) is equivalent to the category K M of left K-modules where K is an H-simple left H-comodule algebra [58]. The Rep(H)-module category structure on K M is represented by a tensor functor from Rep(H) to the category End( K M) of endofunctors of K M. Since End( K M) is equivalent to the category K M K of K-K bimodules [59], we have a tensor functor F K : Rep(H) → K M K . We can use this tensor functor to pull back a K M K symmetric TQFT to a Rep(H) symmetric TQFT. We show in section 3 that a Rep(H) symmetric TQFT corresponding to a Rep(H)-module category K M is obtained as the pullback of a specific K M K symmetric TQFT, which corresponds to the same category K M regarded as a K M K -module category, by a tensor functor F K .
We also describe the pullback in the context of state sum TQFTs in section 4. Here, a state 2 Precisely, c = 1 CFTs can have infinitely many topological defect lines labeled by continuous parameters [20,23,49], whose algebraic structure should be described by a mathematical framework beyond fusion categories. 3 We can also construct any TQFTs with anomalous fusion category symmetries in the same way, see section 4.6. sum TQFT is a TQFT obtained by state sum construction [60], which is a recipe to construct a 2d TQFT from a semisimple algebra.  [61]. The existence of the state sum construction suggests that we can realize the Rep(H) symmetric TQFTs by lattice models. Indeed, the vacua of a state sum TQFT are in one-to-one correspondence with the ground states of an appropriate commuting projector Hamiltonian [62,63]. Specifically, when the input algebra of a state sum TQFT is K, the commuting projector Hamiltonian H is given by where K i := K is the local Hilbert space on the lattice, m K : K ⊗ K → K is multiplication on K, and ∆ K : K → K ⊗ K is comultiplication for the Frobenius algebra structure on K. The diagram in the above equation is the string diagram representation of the linear map h i,i+1 . We find that when K is a left H-comodule algebra, we can define the action of Rep(H) on the lattice Hilbert space H = i K i via the left H-comodule action on K. Here, we need to choose K appropriately so that the Rep(H) action becomes faithful on the lattice. In section 4, we show that the above Hamiltonian has a Rep(H) symmetry by explicitly computing the commutation relation of the Hamiltonian (1.1) and the action of the Rep(H) symmetry. Moreover, we will see that the Rep(H) symmetry action of the lattice model agrees with that of the state sum TQFT when the Hilbert space H is restricted to the subspace spanned by the ground states. This implies that the commuting projector Hamiltonian (1.1) realizes a Rep(H) symmetric TQFT K M.
We also examine the edge modes of SPT phases with Rep(H) symmetry by putting the systems on an interval. The ground states of the commuting projector Hamiltonian (1.1) on an interval are described by the input algebra K itself [64,65]. In particular, for SPT phases, K is isomorphic to the endomorphism algebra End(M ) ∼ = M * ⊗ M of a simple left K-module M . We can interpret M * and M as the edge modes by using the matrix product state (MPS) representation of the ground states. Thus, the edge modes of the Hamiltonian (1.1) for a Rep(H) SPT phase K M become either a left K-module M or a right K-module M * depending on which boundary they are localized to. As a special case, we reproduce the well-known result that the edge modes of an SPT phase with finite group symmetry G have anomalies, which take values in the second group cohomology H 2 (G, U(1)). We note that the edge modes of the Hamiltonian (1.1) are not necessarily minimal: it would be possible to partially lift the degeneracy on the boundaries by adding symmetric perturbations.
Although we will only consider the fixed point Hamiltonians (1.1) in this paper, we can add terms to our models while preserving the Rep(H) symmetry. In general, the lattice models still have the Rep(H) symmetry if the additional terms are H-comodule maps. Since the Hamiltonians with additional terms are generically no longer exactly solvable, one would use numerical calculations to determine the phase diagrams. For this purpose, we need to write the Hamiltonians in the form of matrices by choosing a basis of the lattice Hilbert space H. As a concrete example, we will explicitly compute the action of the Hamiltonian (1.1) with Rep(G) symmetry by choosing a specific basis of H. Here, Rep(G) is the category of representations of a finite group G, which describes the symmetry of G gauge theory.
Before proceeding to the next section, we comment on a relation between the state sum models discussed in this paper and the anyon chain models. 4 As we summarized above, we construct a Rep(H) symmetric commuting projector Hamiltonian of the state sum model by using a left H-comodule algebra K in this paper. On the other hand, we can also construct a Rep((H * ) cop ) symmetric commuting projector Hamiltonian of the anyon chain model by using the same algebra K, 5 where (H * ) cop is the coopposite coalgebra of the dual Hopf algebra H * . The anyon chain with Rep((H * ) cop ) symmetry is a lattice model whose Hilbert space is spanned by fusion trees in Rep((H * ) cop ). The commuting projector Hamiltonian of the anyon chain can be written diagrammatically as where the horizontal edges of the fusion diagrams are labeled by objects in Rep((H * ) cop ). We note that a left H-comodule algebra K is an algebra object in Rep((H * ) cop ). The right diagram in eq. (1.2) can be deformed to a sum of fusion trees via F -moves and hence the Hamiltonian can be explicitly written in terms of F -symbols. The above Hamiltonian has ground states represented by the fusion trees all of whose horizontal edges are labeled by a right K-module M ∈ Rep((H * ) cop ). This suggests, though not prove, that the gapped phase of this anyon chain corresponds to the category of right K-modules in Rep((H * ) cop ), which is a Rep((H * ) cop )-module category. As we will argue in section 4.1, this also suggests that the gapped phase of the anyon chain model constructed from the opposite algebra K op is obtained by the generalized gauging of the state sum model constructed from K, see footnote 14.
The reason why the state sum model (1.1) and the anyon chain model (1.2) have different symmetries despite the similarity between their Hamiltonians is that the symmetry actions are defined differently due to the different structures of their Hilbert spaces. Specifically, the Rep((H * ) cop ) symmetry of the anyon chain model is defined via the fusion of topological defect lines and the horizontal edges, whereas the Rep(H) symmetry of the state sum model is defined via the Hcomodule structure on the algebra K as we will discuss in section 4, see eq. (4.24). Since the state sum models do not have counterparts of horizontal edges of fusion trees, the Rep((H * ) cop ) symmetry does not act on the state sum models. Conversely, since the Rep(H) action (4.24) is not a morphism in Rep((H * ) cop ) and therefore is not given by a fusion diagram in Rep((H * ) cop ), the Rep(H) symmetry does not act on the anyon chains.
The rest of the paper is organized as follows. In section 2, we briefly review some mathematical backgrounds. In section 3, we introduce the notion of pullback of a TQFT and show that every TQFT with non-anomalous fusion category symmetry Rep(H) is obtained by pulling back a K M K symmetric TQFT K M by a tensor functor F K : Rep(H) → K M K . In section 4, we define state sum TQFTs with Rep(H) symmetry and show that they are realized by the commuting projector Hamiltonians (1.1). We emphasize that these Hamiltonians have fusion category symmetries at the level of the lattice models. These lattice realizations enable us to examine the edge modes of Rep(H) SPT phases. We also comment on a generalization to TQFTs and commuting projector Hamiltonians with anomalous fusion category symmetries in the last subsection. In appendix A, we describe state sum TQFTs with fusion category symmetries in the presence of interfaces.

Fusion categories, tensor functors, and module categories
We begin with a brief review of unitary fusion categories, tensor functors, and module categories [66]. A unitary fusion category C is equipped with a bifunctor ⊗ : C × C → C, which is called a tensor product. The tensor product of objects x, y ∈ C is denoted by x ⊗ y. The tensor product (x ⊗ y) ⊗ z of three objects x, y, z ∈ C is related to x ⊗ (y ⊗ z) by a natural isomorphism α x,y,z : (x ⊗ y) ⊗ z → x ⊗ (y ⊗ z) called an associator, which satisfies the following pentagon equation: There is a unit object 1 ∈ C that behaves as a unit of the tensor product, i.e. 1 ⊗ x ∼ = x ⊗ 1 ∼ = x. The isomorphisms l x : 1 ⊗ x → x and r x : x ⊗ 1 → x are called a left unit morphism and a right unit morphism respectively. These isomorphisms satisfy the following commutative diagram: We can always take l x and r x as the identity morphism id x by identifying 1 ⊗ x and x ⊗ 1 with x.
In sections 3 and 4, we assume l x = r x = id x . A unitary fusion category C also has an additive operation ⊕ : C × C → C called a direct sum. An object x ∈ C is called a simple object when it cannot be decomposed into a direct sum of other objects. In particular, the unit object 1 ∈ C is simple. The number of (isomorphism classes of) simple objects is finite, and every object is isomorphic to a direct sum of finitely many simple objects. Namely, for any object x ∈ C, we have an isomorphism x ∼ = i N i a i where {a i } is a set of simple objects and N i is a non-negative integer.
The Hom space Hom(x, y) for any objects x, y ∈ C is a finite dimensional C-vector space equipped with an adjoint † : f ∈ Hom(x, y) → f † ∈ Hom(y, x). The associators, the left unit morphisms, and the right unit morphisms are unitary with respect to this adjoint, i.e. α † x,y,z = α −1 x,y,z , l † x = l −1 x , and r † x = r −1 x . We note that the endomorphism space of a simple object a i is one-dimensional, i.e. End(a i ) := Hom(a i , a i ) ∼ = C.
For every object x ∈ C, we have a dual object x * ∈ C and a pair of morphisms ev L x : x * ⊗x → 1 and coev L x : 1 → x ⊗ x * that satisfy the following relations: A tensor functor F : C → C between fusion categories C and C is a functor equipped with a natural isomorphism J x,y : F (x) ⊗ F (y) → F (x ⊗ y) and an isomorphism φ : 1 → F (1) that satisfy the following commutative diagrams: Here, 1 and 1 are unit objects of C and C respectively. When C and C are unitary fusion categories, we require that J x,y and φ are unitary in the sense that J † x,y = J −1 x,y and φ † = φ −1 . The isomorphism φ can always be chosen as the identity morphism by the identification 1 = F (1).
A module category M over a fusion category C is a category equipped with a bifunctor ⊗ : C × M → M, which represents the action of C on M. For any objects x, y ∈ C and M ∈ M, we have a natural isomorphism m x,y,M : (x ⊗ y)⊗M → x⊗(y⊗M ) called a module associativity constraint that satisfies the following commutative diagram: The action of the unit object 1 ∈ C gives an isomorphism l M : 1⊗M → M called a unit constraint such that the following diagram commutes: A C-module category structure on M can also be represented by a tensor functor from C to the category of endofunctors of M , i.e. F : C → End(M), which is analogous to an action of an algebra on a module. A module category M is said to be indecomposable if it cannot be decomposed into a direct sum of two non-trivial module categories. When we have a tensor functor (F, J, φ) : C → C , we can regard a C -module category M as a C-module category by defining the action of C on M as x⊗M := F (x)⊗ M for x ∈ C and M ∈ M, where ⊗ is the action of C on M. The natural isomorphisms m x,y,M and l M are given by where m and l are the module associativity constraint and the unit constraint for the C -module category structure on M.
An important example of a unitary fusion category is the category K M K of K-K bimodules where K is a finite dimensional semisimple algebra. We review this category in some detail for later convenience. The objects and morphisms of K M K are K-K bimodules and K-K bimodule maps respectively. The monoidal structure on K M K is given by the tensor product over K, which is usually denoted by ⊗ K . To describe the tensor product we first recall that a finite dimensional semisimple algebra K is a Frobenius algebra. Here, an algebra K equipped with multiplication m K : K ⊗ K → K and a unit η K : C → K is called a Frobenius algebra if it is also a coalgebra equipped with comultiplication ∆ K : K → K ⊗ K and a counit K : K → C such that the following Frobenius relation is satisfied: (2.10) In the string diagram notation, the above relation is represented as where each string and junction represent the algebra K and the (co)multiplication respectively. In our convention, we read these diagrams from bottom to top. The comultiplication ∆ K and the counit K can be written in terms of the multiplication m K and the unit η K as follows [67]: (2.12) In the above equation, K * denotes the dual vector space of K. The linear maps ev and coev are the evaluation and coevaluation morphisms of the category of vector spaces. Specifically, we have where {u i } and {u i } are dual bases of K and K * . It turns out that the Frobenius algebra structure given by eq. (2.12) satisfies the following two properties [52]: 6 ∆-separability: = , symmetricity: = . (2.14) The tensor product where the junction of Y 1 (Y 2 ) and K represents a right (left) K-module action. We note that the unit object for the tensor product over K is K itself. The splitting maps of the projector is given by a composition of these splitting maps as We finally notice that the category K M of left K-modules is a K M K -module category, on which K M K acts by the tensor product over K. The module associativity constraint m Y 1 ,Y 2 ,M : is given by the composition of the splitting maps as the associator (2.16): (2.17)

Hopf algebras, (co)module algebras, and smash product
In this subsection, we briefly review the definitions and some basic properties of Hopf algebras. For details, see for example [68][69][70]. We first give the definition. A C-vector space H is called a Hopf algebra if it is equipped with structure maps (m, 1, ∆, , S) that satisfy the following conditions: 1. (H, m, 1) is a unital associative algebra where m : H ⊗ H → H is the multiplication and 1 ∈ H is the unit.
2. (H, ∆, ) is a counital coassociative coalgebra where ∆ : H → H ⊗ H is the comultiplication and : H → C is the counit. 7 3. The comultiplication ∆ is a unit-preserving algebra homomorphism where we denote the multiplication of g and h as gh. The multiplication on H ⊗ H is induced by that on H.
4. The counit is a unit-preserving algebra homomorphism 8 In particular, the antipode S squares to the identity when H is semisimple, i.e. S 2 = id. In the rest of this paper, we only consider finite dimensional semisimple Hopf algebras and do not distinguish between S and S −1 .
When H is a Hopf algebra, the opposite algebra H op is also a Hopf algebra, whose underlying vector space is H and whose structure maps are given by (m op , 1, ∆, , S −1 ). Here, the opposite multiplication m op : H op ⊗H op → H op is defined by m op (g⊗h) = hg for all g, h ∈ H. Similarly, the coopposite coalgebra H cop also becomes a Hopf algebra, whose underlying vector space is H and whose structure maps are given by (m, 1, ∆ cop , , S −1 ). Here, the coopposite comultiplication 9 In the subsequent sections, we will use the string diagram notation where the above conditions 1-5 are represented as follows: (2.21) 7 The comultiplication ∆ for the Hopf algebra structure on a semisimple Hopf algebra H is different from the comultiplication ∆ H for the Frobenius algebra structure on H. The same comment applies to and H . 8 The right-hand side of the second equation of (2.19) is just a number 1 ∈ C, which defers from the unit of H. 9 We use Sweedler's notation for the comultiplication When a left H-module A has an algebra structure that is compatible with the H-module structure, A is called a left H-module algebra. More precisely, a left H-module A with a module action We can also define a left H-comodule algebra similarly. A left H-comodule algebra K is a unital associative algebra whose algebra structure (K, m K , η K ) is compatible with the H-comodule action λ K : K → H ⊗ K in the following sense: (2.28) A left H-comodule algebra K is said to be H-simple if K does not have any proper non-zero ideal I such that λ K (I) ⊂ H ⊗ I. In particular, an H-simple left H-comodule algebra K is semisimple [61]. The left H-comodule action on K is said to be inner-faithful if there is no Given a left H-module algebra A, we can construct a left H cop -comodule algebra A#H called the smash product of A and H. As a vector space, A#H is the same as the tensor product A ⊗ H. The left H cop -comodule action on A#H is defined via the coopposite comultiplication ∆ cop as .
( 2.29) The algebra structure on A#H is given by

Representation categories of Hopf algebras
Every non-anomalous fusion category symmetry is equivalent to the representation category of a Hopf algebra. 10 In this subsection, we describe the representation category of a Hopf algebra and module categories over it following [58]. The representation category Rep(H) of a Hopf algebra H is a category whose objects are left H-modules and whose morphisms are left H-module maps. The tensor product V ⊗ W of left H-modules V and W is given by the usual tensor product over C. The left H-module structure on the tensor product V ⊗ W is defined via the comultiplication ∆. Specifically, if we denote the left H-module action on V ∈ Rep(H) as (2.31) An indecomposable semisimple module category over Rep(H) is equivalent to the category of right A-modules in Rep(H) where A is an H-simple left H-module algebra [72,73]. We denote this module category as ( H M) A . As a module category over Rep(H), the category ( H M) A is equivalent to the category of left A op #H cop -modules [58], which we denote by A op #H cop M:  [58]. The action of Rep(H) on K M is given by the usual tensor product, i.e.

Pullback of fusion category TQFTs by tensor functors
In this section, we show that given a 2d TQFT Q with symmetry C and a tensor functor F : C → C , we can construct a 2d TQFT Q with symmetry C by pulling back the TQFT Q by the tensor functor F . In particular, we can construct any 2d TQFT with non-anomalous fusion category symmetry Rep(H) by pulling back a specific K M K symmetric TQFT by a tensor functor F K : Rep(H) → K M K . We note that the content of this section can also be applied to anomalous fusion category symmetries as well as non-anomalous ones.

TQFTs with fusion category symmetries
We first review the axiomatic formulation of 2d unitary TQFT with fusion category symmetry C following [13]. A 2d TQFT assigns a Hilbert space Z(x) to a spatial circle that has a topological defect x ∈ C running along the time direction. When the spatial circle has multiple topological Figure 1: The Hilbert space on the above spatial circle is given by Z((x ⊗ y) ⊗ z), where the base point is represented by the cross mark in the above figure. We can also assign a Hilbert space to a circle with an arbitrary number of topological defects in a similar way. defects x, y, z, · · · , the Hilbert space is given by Z(((x ⊗ y) ⊗ z) ⊗ · · · ), where the order of the tensor product is determined by the position of the base point on the circle, see figure 1. A 2d TQFT also assigns a linear map to a two-dimensional surface decorated by a network of topological defects. The linear map assigned to an arbitrary surface is composed of the following building blocks, see also figure 2: For unitary TQFTs, the counit and the comultiplication ∆ x,y are the adjoints of the unit η and the multiplication M x,y respectively, i.e. = η † and ∆ x,y = M † x,y . In particular, the counit and the comultiplication ∆ x,y are no longer independent data of a TQFT.
For the well-definedness of the cylinder amplitude, we require that Z(f ) is C-linear in morphisms and preserves the composition of morphisms: Hom(x, y), ∀g ∈ Hom(y, z). (3.2) Thus, a 2d TQFT with fusion category symmetry C gives a functor Z : C → Vec from C to the category of vector spaces. This functor obeys various consistency conditions so that the assignment of Hilbert spaces and linear maps are well-defined. Specifically, a TQFT with fusion category symmetry C is a functor Z : C → Vec equipped with a set of linear maps (X, η, M ) that satisfies the following consistency conditions [13]: 1. Well-definedness of the change of the base point: 2. Naturality of the change of the base point: 3. Associativity of the change of the base point: The 5. Unit constraint: 6. Associativity of the multiplication: 7. Twisted commutativity: 8. Naturality of the multiplication: 9. Uniqueness of the multiplication: 11) where A is a generalized associator that we will define below.

Consistency on the torus:
(3.12) In the last two equations, the generalized associator A p→q : Z(p) → Z(q) is defined as a composition of the change of the base point X and the associator Z(α). We note that the isomorphism A p→q is uniquely determined by p and q [13].
In summary, a 2d unitary TQFT with fusion category symmetry C is a functor Z : C → Vec equipped with a triple (X, η, M ) that satisfies the consistency conditions (3.3)- (3.12). It is shown in [15,19] that 2d unitary TQFTs with fusion category symmetry C are classified by semisimple module categories over C. Namely, each 2d unitary TQFT with symmetry C is labeled by a semisimple C-module category. The TQFT labeled by a C-module category M has the category of boundary conditions described by M [19,74], whose semisimplicity follows from the unitarity of the TQFT [74,75].

Pullback of TQFTs by tensor functors
Let (Z , X , η , M ) be a 2d TQFT with symmetry C . Given a tensor functor (F, J, φ) : C → C , we can construct a 2d TQFT (Z, X, η, M ) with symmetry C as follows: the functor Z : C → Vec is given by the composition Z := Z • F , and the linear maps (X, η, M ) are defined as We can show that the quadruple (Z, X, η, M ) defined as above becomes a 2d TQFT, provided that (Z , X , η , M ) satisfies the consistency conditions (3.3)-(3.12). We will explicitly check some of the consistency conditions for (Z, X, η, M ) below. The other equations can also be checked similarly. Let us begin with eq. (3.3). This equation holds because the right-hand side can be written as where we used the fact that X satisfies eq. (3.3). Equation (3.4) follows from the naturality of J: , ∀g ∈ Hom(y, y ), ∀f ∈ Hom(x, x ). Indeed, if we choose either g or f as the identity morphism and use eq. (3.4) for X , we obtain eq. (3.4) for X. To show eq. (3.5), we note that F (α xyz ) can be written in terms of the associators α F (x),F (y),F (z) of C due to the commutative diagram (2.5) as follows: We also notice that the naturality (3.4) of X implies

(3.19)
By plugging eqs. (3.18) and (3.19) into the left-hand side of eq. (3.5), we find (3.20) The non-degeneracy condition (3.6) for an object x ∈ C follows from that for F (x) ∈ C because 10.6. in [66]. The unit constraint (3.7) is an immediate consequence of the commutative diagram (2.6) and eqs. We can also check the remaining equations similarly. Thus, we find that the quadruple (Z, X, η, M ) becomes a 2d TQFT with symmetry C. We call a TQFT (Z, X, η, M ) the pullback of a TQFT (Z , X , η , M ) by a tensor functor (F, J, φ).
By using the pullback, we can construct all the TQFTs with non-anomalous fusion category symmetry C. 11 To see this, we first recall that every non-anomalous fusion category symmetry C is equivalent to the representation category Rep(H) of a Hopf algebra H. Indecomposable semisimple module categories over Rep(H) are given by the categories K M of left K-modules where K is an H-simple left H-comodule algebra. Accordingly, we have a tensor functor F K : Rep(H) → K M K that represents the Rep(H)-module category structure on K M. Therefore, we can pull back a K M K symmetric TQFT by F K to obtain a Rep(H) symmetric TQFT. Here, we notice that there is a canonical K M K symmetric TQFT labeled by a K M K -module category K M, whose module category structure was discussed in section 2.1. Thus, by pulling back this canonical K M K symmetric TQFT K M by the tensor functor F K : Rep(H) → K M K , we obtain a Rep(H) symmetric TQFT canonically from the data of a Rep(H)-module category K M. This suggests that the TQFT obtained in this way is a Rep(H) symmetric TQFT labeled by a module category K M, or equivalently, this is a Rep(H) symmetric TQFT whose category of boundary conditions is given by K M. In the next section, we will see that this is the case by showing that the action of the Rep(H) symmetry on the boundary conditions of this TQFT is described by the Rep(H)-module action on K M.

State sum TQFTs and commuting projector Hamiltonians
The canonical K M K symmetric TQFT K M is obtained by state sum construction [60] whose input datum is a semisimple algebra K. The K M K symmetry of this TQFT was first discussed in [65]. This symmetry can also be understood from a viewpoint of generalized gauging [8][9][10][11][12][13]37]. In this section, we show that this state sum TQFT actually has Rep(H) symmetry when the input algebra K is a left H-comodule algebra. Specifically, this TQFT is regarded as the pullback of a K M K symmetric TQFT K M by a tensor functor F K : Rep(H) → K M K . We also construct Rep(H) symmetric commuting projector Hamiltonians whose ground states are described by the above state sum TQFTs. These commuting projector Hamiltonians realize all the gapped phases with non-anomalous fusion category symmetries.

State sum TQFTs with defects
We begin with reviewing state sum TQFTs with defects following [65]. We slightly modify the description of topological junctions in [65] so that it fits into the context of TQFTs with fusion category symmetries discussed in section 3.
Let Σ be a two-dimensional surface with in-boundary ∂ in Σ and out-boundary ∂ out Σ. The surface Σ is decorated by a network of topological defects that are labeled by objects of the category K M K . We assume that the junctions of these topological defects are trivalent and labeled by morphisms of K M K . We further assume, as in section 3.1, that the topological defects intersecting the in-boundary (out-boundary) are oriented so that they go into (out of) Σ To assign a linear map to Σ, we first give a triangulation T (Σ) of Σ such that every face p contains at most one trivalent junction and every edge e intersects at most one topological defect. The possible configurations of topological defects on a face p are as follows: Here, topological defects are labeled by K-K bimodules Y, Y 1 , Y 2 , Y 3 ∈ K M K , and trivalent junctions are labeled by bimodule maps We note that all of the above configurations are obtained from configuration (iv) by choosing some of the topological defects as trivial defects or replacing some of the topological defects with their duals. Nevertheless, we distinguish these configurations for convenience. For the triangulated surface T (Σ), we define a linear map Z T (Σ) as [65] Z T (Σ) : The constituents of this linear map are described below.
The vector spaces Z T (∂ in Σ) and Z T (∂ out Σ) The vector space Z T (∂ a Σ) for a = in, out is defined as the tensor product of vector spaces R e assigned to edges e ∈ ∂ a Σ, namely where the vector spaces R e are given as follows: R e := K when e does not intersect a topological defect, Y when e intersects a topological defect Y ∈ K M K .

(4.4)
We recall that the orientation of a topological defect Y on a boundary edge e ∈ ∂ a Σ is uniquely determined by assumption.

The vector space Q(Σ)
Similarly, we define the vector space Q(Σ) as the tensor product of the vector spaces Q (p,e) assigned to flags (p, e) ∈ Σ except for those whose edge e is contained in the in-boundary ∂ in Σ:  where the tensor product is taken over all edges e of Σ except for those on the in-boundary. The linear map P e for each edge e ∈ Σ \ ∂ in Σ is given by P e := ∆ K • η K when e does not intersect a topological defect, coev Y when e intersects a topological defect Y, (4.8) where ∆ K : K → K ⊗ K and η K : C → K are the comultiplication and the unit of the Frobenius algebra K, see section 2.1. The coevaluation map coev Y : C → Y ⊗ Y * is given by the usual embedding analogous to eq. (2.13).

The linear map E(Σ)
Finally, the linear map E(Σ) : Z T (∂ in Σ) ⊗ Q(Σ) → C is again given by the tensor product where the linear map E p for each face p ∈ Σ depends on a configuration of topological defects on p. We have five different configurations (i)-(v) as shown in eq. (4.1), and define the linear map E p for each of them as follows: (4.10) Here, ρ L Y : K ⊗ Y → Y and ρ R Y : Y ⊗ K → Y denote the left and right K-module actions on Y respectively, and π Y 1 ,Y 2 and ι Y 1 ,Y 2 are the splitting maps defined in section 2.1. The are morphisms in the category of K-K bimodules. As we mentioned before, the linear maps for (i)-(iii) and (v) are obtained from that for (iv) with an appropriate choice of Y 1 , Y 2 , Y 3 , and f .
, whose image will be denoted by Z(∂ in Σ). It turns out that Z(∂ in Σ) is mapped to Z(∂ out Σ) by Z T (Σ). Hence, we obtain a linear map Z(Σ) : Z(∂ in Σ) → Z(∂ out Σ) by restricting the domain of the linear map (4.2) to Z(∂ in Σ). We note that the linear map assigned to a cylinder is now the identity map. It is shown in [65] that the assignment of the vector spaces Z(∂ in/out Σ) and the linear map Z(Σ) gives a TQFT with defects. 12 Based on the above definition, we find that the two possible ways to resolve a quadrivalent junction into two trivalent junctions are related by the associator α Y 1 ,Y 2 ,Y 3 defined by eq. (2.16) as follows: = . (4.11) The square in the above equation represents a local patch of an arbitrary triangulated surface. This equation (4.11) implies that the symmetry of the state sum TQFT is precisely described by K M K . To argue that the state sum TQFT obtained above is the canonical K M K symmetric TQFT K M, we first notice that the state sum construction can be viewed as a generalized gauging of the trivial TQFT [37]. Here, the generalized gauging of a TQFT Q with fusion category symmetry C is the procedure to condense a ∆-separable symmetric Frobenius algebra object A ∈ C on a twodimensional surface. This procedure gives rise to a new TQFT Q/A whose symmetry is given by the category A C A of A-A bimodules in C [12,13]. To examine the relation between Q and Q/A in more detail, we consider the categories of boundary conditions of these TQFTs. Let B be the category of boundary conditions of the original TQFT Q. We note that B is the category of right B-modules in C for some ∆-separable symmetric Frobenius algebra object B ∈ C because B is a left C-module category [72,73]. Then, the category of boundary conditions of the gauged TQFT Q/A should be the category of left A-modules in B [19], which is a left A C A -module category. This is because the algebra object A is condensed in the gauged theory and hence a boundary condition in B survives after gauging only when it is a left A-module. 13  . In the case of the state sum TQFT with the input K, the condensed algebra object is K ∈ Vec and the category of boundary conditions of the original TQFT is Vec. Therefore, the category of boundary conditions of the state sum TQFT would be the category K M of left K-modules.
We can also see this more explicitly by computing the action of the K M K symmetry on the boundary states of the state sum TQFT. For this purpose, we first notice that a boundary of the state sum TQFT is equivalent to an interface between the state sum TQFT and the trivial TQFT. Since the trivial TQFT is a state sum TQFT with the trivial input C, interfaces are described by K-C bimodules, or equivalently, left K-modules. The wave function of the boundary state M | corresponding to the boundary condition M ∈ K M is the linear map assigned to a triangulated disk where the outer circle is an in-boundary and the inner circle labeled by M is the interface between the trivial TQFT (shaded region) and the state sum TQFT with the input K (unshaded region). We can compute the linear map assigned to the above disk by using a left K-module M instead of a K-K bimodule Y in eqs. (4.4), (4.6), (4.8), and (4.10) [65], see also appendix A for more details. Specifically, we can express the wave function M | in the form of a matrix product state (MPS) as [75,77] where {e i } is a basis of K, N is the number of edges on the boundary, and T M : K → End(M ) is the K-module action on M . In the string diagram notation, this MPS can be represented as shown in figure 3. We notice that the MPS (4.13) satisfies the additive property due to which it suffices to consider simple modules M j ∈ K M. A topological defect Y ∈ K M K acts on a boundary state M j | by winding around the spatial circle. We denote the wave function of the resulting state by Y · M j |. By giving a specific triangulation of a disk as follows, we can compute the action of Y on the boundary state M j | as where the blue circle and the purple circle represent a topological defect Y ∈ K M K and a boundary condition M j ∈ K M respectively, and N Y ij is a non-negative integer that appears in the direct sum decomposition of Y ⊗ K M j ∼ = i N Y ij M i . We note that the boundary states form a nonnegative integer matrix representation (NIM rep) of the fusion ring of K M K . Equation (4.15) implies that the action of the K M K symmetry on boundary conditions is described by a module category K M. The module associativity constraint (2.17) is also captured in the same way as eq. (4.11). Thus, the category of boundary conditions is given by the K M K -module category K M, which indicates that the state sum TQFT with the input K is a K M K symmetric TQFT K M.

Pullback of state sum TQFTs
When K is a left H-comodule algebra, the K M K symmetric TQFT K M can be pulled back to a Rep(H) symmetric TQFT by a tensor functor F K : Rep(H) → K M K . Accordingly, the symmetry of the state sum TQFT with the input K can be regarded as Rep(H). Specifically, when a two-dimensional surface Σ is decorated by a topological defect network associated with the Rep(H) symmetry, the assignment of the vector spaces (4.4), (4.6) and linear maps (4.8), K when e does not intersect a topological defect, when a topological defect V goes into p across e, F K (V ) * when a topological defect V goes out of p across e. (4.17) when e intersects a topological defect V. (4.18) .    15 Here, we recall that an H-simple left H-comodule algebra is semisimple [61] and hence can be used as an input of the state sum construction.

Commuting projector Hamiltonians
In this subsection, we write down Rep(H) symmetric commuting projector Hamiltonians whose ground states are described by the where the comultiplication ∆ K for the Frobenius algebra structure on K is given by eq. (2.12). The fact that K is a ∆-separable symmetric Frobenius algebra (2.14) guarantees that the linear map h i,i+1 becomes a local commuting projector, i.e. h i,i+1 h j,j+1 = h j,j+1 h i,i+1 and h 2 i,i+1 = h i,i+1 . The local commuting projector h i,i+1 can also be written in terms of a string diagram as where we used the Frobenius relation (2.11). The projector Π to the subspace of H spanned by the ground states of the Hamiltonian (4.21) is given by the composition of the local commuting projectors h i,i+1 for all edges i = 1, 2, · · · , N . This projector Π : H → H can be represented by the following string diagram: (4.23) This coincides with the string diagram representation of the linear map Z T (S 1 × [0, 1]) assigned to a triangulated cylinder S 1 × [0, 1]. Therefore, the ground states of the commuting projector Hamiltonian (4.21) agree with the vacua of the state sum TQFT whose input algebra is K.
We can define the action of the Rep(H) symmetry on the lattice Hilbert space H via the Hcomodule structure on K. Concretely, the adjoint of the action U V : H → H associated with a topological defect V ∈ Rep(H) is given by the following string diagram where χ V ∈ H * is the character of the representation V ∈ Rep(H), which is defined as the trace of the left H-module action on V . 17 The above Rep(H) action obeys the fusion rule of Rep(H), The cyclic symmetry of the character guarantees that the action (4.24) is well-defined on a periodic lattice T (S 1 ). Moreover, this action is faithful since the left H-comodule action on K = A op #H cop is inner-faithful. 18 Let us now show the commutativity of the Rep(H) action (4.24) and the commuting projector Hamiltonian (4.21). It suffices to check that the Rep(H) action commutes with each local commuting projector h i,i+1 . Namely, we need to check = , or equivalently, = . (4.25) The first equality follows from the second equality because K is a left H-comodule algebra. Conversely, we can derive the second equality from the first equality by composing a unit at the bottom of the diagram. To show eq. (4.25), we first notice that the counit given by eq. (2.12) satisfies , (4.26) 17 We note that the Rep(H) action (4.24) does not involve the algebra structure on K, which means that we can define the Rep(H) action on the lattice as long as the local Hilbert space is a left H-comodule. 18 Another choice of K is also possible as long as the Rep(H) symmetry acts faithfully on the lattice Hilbert space.
where we used the left H-comodule action on K * defined in a similar way to eq. (2.32). We note that the above equation relies on the fact that the antipode S of a semisimple Hopf algebra H squares to the identity. Equation (4.26) in turn implies that the isomorphism Φ : K → K * defined in eq. (2.12) is an H-comodule map because This indicates that Φ −1 : K * → K is an H-comodule map as well. Therefore, we have which shows eq. (4.25). We can also compute the action (4.24) of the Rep(H) symmetry on the ground states of the Hamiltonian (4.21). To perform the computation, we recall that the ground states of (4.21) are in one-to-one correspondence with the vacua of the state sum TQFT, and hence can be written as the boundary states (4.13) [74]. The Rep(H) symmetry action U V on a boundary state M | is given by

Examples: gapped phases of finite gauge theories
Let G be a finite group and C[G] be a group algebra. Gapped phases of G gauge theory are labeled by a pair (H, ω) [15] where H is a subgroup of G to which the gauge group G is Higgsed down and ω ∈ H 2 (H, U(1)) is a discrete torsion [80]. The symmetry of G gauge theory is described by Rep(G) := Rep(C[G]), which is generated by the Wilson lines. Therefore, we can realize these phases by the commuting projector Hamiltonians (4.21) where the input algebra K is a left C[G]-comodule algebra. Specifically, the input algebra K for the gapped phase labeled by (H, ω) is given by , where U is a projective representation of H characterized by ω [58]. 19 The action (4.24) of a representation V ∈ Rep(G) is expressed as U † V |a 1 #g 1 , a 2 #g 2 , · · · , a N #g N = χ V (g 1 g 2 · · · g N ) |a 1 #g 1 , a 2 #g 2 , · · · , a N #g N (4.30) for a i ∈ (C[G] ⊗ C[H] End(U )) op and g i ∈ G. In the following, we will explicitly describe the actions of the commuting projector Hamiltonians (4.21) for gapped phases of G gauge theory by choosing a specific basis of K. For simplicity, we will only consider two limiting cases where the gauge group G is not Higgsed at all or completely Higgsed. When G is not Higgsed, the gapped phases of G gauge theory are described by Dijkgraaf-Witten theories [81]. The input algebras K for these phases are given by K = End(U ) op #C[G]. 20 We choose a basis of the algebra K as {E ij #v g | i, j = 1, 2, · · · , dimU, g ∈ G}, where E ij is a dimU × dimU matrix whose (k, l) component is 1 when (k, l) = (i, j) and otherwise 0. If we denote the projective action of G on U by Q : G → End(U ), the multiplication (2.30) on the algebra K is written as (4.31) The Frobenius algebra structure on K is characterized by a pairing where the last term on the right-hand side represents the (j, i) component of Q(g)E kl Q(g) −1 . The above equation implies that Q(g) −1 E ji Q(g)#v g −1 /|G|dimU is dual to E ij #v g with respect to the pairing K • m K , and hence the comultiplication of the unit element 1 K ∈ K is given by Therefore, we can explicitly write down the action of the local commuting projector h : K ⊗K → K ⊗ K defined by eq. (4.22) as (4.34) On the other hand, when G is completely Higgsed, the input algebra K is given by K = C[G] * #C[G]. We choose a basis of K as {v g #v h | g, h ∈ G} where v g ∈ C[G] * denotes the dual basis of v g ∈ C[G]. The multiplication (2.30) on the algebra K is written as where we defined a left C[G]-module action on C[G] * by the left translation ρ(v g )v h := v gh . Since the dual of v g #v h with respect to the Frobenius pairing K • m K is given by v h −1 g #v h −1 /|G|, we have

Edge modes of SPT phases with fusion category symmetries
SPT phases with fusion category symmetry C are uniquely gapped phases preserving the symmetry C. Since anomalous fusion category symmetries do not admit SPT phases, it suffices to consider non-anomalous symmetries C = Rep(H). SPT phases with Rep(H) symmetry are realized by the commuting projector Hamiltonians (4.21) when K = A op #H cop is a simple algebra. 21 These Hamiltonians have degenerate ground states on an interval even though they have unique ground states on a circle. Specifically, it turns out that the ground states on an interval are given by the algebra K [64,65]. Since K is simple, we can write K ∼ = End(M ) ∼ = M * ⊗ M where M is a simple left K-module, which is unique up to isomorphism. We can interpret M * and M as the edge modes localized to the left and right boundaries because the bulk is a uniquely gapped state represented by an MPS (4.13). Indeed, if we choose a basis of the local Hilbert space on an edge e as {|v i e ⊗ |v j e ∈ M * ⊗ M }, we can write the ground states of the commuting projector Hamiltonian (4.21) on an interval as |v i 1 ⊗ |Ω 1,2 ⊗ |Ω 2,3 ⊗ · · · ⊗ |Ω N −1,N ⊗ |v j N , where |Ω e,e+1 := k |v k e ⊗|v k e+1 is the maximally entangled state. This expression indicates that the degrees of freedom of M * and M remain on the left and right boundaries respectively. Therefore, the edge modes of the Hamiltonian It is instructive to consider the case of an ordinary finite group symmetry G. A finite group symmetry G is described by the category Vec G of G-graded vector spaces, which is equivalent to the representation category of a dual group algebra C[G] * . SPT phases with this symmetry are classified by the second group cohomology H 2 (G, U(1)) [74,78,[82][83][84][85][86][87]. An SPT phase labeled by ω ∈ H 2 (G, U(1)) is realized by the commuting projector Hamiltonian (4.21) when A is a twisted group algebra C[G] ω . The edge modes M * of this model become a right (C[G] ω ) op #(C[G] * ) copmodule, which is a left C[G] ω -module in particular. This implies that these edge modes have an anomaly ω of the finite group symmetry G.

Generalization to anomalous fusion category symmetries
The most general unitary fusion category, which may or may not be anomalous, is equivalent to the representation category Rep(H) of a finite dimensional semisimple pseudo-unitary connected weak Hopf algebra H [73,[88][89][90]. As the case of Hopf algebras, any semisimple indecomposable module category over Rep(H) is given by the category K M of left K-modules, where K is an H-simple left H-comodule algebra [91]. We note that an H-simple left H-comodule algebra is semisimple [90,91]. Accordingly, we can construct all the TQFTs K M with anomalous fusion category symmetry Rep(H) by pulling back the state sum TQFT with the input K by a tensor functor F K : Rep(H) → K M K . Moreover, the fact that K is semisimple allows us to write down a commuting projector Hamiltonian in the same way as eq. (4.21). We can also define the action of Rep(H) on the lattice Hilbert space just by replacing a Hopf algebra with a weak Hopf algebra in (4.24). One may expect that these Hamiltonians realize all the gapped phases with anomalous fusion category symmetries. However, since our proof of the commutativity of the Rep(H) action (4.24) and the commuting projector Hamiltonian (4.21) relies on the properties that are specific to a semisimple Hopf algebra, our proof does not work when H is not a Hopf algebra, i.e. when the fusion category symmetry is anomalous. Therefore, we need to come up with another proof that is applicable to anomalous fusion category symmetries. We leave this problem to future work. These junctions are labeled by K 1 -K 3 bimodule maps h ∈ Hom K 1 K 3 (M 1 ⊗ K 2 M 2 , M 3 ) and l ∈ Hom K 1 K 3 (M 3 , M 1 ⊗ K 2 M 2 ), where M 1 ∈ K 1 M K 2 , M 2 ∈ K 2 M K 3 , and M 3 ∈ K 1 M K 3 .
To incorporate these configurations, we need to extend the assignment of the vector spaces and the linear maps (4.16)- (4.19). Specifically, we add the following vector spaces and linear maps: