Gapped domain walls between 2+1D topologically ordered states

The 2+1D topological order can be characterized by the mapping-class-group representations for Riemann surfaces of genus-1, genus-2, etc. In this paper, we use those representations to determine the possible gapped boundaries of a 2+1D topological order, as well as the domain walls between two topological orders. We find that mapping-class-group representations for both genus-1 and genus-2 surfaces are needed to determine the gapped domain walls and boundaries. Our systematic theory is based on the fixed-point partition functions for the walls (or the boundaries), which completely characterize the gapped domain walls (or the boundaries). The mapping-class-group representations give rise to conditions that must be satisfied by the fixed-point partition functions, which leads to a systematic theory. Such conditions can be viewed as bulk topological order determining the (non-invertible) gravitational anomaly at the domain wall, and our theory can be viewed as finding all types of the gapped domain wall given a (non-invertible) gravitational anomaly. We also developed a systematic theory of gapped domain walls (boundaries) based on the structure coefficients of condensable algebras.


I. INTRODUCTION
Topological order is a new kind of order in gapped quantum states of matter beyond Landau symmetry breaking theory. [1,2] In Ref. 3 and 4, it was conjectured that the non-Abelian geometric phases [5] (both the U (1) part and the non-Abelian part) of degenerate ground states generated by the automorphism of Riemann surfaces can completely characterize different topological orders. [3] The non-Abelian geometric phases contain an universal non-Abelian part [3,4] and a path dependent U (1) part [3,6]. The non-Abelian part carries information about the projective representation of mapping class group (MCG) of the space manifold. For torus, the MCG is Γ 1 = SL(2, Z), which is generated by a 90 • rotation and a Dehn twist. The associated non-Abelian geometric phases for such two generators are denoted by S and T , which are unitary matrices. S and T generate a projective representation of MCG SL(2, Z) for torus. The Abelian part of the non-Abelian geometric phases are also important: it is related to the gravitational Chern-Simons term [7][8][9] in the partition function and carries information about the chiral central charge c for the gapless edge excitations. [10,11] The data (S, T, c) is a quite complete description of 2+1D topological orders. However, to obtain a full description of 2+1D topological orders, the modular data for genus-1 surface is not enough, [12][13][14] we must also use the non-Abelian geometric phases (i.e. the mapping class group representations) for genus-2 surfaces. [15] In this paper, we will study the boundary of topological orders, or more generally, the domain wall between two topological orders. We will see how the boundary properties are determined by the bulk topological orders. We would like to consider the following issues: What is the data that allow us to characterize different gapped domain walls between topologically ordered states? How to classify gapped domain walls? Ref. [16][17][18][19] studied those issues for the case of 2+1D Abelian topological orders, using condensable set of bosonic topological excitations. Ref. [20][21][22][23][24][25][26] considered this problem for the more general case of 2+1D non-Abelian topological orders, using boson condensation [27,28] and/or condensable algebra [25]. Some discussions for the boundaries of topological orders beyond 2+1D can be found in Ref. 9, 29, and 30. In particular, it was pointed out that the boundary effective theory of a bulk topological order has a gravitational anomaly. [9,[31][32][33] In fact, the bulk topological order gives an one-to-one classification of gravitational anomaly in one low dimension (realized by the boundary). [9] Thus in some sense types of bosonic topological order = types of bosonic gravitational anomaly in one lower dimension. (Note that bosonic gravitational anomaly is a property of an effective theory that cannot be realized as a local bosonic system with finite cut-off. Two effective theories are said to have the same type of gravitational anomaly if one effective theory can change into another effective theory, possibly via phase transitions in the same dimension. [9]) In this paper, we will rederive the simple results based on the (S, T, c), for the boundaries of general 2+1D non-Abelian topological orders introduced in Ref. 34. Our new approach also allows us to generalize the approach in Ref. 34 to high genus Riemann surface which may lead to a complete description of the gaped boundary of 2+1D topological orders. We try to address the following question: given a 2+1D topological order described by (S, T, c) and the data from higher genus, how to describe and classify different gapped 1+1D boundary phases? If we find that there is no gapped 1+1D boundary for a type of 2+1D topological order, then such a type of topological order must have a gapless boundary. It can also be stated in the following way: given a type of gravitational anomaly determined, how to describe different gapped 1+1D phases? If we find that there is no gapped 1+1D phase for a type of gravitational anomaly, then such a type of gravitational anomaly will require (or ensure) the 1+1D phases to be gapless.

A. Dimensions of fusion spaces
To introduce data that can characterize different gapped domain walls, let us consider a 2D space S 2 . Half of S 2 is occupied by the phase A and the other half by the phase B. Let us assume the domain wall between the topological phases is gapped. We put a type-a topological excitation in the phase A and a type-i topological excitation in the phase B. We denote such a configuration as S 2 BA;ia (see Fig. 1). The ground state space for such a configuration is a fusion space. The dimension of the fusion space (i.e. the ground state degeneracy), denoted as M ia BA ∈ N, is the data that characterizes the gapped domain wall between A and B phases. We may also view M ia BA as the fusion coefficients for the fusion of type-i particle, type-a particle, and the domain wall BA.

B. Weighted wave function overlap
To obtain more data to characterize the domain walls, let us consider the degenerate ground states of the topologically ordered phase A, described by normalized wave function |ψ A I A , where the index I A label different ground states on a closed Riemann surface of genus g. Similarly, we have the degenerate ground states of the topologically ordered phase B: |ψ B I B . A filled genus-2 Riemaniann surface (denoted as Σ fill 2 ), which has three world-lines i, j, and z in the interior.
The degenerate ground states on a closed surface can be obtained as path integral on the "solid surface" (a 3 dimensional manifold whose boundary is the surface) with a world-line of a topological excitation (see Fig. 2). Thus we can label different ground states using the label of those topological excitations. Also, the above construction of degenerate ground states using world-lines give rise to a natural basis for the degenerate ground states. We will refer such a basis as the excitation basis.
For example, on a genus-1 surface, the degenerate ground states are labeled by I B = i for the phase B and I A = a for the phase A. Here i = 1, · · · , N B label the types of the topological excitations in the phase B and a = 1, · · · , N A label the types of the topological excitations in the phase A. For genus-1 surface, those excitation-labeled ground states happen to be orthogonal. We see that the ground state degeneracies on genus-1 surface are given by N A for phase A and N B for phase B.
As another example, on a genus-2 surface, the degenerate ground states are labeled by I B = (i, j, z, µ, ν) where µ = 1, · · · , N iī B;z and ν = 1, · · · , N jj B;z for the phase B. The degenerate ground states are labeled by I A = (a, b, c, α, β) where α = 1, · · · , N aā A;c and β = 1, · · · , N bb A;c for the phase A (see Fig.3). Here i, j, z label the types of the topological excitations in the phase B and a, b, c label the types of the topological excitations in the phase A. N ij B,z are the fusion coefficients of the topological excitations in the phase A and N ab A,c are the fusion coefficients of the topological excitations in the phase B. We see that the ground state degeneracies on genus-2 surface are given by Motivated by the wave function overlap [35][36][37][38] that can characterize different topological orders, here we will use the weighted wave function overlap to characterize different domain walls. We conjecture that [9,39] the weighted overlap of the degenerate ground states on a closed genus-g surface Σ g for topologically ordered phases A and B have the following form where H W is local hermitian operator like a Hamiltonian of a quantum system, A Σg is the area of the surface Σ g and W I B I A BA,g is a topological invariant that characterize the domain between the phases A and B.
In general, W I B I A BA,g depends on the choices of H W which correspond to different choices of domain walls. A concrete calculation of the wave function overlap, W I B I A BA,g , for a simple topological order is presented in Section IV. We will show in next section that the wave function overlap can also be viewed as partition function of the domain wall.
When B is a trivial product state, the index I B is always fixed to be 1, since B has no ground state degeneracy. In this case we simplify by dropping index for trivial phase B and I B . Similarly, we simplify Those are data that describe a gapped boundary of topological order A. We note that W I B I A BA,g are in general complex numbers, whose phase can be changed by a change of phases for the ground states Also W I B I A BA,g may depend on some choices of basis that cannot be fixed. So W I B I A BA,g by themselves are not a topological invariant, and they are not even physical. So we need to find a way mode out those phase and basis dependence. When we said W I B I A BA,g are "topological invariant", we mean that they are topological invariant up to those phase and basis choices.
For genus-1 surface Σ 1 , we may choose the phases of |ψ A 1 and |ψ B 1 to make W 11 BA,1 real and positive. (Here 1 correspond to the trivial particle.) We then choose the phases of |ψ A a to make W 1a BA,1 real and positive. Similarly, we choose the phases of |ψ B i to make W i1 BA,1 real and positive. With such a choice, we find that W I B I A BA,1 = M ia BA , which is the dimension of the fusion space defined in the last subsection (see Section III A for an explanation). For higher genus g > 1, W I B I A BA,g , after moding out some gauge redundancy, are new topological invariants. We hope W I B I A BA,g carry enough information to fully characterize a gapped domain wall.

III. THE CONDITIONS ON THE DOMAIN-WALL DATA
In this section, we are going to derive some conditions on the data that characterize the gapped domain walls. For example we like to show that the dimension of the fusion state M ia BA (which are non-negative integers) and the weighted wave function overlap for torus W ia BA,1 (which can be complex numbers) are actually equal to each other M ia BA = W ia BA,1 . This is quite an amazing relation. To derive those conditions, we need to introduce topological path integral, i.e. the path integral for triangu-lated space-time whose value is re-triangulation invariant. This is done in Appendix A. We also need to introduce an algebraic (i.e. a categorical) approach to evaluate those topological path integrals, which is done in Appendix B. Using those results, we can derive the conditions on the domain-wall data.
First, we like to show M ia BA = W ia BA,1 . Let us consider the topological path integral on space-time i is a disk with a puncture. Such a puncture corresponds to a world-line of a type-i topological excitation wrapping around S 1 . The topological path integral on D 2 i × S 1 only sum over the degrees of freedom in the bulk, and leave the degrees of freedom boundary fixed. 39). First, we like to show that such a wave function |ψ i is automatically normalized. According to Ref. 39 i ×S 1 together along its boundary. S 2 i,ī ×S 1 contains two world-lines of type-i and type-ī.
Let us evaluate the above partition function on S 2 i,ī × S 1 , which turns out to be Z top (S 2 i,ī × S 1 ) = 1: where we have used N iī 1 = Nī i 1 = 1, so α = β = 1 and the αβ only contain one term. k has to be the trivial particle 1, since k lives on a sphere S 2 alone. We have also used Yī i 1,11 Oī i,11 1 = 1 (see appendix B for the definition of Y -move and O-move) and Z top (S 2 ×S 1 ) = 1. So the topological path integral on D 2 i × S 1 gives rise to normalized wave function |ψ i . Now, instead of a path integral on S 2 i,ī ×S 1 which gives us Z top (S 2 i,ī × S 1 ) = 1, let us consider a path integral on S 2 BA;ia × S 1 , where S 2 BA;ia is described by Fig. 1. In space-time, the particle a and i correspond to world-lines wrapping around S 1 which is viewed as a space direction. The boundary between the two hemisphere in Fig. 1 is viewed as another space direction. The topological partition function for space-time S 2 BA;ia × S 1 corresponds to the weighted wave function overlap ψ B i | e −H W |ψ A a with a fine tune choice of H W , which has a form Fig. 1 into three pieces.
Here σ BA is the energy density of the domain wall between the A and B phases, and A Σ1 is the space-time area occupied by the domain wall. For our topological path integral, the energy density of the domain wall σ BA is fine tuned to zero. Thus we actually have In the above, we have regarded the S 1 in S 2 BA;ia ×S 1 as a space direction. Now we regard the S 1 in S 2 BA;ia ×S 1 as the time direction, and S 2 BA;ia in S 2 BA;ia ×S 1 as the space. In this case, the area independent part of the partition function has a different meaning: it becomes the ground state degeneracy on the space It is interesting to see that the weighted wave function overlaps for torus W ia BA,1 (after fine-tuned into topological ones), are always given by non-negative integers in the excitation basis.
When viewed as partition function, W ia BA,1 is a multicomponent partition function (labeled by i and a) on a torus. Such a multi-component partition function describe a gapped theory on the torus (i.e. the domain wall) that has a non-invertible gravitational anomaly. [40] Thus the conditions on W ia BA,1 (see next subsection) are a special case of the modular covariance condition discussed in Ref. 40.

B. Invariance under the MCG action
Next, we divide the space-time S 2 BA;ia × S 1 into three pieces D 2 A;a × S 1 (see Fig. 4a), D 2 B;i × S 1 (see Fig. 4c), and S 1 BA × S 1 × S 1 (see Fig. 4b), where D 2 A;a is a disk occupied by the phase A and the type-a particle, D 2 B;i is a disk occupied by the phase B and the type-i particle, and S 1 BA × S 1 is a cylinder occupied by the A and B phases and the domain wall. We can use the three pieces, D 2 A;a × S 1 , D 2 B;i × S 1 , and S 1 BA × S 1 × S 1 , to asamble the same closed space-time in two ways: (1) we glue D 2 A;a ×S 1 and S 1 BA × S 1 × S 1 directly along S 1 × S 1 boundary, and glue S 1 BA × S 1 × S 1 and D 2 B;i × S 1 with a twist U ∈ Γ 1 = SL(2, Z) for the boundary S 1 × S 1 ; (2) we glue D 2 A;a × S 1 and S 1 BA × S 1 × S 1 with a twistÛ ∈ Γ 1 = SL(2, Z) and glue S 1 BA × S 1 × S 1 and D 2 B;i × S 1 directly.
The two ways to assemble the closed space-time only differ by a re-triangulation, and they must produce the same topological partition function (see Appendix A 4 and A 5). This leads to a matrix relation We notice that the path integral on D 2 A;a × S 1 produce the basis state |a; A in the excitation basis for the phase A on the surface ∂(D 2 A;a × S 1 ). The action of the MCG transformationÛ on |a; A is described by the unitary representation R U A ofÛ for the phase A. Similarly, the path integral on D 2 B;i × S 1 produce the basis state |i; B in the quasiparticle basis for the phase B. The action of the MCG transformationÛ on |i; B is described by the unitary representation R U B ofÛ for the phase B. This is how do we obtain eqn. (7).

C. A consistent condition between the fusions in the bulk and on the wall
In this section, we like to obtain some additional conditions. Let us consider the following fusion process: we bring a type-a excitation in phase A and a type-i excitation in phase B to the domain wall w BA between the two phase and fuse them into an excitation of type s on the domain wall. The corresponding fusion algebra is given by It turns out that where s = 1 represents the trivial type of the excitations on the domain wall. Now let us compute the dimension of the fusion space of S 2 BA with excitations of types a and b in the phase A and excitations of types i and j in the phase B. There are two ways to compute the dimension of the fusion space: (1) we may fuse a and b into c and fuse i and j into k, or (2) we may first fuse a and i into s on the domain wall and fuse i and j intos on the domain, and then fuse s ands into the trivial excitation on the domain wall. The two fusion path should produce the same result, and we obtain The two equations eqn. (9) and eqn. (10) imply the condition eqn. (14). are N A and N B types of topological excitations in the phase A and the phase B, then the ranks of their modular matrices are N A and N B respectively. We find the following necessry conditions for the phase A and the phase B to be connected by a gapped domain wall: [34] there exist non-zero such that Here N denotes the set of non-negative integers. a, b, c, . . . and i, j, k, . . . are indices for the particle types in phases A and B. N ab A;c and N ij B;k are fusion coefficient of phases A and B. We may call Eq.(13) the commuting condition, and Eq.(14) the stable condition. [34] In fact the matrix M BA label a gapped domain wall between phases A and B. Eqns. (11), (12), (13), and (14) is a set of necessary conditions a gapped domain wall M BA must satisfy, i.e., if there is a gapped domain wall, we will have a non-zero M BA that satisfies those conditions. This implies that if there is no non-zero solution of M BA , the domain wall must be gapless. However, it is not clear if those condition are sufficient for a gapped domain wall to exist. In some simple examples, the solutions M BA are in one-to-one correspondence with gapped domain walls. However, for some complicated examples [41], a M BA may correspond to more than one type of gapped domain wall. This indicates that some additional data are needed to completely characterize gapped domain walls.
E. Wave function overlap on genus-2 Riemaniann surfaces and topological path integral In the above, we have developed a theory on the domain wall based on the data W ia BA;1 , the wave function Σ fill 2 without worldlines. Since Ψ i,j,z,µ,ν |Ψ i,j,z,µ,ν is positive, the ambiguous phase factor must be 1. We see that the normalized ground state wave function is given by the topological path integral on Σ fill 2 with the world lines i, j, z: .
This result allows us to express the wave function overlap in terms of path integral on Σ fill 2 Σ fill 2 : Using the same reasoning as in eqn. (7), we find that the rectangular matrix W BA,2 satisfies where R U A,B are the representations of genus-2 MCG Γ 2 for the phase A and B.
The wave function overlap on genus-2 surface W BA,2 and the wave function overlap on genus-1 surface W BA,1 are related: Here W (which gives rise to a S 3 occupied by phase A and B). Also, W i B i A BA,1 is the topological partition function on the space-time obtained by gluing Fig. 8a and Fig. 8b together. It is the topological partition function on S 3 where half of S 3 is occupied by phase A, and the other half by phase B. W j B j A BA,1 is the topological partition function on the space-time obtained by gluing Fig. 8c and  Fig. 7d together, we obtain a spacetime obtained by gluing Fig. 8c and Fig. 8d together.
Thus we haveW The normalized wave function overlap for genus-1 and genus-2 have a simpler relatioñ

IV. THE WAVE FUNCTION OVERLAP IN A LATTICE HAMILTONIAN MODEL
In this section we try to compute the wave function overlap in a concrete lattice Hamiltonian model. Above, in the path integral formulation in Euclidean spacetime, we learned the lesson that a gapped domain wall W between phases A, B (a defect along space direction), after a proper Wick-rotation, becomes an operator e −H W (a defect along the time direction) that determines the weighted wave function overlap. However, in the Hamiltonian formulation there is no natural equivalence between space and time. To apply the previous results, we first try to analyze the physical picture in a lattice model.
Assume that there is a gapped domain wall at x = 0, phase A in the region x < 0 a and phase B in the region x > 0. Clearly the Hamiltonian near x = 0 can not be the same as those far from x = 0. For simplicity, we may assume that for some small positive number ε, the Hamiltonian is uniform in the regions x < −ε and x > ε. On the other hand, in the region −ε < x < ε, the Hamiltonian is uniform along y and t directions, but changes with x. The Wick-rotated version of this picture, is that there are uniform phases A at time t < −ε and B at time t > ε, and during time −ε < t < ε, the Hamiltonian evolves from that of A to that of B. Thus e −H W should correspond to the accumulated evolution operator during −ε < t < ε.
For the domain wall Hamiltonian in the region −ε < x < ε, the requirement is that it should keep the spectrum of the whole system gapped. However, we have no idea how this requirement is translated for e −H W . There is further another subtlety, that phases A, B may not be defined on the same microscopic lattice, in particular, any generalized local unitary transformations can be inserted to deform A or B. In the followings, we try to propose some reasonable assumptions that allows us to calculate the quantity W I B I A BA,g .
• First, to remove the ambiguity of generalized local unitary transformations, we assume that proper generalized local unitary transformations have been applied such that A, B are already on the same microscopic lattice and their Hamiltonians satisfy the following relation; • We focus on the gapped domain walls that are directly induced by anyon condensations. Suppose H ac is the Hermitian operator that forces anyon condensation in A. (If only Abelian anyons are condensed, H ac is simply the sum of hopping string operators of the condensed anyons, that forces a total zero-momentum state of the condensed anyons. We will give more detail later in the toric code example.) We require the Hamiltonian H B of phase B to be: where H A is the Hamiltonian of phase A.
• Then we assume that e −H W acts on the ground state subspace of H A (or H B ) by multiplying a constant factor that may depend on the system size.
Now we look at the example of toric code model [42] which realize a Z 2 topological order: [43,44] The spins are on the links of the lattice. The first term sums over all vertices and star(v) denotes the legs of the vertex v. The second term sums over all plaquettes and ∂p denotes the boundary edges of the plaquette p. There are three types of nontrivial anyons e, m, f : e is created by string operators of the form σ x i along links, m is created by string operators of the form σ z i along links on the dual lattice, and f is the fusion of e and m. Both e and m can be condensed. The term that forces the condensation of e is H ac e = − i σ x i , and that for m is H ac m = − i σ z i . According to the above assumptions, we next calculate the overlap between the ground states of H A and H B = H A + hH ac e/m . It is easy to see that, under the h → +∞ limit, the ground state of H B is simply the spin polarized state ⊗ i

Now we write down the ground states of A on a torus
with the simplest lattice of only three links (see Fig. 9) According to the anyon flux along y direction (measured by string operators winding around x direction), the four ground states are We obtain: which are indeed solutions to (13) and (14) after dropping the prefactors. One reason for the different prefactors of e, m condensations is that the number of excitations the lattice can host is different. e is hosted by vertices while m is hosted by plaquettes and there are two vertices and only one plaquette in Fig. 9. Let us generalize the above calculation to arbitrary system size and arbitrary lattice. We note that the four ground states on a torus for the toric-code model H A are sum of all closed strings formed by |1 's on the links, with fixed even-odd winding numbers in the two directions of torus, which are given by |ee , |eo , |oe , and |oo . The four ground states in the quasiparticle basis are given by The ground states for the two possible B phases are To compute the overlap between |ψ ac e,m with |ψ 1,e,m,f , let N l be the number of links, N v the number vertices and N p the number of plaquettes of the triangulated torus (see Fig. 9). On the torus whose genus g = 1, they satisfy We note that each of |ee , |eo , |oe , and |oo is an equalweight superposition of 2 Np−1 = 2 N l −Nv−1 closed string configurations.
Since |ψ ac m correspond to a single no-string configuration, it only overlaps with |ee (which contains a no-string configuration). We find After removing the area term 2 − Np 2 , we see that the wave function overlaps W 1a BA,1 are given by integers W 1a BA,1 = (1, 0, 1, 0), a = 1, e, m, f , when the B phase is given by m-particle condensation.
Also |ψ ac e have the same overlap 2 −N l /2 with any string configuration. We find After removing the area term 2 − Nv 2 , we see that the wave function overlaps W 1a BA,1 are given by integers W 1a BA,1 = (1, 1, 0, 0), a = 1, e, m, f , when the B phase is given by e-particle condensation.
The above example supports our previous results that the area independent part of wave function overlap on torus are universal and are given by integers.
From a more experimental point of view, suppose there is a small physical system (say realized by a quantum simulator) where we can tune several paremeters to force phase transitions, and the wave function overlaps can be measured (by interference for example). If quantization is observed in the wave function overlaps, it may relate to the universal integer part as above, and such quantization is a sign of topological order and anyon condensation.

V. GAPPED BOUNDARIES OF A 2+1D TOPOLOGICAL ORDER
We can use the results developed in this paper to study the gapped boundaries of 2+1D topological order, for example try to find out how many different gapped boundaries a topological order can have. In this section, we study some simple examples.
A. Boundary of Z2 topological order Let us choose phase A to be Z 2 topological order (denoted by Z 2 ) and phase B to be trivial. A domain wall between them is a boundary of Z 2 topological order. The modular matrices for the Z 2 topological order are given by in the basis (1, e, m, f ). The modular matrices for the trivial topological order are S B = 1, T B = 1. From modular transformation for normalized wave function overlap we find two types of gapped boundaries, characterized by two integer vector solutions of the above equation: Next, let us consider the case of a genus-2 manifold Σ 2 : For MCG(Σ 2 ), there are five generators which are Dehn twists along the closed curves a 1 , b 1 , a 2 , b 2 , and γ, as shown in (39). Denoting the projective representations of these five Dehn twists as T a1 , T b1 , T a2 , T b2 , and T γ respectively, we can use T ai and T bi to construct S i matrix that acts on the left (right) half of Σ 2 , with Then we have the following projective representations of the five generators of MCG(Σ 2 ): where we have denoted T 1 := T a1 , T 2 := T a2 , and T 5 := T γ . Then based on Eq.(20), we havẽ where R Z2 = S 1 , T 1 , S 2 , T 2 , T 5 . It is noted that for an Abelian theory, the basis in Fig.3 can be represented as i j That is, the anyon z in Fig.3 corresponds to the identity anyon 1 now. Then the two anyon loops in (42) are decoupled. With this basis, it is straightforward to check that for R Z2 = S 1 , T 1 , S 2 and T 2 , the solutions to Eq.(41) are simply tensor product of genus-1 solutions (see Eq. (26)):W whereW e Z2,g=1 andW m Z2,g=1 are the genus-1 results as expressed in Eq. (38). However, the solutionsW Z2,g=2 in (43) are illegal, since both phase Z 2 and phase I are homogeneous here and we do not consider the case that e-condensation and m-condensation boundaries coexist. In the following, we show thatW Z2,g=2 are ruled out by considering R Z2 = T 5 in Eq. (41).
For an Abelian theory, since the fusion result of i ⊗j is unique, the basis in (42) can be rewritten as ij z (44) with i ⊗j = z . Then acting T 5 on the basis in (42) or equivalently (44) results in a phase θz = θ z = e 2πis z (see Fig.16). In other words, the matrix T 5 in the basis (42) then T 5 is a diagonal matrix of the form and W Z2,g=2 = 1, 1, 0, 0; 0, 0, 0, 0; 1, 1, 0, 0; 0, 0, 0, 0). (47) One can check explicitly thatW Z2,g=2 · T 5 . Similarly, one can find thatW That is, onlyW B. Boundary of S3 topological order S 3 topological order (described by quantum double of finite group S 3 with fusion rule given by Table I) is more interesting, since it is a non-abelian theory. Let us choose phase A to be S 3 topological order (denoted as S 3 ) and phase B to be trivial (denoted as I). The modular matrices for the S 3 topological order are given by (with basis (1, a 1 , a 2 Eqn. (13) has five solutions with M 11 S3 = 1: It is found that the last solution does not satisfy the stable condition in eqn. (14), i.e. (W So the S 3 topological order has only 4 types of gapped boundaries. In fact, as discussed in the following, without resorting to the stable condition in (14), we show that the fake solutionW (5) S3,g=1 can be ruled out by the genus-2 gapping boundary condition in Eq. (20), i.e., which is a linear condition. Since the S 3 topological order is multiplicity-free (i.e., N ij k ≤ 1), we denote the component ofW S3,g=2 asW i,j,z S3,g=2 (See also Eq. (19)). Here the anyon types i, j, z are used to label the basis vectors in the Hilbert space of degenerate ground states on a genus-2 manifold Σ 2 (see Fig.3). It is noted that bothW i,j,z S3,g=2 and the projective representations R U S3 of MCG(Σ 2 ) depend on the choice of basis vectors (see Appendix D). In the following discussion, we use the basis vectors in Fig.3.
Similar to the previous subsection on Z 2 topological order, we check whether the genus-1 solutions in Eq.(50)  can be embedded in the genus-2 solutions of Eq.(52). Our logic is as follows: First, it is straightforward to find that W , and it is not apparent thatW S3,g=1 are solutions of Eq. (52) for R S3 = T 5 . In this case, a careful study of Eq. (52) is needed. Third, we need to solve for the components In the following, we solve all the componentsW i,j,z

S3,g=2
that are solutions of Eq. (52). The results can be mainly summarized as follows: 1. It is found there are in total 4 sets of independent solutions of the genus-2 condition in Eq.(52), which we denote asW S3,g=2 , with i = 1, 2, 3, 4. They embed the genus-1 solutions of the formW S3,g=1 ) are ruled out by Eq.(52), as summarized in Table.II.

For the 4 sets of independent solutionsW
The nonzero componentsW i,j,z S3,g=2 with z = 1 can be expressed as the product of genus-1 results as S3,g=2 , which embeds the genus-1 solution of the formW S3,g=1 , the condensed anyons are 1, a 2 , and c. One can obtain the nonzero component ofW i,j,z S3,g=2 with z = 1 as The nonzero componentsW i,j,z S3,g=2 with z = 1 can be expressed as the product of genus-1 results as S3,g=2 , which embeds the genus-1 solution of the formW S3,g=1 , the condensed anyons are 1, a 1 , and b. As studied in Appendix D, all the componentsW i,j,z S3,g=2 with z = 1 are zero. The only non-zero components can be considered as the product of genus-1 results asW i,j,z=1 S3,g=2 = (W  with z = 1 are zero, and the only non-zero components correspond to the product of genus-1 results asW i,j,z=1 1. Rule out the fake solution with genus-2 condition As an illustration of how to determine the solutions of the genus-2 condition in Eq.(52), here we give an example on howW To study the effect of T 5 , it is convenient to consider the basis vectors II in (54). Let us focus on the components with i = a 2 and j = b. (Recall that T 5 does not change i and j.) Considering the fusion rule a 2 ⊗ b = b 1 ⊕ b 2 , then in the basis vectors |ψ II;a 2 ,b,z , the anyon type z can only be chosen as b 1 or b 2 , which have non-trivial topological spins [see (48)]. Then, considering R S3 = T 5 in Eq. (52) · ψ I;a 2 ,b,1 |ψ II;a 2 ,b,b 2 +W I;a 2 ,b,z=a 1
In the appendix (see Eq.D20), one can explicitly show thatW I;a 2 ,b,z=a 1
One can refer to Appendix D for all the solutions of Eq.(52) for S 3 topological order.

VI. SUMMARY
In this paper, we develop a systematic approach to the gapped domain walls between two topological orders A and B. (If B is the trivial topological order, the domain wall becomes the boundary of topological order A.) Our systematic approach is based on the topological partition function W I B I A BA,g of the domain wall Σ g , which is a Riemann surface of arbitrary genus g, which is a multicomponent partition function labeled by I A , I B . The multi-component partition function W I B I A BA,g is expected since the domain wall has a non-invertible gravitational anomaly [40] as characterized by topological orders A and B. The topological partition function W I B I A BA,g can also be viewed as the overlap of the degenerate ground states of A and B on Σ g , where I A (I B ) labels the ground states of topological order A on Σ g . This allows us to derive the following linear conditions on the data W I B I A BA,g that characterized the domain walls: where R A (R B ) is the mapping-class-group representation for topological order A (topological order B) for genus-g Riemann surface. Eqn. (59) is a special case of a more general condition proposed in Ref. 40, for the partition function with non-invertible gravitational anomaly. In this paper, through some simple examples, we try to demonstrate the validity of the condition (59) (and the condition in Ref. 40), by showing that the condition gives rise to a classification of gapped domain walls between two topological orders. In particular, we show that the topological partition function W I B I A BA,1 for genus-1 surface, plus the linear condition (59), is not enough. [34] We need to, at least, use the topological partition function W I B I A BA,2 for genus-2 surface and its condition (59), to obtain a correct classification of gapped domain walls.
At moment, we do not known if the topological partition functions W I B I A BA,g for arbitrary genus-g surface can fully characterize the gapped domain wall or not (although we think that, very likely, they do). In Appendix E, using the connection between anyon condensation and domain walls, we develop a classifying theory of gapped domain wall based on the structure coefficients M kc,u ia,jb that describe the condensable algebra in a topological order. We also give the conditions eqn. (E21), eqn. (E26), and eqn. (E28) on the structure coefficients M kc,u ia,jb , so that they can described a gapped domain wall. However, those conditions are non-linear and is very hard to solve.
We see that we have two systematic ways to describe the gapped domain walls between two topological orders. The first approach is based on topological partition functions W I B I A BA,g , which is easier to solve, but not known to fully classify the domain walls. The second approach is based on the structure coefficients M kc,u ia,jb , which classify the gapped domain walls, but is hard to solve. Try to gain a deeper understanding of the two approaches may help us to find an easier way to fully classify gapped domain walls.  tegral. We first triangulate the 3-dimensional space-time to obtain a simplicial complex M 3 (see Fig. 10). Here we assume that all simplicial complexes are of bounded geometry in the sense that the number of edges that connect to one vertex is bounded by a fixed value. Also the number of triangles that connect to one edge is bounded by a fixed value, etc.
In order to define a generic lattice theory on the spacetime complex M 3 , it is important to give the vertices of each simplex a local order. A nice local scheme to order the vertices is given by a branching structure. [45][46][47] A branching structure is a choice of orientation of each edge in the n-dimensional complex so that there is no oriented loop on any triangle (see Fig. 11).
The branching structure induces a local order of the vertices on each simplex. The first vertex of a simplex is the vertex with no incoming edges, and the second vertex is the vertex with only one incoming edge, etc. So the simplex in Fig. 11a has the following vertex ordering: The branching structure also gives the simplex (and its sub simplexes) an orientation denoted by s ij···k = 1, * .
is associated with a tetrahedron, which has a branching structure. If the vertex-0 is above the triangle-123, then the tetrahedron will have an orientation s0123 = * . If the vertex-0 is below the triangle-123, the tetrahedron will have an orientation s0123 = 1. The branching structure gives the vertices a local order: the i th vertex has i incoming edges.

Discrete path integral
In this paper, we will only consider a type of 2+1D path integral that can be constructed from a tensor set T of two real and one complex tensors: ). The complex ten- can be associated with a tetrahedron, which has a branching structure (see Fig.  12). A branching structure is a choice of orientation of each edge in the complex so that there is no oriented loop on any triangle (see Fig. 12). Here the v 0 index is associated with the vertex-0, the e 01 index is associated with the edge-01, and the φ 012 index is associated with the triangle-012. They represents the degrees of freedom on the vertices, edges, and the triangles.
Using the tensors, we can define the path integral on any 3-complex that has no boundary: where v0,··· ;e01,··· ;φ012,··· sums over all the vertex indices, the edge indices, and the face indices, s 0123 = 1 or * depending on the orientation of tetrahedron (see Fig.  12). We believe such type of path integral can realize any 2+1D topological order.

Path integral on space-time with natural boundary
On the complex M 3 with boundary: B 2 = ∂M 3 , the partition function is defined differently: . The boundary (A2) defined above is called a natural boundary of the path integral.
We also note that only the vertices and the edges in the bulk (i.e. not on the But when we glue two boundaries together, those tensors w vi and d

Topological path integral
We notice that the above path integral is defined for any space-time lattice. The partition function Z(M 3 ) depends on the choices of of the space-time lattice. For example, Z(M 3 ) depends on the number of the cells in space-time, which give rise to the leading volume dependent term, in the large space-time limit (i.e. the thermodynamic limit) where V is the space-time volume, is the energy density of the ground state, and Z top (M 3 ) is the volume independent partition function. It was conjectured that the volume independent partition function Z top (M 3 ) in the thermodynamic limit, as a function of closed space-time M 3 , is a topological invariant that can fully characterize topological order. [9,39] So it is very desirable to fine tune the path integral to make the energy density = 0. This can be achieved by fine tuning the tensors w vi and d vivj eij . But we can be better. We can also choose the tensor (w v0 , d v0v1 e01 , C e01e02e03e12e13e23;φ012φ023 v0v1v2v3;φ013φ123 ) to be the fixed-point tensor-set under the renormalization group flow of the tensor network. [48,49] In this case, not only  the volume factor e − V disappears, the volume independent partition function Z top (M 3 ) is also re-triangulation invariant, for any size of space-time lattice. In this case, we refer the path integral as a topological path integral, and denote the resulting partition function as Z top (M 3 ). Z top is also referred as the volume independent the partition function, which is a very important concept, since only volume independent the partition functions correspond to topological invariants. In particular, it was conjectured that such kind of topological path integrals describe all the topological order with gappable boundary. For details, see Ref. 9 and 39.
The invariance of partition function Z under the retriangulation in Fig. 13  ) , on the two sides of the domain wall. Here we will assume that the two tensor sets define topological path integrals in the bulk. The domain wall is defined via a different tensor set for the simplexes that touch the domain wall. Again we can choose the domain wall tensors to make the partition function with domain wall (after summing over the bulk and domain wall degrees of freedom) to be re-triangulation invariant (even for the re-triangulations that involve the domain wall). Therefore, the topological path integral can also be defined for spacetime with domain walls. Different choices of domain wall tensors give rise to different domain walls. Those domain walls can be characterized by the data introduced in Section II.
To find the conditions on those domain-wall data, we also need to use the space-time path integral with worldlines of topological excitations. We denote the resulting partition function as where i, j, k, · · · ∈ {1, 2, · · · , N } label the type of topological excitations, and α, β, γ label the fusion channels (i.e. different choice of actions at the junction of three world-lines). Again, the world line is defined via a different tensor set for the simplexes that touch the world line. We can choose the world line tensors to make the partition function with world line to be re-triangulation invariant (even for the re-triangulations that involve the world line). Therefore, the topological path integral can also be defined for spacetime with world lines. Different choices of world line tensors give rise to different world lines, which are labeled by the types of topological excitations. In this paper, we will only consider those topological path integrals with re-triangulation invariance.
Appendix B: Categorical approach to evaluate topological path integral with world lines

Planar world-lines and unitary m-fusion category
The topological path integrals with world lines (i.e. the re-triangulation invariant path integrals with world lines) Z top can be computed via an algebraic approach (or more precisely an categorical approach). In the following, we will give a brief introduction of such an approach. More details can be found in Ref. 51.
First, let us consider the partition functions with only planar world-line configurations. In this case, we may pretend the space-time to be 2-dimensional (or the spacetime is actually 2-dimensional).
The partition functions with different world-line configurations can be related by some linear relations. For example This is because the above partition functions describe the amplitude of fusion type-i, j, k topological excitations into degenerate type-l topological excitations. The two sides of the equation just correspond to different order of fusion which give rise to the same end product. Thus those amplitudes are related.
Let us consider the fusion of two topological excitations of type-i, j. From far away, the two topological excitations may be viewed as single topological excitation. But such a single topological excitation may correspond to several different topological excitations which happen to have the same energy. For example, two spin-1/2 excitations can be viewed as spin-0 and spin-1 excitations that happen to have the same energy. So to describe the fusion of i and j, we need to introduce N ij k ∈ N, which count the number of the type-k topological excitation which happen to have the same energy that appear in the fusion of type-i, j topological excitations. Note that N ij k may not equal to N ji k . Therefore, we have We call such local change of graph an O-move.
For fixed i, j, We will call such a local change as a Y-move. We can choose We can adjust the action at the triple world-line junction to simplify O jk,αβ i and Y ij k,αβ . After the simplification, where δ jk i = 1 for N jk i > 0 and δ jk i = 0 for N jk i = 0. Also d i are the real and positive solutions from which are called the quantum dimensions of type-k topological excitation. We see that the partition function A(X) for any worldline configurations can be characterized by tensor data (N, N ij k , F ijm,αβ kln,γλ ). However, only certain tensor data (N, N ij k , F ijm,αβ kln,γλ ), that satisfy some conditions can selfconsistently describe partition function A(X). Those conditions form a set of non-linear equations whose variables are N ij k , F ijm,αβ kln,γλ , d i (where d i can be determined by N ij k alone):  (B10) We like to mention that, in the above we did not assume the existence of trivial particle, which fuse with other particles as an identity. We will call such kind of fusion as unitary m-fusion category As an application of the above algebraic structure -the unitary m-fusion category, let us consider two world-lines of type-i and type-j, wrapping around a torus S 1 x × S 1 y in the x-direction. Let W x i and W x j be the string operators that creates the world-lines. Applying the Y-move and then the O-move, and using eqn. (B8) (see Fig. 15), we find that We see that the algebra of the loop operator W i forms a representation of fusion algebra i ⊗ j = k N ij k k.

Presence of trivial particle and unitary fusion category
Now let us assume such a trivial particle type to exist, and denoted it by 1, which satisfies the following fusion rule We also requires that for every i there exists a uniqueī such thatī We can represent a type-1 string by a dash line. By examine the O-move with k = 1: we see that we can remove or add any vertex with dash line without changing Z top .
With the presence of trivial particle type, we can determine the amplitude for a loop of i-string. Using the rule of adding dash lines (the trivial strings) and O-move eqn. (B5), we find Thus a loop of type-i world-line has an amplitude d i .

Non-planar diagram and braided fusion category
We have being considering planar graphs and the related fusion category theory. In this section we will consider non-planar graphs. Since the particles now live in 2-dimensional space (or higher), the fusion of the particles satisfies 16. (Color online) A "self-loop" with canonical framing corresponds to a twist by 2π. A twist by 2π induces a phase e i 2πs i that defines the spin si of the particle.
(a) (b) FIG. 17. The two "self-loops" in (a) are "right-handed" and correspond to the same twist. The two "self-loops" in (b) are "left-handed" and also correspond to the same twist that is opposite to that in (a). and thus So the fusion of 2D particles are commutative (while the fusion of 1D particles may not be commutative).
Here, we also like point out that a world-line of a particle are always framed (i.e. having a shadow world-line running parallel to it). When we draw a graph on a plane, there is canonical framing, obtain by shifting the graphs perpendicular to the plane (see Fig. 16). We have been using such a canonical framing in our previous discussion, and we have omitted drawing the framing. But if we do not use this canonical framing, then we need to draw the framing explicitly, as in Fig. 16.
Let us consider simple string configuration with crossing: a "self-loop" with the canonical framing (see Fig.  16). Such a "self-loop" corresponds to a straight line with a 2π twist, which is equal to a untwisted straight line with a phase e i 2πsi . Here s i is the spin of the type-i topological excitation, which is defined mod 1. We also note that the handness of the "self-loop" determines the direction of the twist (see Fig. 17). As a result, a figure "8" of type-i string has an amplitude e 2π i si d i (see Fig.  18).  There is another ways to fully characterize the domain walls. We note that the gapped 1+1D domain walls can be viewed as 1+1D anomalous topological orders. [9] The 1+1D anomalous topological orders are characterized by unitary fusion categories (UFC) which are described by the following data (1) N ∈ N: the number of types (including the trivial type) of topological excitations on the domain wall. We will use i, j, k, etc, to label the types of topological exitations and use 1 to label the trivial type.
(2) N ij k ∈ N: the fusion coefficients of the topological excitations. (3) F ijm,αβ kln,γλ : the unitary relation between different fusion spaces obtained via different fusion paths.
Those data (N, N ij k , F ijm,αβ kln,γλ ) satisfy Here d i is the largest left eigenvalue of the matrix N i (defined as (N i ) kj = N ij k ). Fusion category (N, N ij k , F ijm,αβ kln,γλ ) and weighted wave function overlap W I B I A BA,g provide two very different ways to characterize the same domain wall. It is amazing that to two sets of data should have a one-to-one correspondence. In particular, 'basis I' will be useful in making a connection to the genus-1 solution by choosing z = 1, and 'basis II' will be useful in studying the effect of Dehn twist operator T 5 (or T γ ) in Fig.39. Within basis II, T 5 will be a diagonal matrix with the diagonal elements corresponding to the topological spin θ z of anyon z.
Before any concrete calculation, it is noted that for the solutions of gapped boundaries, ifW I (II);i,j,z S3,g=2 = 0, then the topological spins of i, j, and z in D1 and D2 must be trivial. This can be understood by considering We denote basis I and basis II in (D1) and (D2) as |ψ I;i,j,z;µ,ν , and |ψ II;i,j,z;µ,ν , respectively. Since the S 3 topological order is multiplicity free, we can write the above basis vectors as |ψ I;i,j,z , and |ψ II;i,j,z .
These two basis vectors are normalized as follows: ψ I;i,j,z |ψ I;i ,j ,z = δ i,i δ j,j δ z,z , ψ II;i,j,z |ψ II;i ,j ,z = δ i,i δ j,j δ z,z . (D5) In addition, they are related to each other as, Here [F ij ij ] (z,z ) is related to the conventional F -matrix as defined in (B1) through the following relation: It is convenient to study the effect of Dehn twist T 5 in basis II, which simply results in a phase factor, i.e., T 5 |ψ II;i,j,z = θ z |ψ II;i,j,z .
Within basis I, one has (D10) Apparently, T 5 is in general not diagonal in basis I. However, if if the theory is abelian, then one has z = 1 and [F ij ij ] 1z = N z ij . Then T 5 is a diagonal matrix with the matrix elements ψ I;i,j,z |T 5 |ψ I;i,j,z = δ 1,z δ 1,z θ z N z ij .
Another interesting case is that if θ z = 1 for all [F ij ij ] (z,z ) = 0, then based on Eq.(D10), one has (D12) For S 3 topological order, all the F matrices have been obtained in Ref. 52 The so-called punctured S matrix can be expressed in terms of F -matrix as (in the multiplicity-free case) based on which we can obtain the punctured S (z) matrix. For our motivation of studying the gapped boundaries, we only need to consider S (z) with θ z = 1 in (D1). Based on Eq.(48) and the fusion rules of i⊗ī in Table I, we only need to check the cases of z = a 1 , a 2 , b.
• For z = b, one can obtain (in the basis c 1 , c, and b): and T (z=b) = diag (−1, 1, 1). For the punctured S (z) and T (z) presented above, one can check explicitly that they satisfied the so-called modular relation Now we are ready to study the solutions of with R = S (z) and T (z) .
• For z = a 1 , since T (z=a 1 ) is diagonal, one can find that W has the form W = (x, y, 0, 0). Then choosing R = S (z=a 1 ) in (D19), one can find that x = y = 0. That is, there is no non-zero solution of (D19) for R = S (z=a 1 ) and T (z=a 1 ) . More explicitly, with the basis in (D1), we haveW • For z = a 2 , since T (z=a 2 ) = diag(−1, 1, 1), we have W = (0, x, y). Then by choosing R = S (z=a 2 ) in (D19), one can find that x = − √ 2y. That is, where y is to be determined by other conditions. With where i ∈ {a 2 , c, c 1 }.

Solutions of genus-2 condition
In this subsection, we solve the genus-2 condition in Eq. (52), and find there are in total 4 sets of independent solutions, which embed the genus-1 solutionsW S3,g=1 with i = 1, 2, 3, 4, respectively.

Solutions of genus-2 condition:
Now let us check the pairings of genus-1 solutions W S3,g=1 with p = 1, 2, 3, 4 respectively in Table III.
The genus-1 solutionW In the following, we will show thatW I;i,j,z=1

S3,g=2
are coupled to certainW I;i,j,z S3,g=2 with z = 1 through the operation R S3 = T 5 . We will frequently use the basis transformation in Eq.(D6). Let us start from i = b and j = c. Then for |ψ I;b,c,z = [F bc bc ] zz |ψ II;b,c,z , where z ∈ {1, b} and z ∈ {c, c 1 }, one can find that Since the action of T 5 on |ψ II;b,c,z is simply T 5 |ψ II;b,c,z = θ z |ψ II;b,c,z , then based on the basis transformation in Eqs.(D6) and (D30) one can find that T 5 in the basis {|ψ I;b,c,1 , |ψ I;b,c,b } has the expression: By solving Eq.(52), one can obtaiñ For the case of i = c and j = b, the discussion is almost the same, and one can find that Next, let us consider the case i = j = c. For |ψ I;c,c,z = [F cc cc ] zz |ψ II;c,c,z , where z, z ∈ {1, a 2 , b, b 1 , b 2 }, one can find that Combining with T 5 |ψ II;c,c,z = θ z |ψ II;c,c,z , one can find the expression of T 5 in the basis {|ψ I;c,c,1 , |ψ I;c,c,a 2 , |ψ I;c,c,b , |ψ I;c,c,b 1 , |ψ I;c,c,b 2 } as follows Then from Eq.(D36), one hasW I;c,c,a 2 S3,g=2 = 0. Based on Eq.(D37), one hasW I;c,c,a 2 S3,g=2 =W I;c,a 2 ,a 2 S3,g=2 =W I;a 2 ,c,a 2 S3,g=2 = W I;a 2 ,a 2 ,a 2 S3,g=2 = 0. Till now, we have obtained certainW I;i,j,z S3,g=2 that em-bedsW S3,g=1 . In particular, the nonzero components as follows: andW I;i,j,z=1 Furthermore, we claim that these are the only non-zero components ofW I;i,j,z S3,g=2 that embedsW That is, for all the non-zero componentsW i,j,z I;S3,g=2 , one can find that i, j, z ∈ {1, b, c}. All the other (116 − 13 = 103) components are zero. This can be checked explicitly as follows.

S3,g=2
which is the tensor product of genus-1 solutions, there are 55 components that are zero. They areW I;i,j,z=1 S3,g=2 such that there ∃ i, j / ∈ {1, b, c}. Second, there are 18 components ofW I;i,j,z S3,g=2 which are zero, with θ z = 1. They correspond toW I;i,j,z S3,g=2 with z = b 1 , b 2 . The rest components with z = 1 and θ z = 1 need more careful study. Let us check them case by case.
-For z = a 1 , we have shown in Eq.(D20) that There are in total 16 components.
-For z = a 2 , as we have analyzed above, all the 9 componentsW I;i,j,z S3,g=2 with z = a 2 are zero. -For z = b, we haveW I;i,j,z=b S3,g=2 = 0, if there ∃ i, j = c 1 . There are in total 5 components.
-For z = c, this is not allowed by the fusion rules.
-For z = a 2 , we haveW I;i,j,z=a 2 S3,g=2 = 0, if there ∃ i, j = c 1 . There are in total 5 components of this kind.
Altogether, for all the 116 genus-2 components W I;i,j,z S3,g=2 that embedW     which are zero, with θ z = 1. They correspond toW I;i,j,z S3,g=2 with z = b 1 , b 2 . The rest components with z = 1 and θ z = 1 can be checked case by case as follows.
-For z = a 1 , we have shown in Eq.(D20) that There are in total 16 components.

S3,g=2
such that there ∃ i, j / ∈ {1, a 1 , a 2 }. Then one can find that forW I;i,j,z S3,g=2 that embed W    It is known that the gapped boundary is closely related to anyon condensation, whose data is fully encoded in the so-called condensable algebra [25] in the unitary modular tensor category (UMTC) that describes the anyons.
Let (C, c) and (D, c) be two 2d topological orders with the same chiral central charge c, where C and D are two UMTC's. We assume that they are Witt equivalent, in other words, they can be connected by a gapped domain wall. There are two equivalent mathematical descriptions of such a gapped domain wall: • A gapped domain wall between (C, c) and (D, c) can be described by a unitary fusion category M, which is equipped with a unitary braided monoidal equivalence where C D is the UMTC describing the topological order obtained by stacking C with the time-reversal of D and Z(M) denotes the Drinfeld center of M.
• A gapped domain wall between (C, c) and (D, c) is uniquely determined by a Lagrangian condensable algebra A in C D. Here Lagrangian means that A is "maximal" in the sense In this case, the gapped domain wall is described by the UFC (C D) A , i.e. the category of right A-modules in C D. A Lagrangian algebra A, as an object in C D, can be decomposed as follows: where Irr(C) and Irr(D) denote the sets of isomorphic classes of simple objects.
In particular, when D is the trivial topological order, the above describes a gapped boundary of C, and A is a Lagrangian algebra in C. The existence of a Lagrangian algebra, for example, in terms of the structure coefficients to be introduced in this section, is a necessary and sufficient condition for the existence of a gapped boundary. The approached presented in the main text, using the invariance of wave function overlaps under the modular transformations (or mapping class group transformations), however, only constitutes necessary conditions. The advantage of the modular invariance approach is that they are all linear equations and much easier to solve, as opposed to the non-linear equations for the structure coefficients such as the associativity condition. For the examples presented in this paper, each of our solutions does correspond to a Lagrangian algebra.
Physically, a condensable algebra is just a composite anyon that becomes the ground state after condensation (i.e. it is condensed). For this to be possible, it is necessary that this composite anyon has special algebraic structures, which can be thought as a generalization of usual commutative associative algebra. We know that after picking a basis, the multiplication of a usual associative algebra can be expressed by the structure coefficients. It is also the case for the condensable algebra. The structure coefficients encode all the "special algebraic structures" of the condensed composite anyon; in particular, they predict the form of the ground-state overlap.
To explain what a "basis" of a composite anyon means in a unitary modular tensor category, we begin by recalling the (orthonormal) basis of an n-dimensional Hilbert space H, which is a set of state vectors |α satisfying To generalize this notion, note that an n-dimensional Hilbert space can be viewed as a composite anyon, that is composed of n trivial anyons, H = ⊕1 ⊕n . The trivial anyon is the tensor unit 1, namely the ground field C, in the tensor category. Moreover, a vector |α can be identified with the embedding linear map q α : C → H, by q α (λ) = λ|α . Similarly, α| can be identified with the projection p α : H → C, by p a (|ψ ) = α|ψ . Now, given a composite anyon where i labels the simple anyons, a basis of A is a set of morphisms ("linear maps" in generic tensor category) We usually require that p i,α A , q A i,α are Hermitian conjugates: It is intuitive to use the following graphs for the basis: Our choice of symbol is to remind the reader of the similarity to the basis of Hilbert spaces (rotate the graph by 90 • anticlockwise). They satisfy similar orthonormal and complete conditions: The only subtle part is that the tensor product and braiding involving nontrivial i is different from usual intuitions from vector spaces, which will be explained below. First, We like to remark that, in general the tensors such as F, R matrices do depend on our choice of basis. However, it is easily verified that the two choices of basis (E12) differ by the same overall factor on both sides of (E13)(E14). Thus, F, R matrices remain the same under the change of basis (E12). For an algebra A in a tensor category, there is a multiplication morphism m : A ⊗ A → A. First take a basis of A as in (E5) The multiplication morphism m is then ?
M kχ,µ iα,jβ is the "structure coefficients" of the algebra. In the category of vector spaces Vec, the object labels i, j, k and the vertex label µ reduce to trivial, and M kχ,µ iα,jβ reduces to structure coefficients of usual associative algebra, with α, β, χ the labels of basis vectors. Again, similar to the usual associative algebra, the structure coefficients depends on the choice of basis, both the basis of A, p i,α A , q A i,α and the vertex basis p k,µ ij . It is easy to write down transformations of M kχ,µ iα,jβ under a change of basis. For example, using the rotatable basis (E12), the corresponding structure coefficients is ?
expressed in terms of embedding basis η = η α q A 1,α , and A unital connected isometric algebra in a unitary tensor category automatically satisfies the Frobenius condition [53]: As a direct corollary, A is self dual and where η 1 is the from the unit morphism η = η 1 q A 1,1 : 1 → A. We choose the phase of η 1 and let Together with (E28), one can permute any pair of iα indices. The structure coefficients can be used to compute the (topological universal part of) overlap between ground states before and after condensation. For simplicity, here we only discuss the case that the phase after condensation is topologically trivial and has a unique ground state. We start with the ground state on a torus. Before condensation, the loops of simple objects form a basis of the ground space on a torus: After condensation, the ground state is, up to an overall factor that depends on the system size, the loop of the corresponding condensable algebra A: Thus we can extract the multiplicity N A i from the ground-state overlap on a torus, which is exactly the universal part of wave function overlap (the labels in the trivial phase is omitted, see eqn. (2) and eqn. (3)) On a closed two-dimensional surface with genus g = 2, one choice of the basis before condensation is given by (E41) Still, after condensation, the ground state is Now, the structure coefficients kick in. After straightforward calculation, we find Therefore, for genus 2, the wave function overlap, up to an overall factor, is given by (the labels in the trivial where we used the punctured S matrix as in (D13). Therefore, the structure coefficients satisfy the (punctured) S invariance: jν S (z) j,ν;i,µ α M jα,ν jα,zβ = α M iα,µ iα,zβ .
For a generic closed surface, as long as an appropriate graph basis such as (E41) is picked, part of the graph will look like (E52), and thus the wave function overlap is invariant under the punctured S transformation. For example, on genus 2, taking the left half of (E41) and applying punctured S transformation, we find that kχ S (z) k,χ;i,µW (k,j,z,χ,ν) C;2 =W (i,j,z,µ,ν) C;2 . (E55) In particular, when z = 1, α M iα,1 iα,11 = η −1 1 N A i , we have the S invariance on torus