Fusing Binary Interface Defects in Topological Phases:The Z/pZ case

Jacob C. Bridgeman,1, ∗ Daniel Barter,2, † and Corey Jones3, ‡

1Centre for Quantum Software and Information,Faculty of Engineering and Information Technology, University of Technology Sydney, Australia

2Mathematical Sciences Institute, Australian National University, Canberra, Australia3Department of Mathematics, The Ohio State University, USA

(Dated: January 25, 2019)

A binary interface defect is any interface between two (not necessarily invertible) domain walls.We compute all possible binary interface defects in Kitaev’s Z/pZ model and all possible fusionsbetween them. Our methods can be applied to any Levin-Wen model. We also give physicalinterpretations for each of the defects in the Z/pZ model. These physical interpretations provide anew graphical calculus which can be used to compute defect fusion.

Topological phases are promising candidate materials for robust encoding, storage and manipulation of quantuminformation[1–4]. Formed by locally interacting degrees of freedom, these quantum systems have emergent globalproperties that are protected against the presence of environmental noise. In addition to the bulk properties, inclusionof defects has been shown to improve the power of these materials from a quantum computational perspective[1, 5–18]. It is therefore important to understand the full theory, including defects. In this paper, we study non-chiral,2-dimensional, long-range-entangled topological phases with defects.

A defect of a topological phase is a region of positive codimension which differs from the ground state. For example,in a 2-dimensional topological phase, domain walls are codimension 1 defects and anyonic excitations are codimension2 defects. Much work has been done on defects in topological phases, for example Refs. [1, 5–28]. In previous work, theterm defect frequently refers to a 0-dimensional interface between two invertible domain walls. To avoid confusion,we shall use the term binary interface defect to refer to any 0-dimensional interface between two, not necessarilyinvertible, domain walls.

This work builds on our previous paper Ref. [29], in which we computed the fusions of all domain walls in Kitaev’sVec(Z/pZ) models[2] (with p prime). In this paper, we compute all possible fusions between all possible binaryinterface defects in the Kitaev Vec(Z/pZ) model. The tools from both Ref. [29] and the present work can be adaptedto more general Levin-Wen models. In the physics literature, defect fusion is often synonymous with symmetrygauging. We compute the fusions even when no gauging exists.

In Ref. [2], Kitaev defined a 2D lattice model associated to any finite groupG with particle-like low energy excitations(known as anyons) parameterized by the simple representations of the Drinfeld double of G. These Kitaev modelsare some of the most well known examples of topological phases, and are of great experimental interest[30, 31].

These models were generalized in Ref. [32] to allow any fusion category C as input. The low energy excitationsof these Levin-Wen models are particle-like and parameterized by simple object of the Drinfeld center Z(C). WhenC = Vec(G), the Levin-Wen model can be transformed into the Kitaev model associated to G with a finite depthquantum circuit, so they describe the same phase of matter. Indeed, the category of representations of the Drinfelddouble of G is equivalent to the Drinfeld center of Vec(G).

There are many interesting examples of fusion categories, so the Levin-Wen construction gives us many interesting2D lattice models. Unfortunately they are too complicated to simulate or study using conventional lattice quantumfield theoretic techniques. Moreover, for most interesting fusion categories, the data required to write down theHamiltonian in the Levin-Wen construction is not known explicitly. For this reason, we need alternative tools tostudy these models.

The renormalization invariant properties of a topological phase is described by a topological quantum field theory(TQFT). In mathematics, a TQFT is a functor from a bordism category into a linear algebraic category. It would becounter-productive for us to give a precise definition here. The reader interested in details is encouraged to consultthe recent survey Ref. [33] which includes complete definitions. In Ref. [34], Barrett and Westbury described how toconstruct a (2+1)D TQFT from a fusion category, generalizing the Turaev-Viro construction from Ref. [35]. It is wellunderstood that the TQFT associated to a fusion category captures the renormalization invariant properties of thecorresponding Levin-Wen model. In this paper, we use TQFTs as studied in mathematics to compute renormalizationinvariant properties of Levin-Wen models which are intractable from the lattice theoretic perspective.

∗ jcbridgeman1@gmail.com† danielbarter@gmail.com‡ cormjones88@gmail.com

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Our work builds on the work of Morrison and Walker in Ref. [36]. If C is a fusion category, morphisms in C can bedescribed by 2-dimensional string diagrams up to isotopy. If Σ is a 2-manifold with boundary, the TQFT associatedto C sends Σ to the vector space of string diagrams from C drawn on Σ modulo local relations. These vector spacesare called Skein modules. The string diagrams are allowed to terminate on boundary components. When Σ has aboundary, the vector space is graded by the object labels on the boundary components, and the graded pieces oftenassemble into an algebraic object. The process of drawing string diagrams from C on Σ also extends to modules overC. Morphisms in these module categories can also be described using string diagrams, and drawing string diagramsfrom module categories on Σ allows us to use diagrammatic techniques to study defects of codimension 1 and 2 in thecorresponding topological phase. The TQFT assigns a category to the annulus, which can be additionally decoratedwith bimodule strings

g h

m

n

m′

n′

: (m,n)→ (m′, n′). (1)

The representations of this category classify codimension 2 defects (binary interface defects in our language). We referthe reader to the recent survey Ref. [37] for more details. In Ref. [36], representations of this annular category arecalled sphere modules. Indeed, when C = Vec(G) and both of the modules are Vec(G), then this category is Moritaequivalent to the Drinfeld double of G. The representations (defects) then correspond to the anyonic excitations ofthe model. In Ref. [19], Kitaev and Kong explain that fusion category bimodules correspond to domain walls andbimodule functors correspond to codimension 2 defects in the Levin-Wen models. In Ref. [17], annular categories,called dube algebras by Williamson, Bultinck and Verstraete, are used to study defects interfacing invertible domainwalls.

Many of the defect fusions which we compute have multiplicity, which correspond to multiple fusion channels. Thesemultiplicities are somewhat mysterious from the physical perspective.

From a mathematical perspective, we are computing decomposition rules for relative tensor product and compositionof bimodule functors. In an upcoming paper [38], we will provide a rigorous proof of this fact using a robust theoryof skeletalization of fusion categories and their bimodules.

What is being computed in this work

Suppose that A, B are 2-dimensional phases of matter and M, N are domain walls between the two phases. Adefect α interfaces between two domain walls

αA B

N

M

. (2)

In this paper, we are interested in defects when A and B are the phase associated to the fusion category Vec(Z/pZ),and all possible ways these defects can be fused. There are two ways in which defects can fuse:

• Horizontally, where the supporting domain walls also fuse

α β

A B C

M N

P Q

→ α⊗ β

A B C

M⊗N

P ⊗Q

→∑R,S,γ

γA C

R

S

. (3)

3

• Vertically, where the defects fuse along a common domain wall

α

βA B→ α β

A B→∑γ

γA B

. (4)

In this work, we present algorithms that can be used to compute both the horizontal and vertical fusions between allbinary interface defects.

Structure of the paper

The paper is structured as follows. In Section I, we describe how to compute idempotent representations of all thebinary interface defects in Kitaev’s Z/pZ model. For a semi-simple category, the Karoubi envelope agrees with thecategory of representations, so these idempotents parameterize representations of the annulus categories. In Section II,we describe the procedure used to compute the horizontal fusion of binary interface defects. Rather than give a formalalgorithm, we describe how to proceed in a sufficiently general example. In Section III, we demonstrate some of thehorizontal fusion computations to elucidate some of the complications. In Section IV, we outline the procedure usedto compute vertical fusion. In Section V, we include some example vertical fusion computations. In Section VI, wegive physical interpretations for all the binary interface defects. The physical interpretations can be used to reproduceall the horizontal and vertical fusion tables, except for the multiplicities, which are still somewhat mysterious fromthe physical perspective. In Section VII, we explain how natural transformations between bimodule categories fit intoour framework. This will be expanded on in future work.

In Appendix A, we tabulate the idempotent representations of all the binary interface defects in Kitaev’s Z/pZmodel. In Appendix B, we tabulate all the inflation data required to compute the horizontal fusions. This data wascomputed as an intermediate step in Ref. [29]. In Appendix C, we tabulate the horizontal fusions and in Appendix D,we tabulate the vertical fusions.

I. CLASSIFYING DEFECTS

For an underlying fusion category C, and given a pair of bimodule categories M,N , the annular categoryAnnM,N (C) is defined as follows. The objects are pairs of simples (m,n) and the morphisms are annular dia-grams as shown in Eqn. 1. Representations of AnnM,N (C) classify binary interface defects, which we denote NM

∣∣.If C is semi-simple, the category of representations is equivalent to the Karoubi envelope Kar(AnnM,N (C)). As inRef. [29], we utilize this equivalence to classify the defects.

Objects of Kar(AnnM,N (C)) are pairs (A, e), where A is an object from AnnM,N (C), and e : A → A is anidempotent annular diagram. The classification of defects is equivalent to construction of inequivalent idempotents.Two idempotent annular morphisms are equivalent if there exists a morphism absorbing the first idempotent on theinside and the second idempotent on the outside.

In the remainder of this section, we show how representative idempotents are constructed for C = Vec(Z/pZ). Thedomain wall labels used are those defined in Ref. [29].

Example I.1 ( TT∣∣)

In many cases, the construction of representative idempotents is a simple task. For example, the basic annulusdiagrams for T

T

∣∣ defects are

g h

(a+ g, b+ h)

(c+ g, d+ h)

(a, b)

(c, d)

. (5)

4

For such a diagram to contribute to an idempotent, the inner and outer bimodule labels must be the same. In thiscase, the only way for this to happen is g = h = 0 (the group identity)

(a, b)

(c, d)

(a, b)

(c, d)

. (6)

It remains to find representatives of the isomorphism classes, where any annulus in Eqn. 5 defines an isomorphismbetween idempotents. Using Eqn. 5, there is no way to change the value of α = c− a or β = d− b. These conservedquantities label the isomorphism classes, and we can pick representatives for each of the p2 classes

(0, 0)

(α, β)

(0, 0)

(α, β)

, (7)

with α, β ∈ Z/pZ.

Example I.2 ( Fr

Xk

∣∣: An unusual representation of Z/pZ)

A more complicated example involves Fr

Xk

∣∣ defects. In this case, the general annulus diagrams are

g h

a+ g + kh

∗

a

∗, (8)

so the diagrams contributing to idempotents are

Mg := −kg g

0

∗

0

∗, (9)

where a = 0 has been chosen using the isomorphism Eqn. 8. Due to the nontrivial associator on Fr, the multiplicationrule for these annuli is

MgMh = g−kg h−kh

∗

0

= ωkghr -k(g+h) g + h

0

∗

= ωkrghMg+h, (10)

where ω = exp( 2πip ). These annuli are therefore forming a twisted representation of Z/pZ with 2-cocycle φ(g, h) =

ωkrgh. Since H2(G,U(1)) ∼= 1, this is equivalent to a linear representation Ug by some 1-cochain

βkr(g)Mg = Ug. (11)

5

One can obtain an explicit formula for βkr

βkr(g) =

ig p = 2

ωkrg22−1

p > 2, (12)

where 2−1 is the multiplicative inverse modulo p. With this explicit cochain, representatives for the inequivalentidempotents can be found

Fr

Xk

∣∣x

=1

p

∑g

ωgxωUg (13)

=1

p

∑g

ωgxβ(g)Mg =:1

p

∑g

Θx,kr(g)Mg. (14)

The equation Θx,a(g + k) = Θx,a(g) Θx,a(k) ωagk implies that the idempotent Fr

Xk

∣∣x

absorbs the diagram

kk0 −k0

0

∗

(15)

up to a global phase.

Example I.3 ( Fr

Fq

∣∣: A genuine projective representation)

A particularly interesting example is given by Fr

Fq

∣∣ defects. Since there is a single object (denoted ∗) in the Fxbimodule, all annulus diagrams potentially contribute to the idempotents

Mg,h := g h

∗

∗

. (16)

Both modules (potentially) have nontrivial associators, leading to the multiplication rule

Mg0,h0Mg1,h1

= h0g0 h1g1

∗

∗

= ω(q−r)h0g1 g0+g1 h0+h1

∗

∗

= ω(q−r)h0g1Mg0+g1,h0+h1. (17)

For q 6= r, this is a nontrivial 2-cocycle for Z/pZ× Z/pZ[39, 40], and therefore these annuli form a nontrivial twistedgroup algebra. There is an algebra isomorphism from this annulus algebra to the p2 dimensional Pauli algebra. Thisisomorphism is defined by

Mg,h 7→ X(q−r)gZh, (18)

where X and Z are Pauli matrices obeying ZX = ωXZ. The Pauli matrices span the full p× p matrix algebra. Upto isomorphism, there is a unique primitive idempotent of this algebra

P =

1 0 · · · 00 0 · · · 0...

.... . .

...0 0 · · · 0

(19)

6

=1

p

∑g

Zg. (20)

Finally, we use the algebra isomorphism Eqn. 18 to obtain the idempotent for the single defect

Fr

Fq

∣∣ =1

p

∑g

g

∗

∗

. (21)

Other idempotents can be found using the techniques outlined here. A full set of representative idempotents isprovided in Table III.

II. THE HORIZONTAL FUSION ALGORITHM

In this section, we explain the algorithm used to compute horizontal defect fusion. Due to the diagrammatic natureof the algorithm and the variety of phenomena that can occur during the computation, we are not going to give aformal specification, suitable for computer implementation. Instead, we shall explain how to proceed by hand in aspecific example. The procedure can vary between examples and we demonstrate this in Section III. This variabilityonly occurs after all the necessary information has been extracted from the correct tables, so for the purpose ofexplaining how to navigate the tables of data which appear in this paper, a specific example is easier to understandthan a general algorithm.

The algorithm proceeds in four key steps:

1. Determine idempotents corresponding to source defects using Table III.

2. Determine idempotent for target defect using Tables IV and III.

3. Inflate the target idempotent to a 4-string annulus using Tables V(a)-V(b).

4. Find a nonzero pants diagram that absorbs the source idempotents on the legs and (inflated) target on thewaist.

Example II.1 ( TXk

∣∣ ⊗ RFs

∣∣)We shall give the step by step procedure for the fusion

TXk

∣∣a⊗ R

Fs

∣∣z. (22)

1. Writing down the source defect idempotents

The first step in the procedure is to look up the idempotents representing the defects. These are found in Table III.For our current example, we have

TXk

∣∣a

=

0

(a, 0)

, RFs

∣∣z

=1

p

∑g

ωgz −g

∗

0

. (23)

7

2. Writing down the target defect idempotent

To decide the defect species resulting from the fusion, we need to look up the relevant domain wall fusions. Theseare contained in Table IV. For our current example, we have

T ⊗Vec(Z/pZ) R = p ·R (24)

Xk ⊗Vec(Z/pZ) Fs = Fk−1s. (25)

From Eqns. (24) and (25), we can read of the target defect up to the labels. In this case it is RFk−1s

∣∣ζ, where ζ is

yet to be determined. From Table III, we find that the defect idempotent is

RFk−1s

∣∣ζ

=1

p

∑γ

ωγζ −γ

∗

0

, (26)

At this stage of the computation, the multiplicities in the domain wall fusion don’t play a role. They show up in theinflation stage.

3. Inflating the target idempotent

In order for the target defect idempotent and the initial defect idempotents to interact with each other on a pair-of-pants, we need to inflate the target idempotent so that it has four vertical strings. The information required todo this is contained in Table V(a)-V(b). These tables contain the information required to explicitly decompose thetensor productsM⊗Vec(Z/pZ)N into simple bimodule categories. In our example, we need the entries correspondingto R→ T ⊗Vec(Z/pZ) R and Fk−1s → Xk ⊗Vec(Z/pZ) Fs. These entries tell how to replace the trivalent vertices in ourtarget idempotent, giving us

1

p

∑γ

ωγζ−γ

0 ∗

(0, v) 0

(27)

The ν appearing corresponds to the multiplicity in the tensor product T ⊗Vec(Z/pZ) R = p ·R.

4. Decorating the pants

At this point, we have extracted all the data we need from the tables. Now we need to find all the pants diagramswhich absorb our source defect idempotents on the legs and our target inflated defect idempotent at the waist. Themost general pair-of-pants which we need to consider looks like:

k0k1

k3 k4

k2

. (28)

8

We use the following equation to bring every horizontal pair-of-pants into the standard form

g0g1

g2 g3

h0

h1

h2h3

m n

p q

=ΩM (h−10 , g−11 )ΩQ(h1, h2)ΩN (h3, (g3h2)−1)ΩQ(h3, g3h2)

ΩP (h0, g1) h0g0g1h3

h−13 g2 g3h2

h1h3

m n

p q

, (29)

where ΩX(•, •) is the associator for the bimodule X[29]. Now, when we insert our source defect idempotents on thelegs and our inflated target idempotent on the belt, we get

1

p2

∑g,γ

ωgz+γζ+sγk3k0

k1

k3−γ − g + k4

k2

0 ∗

(0, ν) 0

(30)

up to a global phase. The term ωsγk3 appears because we need to use the middle associator on Fs to bring the diagraminto the standard form. We have omitted the labels on the leg holes to make the diagram less cluttered. They matchthe labels on the source defect idempotents. To make all the labels match up, we must have

k0 = −ak1 = k−1a

k2 = k−1a− νk3 = ν − k−1a

The transformation g → g+k4 only changes the expression by a phase. The whole expression is zero unless z = ζ+sk3.Rearranging this equation gives

ζ = z + s(k−1a− v). (31)

So we have

TXk

∣∣a⊗ R

Fs

∣∣z

= RFk−1s

∣∣νz+s(k−1a−ν) (32)

As explained above, the superscript ν indexes the multiplicity in the top domain wall fusion.

III. HORIZONTAL DEFECT FUSION

In this section, we present several horizontal defect fusion computations to elucidate some of the complications thatarise.

Example III.1 ( TT∣∣ ⊗ T

T

∣∣)Consider the fusion of two T -T defects T

T

∣∣(a,b)

and TT

∣∣(c,d)

. This case is interesting because there is multiplicity

in the domain wall fusion T ⊗Z/pZ T = p · T . The domain wall fusion and the defect fusion are correlated, thereforewe represent this defect fusion as a p× p-matrix indexed by the components in the decomposition T ⊗Z/pZ T = p · T :[

TT

∣∣(a,b)⊗ T

T

∣∣(c,d)

]µ,ν→ T

T

∣∣(α,β)

. (33)

9

We want to establish the possible values of α and β and the associated multiplicities. We do this by decomposing thetensor product into simple bimodule categories using T → T⊗Vec(Z/pZ). By fixing µ and ν, we can inflate T

T

∣∣(α,β)

to

a 4-string annulus

(0, 0)

(α, β)

.

(0, µ) (0, 0)

(α, ν) (0, β)

, (34)

Next, we can find a pant mapping the pair of 2-string defects to the 4-string defect. The most general form of thepants is

gh

k

l

(0, µ) (0, 0)

(α, ν) (0, β)

=g+l

h−l

k + l

(0, µ) (0, 0)

(α, ν) (0, β)

=µ

−c

(0, µ) (0, 0)

(α, ν = b+ c+ µ)(0, β)

. (35)

From this, we conclude that α = a, β = d and ν − µ ≡ b+ c mod p, which implies that[TT

∣∣(a,b)⊗ T

T

∣∣(c,d)

]µ,ν

= δb+cν−µ · TT∣∣(a,d)

. (36)

Example III.2 (Xk

Xk

∣∣ ⊗ Xl

Xl

∣∣, which includes fusing ‘anyons’)

The bimodule X1 corresponds to Vec(Z/pZ) as a self-bimodule. As such, X1

X1

∣∣ defects correspond to the excita-tions of the underlying topological phase. In the physics literature, these excitations are referred to as anyons.Consider the fusion

Xk

Xk

∣∣(a,x)

⊗ Xl

Xl

∣∣(b,y)→ Xkl

Xkl

∣∣(α,β)

. (37)

We can inflate Xkl

Xkl

∣∣(α,β)

to a 4-string annulus

1

p

∑g

ωgβ klg −g

0

c

. 1

p

∑g

ωgβ klg

lg

−g

−lg

0 0

α 0

. (38)

Next, we can find a pant mapping the 2-string defects to the 4-string defect. We immediately observe that (fornonzero maps) this forces α = a+ kb

1

p3

∑g,h,m

ωgx+hy+mβkg

-g

lh-h

klm

lm−b

-m−lm

0 0

β 0

=1

p3

∑g,h,m

ωgx+hy+mβk(g+lm)

-(g+lm)

l(h+m)-(h+m)

−b

0 0

β 0

. (39)

Making the replacements

g′ = g + lm, h′ = h+m, (40)

we obtain

1

p2

∑g,h

ωgx+hy+mβkg′

-g′lh′

-h′

−b

0 0

β 0

× 1

p

∑m

ωm(β−lx−z). (41)

10

The final sum is zero unless the exponent is 0. Therefore, β = lx+ z. The result of the fusion is

Xk

Xk

∣∣(a,x)

⊗ Xl

Xl

∣∣(b,y)

= Xkl

Xkm

∣∣(a+kb,z+lx)

. (42)

For the special case k = l = 1, we can identify this with the known anyon fusion rule

maex ×mbey = ma+bex+y. (43)

Example III.3 (Xl

Xk

∣∣ ⊗ LL

∣∣)Consider the defect fusion

Xl

Xk

∣∣ ⊗ LL

∣∣(c,z)→ L

L

∣∣(α,ζ)

(44)

First, we inflate LL

∣∣(α,ζ)

to a 4-string idempotent:

1

p

∑h

ωhζ h

0

α

. 1

p

∑h

ωhζ h

l−1h

−k−1h

0 0

0 α

. (45)

In order to find a pair-of-pants absorbing the inflated idempotent at the waist, and Xl

Xk

∣∣ and LL

∣∣(c,z)

on the legs

respectively, we must have α = c. The general pair-of-pants looks like

1

p2

∑g,h

ωgz+hζ

−k(k1 − k−1h)

k1 − k−1h

k3 + g + k−1h

(1− l−1k)(k1 − k−1h)

0 0

0 c

=ωkk1ζ−k1z−k3z

p2

∑g,h

ωgz+hkζkh

−h

g + h

(l−1k − 1)h

0 0

0 c

(46)

which is non-zero for all choices of ζ. Therefore

Xl

Xk

∣∣ ⊗ LL

∣∣(c,z)

= ⊕ζ LL∣∣(c,ζ)

(47)

Example III.4 (Xl

F0

∣∣ ⊗ LL

∣∣)Consider the defect fusion

Xl

F0

∣∣x⊗ L

L

∣∣(c,z)→ L

L

∣∣(α,ζ)

(48)

Inflating LL

∣∣(α,ζ)

to a 4-string idempotent gives

1

p

∑γ

ωγζ γ

0

α

. 1

p2

∑γ,δ

ωδµ+γζ γ

l−1γ

δ

∗ 0

0 α

. (49)

11

The general pair-of-pants absorbing Xl

F0

∣∣x

and LL

∣∣(c,z)

on the legs and the inflated idempotent on the belt is

1

p4

∑g,h,γ,δ

ωgx+hz+δµ+γζγ + g + k0

k1 − l−1g

h

l−1γ + k2

δ

∗ 0

0 α

=1

p4

∑g,h

ωgx+hz+δµ+γζγ + g + k0

k1 − l−1g + δ

h− δ

l−1γ + k2 + δ

∗ 0

0 c

(50)

In order for the objects to match on the green string, we must have k0 = l(k2 − k1). This equation together with thetransformations γ → γ − lk2 and g → g + lk1 let us phase away k0, k1 and k2 giving the following pants:

1

p4

∑g,h,γ,δ

ωgx+hz+δµ+γζγ + g

−l−1g + δ

h− δ

l−1γ + δ

∗ 0

0 c

(51)

If we define

b = l−1γ + δ (52)

c = −l−1g + δ (53)

d = h− δ (54)

then (g, h, γ, δ)→ (b, c, d, γ) is invertible over Z/pZ and we get the following pants:

1

p3

∑b,c,d

ω(lx+z+µ)b−lcx+dzl(b− c)

c

d

b

∗ 0

0 c

× 1

p

∑γ

ω(ζ−x−l−1z−l−1µ)γ (55)

which is nonzero only if ζ = x+ l−1(µ+ z). Therefore we have

Xl

Xk

∣∣ ⊗ LL

∣∣(c,z)→ L

L

∣∣(c,x+l−1(µ+z))

(56)

Example III.5 (Xl

Xk

∣∣ ⊗ Xn

Xm

∣∣)Consider the defect fusion

Xl

Xk

∣∣ ⊗ Xn

Xm

∣∣ → Xln

Xkm

∣∣ km 6= lnXkm

Xkm

∣∣(α,β)

km = ln. (57)

where we assume k 6= l and m 6= n. We will present the calculation for these two cases separately.

12

Case I: km = ln

Inflating Xkm

Xkm

∣∣(α,β)

to a 4-string idempotent gives

1

p

∑γ

ωγβ kmγ −γ

0

α

. 1

p

∑γ

ωγβ kmγ

nγ

−γ

−mγ

0 0

α 0

. (58)

The general pair-of-pants absorbing Xl

Xk

∣∣ and Xn

Xm

∣∣ on the legs and the inflated idempotent on the belt is

1

p

∑γ

ωγβkx1

−x1

mx2−x2

kmγ

nγ + q

−γ−mγ

0 0

α 0

=1

p

∑γ

ωγβk(mγ + x1)

−(mγ + x1)

m(γ + x2)−(γ + x2)

(n−m)γ + q

0 0

α 0

, (59)

where q = (l−1k − 1)x1 − l−1α, x2 = −q(m− n)−1. Making the change of variables γ′ = mγ + x1 eliminate x1 up toa global phase. Therefore, there is a single, nonzero map

Xl

Xk

∣∣ ⊗ Xn

Xm

∣∣ → Xkm

Xkm

∣∣(α,β)

(60)

for all α, β.

Case II: km 6= ln

Inflating Xln

Xkm

∣∣ to a 4-string idempotent gives

0

0

.

0 0

0 0

. (61)

The general pair-of-pants absorbing Xl

Xk

∣∣ and Xn

Xm

∣∣ on the legs and the inflated idempotent on the belt is

kx1

−x1

mx2−x2

(1− kl−1)x1

0 0

0 0

, (62)

where x2 = x1(kl−1−1)(m−n)−1. where q = (l−1k−1)x1−l−1α, x2 = −q(m−n)−1. There are no further constraintson the maps. Therefore, there are p distinct maps between these defects

Xl

Xk

∣∣ ⊗ Xn

Xm

∣∣ = p · Xln

Xkm

∣∣. (63)

13

Example III.6 (Xl

Fq

∣∣ ⊗ Xn

Fs

∣∣, which includes fusing ‘twists’)

A particular set of defects of the Z/2Z Kitaev model identified in Ref. [7], where they were referred to as twists. Inour notation, twists occur at the interface of an X1 and F1 domain wall. In this example, we compute the fusion ofgeneralizations of twists in the Z/pZ Kitaev model.

Consider the defect fusion

Xl

Fq

∣∣x⊗ Xn

Fs

∣∣z→Xln

Xq−1s

∣∣ q−1s 6= lnXln

Xln

∣∣(α,β)

q−1s = ln. (64)

Case I: q−1s 6= ln

When q−1s 6= ln, the fusion looks like

Xl

Fq

∣∣x⊗ Xn

Fs

∣∣z→ Xln

Xq−1s

∣∣, (65)

but we need to check for multiplicity. We can inflateXln

Xq−1s

∣∣ to a 4-string annulus

0

0

. 1

p

∑γ γ

∗ ∗

0 0

. (66)

This implies that the general pair-of-pants absorbing the 2-string defects on the legs and the 4-string defect on thewaist is

1

p3

∑g,h,γ

Θx,−ql(g)Θz,−ns(h)ω−qk0g−sk3hlg + k0

−g + k1

nh+ k3 −h+ k4

k2

γ

∗ ∗

0 0

(67)

=1

p3

∑g,h,γ

Θx,−ql(g)Θz,−ns(h)ω−qk0g−sk3h+sγ(h−k4)lg + k0

γ − g + k1

nh + k3−γ −h+ k4

γ + k2

∗ ∗

0 0

. (68)

For the labels to match, we must have

k0 + lk1 − lk2 = 0 (69)

k3 + nk4 + k2 = 0. (70)

If we make the transformation

g = a− l−1k0 (71)

γ = b− k2 (72)

h = c+ k4 (73)

then we get

1

p3

∑a,b,c

Θx,−ql(a)Θz,−ns(c)ωsbc

la

b− a

c− b −c

b

∗ ∗

0 0

. (74)

14

So there is 1 map

Xl

Fq

∣∣x⊗ Xn

Fs

∣∣z→ Xln

Xq−1s

∣∣. (75)

Case II: q−1s = ln

When q−1s = ln, the fusion looks like

Xl

Fq

∣∣x⊗ Xn

Fs

∣∣z→ Xln

Xln

∣∣(α,ζ)

. (76)

We can inflate Xln

Xln

∣∣(α,ζ)

to a 4-string annulus

1

p

∑γ

ωγζ lnγ −γ

0

α

. 1

p2

∑γ,δ

ωγζ lnγ

nγ

−γ

δ

∗ ∗

α 0

. (77)

Next, we can find the pant that absorbs the 2-string idempotents on the legs and 4-string idempotent on the waist.The most general pant that will absorb the appropriate idempotents is

1

p4

∑g,h,γ,δ

Θx,−ql(g)Θz,−ns(h)ωγζ−hk3s−gk0qlg + k0

−g + k1

nh + k3

−h + k4

lnγ

nγ + k2

−γδ

∗ ∗

α 0

(78)

=1

p4

∑g,h,γ,δ

Θx,−ql(g)Θz,−ns(h)ωγζ−hk3s−gk0q+lnγ(−g+k1)q+δ(γ+h−k4)slg + lnγ + k0

δ − g + k1

−δ + nh + k3

−h − γ + k4

nγ + k2 + δ

∗ ∗

α 0

. (79)

For the labels to match up

k0 + lk1 − lk2 − α = 0 (80)

k3 + nk4 + k2 = 0. (81)

If we make the change of coordinates

a = δ − g (82)

b = nγ + δ (83)

c = −h− γ (84)

then we get

1

p4

∑a,b,c,γ

ωexponent

l(b− a) + k0

a+ k1

−nc− b+ k3c+ k4

b+ k2

∗ ∗

α 0

(85)

15

where

exponent =− 2−1a2lq + k0q(a− b+ γn) + ablq − ax− 2−1b2lq+

lnq(k4(γn− b) + k3(c+ γ) + γk1)− bclnq + bx− 2−1c2ln2q + γζ − cz − γnx− γz

Making the transformation

a→ a− k1 (86)

b→ b− k2 (87)

c→ c− k4 (88)

gives

1

p4

∑a,b,c,γ

ωexponent

l(b− a) + α

a

−nc− bc

b

∗ ∗

α 0

(89)

where

exponent =− 2−1a2lq + ablq + αaq − ax− 2−1b2lq − bclnq−αbq + bx− 2−1c2ln2q − cz + γ(ζ + αnq − nx− z)

This implies that ζ = nx+ z − αnq, so

Xl

Fq

∣∣x⊗ Xn

Fs

∣∣z→ ⊕α Xln

Xln

∣∣(α,z+nx−nqα) (90)

for odd primes. The same calculation for p = 2 yields

X1

F1

∣∣x⊗ X1

F1

∣∣z→ ⊕α X1

X1

∣∣(α,z+x+α)

. (91)

This result was presented in Ref. [7] as σxσz =∑j ej+x+zmj .

Example III.7 (Xl

Fq

∣∣ ⊗ Xm

Xm

∣∣)Xl

Fq

∣∣x⊗ Xm

Xm

∣∣(c,z)→ Xlm

Fqm

∣∣α

(92)

We can inflate Xlm

Fqm

∣∣α

to a 4-string annulus

1

p

∑g

Θα,−qlm2(g) lmg −g

∗

0

. 1

p

∑g

Θα,−qlm2(g) lmg

mg

−g

−mg

∗ 0

0 0

. (93)

Next, we can find a pant mapping the 2-string defects to the 4-string defect

1

p3

∑g,h,k

Θx,−ql(g)Θα,−qlm2(k)ωhz−qlµgl(g+µ)

-(g+µ+c)

mh−h

lmk

mk − c

−k−mk

∗ 0

0 0

(94)

16

=1

p3

∑g,h,k

Θx,−ql(g)Θα,−qlm2(k)ωhz−qlµg−qlmk(g+µ+c)l(g+µ+mk)

-(g+µ+mk+c)

m(h+k)-(h+k)

−c

∗ 0

0 0

(95)

∝ 1

p2

∑g,h

Θx,−ql(g)ωhzlg

-(g + c)

mh-h

−c

∗ 0

0 0

× 1

p

∑k

ωk(α−z−m(x+qlc)). (96)

Therefore

Xl

Fq

∣∣x⊗ Xm

Xm

∣∣(c,z)

= Xlm

Fqm

∣∣z+m(x+qlc)

. (97)

For the special case q = l = m = 1, this recovers the known fusion rule of ‘twists’ with anyons[7, 25, 27]

σx ×mcez = σx+z+c. (98)

IV. THE VERTICAL FUSION ALGORITHM

In this section, we explain the vertical defect fusion algorithm. The structure of this section parallels Section II.Since there is no domain wall fusion, vertical fusion is simpler than horizontal fusion. No inflation is required, so thealgorithm proceeds in three key steps:

1. Determine idempotents corresponding to source defects using Table III.

2. Determine idempotent for target defect using Tables IV and III.

3. Find a nonzero pants diagram that absorbs the source idempotents on the legs and (inflated) target on thewaist.

We shall explain how the algorithm works, step by step, using an example.

Example IV.1 (RR∣∣(a,x)

Fs

R

∣∣z)

Consider the vertical defect fusion

RR

∣∣(a,x)

Fs

R

∣∣z. (99)

The convention for the operator is that the left argument is the bottom defect and the right argument is the topdefect.

1. Writing down the defect idempotents to compose

The first step in the procedure is to look up the idempotents that we are composing. These are found in Table III.In our current example, we have

Fr

R

∣∣z

=1

p

∑h

ωhz −h

0

∗

, RR

∣∣(a,x)

=1

p

∑g

ωgx −g

0

a

. (100)

17

2. Writing down the target idempotent

Next, we need to write down the target defect idempotent. These can be found in Table III.

Fr

R

∣∣ζ

=1

p

∑γ

ωγζ −γ

0

∗

. (101)

3. Decorating the pants

At this point we have extracted all the data we need from the tables. Now we need to find all the pants diagramswhich absorb our source defect idempotents on the legs and our target defect idempotent on the waist. In this case,the, the pants are oriented perpendicular to the horizontal case. The most general pair of pants for any 3 domainwalls M,N ,P is

k2 k3

k0 k1

m

n

p

. (102)

We use the following equation to bring every vertical pair of pants into a standard form

g0 g1g2 g3h0 h1

m

n

p

=ΩM (h−10 , g−11 )ΩN (h−10 , g1)

ΩP (h0, g3)ΩN (h0, g−13 )

h0g0 g1h1

h0g2 g3h1

m

n

p

. (103)

When we insert our source defect idempotents on the legs and our target defect idempotent on the waist, we get

1

p3

∑g,h,γ

ωhz+gx+γζ+sk0h

k2 k3−g−γ

k0 k1−h−γ

0

0

∗

a

. (104)

The variables k1 and k2 can be converted into a global phase by the translations h → h + k1, g → g + k3. For theobject labels to match up, we need k0 = a and k2 = 0. Now, if we make the change of variables,

c = g + γ (105)

b = h+ γ (106)

18

we get

1

p3

∑c,b,γ

ω(b−γ)z+(c−γ)x+γζ+sa(b−γ)

−c

a −b

0

0

∗

. (107)

For this to be non zero, we need to have ζ = x+ z + sa, so

RR

∣∣(a,x)

Fs

R

∣∣z

= Fs

R

∣∣x+z+sa

(108)

V. VERTICAL DEFECT FUSION

In this section, we present several of the more complicated vertical defect fusion computations.

Example V.1 (Xl

Fq

∣∣ Fs

Xl

∣∣)Consider the defect fusion

Xl

Fq

∣∣ Fs

Xl

∣∣ → Fs

Fs

∣∣(α,β)

q = sFs

Fq

∣∣ q 6= s. (109)

Case I: q = s

When q = s, the vertical fusion looks likeXl

Fq

∣∣x Fs

Xl

∣∣z→ Fs

Fs

∣∣(ζ,η)

(110)

The general pants absorbing Fs

Xl

∣∣z

and Xl

Fq

∣∣x

on the legs and Fs

Fs

∣∣(ζ,η)

on the waist looks like

1

p4

∑g,h,γ,δ

ωγζ+δη+sγ(h−g)Θz,ls(h)Θx,−ql(g)

lg+γ −g−δ

lh+γ −h−δ

∗

0

∗

(111)

If we set

a = −g − δ (112)

b = −h− δ (113)

c = lh+ γ (114)

then this pair of pants becomes

1

p4

∑a,b,c,δ

ωexponent

c+l(b−a) a

c b

∗

0

∗

(115)

19

where

exponent = −2−1a2ls+ a(bls+ cs− x)− 2−1b2ls− b(cs− ζl + z) + cζ + δ(η + ζl − x− z). (116)

This implies η = x+ z − lζ, so

Xl

Fq

∣∣x Fs

Xl

∣∣z→ ⊕ζ Fs

Fs

∣∣(ζ,x+z−lζ) (117)

Case II: q 6= s

When q = s, the vertical fusion looks like

Xl

Fq

∣∣x Fs

Xl

∣∣z→ Fs

Fq

∣∣ (118)

All the variables in general pants transform into global phases, so there is no multiplicity.

Example V.2 (Xl

Xk

∣∣ Xm

Xl

∣∣)Consider the defect fusion

Xl

Xk

∣∣ Xm

Xl

∣∣ → Xk

Xk

∣∣(α,β)

k = mXm

Xk

∣∣ k 6= m. (119)

Case I: k = m

The general pants absorbing Xm

Xl

∣∣ and Xl

Xk

∣∣ on the legs and Xk

Xk

∣∣(α,ζ)

on the waist is

1

p

∑γ

ωγζ

kγ+k2 k3−γ

kγ+k0 k1−γ

0

0

α

(120)

where

k0 + kk1 = α (121)

k0 + lk1 − k2 − lk3 = 0 (122)

k2 + kk3 = 0. (123)

The solution to these equations is

k1 = k−1(α− k0)

k2 = lα(k − l)−1 + k0

k3 = −lα(k − l)−1k−1 − k0k−1

so the substitution γ → γ − k−1t transforms k0 into a global phase. Therefore

Xl

Xk

∣∣ Xk

Xl

∣∣ = ⊕α,ζ Xk

Xk

∣∣(α,ζ)

(124)

20

Bimodule label Domain wall Action on particles

T Condenses e on both sides

L Condenses m on left and e on right

R Condenses e on left and m on right

F0 Condenses m on both sides

Xk Xk : eamb 7→ ekamk−1b (moving left to right), where k−1 is taken multiplicatively modulo p

Fq = F1Xq F1 : eamb 7→ ebma

TABLE I. Domain walls on the lattice corresponding to bimodules. Reproduced from Ref. [29].

Case II: k 6= m

The general pant absorbing Xm

Xl

∣∣ and Xl

Xk

∣∣ on the legs and Xm

Xk

∣∣ on the waist is

k2 k3

k0 k1

0

0

0

(125)

where

k0 +mk1 = 0 (126)

k0 + lk1 − k2 − lk3 = 0 (127)

k2 + kk3 = 0. (128)

This system has rank 3, which implies

Xl

Xk

∣∣ Xm

Xl

∣∣ = p · Xm

Xk

∣∣ (129)

VI. PHYSICAL INTERPRETATION OF DEFECTS

Associated to a fusion category C is a topological phase described by its Drinfeld center Z(C). The Levin-Wenprocedure[32] can be used to construct a lattice model which realizes the topological phase in its low energy space.In Ref. [29], we showed how the bimodule categories for Vec(Z/pZ) can be interpreted in terms of boundaries anddomain walls in the physical theory. This data is reproduced in Table I for completeness.

In this section, we discuss a physical interpretation for all defects in Table III. We will also show how the fusionrules in Tables VI(a)-VI(f) and VII can be obtained from the physical theory up to multiplicity. The multiplicitiesremain mysterious from the physical perspective, but they can be computed from the Frobenius-Perron dimensionsof the defects.

The simplest defect to interpret is X1

X1

∣∣(a,x)

. By studying Table IV we observe that X1 is the ‘identity’ domain wall,

corresponding to Vec(Z/pZ) as a bimodule over itself. The only defects of such a domain wall are the anyons of thetheory. Therefore

X1

X1

∣∣(a,x)

7→ exma, (130)

where the p2 particles of the (domain wall free) theory associated to Vec(Z/pZ) are usually denoted exma, witha, x ∈ Z/pZ[2].

21

For other defects, the physical interpretation is found by studying how anyons can be introduced to obtain distinctstates. For example, consider the defects Xl

Xk

∣∣·. If an anyon ma is introduced on the left of the wall and ex is introduced

on the right, they can be split and passed through the upper and lower walls to find the equivalent states

Xk

Xl

exma=

Xk

Xl

mα

ma−α

eβ

ex−β=

Xk

Xl

ma+(kl−1−1)α ex+(kl−1−1)β. (131)

If k = l, then all such states are distinct, so there are p2 defects, corresponding to a basic defect with anyons pushedonto it. For k 6= l, we are free to choose α and β in Eqn. 131, so all states are equivalent, and there is a unique defect.

A very similar computation can be performed for other domain wall interfaces. In the case of Xl

Fq

∣∣·, the computation

is

Fq

Xl

exma=

Fq

Xl

ex+qa, (132)

with no further equivalence possible. Therefore, there are p distinct states corresponding to a base defect with anumber of e particles absorbed from the right. We remark that, just as in the choice of idempotents, there is a choiceof labeling. We could have instead chosen to label by some combination of e and m on the left and right. Thiscorresponds to permuting the labels.

When there is a boundary present, particles can be discarded (condensed) into the boundary. In the case of a TL

∣∣·

defect, we have

ma mb

ex ey

=

mb

, (133)

so there are p distinct defects, corresponding to pinning a number of m defects to the right hand boundary.

The full set of defect interpretations are listed in Table II. We remark that in the case Z/2Z, many of the defectsstudied here are the subject of previous work, for example

F1

X1

∣∣x, X1

F1

∣∣x7→ σx (134)

was referred to as a ‘twist’ in Ref. [7], and defects involving rough/smooth interfaces in Table II were named ‘corners’in Ref. [12].

A. Fusing defects: Horizontal

The physical interpretations from Table II can be used to compute the fusion rules. In this section, we will illustratehow this is done using a few examples.

Example VI.1 (Xk

Xk

∣∣ ⊗ Xl

Xl

∣∣)Consider the fusion Xk

Xk

∣∣(a,x)

⊗ Xl

Xl

∣∣(c,z)

. In the physical theory, we begin by clearing any anyons occurring on

22

T L R F0 Xl Fr

T TT

∣∣(a,b)

= ma mb LT

∣∣a

= ma RT

∣∣a

= ma F0T

∣∣ = XlT

∣∣a

= maFrT

∣∣ =

L TL

∣∣a

= ma LL

∣∣(a,x)

= ex ma RL

∣∣ = F0L

∣∣x

= ex XlL

∣∣= FrL

∣∣x

= ex

R TR

∣∣a

= ma LR

∣∣ = RR

∣∣(a,x)

= ma ex F0R

∣∣x

= ex XlR

∣∣ = FrR

∣∣x

= ex

F0TF0

∣∣ = LF0

∣∣=(x,)

ex RF0

∣∣x

= ex F0F0

∣∣(x,y)

= ex ey XlF0

∣∣x

= exFrF0

∣∣ =

XkTXk

∣∣a

=ma

LXk

∣∣ = RXk

∣∣ = F0Xk

∣∣x

=ex

XkXk

∣∣(a,x)

= ma ex

XlXk

∣∣ =Xk

Xl

FrXk

∣∣x

= ex

FqTFq

∣∣ = LFq

∣∣x

=ex R

Fq

∣∣x

=ex F0

Fq

∣∣ =XlFq

∣∣x

= ex

Fq

Fq

∣∣(x,y)

= ex ey

FrFq

∣∣ =Fq

Fr

TABLE II. Physical interpretation of all defects.

the central region

Xk

Xk

Xl

Xl

exma ez

mc =

Xk

Xk

Xl

Xl

ma+kc ez+lx=

Xkl

Xkl

ma+kc ez+lx. (135)

This recovers the fusion computed using the annular algebra.

Example VI.2 (Xl

Xk

∣∣ ⊗ Xn

Xm

∣∣)A more interesting computation arises when we allow the upper and lower domain walls to differ

Xk

Xl

Xm

Xn

=

Xk

Xl

Xm

Xn

eβ

e−β

mα

m−α

=

Xk

Xl

Xm

Xn

eβ(n−m)mα(l−k)=

Xkm

Xln

eβ(n−m)mα(l−k). (136)

If km 6= ln, then from Eqn. 131, this is equivalent to the base defect for all choices α, β. In the case where km = ln,these states are distinct for each α, β, giving the fusion rule

Xl

Xk

∣∣ ⊗ Xn

Xm

∣∣ =

p · Xln

Xkm

∣∣ km 6= ln

⊕α,β Xkm

Xkm

∣∣(α,β)

km = ln. (137)

23

The coefficient p is not obtained from this computation, but can be calculated using the Frobenius-Perron dimensionscomputed in Section VI C.

Example VI.3 ( Fr

Xk

∣∣ ⊗ Ft

Xm

∣∣)Consider the fusion process

Fr

Xk

∣∣x⊗ Ft

Xm

∣∣z. (138)

In the physical theory, this is computed using

Xk

Fr

Xm

Ft

ex ez =

Xk

Fr

Xm

Ft

eβ

ex−β

m−α

mα

ez =

Xk

Fr

Xm

Ft

ez−tα+m(x−β)mkα+r−1β=

Xkm

Xr−1t

eβ′

mα′, (139)

where α′ = kα + r−1β and β′ = z + mx −mrα′ + (mrk − t)α. If km 6= r−1t, these states are all equivalent to thebase defect following Eqn. 131. In the case km = r−1t, we have

Xkm

Xr−1t

ez+mx−mrαmα, (140)

for any α. The fusion rule is therefore

Fr

Xk

∣∣x⊗ Ft

Xm

∣∣z

=

Xr−1t

Xkm

∣∣ km 6= r−1t

⊕α Xkm

Xkm

∣∣(α,z+m(x−rα)) km = r−1t

, (141)

where the Frobenius-Perron dimensions (Section VI C) can be used to check that no multiplicity is required.

Example VI.4 ( TT∣∣ ⊗ T

T

∣∣)Multiplicity in the domain wall fusion (Table IV) has an effect on the defect fusion. For the fusion

TT

∣∣(a,b)⊗ T

T

∣∣(c,d)

, (142)

the physical pre-fusion state is

mb+c

ma mb mc md . (143)

The central strip supports a p dimensional vector space. The qudit state can be read out by exchanging an e particlebetween the boundaries. The state is changed by inserting an m line vertically. Suppose the strip is in the state mµ.To perform the fusion, we push the inner m particles away from the fusion region

mb+c

ma md . (144)

24

After fusing the central strip, there are still p states. These are now understood as a subspace of a 2 qudit system, withone supported on each of the upper and lower regions. The subspace is spanned by states of the form |mµ〉⊗|mµ+b+c〉.The fusion outcome is therefore [

TT

∣∣(a,b)⊗ T

T

∣∣(c,d)

]µ,ν

= δb+cν−µ · TT∣∣(a,d)

. (145)

Example VI.5 ( TL∣∣ ⊗ T

L

∣∣)As a final example, we show how the fusion

TL

∣∣a⊗ T

L

∣∣b

(146)

can be computed.Physically, this process is represented by

ma mb. (147)

The central strip supports a p dimensional vector space. The qudit state can be read out by exchanging an e particlebetween the boundaries in the upper region. The state is changed by inserting an m line vertically. To perform thefusion, we must push the m particles away from the fusion region

ma

mb . (148)

After fusing the central strip, there are still p states, fully supported on the upper region. The fusion outcome istherefore [

TL

∣∣a× T

L

∣∣b

]µ

= TL

∣∣b

(149)

B. Fusing defects: Vertical

Since the vertical fusions are much simpler than the horizontal ones, we will provide a single example to illustratehow the physical interpretation can be used for the fusion calculation.

Example VI.6 ( Fr

L

∣∣ LFr

∣∣)Consider the fusion

Fr

L

∣∣x LFr

∣∣z. (150)

Physically, the fusion is

exez

=ex+z−rα

erα=

ex+z−rα mα = ex+z−rα mα. (151)

Since any α could have been chosen for this calculation, the fusion rule is

Fr

L

∣∣x LFr

∣∣z

= ⊕α LL∣∣(α,x+z−rα). (152)

25

C. Frobenius-Perron Dimension

The Frobenius-Perron dimension (FPd) of the defects can be computed using the fusion table. For defects a, b andc, the FPd obeys

dadb =∑c

N ca,bdc, (153)

where N ca,b is the multiplicity of the fusion. The FPd of the defects of Vec(Z/pZ) do not depend on the defect label,

only on the domain walls involved. The FPds are

dim(Xl

Xk

∣∣•)

= dim(Fl

Fk

∣∣•)

=

1 k = l

p k 6= l, (154)

dim(Fr

Xk

∣∣•)

= dim(Xk

Fr

∣∣•)

=√p, (155)

dim(W2

W1

∣∣•)

= 0 for all other defects. (156)

VII. NATURAL TRANSFORMATIONS

As explained in Ref. [19], fusion category bimodules correspond to domain walls, and bimodule functors correspondto defects. On the mathematics side, we also have natural transformations. Using the diagrammatic framework fromthis paper, these natural transformations are easy to compute: they are just morphisms in the Karoubi envelope ofthe annular category AnnM,N . Since the annular category is semi-simple, there are no morphisms between distinctobjects in the Karoubi envelope and the endomorphism algebra of any simple object is just C. Interesting naturaltransformations show up when we consider fusion. For example, in the vertical fusion

LR

∣∣ Xm

L

∣∣ = p · Xm

R

∣∣ (157)

There are p distinct pants diagrams

k

−mk k

0

0

(158)

parameterized by k, which absorb LR

∣∣, Xm

L

∣∣ on the legs and Xm

R

∣∣ on the waist. These pants diagrams represent differentnatural transformations. For every defect fusion in this paper, the pants diagram which is computed represents anatural transformation.

A reasonable physical interpretation is that natural transformations capture certain aspects of the renormalizationprocess. When computing defect fusion, we are witnessing an isomorphism between the horizontal or vertical con-catenation of the defects and another defect. Physically, this corresponds to bringing the defects close together andthen locally renormalizing, or zooming out.

VIII. CONCLUSIONS

In this work, we have studied binary interface defects and their fusion. Using string diagrams and the annularcategory, we have classified the full set of defects occurring interfacing a pair of (not necessarily invertible) domainwalls for the tensor category Vec(Z/pZ). Further, we have provided algorithms for computing both horizontal (tensorproduct) and vertical (composition) fusion of arbitrary pairs of defects. For the theory Vec(Z/pZ), we have providedcomplete fusion tables.

We have specialized to Vec(Z/pZ) for simplicity. The framework outlined here is not restricted to this class offusion categories. Of particular interest is the Color code[6, 10] (Vec(Z/2Z× Z/2Z)) due to its importance in quantum

26

computation. Defects between invertible domain walls and the trivial wall (X1 in this paper) were studied in Ref. [18],but the full theory is currently open. Additionally, the tools presented here are expected to be useful for studyingnonabelian theories[15]. Additionally, one could study domain walls and defects between distinct phases, such as theColor code and Vec(Z/4Z), which may prove useful for quantum computing tasks. Although we restrict to binaryinterface defects, generalizations of the techniques developed here can be applied to higher defects such as thoseoccurring at the interface of three or more domain walls. Such defects allow the meeting of many distinct topologicalphases.

In the physics literature, defect fusion is often synonymous with symmetry gauging[17, 25, 41–43]. In this work, wehave computed the fusions without consideration of gauging. It would be extremely useful if the techniques developedin this work can say something about (obstructions to) gauging invertible defects.

ACKNOWLEDGMENTS

This work is supported by the Australian Research Council (ARC) via Discovery Project “Subfactors and sym-metries” DP140100732 and Discovery Project “Low dimensional categories” DP16010347. Support is also providedby the US Army Research Office for Basic Scientific Research via Grant W911NF-17-1-0401. We thank ChristopherChubb for feedback on the draft manuscript.

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28

Appendix A: Defect idempotents

TL

RF0

Xl

Fr

TT T

∣ ∣ (a,b)

=

(0,0

)

(a,b)

L T

∣ ∣ a=(0,0

)

a

R T

∣ ∣ a=(0,0

)

a

F0T

∣ ∣ =(0,0

)

∗

Xl

T

∣ ∣ a=(0,0

)

a

FrT

∣ ∣ =(0,0

)

∗

LT L

∣ ∣ a=0

(0,a

)

L L

∣ ∣ (a,x

)=

1 p

∑ gωgx

g

0a

R L

∣ ∣ =00

F0L

∣ ∣ x=1 p

∑ gωgx

g

0∗

Xl

L

∣ ∣ =00

FrL

∣ ∣ x=1 p

∑ gωgx

g

0∗

RT R

∣ ∣ a=0

(a,0

)

L R

∣ ∣ =00

R R

∣ ∣ (a,x

)=

1 p

∑ gωgx

−g

0a

F0R

∣ ∣ x=1 p

∑ gωgx

−g

0∗

Xl

R

∣ ∣ =00

FrR

∣ ∣ x=1 p

∑ gωgx

−g

0∗

F0

T F0

∣ ∣ =∗

(0,0

)

L F0

∣ ∣ x=1 p

∑ gωgx

g

∗0

R F0

∣ ∣ x=1 p

∑ gωgx

−g

∗0

F0F0

∣ ∣ (x,y

)=

1 p2

∑ g,hωgx+hy

g−h

∗∗

Xl

F0

∣ ∣ x=1 p

∑ gωgx

g−l−

1g

∗0

FrF0

∣ ∣ =1 p

∑ g−g

∗∗

Xk

T Xk

∣ ∣ a=0

(a,0

)

L Xk

∣ ∣ =00

R Xk

∣ ∣ =00

F0X

k

∣ ∣ x=1 p

∑ gωgx

g−k−1g

0∗

Xk

Xk

∣ ∣ (a,x

)=

1 p

∑ gωgx

kg

−g

0a

Xl

Xk

∣ ∣ =00

FrX

k

∣ ∣ x=1 p

∑ gΘx,kr(g

)kg

−g

0∗

Fq

T Fq

∣ ∣ =∗

(0,0

)

L Fq

∣ ∣ x=1 p

∑ gωgx

g

∗0

R Fq

∣ ∣ x=1 p

∑ gωgx

−g

∗0

F0Fq

∣ ∣ =1 p

∑ g−g

∗∗

Xl

Fq

∣ ∣ x=1 p

∑ gΘx,−ql(g)

lg−g

∗0

Fq

Fq

∣ ∣ (x,y

)=

1 p2

∑ g,hωgx+hy

g−h

∗∗

FrFq

∣ ∣ =1 p

∑ g−g

∗∗

TABLE III. Indecomposable idempotents for 2-string annuli of all domain walls, corresponding to defects. For p = 2, Θx,a(g) =

(−1)gxiag, whilst for odd p Θx,a(g) = ωgx+ag22−1

, where 2−1 is the modular inverse of 2.

29

Appendix B: Inflations

When performing the horizontal fusion algorithm, the target idempotent needs to be inflated from a 2-string annulusonto a 4-string annulus. The tables in this section contain the data needed for this process. The procedure used tocompute this data is explained in Ref. [29]. For completeness, Table IV contains the domain wall fusion data fromRef. [29].

⊗Vec(Z/pZ) T L R F0 Xl FrT p · T T p ·R R T RL p · L L p · F0 F0 L F0

R T p · T R p ·R R TF0 L p · L F0 p · F0 F0 L

Xk T L R F0 Xkl Fk−1r

Fq L T F0 R Fql Xq−1r

TABLE IV. Multiplication table for BPR(Vec(Z/pZ)), reproduced from Ref. [29].

T ⊗Vec(Z/pZ) T

g h

(a, b)

g h

(a, µ) (0, b)

g h

(a, b)

g h

(a, ν) (0, b)

T ⊗Vec(Z/pZ) L

g h

(a, 0) b g h

(a, 0) b

T ⊗Vec(Z/pZ) Xl

g h

(a, 0) lb g h

(a, 0) lb

R⊗Vec(Z/pZ) T

g h

a (0, b) g h

a (0, b)

R⊗Vec(Z/pZ) L1p

∑k ω

µk

g

k

h

a b

1p

∑k ω

νk

g

k

h

a b

R⊗Vec(Z/pZ) Fr1p

∑k ω

kbr

g

k

h

a ∗

1p

∑k ω

kr(b−h)g

k

h

a ∗

Xk ⊗Vec(Z/pZ) T

g h

a (0, b) g h

a (0, b)

T

Fq ⊗Vec(Z/pZ) L1p

∑k ω

qak

g

k

h

∗ b

1p

∑k ω

kqa

g

k

h

∗ b

L⊗Vec(Z/pZ) T

g h

a

g h

µ (0, a)

g h

a

g h

ν (0, a)

L⊗Vec(Z/pZ) L

g h

0 a g h

0 a

L⊗Vec(Z/pZ) Xl

g h

0 la g h

0 la

F0 ⊗Vec(Z/pZ) T

g h

∗ (0, a) g h

∗ (0, a)

F0 ⊗Vec(Z/pZ) L1p

∑k ω

kµ

g

k

h

∗ a

1p

∑k ω

kν

g

k

h

∗ a

F0 ⊗Vec(Z/pZ) Fr1p

∑k ω

kar

g

k

h

∗ ∗

1p

∑k ω

k(a−h)rg

k

h

∗ ∗

Xk ⊗Vec(Z/pZ) L

g

−k−1g

h

0 a g

k−1g

h

0 a

L

Fq ⊗Vec(Z/pZ) T

g h

∗ (0, a) g h

∗ (0, a)

TABLE V(a). Inflations (part a). All µ, ν occurring label components of the tensor decomposition

30

T ⊗Vec(Z/pZ) R

g h

a

g h

(a, µ) 0

g h

a

g h

(a, ν) 0

T ⊗Vec(Z/pZ) F0

g h

(a, 0) ∗ g h

(a, 0) ∗

T ⊗Vec(Z/pZ) Fr

g h

(a, 0) ∗ g h

(a, 0) ∗

R⊗Vec(Z/pZ) R

g h

a 0 g h

a 0

R⊗Vec(Z/pZ) F01p

∑k ω

kµ

g

k

h

a ∗

1p

∑k ω

kν

g

k

h

a ∗

R⊗Vec(Z/pZ) Xl

g

lh

h

a 0 g

−lh

h

a 0

Xk ⊗Vec(Z/pZ) R

g h

a 0 g h

a 0

R

Fq ⊗Vec(Z/pZ) F01p

∑k ω

qak

g

k

h

∗ ∗

1p

∑k ω

qak

g

k

h

∗ ∗

L⊗Vec(Z/pZ) R

g h

∗

g h

µ 0

g h

∗

g h

ν 0

L⊗Vec(Z/pZ) F0

g h

0 ∗ g h

0 ∗

L⊗Vec(Z/pZ) Fr

g h

0 ∗ g h

0 ∗

F0 ⊗Vec(Z/pZ) R

g h

∗ 0 g h

∗ 0

F0 ⊗Vec(Z/pZ) F01p

∑k ω

kµ

g

k

h

∗ ∗

1p

∑k ω

kν

g

k

h

∗ ∗

F0 ⊗Vec(Z/pZ) Xl

g

hl

h

∗ 0 g

−hl

h

∗ 0

Xk ⊗Vec(Z/pZ) F0

g

−k−1g

h

0 ∗ g

k−1g

h

0 ∗

F0

Fq ⊗Vec(Z/pZ) R

g h

∗ 0 g h

∗ 0

Xk ⊗Vec(Z/pZ) Xl, m = kl g h

a

g

lh

h

a 0

g h

a

g

−lh

h

a 0

Xm

Fq ⊗Vec(Z/pZ) Fr, m = q−1r 1p

∑k ω

qka

g

k

h

∗ ∗

1p

∑k ω

(qa−rh)kg

k

h

∗ ∗

Xk ⊗Vec(Z/pZ) Fr, n = k−1r g h

∗

ω−nghg

−k−1g

h

0 ∗

g h

∗

g

k−1g

h

0 ∗

Fn

Fq ⊗Vec(Z/pZ) Xl, n = ql

g

lh

h

∗ 0 g

−lh

h

∗ 0

TABLE V(b). Inflations (part b). All µ, ν occurring label components of the tensor decomposition

31

Appendix C: Horizontal fusion outcomes

TT

∣∣(c,d)

LT

∣∣c

RT

∣∣c

F0T

∣∣ XnT

∣∣c

FtT

∣∣TT

∣∣(a,b)

δb+cν−µTT

∣∣µ,ν(a,d)

TT

∣∣µ(a,c)

δb+cν−µRT

∣∣µ,νa

RT

∣∣µa

TT

∣∣µ(a,n−1(b+c+µ))

RT

∣∣µa

LT

∣∣a

δa+cν−µLT

∣∣µ,νd

LT

∣∣µc

δa+cν−µF0T

∣∣µ,ν F0T

∣∣µ LT

∣∣µn−1(a+c+µ)

F0T

∣∣µRT

∣∣a

TT

∣∣µ(a,d)

TT

∣∣µ,ν(a,c)

RT

∣∣µa

RT

∣∣µ,νa

RT

∣∣µa

⊕β TT∣∣µ(a,β)

F0T

∣∣ LT

∣∣µd

LT

∣∣µ,νc

F0T

∣∣µ F0T

∣∣µ,ν F0T

∣∣µ ⊕α LT∣∣µα

XlT

∣∣a

TT

∣∣µ(a+l(c+µ),d)

LT

∣∣µc

RT

∣∣µa+l(c+µ)

F0T

∣∣µ XlnT

∣∣µa+l(c+µ)

Fl−1tT

∣∣µFrT

∣∣ LT

∣∣µd

⊕α TT∣∣µ(α,c)

F0T

∣∣µ ⊕α RT∣∣µα

FnrT

∣∣µ ⊕α Xr−1tT

∣∣µα

TL

∣∣a

δa+cν−µTL

∣∣µ,νd

TL

∣∣µc

δa+cν−µRL

∣∣µ,ν RL

∣∣µ TL

∣∣µn−1(a+c+µ)

RL

∣∣µLL

∣∣(a,x)

δa+cν−µLL

∣∣µ,ν(d,x)

LL

∣∣µ(c,x)

δa+cν−µF0L

∣∣µ,νx

F0L

∣∣µx

LL

∣∣µ(n−1(a+c+µ),x)

F0L

∣∣µx

RL

∣∣ TL

∣∣µd

TL

∣∣µ,νc

RL

∣∣µ RL

∣∣µ,ν RL

∣∣µ ⊕α TL∣∣µα

F0L

∣∣x

LL

∣∣µ(d,x)

LL

∣∣µ,ν(c,x)

F0L

∣∣µx

F0L

∣∣µ,νx

F0L

∣∣µx

⊕α LL∣∣µ(α,x)

XlL

∣∣ TL

∣∣µd

⊕β LL∣∣µ(c,β)

RL

∣∣µ ⊕α F0L

∣∣µα

XlnL

∣∣µ ⊕α Fl−1tL

∣∣µα

FrL

∣∣x

LL

∣∣µ(d,x+r(c+µ))

TL

∣∣µc

F0L

∣∣µx+r(c+µ)

RL

∣∣µ FnrL

∣∣µx+r(c+µ)

Xr−1t

L

∣∣µTR

∣∣a

TT

∣∣ν(a,d)

p · TT∣∣(a,c)

RT

∣∣νa

p · RT∣∣a

⊕β TT∣∣(a,β)

p · RT∣∣a

LR

∣∣ LT

∣∣νd

p · LT∣∣c

F0T

∣∣ν p · F0T

∣∣ ⊕α LT∣∣α

p · F0T

∣∣RR

∣∣(a,x)

TT

∣∣(a,d)

TT

∣∣ν(a,c)

RT

∣∣a

RT

∣∣νa

RT

∣∣a

⊕β TT∣∣(a,β)

F0R

∣∣x

LT

∣∣d

LT

∣∣νc

F0T

∣∣ F0T

∣∣ν F0T

∣∣ ⊕α LT∣∣α

XlR

∣∣ ⊕α TT∣∣(α,d)

p · LT∣∣c

⊕α RT∣∣α

p · F0T

∣∣ ⊕α XlnT

∣∣α

p · Fl−1tT

∣∣FrR

∣∣x

LT

∣∣d

⊕α TT∣∣(α,c)

F0T

∣∣ ⊕α RT∣∣α

FnrT

∣∣ ⊕α Xr−1tT

∣∣α

TF0

∣∣ TL

∣∣νd

p · TL∣∣c

RL

∣∣ν p · RL∣∣ ⊕α TL

∣∣α

p · RL∣∣

LF0

∣∣x

LL

∣∣ν(d,x)

p · LL∣∣(c,x)

F0L

∣∣νx

p · F0L

∣∣x

⊕α LL∣∣(α,x)

p · F0L

∣∣x

RF0

∣∣x

TL

∣∣d

TL

∣∣νc

RL

∣∣ RL

∣∣ν RL

∣∣ ⊕α TL∣∣α

F0F0

∣∣(x,y)

LL

∣∣(d,x)

LL

∣∣ν(c,x)

F0L

∣∣x

F0L

∣∣νx

F0L

∣∣x

⊕α LL∣∣(α,x)

XlF0

∣∣x

TL

∣∣d

⊕β LL∣∣(c,β)

RL

∣∣ ⊕α F0L

∣∣α

XlnL

∣∣ ⊕α Fl−1tL

∣∣α

FrF0

∣∣ ⊕β LL∣∣(d,β)

p · TL∣∣c

⊕α F0L

∣∣α

p · RL∣∣ ⊕α Fnr

L

∣∣α

p · Xr−1tL

∣∣TXk

∣∣a

TT

∣∣ν(a+k(c−ν),d) ⊕α TT

∣∣(α,c)

RT

∣∣νa+k(c−ν) ⊕α RT

∣∣α⊕α TT

∣∣(α,(kn)−1(a+ck−α)) ⊕α RT

∣∣α

LXk

∣∣ LT

∣∣νd

p · LT∣∣c

F0T

∣∣ν p · F0T

∣∣ ⊕α LT∣∣α

p · F0T

∣∣RXk

∣∣ ⊕α TT∣∣(α,d)

⊕α TT∣∣ν(α,c)

⊕α RT∣∣α

⊕α RT∣∣να

⊕α RT∣∣α

⊕α,β TT∣∣(α,β)

F0Xk

∣∣x

LT

∣∣d

LT

∣∣νc

F0T

∣∣ F0T

∣∣ν F0T

∣∣ ⊕α LT∣∣α

XkXk

∣∣(a,x)

TT

∣∣(a+ck,d)

LT

∣∣c

RT

∣∣a+ck

F0T

∣∣ XknT

∣∣a+ck

Fk−1tT

∣∣XlXk

∣∣ ⊕α TT∣∣(α,d)

p · LT∣∣c

⊕α RT∣∣α

p · F0T

∣∣ ⊕α XlnT

∣∣α

p · Fl−1tT

∣∣FrXk

∣∣x

LT

∣∣d

⊕α TT∣∣(α,c)

F0T

∣∣ ⊕α RT∣∣α

FnrT

∣∣ ⊕α Xr−1tT

∣∣α

TFq

∣∣ TL

∣∣νd

p · TL∣∣c

RL

∣∣ν p · RL∣∣ ⊕α TL

∣∣α

p · RL∣∣

LFq

∣∣x

LL

∣∣ν(d,cq−νq+x) ⊕β LL

∣∣(c,β)

F0L

∣∣νcq−νq+x ⊕α

F0L

∣∣α

⊕α LL∣∣(α,cq−nαq+x) ⊕α F0

L

∣∣α

RFq

∣∣x

TL

∣∣d

TL

∣∣νc

RL

∣∣ RL

∣∣ν RL

∣∣ ⊕α TL∣∣α

F0Fq

∣∣ ⊕β LL∣∣(d,β)

⊕β LL∣∣ν(c,β)

⊕α F0L

∣∣α⊕α F0

L

∣∣να

⊕α F0L

∣∣α

⊕α,β LL∣∣(α,β)

XlFq

∣∣x

TL

∣∣d

⊕β LL∣∣(c,β)

RL

∣∣ ⊕α F0L

∣∣α

XlnL

∣∣ ⊕α Fl−1tL

∣∣α

Fq

Fq

∣∣(x,y)

LL

∣∣(d,cq+x)

TL

∣∣c

F0L

∣∣cq+x

RL

∣∣ Fnq

L

∣∣cq+x

Xq−1t

L

∣∣FrFq

∣∣ ⊕β LL∣∣(d,β)

p · TL∣∣c

⊕α F0L

∣∣α

p · RL∣∣ ⊕α Fnr

L

∣∣α

p · Xr−1tL

∣∣TABLE VI(a). Defect fusion table (part a). µ (ν) indexes degeneracy in the bottom (top) domain wall fusion.

32

TL

∣∣c

LL

∣∣(c,z)

RL

∣∣ F0L

∣∣z

XnL

∣∣ FtL

∣∣z

TT

∣∣(a,b)

TT

∣∣ν(a,c)

TT

∣∣(a,c)

RT

∣∣νa

RT

∣∣a

⊕β TT∣∣(a,β)

RT

∣∣a

LT

∣∣a

LT

∣∣νc

LT

∣∣c

F0T

∣∣ν F0T

∣∣ ⊕α LT∣∣α

F0T

∣∣RT

∣∣a

p · TT∣∣(a,c)

TT

∣∣ν(a,c)

p · RT∣∣a

RT

∣∣νa

p · RT∣∣a

⊕β TT∣∣(a,β)

F0T

∣∣ p · LT∣∣c

LT

∣∣νc

p · F0T

∣∣ F0T

∣∣ν p · F0T

∣∣ ⊕α LT∣∣α

XlT

∣∣a⊕α TT

∣∣(α,c)

LT

∣∣c

⊕α RT∣∣α

F0T

∣∣ ⊕α XlnT

∣∣α

Fl−1tT

∣∣FrT

∣∣ p · LT∣∣c

⊕α TT∣∣(α,c)

p · F0T

∣∣ ⊕α RT∣∣α

p · FnrT

∣∣ ⊕α Xr−1tT

∣∣α

TL

∣∣a

TL

∣∣νc

TL

∣∣c

RL

∣∣ν RL

∣∣ ⊕α TL∣∣α

RL

∣∣LL

∣∣(a,x)

LL

∣∣ν(c,x)

LL

∣∣(c,x)

F0L

∣∣νx

F0L

∣∣x

⊕α LL∣∣(α,x)

F0L

∣∣x

RL

∣∣ p · TL∣∣c

TL

∣∣νc

p · RL∣∣ R

L

∣∣ν p · RL∣∣ ⊕α TL

∣∣α

F0L

∣∣x

p · LL∣∣(c,x)

LL

∣∣ν(c,x)

p · F0L

∣∣x

F0L

∣∣νx

p · F0L

∣∣x

⊕α LL∣∣(α,x)

XlL

∣∣ p · TL∣∣c

⊕β LL∣∣(c,β)

p · RL∣∣ ⊕α F0

L

∣∣α

p · XlnL

∣∣ ⊕α Fl−1tL

∣∣α

FrL

∣∣x⊕β LL

∣∣(c,β)

TL

∣∣c

⊕α F0L

∣∣α

RL

∣∣ ⊕α FnrL

∣∣α

Xr−1t

L

∣∣TR

∣∣a

TT

∣∣µ,ν(a,c)

TT

∣∣µ(a,c)

RT

∣∣µ,νa

RT

∣∣µa

⊕β TT∣∣µ(a,β)

RT

∣∣µa

LR

∣∣ LT

∣∣µ,νc

LT

∣∣µc

F0T

∣∣µ,ν F0T

∣∣µ ⊕α LT∣∣µα

F0T

∣∣µRR

∣∣(a,x)

TT

∣∣µ(a,c)

δx+zν−µTT

∣∣µ,ν(a,c)

RT

∣∣µa

δx+zν−µRT

∣∣µ,νa

RT

∣∣µa

TT

∣∣µ(a,t−1(x+z+µ))

F0R

∣∣x

LT

∣∣µc

δx+zν−µLT

∣∣µ,νc

F0T

∣∣µ δx+zν−µF0T

∣∣µ,ν F0T

∣∣µ LT

∣∣µt−1(x+z+µ)

XlR

∣∣ ⊕α TT∣∣µ(α,c)

LT

∣∣µc

⊕α RT∣∣µα

F0T

∣∣µ ⊕α XlnT

∣∣µα

Fl−1tT

∣∣µFrR

∣∣x

LT

∣∣µc

TT

∣∣µ(r−1(x+z+µ),c)

F0T

∣∣µ RT

∣∣µr−1(x+z+µ)

FnrT

∣∣µ Xr−1t

T

∣∣µr−1(x+z+µ)

TF0

∣∣ TL

∣∣µ,νc

TL

∣∣µc

RL

∣∣µ,ν RL

∣∣µ ⊕α TL∣∣µα

RL

∣∣µLF0

∣∣x

LL

∣∣µ,ν(c,x)

LL

∣∣µ(c,x)

F0L

∣∣µ,νx

F0L

∣∣µx

⊕α LL∣∣µ(α,x)

F0L

∣∣µx

RF0

∣∣x

TL

∣∣µc

δx+zν−µTL

∣∣µ,νc

RL

∣∣µ δx+zν−µRL

∣∣µ,ν RL

∣∣µ TL

∣∣µt−1(x+z+µ)

F0F0

∣∣(x,y)

LL

∣∣µ(c,x)

δy+zν−µLL

∣∣µ,ν(c,x)

F0L

∣∣µx

δy+zν−µF0L

∣∣µ,νx

F0L

∣∣µx

LL

∣∣µ(t−1(y+z+µ),x)

XlF0

∣∣x

TL

∣∣µc

LL

∣∣µ(c,l−1(lx+z+µ))

RL

∣∣µ F0L

∣∣µl−1(lx+z+µ)

XlnL

∣∣µ Fl−1tL

∣∣µl−1(lx+z+µ)

FrF0

∣∣ ⊕β LL∣∣µ(c,β)

TL

∣∣µc

⊕α F0L

∣∣µα

RL

∣∣µ ⊕α FnrL

∣∣µα

Xr−1t

L

∣∣µTXk

∣∣a

TL

∣∣νc

TL

∣∣c

RL

∣∣ν RL

∣∣ ⊕α TL∣∣α

RL

∣∣LXk

∣∣ ⊕β LL∣∣ν(c,β)

⊕β LL∣∣(c,β)

⊕α F0L

∣∣να

⊕α F0L

∣∣α

⊕α,β LL∣∣(α,β)

⊕α F0L

∣∣α

RXk

∣∣ p · TL∣∣c

TL

∣∣νc

p · RL∣∣ R

L

∣∣ν p · RL∣∣ ⊕α TL

∣∣α

F0Xk

∣∣x⊕β LL

∣∣(c,β)

LL

∣∣ν(c,k−1(kx+z−ν)) ⊕α

F0L

∣∣α

F0L

∣∣νk−1(kx+z−ν) ⊕α F0

L

∣∣α⊕α LL

∣∣(α,k−1(kx+z−tα))

XkXk

∣∣(a,x)

TL

∣∣c

LL

∣∣(c,k−1(x+z))

RL

∣∣ F0L

∣∣k−1(x+z)

XknL

∣∣ Fk−1tL

∣∣k−1(x+z)

XlXk

∣∣ p · TL∣∣c

⊕β LL∣∣(c,β)

p · RL∣∣ ⊕α F0

L

∣∣α

p · XlnL

∣∣ ⊕α Fl−1tL

∣∣α

FrXk

∣∣x⊕β LL

∣∣(c,β)

TL

∣∣c

⊕α F0L

∣∣α

RL

∣∣ ⊕α FnrL

∣∣α

Xr−1t

L

∣∣TFq

∣∣ ⊕α TT∣∣ν(α,c)

⊕α TT∣∣(α,c)

⊕α RT∣∣να

⊕α RT∣∣α

⊕α,β TT∣∣(α,β)

⊕α RT∣∣α

LFq

∣∣x

LT

∣∣νc

LT

∣∣c

F0T

∣∣ν F0T

∣∣ ⊕α LT∣∣α

F0T

∣∣RFq

∣∣x⊕α TT

∣∣(α,c)

TT

∣∣ν(q−1(x+z−ν),c) ⊕α RT

∣∣α

RT

∣∣νq−1(x+z−ν) ⊕α RT

∣∣α

⊕α TT∣∣(α,t−1(x+z−qα))

F0Fq

∣∣ p · LT∣∣c

LT

∣∣νc

p · F0T

∣∣ F0T

∣∣ν p · F0T

∣∣ ⊕α LT∣∣α

XlFq

∣∣x⊕α TT

∣∣(α,c)

LT

∣∣c

⊕α RT∣∣α

F0T

∣∣ ⊕α XlnT

∣∣α

Fl−1tT

∣∣Fq

Fq

∣∣(x,y)

LT

∣∣c

TT

∣∣(q−1(y+z),c)

F0T

∣∣ RT

∣∣q−1(y+z)

Fnq

T

∣∣ Xq−1t

T

∣∣q−1(y+z)

FrFq

∣∣ p · LT∣∣c

⊕α TT∣∣(α,c)

p · F0T

∣∣ ⊕α RT∣∣α

p · FnrT

∣∣ ⊕α Xr−1tT

∣∣α

TABLE VI(b). Defect fusion table (part b). µ (ν) indexes degeneracy in the bottom (top) domain wall fusion.

33

TR

∣∣c

LR

∣∣ RR

∣∣(c,z)

F0R

∣∣z

XnR

∣∣ FtR

∣∣z

TT

∣∣(a,b)

δb+cν−µTR

∣∣µ,νa

TR

∣∣µa

δb+cν−µRR

∣∣µ,ν(a,z)

RR

∣∣µ(a,z)

TR

∣∣µa

RR

∣∣µ(a,z+t(b+µ))

LT

∣∣a

δa+cν−µLR

∣∣µ,ν LR

∣∣µ δa+cν−µF0R

∣∣µ,νz

F0R

∣∣µz

LR

∣∣µ F0R

∣∣µz+t(a+µ)

RT

∣∣a

TR

∣∣µa

TR

∣∣µ,νa

RR

∣∣µ(a,z)

RR

∣∣µ,ν(a,z)

⊕β RR∣∣µ(a,β)

TR

∣∣µa

F0T

∣∣ LR

∣∣µ LR

∣∣µ,ν F0R

∣∣µz

F0R

∣∣µ,νz

⊕α F0R

∣∣µα

LR

∣∣µXlT

∣∣a

TR

∣∣µa+l(c+µ)

LR

∣∣µ RR

∣∣µ(a+l(c+µ),z)

F0R

∣∣µz

XlnR

∣∣µ Fl−1tR

∣∣µµt+l−1(at)+z

FrT

∣∣ LR

∣∣µ ⊕α TR∣∣µα

F0R

∣∣µz

⊕α RR∣∣µ(α,z)

⊕α FnrR

∣∣µα

Xr−1t

R

∣∣µTL

∣∣a

δa+cν−µTF0

∣∣µ,ν TF0

∣∣µ δa+cν−µRF0

∣∣µ,νz

RF0

∣∣µz

TF0

∣∣µ RF0

∣∣µz+t(a+µ)

LL

∣∣(a,x)

δa+cν−µLF0

∣∣µ,νx

LF0

∣∣µx

δa+cν−µF0F0

∣∣µ,ν(x,z)

F0F0

∣∣µ(x,z)

LF0

∣∣µx

F0F0

∣∣µ(x,z+t(a+µ))

RL

∣∣ TF0

∣∣µ TF0

∣∣µ,ν RF0

∣∣µz

RF0

∣∣µ,νz

⊕α RF0

∣∣µα

TF0

∣∣µF0L

∣∣x

LF0

∣∣µx

LF0

∣∣µ,νx

F0F0

∣∣µ(x,z)

F0F0

∣∣µ,ν(x,z)

⊕β F0F0

∣∣µ(x,β)

LF0

∣∣µx

XlL

∣∣ TF0

∣∣µ ⊕α LF0

∣∣µα

RF0

∣∣µz

⊕α F0F0

∣∣µ(α,z)

⊕α XlnF0

∣∣µα

Fl−1tF0

∣∣µFrL

∣∣x

LF0

∣∣µx+r(c+µ)

TF0

∣∣µ F0F0

∣∣µ(x+r(c+µ),z)

RF0

∣∣µz

FnrF0

∣∣µ Xr−1t

F0

∣∣µx+r(t−1z+µ)

TR

∣∣a

TR

∣∣νa

p · TR∣∣a

RR

∣∣ν(a,z)

p · RR∣∣(a,z)

p · TR∣∣a

⊕β RR∣∣(a,β)

LR

∣∣ LR

∣∣ν p · LR∣∣ F0

R

∣∣νz

p · F0R

∣∣z

p · LR∣∣ ⊕α F0

R

∣∣α

RR

∣∣(a,x)

TR

∣∣a

TR

∣∣νa

RR

∣∣(a,z)

RR

∣∣ν(a,z)

⊕β RR∣∣(a,β)

TR

∣∣a

F0R

∣∣x

LR

∣∣ LR

∣∣ν F0R

∣∣z

F0R

∣∣νz

⊕α F0R

∣∣α

LR

∣∣XlR

∣∣ ⊕α TR∣∣α

p · LR∣∣ ⊕α RR

∣∣(α,z)

p · F0R

∣∣z

p · XlnR

∣∣ ⊕α Fl−1tR

∣∣α

FrR

∣∣x

LR

∣∣ ⊕α TR∣∣α

F0R

∣∣z

⊕α RR∣∣(α,z)

⊕α FnrR

∣∣α

Xr−1t

R

∣∣TF0

∣∣ TF0

∣∣ν p · TF0

∣∣ RF0

∣∣νz

p · RF0

∣∣z

p · TF0

∣∣ ⊕α RF0

∣∣α

LF0

∣∣x

LF0

∣∣νx

p · LF0

∣∣x

F0F0

∣∣ν(x,z)

p · F0F0

∣∣(x,z)

p · LF0

∣∣x

⊕β F0F0

∣∣(x,β)

RF0

∣∣x

TF0

∣∣ TF0

∣∣ν RF0

∣∣z

RF0

∣∣νz

⊕α RF0

∣∣α

TF0

∣∣F0F0

∣∣(x,y)

LF0

∣∣x

LF0

∣∣νx

F0F0

∣∣(x,z)

F0F0

∣∣ν(x,z)

⊕β F0F0

∣∣(x,β)

LF0

∣∣x

XlF0

∣∣x

TF0

∣∣ ⊕α LF0

∣∣α

RF0

∣∣z

⊕α F0F0

∣∣(α,z)

⊕α XlnF0

∣∣α

Fl−1tF0

∣∣FrF0

∣∣ ⊕α LF0

∣∣α

p · TF0

∣∣ ⊕α F0F0

∣∣(α,z)

p · RF0

∣∣z

p · FnrF0

∣∣ ⊕α Xr−1tF0

∣∣α

TXk

∣∣a

TR

∣∣νa+k(c−ν) ⊕α TR

∣∣α

RR

∣∣ν(a+k(c−ν),z) ⊕α RR

∣∣(α,z)

⊕α TR∣∣α

⊕α RR∣∣(α,z+k−1(t(a−α)))

LXk

∣∣ LR

∣∣ν p · LR∣∣ F0

R

∣∣νz

p · F0R

∣∣z

p · LR∣∣ ⊕α F0

R

∣∣α

RXk

∣∣ ⊕α TR∣∣α

⊕α TR∣∣να

⊕α RR∣∣(α,z)

⊕α RR∣∣ν(α,z)

⊕α,β RR∣∣(α,β)

⊕α TR∣∣α

F0Xk

∣∣x

LR

∣∣ LR

∣∣ν F0R

∣∣z

F0R

∣∣νz

⊕α F0R

∣∣α

LR

∣∣XkXk

∣∣(a,x)

TR

∣∣a+ck

LR

∣∣ RR

∣∣(a+ck,z)

F0R

∣∣z

XknR

∣∣ Fk−1tR

∣∣k−1(at)+z

XlXk

∣∣ ⊕α TR∣∣α

p · LR∣∣ ⊕α RR

∣∣(α,z)

p · F0R

∣∣z

p · XlnR

∣∣ ⊕α Fl−1tR

∣∣α

FrXk

∣∣x

LR

∣∣ ⊕α TR∣∣α

F0R

∣∣z

⊕α RR∣∣(α,z)

⊕α FnrR

∣∣α

Xr−1t

R

∣∣TFq

∣∣ TF0

∣∣ν p · TF0

∣∣ RF0

∣∣νz

p · RF0

∣∣z

p · TF0

∣∣ ⊕α RF0

∣∣α

LFq

∣∣x

LF0

∣∣νcq−νq+x ⊕α

LF0

∣∣α

F0F0

∣∣ν(cq−νq+x,z) ⊕α

F0F0

∣∣(α,z)

⊕α LF0

∣∣α

⊕α F0F0

∣∣(α,z+q−1(t(x−α)))

RFq

∣∣x

TF0

∣∣ TF0

∣∣ν RF0

∣∣z

RF0

∣∣νz

⊕α RF0

∣∣α

TF0

∣∣F0Fq

∣∣ ⊕α LF0

∣∣α⊕α LF0

∣∣να⊕α F0

F0

∣∣(α,z)

⊕α F0F0

∣∣ν(α,z)

⊕α,β F0F0

∣∣(α,β)

⊕α LF0

∣∣α

XlFq

∣∣x

TF0

∣∣ ⊕α LF0

∣∣α

RF0

∣∣z

⊕α F0F0

∣∣(α,z)

⊕α XlnF0

∣∣α

Fl−1tF0

∣∣Fq

Fq

∣∣(x,y)

LF0

∣∣cq+x

TF0

∣∣ F0F0

∣∣(cq+x,z)

RF0

∣∣z

Fnq

F0

∣∣ Xq−1t

F0

∣∣x+t−1(qz)

FrFq

∣∣ ⊕α LF0

∣∣α

p · TF0

∣∣ ⊕α F0F0

∣∣(α,z)

p · RF0

∣∣z

p · FnrF0

∣∣ ⊕α Xr−1tF0

∣∣α

TABLE VI(c). Defect fusion table (part c). µ (ν) indexes degeneracy in the bottom (top) domain wall fusion.

34

TF0

∣∣ LF0

∣∣z

RF0

∣∣z

F0F0

∣∣(z,w)

XnF0

∣∣z

FtF0

∣∣TT

∣∣(a,b)

TR

∣∣νa

TR

∣∣a

RR

∣∣ν(a,z)

RR

∣∣(a,w)

TR

∣∣a

⊕β RR∣∣(a,β)

LT

∣∣a

LR

∣∣ν LR

∣∣ F0R

∣∣νz

F0R

∣∣w

LR

∣∣ ⊕α F0R

∣∣α

RT

∣∣a

p · TR∣∣a

TR

∣∣νa

p · RR∣∣(a,z)

RR

∣∣ν(a,w)

⊕β RR∣∣(a,β)

p · TR∣∣a

F0T

∣∣ p · LR∣∣ L

R

∣∣ν p · F0R

∣∣z

F0R

∣∣νw

⊕α F0R

∣∣α

p · LR∣∣

XlT

∣∣a⊕α TR

∣∣α

LR

∣∣ ⊕α RR∣∣(α,z)

F0R

∣∣w

XlnR

∣∣ ⊕α Fl−1tR

∣∣α

FrT

∣∣ p · LR∣∣ ⊕α TR

∣∣α

p · F0R

∣∣z

⊕α RR∣∣(α,w)

⊕α FnrR

∣∣α

p · Xr−1tR

∣∣TL

∣∣a

TF0

∣∣ν TF0

∣∣ RF0

∣∣νz

RF0

∣∣w

TF0

∣∣ ⊕α RF0

∣∣α

LL

∣∣(a,x)

LF0

∣∣νx

LF0

∣∣x

F0F0

∣∣ν(x,z)

F0F0

∣∣(x,w)

LF0

∣∣x

⊕β F0F0

∣∣(x,β)

RL

∣∣ p · TF0

∣∣ TF0

∣∣ν p · RF0

∣∣z

RF0

∣∣νw

⊕α RF0

∣∣α

p · TF0

∣∣F0L

∣∣x

p · LF0

∣∣x

LF0

∣∣νx

p · F0F0

∣∣(x,z)

F0F0

∣∣ν(x,w)

⊕β F0F0

∣∣(x,β)

p · LF0

∣∣x

XlL

∣∣ p · TF0

∣∣ ⊕α LF0

∣∣α

p · RF0

∣∣z

⊕α F0F0

∣∣(α,w)

⊕α XlnF0

∣∣α

p · Fl−1tF0

∣∣FrL

∣∣x⊕α LF0

∣∣α

TF0

∣∣ ⊕α F0F0

∣∣(α,z)

RF0

∣∣w

FnrF0

∣∣ ⊕α Xr−1tF0

∣∣α

TR

∣∣a

TR

∣∣µ,νa

TR

∣∣µa

RR

∣∣µ,ν(a,z)

RR

∣∣µ(a,w)

TR

∣∣µa

⊕β RR∣∣µ(a,β)

LR

∣∣ LR

∣∣µ,ν LR

∣∣µ F0R

∣∣µ,νz

F0R

∣∣µw

LR

∣∣µ ⊕α F0R

∣∣µα

RR

∣∣(a,x)

TR

∣∣µa

δx+zν−µTR

∣∣µ,νa

RR

∣∣µ(a,z)

δx+zν−µRR

∣∣µ,ν(a,w)

RR

∣∣µ(a,n(x+z+µ))

TR

∣∣µa

F0R

∣∣x

LR

∣∣µ δx+zν−µLR

∣∣µ,ν F0R

∣∣µz

δx+zν−µF0R

∣∣µ,νw

F0R

∣∣µn(x+z+µ)

LR

∣∣µXlR

∣∣ ⊕α TR∣∣µα

LR

∣∣µ ⊕α RR∣∣µ(α,z)

F0R

∣∣µw

XlnR

∣∣µ ⊕α Fl−1tR

∣∣µα

FrR

∣∣x

LR

∣∣µ TR

∣∣µr−1(x+z+µ)

F0R

∣∣µz

RR

∣∣µ(r−1(x+z+µ),w)

FnrR

∣∣µn(x+z+µ)

Xr−1t

R

∣∣µTF0

∣∣ TF0

∣∣µ,ν TF0

∣∣µ RF0

∣∣µ,νz

RF0

∣∣µw

TF0

∣∣µ ⊕α RF0

∣∣µα

LF0

∣∣x

LF0

∣∣µ,νx

LF0

∣∣µx

F0F0

∣∣µ,ν(x,z)

F0F0

∣∣µ(x,w)

LF0

∣∣µx

⊕β F0F0

∣∣µ(x,β)

RF0

∣∣x

TF0

∣∣µ δx+zν−µTF0

∣∣µ,ν RF0

∣∣µz

δx+zν−µRF0

∣∣µ,νw

RF0

∣∣µn(x+z+µ)

TF0

∣∣µF0F0

∣∣(x,y)

LF0

∣∣µx

δy+zν−µLF0

∣∣µ,νx

F0F0

∣∣µ(x,z)

δy+zν−µF0F0

∣∣µ,ν(x,w)

F0F0

∣∣µ(x,n(y+z+µ))

LF0

∣∣µx

XlF0

∣∣x

TF0

∣∣µ LF0

∣∣µl−1(lx+z+µ)

RF0

∣∣µz

F0F0

∣∣µ(l−1(lx+z+µ),w)

XlnF0

∣∣µl−1(lx+z+µ)

Fl−1tF0

∣∣µFrF0

∣∣ ⊕α LF0

∣∣µα

TF0

∣∣µ ⊕α F0F0

∣∣µ(α,z)

RF0

∣∣µw

FnrF0

∣∣µ ⊕α Xr−1tF0

∣∣µα

TXk

∣∣a

TF0

∣∣ν TF0

∣∣ RF0

∣∣νz

RF0

∣∣w

TF0

∣∣ ⊕α RF0

∣∣α

LXk

∣∣ ⊕α LF0

∣∣να

⊕α LF0

∣∣α

⊕α F0F0

∣∣ν(α,z)

⊕α F0F0

∣∣(α,w)

⊕α LF0

∣∣α

⊕α,β F0F0

∣∣(α,β)

RXk

∣∣ p · TF0

∣∣ TF0

∣∣ν p · RF0

∣∣z

RF0

∣∣νw

⊕α RF0

∣∣α

p · TF0

∣∣F0Xk

∣∣x⊕α LF0

∣∣α

LF0

∣∣νk−1(kx+z−ν) ⊕α

F0F0

∣∣(α,z)

F0F0

∣∣ν(k−1(kx+z−ν),w)

⊕β F0F0

∣∣(x+(kn)−1(nz−β),β) ⊕α LF0

∣∣α

XkXk

∣∣(a,x)

TF0

∣∣ LF0

∣∣k−1(x+z)

RF0

∣∣z

F0F0

∣∣(k−1(x+z),w)

XknF0

∣∣k−1(x+z)

Fk−1tF0

∣∣XlXk

∣∣ p · TF0

∣∣ ⊕α LF0

∣∣α

p · RF0

∣∣z

⊕α F0F0

∣∣(α,w)

⊕α XlnF0

∣∣α

p · Fl−1tF0

∣∣FrXk

∣∣x⊕α LF0

∣∣α

TF0

∣∣ ⊕α F0F0

∣∣(α,z)

RF0

∣∣w

FnrF0

∣∣ ⊕α Xr−1tF0

∣∣α

TFq

∣∣ ⊕α TR∣∣να

⊕α TR∣∣α

⊕α RR∣∣ν(α,z)

⊕α RR∣∣(α,w)

⊕α TR∣∣α

⊕α,β RR∣∣(α,β)

LFq

∣∣x

LR

∣∣ν LR

∣∣ F0R

∣∣νz

F0R

∣∣w

LR

∣∣ ⊕α F0R

∣∣α

RFq

∣∣x⊕α TR

∣∣α

TR

∣∣νq−1(x+z−ν) ⊕α RR

∣∣(α,z)

RR

∣∣ν(q−1(x+z−ν),w)

⊕α RR∣∣(α,n(x+z−qα)) ⊕α TR

∣∣α

F0Fq

∣∣ p · LR∣∣ L

R

∣∣ν p · F0R

∣∣z

F0R

∣∣νw

⊕α F0R

∣∣α

p · LR∣∣

XlFq

∣∣x⊕α TR

∣∣α

LR

∣∣ ⊕α RR∣∣(α,z)

F0R

∣∣w

XlnR

∣∣ ⊕α Fl−1tR

∣∣α

Fq

Fq

∣∣(x,y)

LR

∣∣ TR

∣∣q−1(y+z)

F0R

∣∣z

RR

∣∣(q−1(y+z),w)

Fnq

R

∣∣n(y+z)

Xq−1t

R

∣∣FrFq

∣∣ p · LR∣∣ ⊕α TR

∣∣α

p · F0R

∣∣z

⊕α RR∣∣(α,w)

⊕α FnrR

∣∣α

p · Xr−1tR

∣∣TABLE VI(d). Defect fusion table (part d). µ (ν) indexes degeneracy in the bottom (top) domain wall fusion.

35

TXm

∣∣c

LXm

∣∣ RXm

∣∣ F0Xm

∣∣z

XmXm

∣∣(c,z)

XnXm

∣∣ FtXm

∣∣z

TT

∣∣(a,b)

TT

∣∣ν(a,m−1(b+c−ν)) ⊕β TT

∣∣(a,β)

RT

∣∣νa

RT

∣∣a

TT

∣∣(a,m−1(b+c))

⊕β TT∣∣(a,β)

RT

∣∣a

LT

∣∣a

LT

∣∣νm−1(a+c−ν) ⊕α LT

∣∣α

F0T

∣∣ν F0T

∣∣ LT

∣∣m−1(a+c)

⊕α LT∣∣α

F0T

∣∣RT

∣∣a

⊕β TT∣∣(a,β)

⊕β TT∣∣ν(a,β)

p · RT∣∣a

RT

∣∣νa

RT

∣∣a

p · RT∣∣a

⊕β TT∣∣(a,β)

F0T

∣∣ ⊕α LT∣∣α

⊕α LT∣∣να

p · F0T

∣∣ F0T

∣∣ν F0T

∣∣ p · F0T

∣∣ ⊕α LT∣∣α

XlT

∣∣a⊕α TT

∣∣(α,(lm)−1(a+cl−α)) ⊕α LT

∣∣α

⊕α RT∣∣α

F0T

∣∣ XlmT

∣∣a+cl

⊕α XlnT

∣∣α

Fl−1tT

∣∣FrT

∣∣ ⊕α LT∣∣α

⊕α,β TT∣∣(α,β)

p · F0T

∣∣ ⊕α RT∣∣α

FmrT

∣∣ p · FnrT

∣∣ ⊕α Xr−1tT

∣∣α

TL

∣∣a

TL

∣∣νm−1(a+c−ν) ⊕α TL

∣∣α

RL

∣∣ν RL

∣∣ TL

∣∣m−1(a+c)

⊕α TL∣∣α

RL

∣∣LL

∣∣(a,x)

LL

∣∣ν(m−1(a+c−ν),x) ⊕α LL

∣∣(α,x)

F0L

∣∣νx

F0L

∣∣x

LL

∣∣(m−1(a+c),x)

⊕α LL∣∣(α,x)

F0L

∣∣x

RL

∣∣ ⊕α TL∣∣α

⊕α TL∣∣να

p · RL∣∣ R

L

∣∣ν RL

∣∣ p · RL∣∣ ⊕α TL

∣∣α

F0L

∣∣x

⊕α LL∣∣(α,x)

⊕α LL∣∣ν(α,x)

p · F0L

∣∣x

F0L

∣∣νx

F0L

∣∣x

p · F0L

∣∣x

⊕α LL∣∣(α,x)

XlL

∣∣ ⊕α TL∣∣α

⊕α,β LL∣∣(α,β)

p · RL∣∣ ⊕α F0

L

∣∣α

XlmL

∣∣ p · XlnL

∣∣ ⊕α Fl−1tL

∣∣α

FrL

∣∣x

⊕α LL∣∣(α,cr−mαr+x) ⊕α TL

∣∣α

⊕α F0L

∣∣α

RL

∣∣ FmrL

∣∣cr+x

⊕α FnrL

∣∣α

Xr−1t

L

∣∣TR

∣∣a

TR

∣∣νa

p · TR∣∣a

⊕β RR∣∣ν(a,β)

⊕β RR∣∣(a,β)

TR

∣∣a

p · TR∣∣a

⊕β RR∣∣(a,β)

LR

∣∣ LR

∣∣ν p · LR∣∣ ⊕α F0

R

∣∣να

⊕α F0R

∣∣α

LR

∣∣ p · LR∣∣ ⊕α F0

R

∣∣α

RR

∣∣(a,x)

TR

∣∣a

TR

∣∣νa

⊕β RR∣∣(a,β)

RR

∣∣ν(a,m(x+z−ν))

RR

∣∣(a,mx+z)

⊕β RR∣∣(a,β)

TR

∣∣a

F0R

∣∣x

LR

∣∣ LR

∣∣ν ⊕α F0R

∣∣α

F0R

∣∣νm(x+z−ν)

F0R

∣∣mx+z

⊕α F0R

∣∣α

LR

∣∣XlR

∣∣ ⊕α TR∣∣α

p · LR∣∣ ⊕α,β RR

∣∣(α,β)

⊕α F0R

∣∣α

XlmR

∣∣ p · XlnR

∣∣ ⊕α Fl−1tR

∣∣α

FrR

∣∣x

LR

∣∣ ⊕α TR∣∣α

⊕α F0R

∣∣α

⊕α RR∣∣(α,m(x+z−rα))

FmrR

∣∣mx+z

⊕α FnrR

∣∣α

Xr−1t

R

∣∣TF0

∣∣ TF0

∣∣ν p · TF0

∣∣ ⊕α RF0

∣∣να

⊕α RF0

∣∣α

TF0

∣∣ p · TF0

∣∣ ⊕α RF0

∣∣α

LF0

∣∣x

LF0

∣∣νx

p · LF0

∣∣x

⊕β F0F0

∣∣ν(x,β)

⊕β F0F0

∣∣(x,β)

LF0

∣∣x

p · LF0

∣∣x

⊕β F0F0

∣∣(x,β)

RF0

∣∣x

TF0

∣∣ TF0

∣∣ν ⊕α RF0

∣∣α

RF0

∣∣νm(x+z−ν)

RF0

∣∣mx+z

⊕α RF0

∣∣α

TF0

∣∣F0F0

∣∣(x,y)

LF0

∣∣x

LF0

∣∣νx

⊕β F0F0

∣∣(x,β)

F0F0

∣∣ν(x,m(y+z−ν))

F0F0

∣∣(x,my+z)

⊕β F0F0

∣∣(x,β)

LF0

∣∣x

XlF0

∣∣x

TF0

∣∣ ⊕α LF0

∣∣α

⊕α RF0

∣∣α

⊕β F0F0

∣∣(x+(lm)−1(mz−β),β)

XlmF0

∣∣x+(lm)−1z

⊕α XlnF0

∣∣α

Fl−1tF0

∣∣FrF0

∣∣ ⊕α LF0

∣∣α

p · TF0

∣∣ ⊕α,β F0F0

∣∣(α,β)

⊕α RF0

∣∣α

FmrF0

∣∣ p · FnrF0

∣∣ ⊕α Xr−1tF0

∣∣α

TXk

∣∣a

TXkm

∣∣νa+k(c−ν) ⊕α TXkm

∣∣α

RXkm

∣∣ν RXkm

∣∣ TXkm

∣∣a+ck

⊕α TXkm

∣∣α

RXkm

∣∣LXk

∣∣ LXkm

∣∣ν p · LXkm

∣∣ ⊕α F0Xkm

∣∣να

⊕α F0Xkm

∣∣α

LXkm

∣∣ p · LXkm

∣∣ ⊕α F0Xkm

∣∣α

RXk

∣∣ ⊕α TXkm

∣∣α

⊕α TXkm

∣∣να

p · RXkm

∣∣ RXkm

∣∣ν RXkm

∣∣ p · RXkm

∣∣ ⊕α TXkm

∣∣α

F0Xk

∣∣x

LXkm

∣∣ LXkm

∣∣ν ⊕α F0Xkm

∣∣α

F0Xkm

∣∣νk−1(kx+z−ν)

F0Xkm

∣∣x+(km)−1z

⊕α F0Xkm

∣∣α

LXkm

∣∣XkXk

∣∣(a,x)

TXkm

∣∣a+ck

LXkm

∣∣ RXkm

∣∣ F0Xkm

∣∣k−1(x+z)

XkmXkm

∣∣(a+ck,mx+z)

XknXkm

∣∣ Fk−1tXkm

∣∣k−1(at)+mx+z

XlXk

∣∣ ⊕α TXkm

∣∣α

p · LXkm

∣∣ p · RXkm

∣∣ ⊕α F0Xkm

∣∣α

XlmXkm

∣∣ ⊕α,β Xkm

Xkm

∣∣(α,β)

km = ln

p · XlnXkm

∣∣ otherwise⊕α Fl−1t

Xkm

∣∣α

FrXk

∣∣x

LXkm

∣∣ ⊕α TXkm

∣∣α

⊕α F0Xkm

∣∣α

RXkm

∣∣ FmrXkm

∣∣ckmr+mx+z

⊕α FnrXkm

∣∣α

⊕α Xkm

Xkm

∣∣(α,mx+z−mrα) km = r−1t

Xr−1t

Xkm

∣∣ otherwise

TFq

∣∣ TFmq

∣∣ν p · TFmq

∣∣ ⊕α RFmq

∣∣να

⊕α RFmq

∣∣α

TFmq

∣∣ p · TFmq

∣∣ ⊕α RFmq

∣∣α

LFq

∣∣x

LFmq

∣∣νcq−νq+x ⊕α LFmq

∣∣α

F0Fmq

∣∣ν F0Fmq

∣∣ LFmq

∣∣cq+x

⊕α LFmq

∣∣α

F0Fmq

∣∣RFq

∣∣x

TFmq

∣∣ TFmq

∣∣ν ⊕α RFmq

∣∣α

RFmq

∣∣νm(x+z−ν)

RFmq

∣∣mx+z

⊕α RFmq

∣∣α

TFmq

∣∣F0Fq

∣∣ ⊕α LFmq

∣∣α

⊕α LFmq

∣∣να

p · F0Fmq

∣∣ F0Fmq

∣∣ν F0Fmq

∣∣ p · F0Fmq

∣∣ ⊕α LFmq

∣∣α

XlFq

∣∣x

TFmq

∣∣ ⊕α LFmq

∣∣α

⊕α RFmq

∣∣α

F0Fmq

∣∣ XlmFmq

∣∣clmq+mx+z

⊕α XlnFmq

∣∣α

⊕α Fmq

Fmq

∣∣(α,mx+z−lmα) qm = l−1t

Fl−1tFmq

∣∣ otherwise

Fq

Fq

∣∣(x,y)

LFmq

∣∣cq+x

TFmq

∣∣ F0Fmq

∣∣ RFmq

∣∣m(y+z)

Fmq

Fmq

∣∣(cq+x,my+z)

Fnq

Fmq

∣∣ Xq−1t

Fmq

∣∣q−1(tx)+my+z

FrFq

∣∣ ⊕α LFmq

∣∣α

p · TFmq

∣∣ p · F0Fmq

∣∣ ⊕α RFmq

∣∣α

FmrFmq

∣∣ ⊕α,β Fmq

Fmq

∣∣(α,β)

qm = rn

p · FnrFmq

∣∣ otherwise⊕α Xr−1t

Fmq

∣∣α

TABLE VI(e). Defect fusion table (part e). µ (ν) indexes degeneracy in the bottom (top) domain wall fusion.

36

TFs

∣∣ LFs

∣∣z

RFs

∣∣z

F0Fs

∣∣ XnFs

∣∣z

FsFs

∣∣(z,w)

FtFs

∣∣TT

∣∣(a,b)

TR

∣∣νa

TR

∣∣a

RR

∣∣ν(a,bs−νs+z) ⊕β RR

∣∣(a,β)

TR

∣∣a

RR

∣∣(a,bs+w)

⊕β RR∣∣(a,β)

LT

∣∣a

LR

∣∣ν LR

∣∣ F0R

∣∣νas−νs+z ⊕α F0

R

∣∣α

LR

∣∣ F0R

∣∣as+w

⊕α F0R

∣∣α

RT

∣∣a

p · TR∣∣a

TR

∣∣νa

⊕β RR∣∣(a,β)

⊕β RR∣∣ν(a,β)

⊕β RR∣∣(a,β)

TR

∣∣a

p · TR∣∣a

F0T

∣∣ p · LR∣∣ L

R

∣∣ν ⊕α F0R

∣∣α

⊕α F0R

∣∣να

⊕α F0R

∣∣α

LR

∣∣ p · LR∣∣

XlT

∣∣a

⊕α TR∣∣α

LR

∣∣ ⊕α RR∣∣(α,z+l−1(s(a−α))) ⊕α F0

R

∣∣α

XlnR

∣∣ Fl−1sR

∣∣l−1(as)+w

⊕α Fl−1tR

∣∣α

FrT

∣∣ p · LR∣∣ ⊕α TR

∣∣α

⊕α F0R

∣∣α

⊕α,β RR∣∣(α,β)

⊕α FnrR

∣∣α

Xr−1s

R

∣∣ p · Xr−1tR

∣∣TL

∣∣a

TF0

∣∣ν TF0

∣∣ RF0

∣∣νas−νs+z ⊕α RF0

∣∣α

TF0

∣∣ RF0

∣∣as+w

⊕α RF0

∣∣α

LL

∣∣(a,x)

LF0

∣∣νx

LF0

∣∣x

F0F0

∣∣ν(x,as−νs+z) ⊕β F0

F0

∣∣(x,β)

LF0

∣∣x

F0F0

∣∣(x,as+w)

⊕β F0F0

∣∣(x,β)

RL

∣∣ p · TF0

∣∣ TF0

∣∣ν ⊕α RF0

∣∣α

⊕α RF0

∣∣να

⊕α RF0

∣∣α

TF0

∣∣ p · TF0

∣∣F0L

∣∣x

p · LF0

∣∣x

LF0

∣∣νx

⊕β F0F0

∣∣(x,β)

⊕β F0F0

∣∣ν(x,β)

⊕β F0F0

∣∣(x,β)

LF0

∣∣x

p · LF0

∣∣x

XlL

∣∣ p · TF0

∣∣ ⊕α LF0

∣∣α

⊕α RF0

∣∣α

⊕α,β F0F0

∣∣(α,β)

⊕α XlnF0

∣∣α

Fl−1sF0

∣∣ p · Fl−1tF0

∣∣FrL

∣∣x

⊕α LF0

∣∣α

TF0

∣∣ ⊕α F0F0

∣∣(α,z+r−1(s(x−α))) ⊕α RF0

∣∣α

FnrF0

∣∣ Xr−1s

F0

∣∣s−1(rw)+x

⊕α Xr−1tF0

∣∣α

TR

∣∣a

⊕β TT∣∣ν(a,β)

⊕β TT∣∣(a,β)

RT

∣∣νa

p · RT∣∣a

⊕β TT∣∣(a,β)

RT

∣∣a

p · RT∣∣a

LR

∣∣ ⊕α LT∣∣να

⊕α LT∣∣α

F0T

∣∣ν p · F0T

∣∣ ⊕α LT∣∣α

F0T

∣∣ p · F0T

∣∣RR

∣∣(a,x)

⊕β TT∣∣(a,β)

TT

∣∣ν(a,s−1(x+z−ν))

RT

∣∣a

RT

∣∣νa

RT

∣∣a

TT

∣∣(a,s−1(x+z))

⊕β TT∣∣(a,β)

F0R

∣∣x

⊕α LT∣∣α

LT

∣∣νs−1(x+z−ν)

F0T

∣∣ F0T

∣∣ν F0T

∣∣ LT

∣∣s−1(x+z)

⊕α LT∣∣α

XlR

∣∣ ⊕α,β TT∣∣(α,β)

⊕α LT∣∣α

⊕α RT∣∣α

p · F0T

∣∣ ⊕α XlnT

∣∣α

Fl−1sT

∣∣ p · Fl−1tT

∣∣FrR

∣∣x

⊕α LT∣∣α

⊕α TT∣∣(α,s−1(x+z−rα))

F0T

∣∣ ⊕α RT∣∣α

FnrT

∣∣ Xr−1s

T

∣∣r−1(x+z)

⊕α Xr−1tT

∣∣α

TF0

∣∣ ⊕α TL∣∣να

⊕α TL∣∣α

RL

∣∣ν p · RL∣∣ ⊕α TL

∣∣α

RL

∣∣ p · RL∣∣

LF0

∣∣x

⊕α LL∣∣ν(α,x)

⊕α LL∣∣(α,x)

F0L

∣∣νx

p · F0L

∣∣x

⊕α LL∣∣(α,x)

F0L

∣∣x

p · F0L

∣∣x

RF0

∣∣x

⊕α TL∣∣α

TL

∣∣νs−1(x+z−ν)

RL

∣∣ RL

∣∣ν RL

∣∣ TL

∣∣s−1(x+z)

⊕α TL∣∣α

F0F0

∣∣(x,y)

⊕α LL∣∣(α,x)

LL

∣∣ν(s−1(y+z−ν),x)

F0L

∣∣x

F0L

∣∣νx

F0L

∣∣x

LL

∣∣(s−1(y+z),x)

⊕α LL∣∣(α,x)

XlF0

∣∣x

⊕α TL∣∣α

⊕α LL∣∣(α,l−1(lx+z−sα))

RL

∣∣ ⊕α F0L

∣∣α

XlnL

∣∣ Fl−1sL

∣∣x+l−1z

⊕α Fl−1tL

∣∣α

FrF0

∣∣ ⊕α,β LL∣∣(α,β)

⊕α TL∣∣α

⊕α F0L

∣∣α

p · RL∣∣ ⊕α Fnr

L

∣∣α

Xr−1s

L

∣∣ p · Xr−1tL

∣∣TXk

∣∣a

TFk−1s

∣∣ν TFk−1s

∣∣ RFk−1s

∣∣ν−νs+k−1(as)+z

⊕α RFk−1s

∣∣α

TFk−1s

∣∣ RFk−1s

∣∣k−1(as)+w

⊕α RFk−1s

∣∣α

LXk

∣∣ ⊕α LFk−1s

∣∣να

⊕α LFk−1s

∣∣α

F0Fk−1s

∣∣ν p · F0Fk−1s

∣∣ ⊕α LFk−1s

∣∣α

F0Fk−1s

∣∣ p · F0Fk−1s

∣∣RXk

∣∣ p · TFk−1s

∣∣ TFk−1s

∣∣ν ⊕α RFk−1s

∣∣α

⊕α RFk−1s

∣∣να

⊕α RFk−1s

∣∣α

TFk−1s

∣∣ p · TFk−1s

∣∣F0Xk

∣∣x⊕α LF

k−1s

∣∣α

LFk−1s

∣∣νk−1(kx+z−ν)

F0Fk−1s

∣∣ F0Fk−1s

∣∣ν F0Fk−1s

∣∣ LFk−1s

∣∣x+k−1z

⊕α LFk−1s

∣∣α

XkXk

∣∣(a,x)

TFk−1s

∣∣ LFk−1s

∣∣k−1(x+z)

RFk−1s

∣∣k−1(as)+z

F0Fk−1s

∣∣ XknFk−1s

∣∣k−1(as)+nx+z

Fk−1sFk−1s

∣∣(k−1(x+z),k−1(as)+w)

Fk−1tFk−1s

∣∣XlXk

∣∣ p · TFk−1s

∣∣ ⊕α LFk−1s

∣∣α

⊕α RFk−1s

∣∣α

p · F0Fk−1s

∣∣ ⊕α XlnFk−1s

∣∣α

Fl−1sFk−1s

∣∣ ⊕α,βFk−1sFk−1s

∣∣(α,β)

k−1s = l−1t

p · Fl−1tFk−1s

∣∣ otherwise

FrXk

∣∣x⊕α LF

k−1s

∣∣α

TFk−1s

∣∣ F0Fk−1s

∣∣ ⊕α RFk−1s

∣∣α

⊕α Fnr

Fnr

∣∣(α,nx+z−knα) k−1s = rn

FnrFk−1s

∣∣ otherwise

Xr−1s

Fk−1s

∣∣w+(kr)−1(s(x+z))

⊕α Xr−1tFk−1s

∣∣α

TFq

∣∣ ⊕α TXq−1s

∣∣να

⊕α TXq−1s

∣∣α

RX

q−1s

∣∣ν p · RXq−1s

∣∣ ⊕α TXq−1s

∣∣α

RX

q−1s

∣∣ p · RXq−1s

∣∣LFq

∣∣x

LX

q−1s

∣∣ν LX

q−1s

∣∣ F0X

q−1s

∣∣νx+s−1(qz)−qν ⊕α F0

Xq−1s

∣∣α

LX

q−1s

∣∣ F0X

q−1s

∣∣s−1(qw)+x

⊕α F0X

q−1s

∣∣α

RFq

∣∣x⊕α TX

q−1s

∣∣α

TX

q−1s

∣∣νq−1(x+z−ν)

RX

q−1s

∣∣ RX

q−1s

∣∣ν RX

q−1s

∣∣ TX

q−1s

∣∣q−1(x+z)

⊕α TXq−1s

∣∣α

F0Fq

∣∣ p · LXq−1s

∣∣ LX

q−1s

∣∣ν ⊕α F0X

q−1s

∣∣α

⊕α F0X

q−1s

∣∣να

⊕α F0X

q−1s

∣∣α

LX

q−1s

∣∣ p · LXq−1s

∣∣XlFq

∣∣x⊕α TX

q−1s

∣∣α

LX

q−1s

∣∣ RX

q−1s

∣∣ ⊕α F0X

q−1s

∣∣α

⊕α Xln

Xln

∣∣(α,nx+z−nqα) q−1s = ln

XlnX

q−1s

∣∣ otherwise

Fl−1sX

q−1s

∣∣w+(lq)−1(s(x+z))

⊕αFl−1tX

q−1s

∣∣α

Fq

Fq

∣∣(x,y)

LX

q−1s

∣∣ TX

q−1s

∣∣q−1(y+z)

F0X

q−1s

∣∣x+s−1(qz)

RX

q−1s

∣∣ Fnq

Xq−1s

∣∣q−1(sx)+ny+z

Xq−1s

Xq−1s

∣∣(q−1(y+z),w+q−1(sx))

Xq−1t

Xq−1s

∣∣FrFq

∣∣ p · LXq−1s

∣∣ ⊕α TXq−1s

∣∣α

⊕α F0X

q−1s

∣∣α

p · RXq−1s

∣∣ ⊕α FnrX

q−1s

∣∣α

Xr−1s

Xq−1s

∣∣ ⊕α,βX

r−1tX

r−1t

∣∣(α,β)

q−1s = r−1t

p · Xr−1tX

q−1s

∣∣ otherwise

TABLE VI(f). Defect fusion table (part f). µ (ν) indexes degeneracy in the bottom (top) domain wall fusion.

37

Appendix D: Vertical fusion outcomes

LT L

∣ ∣ cL L

∣ ∣ (c,z)R L

∣ ∣F0L

∣ ∣ zX

mL

∣ ∣FsL

∣ ∣ zL T

∣ ∣ a⊕αT T

∣ ∣ (α,a+c)

L T

∣ ∣ a+c⊕αR T

∣ ∣ αF0T

∣ ∣⊕αX

mT

∣ ∣ αFsT

∣ ∣L L

∣ ∣ (a,x)

T L

∣ ∣ a+cL L

∣ ∣ (a+c,x

+z)

R L

∣ ∣F0L

∣ ∣ x+zX

mL

∣ ∣FsL

∣ ∣ x+z+sa

L R

∣ ∣⊕αT R

∣ ∣ αL R

∣ ∣⊕α,ζR R

∣ ∣ (α,ζ)

⊕ζF0R

∣ ∣ ζp·X

mR

∣ ∣⊕ζFsR

∣ ∣ ζL F0

∣ ∣ xT F0

∣ ∣L F0

∣ ∣ x+z⊕ζR F0

∣ ∣ ζ⊕ηF0F0

∣ ∣ (x+z,η

)⊕ζX

mF0

∣ ∣ ζFsF0

∣ ∣L X

k

∣ ∣⊕αT X

k

∣ ∣ αL X

k

∣ ∣p·R X

k

∣ ∣⊕ζF0X

k

∣ ∣ ζ ⊕ α

,βX

kX

k

∣ ∣ (α,β)

k=m

p·X

mX

k

∣ ∣k6=m

⊕αFsX

k

∣ ∣ αL Fq

∣ ∣ xT Fq

∣ ∣L Fq

∣ ∣ x+z+qc

⊕ζR Fq

∣ ∣ ζF0Fq

∣ ∣⊕ζX

mFq

∣ ∣ ζ ⊕ α

Fq

Fq

∣ ∣ (x+z,α

)q

=s

FsFq

∣ ∣q6=s

F0

T F0

∣ ∣L F0

∣ ∣ zR F0

∣ ∣ zF0F0

∣ ∣ (z,w

)

Xm

F0

∣ ∣ zFsF0

∣ ∣F0T

∣ ∣⊕α,βT T

∣ ∣ (α,β

)⊕αL T

∣ ∣ α⊕αR T

∣ ∣ αF0T

∣ ∣⊕αX

mT

∣ ∣ αp·F

sT

∣ ∣F0L

∣ ∣ x⊕αT L

∣ ∣ α⊕αL L

∣ ∣ (α,x

+z)

R L

∣ ∣F0L

∣ ∣ x+zX

mL

∣ ∣⊕ζFsL

∣ ∣ ζF0R

∣ ∣ x⊕αT R

∣ ∣ αL R

∣ ∣⊕αR R

∣ ∣ (α,x

+z)

F0R

∣ ∣ x+wX

mR

∣ ∣⊕ζFsR

∣ ∣ ζF0F0

∣ ∣ (x,y

)T F0

∣ ∣L F0

∣ ∣ x+zR F0

∣ ∣ y+zF0F0

∣ ∣ (x+z,y

+w)

Xm

F0

∣ ∣ x+z+m

−1y

FsF0

∣ ∣F0X

k

∣ ∣ x⊕αT X

k

∣ ∣ αL X

k

∣ ∣R X

k

∣ ∣F0X

k

∣ ∣ x+z+k−

1w

⊕ αX

kX

k

∣ ∣ (α,k

(x+z))

k=m

Xm

Xk

∣ ∣k6=m

⊕ζFsX

k

∣ ∣ ζF0Fq

∣ ∣p·T F

q

∣ ∣⊕ζL Fq

∣ ∣ ζ⊕ζR Fq

∣ ∣ ζF0Fq

∣ ∣⊕ζX

mFq

∣ ∣ ζ ⊕ ζ

,ηFq

Fq

∣ ∣ (ζ,η

)q

=s

p·F

sFq

∣ ∣q6=s

Fr

T Fr

∣ ∣L Fr

∣ ∣ zR Fr

∣ ∣ zF0Fr

∣ ∣X

mFr

∣ ∣ zFrFr

∣ ∣ (z,w

)

FsFr

∣ ∣FrT

∣ ∣ ⊕α,βT T

∣ ∣ (α,β

)⊕αL T

∣ ∣ α⊕αR T

∣ ∣ αp·F

0T

∣ ∣⊕αX

mT

∣ ∣ αFrT

∣ ∣p·F

sT

∣ ∣FrL

∣ ∣ x⊕αT L

∣ ∣ α⊕αL L

∣ ∣ (α,x

+z−rα)

R L

∣ ∣⊕ζF0L

∣ ∣ ζX

mL

∣ ∣FrL

∣ ∣ x+z⊕ζFsL

∣ ∣ ζFrR

∣ ∣ x⊕αT R

∣ ∣L R

∣ ∣⊕αR R

∣ ∣ (α,x

+z−rα)⊕ζF0R

∣ ∣ ζX

mR

∣ ∣FrR

∣ ∣ x+w⊕ζFsR

∣ ∣ ζFrF0

∣ ∣p·T F

0

∣ ∣⊕ζL F0

∣ ∣ ζ⊕ζR F0

∣ ∣ ζ⊕ζ,ηF0F0

∣ ∣ (ζ,η

)

∑ αX

mF0

∣ ∣ αFrF0

∣ ∣p·F

sF0

∣ ∣FrX

k

∣ ∣ x⊕αT X

k

∣ ∣ αL X

k

∣ ∣R X

k

∣ ∣⊕ζF0X

k

∣ ∣ ζ ⊕ α

Xk

Xk

∣ ∣ (α,x

+z−rα)

k=m

Xm

Xk

∣ ∣k6=m

FrX

k

∣ ∣ w+x+kz

⊕ζFsX

k

∣ ∣ ζFrFr

∣ ∣ (x,y

)T Fr

∣ ∣L Fq

∣ ∣ x+zR Fq

∣ ∣ y+zF0Fq

∣ ∣X

mFq

∣ ∣ y+z+mx

FrFr

∣ ∣ (x+z,y

+w)

FsFr

∣ ∣FrFq

∣ ∣p·T F

q

∣ ∣⊕ζL Fq

∣ ∣ ζ⊕ζR Fq

∣ ∣ ζp·F

0Fq

∣ ∣⊕ζX

mFq

∣ ∣ ζFrFq

∣ ∣ ⊕ ζ

,ηFq

Fq

∣ ∣ (ζ,η

)q

=s

p·F

sFq

∣ ∣q6=s

TT T

∣ ∣ (c,d)L T

∣ ∣ cR T

∣ ∣ cF0T

∣ ∣X

mT

∣ ∣ cFsT

∣ ∣T T

∣ ∣ (a,b)T T

∣ ∣ (a+c,b+d)

L T

∣ ∣ b+cR T

∣ ∣ a+cF0T

∣ ∣X

mT

∣ ∣ a+c+mb

FsT

∣ ∣T L

∣ ∣ aT L

∣ ∣ a+d⊕βL L

∣ ∣ (a+c,β

)R L

∣ ∣⊕αF0L

∣ ∣ αX

mL

∣ ∣⊕αFsL

∣ ∣ αT R

∣ ∣ aT R

∣ ∣ a+cL R

∣ ∣⊕βR R

∣ ∣ (a+c,β

)⊕αF0R

∣ ∣ αX

mR

∣ ∣⊕αFsR

∣ ∣ αT F0

∣ ∣T F0

∣ ∣⊕αL F0

∣ ∣ α⊕αR F0

∣ ∣ α⊕α,βF0F0

∣ ∣ (α,β)

⊕αX

mF0

∣ ∣ αp·F

sF0

∣ ∣T X

k

∣ ∣ aT X

k

∣ ∣ a+c+kd

L Xk

∣ ∣R X

k

∣ ∣⊕αF0X

k

∣ ∣ α ⊕ β

Xk

Xk

∣ ∣ (a+c,β

)k

=m

Xm

Xk

∣ ∣k6=m

⊕αFsX

k

∣ ∣ αT Fq

∣ ∣T Fq

∣ ∣⊕αL Fq

∣ ∣ α⊕αR Fq

∣ ∣ αp·F

0Fq

∣ ∣⊕αX

mFq

∣ ∣ α ⊕ α

,βFq

Fq

∣ ∣ (α,β)

q=s

p·F

sFq

∣ ∣q6=s

RT R

∣ ∣ cL R

∣ ∣R R

∣ ∣ (c,z)F0R

∣ ∣ zX

mR

∣ ∣FsR

∣ ∣ zR T

∣ ∣ a⊕βT T

∣ ∣ (a+c,β

)⊕αL T

∣ ∣ αR T

∣ ∣ a+cF0T

∣ ∣⊕αX

mT

∣ ∣ αFsT

∣ ∣R L

∣ ∣⊕αT L

∣ ∣ α⊕α,ζL L

∣ ∣ (α,ζ)

R L

∣ ∣⊕ζF0L

∣ ∣ ζp·X

mL

∣ ∣⊕ζFsL

∣ ∣ ζR R

∣ ∣ (a,x)

T R

∣ ∣ a+cL R

∣ ∣R R

∣ ∣ (a+c,x

+z)

F0R

∣ ∣ x+zX

mR

∣ ∣FsR

∣ ∣ x+z+sa

R F0

∣ ∣ xT F0

∣ ∣⊕ζL F0

∣ ∣ ζR F0

∣ ∣ x+z⊕ζF0F0

∣ ∣ (ζ,x+z)

⊕ζX

mF0

∣ ∣ ζFsF0

∣ ∣R X

k

∣ ∣⊕αT X

k

∣ ∣ αp·L X

k

∣ ∣R X

k

∣ ∣⊕ζF0X

k

∣ ∣ ζ ⊕ α

,ζX

kX

k

∣ ∣ (α,ζ)

k=m

p·X

mX

k

∣ ∣k6=m

⊕ζFsX

k

∣ ∣ ζR Fq

∣ ∣ xT Fq

∣ ∣⊕ζL Fq

∣ ∣ ζR Fq

∣ ∣ x+z+qc

F0Fq

∣ ∣⊕ζX

mFq

∣ ∣ ζ ⊕ ζ

Fq

Fq

∣ ∣ (ζ,x+z)

q=s

FsFq

∣ ∣q6=s

Xl

T Xl

∣ ∣ cL X

l

∣ ∣R X

l

∣ ∣F0X

l

∣ ∣ zX

lX

l

∣ ∣ (c,z

)

Xm

Xl

∣ ∣FsX

l

∣ ∣ zX

lT

∣ ∣ a⊕βT T

∣ ∣ (a+c−lβ,β

)⊕αL T

∣ ∣ α⊕αR T

∣ ∣ αF0T

∣ ∣X

mT

∣ ∣ a+c⊕αX

mT

∣ ∣ αFsT

∣ ∣X

lL

∣ ∣⊕αT L

∣ ∣ α⊕α,ζL L

∣ ∣ (α,ζ

)p·R L∣ ∣

⊕ζF0L

∣ ∣ ζX

mL

∣ ∣p·X

mL

∣ ∣⊕ζFsL

∣ ∣ ζX

lR

∣ ∣⊕αT R

∣ ∣ αp·L R∣ ∣

⊕α,ζR R

∣ ∣ (α,ζ

)⊕ζF0R

∣ ∣ ζX

mR

∣ ∣p·X

mR

∣ ∣⊕ζFsR

∣ ∣ ζX

lF0

∣ ∣ xT F0

∣ ∣⊕ζL F0

∣ ∣ ζ⊕ζR F0

∣ ∣ ζ⊕ηF0F0

∣ ∣ (x+z−l−

1η,η

)

Xl

F0

∣ ∣ x+l−1z

⊕ζX

mF0

∣ ∣ ζFsF0

∣ ∣X

lX

l

∣ ∣ (a,x

)T X

k

∣ ∣ a+cL X

k

∣ ∣R X

k

∣ ∣F0X

k

∣ ∣ l−1 x+z

Xl

Xl

∣ ∣ (a+c,x

+z)

Xm

Xl

∣ ∣FsX

k

∣ ∣ x+z+as

Xl

Xk

∣ ∣⊕αT X

k

∣ ∣ αp·L X

k

∣ ∣p·R X

k

∣ ∣⊕ζF0X

k

∣ ∣ ζX

lX

k

∣ ∣ ⊕ α

,ζX

kX

k

∣ ∣ (α,ζ

)k

=m

p·X

mX

k

∣ ∣k6=m

⊕ζFsX

k

∣ ∣ ζX

lFq

∣ ∣ xT Fq

∣ ∣⊕ζL Fq

∣ ∣ ζ⊕ζR Fq

∣ ∣ ζF0Fq

∣ ∣X

mFq

∣ ∣ x+z+qc

⊕ζX

mFq

∣ ∣ ζ ⊕ ζ

Fq

Fq

∣ ∣ (ζ,x

+z−lζ

)q

=s

FsFq

∣ ∣q6=s

TABLE VII. Vertical fusion tables