Symmetry TFT for Subsystem Symmetry

We generalize the idea of symmetry topological field theory (SymTFT) for subsystem symmetry. We propose the 2-foliated BF theory with level $N$ in $(3+1)$d as subsystem SymTFT for subsystem $\mathbb Z_N$ symmetry in $(2+1)$d. Focusing on $N=2$, we investigate various topological boundaries. The subsystem Kramers-Wannier and Jordan-Wigner dualities can be viewed as boundary transformations of the subsystem SymTFT and are included in a larger duality web from the subsystem $SL(2,\mathbb Z_2)$ symmetry of the bulk foliated BF theory. Finally, we construct the condensation defects and twist defects of $S$-transformation in the subsystem $SL(2,\mathbb Z_2)$, from which the fusion rule of subsystem non-invertible operators can be recovered.

One valid construction of fracton models arises from generalizing the ordinary gauge princeples [20] by introducing the tensor gauge theories [21][22][23][24], where gauge fields are tensor representations of the symmetry group. There is another foliation construction [6][7][8][25][26][27] where the spacetime manifold is a foliation of lower dimensional submanifold. The gauge invariant operators have restricted mobility in the foliated directions but are topological in the other directions without foliation. The two constructions are equivalent through the exotic-foliated duality [28,29].
From a symmetry point of view, fracton models are often realized by gauging the subsystem symmetry [26,30] or dipole symmetry [31,32] which generalizes the notion of symmetry by relaxing the topologicalness of the symmetry operators. Therefore, studying these generalized symmetries is of equal importance and will shed light on the underlying structure of fracton models. In this paper, we will focus on the subsystem symmetry. Subsystem symmetry allows symmetry transformations acting on rigid spatial submanifolds and it is sometimes referred to as "gauge-like" symmetry [33][34][35]. However, it should be viewed as a global symmetry rather than gauge symmetry because the subsystem symmetry operator acts nontrivially on the Hilbert space. It is natural to study subsystem symmetry by generalizing corresponding ideas in ordinary global symmetry, like selection rules [36], spontaneously breaking [37][38][39], anomaly inflow [40] and constraints on IR dynamics [11][12][13][14][15]. In particular, we will study the duality web and the generalization of symmetry topological field theory (SymTFT) for subsystem symmetry.
Duality is a powerful tool in theoretical physics, where the two apparently different Lagrangians describe the same theory. Here we focus on (1 + 1)d quantum field theories (QFTs) where the duality web has been revisited recently from the perspective of gauging a discrete symmetry [41][42][43][44][45][46][47]. We are interested in the duality transformation generated by symmetry manipulations such as gauging and stacking invertible phases [48][49][50]. For example, gauging the non-anomalous Z 2 symmetry of (1 + 1)d Ising conformal field theory (CFT) is a self-duality and the corresponding duality defect gives the simplest example of non-invertible symmetry [51,52]. Another famous example is the boson-fermion duality [53][54][55], where the Ising CFT is dual to a free Majorana fermion by first stacking a topological phase given by the Arf-invariant (Kitaev Majorana chain) and then gauging the diagonal Z 2 symmetry. Recently, generalizations of Kramers-Wannier (KW) and Jordan-Wigner (JW) duality has been studied in the context of subsystem symmetry [56,57], where a new subsystem non-invertible symmetry has been found.
SymTFT is another powerful tool that provides a unified picture to study duality transformations and symmetry manipulations [47,[58][59][60][61][62][63][64][65][66][67][68][69][70]. The idea of SymTFT is illustrated in Fig. 1. Given a d-dimensional theory T S with a finite symmetry S, the SymTFT is a (d + 1)-dimensional topological quantum field theory Z(S) that allows a topological boundary B sym S encoding the symmetry S of the original theory T S . The original theory T S can be expressed as an interval compactification of Z(S) with two boundaries. In the condensed matter literature, the similar idea of SymTFT has been proposed as symmetry/topological order correspondence [71,72].
The power of SymTFT is that the information of symmetry S and the dynamics are separately stored in the two boundaries. The left boundary is the topological boundary B sym S supporting the symmetry S and all symmetry manipulations take place on this boundary. The symmetry manipulations are implemented by fusing a co-dimension one symmetry defect of the SymTFT to the topological boundary. The right boundary is the dynamical (physical) boundary B phys T S that depends on the details of T S . As a concrete example, we give a review of the (2 + 1) BF theory as a SymTFT in Appendix A.
In this paper, we will propose a SymTFT for subsystem symmetry. We will focus on subsystem Z 2 symmetry in (2 + 1)d, which is a 2-foliated theory with one- Figure 1: Illustration of the SymTFT. We will get boundary theory after shrinking the slab. When fusing on the boundary, a co-dimension one symmetry defect D in the SymTFT will change the boundary condition, which corresponds to a symmetry manipulation/duality transformation of the boundary theory.
dimensional layers foliated in all spacial directions x, y. The natural candidate for the SymTFT is a theory with the same foliation structure but with an extra topological direction, which turns out to be the 2-foliated BF theory in (3 + 1)d [29,73]. This principle to construct subsystem SymTFT can apply to theories in higher dimensions, like the X-cube model, which we leave for future investigation.
Here is the organization of this paper. In Sec. 2, we review the (2 + 1)d subsystem Z 2 symmetry on the lattice and subsystem KW/JW duality transformation. In Sec. 3, we propose the (3 + 1)d SymTFT for subsystem Z 2 symmetry in (2 + 1)d and study the topological boundary conditions. In Sec. 4, we consider the SL(2, Z 2 ) symmetry of the subsystem SymTFT and the duality web of the boundary theories. In Sec. 5, we construct the condensation defects and twist defects in the bulk SymTFT. Finally, we conclude and point out interesting future directions in Sec. 6.
2 Subsystem symmetry and duality in (2 + 1)d In this section, we will review the subsystem Z 2 symmetry in (2+1)d regularized on a 2d square lattice and the duality transformations including the subsystem Kramers-Wannier (KW) transformation [56] and the subsystem Jordan-Wigner (JW) transformation [57].

Subsystem Z 2 symmetry on lattice
Consider a closed L x × L y square lattice. On each site there is a spin-1/2 state |s⟩ i,j where s = ±1, i = 1, · · · , L x and j = 1, · · · , L y . Denote the Pauli matrices at each site as X i,j , Y i,j , Z i,j and they act on the site in a canonical way The generators of subsystem Z 2 global symmetry are line operators acting on each row and column They satisfy (U x j ) 2 = (U y i ) 2 = 1 and flip the spin of all sites of jth-row or ith-column as illustrated in Fig. 2. We will denote the eigenvalues of U x j , U y i as (−1) u x j , (−1) u y i where u x j , u y i = 0, 1 are Z 2 -valued integers. These L x + L y operators are not independent and they are restricted by the constraint and there are L x + L y − 1 independent symmetry generators.
One can also insert the subsystem Z 2 defects along the time direction (represented by z) as shown in the middle diagram in Fig. 2. If the lattice is infinite, they are implemented by the Z 2 twist operators (e.g. U xz 0j in Fig. 2) on half line The operator U xz 0,j is mobile along the x-direction and is not mobile along the y-direction. Similarly, U yz i,0 is mobile along the y-direction and is not mobile along the x-direction. For periodic lattice, inserting defects on the lattice will twist the boundary condition for each row and column by j , t y i = 0, 1 are twist variables and t xy = 0, 1 is the boundary condition of the twist variables Although there are L x + L y + 1 twist parameters but the Hamiltonian with subsystem Z 2 symmetry depends only on the combinations t x and only L x + L y − 1 twist variables are independent. Given a (2+1)-theory T sub with the subsystem Z 2 symmetry, the eigenvalues of subsystem symmetry and twist boundary conditions will divide the Hilbert space into sectors with Z 2 -valued symmetry-twist labels Here {· · · } denotes the collection of variables for all j = 1, · · · , L y and i = 1, · · · , L x . The symmetry-twist labels have overall constraints With the above constraints, the Hilbert space is divided into 2 2(Lx+Ly−1) different sectors and the partition function for each sector is ) in the following discussion.

Coupling to background field
We can introduce background subsystem Z 2 symmetry gauge field (A z , A xy ) on the lattice. Consider a cubic spacetime lattice M 3 with L x × L y × L z sites and the topological z-direction is the time direction. The space component of the gauge field A xy lives on the xy-plaquette and the time component A z lives on the z-link, as shown in Fig. 3.
x z y (i, j, k + 1) The Z 2 -valued holonomies are regularized by summing the gauge fields along different cycles on the lattice. The holonomy of A z along the time direction is which is highly reducible and we can decompose it into w z,x;j , w z,y;i detecting the insertion of symmetry operator (U x j ) w z,x;j and (U y i ) w z,y;i respectively. The constraint (2.3) on the symmetry operators imposes a gauge redundancy (w z,x;j , w z,y;i ) ∼ (w z,x;j + 1, w z,y;i + 1). (2.11) On the other hand, the holonomy of A xy along x and y directions are (2.12) They detect the insertion of symmetry defects along the z-direction and are the same as the twist variables t x . They obey the same constraint For a generic subsystem Z 2 symmetry background (w z,x;j , w z,y;i , w x;j+ 1 2 , w y;i+ 1 2 ), the partition function is (2.14) It is related to the partition function in the sector with symmetry-twist label (2.9) by a discrete Fourier transformation where the summation over (u y i , u x j ) should obey the constraint (2.8).

Subsystem KW transformation
We can gauge the subsystem Z 2 symmetry by subsystem KW transformation N sub [56] are Pauli operators acting on the dual lattice. After gauging, the dual theoryT sub lives on the dual lattice and has a dual subsystem Z 2 symmetry. The Hilbert space of the dual theoryT sub is similarly divided into sectors labelled by the dual symmetry-twist variables (û x They are related to the symmetry-twist variables (u where symmetry/twist sectors are exchanged as shown in Fig. 4. As before, one hasŵ x;j =t x j ,ŵ y,i =t y i and (ŵ z,x;j+ 1 2 ,ŵ z,y;i+ 1 2 ) are the Fourier partners of (û x ) as in (2.15). Implied by (2.18), the partition function of the dual theoryT sub is related to the partition of the original theory T sub in (2.14) as (2.20) The summation of (w z,x;j , w z,y;i , w x;j+ 1 2 , w y;i+ 1 2 ) should obey the restrictions in (2.11) and (2.13).
Suppose the theory T sub is invariant under the subsystem KW transformation, which meansT sub = T sub . The subsystem KW transformation becomes a symmetry and we can insert the KW operator/defect N sub along a 2-dimensional surface M 2 by gauging half of the spacetime. If M 2 is the x-y plane, N sub is an operator acting on the Hilbert space. The fusion between the symmetry operator N sub and its orientation reversal N sub † is On the other hand, if M 2 is the z-x (or z-y) plane then N sub is a defect twisting the boundary condition. The fusion rule of the subsystem KW defect on the z-x plane is The fusion rules are first derived in [56]. We give an alternative derivation in Appendix C following [61].

Subsystem JW transformation
Besides the subsystem KW transformation that maps a bosonic T sub theory to another bosonic theoryT sub , we also have the subsystem JW transformation that maps the bosonic theory T sub to a fermionic theory T F,sub [57].
The subsystem JW transformation maps Pauli operators X i,j , Y i,j , Z i,j to Majorana fermion operators γ i,j , γ ′ i,j and vice versa. To preserve the standard anticommutation relation among Majorana fermions, one must attach a 1d JW tail (product of Pauli X operators) whose winding directions will lead to different choices of subsystem JW transformation. In Fig. 5, we give examples where the tail winds around the x direction and y direction and we will denote the two fermionic theories after each transformation separately as T F,x,sub and T F,y,sub . For the first choice, the explicit transformation is (2.23) The fermionic theory has subsystem Z 2 fermion parity symmetry (−1) F . Considering the symmetry operators and twists of (−1) F , the Hilbert space is divided into 2 2(Lx+Ly−1) sectors with labels (u ). Using the transformation (2.23), one can work out the mapping between symmetry-twist sectors in the bosonic and fermionic theory (2.24) We can introduce the background fields for subsystem Z 2 fermion parity symmetry (−1) F and define the corresponding holonomy variables as (s z,x;j , s z,y;i , s x;j+ 1 2 , s y;i+ 1 2 ). Similar to the bosonic case, the space direction holonomy has the following identifi- From the sector correspodence (2.24), we can derive the relation between the partition functions of the bosonic theory T sub and the fermionic theory T F,x,sub If the subsystem JW transformation winds along the y direction, we have a different transformation and a different symmetry-twist sector mapping Moreover, one can first perform a JW transformation winds along the x direction and then do an inverse JW transformation winds along the y direction, which ends to another bosonic theory T xy,sub . One can easily check that now the symmetry-twist sector labels (u ′x j , u ′y i , t ′x ) in this new bosonic theory are Combining different subsystem JW transformation, we get a duality web relating two bosonic theoies and two fermionic theories. A simple realization of the duality web starts from the plaquette Ising model (2.29) Applying the subsystem JW transformation winding along x and y direction seper-ately, we get two different plaquette fermion models (2.30) Further applying the inverse subsystem JW transformation along y direction to H Pfer,x , or along x direction to H Pfer,y , we will get another bosonic theory The duality web will be enlarged by further considering the subsystem KW transformation, which is elaborated in Sec. 4.

2-foliated theory as the subsystem SymTFT
In this section, we will give the analogy of SymTFT for subsystem Z N symmetry in (2 + 1)d. The candidate theory is the (3 + 1)d 2-foliated BF theory with level N (3.1) where the foliation is along x, y directions. The theory is topological along the remaining directions z, τ . From the exotic-foliated duality [28,29], we will focus on the dual formulation, the exotic tensor gauge theory (3.2) where the subsystem symmetry is more obvious. We will quantize the theory by picking the topological direction τ as the time direction. After quantization, we will see this theory supports a topological boundary B sym sub with a (2 + 1)d subsystem Z N symmetry. We will explore various bosonic and fermionic topological boundaries of the bulk theory. As an application, the subsystem KW and JW transformations have a subsystem SymTFT interpretation as switching between different topological boundaries.

2-foliated BF theory revisited
The candidate for subsystem SymTFT of our interest is the (3 + 1)d 2-foliated BF theory with level N The first term is a usual 4d BF theory where b is a 2-form gauge field and c is a 1-form gauge field, the second term gives a foliation of 3d BF theories along x 1 , x 2 direction where B 1 , B 2 , C 1 , C 2 are 1-form gauge fields, and the third term is the interaction term that couples the foliated fields and the bulk fields. In the following we will label the coordinates (x 0 , x 1 , x 2 , x 3 ) as (τ, x, y, z).
The 2-foliated BF theory (3.1) is equivalent to the exotic tensor gauge theory [29,73] (3. 2) The foliated-exotic duality is sketched in Appendix B by integrating out some auxiliary fields and redefining the others. In the action (3.2), A = {A τ , A z , A xy } and A = {Â τ ,Â z ,Â xy } are electric and magnetic gauge fields with the following gauge transformations where λ,λ are gauge parameters. The equations of motion for gauge fields A andÂ are In the exotic theory (3.2), there exists a naive SL(2, Z N ) symmetry which is hard to see in the original 2-foliated formulation. We will elaborate more on this SL(2, Z N ) symmetry regularized on the lattice in the next section.
The gauge invariant operators have restricted mobility due to the foliation. There exist the electric/magnetic line operators that are topological in the z-τ plane but cannot move freely along the x, y directions where C z,τ (x, y) is a curve in the z-τ plane and is localized at (x, y) in the ambient space. The exotic theory also has gauge invariant strip operators spanned along x or y directions for electric gauge field A. There are also hat versions for magnetic gauge fieldÂ. Here C x,z,τ (y) is a curve in the x-z-τ plane with fixed y, and C y,z,τ (x) is a curve in the y-z-τ plane with fixed x. The curve C x,z,τ (y) can be deformed in x-z-τ plane but not along y direction and the similar restricted mobility for C y,z,τ (x). The above properties of restricted mobility follow from the equations of motion (3.4) of the gauge fields A andÂ.

Quantization
We can quantize the exotic theory (3.2) by picking τ as the time direction with the Coulomb gauge A τ =Â τ = 0. The action (3.2) becomes with the canonical commutation relations between conjugate fields A andÂ The Gauss laws imply the flat condition.
We will consider the 2-foliated theory (or exotic tensor theory) on a spatial manifold M 3 = T 2 × S 1 , where (x, y) parameterize the torus T 2 and z is the coordinate of S 1 . The gauge invariant operators (3.7),(3.8) restricting to M 3 gives the electric line/strip operators (3.12) and the magnetic line/strip operatorŝ with the following commutation relations where the extra phase exp(±2πi/N ) indicates a mixed t' Hooft anomaly between the two sets of subsystem Z N symmetry generated by the electric and magnetic line/strip operators.

Topological boundaries with subsystem symmetry
In this subsection, we will study the topological boundaries of the exotic theory (3.2), which are also the topological boundaries of the 2-foliated theory because of the foliated-exotic duality. The boundary theory has subsystem Z N symmetry. For simplicity, we will present the case for N = 2 which is straightforward to be extended to general N . We will study the bosonic topological boundaries corresponding to the Dirichlet boundary condition for A andÂ and the fermionic boundary from the subsystem JW transformation on the bosonic boundary. In addition, we will give a bulk-boundary point of view of subsystem KW and JW transformation.
As reviewed in Sec. 2, it is natural to regularize theories with subsystem symmetry on a lattice. On a finite lattice, the Gauss laws impose nontrivial constraints between gauge invariant operators. For example, using the Gauss laws (3.11), the holonomy of electric gauge field A z can be split as where A y (x) and A x (y) are operators only depend on x and y. The split of holonomy (3.17) implies the decomposition of the line operator where W z,y (x), W z,x (y) are two line operators along z-directions that are separately mobile along y and x directions. However, this decomposition is not unique because of the gauge redundancy which leaves dzA z invariant modulo 2π. Both W z,y (x) and W z,x (x) flip the sign under the transformation but the combination W (x, y) is invariant. On the other hand, the strip operators W (x 1 , x 2 ) and W (y 1 , y 2 ) are mobile along z directions with the constraint There are similar gauge redundancy and constraint for magnetic operatorsŴ .

Discretization on a lattice
Discretizing the boundary manifold M 3 as a L x × L y × L z periodic lattice with label {x i , y j , z k }, we have in total 2(L x + L y ) electric operators: line operators W z,y (x i ), W z,x (y j ) and strip operators W (x i , x i+1 ), W (y j , y j+1 ) with i = 1, · · · , L x , j = 1, · · · , L y . On the lattice, the gauge redundancy (3.19) and the constraint (3.20) become and, ) on the dual lattice with similar gauge redundancy and constraint.
The discretized version of the algebras between W andŴ (3.15),(3.16) is Dirichlet boundary condition for gauge field A The gauge redundancy (3.21) and constraint (3.22) for electric operators W are consistent to those satisfied by the holonomies (w z,x;j , w z,y;i , w x;j+ 1 2 , w y,i+ 1 2 ) introduced in (2.10),(2.11),(2.12) and (2.13) with the following correspondence Therefore, we can introduce a canonical basis of the Hilbert space of the 2-foliated BF theory on the boundary M 3 |w⟩ := |w z,x;j , w z,y;i , w x;j+ 1 2 , w y;i+ 1 2 ⟩, (3.27) and the electric operators W are diagonalized as (3.28) This canonical basis (3.27) defines the Dirichlet boundary condition for gauge field A where the values of A are fixed at the boundary.
On the other hand, the magnetic operatorsŴ conjugate to electric operators W will shift the eigenvalues when acting on the state |w⟩ which follows from the algebras (3.23) and (3.24). Because the magnetic operatorsŴ along the spatial/temporal cycle shift the temporal/spatial holonomies w of electric gauge field A, they are identified one-to-one to the subsystem Z 2 symmetry and twist operators in (2.2),(2.4) Therefore, the boundary represented by the |w⟩ basis is a topological boundary supporting the subsystem Z 2 symmetry generated by the magnetic operatorsŴ . The general boundary state |w⟩ with nontrivial W -holonomies is created by acting magnetic operatorsŴ on the vacuum state |0⟩ where all W -holonimies are trivial As a consistency check, the invariance of |w⟩ under the gauge redundancy of magnetic operatorsŴ implies the constraints (2.13) and the constraint among magnetic operatorsŴ requires that the invariance of the state |w⟩ under the gauge transformation (2.11).

Dirichlet boundary condition for gauge fieldÂ
Alternatively, one can consider the dual basis whereŴ operators are diagonalized (3.33) The dual basis (3.32) defines the Dirichlet boundary condition for the gauge fieldÂ. Acting on the state |ŵ⟩, the electric operators W will shift the dual holonomies Therefore, the electric operators W can be identified as the symmetry and twist operators. The boundary state |ŵ⟩ corresponds to a topological boundary supporting the subsystem Z 2 symmetry generated by electric operators W .
The dual state |ŵ⟩ is related to original state |w⟩ via a discrete Fourier transformation, where we introduce M v as the set of Z 2 -valued vector w satisfying the gauge redundancy and constraint, (3.36) The restrictions in (3.36) for w automatically impose restrictions forŵ.

Subsystem KW transformation
Based on the SymTFT picture, we consider the 2-foliated BF theory on the 4dimensional manifold M 3 × [0, 1] where τ is the coordinate of the time interval. The initial state at τ = 0 is the dynamical boundary state |χ⟩ and the final state at τ = 1 is the topological boundary state. Given any (2 + 1)-dimensional theory T sub with a subsystem Z 2 symmetry, we can write down the dynamical boundary state as, (3.37) where the coefficient is the partition function of T sub on M 3 coupled with the subsystem Z 2 symmetry background w.
Choosing |w⟩ as the topological boundary state at τ = 1, one has, which projects back to the partition function of T sub . Alternatively, choosing the dual boundary state |ŵ⟩ at τ = 1 reproduces the partition function of the dual theory The change of boundary conditions in the 2-foliated BF theory recovers the subsystem KW transformation (2.20) between the boundary theories.

Fermionic boundary conditions
Based on the discussion of the subsystem JW transformation in the previous section, we can further consider the fermionic topological state |s⟩ = |s z,x;j , s z,y;i , s x;j+ 1 2 , s y;i+ 1 2 ⟩ and write the partition function of (2+1)d fermionic theory with subsystem symmetry as the path integral ⟨s|χ⟩.
For example, the fermionic topological boundary state corresponding to the fermionic theory T F,x,sub after the subsystem JW transformation (2.26) is and M u,wz the set, (−1) u y i = 1, (w z,x;j , w z,y;i ) ∼ (w z,x;j + 1, w z,y;i + 1) .
(3.41) The fermionic state |s⟩ diagonalizes the electric operators W along the y direction, and the composite operators along x direction made up by the electric operators W sandwiched by a pair of magnetic operatorsŴ nearby The fermionic subsystem Z 2 parity symmetry is generated by magnetic operatorsŴ (3.43) There exists another fermionic topological state |s ′ ⟩ = |s ′ z,x;j , s ′ z,y;i , s ′ x;j+ 1 2 , s ′ y;i+ 1 2 ⟩ which produces the ferrmionic theory T F,y,sub after the subsystem JW transformation along y direction. The fermionic topological state |s ′ ⟩ diagonalizes the line operators, and strip operators, where W z,y (x i ), W (x i , x i+1 ) are sandwiched by a pair ofŴ operators instead. The fermionic subsystem Z 2 parity symmetry is still generated by magnetic operatorsŴ .

Subsystem JW transformation
Consider the subsystem SymTFT with the dynamical boundary state (3.37) at τ = 0 given by the (2 + 1)-dimensional bosonic theory T sub . Implementing the fermionic topological boundaries |s⟩, |s ′ ⟩ at τ = 0 and shrinking the slab gives two fermionic theories T F,x,sub and T F,y,sub whose partition functions are, Z T F,x,sub (s) = ⟨s|χ⟩, Z T F,y,sub (s ′ ) = ⟨s ′ |χ⟩. (3.44) They are related to the bosonic theory T sub by performing the subsystem JW transformations along x and y directions respectively.
4 Subsystem SL(2, Z 2 ) transformation and the duality web In the previous section, we propose the 2-foliated BF theory in (3 + 1)d as the subsystem SymTFT for subsystem Z N symmetry in (2 + 1)d and explore various bosonic and fermionic topological boundaries. In this section, we will see that different topological boundaries are transformed from one to the other via the topological operators associated with the global symmetries of the bulk theory.
In the exotic theory (3.2), we identify a naive 0-form SL(2, Z 2 ) symmetry There should exist corresponding co-dimension one symmetry defects implementing this symmetry. Here, we will mainly focus on the co-dimension one symmetry defects extended along the manifold M ′ 3 parallel to the boundary manifold M 3 such that they act on the Hilbert space as operators. Fusing the topological operators with the boundary implements the SL(2, Z N ) transformation of the boundary theory. We will see the S-transformation generates the subsystem KW transformation, while the T -transformation stacks a phase to the boundary theory. The phase (4.2) is the subsystem symmetry protected topological (SSPT) phase [40] * in (2 + 1)d.
However, the naive SL(2, Z 2 ) transformation (4.1) has ambiguities on the lattice. For example, when we do S-transformation on line operators k A z i,j,k+ 1 2 , the holon- * In [40], the Lagrangian of this SSPT is where the auxiliary field Φ xy guarrentees the flat condition of the gauge field A.
omy of electric gauge field A along the z-direction, one expects to map the electric gauge field A z operators to the nearby magnetic gauge fieldÂ z on the dual lattice. This leads to four inequivalent choices z A z i± 1 2 ,j± 1 2 ,k because the line operators cannot move freely at the x-y plane.
Therefore, in this section, we will formulate the proper S-and T -transformations on the lattice and study their action on the operators and topological states with a focus on N = 2. The proper SL(2, Z 2 ) symmetry transformation after discretization should have the following properties 1. It should be a symmetry of the discretized version of the exotic action (3.2).

It should preserve quantum algebras (3.23) and (3.24).
With the above definition, we find more transformations on the lattice than in the field theory due to the restricted mobility of the operators. Therefore, we will denote the SL(2, Z 2 ) transformation on the lattice as subsystem SL(2, Z 2 ) transformation. Besides recovering the subsystem KW and JW transformations, we will find more duality transformations by implementing the subsystem SL(2, Z 2 ) transformation on the boundary. The whole duality transformations are summarized in the duality web (Fig. 6).

Subsystem S-transformation
The S-transformation in (4.1) will exchange the electric operators {W } with the magnetic operators {Ŵ }. On the lattice, there are two choices for operators along the y direction that preserve the algebras (3.23) and (3.24). Similarly, there are two choices for operators along the x direction: W (y j , y j+1 ) and W z,x (y j ) which are also labelled by ±. In total, there are four choices and we will denote them as S ++ , S +− , S −+ , S −− . The pair (±, ±) stands for different choices for operators depending on x and y. By reflections R x , R y , we can relate the four different subsystem S-transformations. Figure 6: The duality web between four bosonic theories T sub , T xy,sub ,T sub ,T xy,sub and four fermionic theories T F,x,sub , T F,y,sub ,T F,x,sub ,T F,y,sub with susbsytem Z 2 symmetry. The duality transformation is generated by subsystem SL(2, Z 2 ) transformation on the lattice: (1) The subsystem S-transformation implements the subsystem KW transformation. (2) There are four different choices of subsystem T -transformations T ±± which stack a SSPT phase SSPT ±± to the boundary state. In this figure, we omit the index of subsystem T -transformations. There also exist nontrivial compositions of (3) Subsystem JW transformation is a composition of subsystem SL(2, Z 2 ) transformations. For example, the bosonic theory T sub and the fermionic theory T F,x,sub (T F,y,sub ) are related by subsystem JW transformation, which is equivalent to performing S −1 , T 2 +− (T 2 −+ ) and S transformation sequentially.
The subsystem S-transformations will generate the subsystem KW transformation on the boundary by mapping the topological boundary state |w⟩ to the dual state |ŵ⟩. For example, applying S ++ on |w⟩ leads to where we obtain the dual state |ŵ⟩ and we emphasize that the dual holonomiesŵ is the same to the original ones w in values. |0⟩ = S ++ |0⟩ is the vacuum of the dual state and it is the eigenstate of electric operators W with trivial eigenvalues. Different choices of subsystem S-transformation S ±± leads to some rearrangement of the vectorŵ. Without loss of generality, we will consider S ++ in the following discussion and simply denote it as S.

Subsystem T -transformation
For the subsystem T -transformation in (4.1), one needs to dress every magnetic oper-atorŴ with a nearby electric operator W and the choice is not unique. For operators along y direction, the T -transformation choices preserving the quantum algebras are ). There are also four choices in total which is denoted as T ++ , T +− , T −+ , T −− .
The subsystem T -transformations will stack an extra phase when acting on the topological boundary. For example, with the expression (3.31) of the topological boundary state |w⟩, applying T ++ leads to a new topological boundary state where T ++ |0⟩ ∼ |0⟩ because they satisfy the same operators equation (3.28), and we will assume |0⟩ is invariant under the action of T ++ . The phase SSPT ++ (w) := (−1) j w z,x;j w x;j+ 1 2 + i w z,y;i w y;i+ 1 2 is one choice of lattice regularization of the SSPT phase (4.2). In general, acting the subsystem T -transformations T ±± on the topological boundary |w⟩ will stack the phase SSPT ±± (w) := (−1) j w z,x;j w x;j± 1 2 + i w z,y;i w y;i± 1 2 , (4.8) where the four choices of subsystem T -transformations lead to four choices of regularization of SSPT phase (4.2) on the lattice.
In the naive SL(2, Z 2 ) transformation (4.1) of the field theory, acting T -transformation twice is the identity transformation. However, on the lattice, composing different subsystem T -transformations will lead to four distinct operations where the indices follow the sign rule T 2 pp ′ ,qq ′ = T p,q T p ′ ,q ′ , p, q = ±. (4.11) We can further compose the operations in (4.9) with the general rule

Duality web from the subsystem SL(2, Z 2 ) transformation
The duality web (Fig. 6) is generated by implementing the subsystem SL(2, Z 2 ) transformations consecutively. In particular, the subsystem JW transformation is equivalent to performing S −1 , T 2 +− (T 2 −+ ) and S transformation sequentially. This is easy to see in the transformation of the operators. For example, the composition of subsystem T -transformation T 2 +− gives the following transformation where the operators after transformation form the same set of operators that diagonalize the fermion topological state |s⟩ in (3.42) if we exchange the role of W and W . It suggests that if we perform a subsystem KW transformation after stacking the phase SSPT 2 +− (w), we will obtain a new fermionic topological state, where in the second equality we use the relation (4.5 We can rewrite |w ′ ⟩ into |ŵ ′ ⟩ using the KW relation (3.35) and get (4.16) Summing w ′ z,x;j and w ′ z,y;i produces two restrictions, which shows that the dual fermionic state |ŝ⟩ is the subsystem JW transformation of the dual state |ŵ ′ ⟩ resembling (3.40). The above derivation gives the following equivalence acting on the dual state |ŵ⟩. Similarly, we also have 20) therefore the phases SSPT 2 +− and SSPT 2 −+ are both fermionic subsystem SPT phases. † With these identifications, we can generate any path in the duality web. For example, shows that the subsystem KW transformation of SSPT 2 −− |ŵ⟩ leads to the bosonic topological boundary state which is obtained by performing inverse subsystem JW transformation along the y direction after a subsystem JW transformation along the x-direction. From (4.21), the phase SSPT 2 −− is a bosonic phase. Based on the above analysis, we can obtain a duality web as shown in Fig. 6.

Topological defects in the subsystem SymTFT
In this section, we will construct the co-dimensional one symmetry defects generating the SL(2, Z 2 ) 0-form symmetry. It was shown in [61,74,75] that in a (d+1)-dimensonal TFT, such kind of symmetry defects D extending along a co-dimension one hypersurface M d are built by condensing certain types of topological defects L along M d . If the topological defects L generate a q-form symmetry inside M d , the condensation defect D is equivalently understood as gauging the q-form symmetry inside M d which is referred to as 1-gauging of the q-form symmetry. In the appendix A, we give an example of the condensation defect generating the electric-magnetic Z 2 symmetry in (2 + 1)d BF theory with level N .
In particular, we will study the condensation defects, topological defects associated with the SL(2, Z 2 ) symmetry, of the 2-foliated BF theory. We will construct the condensation defects in 2-foliated BF theory along M 3 , a 3d manifold parallel to the boundary, by condensing line/strip operators on M 3 . We will also discuss the twist defects by putting a "Dirichlet" boundary condition for the condensation defects. We will re-derive the subsystem non-invertible fusion rules by the fusion of twist defects. Regarding to the multiple choices of SL(2, Z 2 ) transformation on the lattice, we will focus on S ++ and T ++ and omit the subscript. The discussions of other choices are similar.

Conventions on operators and algebras
For later convenience, we introduce U I andÛ I as the collection of electric and magnetic line/strip operators respectively where I = 1, · · · , 2L x + 2L y . We will use the lattice size integer n ≡ L x + L y for simplicity. In this convention, the quantum algebras (3.23) and (3.24) between electric and magnetic operators has a compact form, where Ω IJ is a 2n × 2n symmetric matrix, We will then formulate the general operators, the algebras between them and their actions on the boundary states. The general operator is parametrized by two 2n-dimensional vectors with Z 2 -valued entries α = (a, b) := (a 1 , a 2 , · · · , a n , b 1 , b 2 , · · · , b n ), α = (â,b) := (â 1 ,â 2 , · · · ,â n ,b 1 ,b 2 , · · · ,b n ).

S-defect
We first study the condensation defect of S-transformation, S-defect. As shown in Fig. 7, it maps the electric operator U I to magnetic operatorÛ I and vice versa. Fusing to the boundary, S-defect maps the boundary state |α⟩ to the dual state |α⟩ with the same holonomyα = α. We will construct the condensation defect on M 3 in the bulk using the bra-ket trick, S = α=α∈Mv |α⟩⟨α|, (5.19) which manifests its action on the state. We useα = α to emphsize that we sum over the dual states |α⟩ that have the holonomy of the state |α⟩. A further check is that this definition (5.19) gives the correct S-transformation on operators We will use the property that the operators K[α,α] with different α andα are orthogonal to each other in the trace to project out the coefficients λ α,α , 24) and the condensation defect is

Twist S-defect
We can consider twist S-defect on a 3d manifold M 3 with a boundary ∂M 3 = M 2 . The boundary can be x-y plane, x-z plane or y-x plane. We impose the "Dirichlet" boundary condition for the defects U IÛI condensing along M 3 . The "Dirichlet" boundary condition is defined as follows. The operators {U IÛI } generate a subsystem Z 2 symmetry along M 3 and we denote the corresponding gauge fields as (A ′z , A ′xy ). For x-y plane we require the x-y component A ′xy to vanish at the boundary. On the other hand, if M 2 is the x-z plane or y-z plane, we will impose ∂ x A ′z = 0 or ∂ y A ′z = 0 separately. The Dirichlet boundary is topological along the normal direction given these boundary conditions and see Appendix C for a detailed discussion about this.
We will denote the twist defect as V 0 (M 2 , M 3 ). As shown in Fig. 8, after shrinking the slab, the twist defect will implement a half-space gauge and create a subsystem KW duality defect N sub . From the fusion of twist defect V 0 (M 2 , M 3 ), the fusion rule of the subsystem KW duality defects can be recovered. We will discuss the case where M 2 is placed on x − y plane and x − z plane in detail. After shinking the slab, the twist defect will create a duality defect as an interface between the original boundary theory T S and the theory D(T sub ) after gauging subsystem Z 2 symmetry.
We will also study the fusions between general operators with twist defects. Because the line operators and strip operators have different restricted mobility, we will discuss about their fusions separately. In our notation, the first n elements of U I ,Û I are strip operators and the rest n elements are line operators. We will write the 2n-dimensional vector α,α into a pair of n-dimensional vector α = (a, b),α = (â,b), (5.28) where a, b labels the strip operators andâ,b labels the line operators. In this notation, for example, the general strip operator is K[(a, 0), (â, 0)].
x-y plane We first consider M 2 to be the x-y plane, for example z = 0. z-direction is topological and the strip operators can move along the z-direction and fuse with the twist defect.
Since The fusion rule between the general strip operators K[(a, 0), (â, We can also discuss the fusion between twist defects and it is sufficient to discuss the fusion between V 0 and V † 0 . Let's put another twist defect V 0 [M 2|ϵ , M 3 ] at z = ϵ and consider the fusion between V 0 [M 2|0 , M 3 ] × V 0 [M 2|ϵ , M 3 ] with ϵ → 0. Here we use M 2|0 (M 2|ϵ ) to emphasize M 2 is located at z = 0 (z = ϵ). Since the condensation defects can also be understood as gauging the (2+1)d subsystem symmetry on M 3 , we can derive the fusion rule in a similar way following the discussion in Appendix C and get, the Euler factor introduced in (C.22). We have a condensation of strip operators that are mobile along z-direction. If we put (5.32) on the top of the topological boundary |0⟩ at τ = 1 where A = 0 at the boundary, then electric strip operators W (x j , x j+1 ), W (y i , y i+1 ) are absorbed into the boundary and we have, x-z plane We then consider M 2 to be the x-z plane, for example at y = 0. Only line operators W z,y (x i ) andŴ z,y (x i+ 1 2 ) can move along y-direction and fuse with the twist defect. Therefore, we will only consider the fusion with the general line operator We then consider the fusion of two twist defects V 0 and V † 0 . Following the Appendix C, the fusion rule is, ) operators that are mobile along y-directions. If we put it on the top of the topological boundary |0⟩ at τ = 1, then W y (x i ) are absorbed into the boundary and we have, which is equivalent to, using the correspondence (3.30) and the relation t y

T -defect
We will now consider T -defect, the condensation defect of the T -transformation, whose action on the operators is, which is a condensation of the electric operator U I only.
Following the discussion of twist S-defect, one can similarly study the twist T -defects. When restricted to the topological boundary, the twist T -defect will create an interface separating two theories with a difference of an SSPT phase.

Conclusion and discussion
In this paper, we initiate the study of the subsystem symmetry and the associated dualities from a bulk SymTFT point of view. To demonstrate this idea, we study the example of 2-foliated BF theory with level N in (3 + 1)d as the subsystem SymTFT of the subsystem Z N symmetry in (2 + 1)d. We analyze the topological boundaries and construct condensation defects of this specific model with N = 2. We interpret the duality transformations of the boundary theory, such as subsystem KW and JW transformation, as the change of topological boundaries which is further implemented by fusing condensation defects of the subsystem SL(2, Z 2 ) symmetry of the bulk subsystem SymTFT on the boundary. On the lattice, the subsystem SL(2, Z 2 ) symmetry has a richer structure than in the field theory. The subsystem T transformation will stack a subsystem SPT phase whose bosonic or fermionic feature depends on the regularization of the lattice. We will leave the detailed study and classification of subsystem SPT phases in the future work. From the subsystem SL(2, Z 2 ) symmetry, we find new dualities among bosonic and fermionic models with subsystem Z 2 symmetry. We summarize the duality web in Fig. 6.
There are many interesting follow-up directions. First, it is natural to extend the study of subsystem SymTFT to other models. For the subsystem Z N symmetry in (2 + 1)d, the subsystem SymTFT is expected to have more diverse topological boundaries that can support subsystem parafermionic structures. Furthermore, we can study models with subsystem symmetry in higher dimensions, for example, the X-cube model [3], where there are fracton excitations. The Z N X-cube model is a 3-foliated theory in (3 + 1)d and the corresponding subsystem SymTFT should be the 3-foliated BF theory with level N in (4 + 1)d where the first term is bulk BF term with 3-form gauge field b and 1-form gauge field c, the second term is the foliated BF term with 2-form gauge field B k and 1-form gauge field C k and the third term is the interaction term. It is interesting to classify the topological boundaries and topological operators of this subsystem SymTFT and explore the duality web of the X-cube model.
Finally, subsystem SymTFT provides a bulk-boundary point of view to study subsystem symmetry. Recently, there are other efforts to study fracton models from bulk-boundary correspondence [76][77][78]. Subsystem SymTFT also provides hints to study fracton statistics [79,80]. The quantume algebras (3.15) and (3.16) resemble the braiding statistics in (2 + 1)d Z N gauge theory. Besides, one more topological direction in the bulk will give fracton (or excitations with other restricted mobility) an extra direction to move, which might lead to interesting braiding structures. We will leave these interesting questions for future study.

A A review on ordinary BF theory as SymTFT
To illustrate the basic idea of SymTFT, we consider a (1 + 1)d theory T Z N with Z N symmetry. The corresponding SymTFT Z(Z N ) is the (2 + 1)d BF theory with level N , whereÂ, A are 1-form gauge fields. It is a Z N gauge theory and is the low energy description of the toric code for N = 2 in the condensed matter literature [81]. Fix a gauge A 0 =Â 0 = 0, the canonical quantization gives, A andÂ are conjugated with each other like position and momentum. For simplicity, we place the BF theory on a spatial torus T 2 , the physical operators are Wilson loops defined as, with Γ ∈ H 1 (T 2 , Z). Since the holonomies of A andÂ are periodic, they are quantized as, The operators satisfy the commutation relation, Let's focus on the partition function Z T Z N of the theory and see how the SymTFT applies. We can introduce a canonical basis of the where a = N A/2π,â = NÂ/2π and we have a,â ∈ H 1 (T 2 , Z N ). The integration γ ∧ a = Γ a gives the holonomy along Γ. The two bases are related by a discrete Fourier transformation, On the other hand, the physical boundary B phys gives a dynamical boundary state |χ⟩ which depends on the partition function of theory T Z N with given the Z N holonomies of Choosing different topological boundary states, the path-integral of the BF theory on the slab gives, agrees with the torus partition function of T Z N and, which is the partition function of the orbifold theory T Z N /Z N , the Kramers-Wannier duaity of T Z N . In other words, the Z N gauging of T Z N can be viewed from the SymTFT as switching the topological boundary state from |a⟩ to |â⟩.
When N = 2, there also exists a topological boundary |s⟩, where s ∈ H 1 (T 2 , Z 2 ) stands for the spin structure, such that JW transformation can be encoded as Z where Arf(s) ≡ s 1 s 2 is the Arf-invariant where s i = Γ i s is the spin structure along Γ icycle (s i = 0 is chosen to be the NS boundary condition). The topological boundary state |s⟩ can also be expressed as, 13) and the transition amplitude ⟨s|χ⟩ is, which gives the partition function of the fermionic theory after JW transformation.
The (2 + 1)d BF theory has a Z 2 symmetry which exchanges the two gauge field, The corresponding symmetry defect D Z 2 [M 2 ] along a surface M 2 can be constructed as, which is a condensation of the defect WŴ −1 along M 2 . If M 2 is a time slice, one can check, and, using the quantum algebra.

B Duality between (3.1) and (3.2)
In this appendix, we sketch the duality between the 2-foliated BF theory (3.1) and the exotic tensor gauge theory (3.2). See also [29]. Begin with the 2-foliated theory (3.1), we split the coordinates (x 0 , x 1 , x 2 , x 3 ) as (τ, x i ) with i = 1, 2, 3 and denote (x 1 , x 2 , x 3 ) as (x, y, z). The action can be written as, for b τ x , b τ y . We can solve b yz , b xz , C y z , C x z and substitute them back to the action. Moreover, integrating b τ z gives, such that we can define A xy = C y x +∂ x c y = C x y +∂ y c x . After renaming other variables, the action is rewritten as, (B.8) which reproduces the exotic tensor gauge theory (3.2).

C Derivation of fusion rule of subsystem KW defects
In this section, we will re-derive the fusion rule between two subsystem KW defects N sub × N sub after the formulation of gauging a subsystem symmetry in a cohomology language ¶ . The derivation is a direct generalization from the fusion of duality defects of guaging 0-form Z N symmetry [61]. For simplicity, we will keep N = 2.

C.1 Conventions
Denote the gauge fields A and its dualÂ for subsystem Z 2 symmetry as the pair, The gauge transformation is, where the action of δ on a function is defined as, δf (x, y, z) = (∂ z f (x, y, z), ∂ x ∂ y f (x, y, z)).

(C.3)
The flatness condition is written as, and one can check δ 2 f (x, y, z) = 0 automatically.
To perform the summation formally, it is useful to introduce the 0-cochain C 0 sub (M 3 ) as the set of functions f on M 3 , 1-cochain C 1 sub (M 3 ) as the set of the pairs g = (g z , g xy ) where g z , g xy are both functions on M 3 , and 2-cochain C 2 sub (M 3 ) as the set of functions denoted by h xyz on M 3 . The coboundary operator δ acts on C * sub as, and it satisfies δ 2 = 0. One can define a product * · * which sends C m sub (M 3 )×C n sub (M 3 ) to C m+n sub (M 3 ) where C m+n sub (M 3 ) with m + n > 2 is defined to be trivial. For example, when one of C * sub is C 0 sub whose elements are functions, the product is the usual multiplication; and for g, g ′ ∈ C 1 sub (M 3 ) one can assign g · g ′ ≡ g xy g ′z + g ′xy g z . Let's consider the cohomology ‖ H * = Z * /B * where Z * (B * ) contains closed (exact) cochains. For example contains scalar functions that only have x or y dependence. Because of the flatness condition, the subsystem gauge field A andÂ belong to H 1 sub (M 3 , Z 2 ), closed 1cochains modulo out the exact 1-cochain (the gauge transformation), and we have where |H 0 sub (M 3 , Z 2 )| is the dimension of the cohomology group H 0 sub (M 3 , Z 2 ). The labels of partition functions are omitted for more concise expressions. The gauge field A will take values in its gauge equivalent class H 1 sub (M 3 , Z 2 ) and the integral is ‖ Strictly speaking, H * are not cohomology groups because the closeness condition is not preserved under the product. For example, given f ∈ H 0 sub (M 3 ) and g ∈ H 1 sub (M 3 ) one can check, which does not vanish. Nevertheless, we do not need this property in the proof. where w andŵ are holonomies of A andÂ. The formal expression (C.8) differs from the regularized one (C.7) by the normalization factor |H 0 sub (M 3 , Z 2 )| (instead of |H 1 sub (M 3 , Z 2 )| = 2 Lx+Ly−1 ) as suggested in [61]. We also need to define relative cohomology H 1 sub (M 3 , M 2 , Z 2 ) where M 2 = ∂M 3 is the boundary where the gauge fields A = (A z , A xy ) should satisfy the "Dirichlet" boundary condition at the boundary M 2 . First, we need to define what the "Dirichlet" boundary condition means for the gauge fields A z and A xy . If M 2 is the x-y plane we can just set A xy = 0 at the boundary. The holonomy of A z dz split into A y (x) and A x (y) and they depend on x and y separately. There also exists a gauge transformation which shifts A y (x) → A y (x) + θ and A x (y) → A x (y) − θ by some constant θ so that A z dz is invariant. If M 2 is the x-z plane we will require A y (x) to be a constant at the boundary and it is gauge equivalent to zero. Therefore we impose ∂ x A z = 0 at the boundary.
In the next subsection, we will consider the KW defects defined by gauging the subsystem Z 2 symmetry in half of the spacetime M 3 with the "Dirichlet" boundary condition imposed at the boundary. To do this, we need to couple the theory to a dynamical subsystem Z 2 gauge theory with flat gauge field (A z , A xy ). The subsystem Z 2 gauge theory can be represented as * * 1 π dxdydz ϕδA = 1 π dxdydz ϕ(∂ z A xy − ∂ x ∂ y A z ), (C.11) where ϕ is a periodic scalar field that serves as a Lagrangian multiplier enforcing (A z , A xy ) to be properly quantized and Z 2 -valued. The Dirichlet boundary condition * * This is a generalization that ordinary q-form Z N gauge theory in D-dimenson can be represented by the BF theory with level N [84][85][86][87], where A q+1 is the (q + 1)-form gauge field and B D−q−2 is the Lagrange multiplier enforcing A q+1 to be Z N -valued.
introduced above is topological along the normal direction. To see this, we need to deform the locus of the boundary slightly and see the variation of the action. For example, if the boundary is x-y plane at z = 0 and we deform it to z = ϵ, the difference can be written as the surface integral at z = 0 and z = ϵ using Stokes theorem z=0 dxdyϕA xy − z=ϵ dxdyϕA xy , (C. 12) which is zero due to the boundary condition A xy = 0. On the other hand, if the boundary is x-z plane at y = 0 and we deform it to y = ϵ, the difference is, 13) and the boundary condition ∂ x A z = 0 is sufficient to set it zero.
We will also see how these choices of boundary conditions give the correct fusion rule in the following derivation.

C.2 Fusion rule of subsystem KW defects
We first consider the case where the defect N sub is along the x-y plane and acts as a symmetry operator. Our strategy, as shown in Fig. 9, is to put two parallel subsystem KW operators with a seperation of ϵ and compute the partition function in the region between two operators. As we take the limit ϵ → 0, we get the fusion of two operators. It is equivalent to performing 1-gauging on a co-dimension one surface [75]. (C.28) The line operator ϵ k=1Â z i+ 1 2 ,j+ 1 2 ,k of the dual fieldÂ z are subsystem symmetry defects. Recall that the line operator can be decomposed into two line operators separately movable along x and y directions and we use the labels [· · · ] x and [· · · ] y to represent them.
We will take the limit ϵ → 0 while fixing the holonomies w. The first line vanishes in the limit. Another point of view is, since A xy | z=0 = A xy | z=ϵ = 0 the holonomies of A xy vanishes and w y,i+ 1 2 = w x,j+ 1 2 = 0 and first line is trivial. Therefore we only need to consider the second line which implies the fusion rule, N sub † ×N sub = w z,y;i ,w z,x;j /∼ (U y i ) w z,y;i (U x j ) w z,x;j = 1 2 , M 2|0 ∪ M 2|ϵ , Z 2 )| = 1 † † and N sub † is normalized as 3 , Z 2 ]N sub . In the sum, we mod out the gauge redundancy ∼ of the holonomies.
Let's then consider the case where the defect N sub is along the x-z plane and acts † † The elements f in H 0 sub (M We can recover (2.22) using w y,i+ 1 2 = t y i + t y i+1 .