KIAS-P23043

Symmetry TFT for Subsystem Symmetry

Weiguang Cao1,2 and Qiang Jia3

1 | Kavli Institute for the Physics and Mathematics of the Universe,
---|---
| University of Tokyo, Kashiwa, Chiba 277-8583, Japan
2 | Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
3 | School of Physics, Korea Institute for Advanced Study, Seoul 02455, Korea

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.

###### Contents

  1. 1 Introduction
  2. 2 Subsystem symmetry and duality in $(2+1)$d
    1. 2.1 Subsystem $\mathbb{Z}_{2}\ $symmetry on lattice
    2. 2.2 Subsystem KW transformation
    3. 2.3 Subsystem JW transformation
  3. 3 2-foliated theory as the subsystem SymTFT
    1. 3.1 2-foliated BF theory revisited
    2. 3.2 Topological boundaries with subsystem symmetry
  4. 4 Subsystem $SL(2,\mathbb{Z}_{2})\ $transformation and the duality web
    1. 4.1 Subsystem $S$-transformation
    2. 4.2 Subsystem $T$-transformation
    3. 4.3 Duality web from the subsystem $SL(2,\mathbb{Z}_{2})\ $transformation
  5. 5 $S$-defects in the subsystem SymTFT
    1. 5.1 Conventions on operators and algebras
    2. 5.2 $S$-defect
    3. 5.3 Twist $S$-defect
  6. 6 Conclusion and discussion
  7. A A review on ordinary BF theory as SymTFT
  8. B Duality between (3.1) and (3.2)
  9. C Derivation of fusion rule of subsystem KW defects
    1. C.1 Conventions
    2. C.2 Fusion rule of subsystem KW defects

##  1 Introduction

Fracton excitation, a new kind of quasiparticle with restricted mobility, has
appeared in new types of exotic phases of matter and received attention from
both condensed matter physics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] and high energy
physics [11, 12, 13, 14, 15]. Fracton phases of matter, originally constructed
as a candidate for quantum memory [1, 2], are famous for their extensive
ground state degeneracy [3, 1, 2], restricted mobility of excitation [3] and
large subleading corrections to the entanglement entropy [16, 17]. More detail
can be found in the reviews [18, 19]. Models with fractons also attract the
interest of field theorists because their low energy effective description
allows discontinuous field configurations and exhibits exotic UV-IR mixing
behavior [11, 12, 13, 14, 15], which challenges our conventional understanding
of field theory.

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 [25, 7, 6, 26, 27, 8] 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.***In [28, 29], the
restricted mobility and UV-IR mixing are also found in rank 2 gauge theory,
resulting from the subsystem higher form symmetry. The two constructions are
equivalent through the exotic-foliated duality [30, 31].

From a symmetry point of view, fracton models are often realized by gauging
the subsystem symmetry [26, 32] or dipole symmetry [33, 34] 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 [35, 36, 37]. 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 [38], spontaneously breaking
[39, 40, 41], anomaly inflow [42] 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 [43, 44, 45, 46,
47, 48, 49]. We are interested in the duality transformation generated by
symmetry manipulations such as gauging and stacking invertible phases [50, 51,
52]. For example, gauging the non-anomalous $\mathbb{Z}_{2}$ 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 [53, 54]. Another famous example is the boson-fermion duality [55,
56, 57], 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 $\mathbb{Z}_{2}$ symmetry. Recently,
generalizations of Kramers-Wannier (KW) and Jordan-Wigner (JW) duality has
been studied in the context of subsystem symmetry [58, 59, 60, 61], 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[62, 63, 64, 65, 66, 67, 68,
69, 70, 49, 71, 72, 73, 74]. The idea of SymTFT is illustrated in Fig. 1.
Given a $d$-dimensional theory $\mathfrak{T}_{\mathcal{S}}$ with a finite
symmetry $\mathcal{S}$, the SymTFT is a $(d+1)$-dimensional topological
quantum field theory $\mathfrak{Z}(\mathcal{S})$ that allows a topological
boundary $\mathfrak{B}^{\textrm{sym}}_{\mathcal{S}}$ encoding the symmetry
$\mathcal{S}$ of the original theory $\mathfrak{T}_{\mathcal{S}}$. The
original theory $\mathfrak{T}_{S}$ can be expressed as an interval
compactification of $\mathfrak{Z}(\mathcal{S})$ with two boundaries. In the
condensed matter literature, the similar idea of SymTFT has been proposed as
symmetry/topological order correspondence [75, 76, 77].

$\langle{\mathfrak{B}^{\textrm{sym}}_{\mathcal{S}}}|$$|{\mathfrak{B}^{\textrm{phys}}_{\mathfrak{T}_{\mathcal{S}}}}\rangle$$\mathfrak{Z}(\mathcal{S})$$\mathfrak{T}_{\mathcal{S}}=\langle\mathfrak{B}^{\textrm{sym}}_{\mathcal{S}}|\mathfrak{B}^{\textrm{phys}}_{\mathfrak{T}_{\mathcal{S}}}\rangle$$\mathcal{D}$$\langle{\mathfrak{B}^{\textrm{sym}}_{\mathcal{S}}}|$$|{\mathfrak{B}^{\textrm{phys}}_{\mathfrak{T}_{\mathcal{S}}}}\rangle$$\langle{\mathfrak{B}^{\textrm{sym}}_{\mathcal{S}}}|\mathcal{D}$$|{\mathfrak{B}^{\textrm{phys}}_{\mathfrak{T}_{\mathcal{S}}}}\rangle$$\mathcal{D}(\mathfrak{T}_{\mathcal{S}})$
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 $\mathcal{D}$ in the SymTFT will change the boundary condition, which
corresponds to a symmetry manipulation/duality transformation of the boundary
theory.

The power of SymTFT is that the information of symmetry $\mathcal{S}$ and the
dynamics are separately stored in the two boundaries. The left boundary is the
topological boundary $\mathfrak{B}^{\textrm{sym}}_{\mathcal{S}}$ supporting
the symmetry $\mathcal{S}$ 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
$\mathfrak{B}^{\textrm{phys}}_{\mathfrak{T}_{\mathcal{S}}}$ that depends on
the details of $\mathfrak{T}_{\mathcal{S}}$. 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 $\mathbb{Z}_{2}$ symmetry in $(2+1)$d, which is a 2-foliated
theory with one-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 [78, 31]. 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 $\mathbb{Z}_{2}$ symmetry on the lattice and subsystem KW/JW duality
transformation. In Sec. 3, we propose the $(3+1)$d SymTFT for subsystem
$\mathbb{Z}_{2}$ symmetry in $(2+1)$d and study the topological boundary
conditions. In Sec. 4, we consider the $SL(2,\mathbb{Z}_{2})$ symmetry of the
subsystem SymTFT and the duality web of the boundary theories. In Sec. 5, we
construct the condensation defects and twist defects of $S$-transformation in
the subsystem $SL(2,\mathbb{Z}_{2})$. 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 $\mathbb{Z}_{2}$ symmetry in
$(2+1)$d regularized on a 2d square lattice and the duality transformations
including the subsystem Kramers-Wannier (KW) transformation [60] and the
subsystem Jordan-Wigner (JW) transformation [61].

###  2.1 Subsystem $\mathbb{Z}_{2}\ $symmetry on lattice

Consider a closed $L_{x}\times L_{y}$ square lattice. On each site there is a
spin-$1/2$ state $|s\rangle_{i,j}$ where $s=\pm 1$, $i=1,\cdots,L_{x}$ and
$j=1,\cdots,L_{y}$. 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

$X_{i,j}|s\rangle_{i,j}=|-s\rangle_{i,j},\quad
Z_{i,j}|s\rangle_{i,j}=s|s\rangle_{i,j}.$ (2.1)

The generators of subsystem $\mathbb{Z}_{2}$ global symmetry are line
operators acting on each row and column

$U_{j}^{x}=\prod_{i=1}^{L_{x}}X_{i,j},\quad
U_{i}^{y}=\prod_{j=1}^{L_{y}}X_{i,j}.$ (2.2)

They satisfy $(U_{j}^{x})^{2}=(U_{i}^{y})^{2}=1$ and flip the spin of all
sites of $j$th-row or $i$th-column as illustrated in Fig. 2. We will denote
the eigenvalues of $U_{j}^{x}$, $U_{i}^{y}$ as
$(-1)^{u_{j}^{x}},(-1)^{u_{i}^{y}}$ where $u_{j}^{x},u_{i}^{y}=0,1$ are
$\mathbb{Z}_{2}$-valued integers. These $L_{x}+L_{y}$ operators are not
independent and they are restricted by the constraint

$\prod_{j=1}^{L_{y}}U^{x}_{j}\prod_{i=1}^{L_{x}}U^{y}_{i}=\prod_{j=1}^{L_{y}}(-1)^{u_{j}^{x}}\prod_{i=1}^{L_{x}}(-1)^{u_{i}^{y}}=1,$
(2.3)

and there are $L_{x}+L_{y}-1$ independent symmetry generators.

One can also insert the subsystem $\mathbb{Z}_{2}$ 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 $\mathbb{Z}_{2}$ twist
operators (e.g. $U^{xz}_{0j}$ in Fig. 2) on half line

$U_{0,j}^{xz}=\prod_{i^{\prime}<0}X_{i^{\prime},j},\quad
U_{i,0}^{yz}=\prod_{j^{\prime}<0}X_{i,j^{\prime}}.$ (2.4)

The operator $U_{0,j}^{xz}$ is mobile along the $x$-direction and is not
mobile along the $y$-direction. Similarly, $U_{i,0}^{yz}$ 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

$|s_{i+L_{x},j}\rangle=|(-1)^{t_{j}^{x}}s_{i,j}\rangle,\quad|s_{i,j+L_{y}}\rangle=|(-1)^{t_{i}^{y}}s_{i,j}\rangle,\quad|s_{i+L_{x},i+L_{y}}\rangle=|(-1)^{t^{xy}+t_{j}^{x}+t_{i}^{y}}s_{i,j}\rangle,$
(2.5)

where $t^{x}_{j},t^{y}_{i}=0,1$ are twist variables and $t^{xy}=0,1$ is the
boundary condition of the twist variables

$t_{i+L_{x}}^{y}=t_{i}^{y}+t^{xy},\quad t_{j+L_{y}}^{x}=t_{j}^{x}+t^{xy}.$
(2.6)

Although there are $L_{x}+L_{y}+1$ twist parameters but the Hamiltonian with
subsystem $\mathbb{Z}_{2}$ symmetry depends only on the combinations
$\mathbf{t}^{x}_{j+\frac{1}{2}},\mathbf{t}^{y}_{i+\frac{1}{2}}$ [61, 60],

$\mathbf{t}^{x}_{j+\frac{1}{2}}\equiv
t^{x}_{j}+t^{x}_{j+1},\quad\mathbf{t}^{y}_{i+\frac{1}{2}}\equiv
t^{y}_{i}+t^{y}_{i+1},\quad\sum_{j=1}^{L_{y}}\mathbf{t}^{y}_{i+\frac{1}{2}}=\sum_{i=1}^{L_{x}}\mathbf{t}^{x}_{j+\frac{1}{2}}=t^{xy},$
(2.7)

and only $L_{x}+L_{y}-1$ twist variables are independent.

$j$$U^{x}_{j}$$i$$U^{y}_{i}$symmetry operator $U^{x}_{j},U^{y}_{i}$$j$defect
operator $U^{x}_{j}$$j$twist operator $U^{xz}_{0,j}$ Figure 2: Examples of
subsystem $\mathbb{Z}_{2}$ symmetry operators, defect operators and twist
operators.

Given a $(2+1)$d theory $\mathfrak{T}_{\textrm{sub}}$ with the subsystem
$\mathbb{Z}_{2}$ symmetry, the eigenvalues of subsystem symmetry and twist
boundary conditions will divide the Hilbert space into sectors with
$\mathbb{Z}_{2}$-valued symmetry-twist labels
$(\\{u^{x}_{j}\\},\\{u^{y}_{i}\\},\\{\mathbf{t}^{x}_{j+\frac{1}{2}}\\},\\{\mathbf{t}^{y}_{i+\frac{1}{2}}\\})$.
Here $\\{\cdots\\}$ denotes the collection of variables for all
$j=1,\cdots,L_{y}$ and $i=1,\cdots,L_{x}$. The symmetry-twist labels have
overall constraints

$\prod_{j=1}^{L_{y}}(-1)^{u^{x}_{j}}\prod_{i=1}^{L_{x}}(-1)^{u^{y}_{i}}=1,\quad\prod_{j=1}^{L_{y}}(-1)^{\mathbf{t}^{x}_{j+\frac{1}{2}}}\prod_{i=1}^{L_{x}}(-1)^{\mathbf{t}^{y}_{i+\frac{1}{2}}}=1.$
(2.8)

With the above constraints, the Hilbert space is divided into
$2^{2(L_{x}+L_{y}-1)}$ different sectors and the partition function for each
sector is

$Z_{\mathfrak{T}_{\textrm{sub}}}[\\{u^{x}_{j}\\},\\{u^{y}_{i}\\},\\{\mathbf{t}^{x}_{j+\frac{1}{2}}\\},\\{\mathbf{t}^{y}_{i+\frac{1}{2}}\\}]=\text{Tr}_{\mathcal{H}_{\mathbf{t}}}\left(\prod_{i=1}^{L_{x}}\frac{1+(-1)^{u^{y}_{i}}U^{y}_{i}}{2}\right)\left(\prod_{j=1}^{L_{y}}\frac{1+(-1)^{u^{x}_{j}}U^{x}_{j}}{2}\right)e^{-\beta
H},$ (2.9)

where $\mathcal{H}_{\mathbf{t}}$ is the Hilbert space of the twist sector with
label
$(\\{\mathbf{t}^{x}_{j+\frac{1}{2}}\\},\\{\mathbf{t}^{y}_{i+\frac{1}{2}}\\})$.

For simplicity, we will write any quartet
$(\\{u^{x}_{j}\\},\\{u^{y}_{i}\\},\\{\mathbf{t}^{x}_{j+\frac{1}{2}}\\},\\{\mathbf{t}^{y}_{i+\frac{1}{2}}\\})$
or doublet
$(\\{\mathbf{t}^{x}_{j+\frac{1}{2}}\\},\\{\mathbf{t}^{y}_{i+\frac{1}{2}}\\})$
as
$(u^{x}_{j},u^{y}_{i},\mathbf{t}^{x}_{j+\frac{1}{2}},\mathbf{t}^{y}_{i+\frac{1}{2}})$
and $(\mathbf{t}^{x}_{j+\frac{1}{2}},\mathbf{t}^{y}_{i+\frac{1}{2}})$ in the
following discussion.

#### Coupling to background field

We can introduce background subsystem $\mathbb{Z}_{2}$ symmetry gauge field
$(A^{z},A^{xy})$ on the lattice. Consider a cubic spacetime lattice $M_{3}$
with $L_{x}\times L_{y}\times L_{z}$ 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$$(i,j,k+1)$$A^{z}_{i,j,k+\frac{1}{2}}$$(i,j,k)$$(i+1,j,k)$$(i+1,j+1,k)$$A^{xy}_{i+\frac{1}{2},j+\frac{1}{2},k}$
Figure 3: Background gauge field for subsystem $\mathbb{Z}_{2}$ symmetry on
lattice.

The $\mathbb{Z}_{2}$-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

$w_{z;i,j}=\sum_{k=1}^{L_{z}}A^{z}_{i,j,k+\frac{1}{2}}=w_{z,x;j}+w_{z,y;i},$
(2.10)

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

$(w_{z,x;j},w_{z,y;i})\sim(w_{z,x;j}+1,w_{z,y;i}+1).$ (2.11)

On the other hand, the holonomy of $A^{xy}$ along $x$ and $y$ directions are

$w_{x;j+\frac{1}{2}}=\sum_{i=1}^{L_{x}}A^{xy}_{i+\frac{1}{2},j+\frac{1}{2},k}=\mathbf{t}^{x}_{j+\frac{1}{2}}\,\quad
w_{y;i+\frac{1}{2}}=\sum_{j=1}^{L_{y}}A^{xy}_{i+\frac{1}{2},j+\frac{1}{2},k}=\mathbf{t}^{y}_{i+\frac{1}{2}}\,.$
(2.12)

They detect the insertion of symmetry defects along the $z$-direction and are
the same as the twist variables
$\mathbf{t}^{x}_{j+\frac{1}{2}},\mathbf{t}^{y}_{i+\frac{1}{2}}$ introduced in
(2.7). They obey the same constraint

$\prod_{j=1}^{L_{y}}(-1)^{w_{x;j+\frac{1}{2}}}\prod_{i=1}^{L_{x}}(-1)^{w_{y;i+\frac{1}{2}}}=1.$
(2.13)

For a generic subsystem $\mathbb{Z}_{2}$ symmetry background
$(w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}})$, the partition
function is

$Z_{\mathfrak{T}_{\textrm{sub}}}[w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}}]=\text{Tr}_{\mathcal{H}_{\mathbf{t}}}\left(\prod_{j=1}^{L_{y}}(U^{x}_{j})^{w_{z,x;j}}\right)\left(\prod_{i=1}^{L_{x}}(U^{y}_{i})^{w_{z,y;i}}\right)e^{-\beta
H}.$ (2.14)

It is related to the partition function in the sector with symmetry-twist
label (2.9) by a discrete Fourier transformation

$\displaystyle
Z_{\mathfrak{T}_{\textrm{sub}}}[w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}}]=\sum_{u^{y}_{i},u^{x}_{j}=0,1}(-1)^{\sum_{i}u^{y}_{i}w_{z,y;i}+\sum_{j}u^{x}_{j}w_{z,x;j}}Z_{\mathfrak{T}_{\textrm{sub}}}[u^{x}_{j},u^{y}_{i},\mathbf{t}^{x}_{j+\frac{1}{2}},\mathbf{t}^{y}_{i+\frac{1}{2}}],$
(2.15)

where
$w_{x;j+\frac{1}{2}}=\mathbf{t}^{x}_{j+\frac{1}{2}},w_{y;i+\frac{1}{2}}=\mathbf{t}^{y}_{i+\frac{1}{2}}$
and the summation over $(u^{y}_{i},u^{x}_{j})$ should obey the constraint
(2.8).

###  2.2 Subsystem KW transformation

We can gauge the subsystem $\mathbb{Z}_{2}$ symmetry by doing a subsystem KW
transformation $\mathcal{N}^{\text{sub}}$ [60] which maps the original lattice
with spin $\\{|s\rangle_{i,j}\\}$ to the dual lattice with spin
$\\{|\hat{s}\rangle_{i+\frac{1}{2},j+\frac{1}{2}}\\}$ living on the plaquette
of the original lattice. In terms of Pauli operators, the explicit
transformation of $\mathcal{N}^{\text{sub}}$ is

$\displaystyle\begin{split}&\mathcal{N}^{\text{sub}}Z_{i,j}Z_{i,j+1}Z_{i+1,j}Z_{i+1,j+1}=\hat{X}_{i+\frac{1}{2},j+\frac{1}{2}}\mathcal{N}^{\text{sub}},\\\
&\mathcal{N}^{\text{sub}}X_{i,j}=\hat{Z}_{i-\frac{1}{2},j-\frac{1}{2}}\hat{Z}_{i+\frac{1}{2},j-\frac{1}{2}}\hat{Z}_{i-\frac{1}{2},j+\frac{1}{2}}\hat{Z}_{i+\frac{1}{2},j+\frac{1}{2}}\mathcal{N}^{\text{sub}},\end{split}$
(2.16)

where
$\hat{X}_{i+\frac{1}{2},j+\frac{1}{2}},\hat{Z}_{i+\frac{1}{2},j+\frac{1}{2}}$
are Pauli operators acting on the dual lattice. After gauging, the dual theory
$\hat{\mathfrak{T}}_{\textrm{sub}}$ lives on the dual lattice and has a dual
subsystem $\mathbb{Z}_{2}$ symmetry. The Hilbert space of the dual theory
$\hat{\mathfrak{T}}_{\textrm{sub}}$ is similarly divided into sectors labelled
by the dual symmetry-twist variables
$(\hat{u}^{x}_{j+\frac{1}{2}},\hat{u}^{y}_{i+\frac{1}{2}},\hat{\mathbf{t}}^{x}_{j},\hat{\mathbf{t}}^{y}_{i})$
with the constraints

$\prod_{j=1}^{L_{y}}(-1)^{\hat{u}^{x}_{j+\frac{1}{2}}}\prod_{i=1}^{L_{x}}(-1)^{\hat{u}^{y}_{i+\frac{1}{2}}}=1,\quad\prod_{j=1}^{L_{y}}(-1)^{\hat{\mathbf{t}}^{x}_{j}}\prod_{i=1}^{L_{x}}(-1)^{\hat{\mathbf{t}}^{y}_{i}}=1.$
(2.17)

They are related to the symmetry-twist variables
$(u^{x}_{j},u^{y}_{i},\mathbf{t}^{x}_{j+\frac{1}{2}},\mathbf{t}^{y}_{i+\frac{1}{2}})$
in the original theory $\mathfrak{T}_{\textrm{sub}}$ as

$\hat{u}^{x}_{j+\frac{1}{2}}=\mathbf{t}^{x}_{j+\frac{1}{2}},\quad\hat{u}^{y}_{i+\frac{1}{2}}=\mathbf{t}^{y}_{i+\frac{1}{2}},\quad\hat{\mathbf{t}}^{x}_{j}=u^{x}_{j},\quad\hat{\mathbf{t}}^{y}_{i}=u^{y}_{i}.$
(2.18)

where symmetry/twist sectors are exchanged as shown in Fig. 4.

$i-1$$i$$i+1$$j-1$$j$$j+1$$u^{y}_{i-1}$$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{\hat{u}^{y}_{i+\frac{1}{2}}}$$\hat{\mathbf{t}}^{y}_{i-1}$$\mathbf{t}^{y}_{i+\frac{1}{2}}$$\hat{u}^{x}_{j+\frac{1}{2}}$${u}^{x}_{j-1}$$\mathbf{t}^{x}_{j+\frac{1}{2}}$$\hat{\mathbf{t}}^{x}_{j-1}$
Figure 4: Mapping of symmetry-twist sectors. The original lattice is in black
while the dual lattice is in red. For example, the symetry variable
$u^{x}_{j-1}$ is mapped to the dual twist variable
$\hat{\mathbf{t}}^{x}_{j-1}=\hat{t}^{x}_{j-\frac{3}{2}}+\hat{t}^{x}_{j-\frac{1}{2}}$.

The holonomy variables of the dual gauge fields $(\hat{A}^{z},\hat{A}^{xy})$
are
$(\hat{w}_{z,x;j+\frac{1}{2}},\hat{w}_{z,y;i+\frac{1}{2}},\hat{w}_{x;j},\hat{w}_{y;i})$,
with the gauge redundancy and constraints

$(\hat{w}_{z,x;j+\frac{1}{2}},\hat{w}_{z,y;i+\frac{1}{2}})\sim(\hat{w}_{z,x;j+\frac{1}{2}}+1,\hat{w}_{z,y;i+\frac{1}{2}}+1),\quad\prod_{j=1}^{L_{y}}(-1)^{\hat{w}_{x;j}}\prod_{i=1}^{L_{x}}(-1)^{\hat{w}_{y;i}}=1.$
(2.19)

As before, one has
$\hat{w}_{x;j}=\hat{\mathbf{t}}^{x}_{j},\hat{w}_{y,i}=\hat{\mathbf{t}}^{y}_{i}$
and $(\hat{w}_{z,x;j+\frac{1}{2}},\hat{w}_{z,y;i+\frac{1}{2}})$ are the
Fourier partners of
$(\hat{u}^{x}_{j+\frac{1}{2}},\hat{u}^{x}_{j+\frac{1}{2}})$ as in (2.15).
Implied by (2.18), the partition function of the dual theory
$\hat{\mathfrak{T}}_{\textrm{sub}}$ is related to the partition of the
original theory $\mathfrak{T}_{\textrm{sub}}$ in (2.14) as

$\displaystyle\begin{split}&Z_{\hat{\mathfrak{T}}_{\textrm{sub}}}[\hat{w}_{z,x;j+\frac{1}{2}},\hat{w}_{z,y;i+\frac{1}{2}},\hat{w}_{x;j},\hat{w}_{y;i}]\\\
&=\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}}=0,1}Z_{\mathfrak{T}_{\textrm{sub}}}[w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}}]\\\
&\times(-1)^{\sum_{i}(\hat{w}_{z,y;i+\frac{1}{2}}w_{y;i+\frac{1}{2}}+\hat{w}_{y;i}w_{z,y;i})+\sum_{j}(\hat{w}_{z,x;j+\frac{1}{2}}w_{x;j+\frac{1}{2}}+\hat{w}_{x;j}w_{z,x;j})}\,.\end{split}$
(2.20)

The summation of
$(w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}})$ should obey
the restrictions in (2.11) and (2.13).

Suppose the theory $\mathfrak{T}_{\textrm{sub}}$ is invariant under the
subsystem KW transformation, which means
$\hat{\mathfrak{T}}_{\textrm{sub}}=\mathfrak{T}_{\textrm{sub}}$. The subsystem
KW transformation becomes a symmetry and we can insert the KW operator/defect
$\mathcal{N}^{\textrm{sub}}$ along a 2-dimensional surface $M_{2}$ by gauging
half of the spacetime. If $M_{2}$ is the $x$-$y$ plane,
$\mathcal{N}^{\text{sub}}$ is an operator acting on the Hilbert space. The
fusion between the symmetry operator $\mathcal{N}^{\text{sub}}$ and its
orientation reversal $\mathcal{N}^{\text{sub}}{}^{\dagger}$ is

$\mathcal{N}^{\text{sub}}{}^{\dagger}\times\mathcal{N}^{\text{sub}}=\frac{1}{2}\prod_{i=1}^{L_{x}}\left(1+(-1)^{\hat{\mathbf{t}}^{y}_{i}}U^{y}_{i}\right)\prod_{j=1}^{L_{y}}\left(1+(-1)^{\hat{\mathbf{t}}^{x}_{j}}U^{x}_{j}\right).$
(2.21)

On the other hand, if $M_{2}$ is the $z$-$x$ (or $z$-$y$) plane then
$\mathcal{N}^{\text{sub}}$ is a defect twisting the boundary condition. The
fusion rule of the subsystem KW defect on the $z$-$x$ plane is

$\mathcal{N}^{\text{sub}}{}^{\dagger}\times\mathcal{N}^{\text{sub}}=\sum_{t^{y}_{i}=0,1}\prod_{i}^{L_{x}}(U^{yz}_{0,i})^{t^{y}_{i}}.$
(2.22)

The fusion rules are first derived in [60]. We give an alternative derivation
in Appendix C following [65].

#### Subsystem KW transformation on one lattice

The subsystem KW transformation (2.16) maps from the lattice to the dual
lattice [60]. We can also define another subsystem KW transformation on one
lattice

$\displaystyle\begin{split}&\bar{\mathcal{N}}^{\text{sub}}Z_{i,j}Z_{i,j+1}Z_{i+1,j}Z_{i+1,j+1}=X_{i+1,j+1}\bar{\mathcal{N}}^{\text{sub}},\\\
&\bar{\mathcal{N}}^{\text{sub}}X_{i,j}=Z_{i,j}Z_{i+1,j}Z_{i,j+1}Z_{i+1,j+1}\bar{\mathcal{N}}^{\text{sub}},\end{split}$
(2.23)

and the fusion rule of
$\bar{\mathcal{N}}^{\text{sub}}\times\bar{\mathcal{N}}^{\text{sub}}$ will mix
with the one-site translation in the diagonal direction $\mathcal{T}$

$\displaystyle\begin{split}\bar{\mathcal{N}}^{\text{sub}}\times\bar{\mathcal{N}}^{\text{sub}}&=\frac{1}{2}\prod_{i=1}^{L_{x}}\left(1+(-1)^{\hat{\mathbf{t}}^{y}_{i}}U^{y}_{i}\right)\prod_{j=1}^{L_{y}}\left(1+(-1)^{\hat{\mathbf{t}}^{x}_{j}}U^{x}_{j}\right)\mathcal{T},\\\
\bar{\mathcal{N}}^{\text{sub}}{}^{\dagger}\times\bar{\mathcal{N}}^{\text{sub}}&=\frac{1}{2}\prod_{i=1}^{L_{x}}\left(1+(-1)^{\hat{\mathbf{t}}^{y}_{i}}U^{y}_{i}\right)\prod_{j=1}^{L_{y}}\left(1+(-1)^{\hat{\mathbf{t}}^{x}_{j}}U^{x}_{j}\right).\end{split}$
(2.24)

This is a natural generalization of the ordinary KW transformation [79] whose
fusion rule on lattice is different from the fusion rule in the continuum
theory by a one-site translation.

###  2.3 Subsystem JW transformation

Besides the subsystem KW transformation that maps a bosonic
$\mathfrak{T}_{\textrm{sub}}$ theory to another bosonic theory
$\hat{\mathfrak{T}}_{\textrm{sub}}$, we also have the subsystem JW
transformation that maps the bosonic theory $\mathfrak{T}_{\textrm{sub}}$ to a
fermionic theory $\mathfrak{T}_{F,\textrm{sub}}$ [61].

The subsystem JW transformation maps Pauli operators $X_{i,j},Y_{i,j},Z_{i,j}$
to Majorana fermion operators $\gamma_{i,j},\gamma^{\prime}_{i,j}$ 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 $\mathfrak{T}_{F,x,\textrm{sub}}$ and
$\mathfrak{T}_{F,y,\textrm{sub}}$.

Figure 5: Subsystem JW transformation winding around $x$ and $y$ directions.

For the first choice, the explicit transformation is

$\displaystyle\begin{split}\gamma_{i,j}&=\left(\prod_{i^{\prime}=1}^{L_{x}}\prod_{j^{\prime}=1}^{j-1}X_{i^{\prime},j^{\prime}}\right)\left(\prod_{i^{\prime}=1}^{i-1}X_{i^{\prime},j}\right)Z_{i,j},\\\
\gamma_{i,j}^{\prime}&=-\left(\prod_{i^{\prime}=1}^{L_{x}}\prod_{j^{\prime}=1}^{j-1}X_{i^{\prime},j^{\prime}}\right)\left(\prod_{i^{\prime}=1}^{i-1}X_{i^{\prime},j}\right)Y_{i,j}.\end{split}$
(2.25)

The fermionic theory has subsystem $\mathbb{Z}_{2}$ fermion parity symmetry
$(-1)^{F}$. Considering the symmetry operators and twists of $(-1)^{F}$, the
Hilbert space is divided into $2^{2(L_{x}+L_{y}-1)}$ sectors with labels
$(u^{x}_{f,j},u^{y}_{f,i},\mathbf{t}^{x}_{f,j+\frac{1}{2}},\mathbf{t}^{y}_{i+\frac{1}{2}})$.
Using the transformation (2.25), one can work out the mapping between
symmetry-twist sectors in the bosonic and fermionic theory

$u^{x}_{f,j}=u^{x}_{f},\quad
u^{y}_{f,i}=u^{y}_{i},\quad\mathbf{t}^{x}_{f,j+\frac{1}{2}}=\mathbf{t}^{x}_{j+\frac{1}{2}}+u^{x}_{j}+u^{x}_{j+1},\quad\mathbf{t}^{y}_{f,i+\frac{1}{2}}=\mathbf{t}^{y}_{i+\frac{1}{2}}.$
(2.26)

We can introduce the background fields for subsystem $\mathbb{Z}_{2}$ fermion
parity symmetry $(-1)^{F}$ and define the corresponding holonomy variables as
$(s_{z,x;j},s_{z,y;i},s_{x;j+\frac{1}{2}},s_{y;i+\frac{1}{2}})$. Similar to
the bosonic case, the space direction holonomy has the following
identification

$s_{x;j+\frac{1}{2}}=\mathbf{t}^{x}_{f,j+\frac{1}{2}},\quad
s_{y;i+\frac{1}{2}}=\mathbf{t}^{y}_{f,i+\frac{1}{2}},$ (2.27)

From the sector correspodence (2.26), we can derive the relation between the
partition functions of the bosonic theory $\mathfrak{T}_{\textrm{sub}}$ and
the fermionic theory $\mathfrak{T}_{F,x,\textrm{sub}}$

$\displaystyle
Z_{\mathfrak{T}_{F,x,\textrm{sub}}}[s_{z,x;j},s_{z,y;i},s_{x;j+\frac{1}{2}},s_{y;i+\frac{1}{2}}]$
$\displaystyle=$
$\displaystyle\sum_{u^{x}_{f},u^{y}_{f}=0,1}(-1)^{\sum_{i}u^{y}_{f,i}s_{z,y;i}+\sum_{j}u^{x}_{f,j}s_{z,x;j}}Z_{\mathfrak{T}_{F,x,\textrm{sub}}}[u^{x}_{f},u^{y}_{f},s_{x;j+\frac{1}{2}},s_{y;i+\frac{1}{2}}]$
$\displaystyle=$
$\displaystyle\sum_{u^{x},u^{y}=0,1}(-1)^{\sum_{i}u^{y}_{i}s_{z,y;i}+\sum_{j}u^{x}_{j}s_{z,x;j}}Z_{\mathfrak{T}_{\textrm{sub}}}[u^{x},u^{y},s_{x;j+\frac{1}{2}}+u^{x}_{j}+u^{x}_{j+1},s_{y;i+\frac{1}{2}}]$
$\displaystyle=$
$\displaystyle\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{u^{x},u^{y},w_{z,x},w_{z,y}=0,1}(-1)^{\sum_{i}u^{y}_{i}(s_{z,y;i}+w_{z,y;i})+\sum_{j}u^{x}_{j}(s_{z,x;j}+w_{z,x;j})}$
$\displaystyle\times
Z_{\mathfrak{T}_{\textrm{sub}}}[w_{z,x;j},w_{z,y;i},s_{x;j+\frac{1}{2}}+u^{x}_{j}+u^{x}_{j+1},s_{y;i+\frac{1}{2}}].$
(2.28)

If the subsystem JW transformation winds along the $y$ direction, we have a
different transformation and a different symmetry-twist sector mapping

$u^{x}_{f,j}=u^{x}_{f},\quad
u^{y}_{f,i}=u^{y}_{i},\quad\mathbf{t}^{x}_{f,j+\frac{1}{2}}=\mathbf{t}^{x}_{j+\frac{1}{2}},\quad\mathbf{t}^{y}_{f,i+\frac{1}{2}}=\mathbf{t}^{y}_{i+\frac{1}{2}}+u^{y}_{i}+u^{y}_{i+1}.$
(2.29)

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
$\mathfrak{T}_{xy,\textrm{sub}}$. One can easily check that now the symmetry-
twist sector labels $(u^{\prime x}_{j},u^{\prime y}_{i},\mathbf{t}^{\prime
x}_{j+\frac{1}{2}},\mathbf{t}^{\prime y}_{i+\frac{1}{2}})$ in this new bosonic
theory are

$u^{\prime x}_{j}=u^{x}_{j},\quad u^{\prime
y}_{i}=u^{y}_{i},\quad\mathbf{t}^{\prime
x}_{j+\frac{1}{2}}=\mathbf{t}^{x}_{j+\frac{1}{2}}+u^{x}_{j}+u^{x}_{j+1},\quad\mathbf{t}^{\prime
y}_{i+\frac{1}{2}}=\mathbf{t}^{y}_{i+\frac{1}{2}}+u^{y}_{i}+u^{y}_{i+1}.$
(2.30)

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

$H_{\text{PlaqIsing}}=-\sum_{i,j}Z_{i,j}Z_{i+1,j}Z_{i,j+1}Z_{i+1,j+1}-h\sum_{i,j}X_{i,j}.$
(2.31)

Applying the subsystem JW transformation winding along $x$ and $y$ direction
seperately, we get two different plaquette fermion models

$\displaystyle\begin{split}H_{\text{Pfer},x}=&\sum_{i,j}\gamma^{\prime}_{i,j}\gamma_{i+1,j}\gamma^{\prime}_{i,j+1}\gamma_{i+1,j+1}+ih\sum_{i,j}\gamma_{i,j}\gamma^{\prime}_{i,j},\\\
H_{\text{Pfer},y}=&\sum_{i,j}\gamma^{\prime}_{i,j}\gamma_{i,j+1}\gamma^{\prime}_{i+1,j}\gamma_{i+1,j+1}+ih\sum_{i,j}\gamma_{i,j}\gamma^{\prime}_{i,j}.\end{split}$
(2.32)

Further applying the inverse subsystem JW transformation along $y$ direction
to $H_{\text{Pfer},x}$, or along $x$ direction to $H_{\text{Pfer},y}$, we will
get another bosonic theory

$H^{\prime}_{bos}=-\sum_{i,j}Z_{i,j}Y_{i+1,j}Y_{i,j+1}Z_{i+1,j+1}-h\sum_{i,j}X_{i,j}.$
(2.33)

The duality web will be enlarged by further considering the subsystem KW
transformation, which is elaborated in Sec. 4.

##  3 2-foliated theory as the subsystem SymTFT

In this section, we will give the analogy of SymTFT for subsystem
$\mathbb{Z}_{N}$ 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 $z,\tau$.
From the exotic-foliated duality [30, 31], 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 $\tau$ as the time direction. After quantization, we will see this
theory supports a topological boundary
$\mathfrak{B}^{\textrm{sym}}_{\textrm{sub}}$ with a $(2+1)$d subsystem
$\mathbb{Z}_{N}$ 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.

###  3.1 2-foliated BF theory revisited

The candidate for subsystem SymTFT of our interest is the $(3+1)$d 2-foliated
BF theory with level $N$

$S_{2\textrm{-foliated}}=\frac{N}{2\pi}\int b\wedge
dc+\sum_{k=1,2}dB^{k}\wedge C^{k}\wedge dx^{k}+\sum_{k=1,2}b\wedge C^{k}\wedge
dx^{k}.$ (3.1)

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 $(\tau,x,y,z)$.

The 2-foliated BF theory (3.1) is equivalent to the exotic tensor gauge theory
[78, 31]

$S_{\textrm{exotic}}=\frac{N}{2\pi}\int\left[A^{\tau}(\partial_{z}\hat{A}^{xy}-\partial_{x}\partial_{y}\hat{A}^{z})-A^{z}(\partial_{\tau}\hat{A}^{xy}-\partial_{x}\partial_{y}\hat{A}^{\tau})-A^{xy}(\partial_{\tau}\hat{A}^{z}-\partial_{z}\hat{A}^{\tau})\right].$
(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),
$A=\\{A^{\tau},A^{z},A^{xy}\\}$ and
$\hat{A}=\\{\hat{A}^{\tau},\hat{A}^{z},\hat{A}^{xy}\\}$ are electric and
magnetic gauge fields with the following gauge transformations

$\displaystyle\begin{split}&A^{\tau}\sim A^{\tau}+\partial_{\tau}\lambda,\quad
A^{z}\sim A^{z}+\partial_{z}\lambda,\quad A^{xy}\sim
A^{xy}+\partial_{x}\partial_{y}\lambda,\\\
&\hat{A}^{\tau}\sim\hat{A}^{\tau}+\partial_{\tau}\hat{\lambda},\quad\hat{A}^{z}\sim\hat{A}^{z}+\partial_{z}\hat{\lambda},\quad\hat{A}^{xy}\sim\hat{A}^{xy}+\partial_{x}\partial_{y}\hat{\lambda},\end{split}$
(3.3)

where $\lambda,\hat{\lambda}$ are gauge parameters. The equations of motion
for gauge fields $A$ and $\hat{A}$ are

$\displaystyle\begin{split}&\partial_{z}A^{\tau}-\partial_{\tau}A^{z}=0,\quad\partial_{\tau}A^{xy}-\partial_{x}\partial_{y}A^{\tau}=0,\quad\partial_{z}A^{xy}-\partial_{x}\partial_{y}A^{z}=0,\\\
&\partial_{z}\hat{A}^{\tau}-\partial_{\tau}\hat{A}^{z}=0,\quad\partial_{\tau}\hat{A}^{xy}-\partial_{x}\partial_{y}\hat{A}^{\tau}=0,\quad\partial_{z}\hat{A}^{xy}-\partial_{x}\partial_{y}\hat{A}^{z}=0.\end{split}$
(3.4)

In the exotic theory (3.2), there exists a naive $SL(2,\mathbb{Z}_{N})$
symmetry

$\displaystyle\begin{split}&S:\quad
A\rightarrow\hat{A},\quad\hat{A}\rightarrow-A,\\\ &T:\quad A\rightarrow
A,\quad\hat{A}\rightarrow\hat{A}+A.\end{split}$ (3.5)

with $S^{2}=C$ the charge conjugation symmetry

$C:\quad A\rightarrow-A,\quad\hat{A}\rightarrow-\hat{A},$ (3.6)

which is hard to see in the original 2-foliated formulation. We will elaborate
more on this $SL(2,\mathbb{Z}_{N})$ 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
$z$-$\tau$ plane but cannot move freely along the $x,y$ directions

$\displaystyle\begin{split}W(C_{z,\tau}(x,y))=&\exp\left(i\oint_{C_{z,\tau}(x,y)}A^{\tau}d\tau+A^{z}dz\right),\\\
\hat{W}(C_{z,\tau}(x,y))=&\exp\left(i\oint_{C_{z,\tau}(x,y)}\hat{A}^{\tau}d\tau+\hat{A}^{z}dz\right),\end{split}$
(3.7)

where $C_{z,\tau}(x,y)$ is a curve in the $z$-$\tau$ 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

$\displaystyle\begin{split}W(x_{1},x_{2},C_{y,z,\tau}(x))=&\exp\left(i\int_{x_{1}}^{x_{2}}dx\oint_{C_{y,z,\tau}(x)}A^{xy}dy+\partial_{x}A^{z}dz+\partial_{x}A^{\tau}d\tau\right),\\\
W(y_{1},y_{2},C_{x,z,\tau}(y))=&\exp\left(i\int_{y_{1}}^{y_{2}}dy\oint_{C_{x,z,\tau}(y)}A^{xy}dx+\partial_{y}A^{z}dz+\partial_{y}A^{\tau}d\tau\right),\end{split}$
(3.8)

for electric gauge field $A$. There are also hat versions for magnetic gauge
field $\hat{A}$. Here $C_{x,z,\tau}(y)$ is a curve in the $x$-$z$-$\tau$ plane
with fixed $y$, and $C_{y,z,\tau}(x)$ is a curve in the $y$-$z$-$\tau$ plane
with fixed $x$. The curve $C_{x,z,\tau}(y)$ can be deformed in $x$-$z$-$\tau$
plane but not along $y$ direction and the similar restricted mobility for
$C_{y,z,\tau}(x)$. The above properties of restricted mobility follow from the
equations of motion (3.4) of the gauge fields $A$ and $\hat{A}$.

### Quantization

We can quantize the exotic theory (3.2) by picking $\tau$ as the time
direction with the Coulomb gauge $A_{\tau}=\hat{A}_{\tau}=0$. The action (3.2)
becomes

$S_{\textrm{exotic}}=\frac{N}{2\pi}\int\left[-A^{xy}(\partial_{\tau}\hat{A}^{z})-A^{z}(\partial_{\tau}\hat{A}^{xy})\right].$
(3.9)

with the canonical commutation relations between conjugate fields $A$ and
$\hat{A}$

$\displaystyle\left[A^{xy}(x,y,z),\hat{A}^{z}(x^{\prime},y^{\prime},z^{\prime})\right]$
$\displaystyle=\frac{2\pi
i}{N}\delta_{3}(x-x^{\prime},y-y^{\prime},z-z^{\prime}),$ (3.10)
$\displaystyle\left[A^{z}(x,y,z),\hat{A}^{xy}(x^{\prime},y^{\prime},z^{\prime})\right]$
$\displaystyle=\frac{2\pi
i}{N}\delta_{3}(x-x^{\prime},y-y^{\prime},z-z^{\prime}).$

The Gauss laws

$\partial_{x}\partial_{y}\hat{A}^{z}-\partial_{z}\hat{A}^{xy}=0,\quad\partial_{x}\partial_{y}A^{z}-\partial_{z}A^{xy}=0,$
(3.11)

imply the flat condition.

We will consider the 2-foliated theory (or exotic tensor theory) on a spatial
manifold $M_{3}=T^{2}\times S^{1}$, 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

$\displaystyle\begin{split}&W(x,y)=\exp\left(i\oint dzA^{z}\right),\\\
&W(x_{1},x_{2})=\exp\left(i\int_{x_{1}}^{x_{2}}dx\oint dyA^{xy}\right),\\\
&W(y_{1},y_{2})=\exp\left(i\int_{y_{1}}^{y_{2}}dy\oint
dxA^{xy}\right),\end{split}$ (3.12)

and the magnetic line/strip operators

$\displaystyle\begin{split}&\hat{W}(x,y)=\exp\left(i\oint
dz\hat{A}^{z}\right),\\\
&\hat{W}(x_{1},x_{2})=\exp\left(i\int_{x_{1}}^{x_{2}}dx\oint
dy\hat{A}^{xy}\right),\\\
&\hat{W}(y_{1},y_{2})=\exp\left(i\int_{y_{1}}^{y_{2}}dy\oint
dx\hat{A}^{xy}\right).\end{split}$ (3.13)

They are $\mathbb{Z}_{N}$ valued operators

$W^{N}=\hat{W}^{N}=1,$ (3.14)

with the following commutation relations

$\displaystyle\begin{split}W(x_{1},x_{2})\hat{W}(x,y)&=\exp(2\pi
i/N)\hat{W}(x,y)W(x_{1},x_{2}),\quad\textrm{if}\ x_{1}<x<x_{2},\\\
W(y_{1},y_{2})\hat{W}(x,y)&=\exp(2\pi
i/N)\hat{W}(x,y)W(y_{1},y_{2}),\quad\textrm{if}\ y_{1}<y<y_{2},\end{split}$
(3.15)

and,

$\displaystyle\begin{split}\hat{W}(x_{1},x_{2})W(x,y)&=\exp(-2\pi
i/N)W(x,y)\hat{W}(x_{1},x_{2}),\quad\textrm{if}\ x_{1}<x<x_{2},\\\
\hat{W}(y_{1},y_{2})W(x,y)&=\exp(-2\pi
i/N)W(x,y)\hat{W}(y_{1},y_{2}),\quad\textrm{if}\ y_{1}<y<y_{2},\end{split}$
(3.16)

where the extra phase $\exp(\pm 2\pi i/N)$ indicates a mixed t’ Hooft anomaly
between the two sets of subsystem $\mathbb{Z}_{N}$ symmetry generated by the
electric and magnetic line/strip operators.

###  3.2 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 $\mathbb{Z}_{N}$ 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 $A$ and $\hat{A}$ 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

$\oint dzA^{z}=\mathcal{A}^{y}(x)+\mathcal{A}^{x}(y),$ (3.17)

where $\mathcal{A}^{y}(x)$ and $\mathcal{A}^{x}(y)$ are operators only depend
on $x$ and $y$. The split of holonomy (3.17) implies the decomposition of the
line operator

$W(x,y)=W_{z,y}(x)W_{z,x}(y),$ (3.18)

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

$\mathcal{A}^{y}(x)\rightarrow\mathcal{A}^{y}(x)+\pi,\quad\mathcal{A}^{x}(y)\rightarrow\mathcal{A}^{x}(y)+\pi,$
(3.19)

which leaves $\oint dzA^{z}$ invariant modulo $2\pi$. 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

$W(x,x+L_{x})=W(y,y+L_{y})=\exp\left(i\oint dxdyA^{xy}\right).$ (3.20)

There are similar gauge redundancy and constraint for magnetic operators
$\hat{W}$.

#### Discretization on a lattice

Discretizing the boundary manifold $M_{3}$ as a $L_{x}\times L_{y}\times
L_{z}$ 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 $i=1,\cdots,L_{x},j=1,\cdots,L_{y}$.
On the lattice, the gauge redundancy (3.19) and the constraint (3.20) become

$(W_{z,y}(x_{i}),W_{z,x}(y_{j}))\sim(-W_{z,y}(x_{i}),-W_{z,x}(y_{j})),$ (3.21)

and,

$\prod_{i=1}^{L_{x}}W(x_{i},x_{i+1})\prod_{j=1}^{L_{y}}W(y_{j},y_{j+1})=1,$
(3.22)

leaving only $2(L_{x}+L_{y}-1)$ operators independent. Similarly, there are
$2(L_{x}+L_{y}-1)$ independent magnetic $\hat{W}$ operators: line operators
$\hat{W}_{z,y}(x_{i+\frac{1}{2}}),\hat{W}_{z,x}(y_{j+\frac{1}{2}})$ and strip
operators
$\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}),\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})$
on the dual lattice with similar gauge redundancy and constraint.

The discretized version of the algebras between $W$ and $\hat{W}$
(3.15),(3.16) is

$\displaystyle\begin{split}W(x_{i},x_{i+1})\hat{W}_{z,y}(x_{i+\frac{1}{2}})&=-\hat{W}_{z,y}(x_{i+\frac{1}{2}})W(x_{i},x_{i+1}),\\\
W(y_{i},y_{i+1})\hat{W}_{z,x}(y_{j+\frac{1}{2}})&=-\hat{W}_{z,x}(y_{j+\frac{1}{2}})W(y_{i},y_{i+1}),\end{split}$
(3.23)

and

$\displaystyle\begin{split}\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})W_{z,y}(x_{i})&=-W_{z,y}(x_{i})\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}),\\\
\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})W_{z,x}(y_{j})&=-W_{z,x}(y_{j})\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}}).\end{split}$
(3.24)

#### 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
$(w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y,i+\frac{1}{2}})$ introduced in
(2.10),(2.11),(2.12) and (2.13) with the following correspondence

$W(x_{i},y_{j})\leftrightarrow(-1)^{w_{z;i,j}},\quad
W_{z,x}(y_{j})\leftrightarrow(-1)^{w_{z,x;j}},\quad
W_{z,y}(x_{i})\leftrightarrow(-1)^{w_{z,y;i}}$ (3.25)

and

$W(y_{j},y_{j+1})\leftrightarrow(-1)^{w_{x;j+\frac{1}{2}}},\quad
W(x_{i},x_{i+1})\leftrightarrow(-1)^{w_{y;i+\frac{1}{2}}}.$ (3.26)

Therefore, we can introduce a canonical basis of the Hilbert space of the
2-foliated BF theory on the boundary $M_{3}$

$|{\mathbf{w}}\rangle:=|{w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}}}\rangle,$
(3.27)

and the electric operators $W$ are diagonalized as

$\left\\{\begin{array}[]{l}W_{z,x}(y_{j})|{\mathbf{w}}\rangle=(-1)^{w_{z,x;j}}|{\mathbf{w}}\rangle\\\
W_{z,y}(x_{i})|{\mathbf{w}}\rangle=(-1)^{w_{z,y;i}}|{\mathbf{w}}\rangle\\\
W(y_{j},y_{j+1})|{\mathbf{w}}\rangle=(-1)^{w_{x;j+\frac{1}{2}}}|{\mathbf{w}}\rangle\\\
W(x_{i},x_{i+1})|{\mathbf{w}}\rangle=(-1)^{w_{y;i+\frac{1}{2}}}|{\mathbf{w}}\rangle\end{array}\right..$
(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 operators $\hat{W}$ conjugate to electric
operators $W$ will shift the eigenvalues when acting on the state
$|{\mathbf{w}}\rangle$

$\left\\{\begin{array}[]{l}\hat{W}(y_{j^{\prime}-\frac{1}{2}},y_{j^{\prime}+\frac{1}{2}})|{\mathbf{w}}\rangle=|{w_{z,x;j}+\delta_{j,j^{\prime}},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}}}\rangle\\\
\hat{W}(x_{i^{\prime}-\frac{1}{2}},x_{i^{\prime}+\frac{1}{2}})|{\mathbf{w}}\rangle=|{w_{z,x;j},w_{z,y;i}+\delta_{i,i^{\prime}},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}}}\rangle\\\
\hat{W}_{z,x}(y_{j^{\prime}+\frac{1}{2}})|{\mathbf{w}}\rangle=|{w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}}+\delta_{j,j^{\prime}},w_{y;i+\frac{1}{2}}}\rangle\\\
\hat{W}_{z,y}(x_{i^{\prime}+\frac{1}{2}})|{\mathbf{w}}\rangle=|{w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}}+\delta_{i,i^{\prime}}}\rangle\end{array}\right.$
(3.29)

which follows from the algebras (3.23) and (3.24). Because the magnetic
operators $\hat{W}$ along the spatial/temporal cycle shift the
temporal/spatial holonomies $\mathbf{w}$ of electric gauge field $A$, they are
identified one-to-one to the subsystem $\mathbb{Z}_{2}$ symmetry and twist
operators in (2.2),(2.4)

$\displaystyle\begin{split}&\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})\leftrightarrow
U^{x}_{j},\quad\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})\leftrightarrow
U^{y}_{i},\\\
&\hat{W}_{z,x}(y_{j+\frac{1}{2}})\leftrightarrow\prod_{j^{\prime}\leq
j}U^{xz}_{0,j^{\prime}},\quad\hat{W}_{z,y}(x_{i+\frac{1}{2}})\leftrightarrow\prod_{i^{\prime}\leq
i}U^{yz}_{0,i^{\prime}}.\end{split}$ (3.30)

Therefore, the boundary represented by the $|{\mathbf{w}}\rangle$ basis is a
topological boundary supporting the subsystem $\mathbb{Z}_{2}$ symmetry
generated by the magnetic operators $\hat{W}$. The general boundary state
$|{\mathbf{w}}\rangle$ with nontrivial $W$-holonomies is created by acting
magnetic operators $\hat{W}$ on the vacuum state $|{0}\rangle$ where all
$W$-holonimies are trivial

$\displaystyle\begin{split}|{\mathbf{w}}\rangle=&\prod_{i}\left(\hat{W}_{z,y}(x_{i+\frac{1}{2}})\right)^{w_{y;i+\frac{1}{2}}}\left(\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})\right)^{w_{z,y;i}}\\\
&\times\prod_{j}\left(\hat{W}_{z,x}(y_{j+\frac{1}{2}})\right)^{w_{x;j+\frac{1}{2}}}\left(\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})\right)^{w_{z,x;j}}|{0}\rangle.\end{split}$
(3.31)

As a consistency check, the invariance of $|{\mathbf{w}}\rangle$ under the
gauge redundancy of magnetic operators $\hat{W}$ implies the constraints
(2.13) and the constraint among magnetic operators $\hat{W}$ requires the
invariance of the state $|{\mathbf{w}}\rangle$ under the gauge transformation
(2.11).

#### Dirichlet boundary condition for gauge field $\hat{A}$

Alternatively, one can consider the dual basis

$|{\hat{\mathbf{w}}}\rangle:=|{\hat{w}_{z,x;j+\frac{1}{2}},\hat{w}_{z,y;i+\frac{1}{2}},\hat{w}_{x;j},\hat{w}_{y;i}}\rangle,$
(3.32)

where $\hat{W}$ operators are diagonalized

$\left\\{\begin{array}[]{l}\hat{W}_{z,x}(y_{j+\frac{1}{2}})|{\hat{\mathbf{w}}}\rangle=(-1)^{\hat{w}_{z,x;j+\frac{1}{2}}}|{\hat{\mathbf{w}}}\rangle\\\
\hat{W}_{z,y}(x_{i+\frac{1}{2}})|{\hat{\mathbf{w}}}\rangle=(-1)^{\hat{w}_{z,y;i+\frac{1}{2}}}|{\hat{\mathbf{w}}}\rangle\\\
\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})|{\hat{\mathbf{w}}}\rangle=(-1)^{\hat{w}_{x;j}}|{\hat{\mathbf{w}}}\rangle\\\
\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})|{\hat{\mathbf{w}}}\rangle=(-1)^{\hat{w}_{y;i}}|{\hat{\mathbf{w}}}\rangle\end{array}\right..$
(3.33)

The dual basis (3.32) defines the Dirichlet boundary condition for the gauge
field $\hat{A}$. Acting on the state $|{\hat{\mathbf{w}}}\rangle$, the
electric operators $W$ will shift the dual holonomies

$\left\\{\begin{array}[]{l}W(y_{j^{\prime}},y_{j^{\prime}+1})|{\hat{\mathbf{w}}}\rangle=|{\hat{w}_{z,x;j+\frac{1}{2}}+\delta_{j,j^{\prime}},\hat{w}_{z,y;i+\frac{1}{2}},\hat{w}_{x;j},\hat{w}_{y;i}}\rangle\\\
W(x_{i^{\prime}},x_{i^{\prime}+1})|{\hat{\mathbf{w}}}\rangle=|{\hat{w}_{z,x;j+\frac{1}{2}},\hat{w}_{z,y;i+\frac{1}{2}}+\delta_{i,i^{\prime}},\hat{w}_{x;j},\hat{w}_{y;i}}\rangle\\\
W_{z,x}(y_{j^{\prime}})|{\hat{\mathbf{w}}}\rangle=|{\hat{w}_{z,x;j+\frac{1}{2}},\hat{w}_{z,y;i+\frac{1}{2}},\hat{w}_{x;j}+\delta_{j,j^{\prime}},\hat{w}_{y;i}}\rangle\\\
W_{z,y}(x_{i^{\prime}})|{\hat{\mathbf{w}}}\rangle=|{\hat{w}_{z,x;j+\frac{1}{2}},\hat{w}_{z,y;i+\frac{1}{2}},\hat{w}_{x;j},\hat{w}_{y;i}+\delta_{i,i^{\prime}}}\rangle\end{array}\right..$
(3.34)

Therefore, the electric operators $W$ can be identified as the symmetry and
twist operators. The boundary state $|{\hat{\mathbf{w}}}\rangle$ corresponds
to a topological boundary supporting the subsystem $\mathbb{Z}_{2}$ symmetry
generated by electric operators $W$.

The dual state $|\hat{\mathbf{w}}\rangle$ is related to original state
$|\mathbf{w}\rangle$ via a discrete Fourier transformation,

$|{\hat{\mathbf{w}}}\rangle=\frac{1}{2^{(L_{x}+L_{y}-1)}}\sum_{\mathbf{w}\in
M_{v}}(-1)^{\sum_{i}(\hat{w}_{z,y;i+\frac{1}{2}}w_{y;i+\frac{1}{2}}+\hat{w}_{y;i}w_{z,y;i})+\sum_{j}(\hat{w}_{z,x;j+\frac{1}{2}}w_{x;j+\frac{1}{2}}+\hat{w}_{x;j}w_{z,x;j})}|{\mathbf{w}}\rangle,$
(3.35)

where we introduce $M_{v}$ as the set of $\mathbb{Z}_{2}$-valued vector
$\mathbf{w}$ satisfying the gauge redundancy and constraint,

$M_{v}=\left\\{\mathbf{w}\Big{|}\prod_{j=1}^{L_{y}}(-1)^{w_{x;j+\frac{1}{2}}}\prod_{i=1}^{L_{x}}(-1)^{w_{y;i+\frac{1}{2}}}=1;(w_{z,x;j},w_{z,y;i})\sim(w_{z,x;j}+1,w_{z,y;i}+1)\right\\}.$
(3.36)

The restrictions in (3.36) for $\mathbf{w}$ automatically impose restrictions
for $\hat{\mathbf{w}}$.

#### Subsystem KW transformation

Based on the SymTFT picture, we consider the 2-foliated BF theory on the
4-dimensional manifold $M_{3}\times[0,1]$ where $\tau$ is the coordinate of
the time interval. The initial state at $\tau=0$ is the dynamical boundary
state $|\chi\rangle$ and the final state at $\tau=1$ is the topological
boundary state. Given any $(2+1)$-dimensional theory
$\mathfrak{T}_{\textrm{sub}}$ with a subsystem $\mathbb{Z}_{2}$ symmetry, we
can write down the dynamical boundary state as,

$|\chi\rangle=\sum_{\mathbf{w}\in
M_{v}}Z_{\mathfrak{T}_{\textrm{sub}}}[\mathbf{w}]|\mathbf{w}\rangle,$ (3.37)

where the coefficient is the partition function of
$\mathfrak{T}_{\textrm{sub}}$ on $M_{3}$ coupled with the subsystem
$\mathbb{Z}_{2}$ symmetry background $\mathbf{w}$.

Choosing $|\mathbf{w}\rangle$ as the topological boundary state at $\tau=1$,
one has,

$Z_{\mathfrak{T}_{\textrm{sub}}}=\langle\mathbf{w}|\chi\rangle,$ (3.38)

which projects back to the partition function of
$\mathfrak{T}_{\textrm{sub}}$. Alternatively, choosing the dual boundary state
$|\hat{\mathbf{w}}\rangle$ at $\tau=1$ reproduces the partition function of
the dual theory

$\displaystyle\begin{split}Z_{\hat{\mathfrak{T}}_{\textrm{sub}}}(\hat{\mathbf{w}})=&\langle\hat{\mathbf{w}}|\chi\rangle\\\
=&\frac{1}{2^{(L_{x}+L_{y}-1)}}\sum_{\mathbf{w}\in
M_{v}}(-1)^{\sum_{i}(\hat{w}_{z,y;i+\frac{1}{2}}w_{y;i+\frac{1}{2}}+\hat{w}_{y;i}w_{z,y;i})+\sum_{j}(\hat{w}_{z,x;j+\frac{1}{2}}w_{x;j+\frac{1}{2}}+\hat{w}_{x;j}w_{z,x;j})}Z_{\mathfrak{T}_{\textrm{sub}}}(\mathbf{w})\end{split}$
(3.39)

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
$|\mathbf{s}\rangle=|s_{z,x;j},s_{z,y;i},s_{x;j+\frac{1}{2}},s_{y;i+\frac{1}{2}}\rangle$
and write the partition function of $(2+1)$d fermionic theory with subsystem
symmetry as the path integral $\langle\mathbf{s}|\chi\rangle$.

For example, the fermionic topological boundary state corresponding to the
fermionic theory $\mathfrak{T}_{F,x,\textrm{sub}}$ after the subsystem JW
transformation (2.3) is

$|\mathbf{s}\rangle=\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{(u,w_{z})\in
M_{u,w_{z}}}(-1)^{\sum_{i}u^{y}_{i}(s_{z,y;i}+w_{z,y;i})+\sum_{j}u^{x}_{j}(s_{z,x;j}+w_{z,x;j})}|w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}}\rangle,$
(3.40)

with
$w_{x;j+\frac{1}{2}}=s_{x;j+\frac{1}{2}}+u^{x}_{j}+u^{x}_{j+1},w_{y;i+\frac{1}{2}}=s_{y;i+\frac{1}{2}}$
and $M_{u,w_{z}}$ the set,

$M_{u,w_{z}}=\left\\{(u^{y}_{i},u^{x}_{j},w_{z,x;j},w_{z,y;i})\Big{|}\prod_{j=1}^{L_{y}}(-1)^{u^{x}_{j}}\prod_{i=1}^{L_{x}}(-1)^{u^{y}_{i}}=1,(w_{z,x;j},w_{z,y;i})\sim(w_{z,x;j}+1,w_{z,y;i}+1)\right\\}.$
(3.41)

The fermionic state $|\mathbf{s}\rangle$ 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
operators $\hat{W}$ nearby

$\left\\{\begin{array}[]{l}\hat{W}_{z,x}(y_{j-\frac{1}{2}})W_{z,x}(y_{j})\hat{W}_{z,x}(y_{j+\frac{1}{2}})|\mathbf{s}\rangle=(-1)^{s_{z,x;j}}|\mathbf{s}\rangle\\\
W_{z,y}(x_{i})|\mathbf{s}\rangle=(-1)^{s_{z,y;i}}|\mathbf{s}\rangle\\\
\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})W(y_{j},y_{j+1})\hat{W}(y_{j+\frac{1}{2}},y_{j+\frac{3}{2}})|\mathbf{s}\rangle=(-1)^{s_{x;j+\frac{1}{2}}}|\mathbf{s}\rangle\\\
W(x_{i},x_{i+1})|\mathbf{s}\rangle=(-1)^{s_{y;i+\frac{1}{2}}}|\mathbf{s}\rangle\end{array}\right.$
(3.42)

The fermionic subsystem $\mathbb{Z}_{2}$ parity symmetry is generated by
magnetic operators $\hat{W}$

$\left\\{\begin{array}[]{l}\hat{W}(y_{j^{\prime}-\frac{1}{2}},y_{j^{\prime}+\frac{1}{2}})|{\mathbf{s}}\rangle=|{s_{z,x;j}+\delta_{j,j^{\prime}},s_{z,y;i},s_{x;j+\frac{1}{2}},s_{y;i+\frac{1}{2}}}\rangle\\\
\hat{W}(x_{i^{\prime}-\frac{1}{2}},x_{i^{\prime}+\frac{1}{2}})|{\mathbf{s}}\rangle=|{s_{z,x;j},s_{z,y;i}+\delta_{i,i^{\prime}},s_{x;j+\frac{1}{2}},s_{y;i+\frac{1}{2}}}\rangle\\\
\hat{W}_{z,x}(y_{j^{\prime}+\frac{1}{2}})|{\mathbf{s}}\rangle=|{s_{z,x;j},s_{z,y;i},s_{x;j+\frac{1}{2}}+\delta_{j,j^{\prime}},s_{y;i+\frac{1}{2}}}\rangle\\\
\hat{W}_{z,y}(x_{i^{\prime}+\frac{1}{2}})|{\mathbf{s}}\rangle=|{s_{z,x;j},s_{z,y;i},s_{x;j+\frac{1}{2}},s_{y;i+\frac{1}{2}}+\delta_{i,i^{\prime}}}\rangle\end{array}\right..$
(3.43)

There exists another fermionic topological state
$|\mathbf{s}^{\prime}\rangle=|s^{\prime}_{z,x;j},s^{\prime}_{z,y;i},s^{\prime}_{x;j+\frac{1}{2}},s^{\prime}_{y;i+\frac{1}{2}}\rangle$
which produces the fermionic theory $\mathfrak{T}_{F,y,\textrm{sub}}$ after
the subsystem JW transformation along $y$ direction. The fermionic topological
state $|\mathbf{s}^{\prime}\rangle$ diagonalizes the line operators,

$W_{z,x}(y_{j}),\quad\hat{W}_{z,y}(x_{i-\frac{1}{2}})W_{z,y}(x_{i})\hat{W}_{z,y}(x_{i+\frac{1}{2}}),$

and strip operators,

$W(y_{j},y_{j+1}),\quad\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})W(x_{i},x_{i+1})\hat{W}(x_{i+\frac{1}{2}},x_{i+\frac{3}{2}}),$

where $W_{z,y}(x_{i}),W(x_{i},x_{i+1})$ are sandwiched by a pair of $\hat{W}$
operators instead. The fermionic subsystem $\mathbb{Z}_{2}$ parity symmetry is
still generated by magnetic operators $\hat{W}$.

#### Subsystem JW transformation

Consider the subsystem SymTFT with the dynamical boundary state (3.37) at
$\tau=0$ given by the $(2+1)$-dimensional bosonic theory
$\mathfrak{T}_{\textrm{sub}}$. Implementing the fermionic topological
boundaries $|\mathbf{s}\rangle,|\mathbf{s}^{\prime}\rangle$ at $\tau=0$ and
shrinking the slab gives two fermionic theories
$\mathfrak{T}_{F,x,\textrm{sub}}$ and $\mathfrak{T}_{F,y,\textrm{sub}}$ whose
partition functions are,

$Z_{\mathfrak{T}_{F,x,\textrm{sub}}}(\mathbf{s})=\langle\mathbf{s}|\chi\rangle,\quad
Z_{\mathfrak{T}_{F,y,\textrm{sub}}}(\mathbf{s}^{\prime})=\langle\mathbf{s}^{\prime}|\chi\rangle.$
(3.44)

They are related to the bosonic theory $\mathfrak{T}_{\textrm{sub}}$ by
performing the subsystem JW transformations along $x$ and $y$ directions
respectively.

##  4 Subsystem $SL(2,\mathbb{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 $\mathbb{Z}_{N}$ 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 $SL(2,\mathbb{Z}_{2})$
symmetry

$\displaystyle\begin{split}&S:\quad
A\rightarrow\hat{A},\quad\hat{A}\rightarrow-A,\\\ &T:\quad A\rightarrow
A,\quad\hat{A}\rightarrow\hat{A}+A.\end{split}$ (4.1)

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 $M_{3}^{\prime}$ 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
$SL(2,\mathbb{Z}_{N})$ transformation of the boundary theory. We will see the
$S$-transformation generates the subsystem KW transformation, while the
$T$-transformation stacks a phase

$\exp(-\frac{iN}{2\pi}\int dxdydzA^{z}A^{xy})$ (4.2)

to the boundary theory. The phase (4.2) is the subsystem symmetry protected
topological (SSPT) phase [42] ††† In [42], the Lagrangian of this SSPT is
$\mathcal{L}_{\text{SSPT}}=\frac{iN}{2\pi}\Phi^{xy}(\partial_{z}A^{xy}-\partial_{x}\partial_{y}A^{z})-\frac{iN}{2\pi}A^{z}A^{xy},$
(4.3) where the auxiliary field $\Phi^{xy}$ guarrentees the flat condition of
the gauge field $A$. in $(2+1)$d.

However, the naive $SL(2,\mathbb{Z}_{2})$ transformation (4.1) has ambiguities
on the lattice. For example, when we do $S$-transformation on line operators
$\sum_{k}A^{z}_{i,j,k+\frac{1}{2}}$, the holonomy 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 field $\hat{A}^{z}$ on the dual
lattice. This leads to four inequivalent choices
$\sum_{z}A^{z}_{i\pm\frac{1}{2},j\pm\frac{1}{2},k}$ because the line operators
cannot move freely at the $x$-$y$ plane. We also need to make a smart choice
to avoid the following inconsistencies.

#### Inconsistency with the quantum algebra

Consider the following choice of regularized $S$-transformation between the
line operators,

$W_{z,y}(x_{i})\leftrightarrow\hat{W}_{z,y}(x_{i+\frac{1}{2}}),\quad
W_{z,x}(y_{j})\leftrightarrow\hat{W}_{z,x}(y_{j+\frac{1}{2}}),$ (4.4)

and strip operators,

$W(x_{i},x_{i+1})\leftrightarrow\hat{W}(x_{i+\frac{1}{2}},x_{i+\frac{3}{2}}),\quad
W(y_{j},y_{j+1})\leftrightarrow\hat{W}(y_{j+\frac{1}{2}},y_{j+\frac{3}{2}}).$
(4.5)

It maps between the site $(i,j)$ and dual site
$(i+\frac{1}{2},j+\frac{1}{2})$. However, the quantum algebras (3.23) and
(3.24) are not preserved under the transformation. For example, consider the
following commutation relation,

$W(x_{i},x_{i+1})\hat{W}_{z,y}(x_{i+\frac{1}{2}})=-\hat{W}_{z,y}(x_{i+\frac{1}{2}})W(x_{i},x_{i+1}).$
(4.6)

If we apply the $S$-transformation given above, we have,

$\hat{W}(x_{i+\frac{1}{2}},x_{i+\frac{3}{2}})W_{z,y}(x_{i})=-W_{z,y}(x_{i})\hat{W}(x_{i+\frac{1}{2}},x_{i+\frac{3}{2}})$
(4.7)

which is clearly wrong because the nontrivial phase only appears after the
exchange of electric operators and magnetic operators with intersection.

#### Inconsistency with the topological property

For another choice, we can keep (4.4) and modify (4.5) to,

$W(x_{i},x_{i+1})\leftrightarrow\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}),\quad
W(y_{j},y_{j+1})\leftrightarrow\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}}),$
(4.8)

and we will denote this choice as $\widetilde{S}$. It is straightforward to
check $\widetilde{S}$ preserve the quantum algebras (3.23) and (3.24) and it
satisfies $\widetilde{S}^{2}=1$. However, $\widetilde{S}$ assumes different
site transformations for line operators and strip operators: it maps $(i,j)$
to $(i+\frac{1}{2},j+\frac{1}{2})$ for line operators and to
$(i-\frac{1}{2},j-\frac{1}{2})$ for strip operators. This is inconsistent with
the fact that we can bend the strip operators to a pair of line operators.

$\widetilde{S}$ will generate the subsystem KW transformation on the boundary
by mapping the topological boundary state $|\mathbf{w}\rangle$ to the dual
state $|\hat{\mathbf{w}}\rangle$. For example, applying $\widetilde{S}$ on
$|\mathbf{w}\rangle$ leads to

$\displaystyle\begin{split}\widetilde{S}|{\mathbf{w}}\rangle=&\
\widetilde{S}\left(\prod_{i}\left(\hat{W}_{z,y}(x_{i+\frac{1}{2}})\right)^{w_{y;i+\frac{1}{2}}}\left(\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}))\right)^{w_{z,y;i}}\right.\\\
&\
\left.\times\prod_{j}\left(\hat{W}_{z,x}(y_{j+\frac{1}{2}})\right)^{w_{x;j+\frac{1}{2}}}\left(\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})\right)^{w_{z,x;j}}|{0}\rangle\right)\\\
=&\
\prod_{i}\left(W_{z,y}(x_{i})\right)^{w_{y;i+\frac{1}{2}}}\left(W(x_{i},x_{i+1})\right)^{w_{z,y;i}}\\\
&\
\times\prod_{j}\left(W_{z,x}(y_{j})\right)^{w_{x;j+\frac{1}{2}}}\left(W(y_{j},y_{j+1})\right)^{w_{z,x;j}}|{\hat{0}}\rangle\\\
=&\ |{\hat{\mathbf{w}}}\rangle,\end{split}$ (4.9)

where the dual holonomies $\hat{\mathbf{w}}$ is the same to the original ones
$\mathbf{w}$ in value,

$\hat{w}_{z,x;j+\frac{1}{2}}=w_{z,x;j},\quad\hat{w}_{z,y;i+\frac{1}{2}}=w_{z,y;i},\quad\hat{w}_{x;j}=w_{x,j+\frac{1}{2}},\quad\hat{w}_{y;i}=w_{y,i+\frac{1}{2}}.$
(4.10)

Here $|\hat{0}\rangle=\widetilde{S}|0\rangle$ is the vacuum of the dual state
and it is the eigenstate of the operators $\hat{W}$ with trivial eigenvalues.

#### Subsystem $SL(2,\mathbb{Z}_{2}$) transformation on the lattice

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 $SL(2,\mathbb{Z}_{2})$ symmetry
transformation after discretization should have the following properties

  1. 1.

It should be a _symmetry_ of the discretized version of the exotic action
(3.2) and preserve quantum algebras (3.23) and (3.24).

  2. 2.

It should be consistent with the topological property of the operators, for
example, the bending of strip operators in $z$ direction (3.8).‡‡‡Thanks to
Wilbur Shirley for raising this issue to us.

We will denote the $SL(2,\mathbb{Z}_{2})$ transformation on the lattice as
_subsystem_ $SL(2,\mathbb{Z}_{2})$ transformation. Besides recovering the
subsystem KW and JW transformations, we will find more duality transformations
by implementing the subsystem $SL(2,\mathbb{Z}_{2})$ transformation on the
boundary. The whole duality transformations are summarized in the duality web
(Fig. 6).

Figure 6: The duality web between four bosonic theories
$\mathfrak{T}_{\textrm{sub}},\mathfrak{T}_{xy,\textrm{sub}},\hat{\mathfrak{T}}_{\textrm{sub}},\hat{\mathfrak{T}}_{xy,\textrm{sub}}$
and four fermionic theories
$\mathfrak{T}_{F,x,\textrm{sub}},\mathfrak{T}_{F,y,\textrm{sub}},\hat{\mathfrak{T}}_{F,x,\textrm{sub}},\hat{\mathfrak{T}}_{F,y,\textrm{sub}}$
with susbsytem $\mathbb{Z}_{2}$ symmetry. The duality transformation is
generated by subsystem $SL(2,\mathbb{Z}_{2})$ transformation on the lattice:
(1) The subsystem $S$-transformation implements the subsystem KW
transformation. (2) There exist nontrivial compositions of $T$-transformations
$T^{2}_{+-},T^{2}_{-+},T^{2}_{--}$ generate the phase
$\text{SSPT}_{+-},\text{SSPT}_{-+},\text{SSPT}_{--}$. (3) Subsystem JW
transformation is a composition of subsystem $SL(2,\mathbb{Z}_{2})$
transformations. For example, the bosonic theory $\mathfrak{T}_{\textrm{sub}}$
and the fermionic theory
$\mathfrak{T}_{F,x,\textrm{sub}}(\mathfrak{T}_{F,y,\textrm{sub}})$ are related
by subsystem JW transformation, which is equivalent to performing $S^{-1}$,
$T^{2}_{+-}(T^{2}_{-+})$ and $S$ transformation sequentially.

###  4.1 Subsystem $S$-transformation

The proper $S$-transformation on the lattice is implemented by changing the
double-headed arrows in (4.4) and (4.5) to one-headed arrows and let the
$S$-transformation maps the dual site $(i+\frac{1}{2},j+\frac{1}{2})$ to
$(i+1,j+1)$ for line operators,

$\hat{W}_{z,y}(x_{i+\frac{1}{2}})\rightarrow
W_{z,y}(x_{i+1}),\quad\hat{W}_{z,x}(y_{j+\frac{1}{2}})\rightarrow
W_{z,x}(y_{j+1}),$ (4.11)

and for strip operators,

$\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})\rightarrow
W(x_{i},x_{i+1}),\quad\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})\rightarrow
W(y_{j},y_{j+1}),$ (4.12)

such that $S^{2}=\mathcal{T}$ is the translation
$\mathcal{T}:(i,j)\rightarrow(i+1,j+1)$ on lattice. This choice preserves the
quantum algebra and is also consistent with the bending of operators at the
expense of giving up $S^{2}=1$. Similarly, it will also generate the subsystem
KW transformation on the boundary by mapping the topological boundary state
$|\mathbf{w}\rangle$ to the dual state
$|\hat{\mathbf{w}}\rangle=S|\mathbf{w}\rangle$ as

$\hat{w}_{z,x;j+\frac{1}{2}}=w_{z,x;j},\quad\hat{w}_{z,y;i+\frac{1}{2}}=w_{z,y;i},\quad\hat{w}_{x;j}=w_{x,j-\frac{1}{2}},\quad\hat{w}_{y;i}=w_{y,i-\frac{1}{2}},$
(4.13)

and also

$\displaystyle S^{2}|\mathbf{w}\rangle=\mathcal{T}|\mathbf{w}\rangle,$ (4.14)

where $\mathcal{T}$ will shift the holonomies as

$w_{z,x;j}\rightarrow w_{z,x;j-1},\quad w_{z,y;i}\rightarrow w_{z,y;i-1},\quad
w_{x;j+\frac{1}{2}}\rightarrow w_{x;j-\frac{1}{2}},\quad
w_{y;i+\frac{1}{2}}\rightarrow w_{y;i-\frac{1}{2}}.$ (4.15)

As a summary, we have two possible definitions of $S$-transformation on
lattice denoted as $\widetilde{S}$ and $S$ and both of them preserve the
quantum algebra. The first one satisfies the naive relation
$\widetilde{S}^{2}=1$ but is not consistent with the bending of operators.
Therefore it is not a suitable choice on the lattice. We will denote
$\widetilde{S}$ as _field theory $S$-transformation_ since it implements the
naive $S$-transformation of the exotic tensor theory in (4.1). The second one
is consistent with the bending on the lattice but $S^{2}$ is a translation
$\mathcal{T}$ instead. We will also denote $S$ as _lattice S-transformation_.
Both of them will map $|\mathbf{w}\rangle$ to dual state
$|\hat{\mathbf{w}}\rangle$ with different assignments of $\hat{\mathbf{w}}$.

###  4.2 Subsystem $T$-transformation

For the subsystem $T$-transformation in (4.1), one needs to dress every
magnetic operator $\hat{W}$ with a nearby electric operator $W$. Again we need
to avoid the following inconsistencies.

#### Inconsistency with the quantum algebra

Naively one could have for example,

$\hat{W}_{z,y}(x_{i+\frac{1}{2}})\rightarrow\hat{W}_{z,y}(x_{i+\frac{1}{2}})W_{z,y}(x_{i+1}),\quad\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})\rightarrow\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})W(x_{i},x_{i+1}),$
(4.16)

where the $W$-operators are on the right of $\hat{W}$-operators. However, just
as the $S$-transformation cases the quantum algebras (3.23) and (3.24) are not
preserved and $T$-transformation is not a good symmetry on the lattice.

#### Inconsistency with the topological property

One can try to modify the transformation (4.16) in a way consistent with the
algebra

$\displaystyle\begin{split}&+:\quad\hat{W}_{z,y}(x_{i+\frac{1}{2}})\rightarrow
W_{z,y}(x_{i})\hat{W}_{z,y}(x_{i+\frac{1}{2}}),\quad\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})\rightarrow\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})W(x_{i},x_{i+1}),\\\
&-:\quad\hat{W}_{z,y}(x_{i+\frac{1}{2}})\rightarrow\hat{W}_{z,y}(x_{i+\frac{1}{2}})W_{y}(x_{i+1}),\quad\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})\rightarrow
W(x_{i-1},x_{i})\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}}),\end{split}$
(4.17)

and there are two similar choices for operators depending on $y$:
$\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})$ and
$\hat{W}_{x}(y_{j+\frac{1}{2}})$. In total, we have four choices and they are
denoted as $T_{++},T_{+-},T_{-+},T_{--}$. However, none of the four choices
$T_{\pm\pm}$ are compatible with the bending of operators. Nevertheless, we
will elaborate on their actions because they are useful later when we consider
the proper $T^{2}$ transformation on the lattice.

The transformations (4.17) will stack an extra phase when acting on the
topological boundary. For example, with the expression (3.31) of the
topological boundary state $|\mathbf{w}\rangle$, applying $T_{++}$ leads to a
new topological boundary state

$\displaystyle\begin{split}T_{++}|{\mathbf{w}}\rangle=&\
T_{++}\prod_{i}\left(\hat{W}_{z,y}(x_{i+\frac{1}{2}})\right)^{w_{y;i+\frac{1}{2}}}\left(\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})\right)^{w_{z,y;i}}\\\
&\
\times\prod_{j}\left(\hat{W}_{z,x}(y_{j+\frac{1}{2}})\right)^{w_{x;j+\frac{1}{2}}}\left(\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})\right)^{w_{z,x;j}}|{0}\rangle\\\
=&\
\prod_{i}\left(W_{z,y}(x_{i})\hat{W}_{z,y}(x_{i+\frac{1}{2}})\right)^{w_{y;i+\frac{1}{2}}}\left(\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})W(x_{i},x_{i+1})\right)^{w_{z,y;i}}\\\
&\
\times\prod_{j}\left(W_{z,x}(y_{j})\hat{W}_{z,x}(y_{j+\frac{1}{2}})\right)^{w_{x;j+\frac{1}{2}}}\left(\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})W(y_{j},y_{j+1})\right)^{w_{z,x;j}}T_{++}|{0}\rangle\\\
=&\
(-1)^{\sum_{j}w_{z,x;j}w_{x;{j+\frac{1}{2}}}+\sum_{i}w_{z,y;i}w_{y;i+\frac{1}{2}}}|\mathbf{w}\rangle\end{split}$
(4.18)

where $T_{++}|0\rangle\sim|0\rangle$ because they satisfy the same operators
equation (3.28), and we will assume $|0\rangle$ is invariant under the action
of $T_{++}$. In general, acting $T_{\pm\pm}$ on the topological boundary
$|\mathbf{w}\rangle$ will stack the phase

$(-1)^{\sum_{j}w_{z,x;j}w_{x;{j\pm\frac{1}{2}}}+\sum_{i}w_{z,y;i}w_{y;i\pm\frac{1}{2}}}.$
(4.19)

As we mentioned before, $T_{\pm\pm}$ are not good transformations on lattice
and we should not take those phases seriously.

#### Proper $T^{2}$ transformation on the lattice

When we compose different $T_{\pm\pm}$ on the lattice, there exist $T^{2}$
-transformations which are consistent with both the algebras and the bending.
In the naive $SL(2,\mathbb{Z}_{2})$ transformation (4.1) of the field theory,
acting $T$-transformation twice is the identity transformation. However, on
the lattice, composing different $T$-transformations will lead to four
distinct operations

$\displaystyle\begin{split}T^{2}_{++}&\equiv
T_{++}T_{++}=T_{--}T_{--}=T_{+-}T_{+-}=T_{-+}T_{-+},\\\ T^{2}_{--}&\equiv
T_{++}T_{--}=T_{--}T_{++}=T_{+-}T_{-+}=T_{-+}T_{+-},\\\ T^{2}_{+-}&\equiv
T_{++}T_{+-}=T_{--}T_{-+}=T_{+-}T_{++}=T_{-+}T_{--},\\\ T^{2}_{-+}&\equiv
T_{++}T_{-+}=T_{--}T_{+-}=T_{+-}T_{--}=T_{-+}T_{++},\end{split}$ (4.20)

where the indices follow the sign rule

$T^{2}_{pp^{\prime},qq^{\prime}}=T_{p,q}T_{p^{\prime},q^{\prime}},\quad
p,q=\pm.$ (4.21)

We can also write down the transformation of operators for $T^{2}_{\pm\pm}$.
$T^{2}_{++}$ is the identity transformation and $T^{2}_{--}$ is realized by,

$T^{2}_{--}:\quad\left\\{\begin{array}[]{l}\hat{W}_{z,y}(x_{i+\frac{1}{2}})\rightarrow
W_{z,y}(x_{i})\hat{W}_{z,y}(x_{i+\frac{1}{2}})W_{z,y}(x_{i+1})\\\
\hat{W}_{z,x}(y_{j+\frac{1}{2}})\rightarrow
W_{z,x}(y_{j})\hat{W}_{z,x}(y_{j+\frac{1}{2}})W_{z,x}(y_{j+1})\\\
\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})\rightarrow
W(x_{i-1},x_{i})\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})W(x_{i},x_{i+1})\\\
\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})\rightarrow
W(y_{j-1},y_{j})\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})W(y_{j},y_{j+1})\\\
\end{array}\right.$ (4.22)

where all $\hat{W}$ operators are sandwiched by a pair of $W$ operators in a
symmetric way. The other two choices $T^{2}_{+-}$ and $T^{2}_{-+}$ are given
by,

$T^{2}_{+-}:\quad\left\\{\begin{array}[]{l}\hat{W}_{z,y}(x_{i+\frac{1}{2}})\rightarrow\hat{W}_{z,y}(x_{i+\frac{1}{2}})\\\
\hat{W}_{z,x}(y_{j+\frac{1}{2}})\rightarrow
W_{z,x}(y_{j})\hat{W}_{z,x}(y_{j+\frac{1}{2}})W_{z,x}(y_{j+1})\\\
\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})\rightarrow\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})\\\
\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})\rightarrow
W(y_{j-1},y_{j})\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})W(y_{j},y_{j+1})\\\
\end{array}\right.$ (4.23)

and,

$T^{2}_{-+}:\quad\left\\{\begin{array}[]{l}\hat{W}_{z,y}(x_{i+\frac{1}{2}})\rightarrow
W_{z,y}(x_{i})\hat{W}_{z,y}(x_{i+\frac{1}{2}})W_{z,y}(x_{i+1})\\\
\hat{W}_{z,x}(y_{j+\frac{1}{2}})\rightarrow\hat{W}_{z,x}(y_{j+\frac{1}{2}})\\\
\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})\rightarrow
W(x_{i-1},x_{i})\hat{W}(x_{i-\frac{1}{2}},x_{i+\frac{1}{2}})W(x_{i},x_{i+1})\\\
\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})\rightarrow\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})\\\
\end{array}\right.$ (4.24)

where only part of $\hat{W}$ operators are sandwiched by $W$ operators.
Obviously, we have $T^{2}_{-+}T^{2}_{+-}=T^{2}_{--}$ and they all satisfy
$\left(T^{2}_{\pm\pm}\right)^{2}=1$. The corresponding subsystem symmetric
protected topological (SSPT) phases are

$\displaystyle\begin{split}\text{SSPT}_{++}(\mathbf{w})=&\ 1,\\\
\textrm{SSPT}_{+-}(\mathbf{w})=&\
(-1)^{\sum_{j}w_{z,x;j}(w_{x;{j-\frac{1}{2}}}+w_{x;{j+\frac{1}{2}}})},\\\
\textrm{SSPT}_{-+}(\mathbf{w})=&\
(-1)^{\sum_{i}w_{z,y;i}(w_{y;i-\frac{1}{2}}+w_{y;i+\frac{1}{2}})},\\\
\textrm{SSPT}_{--}(\mathbf{w})=&\
(-1)^{\sum_{j}w_{z,x;j}(w_{x;{j-\frac{1}{2}}}+w_{x;{j+\frac{1}{2}}})+\sum_{i}w_{z,y;i}(w_{y;i-\frac{1}{2}}+w_{y;i+\frac{1}{2}})}.\end{split}$
(4.25)

###  4.3 Duality web from the subsystem $SL(2,\mathbb{Z}_{2})\ $transformation

The duality web (Fig. 6) is generated by implementing the subsystem
$SL(2,\mathbb{Z}_{2})$ transformations $S$ and $T^{2}$ consecutively. In
particular, the subsystem JW transformation is equivalent to performing $S$,
$T^{2}_{+-}(\textrm{or}\ T^{2}_{-+})$ and $S^{-1}$ transformation
sequentially. This is easy to see in the transformation of the operators.
Begin with the bosonic state $|\mathbf{w}\rangle$ which are eigenstates of $W$
operators. If we do an $S$-transformation the roles of $W$ and $\hat{W}$ are
exchanged and we get the dual state $|\hat{\mathbf{w}}\rangle$ which are
eigenstates of $\hat{W}$ operators. Then applying $T^{2}_{+-}$ we find
$W_{z,y}(x_{i})$ and $W(x_{i},x_{i+1})$ should be sandwiched by a pair of
$\hat{W}$ operators according to (4.23) (notice that the roles of $W$ and
$\hat{W}$ have been exchanged due to the $S$ transformation). It will stack
the phase $\textrm{SSPT}_{+-}(\hat{\mathbf{w}})$ to
$|\hat{\mathbf{w}}\rangle$. If we do another $S^{-1}$ transformation then we
will obtain some states which are the eigenstates of line operators,

$\hat{W}_{z,x}(y_{j-\frac{1}{2}})W_{z,x}(y_{j})\hat{W}_{z,x}(y_{j+\frac{1}{2}}),\quad
W_{z,y}(x_{i}),$ (4.26)

and strip operators,

$\hat{W}(y_{j-\frac{1}{2}},y_{j+\frac{1}{2}})W(y_{j},y_{j+1})\hat{W}(y_{j+\frac{1}{2}},y_{j+\frac{3}{2}}),\quad
W(x_{i},x_{i+1}).$ (4.27)

According to (3.42) they are the same set of operators that diagonalize the
fermion topological state $|\mathbf{s}\rangle$. Therefore we have,

$|\mathbf{s}\rangle=S^{-1}T^{2}_{+-}S|\mathbf{w}\rangle,$ (4.28)

with $\mathbf{s}=\mathbf{w}$. By similar argument, if we perform $T^{2}_{+-}$
and $S$ transformation sequentially the resulting states are eigenvalues of
(4.26) and (4.27) with $W,\hat{W}$ exchanged. We obtain a new fermionic
topological state,

$|\hat{\mathbf{s}}\rangle\equiv ST^{2}_{+-}|{\mathbf{w}}\rangle$ (4.29)

where the relation between $\hat{\mathbf{s}}$ and $w$ are suggested in (4.13).
They are the JW transformations of the dual state $|\hat{\mathbf{w}}\rangle$.

Let’s check explicitly the state $|\hat{\mathbf{s}}\rangle$ can be written as
a JW transformation of $|\hat{\mathbf{w}}\rangle$ by summing over all sectors
of the dual bosonic state with proper phases. Following the definition, we
stack the phase $\text{SSPT}_{+-}(\mathbf{w}^{\prime})$ on the state
$|\mathbf{w}^{\prime}\rangle$ introduced in (3.27) and consider the KW
transformation given by,

$\displaystyle\begin{split}|\hat{\mathbf{s}}\rangle=&\
\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{\mathbf{w}^{\prime}\in
M_{v}}(-1)^{\sum_{i}(\hat{s}_{z,y;i+\frac{1}{2}}w^{\prime}_{y;i+\frac{1}{2}}+\hat{s}_{y;i}w^{\prime}_{z,y;i})+\sum_{j}(\hat{s}_{z,x;j+\frac{1}{2}}w^{\prime}_{x;j+\frac{1}{2}}+\hat{s}_{x;j}w^{\prime}_{z,x;j})}\\\
&\times(-1)^{\sum_{j}w^{\prime}_{z,x;j}(w^{\prime}_{x;{j-\frac{1}{2}}}+w^{\prime}_{x;{j+\frac{1}{2}}})}|\mathbf{w}^{\prime}\rangle.\end{split}$
(4.30)

We can rewrite $|\mathbf{w}^{\prime}\rangle$ into
$|\hat{\mathbf{w}}^{\prime}\rangle$ using the KW relation (3.35) and get

$\displaystyle\begin{split}|\hat{\mathbf{s}}\rangle=&\
\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{\mathbf{w}^{\prime}\in
M_{v}}(-1)^{\sum_{i}(\hat{s}_{z,y;i+\frac{1}{2}}w^{\prime}_{y;i+\frac{1}{2}}+\hat{s}_{y;i}w^{\prime}_{z,y;i})+\sum_{j}(\hat{s}_{z,x;j+\frac{1}{2}}w^{\prime}_{x;j+\frac{1}{2}}+\hat{s}_{x;j}w^{\prime}_{z,x;j})}\\\
&\times(-1)^{\sum_{j}w^{\prime}_{z,x;j}(w^{\prime}_{x;{j-\frac{1}{2}}}+w^{\prime}_{x;{j+\frac{1}{2}}})}\\\
&\times\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{\mathbf{\hat{w}}^{\prime}\in
M_{v}}(-1)^{\sum_{i}(\hat{w}^{\prime}_{z,y;i+\frac{1}{2}}w^{\prime}_{y;i+\frac{1}{2}}+\hat{w}^{\prime}_{y;i}w^{\prime}_{z,y;i})+\sum_{j}(\hat{w}^{\prime}_{z,x;j+\frac{1}{2}}w^{\prime}_{x;j+\frac{1}{2}}+\hat{w}^{\prime}_{x;j}w^{\prime}_{z,x;j})}|\hat{\mathbf{w}}^{\prime}\rangle.\end{split}$
(4.31)

Summing $w^{\prime}_{z,x;j}$ and $w^{\prime}_{z,y;i}$ produces two
restrictions,

$\hat{w}^{\prime}_{x;j}=\hat{s}_{x;j}+w^{\prime}_{x;{j-\frac{1}{2}}}+w^{\prime}_{x;{j+\frac{1}{2}}},\quad\hat{w}^{\prime}_{y;i}=\hat{s}_{y;i}.$
(4.32)

After relabelling $\hat{u}^{x}_{j+\frac{1}{2}}\equiv
w^{\prime}_{x;j+\frac{1}{2}},\hat{u}^{y}_{i+\frac{1}{2}}\equiv
w^{\prime}_{y,i+\frac{1}{2}}$, (4.31) becomes

$|\hat{\mathbf{s}}\rangle=\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{\hat{u},\hat{w}^{\prime}_{z}\in
M_{\hat{u},\hat{w}^{\prime}_{z}}}(-1)^{\sum_{i}(\hat{s}_{z,y;i+\frac{1}{2}}+\hat{w}^{\prime}_{z,y;i+\frac{1}{2}})\hat{u}^{y}_{i+\frac{1}{2}}+(\hat{s}_{z,x;j+\frac{1}{2}}+\hat{w}^{\prime}_{z,x;j+\frac{1}{2}})\hat{u}^{x}_{j+\frac{1}{2}}}|\hat{\mathbf{w}}^{\prime}\rangle$
(4.33)

which shows that the dual fermionic state $|\hat{\mathbf{s}}\rangle$ is the
subsystem JW transformation of the dual state
$|{\hat{\mathbf{w}}^{\prime}}\rangle$ resembling (3.40).

As a summary, begin with a bosoinic state $|\mathbf{w}\rangle$ the JW
transformation can be written as

$S^{-1}T^{2}_{+-}S=\text{JW}_{x},\quad S^{-1}T^{2}_{-+}S=\text{JW}_{y}$ (4.34)

acting on the state $|\mathbf{w}\rangle$ and therefore the phases
$\textrm{SSPT}_{+-}$ and $\textrm{SSPT}_{-+}$ are both fermionic subsystem SPT
phases.§§§We thank Kantaro Ohmori and Yunqin Zheng for pointing out this to
us. With these identifications, we can generate other path in the duality web.
For example,

$\displaystyle\begin{split}(\text{JW}_{y})^{-1}\text{JW}_{x}=S^{-1}(T^{2}_{-+})^{-1}SS^{-1}T^{2}_{+-}S=S^{-1}T^{2}_{-+}T^{2}_{+-}S=S^{-1}T^{2}_{--}S\end{split}$
(4.35)

shows that the subsystem KW transformation of
$\textrm{SSPT}_{--}|\mathbf{w}\rangle$ 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 (4.35), the phase $\textrm{SSPT}^{2}_{--}$ is a
bosonic phase.

Based on the above analysis, we can obtain a duality web as shown in Fig. 6.

##  5 $S$-defects in the subsystem SymTFT

In this section, we will construct the co-dimensional one symmetry defects
generating the $SL(2,\mathbb{Z}_{2})$ 0-form symmetry with a focus on
$S$-defect. It was shown in [80, 81, 65] that in a $(d+1)$-dimensonal TFT,
such kind of symmetry defects $\mathcal{D}$ extending along a co-dimension one
hypersurface $M_{d}$ are built by condensing certain types of topological
defects $\mathcal{L}$ along $M_{d}$. If the topological defects $\mathcal{L}$
generate a $q$-form symmetry inside $M_{d}$, the condensation defect
$\mathcal{D}$ 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 $\mathbb{Z}_{2}$ symmetry in $(2+1)$d BF theory with
level $N$.

As discussed in the previous section, the proper $S$-transformation defined on
the lattice satisfies $S^{2}=\mathcal{T}$ where $\mathcal{T}$ is the
translation $(i,j)\rightarrow(i+1,j+1)$ on the lattice. 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.

###  5.1 Conventions on operators and algebras

For later convenience, we introduce $U_{I}$ and $\hat{U}_{I}$ as the
collection of electric and magnetic line/strip operators respectively

$U_{I}=\left\\{\begin{array}[]{l}W(y_{I},y_{I+1})\quad I=1,\cdots,L_{y}\\\
W(x_{I-L_{y}},x_{I-L_{y}+1})\quad I=L_{y}+1,\cdots,L_{y}+L_{x}\\\
W_{z,x}(y_{I-L_{x}-L_{y}})\quad I=L_{y}+L_{x}+1,\cdots,2L_{y}+L_{x}\\\
W_{z,y}(x_{I-L_{x}-2L_{y}})\quad
I=2L_{y}+L_{x}+1,\cdots,2L_{y}+2L_{x}\end{array}\right.$ (5.1)
$\hat{U}_{I}=\left\\{\begin{array}[]{l}\hat{W}(y_{I-\frac{1}{2}},y_{I+\frac{1}{2}})\quad
I=1,\cdots,L_{y}\\\
\hat{W}(x_{I-L_{y}-\frac{1}{2}},x_{I-L_{y}+\frac{1}{2}})\quad
I=L_{y}+1,\cdots,L_{y}+L_{x}\\\
\hat{W}_{z,x}(y_{I-L_{x}-L_{y}+\frac{1}{2}})\quad
I=L_{y}+L_{x}+1,\cdots,2L_{y}+L_{x}\\\
\hat{W}_{z,y}(x_{I-L_{x}-2L_{y}+\frac{1}{2}})\quad
I=2L_{y}+L_{x}+1,\cdots,2L_{y}+2L_{x}\end{array}\right.$ (5.2)

where $I=1,\cdots,2L_{x}+2L_{y}$. We will use the lattice size integer
$n\equiv L_{x}+L_{y}$ for simplicity. In this convention, the quantum algebras
(3.23) and (3.24) between electric and magnetic operators has a compact and
symmetric form,

$U_{I}\hat{U}_{J}=-\Omega_{IJ}\hat{U}_{J}U_{I},$ (5.3)

where $\Omega_{IJ}$ is a $2n\times 2n$ symmetric matrix,

$\Omega_{IJ}=\left(\begin{array}[]{cc}0&I_{n\times n}\\\ I_{n\times
n}&0\end{array}\right).$ (5.4)

We will then formulate the general operators, the algebras between them and
their actions on the boundary states. The general operator

$K[\alpha,\hat{\alpha}]:=\prod_{I=1}^{2n}U^{\alpha_{I}}_{I}\prod_{J=1}^{2n}\hat{U}^{\hat{\alpha}_{J}}_{J}$
(5.5)

is parametrized by two $2n$-dimensional vectors with $\mathbb{Z}_{2}$-valued
entries

$\displaystyle\begin{split}&\alpha=(a,b):=(a_{1},a_{2},\cdots,a_{n},b_{1},b_{2},\cdots,b_{n}),\\\
&\hat{\alpha}=(\hat{a},\hat{b}):=(\hat{a}_{1},\hat{a}_{2},\cdots,\hat{a}_{n},\hat{b}_{1},\hat{b}_{2},\cdots,\hat{b}_{n}).\end{split}$
(5.6)

From the quantum algebra (5.3), the general operators
$K[\alpha,\hat{\alpha}],K[\alpha^{\prime},\hat{\alpha}^{\prime}]$ have the
following fusion rule

$\displaystyle
K[\alpha,\hat{\alpha}]K[\alpha^{\prime},\hat{\alpha}^{\prime}]=(-1)^{-\hat{\alpha}\cdot\alpha^{\prime}}K[\alpha+\alpha^{\prime},\hat{\alpha}+\hat{\alpha}^{\prime}],$
(5.7)

together with the commutation algebra

$K[\alpha,\hat{\alpha}]K[\alpha^{\prime},\hat{\alpha}^{\prime}]=(-1)^{\alpha\cdot\hat{\alpha}^{\prime}-\hat{\alpha}\cdot\alpha^{\prime}}K[\alpha^{\prime},\hat{\alpha}^{\prime}]K[\alpha,\hat{\alpha}].$
(5.8)

where the symmetric inner product between two $2n$-dimensional vectors is
defined by

$\alpha\cdot\hat{\alpha}=\sum_{IJ}\Omega_{IJ}\alpha_{I}\hat{\alpha}_{J}.$
(5.9)

Here are some comments about the parameters $\alpha,\hat{\alpha}$. Due to the
gauge redundancy (3.21) and constraint (3.22) among the electric operators, as
well as their magnetic counterparts, these parameters have the identification

$a\sim a+1,\quad\hat{a}\sim\hat{a}+1$ (5.10)

of the $n$-dimensional vectors $a$ and $\hat{a}$ and the constraint

$\sum_{i=1}^{n}b_{i}=\sum_{i=1}^{n}\hat{b}_{i}=0$ (5.11)

among the $n$-dimensional vectors $b$ and $\hat{b}$. The inner product (5.9)
is invariant under the gauge transformation (5.10) providing the constraints
(5.11).

On the boundary manifold $M_{3}$, we can identify the vector $\alpha$ with the
holonomies

$\alpha\equiv(w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}}),$
(5.12)

Similarly, we can define $\hat{\alpha}$ as the dual parameter of $\alpha$,
which is

$\hat{\alpha}\equiv(\hat{w}_{z,x;j+\frac{1}{2}},\hat{w}_{z,y;i+\frac{1}{2}},\hat{w}_{x;j},\hat{w}_{y;i}).$
(5.13)

The lattice $S$-defect will map $|\alpha\rangle=|a,b\rangle$ to the dual state
$|\hat{\alpha}\rangle=|\hat{a},\hat{b}\rangle$ with

$\hat{a}=a,\quad\hat{b}\equiv(b_{L_{y}},b_{1},\cdots,b_{L_{y}-1},b_{L_{x}+L_{y}},b_{L_{y}+1},\cdots,b_{L_{x}+L_{y}-1})\equiv
b_{T},$ (5.14)

which is equivalent to (4.13). In the rest of the paper, we will use $b_{T}$
to denote the shifted vector introduced above. On the other hand, the field
theory $S$-defect maps $|\alpha\rangle=|a,b\rangle$ to the dual state
$|\hat{\alpha}\rangle=|\hat{a},\hat{b}\rangle$ with $\hat{a}=a$ and
$\hat{b}=b$.

From now on, we will use $|{\alpha}\rangle$ and $|{\hat{\alpha}}\rangle$ for
the boundary state and its subsystem KW dual. We can rewrite the actions of
electric and magnetic operators on topological boundary states
(3.28),(3.29),(3.33),(3.34) as

$K[\gamma,0]|\alpha\rangle=(-1)^{\gamma\cdot\alpha}|\alpha\rangle,\quad
K[0,\gamma]|\alpha\rangle=|\alpha+\gamma\rangle.$ (5.15)
$K[0,\hat{\gamma}]|\hat{\alpha}\rangle=(-1)^{\hat{\gamma}\cdot\hat{\alpha}}|\hat{\alpha}\rangle,\quad
K[\hat{\gamma},0]|\hat{\alpha}\rangle=|\hat{\alpha}+\hat{\gamma}\rangle,$
(5.16)

together with the subsystem KW transformation (3.35) between the boundary
state $|{\alpha}\rangle$ and its dual $|{\hat{\alpha}}\rangle$ as

$|\hat{\alpha}\rangle=\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{\alpha\in
M_{v}}(-1)^{\hat{\alpha}\cdot\alpha}|\alpha\rangle,$ (5.17)

where $M_{v}$ denote the set of $2n$-dimensional vectors satisfying the
restrictions,

$M_{v}=\left\\{\alpha=(a,b)|a\sim a+1,\sum_{i=1}^{n}b_{i}=0\right\\}.$ (5.18)

With the orthogonality $\langle\beta|\alpha\rangle=\delta_{\alpha\beta}$, one
has

$\langle\beta|\hat{\alpha}\rangle=\frac{1}{2^{L_{x}+L_{y}-1}}(-1)^{\hat{\alpha}\cdot\beta}.$
(5.19)

The inverse transformation is,

$|\alpha\rangle=\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{\hat{\alpha}\in
M_{v}}(-1)^{\hat{\alpha}\cdot\alpha}|\hat{\alpha}\rangle,$ (5.20)

and for consistency, we should have the orthogonality relation,

$\frac{1}{2^{2(L_{x}+L_{y}-1)}}\sum_{\alpha\in
M_{v}}(-1)^{\alpha\cdot\beta}=\delta_{\beta,0}.$ (5.21)

This is not obviously true because $\alpha,\beta$ are not free and they should
satisfy the restrictions given above. Before ending this section, let us check
this relation explicitly. Decompose $\alpha=(a,b),\beta=(c,d)$, one has,

$\frac{1}{2^{2(L_{x}+L_{y}-1)}}\sum_{(a,b)\in M_{v}}(-1)^{a\cdot
d}(-1)^{b\cdot c}.$ (5.22)

In the first factor $(-1)^{a\cdot d}$ we have $a\sim a+1$ and $\sum d=0$.
Therefore we can relax the restriction $a\sim a+1$ and write,

$\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{a\sim a+1}(-1)^{a\cdot
d}=\frac{1}{2^{L_{x}+L_{y}-1}}\times\frac{1}{2}\sum_{a}(-1)^{a\cdot
d}=\delta_{d,0}.$ (5.23)

In the second factor $(-1)^{b\cdot c}$ we have $\sum b=0$ and $c\sim c+1$. We
can also relax the restriction $\sum b=0$ by adding an Lagrangian multiplier
$\lambda\in\mathbb{Z}_{2}$,

$\displaystyle\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{b|\sum b=0}(-1)^{b\cdot c}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{L_{x}+L_{y}-1}}\times\frac{1}{2}\sum_{\lambda}(-1)^{\lambda(\sum
b)}\sum_{b}(-1)^{b\cdot c}$ $\displaystyle=$
$\displaystyle\frac{1}{2^{L_{x}+L_{y}}}\sum_{\lambda}\sum_{b}(-1)^{b\cdot(c+\lambda)}=\delta_{c,0}+\delta_{c+1,0},$
(5.24)

where in the last line $\lambda$ is understood as the constant vector
$(\lambda,\cdots,\lambda)$. This is also consistent with the fact $c\sim c+1$.
Combined everything together we have proven the orthogonality relation.

$U_{I}$$=$$\hat{U}_{I}$$S$-defect Figure 7: Action of $S$-defect on line/strip
operators

###  5.2 $S$-defect

In the last section, we discuss two kinds of $S$-transformation, the lattice
$S$-transformation $S$ and the field theory $S$-transformation
$\widetilde{S}$. We will discuss both lattice $S$-defect and field theory
$S$-defect in the following.

As shown in Fig. 7, the lattice $S$-defect (or field theory $S$-defect) maps
the electric operator $U_{I}$ to magnetic operator $\hat{U}_{I}$ and vice
versa. Fusing to the boundary, it maps the boundary state to its KW dual. As
we mentioned before, the lattice $S$-defect will map
$|\alpha\rangle\equiv|a,b\rangle$ to the dual state
$|\hat{\alpha}\rangle=|\hat{a},\hat{b}\rangle$ with $\hat{a}=a$ and
$\hat{b}=b_{T}$ where $b_{T}$ is related to $b$ as,

$b_{T}\equiv(b_{L_{y}},b_{1},\cdots,b_{L_{y}-1},b_{L_{x}+L_{y}},b_{L_{y}+1},\cdots,b_{L_{x}+L_{y}-1}).$
(5.25)

We will construct the condensation defect on $M_{3}$ in the bulk using the
bra-ket trick,

$S=\sum_{(\hat{a},\hat{b})=(a,b_{T})\in M_{v}}|\hat{a},\hat{b}\rangle\langle
a,b|,$ (5.26)

which manifests its action on the state. We use $(\hat{a},\hat{b})=(a,b_{T})$
to emphasize that the values of the holonomies $\hat{b}$ of the dual states
are related to the holonomies $b$ of the original state via a shift. In the
following, we will omit the $M_{v}$ notation and assume every vector
$\alpha=(a,b)$ or $\hat{\alpha}=(\hat{a},\hat{b})$ should satisfy the
constraints automatically. Notice that $S^{\dagger}S=1$ by construction.

Before moving on, it is convenient to decompose the $S$-operators as,

$S=\widetilde{\mathcal{T}}\widetilde{S},$ (5.27)

where $\widetilde{\mathcal{T}}$ and $\widetilde{S}$ are defined as,

$\widetilde{\mathcal{T}}=\sum_{(\hat{a},\hat{b})}|\hat{a},\hat{b}_{T}\rangle\langle\hat{a},\hat{b}|,\quad\widetilde{S}=\sum_{(a,b)=(\hat{a},\hat{b})}|\hat{a},\hat{b}\rangle\langle
a,b|,$ (5.28)

where $\hat{b}_{T}$ is defined as the hat version of (5.25). One can check

$\displaystyle\widetilde{T}\widetilde{S}$
$\displaystyle=\sum_{(\hat{a}^{\prime},\hat{b}^{\prime})}\sum_{(a,b)=(\hat{a},\hat{b})}|\hat{a}^{\prime},\hat{b}^{\prime}_{T}\rangle\langle\hat{a}^{\prime},\hat{b}^{\prime}|\hat{a},\hat{b}\rangle\langle
a,b|$
$\displaystyle=\sum_{(\hat{a}^{\prime},\hat{b}^{\prime})}\sum_{(a,b)=(\hat{a},\hat{b})}\delta_{\hat{a}^{\prime},\hat{a}}\delta_{\hat{b}^{\prime},\hat{b}}|\hat{a}^{\prime},\hat{b}^{\prime}_{T}\rangle\langle
a,b|$
$\displaystyle=\sum_{(\hat{a},\hat{b})=(a,b)}|\hat{a},\hat{b}_{T}\rangle\langle
a,b|=S.$ (5.29)

Here $\widetilde{S}$ is the field theory $S$-defect which implements the field
theory $S$-transformation defined in (4.9) in the previous section and it
satisfies

$U_{I}\widetilde{S}=\widetilde{S}\hat{U}_{I},\quad\hat{U}_{I}\widetilde{S}=\widetilde{S}U_{I},$
(5.30)

using (5.7) and (5.8). Moreover, one can check $\widetilde{S}^{2}$ is identity

$\displaystyle\widetilde{S}^{2}$
$\displaystyle=\sum_{(a,b)=(\hat{a},\hat{b}),(c,d)=(\hat{c},\hat{d})}|\hat{a},\hat{b}\rangle\langle
a,b|\hat{c},\hat{d}\rangle\langle c,d|$
$\displaystyle=\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{(a,b)=(\hat{a},\hat{b}),(c,d)=(\hat{c},\hat{d})}(-1)^{\hat{c}\cdot
b+\hat{d}\cdot a}|\hat{a},\hat{b}\rangle\langle c,d|$
$\displaystyle=\sum_{(c,d)}|c,d\rangle\langle c,d|\equiv\text{I},$ (5.31)

where we use the subsystem KW transformation (5.17) in the second line. On the
other hand, the operator $\widetilde{\mathcal{T}}$ will shift the holonomies
such that the combination $\widetilde{\mathcal{T}}\widetilde{S}$ implements
the transformation (4.13). To be concrete, let’s evaluate and check
$(\widetilde{\mathcal{T}}\widetilde{S})^{2}=\mathcal{T}$ where $\mathcal{T}$
is the translation. First, we have

$\sum_{(\hat{a},\hat{b})}|\hat{a},\hat{b}_{T}\rangle\langle\hat{a},\hat{b}|\widetilde{S}=\sum_{(a,b)=(\hat{a},\hat{b})}|\hat{a},\hat{b}_{T}\rangle\langle
a,b|.$ (5.32)

Moreover we have the following identities

$\sum_{(\hat{a},\hat{b})}|\hat{a},\hat{b}_{T}\rangle\langle\hat{a},\hat{b}|=\sum_{(a,b)}|a_{T},b\rangle\langle
a,b|,\quad\sum_{(\hat{a},\hat{b})}|\hat{a}_{T},\hat{b}\rangle\langle\hat{a},\hat{b}|=\sum_{(a,b)}|a,b_{T}\rangle\langle
a,b|,$ (5.33)

and also

$\sum_{(\hat{a},\hat{b})}|\hat{a}_{T},\hat{b}_{T}\rangle\langle\hat{a},\hat{b}|=\sum_{(a,b)}|a_{T},b_{T}\rangle\langle
a,b|.$ (5.34)

It is easy to check them by acting both sides on a state $|a,b\rangle$. For
example, for the first identity, acting the LHS on $|a,b\rangle$ we get

$\displaystyle\sum_{\hat{a},\hat{b}}|\hat{a},\hat{b}_{T}\rangle\langle\hat{a},\hat{b}|a,b\rangle$
$\displaystyle=$
$\displaystyle\sum_{\hat{a},\hat{b}}|\hat{a},\hat{b}_{T}\rangle\frac{1}{2^{L_{x}+L_{y}-1}}(-1)^{\hat{a}\cdot
b+\hat{b}\cdot a}$ $\displaystyle=$
$\displaystyle\sum_{\hat{a},\hat{b}}|\hat{a},\hat{b}_{T}\rangle\frac{1}{2^{L_{x}+L_{y}-1}}(-1)^{\hat{a}\cdot
b+\hat{b}_{T}\cdot a_{T}}=|a_{T},b\rangle,$ (5.35)

where we use the subsystem KW transformation and the fact $\hat{b}_{T}\cdot
a_{T}=\hat{b}\cdot a$. It exactly matches the result obtained by acting the
RHS on $|a,b\rangle$. Using those identities we can examine

$\sum_{(\hat{a},\hat{b})}\widetilde{S}|\hat{a}_{T},\hat{b}\rangle\langle\hat{a},\hat{b}|=\sum_{(a,b)}\widetilde{S}|a,b_{T}\rangle\langle
a,b|=\sum_{(\hat{a},\hat{b})=(a,b)}|\hat{a},\hat{b}_{T}\rangle\langle a,b|.$
(5.36)

Therefore we have the following commutation relation

$\sum_{(\hat{a},\hat{b})}|\hat{a},\hat{b}_{T}\rangle\langle\hat{a},\hat{b}|\widetilde{S}=\sum_{(\hat{a},\hat{b})}\widetilde{S}|\hat{a}_{T},\hat{b}\rangle\langle\hat{a},\hat{b}|$
(5.37)

and we have

$\displaystyle\left(\sum_{\hat{a},\hat{b}}|\hat{a},\hat{b}_{T}\rangle\langle\hat{a},\hat{b}|\widetilde{S}\right)^{2}=$
$\displaystyle\sum_{\hat{a}^{\prime},\hat{b}^{\prime}}\sum_{\hat{a},\hat{b}}|\hat{a}^{\prime},\hat{b}^{\prime}_{T}\rangle\langle\hat{a}^{\prime},\hat{b}^{\prime}|\widetilde{S}|\hat{a},\hat{b}_{T}\rangle\langle\hat{a},\hat{b}|\widetilde{S}$
$\displaystyle=$
$\displaystyle\sum_{\hat{a}^{\prime},\hat{b}^{\prime}}\sum_{\hat{a},\hat{b}}|\hat{a}^{\prime},\hat{b}^{\prime}_{T}\rangle\langle\hat{a}^{\prime},\hat{b}^{\prime}|\widetilde{S}^{2}|\hat{a}_{T},\hat{b}\rangle\langle\hat{a},\hat{b}|$
$\displaystyle=$
$\displaystyle\sum_{(\hat{a},\hat{b})}|\hat{a}_{T},\hat{b}_{T}\rangle\langle\hat{a},\hat{b}|=\sum_{(a,b)}|a_{T},b_{T}\rangle\langle
a,b|\equiv\mathcal{T}$ (5.38)

where $\mathcal{T}$ is an operator which shifts $(a,b)$ to $(a_{T},b_{T})$.
Remember that $a=(w_{z,x;j},w_{z,y;i})$ and
$b=(w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}})$ so that $\mathcal{T}$ shifts the
holonomies as,

$w_{z,x;j}\rightarrow w_{z,x;j-1},\quad w_{z,y;i}\rightarrow w_{z,y;i-1},\quad
w_{x;j+\frac{1}{2}}\rightarrow w_{x;j-\frac{1}{2}},\quad
w_{y;i+\frac{1}{2}}\rightarrow w_{y;i-\frac{1}{2}}.$ (5.39)

which indicates $\mathcal{T}$ is the translation operator.

To write the lattice $S$-defect as a condensation of the line/strip operators,
we parameterize it as,

$S=\sum_{\alpha,\hat{\alpha}\in
M_{v}}\lambda_{\alpha,\hat{\alpha}}K[\alpha,\hat{\alpha}].$ (5.40)

We will use the property that the operators $K[\alpha,\hat{\alpha}]$ with
different $\alpha=(a,b)$ and $\hat{\alpha}=(\hat{a},\hat{b})$ are orthogonal
to each other in the trace

$\textrm{Tr}\left(K[\alpha,\hat{\alpha}]^{\dagger}K[\alpha^{\prime},\hat{\alpha}^{\prime}]\right):=\sum_{\gamma}\langle{\gamma}|K[\alpha,\hat{\alpha}]^{\dagger}K[\alpha^{\prime},\hat{\alpha}^{\prime}]|{\gamma}\rangle=2^{2(L_{x}+L_{y}-1)}\delta_{\alpha,\alpha^{\prime}}\delta_{\hat{\alpha},\hat{\alpha}^{\prime}}$
(5.41)

to project out the coefficients $\lambda_{\alpha,\hat{\alpha}}$,

$\displaystyle\lambda_{\alpha,\hat{\alpha}}=$
$\displaystyle\frac{1}{2^{2(L_{x}+L_{y}-1)}}\textrm{Tr}\left(K[\alpha,\hat{\alpha}]^{\dagger}\sum_{(c,d)=(\hat{c},\hat{d})}(|\hat{c},\hat{d}_{T}\rangle)\langle
c,d|\right)$ $\displaystyle=$
$\displaystyle\frac{1}{2^{2(L_{x}+L_{y}-1)}}\sum_{(c,d)=(\hat{c},\hat{d})}\langle
c,d|K[0,(\hat{a},\hat{b})]K[(a,b),0]|\hat{c},\hat{d}_{T}\rangle$
$\displaystyle=$
$\displaystyle\frac{1}{2^{2(L_{x}+L_{y}-1)}}\sum_{(c,d)=(\hat{c},\hat{d})}(-1)^{\hat{a}\cdot(\hat{d}_{T}+b)+\hat{b}\cdot(\hat{c}+a)}\langle
c,d|\hat{c}+a,\hat{d}_{T}+b\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{2^{3(L_{x}+L_{y}-1)}}\sum_{(c,d)}(-1)^{(\hat{a}+c)\cdot(b+d_{T})+(\hat{b}+d)\cdot(a+c)}$
$\displaystyle=$
$\displaystyle\frac{1}{2^{3(L_{x}+L_{y}-1)}}\sum_{(c,d)}(-1)^{((c,d)+\hat{\alpha})\cdot((c,d_{T})+\alpha)},$
(5.42)

and the condensation defect can be written in the compact form,

$\displaystyle S$
$\displaystyle=\frac{1}{2^{3(L_{x}+L_{y}-1)}}\sum_{(c,d),\alpha,\hat{\alpha}}(-1)^{((c,d)+\hat{\alpha})\cdot((c,d_{T})+\alpha)}K[\alpha,\hat{\alpha}]$
$\displaystyle=\frac{1}{2^{3(L_{x}+L_{y}-1)}}\sum_{(c,d),(a,b),(\hat{a},\hat{b})}(-1)^{(c+\hat{a},d+\hat{b})\cdot(c+a,d_{T}+b)}K[(a,b),(\hat{a},\hat{b})]$
$\displaystyle=\frac{1}{2^{3(L_{x}+L_{y}-1)}}\sum_{(c,d),(a,b),(\hat{a},\hat{b})}(-1)^{(c+\hat{a},\hat{b})\cdot(c+a,b)}K[(a,b+d_{T}),(\hat{a},\hat{b}+d)]$
$\displaystyle=\frac{1}{2^{2(L_{x}+L_{y}-1)}}\sum_{a,\hat{a},b,d}(-1)^{(a+\hat{a})\cdot
b}K[(a,b+d_{T}),(\hat{a},b+d)].$ (5.43)

It is also illuminating to write down the condensation defect for
$\widetilde{S}$, which is the field theory $S$-defect, and
$\widetilde{\mathcal{T}}$. Using the same method, one can obtain

$\displaystyle\widetilde{S}=\frac{1}{2^{3(L_{x}+L_{y}-1)}}\sum_{\gamma,\alpha,\hat{\alpha}}(-1)^{\gamma\cdot(\alpha+\hat{\alpha})}K[\alpha,\hat{\alpha}]=\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{\alpha}K[\alpha,\alpha].$
(5.44)

Expanding $K[\alpha,\alpha]$ using $U_{I},\hat{U}_{J}$ gives

$\widetilde{S}=\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{\alpha}\prod_{I=1}^{2n}U^{\alpha_{I}}_{I}\prod_{J=1}^{2n}\hat{U}^{\alpha_{J}}_{J}=\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{\alpha}(-1)^{\frac{\alpha\cdot\alpha}{2}}\prod_{I=1}^{2n}(U_{I}\hat{U}_{I})^{\alpha_{I}},$
(5.45)

which is a condensation of all possible insertions of operators
$U_{I}\hat{U}_{I}$ built from line and strip operators. The orientation
reversal of the field theory $S$-defect is itself
$\widetilde{S}^{\dagger}=\widetilde{S}$ and it is also unitary:
$\widetilde{S}^{\dagger}\widetilde{S}=1$. Similarly, one has

$\displaystyle\widetilde{\mathcal{T}}=\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{\hat{a},b}(-1)^{\hat{a}\cdot
b}K[(0,b_{T}+b),(\hat{a},0)].$ (5.46)

One can also check the fusion between $\widetilde{\mathcal{T}}$ and
$\widetilde{S}$ straightforwardly,

$\displaystyle\widetilde{\mathcal{T}}\times\widetilde{S}$
$\displaystyle=\frac{1}{2^{2(L_{x}+L_{y}-1)}}\sum_{a,\hat{a},d,b}(-1)^{\hat{a}\cdot
d}K[(0,d_{T}+d),(\hat{a},0)]K[(a,b),(a,b)]$
$\displaystyle=\frac{1}{2^{2(L_{x}+L_{y}-1)}}\sum_{a,\hat{a},d,b}(-1)^{\hat{a}\cdot(d+b)}K[(a,b+d_{T}+d),(a+\hat{a},b)]$
$\displaystyle=\frac{1}{2^{2(L_{x}+L_{y}-1)}}\sum_{a,\hat{a},d,b}(-1)^{(\hat{a}+a)\cdot
b}K[(a,b+d_{T}),(\hat{a},b+d)]=S,$ (5.47)

which reproduces the lattice $S$-defect.

###  5.3 Twist $S$-defect

In this subsection, we will consider twist $S$-defect on a 3d manifold $M_{3}$
with a boundary $\partial M_{3}=M_{2}$, where the boundary can be $x$-$y$
plane, $x$-$z$ plane or $y$-$x$ plane. In the last section, we consider both
lattice $S$-defect $S$ and field theory $S$-defect $\widetilde{S}$, and they
are related by $S=\widetilde{\mathcal{T}}\times\widetilde{S}$. Here
$\widetilde{S}$ satisfies $\widetilde{S}^{2}=1$ and $\widetilde{\mathcal{T}}$
shifts the holonomies and makes $S^{2}=\mathcal{T}$ a translation along
$x$-$y$ directions on the lattice.

We will first discuss the twist defects for the field theory $S$-defect
$\widetilde{S}$ and then move to the twist defect for the lattice $S$-defect
$S$. Also, we will mainly focus on the case where the boundary $M_{2}$ is the
$x$-$y$ plane. Since the lattice $S$-defect involves a translation along
$x$-$y$ plane, the discussions of the corresponding twist defects located at a
fixed $x$ or $y$ are subtle and we will not consider them. Nevertheless, one
can still consider the cases where $M_{2}$ is $x$-$z$ plane or $y$-$z$ plane
for the field theory $S$-defect in a similar way.

From (5.45) we see the field theory $S$-defect is a condensation of the
operator $U_{I}\hat{U}_{I}$ where $U_{I},\hat{U}_{I}$ are related to
line/strip operators according to (5.1) and (5.2). We impose the Dirichlet
boundary condition for the defects $U_{I}\hat{U}_{I}$ condensing along
$M_{3}$. The Dirichlet boundary condition is defined as follows. The operators
$\\{U_{I}\hat{U}_{I}\\}$ generate a subsystem $\mathbb{Z}_{2}$ symmetry along
$M_{3}$ and we denote the corresponding gauge fields as $(A^{\prime
z},A^{\prime xy})$. For $x$-$y$ plane we require the $x$-$y$ component
$A^{\prime xy}$ to vanish at the boundary. The Dirichlet boundary is
topological along the normal direction given these boundary conditions and see
Appendix C for a detailed discussion about this.

$M_{2}$$M_{3}$$\widetilde{V}_{0}(M_{2},M_{3})$$\langle{\mathfrak{B}^{\textrm{sym}}_{\text{sub}}}|$$|{\mathfrak{B}^{\textrm{phys}}_{\mathfrak{T}_{\text{sub}}}}\rangle$$\mathcal{D}(\mathfrak{T}_{\text{sub}})$$\mathfrak{T}_{\text{sub}}$$\mathcal{N}^{\text{sub}}$
Figure 8: Twist $S$-defect $\widetilde{V}_{0}(M_{2},M_{3})$ on $M_{3}$ with a
boundary $M_{2}$. After shinking the slab, the twist defect will create a
duality defect as an interface between the original boundary theory
$\mathfrak{T}_{\mathcal{S}}$ and the theory
$\mathcal{D}(\mathfrak{T}_{\text{sub}})$ after gauging subsystem
$\mathbb{Z}_{2}$ symmetry.

We will denote the twist defect as $\widetilde{V}_{0}(M_{2},M_{3})$ with
$M_{2}$ the $x$-$y$ plane, for example at $z=0$ . 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 $\mathcal{N}^{\text{sub}}$. The
$z$-direction is topological and the strip operators can move along the
$z$-direction and fuse with the twist defect. Since $M_{2}$ satisfies the
Dirichlet boundary condition $A^{\prime xy}=0$, the strip operators
$K[(a,0),(a,0)]$ can be absorbed by the twist defect,

$K[(a,0),(a,0)]\times\widetilde{V}_{0}[M_{2},M_{3}]=\widetilde{V}_{0}[M_{2},M_{3}],$
(5.48)

where we split the $2n$-dimensional vector $\alpha,\hat{\alpha}$ into a pair
of $n$-dimensional vector

$\alpha=(a,b),\quad\hat{\alpha}=(\hat{a},\hat{b}),$ (5.49)

such that $a,\hat{a}$ label the strip operators and $b,\hat{b}$ label the line
operators. On the other hand, since $K[(a,0),(0,0)]$ and $K[(0,0),(a,0)]$ do
not belong to the condensate, fusing them with $\widetilde{V}_{0}$ produces
new twist defects,

$\widetilde{V}_{a}[M_{2},M_{3}]:=K[(a,0),(0,0)]\times\widetilde{V}_{0}[M_{2},M_{3}]=K[(0,0),(a,0)]\times\widetilde{V}_{0}[M_{2},M_{3}].$
(5.50)

In general, the fusion rule between the strip operators $K[(a,0),(\hat{a},0)]$
with $\widetilde{V}_{a^{\prime}}[M_{2},M_{3}]$ is,

$K[(a,0),(\hat{a},0)]\times\widetilde{V}_{a^{\prime}}[M_{2},M_{3}]=\widetilde{V}_{a^{\prime}+a+\hat{a}}[M_{2},M_{3}].$
(5.51)

We can also discuss the fusion between twist defects and it is sufficient to
discuss the fusion between $\widetilde{V}_{0}$ and
$\widetilde{V}^{\dagger}_{0}$. Let’s put another twist defect
$\widetilde{V}_{0}[M_{2|\epsilon},M_{3}]$ at $z=\epsilon$ and consider the
fusion between
$\widetilde{V}_{0}[M_{2|0},M_{3}]\times\widetilde{V}_{0}[M_{2|\epsilon},M_{3}]$
with $\epsilon\rightarrow 0$. Here we use $M_{2|0}$ ($M_{2|\epsilon}$) to
emphasize $M_{2}$ is located at $z=0$ ($z=\epsilon$). 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,

$\displaystyle\lim_{\epsilon\rightarrow
0}\widetilde{V}^{\dagger}_{0}(M_{2}|_{z=0},M_{3}^{z\geq
0})\times\widetilde{V}_{0}(M_{2}|_{z=\epsilon},M_{3}^{z\geq\epsilon})$
$\displaystyle=$
$\displaystyle\frac{1}{2}\prod_{i=1}^{L_{x}}(1+W(y_{i},y_{i+1})\hat{W}(y_{i-\frac{1}{2}},y_{i+\frac{1}{2}}))\prod_{j=1}^{L_{y}}(1+W(x_{j},x_{j+1})\hat{W}(x_{j-\frac{1}{2}},x_{j+\frac{1}{2}})).$
(5.52)

where
$\widetilde{V}^{\dagger}_{0}=\chi[M_{3}^{z\geq\epsilon},\mathbb{Z}_{2}]\widetilde{V}_{0}$
with $\chi[M_{3}^{z\geq\epsilon},\mathbb{Z}_{2}]$ the Euler factor introduced
in (C.22). We have a condensation of strip operators that are mobile along
$z$-direction. If we put (5.3) on the top of the topological boundary
$|{0}\rangle$ at $\tau=1$ 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,

$\widetilde{V}_{0}[M_{2},M_{3}]^{\dagger}\times\widetilde{V}_{0}[M_{2},M_{3}]|_{\tau=1}=\frac{1}{2}\prod_{i=1}^{L_{x}}(1+U^{y}_{i})\prod_{j=1}^{L_{y}}(1+U^{x}_{j}),$
(5.53)

where $U^{y}_{i}=\hat{W}(y_{i-\frac{1}{2}},y_{i+\frac{1}{2}})$ and
$U^{x}_{j}=\hat{W}(x_{j-\frac{1}{2}},x_{j+\frac{1}{2}})$ are generators for
subsystem $\mathbb{Z}_{2}$ symmetry at the boundary. The fusion of twist
defects (5.53) recovers the fusion rules of the subsystem KW operators in the
untwisted sector[60]¶¶¶For twisted sectors, we need to consider more general
Dirichlet boundaries of the condensation defects such that corresponding gauge
field $A^{\prime xy}$ is a non-zero fixed value at the boundary. . On the
other hand, if we put it on the top of the dual boundary $|\hat{0}\rangle$
where $\hat{A}=0$, then magnetic strip operators $\hat{W}$ are absorbed
instead and electric strip operators $W$ serve as the symmetry generators.
(5.53) becomes the fusion rule in the untwisted sector of the dual theory.

We then move to the twist defect $V_{0}$ for the lattice $S$-defect. Put two
twist defect $V_{0}[M_{2|0},M_{3}]$ and $V_{0}[M_{2|\epsilon},M_{3}]$ at $z=0$
and $z=\epsilon$, the fusion rule is similarly obtained by sending
$\epsilon\rightarrow 0$. There are two differences here. First, since
$S^{2}=\mathcal{T}$ is the translation, the spatial lattice at $z>\epsilon$ is
related to that at $z<0$ by the translation $\mathcal{T}$. Second, we need to
impose proper Dirichlet boundary condition at the boundary $M_{2}=\partial
M_{3}$. Recall that the condensation defect for $S$ is,

$\displaystyle
S=\frac{1}{2^{2(L_{x}+L_{y}-1)}}\sum_{a,\hat{a},b,d}(-1)^{(a+\hat{a})\cdot
b}K[(a,b+d_{T}),(\hat{a},b+d)].$ (5.54)

We will assume the Dirichlet boundary condition is defined such that only the
strip operators which are mobile along $z$-direction survive in the limit
$\epsilon\rightarrow 0$, then one expects the fusion is,

$\frac{1}{2^{2(L_{x}+L_{y}-1)}}\sum_{a,\hat{a}}K[(a,0),(0,0)]K[(0,0),(\hat{a},0)].$
(5.55)

If we put it on the top of the topological boundary $|0\rangle$ or
$|\hat{0}\rangle$ at $\tau=1$, one of the two $K$-operators will be absorbed
and we get the same result as before. Combined with the translation
$\mathcal{T}$, we have the similar fusion rule,

$V_{0}[M_{2},M_{3}]\times
V_{0}[M_{2},M_{3}]|_{\tau=1}=\frac{1}{2}\prod_{i=1}^{L_{x}}(1+U^{y}_{i})\prod_{j=1}^{L_{y}}(1+U^{x}_{j})\times\mathcal{T}.$
(5.56)

One can also consider the fusion between $V^{\dagger}$ and $V$,

$V_{0}[M_{2},M_{3}]^{\dagger}\times
V_{0}[M_{2},M_{3}]|_{\tau=1}=\frac{1}{2}\prod_{i=1}^{L_{x}}(1+U^{y}_{i})\prod_{j=1}^{L_{y}}(1+U^{x}_{j}).$
(5.57)

where we do not have the translation operator $\mathcal{T}$ on the RHS since
$S^{\dagger}S=1$.

##  6 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 $\mathbb{Z}_{N}$ 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
$SL(2,\mathbb{Z}_{2})$ symmetry of the bulk subsystem SymTFT on the boundary.
On the lattice, the subsystem $SL(2,\mathbb{Z}_{2})$ 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
$SL(2,\mathbb{Z}_{2})$ symmetry, we find new dualities among bosonic and
fermionic models with subsystem $\mathbb{Z}_{2}$ 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
$\mathbb{Z}_{N}$ 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 $\mathbb{Z}_{N}$ 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

$S_{3\textrm{-foliated}}=\frac{N}{2\pi}\int b\wedge
dc+\sum_{k=1,2,3}dB^{k}\wedge C^{k}\wedge dx^{k}+\sum_{k=1,2,3}b\wedge
C^{k}\wedge dx^{k}.$ (6.1)

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 [82, 83, 84, 85]. Subsystem SymTFT also
provides hints to study fracton statistics[86, 87]. The quantume algebras
(3.15) and (3.16) resemble the braiding statistics in $(2+1)$d
$\mathbb{Z}_{N}$ 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.

## Acknowledgements

We thank Linhao Li, Kantaro Ohmori, Masahito Yamazaki, Shutaro Shimamura and
Yunqin Zheng for the discussion when this project is in the early stage. W.C.
is supported by the Global Science Graduate Course (GSGC) program of the
University of Tokyo and the JSPS KAKENHI grant numbers JP19H05810, JP20H01896,
20H05860, JP22J21553 and JP22KJ1072. W.C. also acknowledges th USTEP exchange
program of the University of Tokyo and the FUTI scholarships for mid-to long-
term studies from Friends of UTokyo, Inc. Q.J. is supported by Korea Institute
for Advanced Study (KIAS) Grant PG080802.

##  Appendix A A review on ordinary BF theory as SymTFT

To illustrate the basic idea of SymTFT, we consider a $(1+1)$d theory
$\mathfrak{T}_{\mathbb{Z}_{N}}$ with $\mathbb{Z}_{N}$ symmetry. The
corresponding SymTFT $\mathfrak{Z}(\mathbb{Z}_{N})$ is the $(2+1)$d BF theory
with level $N$,

$S_{BF}=\frac{N}{2\pi}\int\hat{A}\wedge dA,$ (A.1)

where $\hat{A},A$ are 1-form gauge fields. It is a $\mathbb{Z}_{N}$ gauge
theory and is the low energy description of the toric code for $N=2$ in the
condensed matter literature [88]. Fix a gauge $A_{0}=\hat{A}_{0}=0$, the
canonical quantization gives,

$\left[A_{i}(x,y),\hat{A}_{j}(x^{\prime},y^{\prime})\right]=\frac{2\pi
i}{N}\epsilon_{ij}\delta(x-x^{\prime},y-y^{\prime}).$ (A.2)

$A$ and $\hat{A}$ 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,

$W[\Gamma]=\exp\left(i\oint_{\Gamma}A\right),\quad\hat{W}[\Gamma]=\exp\left(i\oint_{\Gamma}\hat{A}\right),$
(A.3)

with $\Gamma\in H_{1}(T^{2},\mathbb{Z})$. Since the holonomies of $A$ and
$\hat{A}$ are periodic, they are quantized as,

$\frac{N}{2\pi}\oint_{\Gamma}A=0,1,\cdots,N-1,\quad\frac{N}{2\pi}\oint_{\Gamma}\hat{A}=0,1,\cdots,N-1,$
(A.4)

and $W^{N}[\Gamma]=\hat{W}^{N}[\Gamma]=1$. The operators satisfy the
commutation relation,

$W[\Gamma_{i}]\hat{W}[\Gamma_{j}]=\omega^{-\int\gamma_{i}\wedge\gamma_{j}}\hat{W}[\Gamma_{j}]W[\Gamma_{i}],$
(A.5)

where $\gamma\in H^{1}(T^{2},\mathbb{Z})$ is the Poincare dual of $\Gamma$
defined as $\int_{\Gamma}\cdots=\int\gamma\wedge\cdots$.

Let’s focus on the partition function $Z_{\mathfrak{T}_{\mathbb{Z}_{N}}}$ of
the theory and see how the SymTFT applies. We can introduce a canonical basis
of the Hilbert space of the BF model on $T^{2}$ where either $W[\Gamma]$ or
$\hat{W}[\Gamma]$ are diagonalized. The two different choices give two kinds
of topological boundary states $\mathfrak{B}^{\textrm{sym}}_{\mathbb{Z}_{N}}$
written as a boundary state,

  * •

Dirichlet boundary state $|a\rangle$ for $A$,

$\left\\{\begin{array}[]{l}W[\Gamma]|a\rangle=\omega^{\int\gamma\wedge
a}|a\rangle,\\\ \hat{W}[\Gamma]|a\rangle=|a-\gamma\rangle,\end{array}\right.$
(A.6)

  * •

Neumann boundary state $|\hat{a}\rangle$ for $A$,

$\left\\{\begin{array}[]{l}\hat{W}[\Gamma]|\hat{a}\rangle=\omega^{\int\gamma\wedge\hat{a}}|\hat{a}\rangle,\\\
W[\Gamma]|\hat{a}\rangle=|\hat{a}-\gamma\rangle,\end{array}\right.$ (A.7)

where $a=NA/2\pi,\hat{a}=N\hat{A}/2\pi$ and we have $a,\hat{a}\in
H^{1}(T^{2},\mathbb{Z}_{N})$. The integration $\int\gamma\wedge
a=\int_{\Gamma}a$ gives the holonomy along $\Gamma$. The two bases are related
by a discrete Fourier transformation,

$|\hat{a}\rangle=\frac{1}{2}\sum_{a\in
H^{1}(T^{2},\mathbb{Z}_{N})}\omega^{\int a\wedge\hat{a}}|a\rangle.$ (A.8)

On the other hand, the physical boundary
$\mathfrak{B}^{\textrm{phys}}_{\mathfrak{T}_{\mathbb{Z}_{N}}}$ gives a
dynamical boundary state $|\chi\rangle$ which depends on the partition
function of theory $\mathfrak{T}_{\mathbb{Z}_{N}}$ with given the
$\mathbb{Z}_{N}$ holonomies of $\mathfrak{T}_{\mathbb{Z}_{N}}$

$|\chi\rangle=\sum_{a\in
H^{1}(T^{2},\mathbb{Z}_{N})}Z_{\mathfrak{T}_{\mathbb{Z}_{N}}}[a]|a\rangle.$
(A.9)

Choosing different topological boundary states, the path-integral of the BF
theory on the slab gives,

$Z[a]=\langle a|e^{iHt}|\chi\rangle=\langle a|\chi\rangle,\quad
Z[\hat{a}]=\langle\hat{a}|e^{iHt}|\chi\rangle=\langle\hat{a}|\chi\rangle.$
(A.10)

where $Z[a]=Z_{\mathfrak{T}_{\mathbb{Z}_{N}}}[a]$ agrees with the torus
partition function of $\mathfrak{T}_{\mathbb{Z}_{N}}$ and,

$Z[\hat{a}]=\frac{1}{2}\sum_{a}\omega^{\int\hat{a}\wedge
a}Z_{\mathfrak{T}_{\mathbb{Z}_{N}}}[a]\equiv
Z_{\mathfrak{T}_{\mathbb{Z}_{N}}/\mathbb{Z}_{N}}[\hat{a}],$ (A.11)

which is the partition function of the orbifold theory
$\mathfrak{T}_{\mathbb{Z}_{N}}/\mathbb{Z}_{N}$, the Kramers-Wannier duaity of
$\mathfrak{T}_{\mathbb{Z}_{N}}$. In other words, the $\mathbb{Z}_{N}$ gauging
of $\mathfrak{T}_{\mathbb{Z}_{N}}$ can be viewed from the SymTFT as switching
the topological boundary state from $|a\rangle$ to $|\hat{a}\rangle$.

When $N=2$, there also exists a topological boundary $|s\rangle$, where $s\in
H^{1}(T^{2},\mathbb{Z}_{2})$ stands for the spin structure, such that JW
transformation can be encoded as $Z_{F}[s]=\langle s|\chi\rangle$∥∥∥For
general $N$, there is a generalized JW transformation that leads to
parafermion theories, see[89, 90, 49].. The states $|s\rangle$ are eigenstates
of the operators $W_{F}[\Gamma]\equiv W[\Gamma]\hat{W}[\Gamma]$ and satisfy,

$\left\\{\begin{array}[]{l}W_{F}[\Gamma]|s\rangle=(-1)^{\textrm{Arf}(s+\gamma)-\textrm{Arf}(s)}|s\rangle\\\
\hat{W}[\Gamma]|s\rangle=|s+\gamma\rangle\end{array}\right.$ (A.12)

where $\textrm{Arf}(s)\equiv s_{1}s_{2}$ is the Arf-invariant where
$s_{i}=\int_{\Gamma_{i}}s$ is the spin structure along $\Gamma_{i}$-cycle
($s_{i}=0$ is chosen to be the NS boundary condition). The topological
boundary state $|s\rangle$ can also be expressed as,

$|s\rangle=\frac{1}{2}\sum_{a\in
H^{1}(T^{2},\mathbb{Z}_{2})}(-1)^{\textrm{Arf}(s+a)}|a\rangle,$ (A.13)

and the transition amplitude $\langle s|\chi\rangle$ is,

$Z_{F}[s]=\langle s|\chi\rangle=\frac{1}{2}\sum_{a\in
H^{1}(T^{2},\mathbb{Z}_{2})}(-1)^{\textrm{Arf}(s+a)}Z_{\mathfrak{T}_{\mathbb{Z}_{N}}}[a]$
(A.14)

which gives the partition function of the fermionic theory after JW
transformation.

The $(2+1)$d BF theory has a $\mathbb{Z}_{2}$ symmetry which exchanges the two
gauge field,

$A\rightarrow\hat{A},\quad\hat{A}\rightarrow A.$ (A.15)

The corresponding symmetry defect $\mathcal{D}_{\mathbb{Z}_{2}}[M_{2}]$ along
a surface $M_{2}$ can be constructed as,

$\mathcal{D}_{\mathbb{Z}_{2}}=\frac{1}{\sqrt{|H^{1}(M_{2},\mathbb{Z}_{N})|}}\sum_{\Gamma\in
H^{1}(M_{2},\mathbb{Z}_{N})}W[\Gamma]\hat{W}[\Gamma]^{-1},$ (A.16)

which is a condensation of the defect $W\hat{W}^{-1}$ along $M_{2}$. If
$M_{2}$ is a time slice, one can check,

$D_{\mathbb{Z}_{2}}[M_{2}]W[\Gamma]=\hat{W}[\Gamma]D_{\mathbb{Z}_{2}}[M_{2}],\quad
D_{\mathbb{Z}_{2}}[M_{2}]\hat{W}[\Gamma]=W[\Gamma]D_{\mathbb{Z}_{2}}[M_{2}],$
(A.17)

and,

$D_{\mathbb{Z}_{2}}[M_{2}]\times D_{\mathbb{Z}_{2}}[M_{2}]=1.$ (A.18)

using the quantum algebra.

##  Appendix 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 [31]. Begin with the
2-foliated theory (3.1),

$S_{2\textrm{-foliated}}=\frac{N}{2\pi}\int\sum_{k=1,2}(dB^{k}+b)\wedge
C^{k}\wedge dx^{k}+b\wedge dc,$ (B.1)

we split the coordinates $(x^{0},x^{1},x^{2},x^{3})$ as $(\tau,x^{i})$ with
$i=1,2,3$ and denote $(x^{1},x^{2},x^{3})$ as $(x,y,z)$. The action can be
written as,

$\displaystyle S_{2\textrm{-foliated}}=$ $\displaystyle\frac{N}{2\pi}\int
d\tau
d^{3}x\Big{[}\epsilon^{ijk}(-B^{x}_{i}\partial_{\tau}C^{x}_{j}\delta^{x}_{k}-B^{y}_{i}\partial_{\tau}C^{y}_{j}\delta^{y}_{k}+\frac{1}{2}b_{ij}\partial_{\tau}c_{k})$
$\displaystyle+\epsilon^{ijk}C^{x}_{\tau}(\partial_{i}B^{x}_{j}+\frac{1}{2}b_{ij})\delta^{x}_{k}+\epsilon^{ijk}C^{y}_{\tau}(\partial_{i}B^{y}_{j}+\frac{1}{2}b_{ij})\delta^{y}_{k}+\frac{1}{2}\epsilon^{ijk}c_{\tau}\partial_{i}b_{jk}$
$\displaystyle+\epsilon^{ijk}b_{\tau
i}(\partial_{j}a_{k}+C^{x}_{j}\delta^{x}_{k}+C^{y}_{j}\delta^{y}_{k})+\epsilon^{ijk}B^{x}_{\tau}\partial_{i}C^{x}_{j}\delta^{x}_{k}+\epsilon^{ijk}B^{y}_{\tau}\partial_{i}C^{y}_{j}\delta^{y}_{k}\Big{]},$
(B.2)

up to total derivatives. Integrate $C^{x}_{\tau},C^{y}_{\tau},b_{\tau
x},b_{\tau y}$, one gets

$b_{yz}=\partial_{z}B^{x}_{y}-\partial_{y}B^{x}_{z},\quad
b_{xz}=\partial_{z}B^{y}_{x}-\partial_{x}B^{y}_{z},$ (B.3)

for $C^{x}_{\tau},C^{y}_{\tau}$ and

$C^{y}_{z}=\partial_{y}c_{z}-\partial_{z}c_{y},\quad
C^{x}_{z}=\partial_{x}c_{z}-\partial_{z}c_{x},$ (B.4)

for $b_{\tau x},b_{\tau y}$. We can solve $b_{yz},b_{xz},C^{y}_{z},C^{x}_{z}$
and substitute them back to the action. Moreover, integrating $b_{\tau z}$
gives,

$C^{y}_{x}+\partial_{x}c_{y}=C^{x}_{y}+\partial_{y}c_{x},$ (B.5)

such that we can define
$A^{xy}=C^{y}_{x}+\partial_{x}c_{y}=C^{x}_{y}+\partial_{y}c_{x}$. After
renaming other variables,

$A^{\tau}\equiv c_{\tau},\quad A^{z}\equiv c_{z},$ (B.6)
$\hat{A}^{xy}=\partial_{x}B^{x}_{y}-\partial_{y}B^{y}_{x}+b_{xy},\quad\hat{A}^{\tau}=B^{x}_{\tau}-B^{y}_{\tau},\quad\hat{A}^{z}=B^{x}_{z}-B^{y}_{z},$
(B.7)

the action is rewritten as,

$S_{\textrm{exotic}}=\frac{N}{2\pi}\int\left[A^{\tau}(\partial_{z}\hat{A}^{xy}-\partial_{x}\partial_{y}\hat{A}^{z})-A^{z}(\partial_{\tau}\hat{A}^{xy}-\partial_{x}\partial_{y}\hat{A}^{\tau})-A^{xy}(\partial_{\tau}\hat{A}^{z}-\partial_{z}\hat{A}^{\tau})\right].$
(B.8)

which reproduces the exotic tensor gauge theory (3.2).

##  Appendix C Derivation of fusion rule of subsystem KW defects

In this section, we will re-derive the fusion rule between two subsystem KW
defects $\mathcal{N}^{\textrm{sub}}\times\mathcal{N}^{\textrm{sub}}$ after the
formulation of gauging a subsystem symmetry in a cohomology language******The
original derivation on lattice can be found in [60].. The derivation is a
direct generalization from the fusion of duality defects of guaging 0-form
$\mathbb{Z}_{N}$ symmetry [65]. For simplicity, we will keep $N=2$.

###  C.1 Conventions

Denote the gauge fields $A$ and its dual $\hat{A}$ for subsystem
$\mathbb{Z}_{2}$ symmetry as the pair,

$A=(A^{z},A^{xy}),\quad\hat{A}=(\hat{A}^{z},\hat{A}^{xy}).$ (C.1)

The gauge transformation is,

$A\rightarrow
A+\delta\lambda,\quad\hat{A}\rightarrow\hat{A}+\delta\hat{\lambda},$ (C.2)

where the action of $\delta$ on a function is defined as,

$\delta f(x,y,z)=(\partial_{z}f(x,y,z),\partial_{x}\partial_{y}f(x,y,z)).$
(C.3)

The flatness condition is written as,

$\delta A\equiv\partial_{x}\partial_{y}A^{z}-\partial_{z}A^{xy}=0,$ (C.4)

and one can check $\delta^{2}f(x,y,z)=0$ automatically.

To perform the summation formally, it is useful to introduce the 0-cochain
$C^{0}_{\textrm{sub}}(M_{3})$ as the set of functions $f$ on $M_{3}$,
1-cochain $C^{1}_{\textrm{sub}}(M_{3})$ 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 $C^{2}_{\textrm{sub}}(M_{3})$ as the set of functions denoted by
$h^{xyz}$ on $M_{3}$. The coboundary operator $\delta$ acts on
$C^{*}_{\textrm{sub}}$ as,

$\delta f=(\partial_{z}f,\partial_{x}\partial_{y}f),\quad\delta
g=\partial_{x}\partial_{y}g^{z}-\partial_{z}g^{xy},\quad\delta h^{xyz}=0,$
(C.5)

and it satisfies $\delta^{2}=0$. One can define a product $*\cdot*$ which
sends $C^{m}_{\textrm{sub}}(M_{3})\times C^{n}_{\textrm{sub}}(M_{3})$ to
$C^{m+n}_{\textrm{sub}}(M_{3})$ where $C^{m+n}_{\textrm{sub}}(M_{3})$ with
$m+n>2$ is defined to be trivial. For example, when one of
$C^{*}_{\textrm{sub}}$ is $C^{0}_{\textrm{sub}}$ whose elements are functions,
the product is the usual multiplication; and for $g,g^{\prime}\in
C^{1}_{\textrm{sub}}(M_{3})$ one can assign $g\cdot g^{\prime}\equiv
g^{xy}g^{\prime z}+g^{\prime xy}g^{z}$.

Let’s consider the cohomology†††††† Strictly speaking, $H^{*}$ are not
cohomology groups because the closeness condition is not preserved under the
product. For example, given $f\in H^{0}_{\textrm{sub}}(M_{3})$ and $g\in
H^{1}_{\textrm{sub}}(M_{3})$ one can check,  $\delta(f\cdot
g)=\partial_{x}\partial_{y}(fg^{z})-\partial_{z}(fg^{xy})=\partial_{x}f\partial_{y}g^{z}+\partial_{y}f\partial_{x}g^{z},$
which does not vanish. Nevertheless, we do not need this property in the
proof. $H^{*}=Z^{*}/B^{*}$ where $Z^{*}(B^{*})$ contains closed (exact)
cochains. For example

$H^{0}_{\textrm{sub}}(M_{3},\mathbb{Z}_{2})=Z^{0}_{\textrm{sub}}(M_{3},\mathbb{Z}_{2}),$
(C.6)

contains scalar functions that only have $x$ or $y$ dependence. Because of the
flatness condition, the subsystem gauge field $A$ and $\hat{A}$ belong to
$H^{1}_{\textrm{sub}}(M_{3},\mathbb{Z}_{2})$, closed 1-cochains modulo out the
exact 1-cochain (the gauge transformation), and we have
$|H^{1}_{\textrm{sub}}(M_{3},\mathbb{Z}_{2})|=|M_{v}|=2^{2(L_{x}+L_{y}-1)}$,
where the dimension of $H^{1}_{\textrm{sub}}(M_{3},\mathbb{Z}_{2})$ is equal
to the number gauge invariant holonomies $\mathbf{w}$.

Now we will take a formal, continuous route and only make it discrete at the
final step. For example, the subsystem KW transformation

$\displaystyle\begin{split}&Z_{\hat{\mathfrak{T}}_{\textrm{sub}}}[\hat{w}_{z,x;j+\frac{1}{2}},\hat{w}_{z,y;i+\frac{1}{2}},\hat{w}_{x;j},\hat{w}_{y;i}]\\\
&=\frac{1}{2^{L_{x}+L_{y}-1}}\sum_{w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}}=0,1}Z_{\mathfrak{T}_{\textrm{sub}}}[w_{z,x;j},w_{z,y;i},w_{x;j+\frac{1}{2}},w_{y;i+\frac{1}{2}}]\\\
&\times(-1)^{\sum_{i}(\hat{w}_{z,y;i+\frac{1}{2}}w_{y;i+\frac{1}{2}}+\hat{w}_{y;i}w_{z,y;i})+\sum_{j}(\hat{w}_{z,x;j+\frac{1}{2}}w_{x;j+\frac{1}{2}}+\hat{w}_{x;j}w_{z,x;j})}\end{split}$
(C.7)

will be written formally as,

$Z[\hat{A}]=\frac{1}{|H^{0}_{\textrm{sub}}(M_{3},\mathbb{Z}_{2})|}\sum_{A\in
H^{1}_{{\textrm{sub}}}(M_{3},\mathbb{Z}_{2})}Z[A](-1)^{\int_{M_{3}}A\cdot\hat{A}},$
(C.8)

where $|H^{0}_{\textrm{sub}}(M_{3},\mathbb{Z}_{2})|$ is the dimension of the
cohomology group $H^{0}_{\textrm{sub}}(M_{3},\mathbb{Z}_{2})$. The labels of
partition functions are omitted for more concise expressions. The gauge field
$A$ will take values in its gauge equivalent class
$H^{1}_{{\textrm{sub}}}(M_{3},\mathbb{Z}_{2})$ and the integral is regularized
by the sum of holonomies,

$\displaystyle\begin{split}\int_{M_{3}}A\cdot\hat{A}=&\int_{M_{3}}(A^{xy}\hat{A}^{z}+A^{z}\hat{A}^{xy})\\\
=&\sum_{i}(\hat{w}_{z,y;i+\frac{1}{2}}w_{y;i+\frac{1}{2}}+\hat{w}_{y;i}w_{z,y;i})+\sum_{j}(\hat{w}_{z,x;j+\frac{1}{2}}w_{x;j+\frac{1}{2}}+\hat{w}_{x;j}w_{z,x;j}),\end{split}$
(C.9)

where $w$ and $\hat{w}$ are holonomies of $A$ and $\hat{A}$. The formal
expression (C.8) differs from the regularized one (C.7) by the normalization
factor $|H^{0}_{\textrm{sub}}(M_{3},\mathbb{Z}_{2})|$ (instead of
$\sqrt{|H^{1}_{\textrm{sub}}(M_{3},\mathbb{Z}_{2})|}=2^{L_{x}+L_{y}-1}$) as
suggested in [65].

We also need to define relative cohomology
$H^{1}_{\textrm{sub}}(M_{3},M_{2},\mathbb{Z}_{2})$ where $M_{2}=\partial
M_{3}$ 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 $\int A^{z}dz$ split into
$\mathcal{A}^{y}(x)$ and $\mathcal{A}^{x}(y)$ and they depend on $x$ and $y$
separately. There also exists a gauge transformation which shifts
$\mathcal{A}^{y}(x)\rightarrow\mathcal{A}^{y}(x)+\theta$ and
$\mathcal{A}^{x}(y)\rightarrow\mathcal{A}^{x}(y)-\theta$ by some constant
$\theta$ so that $\int A^{z}dz$ is invariant. If $M_{2}$ is the $x$-$z$ plane
we will require $\mathcal{A}^{y}(x)$ to be a constant at the boundary and it
is gauge equivalent to zero. Therefore we impose $\partial_{x}A^{z}=0$ at the
boundary.

In the next subsection, we will consider the KW defects defined by gauging the
subsystem $\mathbb{Z}_{2}$ 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 $\mathbb{Z}_{2}$ gauge theory with
flat gauge field $(A^{z},A^{xy})$. The subsystem $\mathbb{Z}_{2}$ gauge theory
can be represented as‡‡‡‡‡‡This is a generalization that ordinary $q$-form
$\mathbb{Z}_{N}$ gauge theory in $D$-dimenson can be represented by the BF
theory with level $N$[91, 92, 93, 94],  $\frac{N}{2\pi}\int
d^{D}xB^{D-q-2}dA^{q+1}$ (C.10) 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
$\mathbb{Z}_{N}$-valued.

$\frac{1}{\pi}\int dxdydz\ \phi\delta A=\frac{1}{\pi}\int dxdydz\
\phi(\partial_{z}A^{xy}-\partial_{x}\partial_{y}A^{z}),$ (C.11)

where $\phi$ is a periodic scalar field that serves as a Lagrangian multiplier
enforcing $(A^{z},A^{xy})$ to be properly quantized and
$\mathbb{Z}_{2}$-valued. The Dirichlet boundary condition 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
$z=\epsilon$, the difference can be written as the surface integral at $z=0$
and $z=\epsilon$ using Stokes theorem

$\int_{z=0}dxdy\phi A^{xy}-\int_{z=\epsilon}dxdy\phi A^{xy},$ (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 $y=\epsilon$, the
difference is,

$-\int_{y=0}dxdz\phi\partial_{x}A^{z}+\int_{y=\epsilon}dxdz\phi\partial_{x}A^{z},$
(C.13)

and the boundary condition $\partial_{x}A^{z}=0$ 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 $\mathcal{N}^{\textrm{sub}}$ is