# Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase

Xinping Yang  Meng Cheng  Department of Physics, Yale University, New Haven,
CT 06520-8120, USA

###### Abstract

In this work we study gapped boundary states of $\mathbb{Z}_{N}$ bosonic
symmetry-protected topological (SPT) phases in 4+1d, which are characterized
by mixed $\mathbb{Z}_{N}$-gravity response, and the closely related phases
protected by $C_{N}$ rotation symmetry. We show that if
$N\notin\\{2,4,8,16\\}$, any symmetry-preserving boundary theory is
necessarily gapless for the root SPT state. We then propose a 3+1d
$\mathbb{Z}_{2}$ gauge theory coupled to fermionic matter as a candidate
boundary theory for $N=2,4,8,16$, where the anomalous symmetry is implemented
by invertible topological defects obtained from gauging 2+1d chiral
topological superconductors. For the $C_{N}$ case, we present an explicit
construction for the boundary states for $N=2,4,8,16$, and argue that the
construction fails for other values of $N$.

##  I Introduction

The concept of bulk-boundary correspondence is fundamental to the theory of
topological phases. It is most well-understood when the bulk is an invertible
symmetry-protected topological (SPT) phase, where the boundary has an ’t Hooft
anomaly of the symmetry group that protects the bulk SPT phase Chen  _et al._
(2013). The presence of a ’t Hooft anomaly puts nontrivial constraints on the
low-energy dynamics, and in particular excludes a trivially gapped symmetric
ground state. More generally, any low-energy theory realized in the system
must have the given anomaly. Generally, three options are possible for the
boundary theory: gapless, symmetry breaking or a symmetry-preserving
topologically ordered phase (when the boundary has spatial dimension $D$
greater than 1).

The last possibility, namely the boundary forming a symmetry-enriched
topological (SET) phase Vishwanath and Senthil (2013); Chen  _et al._ (2015)
with the ’t Hooft anomaly, has been extensively investigated in the past few
years. In particular for $D=2$, general theories of SET phases in both bosonic
and fermionic systems have been formulated Barkeshli  _et al._ (2019); Teo
_et al._ (2015); Tarantino  _et al._ (2016); Bulmash and Barkeshli (2022a, b);
Aasen  _et al._ (2021). Systematic methods to compute ’t Hooft anomalies given
a SET phase have been given. It is also known that certain ’t Hooft anomalies
can not be matched by any SET in $D=2$, thus any symmetry-preserving theory
must be gapless. Known examples of “symmetry-enforced gaplessness” involve
continuous and anti-unitary symmetry group 111There are also examples
involving finite higher-form symmetries, see e.g. [Cordova and Ohmori, 2019].,
such as a bosonic anomaly for
$\mathrm{SO}(5)\times\mathbb{Z}_{2}^{\mathsf{T}}$ symmetry Wang  _et al._
(2017); Geraedts  _et al._ (2017).

The focus of this work is $D=3$, where a full theory of SET phases is not
available yet. A necessary ingredient of such a theory is a complete
understanding of the structure of ’t Hooft anomaly. It is well-known that ’t
Hooft anomalies are classified by SPT phases in one dimension higher.
Interestingly, in (4+1)d there is a class of bosonic SPT phases protected by
unitary symmetry, that goes beyond the well-known “group-cohomology”
classification. Such “beyond-cohomology” SPT phases can be understood as
decorating lower-dimensional invertible topological phases on symmetry
defects. The physical characterization of these beyond-cohomology SPT phases
turns out to be rather subtle. An argument based on topological quantum field
theory (TQFT) consideration suggests that for $N=2$, the nontrivial phase can
be characterized by the ground state having an odd $\mathbb{Z}_{2}$ charge
when put on the $\mathbb{CP}^{2}$ manifold. However, a commuting-projector
model Hamiltonian, unitarily equivalent to a group-cohomology SPT model in
flat space, can also exhibit the same phenomenon Fidkowski  _et al._ (2020a).
Therefore a definite invariant of the phase requires considering the boundary
anomaly (or the closely related defect decoration) Fidkowski  _et al._ (2019).

We will study the boundary theory of the root $\mathbb{Z}_{N}$ beyond-
cohomology SPT phase, using two complementary points of view. First of all, we
will use a theorem proven by Cordova and Ohmori to show that if $N$ does not
divide $48$, then there can not be any symmetry-preserving TQFT boundary
states. We then propose a boundary TQFT for the allowed values of
$N=2,4,8,16$: a 3+1d $\mathbb{Z}_{2}$ gauge theory with a fermionic
$\mathbb{Z}_{2}$ charge (which will be referred to as a fermionic
$\mathbb{Z}_{2}$ gauge theory from now on). However, given that solvable
models for $\mathbb{Z}_{N}$ BC SPT phases are still lacking for $N>2$, for
explicit constructions of the boundary state we turn to a different but
related system, that is a 4+1d SPT phase protected by $C_{N}$ rotation
symmetry. Following the dimensional reduction approach Song  _et al._ (2017),
we provide explicit constructions of gapped boundary topological orders for
$N=2,4,8$ and show that they preserve the boundary $C_{N}$ symmetry. However,
surpringly we find that the construction fails for $N=16$ but a slight
modification yields a similar boundary state.

##  II Bosonic $\mathbb{Z}_{N}$ SPT in (4+1)d

First we review the classification of (4+1)d bosonic SPT phases, following
Xiong (2018); Gaiotto and Johnson-Freyd (2019). Let $G$ be a compact unitary
group. The “group-cohomology” SPT phases are classified by
$\mathcal{H}^{5}[G,\mathrm{U}(1)]$, and the “beyond-cohomology” SPT phases are
classified by $\mathcal{H}^{2}[G,\mathbb{Z}]$, as we will argue below. The
total group of SPT phases is an extension of $\mathcal{H}^{2}[G,\mathbb{Z}]$
by $\mathcal{H}^{5}[G,\mathrm{U}(1)]$. Additional work is needed to determine
the group structure, which we do for $G=\mathbb{Z}_{N}$ in Appendix A. For
finite $G$, exactly solvable models (either in the form of a state-sum TQFT,
or commuting-projector Hamiltonian) are known for such phases Chen  _et al._
(2013).

The “beyond-cohomology” SPT phases can be understood as decorating (2+1)d
nontrivial invertible states on junctions of symmetry defects Chen  _et al._
(2014); Wang  _et al._ (2021). Recall that invertible phases in (2+1)d form a
$\mathbb{Z}$ group, generated by the so-called $E_{8}$ state with chiral
central charge $c_{-}=8$, with the simplest edge theory being a chiral
$(E_{8})_{1}$ conformal field theory (CFT). We then consider decorating the
codimension-2 tri-junctions of symmetry defects, which are surfaces in 4D
space, by invertible states. A tri-junction is labeled by a pair of group
elements $\mathbf{g,h}\in G$, fusing $\mathbf{g,h}$ defects into a
$\mathbf{gh}$ defect. So the decoration pattern is parametrized by an integer-
valued function $n(\mathbf{g,h})\in\mathbb{Z}$. In order to get a short-range
entangled state, it is necessary that the decorated states on defect
configurations which can be locally deformed to each other are (adiabatically)
equivalent. Applying this condition to a junction that fuses three defects
$\mathbf{g,h,k}$ into $\mathbf{ghk}$, it follows that $n$ must be a 2-cocycle.
In addition, one can change $n(\mathbf{g,h})\rightarrow
n(\mathbf{g,h})+m(\mathbf{g})+m(\mathbf{h})-m(\mathbf{gh})$ by creating a pair
of invertible states labeled by $m(\mathbf{g})$ and $-m(\mathbf{g})$ on the
$\mathbf{g}$ defect and moving them to the adjacent tri-junctions. This
procedure does not change the underlying phase of matter, thus $n$ is defined
up to 1-coboundary and the classification is given by the second group
cohomology $\mathcal{H}^{2}[G,\mathbb{Z}]$.

For compact (including finite) groups
$\mathcal{H}^{2}[G,\mathbb{Z}]=\mathcal{H}^{1}[G,\mathrm{U}(1)]$, so it is
enough to understand the $G=\mathbb{Z}_{N}$ case, which will be the focus of
this work. It is however instructive to start from $G=\mathrm{U}(1)$ and then
break it down to $\mathbb{Z}_{N}$. In that case, the group-cohomology SPT
phases, classified by
$\mathcal{H}^{5}[\mathrm{U}(1),\mathrm{U}(1)]=\mathbb{Z}$, can be
characterized by the (4+1)d quantum Hall response. The beyond-cohomology
phases
($\mathcal{H}^{2}[\mathrm{U}(1),\mathbb{Z}]=\mathcal{H}^{1}[\mathrm{U}(1),\mathrm{U}(1)]=\mathbb{Z}$)
can be constructed as follows: suppose the $\mathrm{U}(1)$ symmetry is
spontaneously broken so the system is in a superfluid phase. In a (4+1)d
superfluid, the vortices are codimension-2 defects (i.e. spatially they are
surfaces). It is well-known that by proliferating the vortices one can restore
the $\mathrm{U}(1)$ symmetry and enter an insulating state. To create a
nontrivial BC SPT state, the vortices are decorated by (2+1)d invertible
states.

A key question here is what kind of (2+1)d invertible states can be decorated
consistently on the $\mathrm{U}(1)$ vortex surfaces. Naively one might think
that $E_{8}$ states can be decorated on vortex sheets. But it is not obvious
that such a decoration is consistent. While we do not have a direct way to
check the consistency at the level of a wavefunction, it is useful to consider
the following QFT arugment: assuming that the system can be described by a
relativistic field theory, then we can study the theory on a general curved
manifold (in Euclidean spacetime) and its response to background
$\mathrm{U}(1)$ gauge field. Consider the partition function
$\mathcal{Z}(M_{5},A)$ of the theory defined on a closed 5-manifold $M_{5}$
equipped with a $\mathrm{U}(1)$ background gauge field $A$. To write down the
response, it is convenient to introduce a 6-dimensional manifold $B_{6}$,
whose boundary is $M_{5}$, and the gauge field $A$ is also extended to
$B_{6}$. The BC SPT phase is characterized by the following topological term
Wang  _et al._ (2015); Lapa  _et al._ (2017); Wen (2015):

$\mathcal{Z}(M_{5},A)=\exp\left(ik\int_{B_{6}}{F}\wedge p_{1}\right).$ (1)

Here $F=dA$ is the field strength, and $p_{1}$ is the Pontryagin class of the
tangent bundle of the manifold. Formally, the action can also be written as
$k\int_{M_{5}}A\wedge p_{1}$. To understand the physical meaning of this
action, first observe that $e^{2\pi ik\int p_{1}}$ defines an invertible
theory in (3+1)d. 222For a closed 4-manifold,
$\int_{M_{4}}{p_{1}}=3{\sigma}(M_{4})$ where $\sigma(M_{4})$ is the signature
of the 4-manifold. The partition function $e^{2\pi ik\int p_{1}}=e^{6\pi
ik\sigma}=e^{\frac{2\pi i}{8}\cdot 24k\cdot\sigma}$,  When the manifold has
(2+1)d boundary, the theory reduces to a gravitational Chern-Simons term on
the boundary with chiral central charge $24k$. Thus we identify $24k$ as the
chiral central charge of the invertible theory decorated on vortex surfaces
Wen (2015).

On the other hand, in order for the topological term to be well-defined, it
can not depend on how the 5-manifold is extended. In other words, the right-
hand side of Eq. (1) evaluated on any closed 6-manifold must give $1$, which
requires $k$ to be an integer 333This is because $\int\frac{{F}}{2\pi}\wedge
p_{1}$ is always an integer. The argument suggests that one can only decorate
minimally $c_{-}=24$ invertible states (i.e. three copies of $E_{8}$ states)
on vortex surfaces. A related fact is that the ground state wavefunctions of
such an invertible state with $c_{-}$ a multiple of 24 on any closed surfaces
are completely invariant under modular transformation (no additional phase
factor). Such modular invariance may be required for a consistent decoration.
It is worth emphasizing that we have assumed relativistic symmetry in this
argument, so it is not entirely clear that they apply to gapped phases in non-
relativistic systems, such as lattice models.

For $\mathbb{Z}_{N}$ symmetry with $3\nmid N$, the response action is

$\mathcal{Z}(M_{5},A)=\exp\left(\frac{2\pi ik}{N}\int_{M_{5}}A\cup
p_{1}\right),k\in\mathbb{Z}_{N}.$ (2)

Here $A$ is the background $\mathbb{Z}_{N}$ gauge field (valued in
$\mathbb{Z}/N\mathbb{Z}$). The $3\mid N$ case requires a separate treatment.
For example, when $N=3$ it is known tht the action (2) turns out to be
equivalent to that of a $\mathbb{Z}_{3}$ group-cohomology SPT phase 444We
thank C.-M. Jian for crucial discussions on this point.. In fact, the
$\mathbb{Z}_{3}$ SPT phases is classified by $\mathbb{Z}_{9}$ Wan and Wang
(2019), where the generator is the root BC SPT phase. Further discussions of
the $N=3$ case can be found in Appendix. B.

In the Hamiltonian formalism, one can interpret this action as follows: let
$M_{5}=M_{4}\times S^{1}$, where $M_{4}$ is a closed 4-manifold. We also
assume there is a unit $\mathbb{Z}_{N}$ holonomy along $S^{1}$. For $3\nmid
N$, the partition function on $M_{5}$ evaluates to $e^{\frac{6\pi
ik}{N}\sigma(M_{4})}$. The physical interpretation is that the ground state on
a closed 4-manifold $M_{4}$ has $\mathbb{Z}_{N}$ charge $3k\sigma(M_{4})$ mod
$N$. However the ground state charge becomes ambiguious away from the pure
TQFT limit. The sublety was recently examplified in the generalized double
semion model Fidkowski  _et al._ (2020b), which is an exactly solvable
(commuting-projector) lattice model with the same ground state property as the
$\mathbb{Z}_{2}$ BC SPT phase on any closed 4-manifold, but on the other hand
is locally equivalent to a group-cohomology $\mathbb{Z}_{2}$ SPT model.

Let us now discuss an alternative characterization of the $\mathbb{Z}_{N}$ BC
SPT phases, which turn out to be useful for studying boundary anomaly.
Formally, 2-cocycles in $\mathcal{H}^{2}[\mathbb{Z}_{N},\mathbb{Z}]$ can be
written as $n(a,b)=\frac{k}{N}(a+b-[a+b]_{N})$, where
$a,b\in\\{0,1,\dots,N-1\\}$ denote the elements of $\mathbb{Z}_{N}$
additively, and $[x]_{N}$ means $x$ mod $N$. $k$ takes values in
$0,1,\dots,N-1$. From this explicit expression of $n$, one can show that
fusing $N$ of the fundamental $\mathbb{Z}_{N}$ defects should yield a (2+1)d
invertible state labeled by $n(1,1)+n(2,1)+\cdots+n(N-1,1)=k$ (so chiral
central charge $24k$). This result can also be understood from the U(1) case,
where the U(1) vortices are decorated by $c_{-}=24k$ invertible states. Once
the U(1) symmetry is broken down to $\mathbb{Z}_{N}$, a U(1) vortex should be
viewed as a junction fusing $N$ fundamental $\mathbb{Z}_{N}$ defects together,
hence the same decoration pattern.

###  II.1 Lattice model realizations

Given the subtleties in the TQFT classification of SPT phases, it is highly
desirable to find microscopic constructions of the nontrivial states. For the
$N=2$ case, an exactly solvable model was given in [Fidkowski  _et al._ ,
2019] (see also [Chen and Hsin, 2021] for an alternative construction, which
applies to $N=4$ as well.). The key observation there is the following:
because $p_{1}\equiv w_{2}^{2}\,(\text{mod }2)$, we can interpret the action
in Eq. (2) as $\mathbb{Z}_{2}$ domain walls being decorated with the
“$w_{2}^{2}$” SPT states, which are realized as the ground state of the so-
called 3-fermion Walker-Wang (WW) model Burnell  _et al._ (2014). The key
ingredient in this construction is a quantum cellular automata (QCA), or a
locality-preserving unitary, that disentangles the 3-fermion WW state. In
addition, the QCA exactly squares to $1$. With such a QCA, a wavefunction of a
consistent, equal-weight superposition of decorated domain wall states, as
well as a commuting projector parent Hamiltonian, can be written down.

It is not clear whether similar constructions can be generalized to other
$N>2$ cases. If we simply generalize the construction in [Fidkowski  _et al._
, 2019], according to the action in (2), the $\mathbb{Z}_{N}$ domain wall is
decorated by a (3+1)d gapped state whose partition function is given by
$e^{\frac{2\pi i}{N}\int p_{1}}$. In addition, the disentangling QCA for this
state must have order $N$. While a complete topological classification of QCAs
in (3+1)d is still unknown, there is a growing body of evidences Haah  _et
al._ (2018); Haah (2021); Shirley  _et al._ (2022) suggesting that they are
classified by the Witt group of (2+1)d modular tensor categories (MTC), which
are mathematical theories describing the universal bulk properties of
topological phases. We review the definition of Witt group in Appendix C.
Conjecturally, a QCA that disentangles a WW model with the input MTC in a
nontrivial Witt class is topologically nontrivial. The partition function for
the WW model is $e^{\frac{2\pi ic_{-}}{24}\cdot p_{1}}$, where $c_{-}$ is the
chiral central charge of the input MTC. Assuming that this conjectured
classification of QCA is correct, we conclude that in order to generalize the
construction to $\mathbb{Z}_{N}$, we would need to find a topological phase
with chiral central charge $\frac{48}{N}$ mod 8 and with order $N$ in the Witt
group.

However, it is also known that the structure of the Witt group is highly
constrained Drinfeld  _et al._ (2010); Davydov  _et al._ (2010, 2011): the
order of elements in the Witt group can not be any odd integer. In fact, the
only possible finite values are $2^{n}$ with $1\leq n\leq 5$ (there are
obviously elements of infinite order). This immediately rules out any odd
$N>3$ in the construction.

Even for $N$ that divides 48, when there exist order-$N$ elements in the Witt
group, there is a further constraint. If the MTC has an order $N$ element in
the Witt group, the corresponding QCA conjecturally is also of order $N$,
meaning that $N$-th power of the QCA is a finite-depth local unitary circuit.
However, for the construction to work, the $N$-th power needs to be exactly
$1$. So far this has only been done for the 3-fermion QCA with $N=2$, and to
the best of our knowledge, no other known examples of QCA satisfy this
property.

###  II.2 Constraint on boundary topological order

Now we turn to the characterization of (3+1)d boundary states of the BC
$\mathbb{Z}_{N}$ SPT phases.

Recently, within the mathematical framework of TQFTs, [Cordova and Ohmori,
2019; Córdova and Ohmori, 2020] established a necessary condition for a (3+1)d
’t Hooft anomaly to be saturated by a symmetry-preserving TQFT: the
corresponding (4+1)d SPT topological partition function must evaluate to 1 on
$K_{3}\times S^{1}$ (with any choice of background gauge field). Here $K_{3}$
is a closed simply-connected 4-manifold with signature $16$. In other words,
if one can find a configuration of gauge field such that the partition
function yields a phase factor different from 1 on $K_{3}\times S^{1}$, then
the SPT phase can not have a symmetry-preserving TQFT boundary.

Let us use the criterion to study the $\mathbb{Z}_{N}$ BC SPT phase, whose
partition function is given by Eq. (2). If we require the partition function
to be $1$ on $K_{3}\times S^{1}$, for $3\nmid N$ using (2) find

$\frac{48k}{N}\in\mathbb{Z}.$ (3)

For $k=1$, it means a symmetry-preserving TQFT boundary is possible only for

$N=2,4,8,16.$ (4)

The case of $3\mid N$ is more delicate. We provide an argument that the
partition function on $K_{3}\times S^{1}$ is not 1 for the root $N=3$ BC SPT
phase in the Appendix B, and hence there can not be a symmetry-preserving
boundary TQFT. Interestingly, even though the root phase does not allow
symmetry-preserving gapped boundary, 3 copies of the root phase is equivalent
to a group-cohomology SPT phase, which can have symmetric gapped boundary. We
believe the same is true for other $3\mid N$.

Ref. [Fidkowski  _et al._ , 2019] constructed a fermionic $\mathbb{Z}_{2}$
gauge theory on the boundary of the $N=2$ BC SPT state. Here a fermionic
$\mathbb{Z}_{2}$ gauge theory refers to a $\mathbb{Z}_{2}$ gauge theory with
fermionic $\mathbb{Z}_{2}$ gauge charges. We will argue that the same theory
can realize the anomaly for $N=4,8,16$ as well. To this end we first discuss
the symmetries of this theory.

###  II.3 Symmetry in the (3+1)d fermionic $\mathbb{Z}_{2}$ gauge theory

We now show that a fermionic $\mathbb{Z}_{2}$ gauge theory in (3+1)d has an
anomalous $\mathbb{Z}_{16}$ (0-form) global symmetry Johnson-Freyd (2020).

First we review the low-energy excitations of the $\mathbb{Z}_{2}$ gauge
theory. There are two elementary types of excitations: a fermionic
$\mathbb{Z}_{2}$ particle, and a $\mathbb{Z}_{2}$ flux loop (denoted by
$m_{2}$ below, where the subscript 2 is the codimension). In addition, it is
useful to introduce an invertible line defect as follow: We can think of the
theory as a system of fermions in a gapped trivial state coupled to a
$\mathbb{Z}_{2}$ gauge field. We insert a Majorana chain in the ungauged
fermion system, and then gauge the $\mathbb{Z}_{2}$ fermion parity. The
Majorana chain then becomes an invertible topological defect of codimension 2,
which will be denoted by $p_{2}$.

We now construct the codimension-1 invertible topological defects that
implement the $\mathbb{Z}_{16}$ 0-form symmetry. To construct the defect, we
insert a 2+1d chiral topological superconductor (TSC) of Chern number $\nu$
(equivalent to $\nu$ copies of $p+ip$ superconductors) into the fermionic
theory before gauging, and then couple the system to a $\mathbb{Z}_{2}$ gauge
field. This way we obtain an invertible topological defect of codimension 1 in
the fermionic $\mathbb{Z}_{2}$ gauge theory 555One subtlety is that because of
the chiral central charge, strictly speaking the defect is not completely
topological, in the same sense as the (2+1)d Chern-Simons theory is not a
topological field theory due to the framing anomaly. For example, the
partition function with defect insertions depend on the geometry. However,
since the dependence comes in as a phase factor, it does not affect the
symmetry action., which defines a 0-form symmetry. From this construction
naively it seems that the defect is labeled by the integer $\nu$ and fusion of
two defects of $\nu_{1}$ and $\nu_{2}$ results in a defect of
$\nu_{1}+\nu_{2}$. However, we will argue that $\nu$ is defined mod 16. This
is because before gauging, a $\nu=16$ TSC is topologically equivalent to a
$E_{8}$ state stacked with completely trivial gapped fermions. Since this
equivalence can be generated by adiabatic evolution with a gapped local
Hamiltonian preserving fermion parity, it is expected that the equivalence is
preserved after gauging. In other words, a $\nu=16$ defect is equivalent to a
$E_{8}$ state. Since the $E_{8}$ state is purely bosonic and decoupled from
the $\mathbb{Z}_{2}$ gauge theory, when viewed as a topological defect it can
only act on the $\mathbb{Z}_{2}$ gauge theory trivially. Therefore the
faithful symmetry group generated by these TSC defects is $\mathbb{Z}_{16}$.
However, the fact that 16 $\nu=1$ defects fuse to a $E_{8}$ state suggests
that the $\mathbb{Z}_{16}$ symmetry is anomalous. Indeed this is exactly what
should happen on the boundary of a $\mathbb{Z}_{16}$ BC SPT state 666Naively,
16 $\nu=1$ defects fuse to a $E_{8}$ state with $c_{-}=8$, but fusing 16
$\mathbb{Z}_{16}$ defects in the bulk leaves an invertible state with
$c_{-}=24k$. However, it should be kept in mind that the chiral central charge
of the defect in the (3+1)d theory is only defined mod 8, so more precisely 16
$\nu=1$ defects fuse to an invertible state with $c_{-}=8(16m+1)$, where $m$
is an integer. By choosing e.g. $m=2$, we have $k=11$ in the bulk..

Figure 1: Illustration of the 0-form symmetry action on a flux loop.

Let us examine how the symmetry acts on various objects in the theory. It is
evident that the TSC defect does not act on the fermionic particle, and the
nontrivial action only happens on the flux loops. To see what is going on when
a flux loop $m_{2}$ passes through the domain wall, we note that the process
is equivalent to wrapping the topological superconductor around the flux loop.
It is a well-known fact that when $\nu$ is odd, the topological superconductor
when wrapped on a cylinder with anti-periodic boundary condition for the
fermions (i.e. with a $\pi$ flux threading the cylinder) is equivalent to a
Majorana chain Read and Green (2000). See Fig. 1 for an illustration of the
symmetry action. Thus we find that $m_{2}\rightarrow m_{2}p_{2}$ when passing
through a TSC defect with an odd $\nu$ Johnson-Freyd (2020).

When $\nu$ is even, the symmetry does not change the type of the $m_{2}$ loop.
Instead, let us consider a Hopf link of two flux loops, and pass the link
through the domain wall. During this process, the worldlines of the four
intersection points of the link with the domain wall precisely trace out the
Hopf link, which correspond to a full braiding between two of them and result
in a phase factor $\pm e^{\frac{i\pi\nu}{4}}$. Here the sign ambiguity $\pm$
comes from possible fermions attached to the flux loops. Notice that this
characterization only applies to $\nu\equiv 2$ mod $4$.

More generally, we can consider a “three-loop braiding” process Wang and Levin
(2014), where two flux loops are linked to a base loop of the defect. The
exchange statistics of the two flux loops is $e^{\frac{i\pi\nu}{8}}$, which
can distinguish all different $\nu$ mod 16.

##  III $C_{N}$ SPT in (4+1)d and the (3+1)d boundary

A drawback of our discussions of the $\mathbb{Z}_{N}$ BC SPT phases is that it
is entirely based on topological partition functions, and at the moment we do
not have a concrete microscopic model for them. In this section we turn to a
different but closely related symmetry $C_{N}$, the point group of $N$-fold
rotations, and study SPT phases protected by this symmetry. In general,
spatial SPT states (i.e. those protected by spatial symmetries) can be
classified and explicitly constructed using the block construction Song  _et
al._ (2017). The reason that we consider $C_{N}$ SPT phases is because there
is a one-to-one correspondence between them and the $\mathbb{Z}_{N}$ SPT
phases, known as the crystalline correspondence principle Thorngren and Else
(2018). We will come back to the physical interpretation of the correspondence
later.

Let us carry out the block construction for $C_{N}$ symmetry in (4+1)d. By the
definition of SPT phases, the bulk state can be disentangled everywhere except
on the rotation “axis”, which is two-dimensional in 4D, and the $C_{N}$
symmetry reduces to a $\mathbb{Z}_{N}$ internal symmetry on the rotation axis.
Now there are two possibilities: a $\mathbb{Z}_{N}$ group-cohomology SPT
phase, or a $E_{8}$ state, on which the $\mathbb{Z}_{N}$ does not act. The
latter corresponds to the generator of the $\mathbb{Z}_{N}$ beyond-cohomology
SPT phases. In Appendix A we compute the group structure of the $C_{N}$ SPT
phases, which turns out to be $\mathbb{Z}_{N}\times\mathbb{Z}_{N}$ for $3\nmid
N$, and $\mathbb{Z}_{3N}\times\mathbb{Z}_{N/3}$ for $3\mid N$ where
$\mathbb{Z}_{3N}$ is generated by the root BC SPT phase.

Figure 2: Illustration of the construction for a gapped state on the boundary
of (4+1)d $C_{N}$ BCSPT state.

Let us now elaborate on the relation between the $\mathbb{Z}_{N}$ SPT phases
and the $C_{N}$ ones. Starting from a $\mathbb{Z}_{N}$ SPT phase, we create a
symmetry-breaking state in the following way: insert $N$ copies of
$\mathbb{Z}_{N}$ domain walls in a $C_{N}$ symmetric configuration. This
system breaks $\mathbb{Z}_{N}$ and $C_{N}$, but preserves the diagonal
subgroup called $C_{N}^{\prime}$. At the rotation center, the $N$ domain walls
fuse together to a codimension-2 defect, which is the state that lives at the
rotation center. Therefore, from a $\mathbb{Z}_{N}$ SPT state we can always
construct a $C_{N}$ SPT state. In the other direction, in a continuum QFT with
continuous spatial symmetry (i.e. SO($D$) and translations, where $D$ is the
spatial dimension), the $C_{N}$ rotation can always be written as a
$\mathbb{Z}_{N}$ internal symmetry transformation combined with the
corresponding rotation in SO$(D)$. It is then expected that the theory with
the $\mathbb{Z}_{N}$ symmetry is in the corresponding $\mathbb{Z}_{N}$ SPT
phase.

We note that in this construction, the rotation center can be decorated with
any (2+1)d invertible topological phase, and unlike the internal
$\mathbb{Z}_{N}$ case there is no need to impose the $c_{-}=24k$ condition.

We choose a 3D boundary perpendicular to the rotation axis, so the boundary is
invariant under rotation. The rotation plane in the bulk terminates as the 1D
axis on the boundary. Since there is an $E_{8}$ state on the plane, the 1D
axis carries the corresponding $(E_{8})_{1}$ chiral edge mode. In the
following we assume that the axis is the $z$ axis.

In order to create a fully gapped (3+1)d boundary, we use the following
construction: choose $N$ half planes all terminating at the $z$ axis, the
positions of which are related to each other by $C_{N}$ rotation. For example,
one of them could be the plane defined by $y=0,x\geq 0$, and the others are
obtained by $C_{N}$ rotations. On each plane we place a 2D chiral topological
phase $\mathcal{B}$. Again all of them are related by $C_{N}$. At the 1D
rotation center, we have $N$ edge modes from the topological phases on the
half-plane blocks and the $(E_{8})_{1}$ CFT from the (4+1)d bulk. The setup is
illustrated in Fig. 2 for $N=4$. We require that these edge modes together can
be gapped out while preserving the $C_{N}$ symmetry. In other words, the $N$
blocks have a $C_{N}$-preserving gapped boundary to vacuum. We can fold the
$N$ layers into one topological phase denoted by $\mathcal{B}^{\boxtimes N}$,
where the $C_{N}$ symmetry becomes the $\mathbb{Z}_{N}$ cyclic permutations. A
similar method was used in Ref. [Qi  _et al._ , 2019] to study (2+1)d
topological phases enriched by reflection symmetry, which is essentially the
$N=2$ case.

To summarize, our construction requires topological phases that satisfy the
following conditions:

  1. 1.

The topological phase $\mathcal{B}$ has a chiral central charge
$c_{-}=\frac{8}{N}$.

  2. 2.

$N$ layers of $\mathcal{B}$ can have a fully gapped edge to a $E_{8}$ state.

  3. 3.

The gapped edge preserves the $\mathbb{Z}_{N}$ cyclic permutation symmetry of
the $N$ layers.

We will say $\mathcal{B}$ is _$N$ -gappable_ if all the conditions are
satisfied.

Therefore, the construction reduces to finding a $\mathbb{Z}_{N}$ symmetry-
preserving gapped boundary (to a $E_{8}$ state) of $N$ copies of
$\mathcal{B}$. Below we will study this problem using the mathematical
framework of modular tensor category (MTC), also known as the anyon theory in
physics literature Kitaev (2006). In this formalism, a topologically ordered
phase in (2+1)d is fully described in terms of the universal data of the low-
energy quasi-particle excitations, i.e. the anyons. The universal data
describe the fusion and braiding properties of the anyons. Alternatively, this
collection of data also suffices to specify the (2+1)d TQFT associated with
the topological phase. We should note that the MTC description does not fully
determine the edge property, i.e. the chiral central charge $c_{-}$. In fact,
it can be shown that the MTC (or the anyon theory) determines $c_{-}$ mod 8.
Physically the ambiguity precisely comes from stacking $E_{8}$ states, which
does not affect the anyon excitations but can change $c_{-}$ by integer
multiples of 8.

Gapped boundaries of a topological phase can also be described in this
formalism Kitaev and Kong (2012); Eliens  _et al._ (2014); Levin (2013); Kong
(2014); Lan  _et al._ (2015); Neupert  _et al._ (2016); Cong  _et al._ (2017).
Each gapped boundary is associated to a unique (composite) anyon object, which
determines which anyons can condense on the boundary. This object is called
the Lagrangian algebra, denoted by $\mathcal{A}$ below. For Abelian anyons,
the condensed anyons form a Lagrangian subgroup Levin (2013). In this case, it
is relatively simple to state the condition for a set of anyons to condense:
they must all be bosons, and the mutual braiding statistics between them must
all be trivial. In addition, the number of anyons in the Lagrangian group must
be the square root of the total number of anyons. The definition of the
algebra in the general case is reviewed in Appendix D.

We can now define the notion of Witt equivalence for topological phases. Two
topological phases $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ are Witt
equivalent, if $\mathcal{B}_{1}\boxtimes\overline{\mathcal{B}}_{2}$ has a
fully gapped interface to an invertible state (i.e. some copies of $E_{8}$
states). Here $\boxtimes$ denotes the operation of stacking two systems, and
$\overline{\mathcal{B}}_{2}$ is the mirror image of $\mathcal{B}_{2}$. In
other words, there is a gapped interface between $\mathcal{B}_{1}$ and
$\mathcal{B}_{2}$ as long as we are allowed to freely stack copies of $E_{8}$
states. Mathematically, two MTCs $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ are
Witt equivalent if $\mathcal{B}_{1}\boxtimes\overline{\mathcal{B}}_{2}$ has a
Lagrangian algebra. The Witt equivalence classes of MTCs form an Abelian
group, known as the Witt group of MTCs. We have already discussed the Witt
group in the context of QCAs in (3+1)d.

With these definitions, the first two conditions of $N$-gappability implies
that the MTC $\mathcal{B}$ has a minimal order $N$ in the Witt group. However,
as already mentioned in Sec. II.1, the structure theorems of the Witt group
implies that $N=2^{n}$ with $1\leq n\leq 5$. Therefore, the construction does
not work for any other values of $N$.

In our problem there is also a global symmetry $G$, i.e. the $\mathbb{Z}_{N}$
group of cyclic permutations of the $N$ layers, and the gapped boundary must
preserve this global symmetry. Therefore it is necessary to incorporate global
symmetry in the categorical formalism. First of all, the global symmetry can
act on the anyons in nontrivial ways. In this case, the action is given by the
layer permutation. There are more subtle aspects of symmetry actions in the
topological phase but they do not occur in our system, so we will not go into
details. For more details on symmetry-enriched topological phases in (2+1)d,
see Ref. [Barkeshli  _et al._ , 2019].

Next, we will need to understand whether a given gapped boundary, or the
Lagrangian algebra, can preserve the global symmetry. Clearly, the set of
condensed anyons must be invariant under the global symmetry (layer
permutations in our case), otherwise the symmetry is explicitly broken. For a
condensed anyon $a$, we define $G_{a}$ as the subgroup of $G$ that keeps $a$
invariant. Then the condensed anyon $a$ should carry a well-defined charge
under the symmetry group $G_{a}$. Here by “charge” we mean a one-dimensional
representation of $G_{a}$, i.e. a homomorphism from $G_{a}$ to U(1). Different
choices of these charges correspond to different types of symmetry-preserving
gapped boundary conditions. Again we note that here our description is
heuristic, and a more precise formulation is given in Appendix D.

To understand the physical consequence of the symmetry charges of the
condensed anyons, it is useful to consider the nature of the gapped boundary.
By definition, the gapped boundary is the interface between the topological
phase and an invertible state, which is often taken to be the vacuum. However,
in the presence of global symmetry, the invertible state could be a nontrivial
SPT state. The nature of this trivial state is determined by the symmetry
charges of the condensate Cheng and Williamson (2020); Qi  _et al._ (2019).
Here we are mainly interested in the case where the invertible state has no
nontrivial SPT order. Following Ref. [Qi  _et al._ , 2019], we study the
problem by gauging the $\mathbb{Z}_{N}$ symmetry in the
$\mathcal{B}^{\boxtimes N}$ theory, the result of which is denoted by
$[\mathcal{B}^{\boxtimes N}]/\mathbb{Z}_{N}$. The interface then becomes one
between the gauged theory $[\mathcal{B}^{\boxtimes N}]/\mathbb{Z}_{N}$, and a
$\mathbb{Z}_{N}$ gauge theory obtained from gauging the $\mathbb{Z}_{N}$ SPT
state (possibly stacked with an $E_{8}$ state). After gauging, the Lagrangian
algebra in $\mathcal{B}^{\boxtimes N}$ is “lifted” to one in
$[\mathcal{B}^{\otimes N}]/\mathbb{Z}_{N}$, however the lifting requires
additional data, i.e. the symmetry charges of the condensed anyons. Once the
Lagrangian algebra and its lifting are given, one can apply the theory of
anyon condensation to determine the nature of the $\mathbb{Z}_{N}$ gauge
theory after condensation. We leave the details of the derivations in Appendix
E.

Below we apply this theory to several examples. In particular, we will
consider a family of anyon theories known as Kitaev’s 16-fold ways Kitaev
(2006). They can be described as (2+1)d fermionic topological superconductors
with Chern number $\nu$ coupled to $\mathbb{Z}_{2}$ gauge field, or as
Spin$(\nu)_{1}$ Chern-Simons theory. We will show that Spin$(\nu)_{1}$ is
$\frac{16}{(16,\nu)}$-gappable when $\nu$ is even. However, we find that
Spin$(\nu)_{1}$ is _not_ 16-gappable when $\nu$ is odd. Instead, we find a
different but closely related construction for the $C_{16}$ case.

###  III.1 Spin$(2n)_{1}$

In this section we will consider Spin$(2n)_{1}$ theories. First we review the
basic properties of these MTCs. Spin$(2n)_{1}$ has four anyons:
$1,\psi,v,v^{\prime}=v\times\psi$. $\psi$ is a fermion and satisfies
$\psi^{2}=1$. $v$ can be viewed as a fermion parity flux since the braiding
phase between $v$ and $\psi$ is $M_{v\psi}=-1$. The topological twist factor
of $v$ is $\theta_{v}=e^{\frac{i\pi n}{4}}$. The chiral central charge is
$c_{-}=n$. The order of Spin$(2n)_{1}$ in the Witt group is
$r=\frac{8}{\text{gcd}(8,n)}$.

We will show that the Spin$(2n)_{1}$ MTCs are $r$-gappable.

First, we construct the Lagrangian subgroup for Spin$(2n)_{1}^{\boxtimes r}$,
preserving the $\mathbb{Z}_{r}$ cyclic permutation symmetry. We label the
anyons by a $r$-tuple $(a_{1},a_{2},\cdots,a_{r})$. Since all Spin$(2n)_{1}$
theories have total quantum dimension $\mathcal{D}=2$, the Lagrangian subgroup
has to have size $2^{r}$. Consider all bosons of the form

$\mathcal{A}_{0}=\\{(a_{1},a_{2},\cdots,a_{r})|a_{i}\in\\{\mathds{1},\psi\\}\\},$
(5)

with an even number of $\psi$’s in the tuple. We will refer those as fermion
bound states. There are $2^{r-1}$ such bosons. Then we fuse $(v,v,\cdots,v)$
with the fermion bound states, which is equivalent to replacing an even number
of $v$’s with $v^{\prime}=v\psi$. Together they generate a Lagrangian subgroup
consist of $2^{r}$ bosons, and it is straightforward to check that they have
trivial mutual braiding statistics, thus forming a Lagrangian subgroup. Cleary
this subgroup is invariant under the $\mathbb{Z}_{r}$ symmetry.

As described in the beginning of this section, we need to examine finer
structures of the Lagrangian algebra under the symmetry action. We perform
these calculations carefully for two representative values of $n=1,2$ in
Appendix D. As discussed above, to determine the nature of the condensed phase
it is necessary to know the symmetry charges of the condensed anyons. We find
that a consistent choice is to have all the $\mathbb{Z}_{r}$-invariant anyons
(i.e. $(\psi,\cdots,\psi)$ and $(v,\cdots,v)$) carry trivial charges under
$\mathbb{Z}_{r}$. With this choice, we can apply the results in Appendix E to
show that the condensation leads to an $E_{8}$ state where the
$\mathbb{Z}_{r}$ symmetry acts trivially. Together we have established that
there is $\mathbb{Z}_{r}$-symmetric gapped interface between
Spin$(2n)_{1}^{\boxtimes r}$ to an $E_{8}$ state where the $\mathbb{Z}_{r}$
symmetry acts trivially, so Spin$(2n)_{1}$ is $r$-gappable.

Starting from this boundary state, we can now construct a $\mathbb{Z}_{2}$
gauge theory in the following way: stack on the boundary a (3+1)d fermionic
$\mathbb{Z}_{2}$ gauge theory where the $C_{r}$ symmetry acts trivially
(besides the coordinate transformation). Now on each of the Spin$(2n)_{1}$
layer, we drive a condensation of the bound state of the emergent fermion in
the $\mathbb{Z}_{2}$ gauge theory and the $\psi$ in Spin$(2n)_{1}$. In other
words, the $\psi$ in Spin$(2n)_{1}$ layers are all identified with the fermion
in the $\mathbb{Z}_{2}$ gauge theory. Consequently, $v$ or $v^{\prime}$ anyons
are attached to the $\mathbb{Z}_{2}$ flux lines. There are no separate anyons
confined on the Spin$(2n)_{1}$ layers anymore, so these layers become
invertible defects embedded in the fermionic $\mathbb{Z}_{2}$ gauge theory.
These defects are precisely the TSC defects introduced in Sec. II.3, since the
Spin$(2n)_{1}$ can be thought of as coupling a TSC of Chern number $2n$ to a
$\mathbb{Z}_{2}$ gauge field, and what we just did is to “Higgs” the emergent
$\mathbb{Z}_{2}$ gauge field in Spin$(2n)_{1}$ with that of the (3+1)d gauge
theory. Notice that the kind of condensation transitions on the Spin$(2n)_{1}$
layers can be driven by interactions that preserve the $C_{r}$ symmetry, and
it is expected that there is no spontaneous symmetry breaking. As a result,
the new $\mathbb{Z}_{2}$ gauge theory obtained this way still has the $C_{r}$
symmetry with the same anomaly.

###  III.2 Spin$(2n+1)_{1}$

Let us now turn to Spin$(2n+1)_{1}$ theories. Recall that the Spin$(2n+1)_{1}$
MTC has three types of anyons $\mathds{1},\sigma$ and $\psi$, where $\psi$ is
a fermion and $\sigma$ is a non-Abelian anyon that satisfies
$\sigma\times\sigma=\mathds{1}+\psi$ and with a topological twist factor
$\theta_{\sigma}=e^{\frac{i\pi(2n+1)}{8}}$.

We first enumerate all bosons in $[\text{Spin}(2n+1)_{1}]^{\boxtimes 16}$.
There are $2^{15}$ Abelian bosons, which are bound states with an even number
of fermions. There is also a non-Abelian boson $(\sigma,\cdots,\sigma)$, with
quantum dimension $(\sqrt{2})^{16}=256$. We can form the following Lagrangian
algebra:

$\mathcal{A}=\sum_{a\in\mathcal{A}_{0}}a+128(\sigma,\sigma,\cdots,\sigma).$
(6)

Unfortunately, due to the large multiplicity $128$, we are not able to obtain
an explicit structure of the condensate. Therefore we adopt a different
approach here. We will show that in fact it is not possible to condense
$\mathcal{A}$ in $[\text{Spin}(2n+1)_{1}]^{\boxtimes 16}$ without breaking the
$\mathbb{Z}_{16}$ symmetry. This can be easily proven using the anyon
condensation theory, see Appendix E. Below we provide a more physical
argument, which will also suggest a way to fix the problem.

We first condense the Abelian subgroup $\mathcal{A}_{0}$ of $\mathcal{A}$,
resulting in a $\mathbb{Z}_{2}$ toric code (TC) phase. It is easy to see the
only deconfined anyons are the Abelian anyons, and $(\sigma,\dots,\sigma)$.
All the Abelian bosons are already condensed, and all the (Abelian) fermions
are identified and become the same $\psi$ of the toric code phase. While
$(\sigma,\cdots,\sigma)$ is deconfined, it is invariant under fusion with any
of the condensed bosons, so it must split into direct sums of $e$ and $m$.

In the following denote $g$ as the cyclic permutation of the 16 layers.
Namely, $g$ generates the $\mathbb{Z}_{16}$ symmetry group. The
$\mathbb{Z}_{16}$ symmetry should be preserved by the condensation, so the
$\mathbb{Z}_{2}$ TC is enriched by the $\mathbb{Z}_{16}$ symmetry. Following
the general classification Barkeshli  _et al._ (2019), first we need to know
how the generator $g$ permutes anyon types. Given that the permutation must
preserve the fusion and braiding properties of anyons, there are two
possibilities for the $\mathbb{Z}_{2}$ TC: either $g$ does not permute, or $g$
swaps $e$ and $m$. In the latter case, there is no symmetric gapped boundary.
This is because the only Lagrangian subgroups for the $\mathbb{Z}_{2}$ TC are
$\mathds{1}+e$ and $\mathds{1}+m$ and neither of them is invariant under
$e\leftrightarrow m$.

To determine the action of $g$, we take a slightly different approach. The
$\mathbb{Z}_{2}$ toric code can be obtained by gauging the fermion parity of
16 copies of $p_{x}+ip_{y}$ superconductors. The $g$ symmetry again permutes
the 16 layers cyclically. In this construction, the $e$ and $m$ anyons
correspond to a fermion parity flux. The fermion parity flux binds 16 Majorana
zero modes $\gamma_{i}$ for $i=1,2,\dots,16$, one from each layer. The local
fermion parity of the flux is thus $P=\prod_{i}\gamma_{i}$. After gauging, the
fermion parity flux with even and odd local fermion parity ($P=\pm 1$) become
the $e$ and $m$ anyons. Now under the $g$ action,
$\gamma_{i}\rightarrow\gamma_{[i+1]_{16}}$, so we find

$P=\prod_{i}\gamma_{i}\rightarrow\gamma_{2}\cdots\gamma_{16}\gamma_{1}=-P.$
(7)

Therefore, the $g$ symmetry flips the fermion parity of the flux, which
becomes the $e\leftrightarrow m$ symmetry after gauging. As a result, there is
no symmetric gapped boundary.

We can fix this issue by modifying the transformation of the Majorana modes to
the following form:

$\begin{gathered}\gamma_{i}\rightarrow\gamma_{i+1},i=1,\dots,15\\\
\gamma_{16}\rightarrow-\gamma_{1}.\end{gathered}$ (8)

Under this transformation, the local parity $P$ remains unchanged. However, it
then follows that $g^{16}$ acts as $\gamma_{i}\rightarrow-\gamma_{i}$ for all
$i$, i.e. $g^{16}=(-1)^{N_{f}}$, so the fermion $\psi$ transforms projectively
under $g$. Since $\psi=e\times m$, one of $e$ or $m$ should transform
linearly, and thus can condense without breaking the $\mathbb{Z}_{16}$
symmetry.

So we are left with two options: either the $g$ symmetry swaps $e$ and $m$, or
have $g^{16}=-1$ on the $\psi$ anyon. We now describe a (3+1)d boundary theory
of the root $C_{16}$ SPT phase. When constructing the $\mathbb{Z}_{2}$ toric
code phase from 16 copies of $p+ip$ superconductors, we place the 16 copies in
a $C_{16}$-invariant configuration, all terminating at the rotation axis. Then
we gauge the fermion parity of the entire 3D system to obtain a fermionic
$\mathbb{Z}_{2}$ gauge theory. We have argued that there should exist a
$C_{16}$-symmetric gapped boundary of the $\mathbb{Z}_{2}$ toric code phase to
an invertible state, and we conjecture that the same is true when the 3D
fermion parity is gauged. However we are not able to precisely determine the
nature of the invertible state besides its chiral central charge $c_{-}=8$. So
this (3+1)d fermionic $\mathbb{Z}_{2}$ gauge theory may exist on the boundary
of the root $C_{16}$ BC SPT state, possibly stacked with another (in-
cohomology) SPT state.

##  IV Discussions and conclusions

In this work we have studied symmetry-preserving gapped boundary states for
(4+1)d BC SPT phases protected by $\mathbb{Z}_{N}$ and $C_{N}$ symmetries. We
show that for $N\notin\\{2,4,8,16\\}$, no such boundary states exist for the
root $\mathbb{Z}_{N}$ SPT phases. We then propose that for $N=2,4,8,16$ a
candidate boundary topological order is a fermionic $\mathbb{Z}_{2}$ gauge
theory, where the anomalous symmetry is generated by topological
superconductor defects. We provide explicit constructions of the boundary
theory for $C_{N}$ SPT phases for $N=2,4,8,16$.

One immediate question left open from our analysis is the 32-gappability of
Spin$(2n+1)_{2n+1}$. Given that the simplest of the series, SU(2)6, already
has 7 anyon types, it is challenging to classify the Lagrangian algebras in
SU(2)${}_{6}^{\boxtimes 32}$. We conjecture that Spin$(2n+1)_{2n+1}$ is not
32-gappable.

An interesting question for future works is to construct possible gapless
boundary theories for general $N$, or even the $\mathrm{U}(1)$ symmetry group.
For $N=2$, a gapless boundary theory was constructed in Ref. [Huang, 2020].

It will also be interesting to clarify the relation between the bosonic
$\mathbb{Z}_{N}$ (or $C_{N}$) SPT phases and the fermionic ones. The fermionic
phases can be realized by non-interacting fermions and the natural boundary
states are Weyl fermions. One can imagine that certain fermionic phases are
actually adiabatically connected to a bosonic one with trivial gapped
fermions. In fact, this provides a possible route to construct gapless
boundary states for many values of $N$ if the bosonic phase can be “embedded”
into a non-interacting fermionic one. This is the case for all odd $N$, so a
possible boundary theory is obtained by gauging fermion parity in a (3+1)d
Weyl fermion.

We have discussed the 0-form symmetry in a fermionic $\mathbb{Z}_{2}$ gauge
theory in (3+1)d. The full symmetry group includes $\mathbb{Z}_{2}$ 1-form and
2-form symmetries, and together with the $\mathbb{Z}_{16}$ 0-form symmetry
they are expected to form a 3-group Barkeshli  _et al._ (2022a). The 1- and
2-form symmetries and their anomalies are analyzed in Ref. [Barkeshli  _et
al._ , 2022b]. It is important to fully understand the structure and the
anomaly of the 3-group. In addition, one can also consider non-invertible
defects to get an even richer structure (conjecturally a fusion 3-category).

##  V Acknowledgement

M.C. thanks Z. Bi, Y.A. Chen, T. Ellison and S.-J. Huang for helpful
conversations. M.C. is particulrly grateful for C.-M. Jian for discussions on
the effective response action of (4+1)d BC SPTs, and Juven Wang on the group
structure of the classification. M.C. would like to acknowledge NSF for
support under award number DMR-1846109.

##  Appendix A Group structure of $C_{N}$ SPT phases

In the following we denote the root BC SPT phase by
$x\in\\{0,1,\cdots,N-1\\}$, and the root group-cohomology SPT phase by
$y\in\\{0,1,\cdots,N-1\\}$. Stacking of phases is denoted additively and $0$
represents the trivial phase. We then have $Ny=0$.

To determine the group structure of $C_{N}$ SPT phases, we observe that in the
block construction Song  _et al._ (2017), a state $|\psi\rangle$ with $N$
copies of $E_{8}$ states, all parallel to the rotation center and arranged in
a $C_{N}$-symmetric configuration is actually adiabatically connected to a
trivial state. On the (3+1)d boundary, we have $N$ copies of $(E_{8})_{1}$
CFTs, where the $C_{N}$ acts as $\mathbb{Z}_{N}$ cyclic permutations.
Apparently, it has the same mixed-gravitational anomaly as the boundary of the
$x^{N}$ phase. On the other hand, it can also have a pure $\mathbb{Z}_{N}$
anomaly. Recall that ’t Hooft anomalies of a $\mathbb{Z}_{N}$ symmetry are
classified by $\mathcal{H}^{3}[\mathbb{Z}_{N},\mathrm{U}(1)]=\mathbb{Z}_{N}$,
so they can be labeled by an integer $\omega\in\mathbb{Z}/N\mathbb{Z}$. It is
known that the $\mathbb{Z}_{N}$ anomaly $\omega_{N}$ for $N$ copies of
$(E_{8})_{1}$ CFTs is given by Bischoff (2020)

$\omega_{N}=\begin{cases}0&3\nmid N\\\ \frac{N}{3}&3\mid N\end{cases}.$ (9)

Therefore, the triviality of $|\psi\rangle$ implies $Nx+\omega_{N}y=0$ mod
$N$.

For $3\nmid N$, we have $Nx=0$, so the group is $\mathbb{Z}_{N}^{2}$.

For $3\mid N$, we find $Nx+\frac{N}{3}y=0$ mod $N$, and it follows that $x$
generates a $\mathbb{Z}_{3N}$ subgroup, and the group structure is
$\mathbb{Z}_{3N}\times\mathbb{Z}_{N/3}$.

Special cases of the classification (for $N=2,3,4,8$) have been been obtained
in Ref. [Wan and Wang, 2019].

##  Appendix B Effective response action for $\mathbb{Z}_{3}$ BC SPT phases

The action in Eq. (2) fails to describe a BC SPT phase for $N=3$. The
cobordism classification gives a $\mathbb{Z}_{9}$ classification for
$\mathbb{Z}_{3}$ SPT phases in (4+1)d, where the generator is the root BC SPT
phase. This is consistent with the $C_{3}$ analysis done in Appendix A. The
partition function given in Eq. (2) however only evaluates to a third root of
unity. In fact, it was pointed out in [Wan and Wang, 2019] that the response
action for the root BC phase is the Postnikov square
${\mathfrak{P}}_{3}(\beta_{3}a)$, where $\beta_{3}$ is the Bockstein
homomorphism. For a definition of ${\mathfrak{P}}_{3}$, see [Wan and Wang,
2019].

In order to calculate the partition function of the root $\mathbb{Z}_{3}$ SPT
phase on $K_{3}\times S^{1}$, we make use of the fact that the classification
of $\mathbb{Z}_{3}\times\mathbb{Z}_{2}^{f}$ fermionic SPT phases is the same
as the bosonic one, with the same topological term Wan and Wang (2019). In
other words, all such fermionic SPT phases are essentially bosonic. However,
the fermionic SPT phases in this case can all be realized using massive Dirac
fermions and we can now compute the partition functions in terms of $\eta$
invariant García-Etxebarria and Montero (2019). We find that the partition
function on $K_{3}\times S^{1}$ with a nontrivial $\mathbb{Z}_{3}$ holonomy is
$e^{\frac{2\pi i}{3}}$.

##  Appendix C Witt group of MTCs

In this section we will review the definition of the Witt group, as well as
some known facts about it.

First we define the notion of Witt equivalence between two (2+1)d topological
phases. Two topological phases $\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ are
Witt equivalent, if $\mathcal{B}_{1}\boxtimes\overline{\mathcal{B}}_{2}$ has a
fully gapped interface to an invertible state (i.e. some copies of $E_{8}$
states). Here $\overline{\mathcal{B}}_{2}$ is the mirror image of
$\mathcal{B}_{2}$. In other words, there is a gapped interface between
$\mathcal{B}_{1}$ and $\mathcal{B}_{2}$ as long as we are allowed to freely
stack copies of $E_{8}$ states. Mathematically, two MTCs $\mathcal{B}_{1}$ and
$\mathcal{B}_{2}$ are Witt equivalent if
$\mathcal{B}_{1}\boxtimes\overline{\mathcal{B}}_{2}$ is a quantum double
(Drinfeld center of some fusion category).

Below we review known results about the torsion subgroup of the Witt group,
particularly for Abelian MTCs. We adopt notations in Ref. [Bonderson, 2012]
for MTCs.

The Witt group of all Abelian MTCs, denoted by $\mathcal{W}_{\text{pt}}$
following the notation in Ref. [Davydov  _et al._ , 2011], has the following
decomposition:

$\mathcal{W}_{\text{pt}}=\bigoplus_{p\text{
prime}}\mathcal{W}_{\text{pt}}(p).$ (10)

For each prime $p$, the $p$-subgroup $\mathcal{W}_{\text{pt}}(p)$ is given by:

$p=2$

: $\mathcal{W}_{\text{pt}}(2)=\mathbb{Z}_{8}\times\mathbb{Z}_{2}$. Here
$\mathbb{Z}_{8}$ is generated by the semion theory $\mathbb{Z}_{2}^{(1/2)}$,
and $\mathbb{Z}_{2}$ is generated by
$\mathbb{Z}_{2}^{(1/2)}\times\overline{\mathbb{Z}_{4}^{(1/2)}}$.

$p\equiv 1\,(\text{mod }4)$

: $\mathcal{W}_{\text{pt}}(p)=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. One
generator can be chosen as $\mathbb{Z}_{p}^{(1)}$, and the other
$\mathbb{Z}_{p}^{(k)}$ where $k$ is a quadratic non-residue mod $p$.

$p\equiv 3\,(\text{mod }4)$

: $\mathcal{W}_{\text{pt}}(p)=\mathbb{Z}_{4}$. The generator could be any
$\mathbb{Z}_{p}^{(n)}$ theory for $1\leq n<p$.

Another important example is Kitaev’s 16-fold way: the Ising MTC generates a
$\mathbb{Z}_{16}$ group, which contains order-$2,4,8$ subgroups.

There also exists infinitely many order-$32$ elements in the Witt group. They
are represented by the Spin$(2n+1)_{2n+1}$ Chern-Simons theories, where $n\geq
1$. The simplest one of them is $\text{Spin}(3)_{3}\simeq\text{SU}(2)_{6}$. It
is the “square root” of an Ising Witt class: two copies of Spin$(2n+1)_{2n+1}$
is Witt equivalent to Spin($(2n+1)^{2})_{1}$.

We will now examine the $N$-gappability of the examples mentioned above.

###  C.1 $\mathbb{Z}_{p}^{(n)}$

We start from the $\mathbb{Z}_{p}^{(n)}$ theories, where $p$ is an odd prime.

First, we consider $p\equiv 3$ mod $4$, and $\mathbb{Z}_{p}^{(n)}$ MTCs have
order 4 in the Witt group. We now show that they are not $4$-gappable.

Label anyon in four copies of $\mathbb{Z}_{p}^{(n)}$ by
$a=(a_{1},a_{2},a_{3},a_{4})$, where $a_{i}\in\\{0,1,\dots,p-1\\}$. The most
general form of a cyclic permutation is generated by

$g(a)=(s_{1}a_{4},s_{2}a_{1},s_{3}a_{2},s_{4}a_{3}),$ (11)

where $s_{i}=\pm 1$. Basically, this is a “bare” permutation that takes $a$ to
$(a_{4},a_{1},a_{2},a_{3})$, combined with a topological symmetry of each of
the $\mathbb{Z}_{p}^{(n)}$ layer. It is known that the only nontrivial
topological symmetry of the $\mathbb{Z}_{p}^{(n)}$ MTC is the charge
conjugation $a\rightarrow-a$.

In our setup, we require $g^{4}$ is the identity. Under repeated actions of
$g$ we have:

$\begin{split}(a_{1},a_{2},a_{3},a_{4})&\rightarrow(s_{1}a_{4},s_{2}a_{1},s_{3}a_{2},s_{4}a_{3})\\\
&\rightarrow(s_{1}s_{4}a_{3},s_{2}s_{1}a_{4},s_{3}s_{2}a_{1},s_{4}s_{3}a_{2})\\\
&\rightarrow(s_{1}s_{4}s_{3}a_{2},s_{2}s_{1}s_{4}a_{3},s_{3}s_{2}s_{1}a_{4},s_{4}s_{3}s_{2}a_{1})\\\
&\rightarrow s_{1}s_{2}s_{4}s_{4}(a_{1},a_{2},a_{3},a_{4}).\end{split}$ (12)

So we must have $s_{1}s_{2}s_{3}s_{4}=1$, otherwise $g^{4}$ is the global
charge conjugation, which acts nontrivially in $\mathbb{Z}_{p}$ theories.

If $a$ is in the Lagrangian subgroup and the subgroup preserves the
$\mathbb{Z}_{4}$ symmetry, then $a,g(a),g^{2}(a),g^{3}(a)$ must form a
condensable subset. Then they must all be bosons, which gives

$a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}\equiv 0\ \text{mod }p.$ (13)

In addition, they must have trivial mutual braiding statistics, which lead to

$\begin{gathered}s_{2}a_{1}a_{2}+s_{3}a_{2}a_{3}+s_{4}a_{3}a_{4}+s_{1}a_{4}a_{1}\equiv
0\ \text{mod }p\\\
(s_{1}s_{4}+s_{2}s_{3})a_{1}a_{3}+(s_{1}s_{2}+s_{3}s_{4})a_{2}a_{4}\equiv 0\
\text{mod }p.\end{gathered}$ (14)

The last equation simplifies to
$2(s_{1}s_{4}a_{1}a_{3}+s_{1}s_{2}a_{2}a_{4})\equiv 0$ mod $p$, and since $2$
is invertible mod $p$, we have

$(s_{1}s_{4}a_{1}a_{3}+s_{1}s_{2}a_{2}a_{4})\equiv 0\ \text{mod }p.$ (15)

Since $s_{1}s_{2}s_{3}s_{4}=1$, we can represent
$s_{1}=c_{4}c_{1},s_{2}=c_{1}c_{2},s_{3}=c_{2}c_{3},s_{4}=c_{3}c_{4}$, and
define $b_{i}=c_{i}a_{i}$. The equations are simplified to

$\begin{gathered}b_{1}^{2}+b_{2}^{2}+b_{3}^{2}+b_{4}^{2}\equiv 0\ \text{mod
}p,\\\ b_{1}b_{2}+b_{2}b_{3}+b_{3}b_{4}+b_{4}b_{1}\equiv 0\ \text{mod }p,\\\
b_{1}b_{3}+b_{2}b_{4}\equiv 0\ \text{mod }p.\end{gathered}$ (16)

Together they imply $(b_{1}+b_{2}+b_{3}+b_{4})^{2}\equiv 0$ mod $p$. Then the
second relation leads to

$\begin{split}b_{1}b_{2}+b_{2}b_{3}+b_{3}b_{4}+b_{4}b_{1}&=(b_{2}+b_{4})(b_{1}+b_{3})\\\
&\equiv-(b_{1}+b_{3})^{2}\equiv 0\ \text{mod }p,\end{split}$

Then the only condition remaining is $b_{1}^{2}+b_{2}^{2}\equiv 0$ mod $p$.

One can show that there is no solution for $p\equiv 3\ (\text{mod }4)$. To
show this is the case, we write $b_{1}^{2}\equiv q$, so $b_{2}^{2}\equiv-q$.
In other words, both $q$ and $-q$ are quadratic residues of $p$. Compute the
Legendre symbols:

$\left(\frac{-q}{p}\right)=\left(\frac{-1}{p}\right)\left(\frac{q}{p}\right)=(-1)^{\frac{p-1}{2}}\left(\frac{q}{p}\right)=-\left(\frac{q}{p}\right).$
(17)

So it is impossible to have both $q$ and $-q$ being quadratic residues when
$p\equiv 3\ (\text{mod }4)$.

Let us now turn to $p\equiv 1\ (\text{mod }4)$, and the
$\mathbb{Z}_{p}^{(n))}$ theories have order 2 in the Witt group. We will show
that they are not $2$-gappable. Using the same argument, $(a_{1},a_{2})$ and
$(s_{1}a_{2},s_{2}a_{1})$ should form a condensable subgroup, which requires

$a_{1}^{2}+a_{2}^{2}\equiv 0\ \text{mod }p,(s_{1}+s_{2})a_{1}a_{2}\equiv 0\
\text{mod }p.$ (18)

Thus we need to have $s_{1}=-s_{2}$. However, under this permutation
$(a_{1},a_{2})\rightarrow(s_{1}a_{2},s_{2}a_{1})\rightarrow
s_{1}s_{2}(a_{1},a_{2})$, if $s_{1}s_{2}=-1$ then $g^{2}$ is equal to the
charge conjugation.

###  C.2 $\mathbb{Z}_{2}^{(\frac{1}{2})}$

$\mathbb{Z}_{2}^{(\frac{1}{2})}$ has order 8 in the Witt group. A Lagrangian
subgroup in 8 copies of $\mathbb{Z}_{2}^{(\frac{1}{2})}$ should have dimension
$16$, so generated by four bosons.

We first find all bosons, such that the its image under $\mathbb{Z}_{8}$ form
a condensable subgroup. It turns out that there are $8$ such bosons, and they
form $\mathbb{Z}_{2}^{3}$ group generated by
$(1,1,0,0,1,1,0,0),(0,1,1,0,0,1,1,0),(0,0,1,1,0,0,1,1)$. So it is impossible
for find a Lagrangian subgroup invariant under $\mathbb{Z}_{8}$.
Interestingly, if we actually condense this $\mathbb{Z}_{2}^{3}$ subgroup, we
find a $\mathbb{Z}_{2}$ toric code, and the $\mathbb{Z}_{8}$ generator acts as
electromagnetic duality in this theory. Therefore, we can not further condense
bosons without breaking the $\mathbb{Z}_{8}$ symmetry.

##  Appendix D Algebraic description of symmetry-preserving anyon condensation

We review the algebraic theory of gapped boundaries of a two-dimensional
topological phase Kong (2014); Kitaev and Kong (2012); Eliens  _et al._
(2014); Lan  _et al._ (2015); Neupert  _et al._ (2016); Cong  _et al._ (2017),
closely following the formulation in [Cong  _et al._ , 2017] and [Cheng and
Williamson, 2020]. We extensively use the language of unitary modular tensor
category (UMTC) for (2+1)d topological phases. A brief summary of UMTC in this
context can be found in the appendix of [Cheng and Williamson, 2020].

A gapped boundary corresponds to a Lagrangian algebra of the bulk MTC.
Physically the Lagrangian algebra indicates which bulk anyons are condensed on
the boundary Kong (2014); Levin (2013).

The mathematical theory of the gapped boundary takes into account the local
process of annihilating a condensable anyon $a$ on the boundary. Similar to
fusion/splitting spaces, we associate a vector space for local operators that
annihilate $a$, denoted as $V^{a}$, with basis vector $|a;\mu\rangle$. The
dimension of this vector space is the “multiplicity” $n_{a}$ of $a$ in the
Lagrangian algebra. Obviously we must have $n_{1}=1$.

Diagrammatically, the condensation process is represented by an anyon line
terminating on a wall representing the boundary. We also attach a label at the
termination point which represents the state of the boundary condensation
space. When $n_{a}=1$ it can be suppressed.

Figure 3: A diagrammatic representation of the M symbol.

An important property of the algebra is the following “$M$ symbol”:

$|a;\mu\rangle|b;\nu\rangle=\sum_{c,\lambda}[M^{ab}_{c}]^{\mu\nu}_{\lambda}|c;\lambda\rangle.$
(19)

The definition is illustrated diagrammatically in Fig. 3.

Next we impose consistency conditions on the $M$ symbols. We can apply $M$
moves to three anyon lines terminating $a,b,c$ on the boundary, but in
different orders, which leads to a variation of the pentagon equation:

$\sum_{e,\sigma}[M^{ab}_{e}]^{\mu\nu}_{\sigma}[M^{ec}_{d}]^{\sigma\lambda}_{\delta}[F^{abc}_{d}]_{ef}=\sum_{\psi}[M^{af}_{d}]^{\mu\psi}_{\delta}[M^{bc}_{f}]^{\nu\lambda}_{\psi}$
(20)

The $M$ symbols also have gauge degrees of freedom, originating from the basis
transformation of the boundary condensation space $V^{a}$:
$\widetilde{|a;\mu\rangle}=\Gamma^{a}_{\mu\nu}|a;\nu\rangle$, where
$\Gamma^{a}_{\mu\nu}$ is a unitary transformation. The $M$ symbol becomes

$[\tilde{M}^{ab}_{c}]^{\mu\nu}_{\lambda}=\sum_{\mu^{\prime},\nu^{\prime},\lambda^{\prime}}\Gamma^{a}_{\mu\mu^{\prime}}\Gamma^{b}_{\nu\nu^{\prime}}[M^{ab}_{c}]^{\mu^{\prime}\nu^{\prime}}_{\lambda^{\prime}}[\Gamma^{c}]_{\lambda^{\prime}\lambda}^{-1}.$
(21)

$M$ symbols are affected by the gauge transformation of bulk fusion space as
well.

It is convenient to fix the gauge for the following symbols:

$[M^{1a}_{a}]^{\mu}_{\nu}=[M^{a1}_{a}]^{\mu}_{\nu}=\delta_{\mu\nu}.$ (22)

Braiding puts further constraints on the $M$ symbols. Since the anyons
condense on the boundary, it should not matter in which order the anyon lines
terminate on the boundary.

$[M^{ba}_{c}]^{\nu\mu}_{\lambda}R^{ab}_{c}=[M^{ab}_{c}]^{\mu\nu}_{\lambda}.$
(23)

There is a similar condition for the inverse braiding.

It was shown in Ref. [Cong  _et al._ , 2017] that these conditions are
equivalent to the mathematical definition of a commutative, connected and
separable Frobenius algebra $\mathcal{A}=\bigoplus_{a}n_{a}a$ in a braided
tensor category, with the algebra morphism
$\mathcal{A}\times\mathcal{A}\rightarrow\mathcal{A}$ precisely given by the
$M$ symbol.

###  D.1 Symmetry-preserving condensation

We now give a precise definition of anyon condensation that preserves the
global symmetry Bischoff  _et al._ (2019). This builds on top of the algebraic
theory of (2+1)d symmetry-enriched topological phases, known as the $G$-graded
braided tensor category, reviewed in [Cheng and Williamson, 2020]. For a more
complete account see [Barkeshli  _et al._ , 2019]. A key fact that we will use
is that in a topological phase enriched by symmetry group $G$, the symmetry
action on anyons is fully specified by the following data:
$\rho_{\mathbf{g}},U_{\mathbf{g}}(a,b;c)$ and $\eta_{a}(\mathbf{g,h})$. Here
$\rho_{\mathbf{g}}$ denotes a permutation of anyon labels, and in the
following we write ${}^{\mathbf{g}}a\equiv\rho_{\mathbf{g}}(a)$.
$U_{\mathbf{g}}(a,b;c)$ are unitary transformations acting on the fusion
spaces. $\eta_{a}(\mathbf{g,h})$ are phase factors that describe projective
symmetry transformations on individual anyons. $\rho,U$ and $\eta$ need to
satisfy consistency conditions given in [Barkeshli  _et al._ , 2019]. Again we
refer to [Barkeshli  _et al._ , 2019] for their definitions and properties. In
the present case of the $\mathcal{B}^{\boxtimes N}$ MTC, $\rho$ is the
$\mathbb{Z}_{N}$ cyclic permutation, and $U$ and $\eta$ are both $1$ if we use
the natural gauge choice where the $F$ and $R$ symbols of
$\mathcal{B}^{\boxtimes N}$ are simply given by the Cartesian products of
those of $\mathcal{B}$.

In the following $a,b,c,\dots$ denote anyons in the condensate, unless
otherwise specified. We assume $n_{a}=1$ whenever $a$ belongs to the
condensate, so we omit the index for the boundary condensation space. Since
the boundary is fully gapped and symmetric, we can posit that for each
$\mathbf{g}$ there exists at least one $\mathbf{g}$-defect that can be
absorbed without creating any additional excitations on the boundary.

Figure 4: A diagrammatic representation of the $\chi$ symbol.

Here $\chi_{a}(\mathbf{g})$ is a phase factor. Physically,
$\chi_{a}(\mathbf{g})$ encodes the $\mathbf{g}$ action on the condensed anyon.
When there is more than one condensation channel, $\chi$ should be replaced by
a unitary transformation acting on the condensation space.

If we slide a vertex which splits a $\mathbf{gh}$-defect to $\mathbf{g}$\- and
$\mathbf{h}$-defects over a boundary vertex, we find

$\eta_{a}(\mathbf{g,h})=\frac{\chi_{a}(\mathbf{gh})}{\chi_{{}^{\bar{\mathbf{g}}}a}(\mathbf{h})\chi_{a}(\mathbf{g})}.$
(24)

We can also consider fusion of condensable anyons on a boundary. For $a,b,c$
in the condensate, sliding a $\mathbf{g}$ line over the diagrammatic equation
that defines $M$ symbol, one finds

$M^{{}^{\bar{\mathbf{g}}}a,{}^{\bar{\mathbf{g}}}b}_{{}^{\bar{\mathbf{g}}}c}U_{\mathbf{g}}(a,b;c)=M^{ab}_{c}\frac{\chi_{a}(\mathbf{g})\chi_{b}(\mathbf{g})}{\chi_{c}(\mathbf{g})}.$
(25)

We believe that these two conditions Eqs. (24) and (25) are sufficient and
necessary for the condensation to preserve symmetry. Mathematically,
$\chi_{a}(\mathbf{g})$ defines an algebraic isomorphism for each $\mathbf{g}$.
The consistency conditions guarantee that one has a $G$-equivariant algebra
structure on $\mathcal{L}$ Bischoff  _et al._ (2019).

Note that for a $\mathbf{g}$-invariant anyon $a$, $\chi_{a}(\mathbf{g})$ can
be interpreted as the $\mathbf{g}$ charge carried by $a$. In particular, it
means that
$\eta_{a}(\mathbf{g,h})=\frac{\chi_{a}(\mathbf{gh})}{\chi_{a}(\mathbf{g})\chi_{a}(\mathbf{h})}$.
Below, in all our examples the bulk has no symmetry fractionalization so we
make the canonical choice that $\eta_{a}(\mathbf{g,h})=1$.

###  D.2 Condensation in Spin$(2n)_{1}$ layers

Below we focus on the cases of Spin$(2n)_{1}^{\boxtimes r}$. We will only
study the $n=1,2$ cases. $n=4$ has been treated in Ref. [Qi  _et al._ , 2019],
and the other values of $n$ are similar. We start from a few general results
that apply to all $n$. As before, anyons are denoted by a $r$-tuple.

Denote the Lagragian subgroup as $\mathcal{A}$ for the system, which has two
types of group elements. The subgroup $\mathcal{A}_{0}$ consists of all bosons
$\\{\mathbf{a}_{0},\mathbf{b}_{0},\dots\\}\in\mathcal{A}_{0}$ made of an even
number of fermions while the rest of the group elements are expressed as
$\mathbf{v}+\mathbf{a}_{0}$, where $\mathbf{v}$ is the fermion parity flux.

We observe that $F^{\mathbf{a}_{0},\mathbf{b}_{0},\mathbf{c}_{0}}=1$. So
$M^{\mathbf{a}_{0},\mathbf{b}_{0}}$ forms a 2-cocycle over $\mathcal{A}_{0}$.
We will postulate that $M^{\mathbf{a}_{0},\mathbf{b}_{0}}=1$ in our solutions.

To simplify Eqn. 20, we make the following gauge choices: By using the
$\Gamma^{\mathbf{v}+\mathbf{a}_{0}}$ gauge freedoms, we set
$M^{\mathbf{v},\mathbf{a}_{0}}=1$. Then using $\Gamma^{\mathbf{v}}$ we set
$M^{\mathbf{v},\mathbf{v}}=1$. From the consistency equations we find the
following expressions:

$\begin{split}M^{\mathbf{v}+\mathbf{a}_{0},\mathbf{b}_{0}}&=F^{\mathbf{v},\mathbf{a}_{0},\mathbf{b}_{0}},\\\
M^{\mathbf{a}_{0},\mathbf{v}+\mathbf{b}_{0}}&=\frac{M^{\mathbf{a}_{0}+\mathbf{b}_{0},\mathbf{v}}}{M^{\mathbf{b}_{0},\mathbf{v}}},\\\
M^{\mathbf{v}+\mathbf{a}_{0},\mathbf{v}+\mathbf{b}_{0}}&=M^{\mathbf{a}_{0},\mathbf{v}}.\end{split}$
(26)

Now $M^{\mathbf{a}_{0},\mathbf{v}}$ has to satisfy

$M^{2\mathbf{v}+\mathbf{a}_{0}+\mathbf{b}_{0}+\mathbf{c}_{0},\mathbf{v}}M^{\mathbf{b}_{0},\mathbf{v}}F^{\mathbf{v}+\mathbf{a}_{0},\mathbf{v}+\mathbf{b}_{0},\mathbf{v}+\mathbf{c}_{0}}=\\\
M^{\mathbf{a}_{0},\mathbf{v}}M^{\mathbf{c}_{0},\mathbf{v}}F^{\mathbf{v},\mathbf{a}_{0},2\mathbf{v}+\mathbf{b}_{0}+\mathbf{c}_{0}}.$
(27)

We can write a general solution of the consistency equations
$M^{\mathbf{a,b}}=M_{0}^{\mathbf{a,b}}\omega^{\mathbf{a,b}}$, consisting of a
special solution $M_{0}^{\mathbf{a},\mathbf{b}}$ of Eqn. 27 and a group
2-cocycle $\omega^{\mathbf{a},\mathbf{b}}$ over $\mathcal{A}$. Then $\omega$
will be fixed through Eqn. 23.

###  D.3 $\mathbb{Z}_{4}$-symmetric Lagrangian algebra of
Spin$(4)_{1}^{\boxtimes 4}$

We denote the anyons in Spin$(4)_{1}^{\boxtimes 4}$ by
$\mathbf{a}=(\vec{a}_{1},\vec{a}_{2},\vec{a}_{3},\vec{a}_{4})$, where
$\vec{a}_{i}=(a_{i}^{1}\;a_{i}^{2})$ are defined mod 2. The F and R symbols
are given by

$\begin{split}F^{\mathbf{a},\mathbf{b},\mathbf{c}}&=\exp\left(\frac{i\pi}{2}\sum_{i=1}^{4}\vec{a}_{i}\cdot(\vec{b}_{i}+\vec{c}_{i}-[\vec{b}_{i}+\vec{c}_{i}]_{2})\right),\\\
R^{\mathbf{a,b}}&=\exp\left[\frac{i\pi}{2}\sum_{i=1}^{4}\vec{a}_{i}\cdot\vec{b}_{i}\right].\end{split}$
(28)

The $\mathbb{Z}_{4}$ symmetry generator $\mathbf{g}$ acts on the anyons in the
following way:

$\rho:(\vec{a}_{1},\vec{a}_{2},\vec{a}_{3},\vec{a}_{4})\rightarrow(\vec{a}_{2},\vec{a}_{3},\vec{a}_{4},\vec{a}_{1}).$
(29)

It is obvious that F and R symbols are invariant under $\rho$:

$F^{\rho(\mathbf{a}),\rho(\mathbf{b}),\rho(\mathbf{c})}=F^{\mathbf{a,b,c}},R^{\rho(\mathbf{a}),\rho(\mathbf{b})}=R^{\mathbf{a,b}}.$
(30)

Therefore the corresponding $U$ symbols are all 1. We can then set all the
$\eta$ symbols to 1 as well.

It will also be convenient to pick a set of generators for the Lagrangian
subgroup. We will use the following set:

$\begin{split}&\mathbf{v}_{1}=(1\;0,1\;0,1\;0,1\;0)\\\
&\mathbf{v}_{2}=(1\;1,1\;1,0\;0,0\;0)\\\
&\mathbf{v}_{3}=(0\;0,1\;1,1\;1,0\;0)\\\
&\mathbf{v}_{4}=(0\;0,0\;0,1\;1,1\;1)\end{split}$ (31)

Together they form a $\mathcal{A}=\mathbb{Z}_{2}^{4}$ group. Then any group
element $\mathbf{a}$ can be expanded in terms of the generators:
$\mathbf{a}=\sum_{i=1}^{4}\tilde{a}_{i}\mathbf{v}_{i}$.

Following the procedure in Section D.2, we find the special solution
$M_{0}^{\mathbf{a},\mathbf{b}}$:

$\begin{split}&M_{0}^{\mathbf{a}_{0},\mathbf{v}}=e^{\frac{i\pi}{2}\mathbf{v}\cdot\mathbf{a}_{0}},\end{split}$
(32)

and the group 2-cocycle $\omega^{\mathbf{a},\mathbf{b}}$:

$\omega^{\mathbf{a,b}}=e^{i\pi(\tilde{a}_{2}\tilde{b}_{3}+\tilde{a}_{3}\tilde{b}_{4})}.$
(33)

We find the following solution of $\chi$:

$\chi_{\mathbf{a}}(\mathbf{g})=e^{i\pi(1-\tilde{a}_{4})(\tilde{a}_{2}+\tilde{a}_{3})}.$
(34)

and

$\chi_{\mathbf{a}}(\mathbf{g}^{2})=e^{i\pi[(1-\tilde{a}_{4})\tilde{a}_{3}+(1-\tilde{a}_{3})\tilde{a}_{2}]}.$
(35)

One can easily verify that all $\mathbf{g}$-invariant anyons have $\chi=1$.
The same is true for all $\mathbf{g}^{2}$-invariant anyons.

###  D.4 $\mathbb{Z}_{8}$-symmetric Lagrangian algebra of
$\mathrm{U}(1)_{4}^{\boxtimes 8}$

We denote the anyons in $\mathrm{U}(1)_{4}^{\boxtimes 8}$ by a 8-tuple
$\mathbf{a}=(a_{1},a_{2},\dots,a_{8})$, where $a_{i}\in\\{0,1,2,3\\}$ are
defined mod 4. Since the theory is Abelian, fusion will be denoted additively.
We will use short-hand notations for the F and R symbols
$(F^{\mathbf{a,b,c}}_{\mathbf{a+b+c}})_{\mathbf{a+b,b+c}}\equiv
F^{\mathbf{a,b,c}},R^{\mathbf{a,b}}_{\mathbf{a+b}}=R^{\mathbf{a,b}}$. They are
given by

$\begin{split}F^{\mathbf{a,b,c}}&=\exp\left(i\pi\mathbf{a}\cdot\frac{\mathbf{b+c-[b+c]_{4}}}{4}\right),\\\
R^{\mathbf{a,b}}&=\exp\left(\frac{i\pi}{4}\mathbf{a\cdot
b}\right).\end{split}$ (36)

The $\mathbb{Z}_{8}$ symmetry generator $\mathbf{g}$ acts on the anyons in the
following way:

$\rho:(a_{1},a_{2},\dots,a_{8})\rightarrow(a_{2},a_{3},\dots,a_{1}).$ (37)

It is obvious that F and R symbols are invariant under $\rho$:

$F^{\rho(\mathbf{a}),\rho(\mathbf{b}),\rho(\mathbf{c})}=F^{\mathbf{a,b,c}},R^{\rho(\mathbf{a}),\rho(\mathbf{b})}=R^{a,b}.$
(38)

Therefore the corresponding $U$ symbols are all 1. We can then set all the
$\eta$ symbols to 1 as well.

The Lagrangian subgroup is generated by the following anyons:

$\begin{split}&\mathbf{v}_{1}=(1,1,1,1,1,1,1,1)\\\
&\mathbf{v}_{2}=(2,2,0,0,0,0,0,0)\\\ &\mathbf{v}_{3}=(0,2,2,0,0,0,0,0)\\\
&\mathbf{v}_{4}=(0,0,2,2,0,0,0,0)\\\ &\mathbf{v}_{5}=(0,0,0,2,2,0,0,0)\\\
&\mathbf{v}_{6}=(0,0,0,0,2,2,0,0)\\\
&\mathbf{v}_{7}=(0,0,0,0,0,2,2,0).\end{split}$ (39)

Together they generate a $\mathcal{A}=\mathbb{Z}_{2}^{6}\times\mathbb{Z}_{4}$
group. We find the solution to be

$\begin{split}&M_{0}^{\mathbf{a}_{0},\mathbf{v}}=e^{\frac{i\pi}{4}\mathbf{v}\cdot\mathbf{a}_{0}},\\\
\end{split}$ (40)

and

$\omega^{\mathbf{a,b}}=e^{i\pi(\tilde{a}_{2}\tilde{b}_{3}+\tilde{a}_{3}\tilde{b}_{4}+\tilde{a}_{4}\tilde{b}_{5}+\tilde{a}_{5}\tilde{b}_{6}+\tilde{a}_{6}\tilde{b}_{7})}.$
(41)

The solution for $\chi$ is given by

$\chi_{\mathbf{a}}(\mathbf{g})=e^{i\pi(1-\tilde{a}_{7})(\tilde{a}_{2}+\tilde{a}_{4}+\tilde{a}_{6})}.$
(42)

In this case, the only $\mathbf{g}$-invariant anyons are generated by
$\mathbf{v}_{1}$, so obviously $\chi_{\mathbf{v}_{1}}(\mathbf{g})=1$. The same
is true for other symmetry transformations.

##  Appendix E Gauging $\mathbb{Z}_{N}$ cyclic permutation symmetry

We describe the $\mathbb{Z}_{N}$ gauging of $\mathcal{B}^{\boxtimes N}$, where
the $\mathbb{Z}_{N}$ generator $g$ acts as

$(a_{1},a_{2},\cdots,a_{N})\rightarrow(a_{N},a_{1},\cdots,a_{N-1}).$ (43)

We will denote the gauged theory by $\mathcal{U}$. Anyons in $\mathcal{U}$
will be labeled by $(x_{g^{r}},\chi)$, where $g^{r}$ is the symmetry flux, and
$\chi$ is an irreducible representation of the stabilizer group. Here since
$G$ is Abelian we can view $\chi$ as a group homomorphism from the stabilizer
group to $\mathrm{U}(1)$. For example, if $r=0$, then $x$ runs through all
orbits of anyons $[\mathbf{b}]$ in $\mathcal{B}^{\boxtimes N}$ under $G$,
where $\mathbf{b}=(b_{1},b_{2},\cdots,b_{N})$ is a representative element of
the orbit. Its stabilizer group, i.e. the subgroup of $G$ that keeps the anyon
$\mathbf{b}$ invariant, is denoted by $G_{\mathbf{b}}$. When the stabilizer
group is strictly smaller than $G$, we call
$([\mathbf{b}]_{1},\chi_{\mathbf{b}})$ a superposition anyon. In general, the
label $x$ can actually be chosen as an element in
$\mathcal{B}^{\boxtimes(N,r)}$.

Among all $(x_{g^{r}},\chi)$ there is one with the minimal quantum dimension,
which we will call $(\mathds{1}_{g^{r}},\chi)$ (for $r=0$,
$(\mathds{1}_{1},1)$ is the identity anyon). We have the following fusion
rule:

$(a_{1},a_{2},\cdots,a_{(N,r)},\mathds{1},\cdots,\mathds{1})\times(\mathds{1}_{g},\chi)=(\mathbf{a}_{g},\chi).$
(44)

Here $\mathbf{a}=(a_{1},a_{2},\cdots,a_{(N,r)})$. Therefore
$d_{(\mathbf{a}_{g^{r}},\chi)}=d_{\mathbf{a}}d_{(\mathds{1}_{g^{r}},1)}$.

To compute $d_{(\mathds{1}_{g^{r}},1)}$, we use the fact that the total
quantum dimension squared of all $g^{r}$ defects must be equal to
$D_{\mathcal{B}}^{2}$ Barkeshli  _et al._ (2019). That is,

$\sum_{\mathbf{a}\in\mathcal{B}^{\boxtimes(N,r)}}d_{\mathbf{a}}^{2}d_{\mathds{1}_{g^{r}}}^{2}=D_{\mathcal{B}}^{2(N,r)}d_{\mathds{1}_{g^{r}}}^{2}=D_{\mathcal{B}}^{2N}.$

Thus $d_{\mathds{1}_{g^{r}}}=D_{\mathcal{B}}^{N-(N,r)}$.

We also have the topological twist factors for the fluxes:

$\theta_{(\mathbf{a}_{g^{r}},\chi)}=\theta_{\mathbf{a}_{g^{r}}}\chi(g^{r}).$
(45)

Here $\theta_{\mathbf{a}_{g^{r}}}$ is the topological twist factor of the
$\mathbf{a}_{g^{r}}$ defect. It can be chosen to be
$\theta_{\mathbf{a}_{g^{r}}}=\theta_{\mathbf{a}}^{\frac{(N,r)}{N}}$, but we do
not need its value in this section.

Now we consider the S matrix elements between fluxes and anyons. The entries
of S matix could be computed from the topological spins and the fusion
coefficients:

$S_{x,y}=\frac{1}{D}\sum_{z}N^{xy}_{z}d_{z}\frac{\theta_{x}\theta_{y}}{\theta_{z}}.$
(46)

We will use this to derive an important property of
$S^{\mathcal{U}}_{(\mathbf{a}_{g^{r}},\chi_{a}),([\mathbf{b}]_{1},\chi_{b})}$,
where $[\mathbf{b}]_{1}$ denotes a superposition anyon (i.e. $G_{\mathbf{b}}$
is smaller than $G$). First we consider the fusion between a superposition
anyon $([\mathbf{b}]_{1},\chi_{b})$ and a bare charge $(\mathds{1}_{1},\chi)$,
where $\chi$ is a 1D rep. of $\mathbb{Z}_{N}$. Following Ref. [Barkeshli  _et
al._ , 2019], we have

$([\mathbf{b}]_{1},\chi_{b})\times(\mathds{1}_{1},\chi)=([\mathbf{b}]_{1},\chi_{b}\cdot\chi|_{G_{\mathbf{b}}}).$
(47)

Here $\chi|_{G_{\mathbf{b}}}$ is the restriction of $\chi$ to
$G_{\mathbf{b}}$.

Suppose $\chi|_{G_{\mathbf{b}}}$ is trivial. For such a $\chi$, using 47 we
get:

$([\mathbf{b}]_{1},\chi_{b})\times(\mathds{1}_{1},\chi)=([\mathbf{b}]_{1},\chi_{b}),$
(48)

which implies that:

$(\mathbf{a}_{g^{r}},\chi_{a})\times([\mathbf{b}]_{1},\chi_{b})\times(\mathds{1}_{1},\chi)=(\mathbf{a}_{g^{r}},\chi_{a})\times([\mathbf{b}]_{1},\chi_{b}).$
(49)

Let us define ${\Lambda}_{b}$ as the group of $\chi$’s which restrict to
identity on $G_{\mathbf{b}}$. It is easy to see that $\Lambda_{b}$ is all the
irreps on $G/G_{b}$. Then Eq. (49) implies

$(\mathbf{a}_{g^{r}},\chi_{a})\times([\mathbf{b}]_{1},\chi_{b})=\sum_{\mathbf{c}_{g^{r}}}N_{\mathbf{a}_{g^{r}},[\mathbf{b}]_{1}}^{\mathbf{c}_{g^{r}}}\sum_{\chi\in\Lambda_{b}}(\mathbf{c}_{g^{r}},\chi_{c}\chi).$
(50)

While $N_{\mathbf{a}_{g^{r}},[\mathbf{b}]_{1}}^{\mathbf{c}_{g^{r}}}$ and
$\chi_{c}$ are undetermined, they are not needed for our purpose.

Using the definition (46), we obtain

$\begin{split}&S^{\mathcal{U}}_{(\mathbf{a}_{g^{r}},\chi_{a}),([\mathbf{b}]_{1},\chi_{b})}\\\
&=\frac{\theta_{(\mathbf{a}_{g^{r}},\chi_{a})}\theta_{([\mathbf{b}]_{1},\chi_{b})}}{{D_{\mathcal{U}}}}\sum_{\mathbf{c}_{g^{r}}}N_{\mathbf{a}_{g^{r}},[\mathbf{b}]_{1}}^{\mathbf{c}_{g^{r}}}d_{(\mathbf{c}_{g^{r}},\chi_{c})}\sum_{\chi\in\Lambda_{b}}\theta_{(\mathbf{c}_{g^{r}},\chi_{c}\chi)}^{-1}.\end{split}$
(51)

Let us consider the sum over $\chi\in\Lambda_{b}$, which can be simplified
further to

$\begin{split}\sum_{\chi\in\Lambda_{b}}\theta_{(\mathbf{c}_{g^{r}},\chi_{c}\chi)}^{-1}=\theta_{(\mathbf{c}_{g^{r}},\chi_{c})}^{-1}\sum_{\chi\in\Lambda_{b}}\chi^{-1}(g^{r}).\end{split}$
(52)

First, we assume $g^{r}\in G_{b}$. By the definition of $\Lambda_{b}$,
$\chi(g^{r})=1$ so the sum evaluates to $|\Lambda_{b}|$. Then if $g^{r}\notin
G_{b}$, the Schur’s orthogonality theorem applied to $G/G_{b}$ shows that the
sum should be 0. To summarize, we have shown that

$S^{\mathcal{U}}_{(\mathbf{a}_{g^{r}},\chi_{a}),([\mathbf{b}]_{1},\chi_{b})}\propto\delta_{g^{r}\in
G_{\mathbf{b}}}.$ (53)

Let us now consider what happens when $g^{r}\in G_{\mathbf{b}}$. It implies
that $b$ has a period $(N,r)$, i.e. write $b=(b_{1},b_{2},\dots,b_{N})$, then
$b_{i}=b_{i+(N,r)\text{ mod }N}$. Thus for this caculation, we can group
$(N,r)$ consecutive layers into one “layer”, with total number of layers
$\tilde{N}=\frac{N}{(N,r)}$. It is also convenient to define
$\tilde{\mathbf{b}}=(b_{1},b_{2},\cdots,b_{(N,r)})\in\tilde{\mathcal{B}}=\mathcal{B}^{\boxtimes(N,r)}$.
Therefore, the problem reduces to computing the $S$ matrix between a
$(\mathbf{a}_{g^{r}},1)$ defect, and
$[\mathbf{b}]_{1}=(\tilde{\mathbf{b}},\dots,\tilde{\mathbf{b}})$. We will
account for the charges later.

Now we use a geometric picture to compute the S matrix. A cylinder with a
$\mathbf{a}_{{g}^{{r}}}$ defect line can be viewed as a $\tilde{N}$-layer
cylinder, with a $g^{r}$ cyclic permutation branch cut, which is topologically
equivalent to a single-“layer” cylinder with the topological order described
by $\tilde{\mathcal{B}}$. On the cylinder there is a Wilson line of the
$\mathbf{a}$ anyon. We then compute the eigenvalue of the Wilson loop of
$([\mathbf{b}]_{1},1)$ along the other non-contractible cycle in two ways.
Note that in the alternative picture, the Wilson loop of
$([\mathbf{b}_{1},1])$ becomes a Wilson loop of $\tilde{\mathbf{b}}$. We thus
have:

$\frac{S^{\mathcal{U}}_{(\mathbf{a}_{g^{r}},1),([\mathbf{b}]_{1},1)}}{S^{\mathcal{U}}_{(\mathbf{a}_{g^{r}},1),(\mathds{1}_{1},1)}}=\frac{S^{\tilde{\mathcal{B}}}_{\mathbf{a}\tilde{\mathbf{b}}}}{S^{\tilde{\mathcal{B}}}_{\mathbf{a}\mathds{1}}}$
(54)

where
$S^{\mathcal{U}}_{(\mathbf{a}_{g^{r}},1),(\mathds{1}_{1},1)}=\frac{d_{(\mathbf{a}_{g^{r}},1)}}{D_{\mathcal{U}}}$,
$S^{\mathcal{B}}_{a\mathds{1}}=\frac{d_{\mathbf{a}}}{D_{\tilde{\mathcal{B}}}}=\frac{d_{\mathbf{a}}}{D_{\mathcal{B}}^{(N,r)}}$.
Using $d_{(\mathbf{a}_{g^{r}},1)}=D_{\mathcal{B}}^{N-(N,r)}d_{\mathbf{a}}$ and
$D_{\mathcal{U}}=ND_{\mathcal{B}}^{N}$, we have:

$\begin{split}S^{\mathcal{U}}_{(\mathbf{a}_{g^{r}},1),([\mathbf{b}]_{1},1)}&=\frac{d_{\mathbf{a}}D_{B}^{N-(N,r)}}{ND_{\mathcal{B}}^{N}}\frac{D_{\mathcal{B}}^{(N,r)}}{d_{\mathbf{a}}}S^{\tilde{\mathcal{B}}}_{\mathbf{a}\tilde{\mathbf{b}}}\\\
&=\frac{1}{N}S^{\tilde{\mathcal{B}}}_{\mathbf{a}\tilde{\mathbf{b}}}.\end{split}$
(55)

Including the contribution from the charge $\chi_{b}$, we find

$S^{\mathcal{U}}_{(\mathbf{a}_{g^{r}},\chi_{a}),([\mathbf{b}]_{1},\chi_{b})}=\frac{1}{N}\chi_{b}(g^{r})S^{\tilde{\mathcal{B}}}_{\mathbf{a}\tilde{\mathbf{b}}}\delta_{g^{r}\in
G_{\mathbf{b}}}.$ (56)

Now we study gapped boundaries of $\mathcal{B}^{\boxtimes N}$ from the gauging
perspective. Suppose we have a Lagrangian algebra $\mathcal{A}_{0}$ of
$\mathcal{B}^{\boxtimes N}$. It can be “lifted” to a condensable algebra
$\mathcal{A}$ in the gauged theory $\mathcal{U}$. Namely, we can first break
$\mathcal{A}_{0}$ into orbits under $G$, and each orbit becomes a single anyon
after gauging.

$\mathcal{A}_{0}=\sum_{[\mathbf{a}]}n_{\mathbf{a}}\sum_{\mathbf{b}\in[\mathbf{a}]}\mathbf{b}.$
(57)

Here $[\mathbf{a}]$ is an orbit under $G$, with $\mathbf{a}$ being a
representative element. After gauging, it becomes

$\mathcal{A}=\sum_{[\mathbf{a}]}n_{\mathbf{a}}([\mathbf{a}]_{1},\chi_{\mathbf{a}}).$
(58)

Roughly speaking, $\chi_{\mathbf{a}}$ is the symmetry charge carried by the
$\mathbf{a}$ anyon when it condenses. The assignment of $\chi_{\mathbf{a}}$ is
not arbitrary. In fact, because $U$ and $\eta$ are all $1$ in our example, the
$\chi_{\mathbf{a}}$’s are precisely those defined in Sec. D.1.

By definition, condensing $\mathcal{A}_{0}$ in $\mathcal{B}^{\boxtimes N}$
results in a trivial theory (the Vec MTC). If the condensation preserves the
$\mathbb{Z}_{N}$ symmetry, condensing $\mathcal{A}$ in $\mathcal{U}$ should
result in a deconfined $\mathbb{Z}_{N}$ gauge theory, denoted by
$\mathcal{D}$. In the same convention, anyons in $\mathcal{D}$ will be denoted
by $(g^{r},\chi)$. The vacuum is $\mathds{1}\equiv(1,1)$. Note that depending
on the choice of the charges $\chi_{\mathbf{a}}$’s, the $\mathbb{Z}_{N}$ gauge
theory may be twisted. We are in particular interested in whether there exists
a choice of $\chi$’s such that the $\mathbb{Z}_{N}$ gauge theory is not
twisted.

We briefly recall a few basic facts about anyon condensation from a MTC
$\mathcal{U}$ to $\mathcal{D}$. It is useful to think of the anyon
condensation as defining a gapped interface between the theories $\mathcal{U}$
and $\mathcal{D}$. To fully describe this process, it is also necessary to
define the theory of (possibly confined) excitations on the interface, called
$\mathcal{T}$. $\mathcal{D}$ is the “deconfined” subcategory of $\mathcal{T}$.

The relations between the three theories are encoded in the restriction map
$r$ and the lifting map $l$. An anyon $\alpha$ in the $\mathcal{U}$ theory can
be “restricted” to the $\mathcal{T}$ theory:

$r(\alpha)=\sum_{t\in\mathcal{T}}n_{\alpha,t}t.$ (59)

where the $n_{\alpha,t}$’s are non-negative integers. On the other hand, a
particle in the $\mathcal{T}$ can be lifted back to the $\mathcal{U}$ theory:

$l(t)=\sum_{\alpha\in\mathcal{U}}n_{\alpha,t}\alpha.$ (60)

Clearly $l(\mathds{1})$ is the Lagrangian algebra $\mathcal{A}$. We are mostly
interested in $n_{\alpha,t}$’s when $t\in\mathcal{D}$.

The integers $n_{\alpha,t}$ should satisfy various constraints. We will only
need the following condition: the restriction/lifting maps should commute with
the modular matrices of $\mathcal{U}$ and $\mathcal{D}$. More specifically,
for $\alpha\in\mathcal{U}$ and $t\in\mathcal{D}$, we have

$\sum_{\beta\in\mathcal{U}}S_{\alpha\beta}n_{\beta,t}=\sum_{s\in\mathcal{D}}n_{\alpha,s}S_{st}.$
(61)

Secondly, for a given $t\in\mathcal{U}$, all $\alpha$’s with $n_{\alpha,t}\neq
0$ share the same topological twist factor and $\theta_{\alpha}=\theta_{t}$.

We will denote

$l((g,1))=\sum_{a\in\mathcal{B}}w_{g}(a)(a_{g},1),$ (62)

where $w_{g}(a)$ is a non-negative integer. We shall use (61) to constrain
$w_{g}(a)$, which can then determine the topological twist factor of $(g,1)$.

Setting $\alpha=(a_{g},1)$, $t=\mathds{1}$ in (61), we have the left-hand side
given by

$\sum_{([\mathbf{b}]_{1},\chi_{b})\in\mathcal{A}}S^{\mathcal{U}}_{(a_{g},1),([\mathbf{b}]_{1},\chi_{\mathbf{b}})}n_{\mathbf{b}}=\frac{1}{N}\sum_{\mathbf{b}=g(\mathbf{b})}\chi_{\mathbf{b}}S_{ab}^{\mathcal{B}}n_{\mathbf{b}}.$
(63)

The RHS is $\frac{1}{N}w_{g}(a)$. So we find

$\sum_{\mathbf{b}=g(\mathbf{b})}\chi_{\mathbf{b}}S^{\mathcal{B}}_{ab}n_{\mathbf{b}}=w_{g}(a).$
(64)

Here $\mathbf{b}=g(\mathbf{b})$ means $\mathbf{b}=(b,b,\cdots,b)$. Similar
relations can be derived for other $g^{r}$. Given that the S matrix elements
are generally not integers, it is a nontrivial consistency check that the sum
on the left-hand side yields a non-negative integer.

Let us now apply the relation to $\mathcal{B}=\text{Spin}(2n)_{1}$ theory. In
this case, $b$ runs over all anyons in $\mathcal{B}$ and with
$n_{\mathbf{b}}=1$ we have

$\sum_{b\in\mathcal{B}}S_{ab}^{\mathcal{B}}\chi_{\mathbf{b}}=w_{g}(a).$ (65)

We have shown in that $\chi_{\mathbf{b}}=1$ for all $\mathbf{b}=g(\mathbf{b})$
is allowed for $n=1,2$ and $4$, and in fact for all other values of $n$. With
this choice, the unitarity of S matrix implies that
$w_{g}(a)=2\delta_{a,\mathds{1}}$. Therefore,
$\theta_{(g,1)}=\theta_{(\mathds{1}_{g},1)}=1$ and the $\mathbb{Z}_{N}$ gauge
theory is not twisted.

In the case of the Ising theory with $N=16$, the LHS becomes

$\begin{split}\frac{1}{2}(1+128\sqrt{2}\chi_{\sigma}+\chi_{\psi})=w_{g}(\mathds{1}),\\\
\frac{1}{2}(1-128\sqrt{2}\chi_{\sigma}+\chi_{\psi})=w_{g}(\psi),\\\
\frac{1}{\sqrt{2}}(1-\chi_{\psi})=w_{g}(\sigma).\end{split}$ (66)

Regardless of the choice of $\chi_{\sigma}$ and $\chi_{\psi}$,
$w_{g}(\mathds{1})$ and $w_{g}(\psi)$ can not be an integer, and the only
possible integer value for $w_{g}(\sigma)$ is 0. We conclude that such a
symmetry-preserving condensation in Ising⊠16 does not exist. This confirms the
more heuristic argument in Sec. III.2.

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