Condensed Matter Theory Center and Joint Quantum Institute, Department of
Physics, University of Maryland, College Park, Maryland 20472 USA

# Symmetry-preserving boundary of (2+1)D fractional quantum Hall states

Ryohei Kobayashi

###### Contents

  1. 1 Introduction
    1. 1.1 Summary of results
    2. 1.2 Organization of paper
  2. 2 Gapped boundary of bosonic topological phases
    1. 2.1 Review: Lagrangian algebra anyons
    2. 2.2 Symmetry-preserving gapped boundary
    3. 2.3 Example: $G=\mathrm{U}(1)$
    4. 2.4 Application: bosonic Abelian TQFT with U(1) symmetry
  3. 3 Gapped boundary of (2+1)D fermionic topological phases
    1. 3.1 Lagrangian algebra anyon for fermionic phases
    2. 3.2 Modular properties of gapped boundary
    3. 3.3 Stacking with toric code $D(\mathbb{Z}_{2})$
    4. 3.4 Additional properties of Lagrangian algebra anyon
    5. 3.5 U(1)f symmetry-preserving gapped boundary
  4. 4 Higher version of $c_{-}$ and Hall conductivity
  5. 5 Discussions
  6. A Review of (2+1)D anyon systems
    1. A.1 Notations of BTC
    2. A.2  Modular and super-modular tensor categories
    3. A.3 Modular extension of super-modular category
  7. B Review on global symmetry of (2+1)D topological phases
    1. B.1  Topological symmetry and braided auto-equivalence
    2. B.2  Global symmetry and symmetry fractionalization: bosonic case
    3. B.3  Fermionic symmetries
  8. C Review of spin TQFT
  9. D $S$-matrix action on boundary states of spin TQFT

##  1 Introduction

Fractional quantum Hall (FQH) states are the most well-studied class of
topological phases, since their experimental discovery in 1982 [1]. A FQH
state hosts a topological order in (2+1)D with U(1) global symmetry,
characterized by the existence of fractionally charged quasiparticles called
anyons [2, 3].

The boundary of a FQH state has a gapless edge state, protected by non-zero
quantized electric Hall conductivity $\sigma_{H}$ in the bulk. This gapless
edge state cannot be gapped out while preserving U(1) global symmetry. This
protection of gapless edge mode is understood as a consequence of perturbative
U(1) anomaly on (1+1)D boundary theory present when $\sigma_{H}\neq 0$, which
cannot be matched by gapped degrees of freedom.

Even in the absence of global symmetry, the boundary of a (2+1)D topological
ordered state is enforced to be gapless when its chiral central charge $c_{-}$
is nonzero. This is the most fundamental obstruction to having a gapped edge
state, and understood as a result of gravitational anomaly on the (1+1)D
boundary theory present when $c_{-}\neq 0$. For example, the Moore-Read state
[4, 5], the most well-known example of a non-Abelian FQH state, carries
$c_{-}=3/2$, and then its boundary cannot be gapped out even when we forget
about U(1) global symmetry.

The electric and thermal Hall conductivity $\sigma_{H}$ and $c_{-}$ give
important obstructions to a gapped edge state of (2+1)D topological order, but
do not cover complete obstructions to a gapped edge state. Actually, even in
the absence of global symmetry where we do not have $\sigma_{H}$, the
condition $c_{-}=0$ alone does not guarantee the existence of a gapped
boundary. For example, it is known that
$\mathrm{U}(1)_{2}\times\mathrm{U}(1)_{-4}$ Chern-Simons theory does not admit
a gapped boundary condition, though this theory carries $c_{-}=0$. The gapped
boundary condition of a bosonic topological phase in (2+1)D without global
symmetry has been extensively studied in [6, 7, 8, 9, 10, 11, 12, 13], and
algebraically characterized by the Lagrangian algebra anyon of a modular
tensor category [14, 15, 16, 17]. Physically, the Lagrangian algebra
represents the set of condensed anyons on the boundary, where a gapped
boundary is generally obtained by performing anyon condensation [18, 19]. The
obstruction to the gapped boundary is most generally understood as the absence
of the Lagrangian algebra.

Though it is not practically easy to see if a given general bosonic
topological order admits the Lagrangian algebra or not, it is known that the
obstructions to gapped boundary can be partially captured by easily computable
quantities called higher central charge $\xi_{n}$ labeled by a positive
integer $n$ [20, 21]. That is, for a given bosonic (2+1)D topologically
ordered state, one can compute these obstructions to gapped boundary by a
simple formula in terms of the properties of anyons,

$\displaystyle\xi_{n}=\frac{\sum_{a}d_{a}^{2}\theta_{a}^{n}}{|\sum_{a}d_{a}^{2}\theta_{a}^{n}|}$
(1.1)

where the sum is over all anyons in the topological order. $d_{a}$ is quantum
dimension, and $\theta_{a}$ is topological twist (i.e., self-statistics) of an
anyon $a$. When $n=1$, $\xi_{1}$ gives chiral central charge of the bosonic
topological phase modulo 8 [22],

$\displaystyle\xi_{1}=\frac{\sum_{a}d_{a}^{2}\theta_{a}}{|\sum_{a}d_{a}^{2}\theta_{a}|}=e^{\frac{2\pi
i}{8}c_{-}},$ (1.2)

$\\{\xi_{n}\\}$ hence provide higher generalizations of the chiral central
charge $c_{-}$. In general, one can only obtain $c_{-}$ mod 8 for a given data
of the anyons. This is because there exists a (2+1)D bosonic invertible phase
called $E_{8}$ state which does not carry anyons and has $c_{-}=8$ [23].
Hence, one can shift $c_{-}$ by integer multiple of 8 by stacking a copy of
$E_{8}$ phases on the topological order, without changing the data of anyons.
Regarding this ambiguity by $8\mathbb{Z}$, the formula (1.2) completely
determines $c_{-}$ mod 8 using the properties of anyons. The formula (1.2) is
sometimes called the Gauss-Milgram formula.

Not all $\xi_{n}$ for $n\in\mathbb{Z}$ correspond to obstructions to gapped
boundary. To be precise, [14] proved that $\xi_{n}=1$ for all $n$ such that
$\gcd(n,N_{\mathrm{FS}})=1$ give necessary conditions for admitting a gapped
boundary. Here, $N_{\mathrm{FS}}$ is called the Frobenius-Schur exponent,
which is defined as a smallest positive integer such that
$\theta_{a}^{N_{\mathrm{FS}}}=1$ for all anyons $a$. These quantities
$\\{\xi_{n}\\}$ for $n>1$ provide obstructions to gapped boundary beyond
$c_{-}$. For example, we have $\xi_{3}=-1$ for
$\mathrm{U}(1)_{2}\times\mathrm{U}(1)_{-4}$ Chern-Simons theory, which shows
that $\mathrm{U}(1)_{2}\times\mathrm{U}(1)_{-4}$ does not admit a gapped
boundary even though $c_{-}=0$.

In the presence of global symmetry, one can ask if a given (2+1)D topological
order admits a gapped boundary preserving the symmetry. In this paper, we
investigate obstructions to gapped boundaries in (2+1)D topological phases
with global symmetry, for both bosonic and fermionic phases. For bosonic
topological phases, we show that the global symmetry puts additional
constraints on the Lagrangian algebra required for preserving global symmetry,
which gives rise to further obstructions to gapped boundary. For example, the
electric Hall conductivity $\sigma_{H}$ is regarded as such an additional
obstruction that arises by enriching the phase with $\mathrm{U}(1)$ global
symmetry.

For fermionic topological phases, the algebraic formulation of gapped boundary
in terms of the Lagrangian algebra has not been developed yet, even in the
absence of the global symmetry. We first extend the formalism of the
Lagrangian algebra valid for fermionic topological phases, and then study its
symmetry enrichment mainly focusing on the U(1)f symmetry, i.e., U(1) symmetry
charging fermions. In particular, we derive new obstructions to symmetry-
preserving gapped boundary beyond $\sigma_{H}$ and $c_{-}$, for a fermionic
phase with U(1)f symmetry.

###  1.1 Summary of results

Here we summarize the results of the paper. First, we study the (2+1)D bosonic
topological phases with global symmetry $G$. In general, global symmetry of a
(2+1)D topological ordered phase is characterized by symmetry
fractionalization on anyons [24]. Roughly speaking, the symmetry
fractionalization means that the global symmetry acts projectively on anyons.
An example of symmetry fractionalization is found in FQH states where anyons
carry fractional charge under U(1) symmetry.

Then, we derive several constraints on the symmetry fractionalization data of
anyons required for the existence of symmetry-preserving gapped boundary. We
basically show that symmetry fractionalization data for Lagrangian algebra
anyons (i.e., condensed anyons on the boundary) must be trivial, in order to
realize a symmetry-preserving gapped boundary. For example, in the case of
U(1) symmetry, we show that the Lagrangian algebra anyons must carry trivial
fractional charge.

Based on the constraints on symmetry fractionalization data of the Lagrangian
algebra anyons, we can generally show that some specific set of anyons must be
condensed to realize a symmetry-preserving gapped boundary. For example, in
the case of U(1) symmetry, a special anyon $v$ called a vison must be
condensed. This in particular means that $v$ must be a boson, $\theta_{v}=1$.
Here, it is known that $\theta_{v}$ computes the electric Hall conductivity as
[25, 26, 27]

$\displaystyle\theta_{v}=e^{i\pi{\sigma}_{H}},$ (1.3)

where we define the Hall conductivity by the electromagnetic response locally
given by the Chern-Simons action $-\frac{\sigma_{H}}{4\pi}A\mathrm{d}A$.

So, the constraint $\theta_{v}=1$ corresponds to vanishing electric Hall
conductivity. As an application, we show that in (2+1)D bosonic Abelian
topological phases with U(1) symmetry, $\theta_{v}=1$ together with
$\xi_{n}=1$ for all $n$ such that
$\gcd(n,\frac{N_{\mathrm{FS}}}{\gcd(n,N_{\mathrm{FS}})})=1$ give necessary and
sufficient conditions for symmetry-preserving gapped boundary. This
generalizes the result in [14] for bosonic Abelian topological phases to the
case with U(1) global symmetry.

Next, we investigate gapped boundary condition for (2+1)D fermionic
topological phases. We first develop a general algebraic framework for gapped
boundary of fermionic topological phases, by extending the formalism of the
Lagrangian algebra anyon for bosonic phases to fermionic cases. 111In this
paper, we restrict ourselves to fermionic phases that no anyons $\sigma$
carrying vortex of $\mathbb{Z}_{2}^{f}$ fermion parity symmetry satisfy
$\sigma\times\psi=\sigma$, where $\psi$ is a fermion physically regarded as an
electron. When we are interested in fermionic phases with $\mathrm{U}(1)^{f}$
symmetry which is main interest in this paper, there exists no such anyons
since the assignment of U(1) charge on anyons cannot be consistent with the
fusion rule $\sigma\times\psi=\sigma$. So, this assumption does not lose
generality in that case. We also make a generalization of the formalism to the
case of symmetry-preserving gapped boundary, mainly focusing on the case with
$\mathrm{U}(1)^{f}$ symmetry.

As an application of the above formalism for fermionic phases, we propose new
obstructions to symmetry-preserving gapped boundary of (2+1)D fermionic
topological phases with $\mathrm{U}(1)^{f}$ symmetry, given by

$\displaystyle\zeta_{n}:=\frac{\sum_{a\in\mathcal{C}}e^{i\pi
Q_{a}}d_{a}^{2}\theta_{a}^{n}}{|\sum_{a\in\mathcal{C}}e^{i\pi
Q_{a}}d_{a}^{2}\theta_{a}^{n}|}$ (1.4)

where $Q_{a}$ is fractional charge of an anyon $a$, and $\mathcal{C}$ is a
super-modular tensor category that characterizes the properties of anyons in a
fermionic phase. We show that $\zeta_{n}=1$ for all $n$ such that
$\gcd(n,N_{\mathrm{FS}})=1$ give necessary conditions to symmetry-preserving
gapped boundary.

The above quantities $\\{\zeta_{n}\\}$ are regarded as higher versions of
$c_{-}$ and $\sigma_{H}$, in the sense that $\\{\zeta_{n}\\}$ provide
obstructions to symmetry-preserving gapped boundaries beyond $c_{-}$ and
$\sigma_{H}$. For $\mathrm{U}(1)^{f}$-symmetric fermionic topological phases,
$c_{-}$ and $\sigma_{H}$ are given by [27]

$\displaystyle e^{-2\pi
ic_{-}}=(\sqrt{2}\mathcal{D})^{8}\cdot\frac{\sum_{a\in\mathcal{C}}e^{3i\pi
Q_{a}}d_{a}^{2}\theta_{a}}{\left(\sum_{a\in\mathcal{C}}e^{i\pi
Q_{a}}d_{a}^{2}\theta_{a}\right)^{9}}$ (1.5) $\displaystyle e^{-2\pi
i{\sigma}_{H}}=\frac{\sum_{a\in\mathcal{C}}e^{3i\pi
Q_{a}}d_{a}^{2}\theta_{a}}{\sum_{a\in\mathcal{C}}e^{i\pi
Q_{a}}d_{a}^{2}\theta_{a}}$ (1.6)

and $\zeta_{n}$ cannot be expressed by any multiplication of $e^{-2\pi
ic_{-}}$ and $e^{-2\pi i\sigma_{H}}$. As we can see in (1.5) and (1.6), one
can only obtain $c_{-}$ and $\sigma_{H}$ mod 1 for a given data of super-
modular category. This is because there exists a (2+1)D fermionic invertible
phase characterized by Spinc Chern-Simons theory characterized by the theta
term [28]

$\displaystyle\int\left(\frac{2\pi}{192\pi^{2}}\mathrm{Tr}(R\wedge
R)-\frac{2\pi}{8\pi^{2}}F\wedge F\right)$ (1.7)

which carries $c_{-}=\sigma_{H}=1$.

###  1.2 Organization of paper

This paper is organized as follows. In Sec. 2, after reviewing an algebraic
formalism of gapped boundary of bosonic phases called the Lagrangian algebra
anyon, we make a generalization of the framework to symmetry-enriched
topological phases. We then derive a necessary and sufficient condition for
U(1)-preserving gapped boundary in the case of bosonic Abelian topological
order with U(1) symmetry. In Sec. 3, we move to fermionic topological phases,
and describe a generic theory for anyon condensation in fermionic non-Abelian
topological order. After studying various properties of fermionic gapped
boundary, we discuss the case with $\mathrm{U}(1)^{f}$ symmetry. In Sec. 4, we
derive obstructions to $\mathrm{U}(1)^{f}$-preserving gapped boundary of
fermionic phases with $\mathrm{U}(1)^{f}$ symmetry. Review of concepts used in
this paper and detailed calculations are relegated to appendices.

##  2 Gapped boundary of bosonic topological phases

###  2.1 Review: Lagrangian algebra anyons

We begin with a brief review for some properties of gapped boundary of a
bosonic topological phase without global symmetry. See [14] for detailed
descriptions.

Gapped boundary of a (2+1)D bosonic topological quantum field theory (TQFT) is
algebraically described by an object called a Lagrangian algebra anyon [14,
15, 16, 17]. The idea is that when a TQFT admits a topological gapped boundary
condition, we consider cutting out a solid cylinder from a spacetime
3-manifold. We introduce a gapped boundary condition on the boundary of the
resulting manifold, getting a cylinder of gapped boundary. We shrink the
radius of the cylinder of a gapped boundary, then it eventually becomes a
topological line operator of a TQFT, see Fig. 1. So, the tube of the gapped
boundary after shrinking is expressed as a sum of simple anyons in a modular
tensor category $\mathcal{C}$,

$\displaystyle\mathcal{L}=\bigoplus_{a\in\mathcal{C}}Z_{0a}a,$ (2.1)

with non-negative integers $Z_{0a}$. This object $\mathcal{L}$ is called a
Lagrangian algebra anyon. Since one can cap off the tube on the top of it and
introduce gapped boundary on the cap, the tube of a gapped boundary can end at
a point. It means that $\mathrm{Hom}(\mathcal{L},1)$ is not empty, and hence
$Z_{00}>0$. See Fig. 1. In general, the Lagrangian algebra anyon with
$Z_{00}>1$ is known to decompose into the sum of those with $Z_{00}=1$. The
simple gapped boundary condition is hence described by the Lagrangian algebra
anyon satisfying $Z_{00}=1$.

Figure 1: One can prepare a cylinder of a gapped boundary, then shrinking it
results in a line operator. One can cap off the cylinder by a gapped boundary,
so the line operator can end at a point, which means that
$\mathrm{Hom}(\mathcal{L},1)$ is not empty.

When $Z_{0a}>0$ for some anyon $a\in\mathcal{C}$,
$\mathrm{Hom}(\mathcal{L}\times a,\mathcal{L})$ is not empty since $Z_{00}>0$.
This implies that the Wilson line of $a$ can end on the tube of gapped
boundary, meaning that $a$ is condensed on the boundary. So, anyons with
$Z_{0a}>0$ is physically regarded as a set of condensed anyons.

$\mathcal{L}$ satisfies nice properties under modular $S,T$ transformations.
The vector $\\{Z_{0a}\\}$ turns out to be an eigenvector of modular $S$ and
$T$ matrices:

$\displaystyle\sum_{b\in\mathcal{C}}S_{ab}Z_{0b}=Z_{0a},\quad\sum_{b\in\mathcal{C}}T_{ab}Z_{0b}=Z_{0a}$
(2.2)

Since $S$ and $T$ of modular tensor category $(ST)^{3}=e^{\frac{2\pi
i}{8}c_{-}}S^{2}$, the existence of the Lagrangian algebra anyon with (2.2)
implies $c_{-}=0$ mod $8$.

We can consider a fusion space of the Lagrangian algebra anyon
$V^{\mathcal{L}\mathcal{L}}_{\mathcal{L}}$ by taking a junction of three tubes
of gapped boundary. We can then talk about the $F$\- and $R$-move of tubes
with junctions, which turn out to be trivial:

$\displaystyle(F^{{\mathcal{L}}{\mathcal{L}}{\mathcal{L}}}_{{\mathcal{L}}})_{{\mathcal{L}},{\mathcal{L}}}\cdot{\left|{{\mu}}\right>}\otimes{\left|{{\mu}}\right>}={\left|{{\mu}}\right>}\otimes{\left|{{\mu}}\right>}$
(2.3) $\displaystyle
R^{{\mathcal{L}}{\mathcal{L}}}_{\mathcal{L}}{\left|{\mu}\right>}={\left|{\mu}\right>}$
(2.4)

###  2.2 Symmetry-preserving gapped boundary

Here, we consider a (2+1)D TQFT with global symmetry $G$, and study gapped
boundary condition that preserves the global symmetry of the bulk. We assume
that the background gauge field is realized by a network of codimension-1
symmetry defects. A symmetry defect in the bulk can transversally end on the
gapped boundary at a line, and the ending line then defines a symmetry defect
of the gapped boundary theory.

First, we can cut out a tube from a 3-manifold and introduce the gapped
boundary condition on the boundary. We then consider a symmetry defect across
the carved tube, see Fig. 2. In this setup, since the boundary condition
should be invariant under the symmetry action, we require that the symmetry
action leaves the Lagrangian algebra anyon invariant,

$\displaystyle Z_{0a}=Z_{0,\rho_{\mathbf{g}}(a)}\quad\text{for $\mathbf{g}\in
G$}.$ (2.5)

Figure 2: A tube of gapped boundary passes through a symmetry defect.

To derive a further constraint on the Lagrangian algebra anyon, we argue that
gauge transformations of flat background $G$ gauge field in the bulk-boundary
system leave the partition function invariant. We verify this statement based
on an similar argument to Ref. [29], which showed that a theory must be free
of ’t Hooft anomaly if it admits a symmetry-preserving boundary condition.
Following Ref. [29], we put several axioms about the properties of background
gauge transformations in the bosonic phases:

  1. 1.

The partition function is invariant under smooth isotopies of symmetry
defects.

  2. 2.

The partition function is invariant under introducing or removing spherical
components of the symmetry defects whose interior does not contain any defects
or other operator insertions.

  3. 3.

We can perform a recombination of the symmetry defects by Pachner moves on
either bulk or boundary, and it only has the effect of shifting the partition
function by a phase.

Then, we think of creating a bubble of $G$ symmetry defects in 3D bulk given
in a following way. First, consider a 3-sphere $S^{3}$ composed of five
3-simplices, regarded as a boundary of a 4-simplex. The bubble of symmetry
defects is given by the Poincaré dual of flat background $G$-gauge field on
the triangulated $S^{3}$ (see Fig. 3), stereographically projected onto a
3-ball $D^{3}$. It can be checked that this bubble can be eliminated from the
bulk by performing a single Pachner move in 3D, up to other moves pushing the
diagram into the boundary without producing a phase [29].

Figure 3: Poincaré dual of background $G$-gauge field on a 3-simplex is
represented by yellow sheets, which are regarded as codimension-1 symmetry
defects. Background gauge field on a 3-simplex is characterized by group
elements $\mathbf{g}_{i}$ assigned on each 0-simplex $i$. That is, a defect on
the dual of a 1-simplex $\langle ij\rangle$ is given by
$\mathbf{g}_{i}^{-1}\mathbf{g}_{j}$. A 3-sphere $S^{3}$ consists of five
3-simplices regarded as a boundary of a 4-simplex, and a bubble of a symmetry
defect on $S^{3}$ is constructed by connecting up the symmetry defects on each
3-simplex along 2-simplices.

The phase produced by a Pachner move is expressed as a function
$\omega:G^{5}\to\mathrm{U}(1)$ where $G^{5}$ denotes group elements on five
0-simplices on $S^{3}$ that characterizes a background gauge fields on
$S^{3}$. $\omega$ turns out to take a value in a 4-cocycle
$Z^{4}(BG,\mathrm{U}(1))$, and characterizes a ’t Hooft anomaly of the (2+1)D
bulk [30]. Then, it has been shown in [29] that the ’t Hooft anomaly must be
trivial when the bulk admits a symmetry-preserving gapped boundary, so we set
$\omega=0$. It means that the bubble can be created without producing any
phase.

Then, we eliminate the bubble by pushing it into the boundary. It involves a
Pachner move on boundary, every time a codimension-3 junction of defects at
the center of a 3-simplex gets absorbed in the boundary, see Fig. 4. We write
the phase produced by a boundary Pachner move as
$\alpha:G^{4}\to\mathrm{U}(1)$, as a function of group labels on the four
vertices of a 3-simplex. The overall phase factor to eliminate the bubble is
given by $\delta\alpha$ evaluated on a 4-simplex.

Figure 4: At the center of a 3-simplex, there is a codimension-3 junction of
symmetry defects (a red starred point). When this junction gets absorbed by
the boundary, it causes the Pachner move of symmetry defects on the boundary.

Since $\delta\alpha=0$ due to the vanishing of ’t Hooft anomaly in the bulk,
$\alpha$ takes its value in $Z^{3}(BG,\mathrm{U}(1))$. When the gauge
transformation on the boundary shifts the partition function by a phase
$\alpha\in Z^{3}(BG,\mathrm{U}(1))$, one can cancel the phase ambiguity by
coupling the bulk with a (2+1)D bosonic SPT phase characterized by the
3-cocycle $\alpha$. This process makes the bulk-boundary system gauge
invariant without modifying the boundary condition, or the data of symmetry-
enriched topological phase in the bulk (i.e., anyons and their symmetry
fractionalization). Hence, one can assume without loss of generality that
gauge transformations of flat background $G$ gauge field in the bulk-boundary
system leave the partition function invariant.

Now, let us consider a junction of three symmetry defects
$\mathbf{g},\mathbf{h},\mathbf{gh}\in G$ in the 3-manifold, then carve out a
tube piercing the symmetry defects, as shown in Fig. 5. We introduce the
gapped boundary condition on the boundary of the tube, where symmetry defects
in 3D end on symmetry defects on the boundary theory.

Then, by performing background gauge transformation, one can move the junction
of symmetry defects across the tube. By shrinking the tube during the process
of the gauge transformations, one can see that symmetry fractionalization on
the Lagrangian algebra anyon is trivial (see Appendix B for a review of
symmetry fractionalization),

$\displaystyle\eta_{\mathcal{L}}(\mathbf{g},\mathbf{h})=1.$ (2.6)

Figure 5: A junction of symmetry defects can pass through a tube of gapped
boundary by background gauge transformation, realized by combination of a
couple of Pachner moves together with introducing a trivial spherical bubble
of a symmetry defect on the boundary.

Also, consider cutting out the junction of three tubes and introducing the
gapped boundary condition on its boundary, see Fig. 6. Then, by performing
background gauge transformation across the junction, we obtain

$\displaystyle U_{\mathbf{g}}(\mathcal{L},\mathcal{L};\mathcal{L})=1,$ (2.7)

which means that $G$ symmetry acts trivially on the fusion space
$V^{\mathcal{L},\mathcal{L}}_{\mathcal{L}}$.

Figure 6: A symmetry defect $\mathbf{g}\in G$ can pass through a junction of
gapped boundaries by background gauge transformation, realized by a single
Pachner move on the boundary.

In particular, suppose that we can fix a gauge as $U_{\mathbf{g}}(a,b;c)=1$
for any $a,b,c\in\mathcal{C}$ with $N^{ab}_{c}>0$ and $\mathbf{g}\in G$, and
consider a symmetry-preserving gapped boundary in this fixed gauge. We can
then express symmetry fractionalization as
$\eta_{a}(\mathbf{g},\mathbf{h})=M_{a,\mathfrak{t}(\mathbf{g},\mathbf{h})}$
using some Abelian anyon $\mathfrak{t}(\mathbf{g},\mathbf{h})\in\mathcal{C}$,
which gives an element of $Z^{2}_{\rho}(BG,\mathcal{B})$ where $\mathcal{B}$
is a group of Abelian anyons in $\mathcal{C}$. Then, by using the property
$SZ=Z$ of the Lagrangian algebra anyon we have

$\displaystyle\sum_{b\in\mathcal{A}}S_{\mathfrak{t}(\mathbf{g},\mathbf{h}),b}Z_{0b}=Z_{0,\mathfrak{t}(\mathbf{g},\mathbf{h})}$
(2.8)

where $\mathcal{A}$ is the set of anyons with $Z_{0a}>0$. Since
$S_{\mathfrak{t}(\mathbf{g},\mathbf{h}),b}=\frac{d_{b}M^{*}_{\mathfrak{t}(\mathbf{g},\mathbf{h}),b}}{\mathcal{D}}$
and
$M_{b,\mathfrak{t}(\mathbf{g},\mathbf{h})}=\eta_{b}(\mathbf{g},\mathbf{h})=1$
when $b\in\mathcal{A}$, (2.8) is rewritten as

$\displaystyle
Z_{0,\mathfrak{t}(\mathbf{g},\mathbf{h})}=\frac{1}{\mathcal{D}}\sum_{b\in\mathcal{A}}d_{b}Z_{0b}=Z_{00}>0,$
(2.9)

so we have $Z_{0,\mathfrak{t}(\mathbf{g},\mathbf{h})}>0$. Due to $TZ=Z$, we
can see that $\mathfrak{t}(\mathbf{g},\mathbf{h})$ must be a boson,
$\theta_{\mathfrak{t}(\mathbf{g},\mathbf{h})}=1$. More precisely, for a given
symmetry fractionalization class $[\mathfrak{t}]\in
H_{\rho}^{2}(BG,\mathcal{B})$, (2.6) should be understood as requiring the
existence of natural isomorphism that picks a specific cocycle representative
of $[\mathfrak{t}]$, so that $\eta_{a}(\mathbf{g},\mathbf{h})=1$ for
$a\in\mathcal{A}$. The statement
$\theta_{\mathfrak{t}(\mathbf{g},\mathbf{h})}=1$ is valid after performing
such a natural isomorphism so that $\eta$ satisfies (2.6).

Here we comment on the simplicity of the Lagrangian algebra anyon in the
presence of the global symmetry. Though a gapped boundary condition with
$Z_{00}>1$ always decomposes into simple ones with $Z_{00}=1$ in the absence
of the global symmetry, the symmetry-preserving boundary with $Z_{00}>0$ does
not necessarily decompose into simple symmetric ones. In fact, for a given
simple symmetry-breaking gapped boundary that violates (2.5), one can sum over
the orbit of $G$ symmetry actions on it to obtain a $G$-symmetric boundary
condition, which does not decompose into symmetry-preserving ones in general.
However, when $G$ does not permute the anyons where (2.5) is satisfied, the
constraints (2.6), (2.7) are obviously preserved for each decomposed component
of the Lagrangian algebra anyons. Hence, we conjecture that the symmetric
boundary condition with $Z_{00}>1$ always decomposes into simple symmetric
ones with $Z_{00}=1$ when $[\rho]$ is trivial.

###  2.3 Example: $G=\mathrm{U}(1)$

When $G=\mathrm{U}(1)$, the symmetry does not permute anyons. Symmetry
fractionalization is then characterized by an assignment of fractional charge
defined as follows. For a given set of $\\{\eta,U\\}$, one can define
fractional charges of anyons that characterize the U(1) symmetry
fractionalization as follows. For a fixed anyon $a$, let $n$ be the smallest
integer such that $a^{n}$ contains the identity as a fusion product. Choose a
sequence of anyons $a,a^{2},\dots a^{n}=1$ such that $a\times a^{k}$ contains
$a^{k+1}$ as a fusion product. Then define a fractional charge
$Q_{a}\in\mathbb{R}/\mathbb{Z}$ as [31]

$\displaystyle e^{2\pi
iQ_{a}}:=\prod_{m=1}^{n-1}\eta_{a}\left(\frac{1}{n},\frac{m}{n}\right)U_{\frac{1}{n}}(a,a^{m};a^{m+1}),$
(2.10)

where the elements of $\mathrm{U}(1)=\mathbb{R}/\mathbb{Z}$ is labeled by the
numbers in $[0,1)$. One can check that the quantity $e^{2\pi iQ_{a}}$ is
gauge-invariant. Since the map $[\rho]$ is trivial, we can fix a gauge where
$U=1$, 222For modular tensor categories with the trivial symmetry action
$[\rho]$, the existence of the gauge with $U=1$ is rigorously shown in Ref.
[26] when all fusion coefficients satisfy $N^{c}_{a,b}\in\\{0,1\\}$. For the
case with generic fusion coefficients, it is a conjecture given in Ref. [24].
so that $\eta$ satisfies ${\eta_{c}({\bf g},{\bf h})}=\eta_{a}({\bf g},{\bf
h})\eta_{b}({\bf g},{\bf h})$ when $N^{c}_{ab}>0$. One can then see that

$\displaystyle e^{2\pi iQ_{a}}e^{2\pi iQ_{b}}=e^{2\pi iQ_{c}}\quad\text{when
$N_{a,b}^{c}\neq 0$.}$ (2.11)

In the gauge where $U=1$, we can write the phases $\eta_{a}({\bf g},{\bf h})$
as

$\displaystyle\eta_{a}({\bf g},{\bf h})=M_{a,\mathfrak{t}({\bf g},{\bf h})},$
(2.12)

for Abelian anyon $\mathfrak{t}\in Z^{2}(B\mathrm{U}(1),\mathcal{B})$ where
$\mathcal{B}$ is the set of Abelian anyons in $\mathcal{C}$. A representative
2-cocycle $\mathfrak{t}$ is given by $\mathfrak{t}({\bf g},{\bf h})=v^{{\bf
g}+{\bf h}-[{\bf g}+{\bf h}]},$ where $v\in\mathcal{B}$ is referred to as the
vison, $\mathbf{g},\mathbf{h}\in\mathbb{R}/\mathbb{Z}$ takes the values in
$[0,1)$, and $[\mathbf{g}+\mathbf{h}]$ means the sum mod 1. Using the vison
$v$, the fractional charge can be rewritten as [32]

$\displaystyle e^{2\pi iQ_{a}}=M_{a,v}.$ (2.13)

Then, suppose that a bosonic TQFT with U(1) symmetry admits a U(1) symmetry-
preserving gapped boundary, after setting the gauge $U=1$ by a suitable
natural isomorphism. Then, (2.6) implies that

$\displaystyle e^{2\pi iQ_{a}}=1\quad\text{when $a\in\mathcal{A}$.}$ (2.14)

Hence, the vison $v$ has trivial mutual braiding with all $a\in\mathcal{A}$,
$M_{a,v}=1$. Using the property of Lagrangian algebra anyon $SZ=Z$ and
$S_{v,b}=\frac{d_{b}M^{*}_{v,b}}{\mathcal{D}}$, we obtain

$\displaystyle
Z_{0v}=\sum_{b\in\mathcal{A}}S_{v,b}Z_{0b}=\frac{1}{\mathcal{D}}\sum_{b\in\mathcal{A}}d_{b}Z_{0b}=Z_{00},$
(2.15)

so we have $Z_{0v}>0$. According to $TZ=Z$, we immediately see that the vison
$v$ must be a boson, $\theta_{v}=1$.

###  2.4 Application: bosonic Abelian TQFT with U(1) symmetry

As a straightforward application, we study an Abelian (2+1)D bosonic TQFT with
U(1) symmetry. We prove the following statement:

######  Theorem 1.

A $c_{-}=0$ bosonic Abelian TQFT with U(1) symmetry has a U(1) symmetric
gapped boundary if and only if the vison $v$ in (2.13) has $\theta_{v}=1$, and
the higher central charge $\xi_{n}$ defined as

$\displaystyle\xi_{n}:=\frac{\sum_{a\in\mathcal{C}}\theta_{a}^{n}}{|\sum_{a\in\mathcal{C}}\theta_{a}^{n}|}$
(2.16)

becomes 1 for all $n$ such that
$\gcd(n,\frac{N_{\mathrm{FS}}}{\gcd(n,N_{\mathrm{FS}})})=1$.

Here, $N_{\mathrm{FS}}$ is the Frobenius-Schur exponent defined as the
smallest positive integer such that $\theta_{a}^{N_{\mathrm{FS}}}=1$ for all
$a\in\mathcal{C}$.

###### Proof.

It was shown in [14] that the TQFT has a gapped boundary only if $\xi_{n}=1$
for all $n$ with $\gcd(n,\frac{N_{\mathrm{FS}}}{\gcd(n,N_{\mathrm{FS}})})=1$,
and we have shown that $\theta_{v}=1$ is necessary for symmetry-preserving
gapped boundary. So the “only if” part is done. We prove the “if” part. To see
this, we use the lemma shown in [14].

######  Lemma 2.

An Abelian TQFT $\mathcal{C}$ with Frobenius-Schur exponent $N_{\mathrm{FS}}$
admits a factorization

$\displaystyle\mathcal{C}=\mathcal{C}_{p_{1}}\times\mathcal{C}_{p_{2}}\times\dots\times\mathcal{C}_{p_{k}}$
(2.17)

where $\mathcal{C}_{p_{i}}$ are TQFTs labelled by distinct primes $p_{i}$,
such that the number of anyons in $\mathcal{C}_{p_{i}}$ is a positive integer
power of $p_{i}$. If we denote the Frobenius-Schur exponents of
$\mathcal{C}_{p_{i}}$ by $N_{i}$, then $N_{\mathrm{FS}}=N_{1}N_{2}\dots
N_{k}$. ∎

Assume that $\xi_{n}=1$ for all $n$ such that
$\gcd(n,\frac{N_{\mathrm{FS}}}{\gcd(n,N_{\mathrm{FS}})})=1$. This implies that
$\xi_{n}(\mathcal{C}_{p_{r}})=1$ for all $n$ such that $\gcd(n,p_{r})=1$ for
each $r$ [14]. Then, one can express the vison by fusion of anyons of each
factorized theory, as $v=v_{1}\times v_{2}\times\dots\times v_{k}$. For each
$v_{r}$, the twist has the form of $\theta_{v_{r}}=\exp(\frac{2\pi
is}{p_{i}^{t}})$ for some non-negative $s,t\in\mathbb{Z}$. Since
$\theta_{v}=1$, we have $\prod_{r}\theta_{v_{r}}=1$, and this implies that
$\theta_{v_{r}}=1$ for each $r$.

By using the vison, the fractional U(1) charge of an anyon $a$ can be
expressed as the mutual statistics between $a$ and $v$,

$\displaystyle e^{2\pi iQ_{a}}=M_{a,v}.$ (2.18)

From this, we see that $Q_{v_{r}}=0$ mod 1 because
$M_{v,v_{r}}=M_{v_{r},v_{r}}=\theta_{v_{r}}^{2}=1$. Since the bosons
$\\{v_{r}\\}$ have the trivial mutual statistics with each other, one can
condense the anyons $\\{v_{r}\\}$. This condensation satisfies the constraint
(2.6), and preserves the U(1) symmetry. This can be understood from QFT
perspective; the line of $v_{r}$ generates a 1-form symmetry which is free of
mixed ’t Hooft anomaly between the U(1) symmetry due to $Q_{v}=0$, then one
can gauge this non-anomalous 1-form symmetry without breaking the U(1)
symmetry.

The higher central charge $\xi_{n}(\mathcal{C}_{p_{r}})$ with
$\gcd(n,p_{r})=1$ is invariant under condensation of $v_{r}$. This can be
shown from the fact that $\xi_{n}$ is given by the phase of the partition
function of Reshetikhin-Turaev theory on the lens space $L(n;1)$ [33, 34, 14]
(without background U(1) gauge field),

$\displaystyle
Z_{\mathrm{RT}}(L(n;1);\mathcal{C}_{p_{r}})=\frac{1}{\mathcal{D}^{2}}\sum_{a\in\mathcal{C}_{p_{r}}}\theta_{a}^{n}.$
(2.19)

Since the lens space $L(n;1)$ has the trivial background gauge field for the
1-form symmetry generated by the line of $v_{r}$ when $\gcd(n,p_{r})=1$,
$Z_{\mathrm{RT}}(L(n;1);\mathcal{C}_{p_{r}})$ also gives the partition
function of the theory after condensation of $v_{r}$, which shows the
invariance of $\xi_{n}$ under condensation of $v_{r}$.

Let us write the theory after condensation of $\\{v_{r}\\}$ as

$\displaystyle\mathcal{C}^{\prime}=\mathcal{C}^{\prime}_{p_{1}}\times\mathcal{C}^{\prime}_{p_{2}}\times\dots\times\mathcal{C}^{\prime}_{p_{k}}.$
(2.20)

Due to the invariance of higher central charge under condensation of
$\\{v_{r}\\}$, we have $\xi_{n}(\mathcal{C}^{\prime}_{p_{r}})=1$ for all $n$
such that $\gcd(n,p_{r})=1$. This implies that each
$\mathcal{C}^{\prime}_{p_{r}}$ has a gapped boundary, as shown in [35]. After
condensing $v_{r}$, the anyons $a\in\mathcal{C}^{\prime}_{p_{r}}$ has trivial
U(1) charge $Q_{a}=0$. So any gapped boundary condition of
$\mathcal{C}^{\prime}_{p_{r}}$ satisfies the constraint (2.6) and preserves
U(1) symmetry, since gauging 1-form symmetry free of mixed ’t Hooft anomaly
does not break U(1) symmetry. Thus, we have shown that $\mathcal{C}^{\prime}$
admits a gapped boundary, which completes the proof. ∎

##  3 Gapped boundary of (2+1)D fermionic topological phases

###  3.1 Lagrangian algebra anyon for fermionic phases

We provide a generic theory for topological gapped boundary of (2+1)D
topological phases, which are effectively described by spin TQFT. For non-
Abelian spin TQFTs, we make a generalization of Lagrangian algebra anyon for
bosonic phases to fermionic cases, and express the gapped boundary condition
of spin TQFT in algebraic form. In general, (2+1)D spin TQFT is described in
terms of a super-modular tensor category $\mathcal{C}$ together with its
minimal modular extension $\breve{\mathcal{C}}$. The basic description of spin
TQFT are reviewed in Appendix C.

Let us assume that a spin TQFT has a gapped topological boundary condition.
For a given 3-manifold, one can cut out a solid torus from it, and equip the
resulting 3-manifold with a spin structure. We then introduce a gapped
boundary condition on the boundary torus $T^{2}$. We can think of shrinking
the radius of the tube of gapped boundary into a line, then the tube is
eventually described by a line operator of spin TQFT. The expression of the
line operator depends on the spin structure of the boundary torus $T^{2}$, and
generally described by an object of a minimal modular extension
$\breve{\mathcal{C}}$ of a super-modular tensor category $\mathcal{C}$.

Figure 7: One can shrink the tube of gapped boundary into a line, and the
resulting object depends on spin structure along the meridian.

We write spin structure on $T^{2}$ in the form of $T^{2}_{\mu,\lambda}$, where
$\mu$ is spin structure on the meridian of the tube, and $\lambda$ is that on
the longitude. We also write the minimal modular extension
$\breve{\mathcal{C}}=\breve{\mathcal{C}}_{\mathrm{NS}}\oplus\breve{\mathcal{C}}_{\mathrm{R}}$
with $\breve{\mathcal{C}}_{\mathrm{NS}}=\mathcal{C}$ the NS sector.

To make an algebraic description of gapped boundary, it is convenient to
consider a boundary state ${\left|{\mathcal{L}_{\mu,\lambda}}\right>}$ on a
space $T^{2}_{\mu,\lambda}$ given by considering a spacetime
$T^{2}_{\mu,\lambda}\times[0,1]$, and then introducing a gapped boundary
condition on $T^{2}_{\mu,\lambda}\times\\{1\\}$. This defines a state
${\left|{\mathcal{L}_{\mu,\lambda}}\right>}$ on
$T^{2}_{\mu,\lambda}\times\\{0\\}$. Equivalently, one can also regard the
geometry as a solid torus $D^{2}\times S^{1}$, with a thin solid torus cut out
at the center of $D^{2}$. So, the boundary state is thought of as an insertion
of a line operator in $D^{2}\times S^{1}$ along $S^{1}$ obtained by a thin
tube of gapped boundary.

Based on the description of Hilbert space for spin TQFT explained in Appendix
C, the boundary state with each spin structure $(\mu,\lambda)$ is expressed as
follows:

$\displaystyle\begin{split}{\left|{\mathcal{L}_{\mathrm{NS},\mathrm{NS}}}\right>}&=\sum_{[a]\in\breve{\mathcal{C}}_{\mathrm{NS}}/\\{1,\psi\\}}Z^{\mathrm{NS}}_{0[a]}({\left|{a}\right>}+{\left|{a\times\psi}\right>})\\\
{\left|{\mathcal{L}_{\mathrm{NS},\mathrm{R}}}\right>}&=\sum_{[a]\in\breve{\mathcal{C}}_{\mathrm{NS}}/\\{1,\psi\\}}Z^{\mathrm{NS}}_{0[a]}({\left|{a}\right>}-{\left|{a\times\psi}\right>})\\\
\end{split}$ (3.1)
$\displaystyle\begin{split}{\left|{\mathcal{L}_{\mathrm{R},\mathrm{NS}}}\right>}&=\sum_{[a]\in\breve{\mathcal{C}}_{\mathrm{R}}/\\{1,\psi\\}}Z^{\mathrm{R}}_{0[a]}({\left|{a}\right>}+{\left|{a\times\psi}\right>})\\\
{\left|{\mathcal{L}_{\mathrm{R},\mathrm{R}}}\right>}&=\sum_{[a]\in\breve{\mathcal{C}}_{\mathrm{R}}/\\{1,\psi\\}}Z^{\mathrm{R}}_{0[a]}({\left|{a}\right>}-{\left|{a\times\psi}\right>})\\\
\end{split}$ (3.2)

with $Z^{\mathrm{NS}}_{0[a]},Z^{\mathrm{R}}_{0[a]}$ set to be non-negative
integers. The reasoning for the expression of the state is that spin structure
along a closed curve is measured by inserting a Wilson line of the transparent
fermion $\psi$ of $\mathcal{C}$ along the curve. For example, the $\psi$ line
along the longitude acts by phase $+1$ (resp. $-1$) when spin structure along
the longitude is NS (resp. R), hence the state is expressed in the form of
${\left|{a}\right>}+{\left|{a\times\psi}\right>}$ (resp.
${\left|{a}\right>}-{\left|{a\times\psi}\right>}$).

For simplicity, we assumed that the R sector
$\breve{\mathcal{C}}_{\mathrm{R}}$ does not contain an anyon $\sigma$ with
fusion rule $\sigma\times\psi=\sigma$ (which is called a “q-type” object). For
example, when we are interested in spin TQFT with $\mathrm{U}(1)^{f}$ symmetry
which is main interest in this paper, there exists no q-type object in
$\breve{\mathcal{C}}_{\mathrm{R}}$ since the assignment of U(1) charge on
anyons cannot be consistent with the fusion rule $\sigma\times\psi=\sigma$.
So, this assumption does not lose generality in that case. The Hilbert space
in the presence of q-type objects are illustrated in [36], and it would be
interesting to study boundary conditions of such spin TQFTs.

###  3.2 Modular properties of gapped boundary

The boundary states (3.1), (3.2) have several important properties under the
$S,T$ transformation. Firstly, since the boundary is gapped and topological,
the diffeomorphism acting on the gapped boundary at
$T^{2}_{\mu,\lambda}\times\\{1\\}$ must not change the boundary state
${\left|{\mathcal{L}_{\mu,\lambda}}\right>}$. Since the torus is equipped with
spin structure, the diffeomorphism here means the mapping class group of
$T^{2}_{\mu,\lambda}$ leaving spin structure invariant.

For the case of $\mu=\mathrm{NS}$, the Dehn twist $T$ along the meridian
exchanges the spin structure as
$T^{2}_{\mathrm{NS},\mathrm{NS}}\leftrightarrow
T^{2}_{\mathrm{NS},\mathrm{R}}$, so we have
$T^{2}{\left|{\mathcal{L}_{\mathrm{NS},\mathrm{NS}}}\right>}={\left|{\mathcal{L}_{\mathrm{NS},\mathrm{NS}}}\right>}$.
This implies that an anyon that appears in the summand of
${\left|{\mathcal{L}_{\mathrm{NS},\mathrm{NS}}}\right>}$ must be either a
boson or a fermion, since Dehn twist acts on Wilson lines as
${\left|{a}\right>}\to\theta_{a}{\left|{a}\right>}$. For convenience, we write
$\mathcal{A}_{\mathrm{NS}}=\mathcal{A}_{\mathrm{NS}}^{1}\sqcup\mathcal{A}_{\mathrm{NS}}^{\psi}$
as a set of anyons that appear in ${\left|{\mathcal{L}_{\mathrm{NS}}}\right>}$
(i.e., all $a,a\times\psi\in\breve{\mathcal{C}}_{\mathrm{NS}}$ with
$Z_{0a}^{\mathrm{NS}}\neq 0$).
$\mathcal{A}_{\mathrm{NS}}^{1}\subset\mathcal{A}_{\mathrm{NS}}$ is the set of
bosons, and $\mathcal{A}_{\mathrm{NS}}^{\psi}\subset\mathcal{A}_{\mathrm{NS}}$
is the set of fermions. Note that
$\mathcal{A}_{\mathrm{NS}}^{1}\cdot\psi=\mathcal{A}_{\mathrm{NS}}^{\psi}$.

Because of
$T{\left|{\mathcal{L}_{\mathrm{NS},\mathrm{NS}}}\right>}={\left|{\mathcal{L}_{\mathrm{NS},\mathrm{R}}}\right>}$,
the sum over a class of anyons $[a]=\\{a,a\times\psi\\}$ in (3.1) is fixed to
be the sum over bosons $\mathcal{A}_{\mathrm{NS}}^{1}$. We then have

$\displaystyle\begin{split}{\left|{\mathcal{L}_{\mathrm{NS},\mathrm{NS}}}\right>}=\sum_{a\in\mathcal{A}_{\mathrm{NS}}^{1}}Z^{\mathrm{NS}}_{0a}({\left|{a}\right>}+{\left|{a\times\psi}\right>})\\\
{\left|{\mathcal{L}_{\mathrm{NS},\mathrm{R}}}\right>}=\sum_{a\in\mathcal{A}_{\mathrm{NS}}^{1}}Z^{\mathrm{NS}}_{0a}({\left|{a}\right>}-{\left|{a\times\psi}\right>})\\\
\end{split}$ (3.3)

where we define
$Z_{0a}^{\mathrm{NS}}=Z_{0,a\times\psi}^{\mathrm{NS}}=Z_{0[a]}^{\mathrm{NS}}$.

For the case of $\mu=\mathrm{R}$, the Dehn twist along the meridian leaves the
spin structure invariant, so we have
$T{\left|{\mathcal{L}_{\mathrm{R},*}}\right>}={\left|{\mathcal{L}_{\mathrm{R},*}}\right>}$.
This means that an anyon that appears in the summand of
${\left|{\mathcal{L}_{\mathrm{R},*}}\right>}$ must be a boson. For
convenience, we again write
$\mathcal{A}_{\mathrm{R}}=\mathcal{A}_{\mathrm{R}}^{e}\sqcup\mathcal{A}_{\mathrm{R}}^{m}$
as a set of anyons that appear in ${\left|{\mathcal{L}_{\mathrm{R}}}\right>}$
(i.e., all $a,a\times\psi\in\breve{\mathcal{C}}_{\mathrm{R}}$ with
$Z_{0a}^{\mathrm{R}}\neq 0$). The decomposition
$\mathcal{A}_{\mathrm{R}}^{e}\sqcup\mathcal{A}_{\mathrm{R}}^{m}$ is taken to
satisfy $\mathcal{A}_{\mathrm{R}}^{e}\times\psi=\mathcal{A}_{\mathrm{R}}^{m}$.
We note that there is no canonical way to perform the decomposition into
$\mathcal{A}_{\mathrm{R}}^{e}$ and $\mathcal{A}_{\mathrm{R}}^{m}$ unlike the
case of NS sector, since all anyons of $\mathcal{A}_{\mathrm{R}}$ are bosons.
We have

$\displaystyle\begin{split}{\left|{\mathcal{L}_{\mathrm{R},\mathrm{NS}}}\right>}&=\sum_{a\in\mathcal{A}_{\mathrm{R}}^{e}}Z^{\mathrm{R}}_{0a}({\left|{a}\right>}+{\left|{a\times\psi}\right>})\\\
{\left|{\mathcal{L}_{\mathrm{R},\mathrm{R}}}\right>}&=\sum_{a\in\mathcal{A}_{\mathrm{R}}^{e}}Z^{\mathrm{R}}_{0a}({\left|{a}\right>}-{\left|{a\times\psi}\right>})\\\
\end{split}$ (3.4)

where we define
$Z_{0a}^{\mathrm{R}}=Z_{0,a\times\psi}^{\mathrm{R}}=Z_{0[a]}^{\mathrm{R}}$.

Next, we discuss the property of the boundary states under the $S$
transformation. In the spacetime manifold $S^{3}$, let us consider a Hopf link
between the thin tube of gapped boundary and a line operator $x$, as described
in Fig. 8. This Hopf link can be evaluated in two ways. On one hand, the
amplitude of the Hopf link is computed by the modular $S$-matrix of a modular
category $\breve{\mathcal{C}}$ between two states on $T^{2}$, the boundary
state ${\left|{\mathcal{L}}\right>}$ and ${\left|{x}\right>}$ respectively. On
the other hand, one can enlarge the radius of the boundary tube, which makes
the geometry $D^{2}\times S^{1}$ with the gapped boundary on $T^{2}$. It then
reduces to a partition function on $D^{2}\times S^{1}$ with a $x$ line
inserted along $S^{1}$. Comparing these two distinct expressions put a
nontrivial constraint on the vectors $Z_{0a}$.

Figure 8: Hopf link between the tube $\mathcal{L}$ of a gapped boundary and an
anyon, which is topologically equivalent to the partition function on
$D^{2}\times S^{1}$.

The analysis of $S$-matrix action on boundary states requires a lengthy
discussion done by cases for spin structure of the spatial torus, and the
detail is relegated to Appendix D. Here we summarize the $S$-matrix actions on
$Z_{0a}$ obtained as follows:

$\displaystyle\begin{split}\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{1}}2S_{ab}Z_{0b}^{\mathrm{NS}}&=Z_{0a}^{\mathrm{NS}}\quad\text{for
$a\in\breve{\mathcal{C}}_{\mathrm{NS}}$,}\qquad\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{\psi}}2S_{ab}Z_{0b}^{\mathrm{NS}}=Z_{0a}^{\mathrm{NS}}\quad\text{for
$a\in\breve{\mathcal{C}}_{\mathrm{NS}}$}\end{split}$ (3.5)

$\displaystyle\begin{split}\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{1}}2S_{ab}Z_{0b}^{\mathrm{NS}}&=Z_{0a}^{\mathrm{R}}\quad\text{for
$a\in\breve{\mathcal{C}}_{\mathrm{R}}$,}\qquad\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{\psi}}2S_{ab}Z_{0b}^{\mathrm{NS}}=-Z_{0a}^{\mathrm{R}}\quad\text{for
$a\in\breve{\mathcal{C}}_{\mathrm{R}}$}\end{split}$ (3.6)

$\displaystyle\begin{split}\sum_{b\in\mathcal{A}_{\mathrm{R}}^{e}}2S_{ab}Z_{0b}^{\mathrm{R}}&=Z_{0a}^{\mathrm{NS}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{NS}}^{1}$},\qquad\sum_{b\in\mathcal{A}_{\mathrm{R}}^{e}}2S_{ab}Z_{0b}^{\mathrm{R}}=-Z_{0a}^{\mathrm{NS}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{NS}}^{\psi}$}\\\
\sum_{b\in\mathcal{A}_{\mathrm{R}}^{m}}2S_{ab}Z_{0b}^{\mathrm{R}}&=Z_{0a}^{\mathrm{NS}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{NS}}^{1}$}\qquad\sum_{b\in\mathcal{A}_{\mathrm{R}}^{m}}2S_{ab}Z_{0b}^{\mathrm{R}}=-Z_{0a}^{\mathrm{NS}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{NS}}^{\psi}$}\\\ \end{split}$ (3.7)

(3.7) also holds when $a$ is not an element of $\mathcal{A}_{\mathrm{NS}}$,
where both lhs and rhs becomes zero.

$\displaystyle\begin{split}\sum_{b\in\mathcal{A}_{\mathrm{R}}^{e}}2S_{ab}Z_{0b}^{\mathrm{R}}&=Z_{0a}^{\mathrm{R}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{R}}^{e}$,}\qquad\sum_{b\in\mathcal{A}_{\mathrm{R}}^{e}}2S_{ab}Z_{0b}^{\mathrm{R}}=-Z_{0a}^{\mathrm{R}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{R}}^{m}$}\\\
\sum_{b\in\mathcal{A}_{\mathrm{R}}^{m}}2S_{ab}Z_{0b}^{\mathrm{R}}&=-Z_{0a}^{\mathrm{R}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{R}}^{e}$,}\qquad\sum_{b\in\mathcal{A}_{\mathrm{R}}^{m}}2S_{ab}Z_{0b}^{\mathrm{R}}=Z_{0a}^{\mathrm{R}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{R}}^{m}$}\\\ \end{split}$ (3.8)

(3.8) also holds when $a$ is not an element of $\mathcal{A}_{\mathrm{R}}$,
where both lhs and rhs becomes zero.

###  3.3 Stacking with toric code $D(\mathbb{Z}_{2})$

In the above study of the $S$-matrix action on boundary states, we obtained a
bunch of equations in the form of $2SZ=\pm Z$ for each spin structure. We can
express the above many equations in a much simpler way, with help of toric
code $D(\mathbb{Z}_{2})$ with anyons $\\{1,\psi,e,m\\}$ and $S$-matrix
$S^{\mathrm{tc}}$. The above properties of the boundary states under the
action of $S$-matrix (3.5) (3.6) (3.7) (3.8) are then neatly summarized as

$\displaystyle\begin{split}\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{x}}4S_{ab}(S^{\mathrm{tc}})_{xy}Z_{0b}^{\mathrm{NS}}&=Z_{0a}^{\mathrm{NS}}\quad\text{for
$a\in\mathcal{A}_{\mathrm{NS}}^{y}$}\\\
\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{x}}4S_{ab}(S^{\mathrm{tc}})_{xy}Z_{0b}^{\mathrm{NS}}&=Z_{0a}^{\mathrm{R}}\quad\text{for
$a\in\mathcal{A}_{\mathrm{R}}^{y}$}\\\
\sum_{b\in\mathcal{A}_{\mathrm{R}}^{x}}4S_{ab}(S^{\mathrm{tc}})_{xy}Z_{0b}^{\mathrm{R}}&=Z_{0a}^{\mathrm{NS}}\quad\text{for
$a\in\mathcal{A}_{\mathrm{NS}}^{y}$}\\\
\sum_{b\in\mathcal{A}_{\mathrm{R}}^{x}}4S_{ab}(S^{\mathrm{tc}})_{xy}Z_{0b}^{\mathrm{R}}&=Z_{0a}^{\mathrm{R}}\quad\text{for
$a\in\mathcal{A}_{\mathrm{R}}^{y}$}\\\ \end{split}$ (3.9)

with $x,y\in D(\mathbb{Z}_{2})$. Then, let us define a new set of anyons
$\widetilde{\mathcal{A}}_{\mathrm{NS}}\subset\breve{\mathcal{C}}_{\mathrm{NS}}\boxtimes
D(\mathbb{Z}_{2})$ as

$\displaystyle\widetilde{\mathcal{A}}_{\mathrm{NS}}:=\\{(a,1)|a\in{\mathcal{A}}_{\mathrm{NS}}^{1}\\}\sqcup\\{(a,\psi)|a\in{\mathcal{A}}_{\mathrm{NS}}^{\psi}\\}$
(3.10)

Here $\boxtimes$ denotes the Deligne product, which physically means stacking
two independent theories.

We also define
$\widetilde{\mathcal{A}}_{\mathrm{R}}\subset\breve{\mathcal{C}}_{\mathrm{R}}\boxtimes
D(\mathbb{Z}_{2})$ as

$\displaystyle\widetilde{\mathcal{A}}_{\mathrm{R}}:=\\{(a,e)|a\in{\mathcal{A}}_{\mathrm{R}}^{e}\\}\sqcup\\{(a,m)|a\in{\mathcal{A}}_{\mathrm{R}}^{m}\\}$
(3.11)

We can then write the equations in terms of $S$-matrix of
$\breve{\mathcal{C}}\boxtimes D(\mathbb{Z}_{2})$ as

$\displaystyle\begin{split}\sum_{b\in\widetilde{\mathcal{A}}_{\mathrm{NS}}}4(S^{\breve{\mathcal{C}}\boxtimes
D(\mathbb{Z}_{2})})_{ab}Z_{0b}^{\mathrm{NS}}&=2Z_{0a}^{\mathrm{NS}}\quad\text{for
$a\in\tilde{\mathcal{A}}_{\mathrm{NS}}$}\\\
\sum_{b\in\widetilde{\mathcal{A}}_{\mathrm{NS}}}4(S^{\breve{\mathcal{C}}\boxtimes
D(\mathbb{Z}_{2})})_{ab}Z_{0b}^{\mathrm{NS}}&=2Z_{0a}^{\mathrm{R}}\quad\text{for
$a\in\tilde{\mathcal{A}}_{\mathrm{R}}$}\\\
\sum_{b\in\widetilde{\mathcal{A}}_{\mathrm{R}}}4(S^{\breve{\mathcal{C}}\boxtimes
D(\mathbb{Z}_{2})})_{ab}Z_{0b}^{\mathrm{R}}&=2Z_{0a}^{\mathrm{NS}}\quad\text{for
$a\in\tilde{\mathcal{A}}_{\mathrm{NS}}$}\\\
\sum_{b\in\widetilde{\mathcal{A}}_{\mathrm{R}}}4(S^{\breve{\mathcal{C}}\boxtimes
D(\mathbb{Z}_{2})})_{ab}Z_{0b}^{\mathrm{R}}&=2Z_{0a}^{\mathrm{R}}\quad\text{for
$a\in\tilde{\mathcal{A}}_{\mathrm{R}}$}\\\ \end{split}$ (3.12)

Finally, by writing the combined vector $\widetilde{Z}:=Z_{\mathrm{NS}}\oplus
Z_{\mathrm{R}}$ and
$\widetilde{\mathcal{A}}:=\widetilde{\mathcal{A}}_{\mathrm{NS}}\sqcup\widetilde{\mathcal{A}}_{\mathrm{R}}$,
we obtain

$\displaystyle\sum_{b\in\widetilde{\mathcal{A}}}(S^{\breve{\mathcal{C}}\boxtimes
D(\mathbb{Z}_{2})})_{ab}\widetilde{Z}_{0b}$
$\displaystyle=\widetilde{Z}_{0a}\quad\text{for
$a\in\widetilde{\mathcal{A}}$}$ (3.13)

and it is not hard to see that the lhs becomes zero when $a$ is not an element
of $\widetilde{\mathcal{A}}$. That is, the vector $\widetilde{Z}$ gives an
eigenvector of the $S$-matrix of $\breve{\mathcal{C}}\boxtimes
D(\mathbb{Z}_{2})$, $S\widetilde{Z}=\widetilde{Z}$. Since the anyons in
$\widetilde{\mathcal{A}}$ are all bosons, we also have
$T\widetilde{Z}=\widetilde{Z}$. So, a newly constructed object
$\widetilde{\mathcal{L}}:=\sum_{a\in\widetilde{\mathcal{A}}}\widetilde{Z}_{0a}a$
satisfies the crucial property for the Lagrangian algebra anyon of a bosonic
TQFT $\breve{\mathcal{C}}\boxtimes D(\mathbb{Z}_{2})$. Hence, we expect that
$\widetilde{Z}$ describes a bosonic gapped boundary of a bosonic modular
theory $\breve{\mathcal{C}}\boxtimes D(\mathbb{Z}_{2})$.

Physically, the gapped boundary of a fermionic theory $\mathcal{C}$ is
regarded as a gapped interface between $\mathcal{C}$ and a trivial fermionic
invertible phase. By gauging fermion parity $\mathbb{Z}_{2}^{f}$ of the whole
system in the presence of the gapped boundary, we obtain a bosonic gapped
interface between $\breve{\mathcal{C}}$ and a $\mathbb{Z}_{2}$ toric code
$D(\mathbb{Z}_{2})$, where the toric code $D(\mathbb{Z}_{2})$ corresponds to
gauging a trivial fermionic invertible phase. Since
$D(\mathbb{Z}_{2})=\overline{D(\mathbb{Z}_{2})}$, it implies the existence of
the gapped boundary for a bosonic theory $\breve{\mathcal{C}}\boxtimes
D(\mathbb{Z}_{2})$. In the next subsection, we confirm this physical intuition
by checking the other additional properties of Lagrangian algebra anyon.

###  3.4 Additional properties of Lagrangian algebra anyon

Here we study the additional constraint of the anyon
$\widetilde{\mathcal{L}}=\sum_{a\in\widetilde{\mathcal{A}}}\widetilde{Z}_{0a}a$
satisfied by Lagrangian algebra anyon represented in (2.3), (2.4). The first
constraint involves $F$-move of anyons, which is obtained by cutting out the
junction of tubes from the spacetime and introducing the gapped boundary, as
shown in Fig. 9. Note that each tube of gapped boundary is equipped with spin
structure along its meridian, and write the corresponding object as

$\displaystyle\mathcal{L}_{\mathrm{NS}}=\bigoplus_{a\in\mathcal{A}_{\mathrm{NS}}}Z_{0a}^{\mathrm{NS}}a,\quad\mathcal{L}_{\mathrm{R}}=\bigoplus_{a\in\mathcal{A}_{\mathrm{R}}}Z_{0a}^{\mathrm{R}}a$
(3.14)

Since the $F$-move can be realized by topological deformation of gapped
boundary, $F$-move acts trivially on the tube of gapped boundary. So, writing
${\left|{\mu^{pq}_{r}}\right>}\in
V^{\mathcal{L}_{p}\mathcal{L}_{q}}_{\mathcal{L}_{r}}$ as a fusion vector of
three tubes of gapped boundaries, we have

$\displaystyle(F^{\mathcal{L}_{p}\mathcal{L}_{q}\mathcal{L}_{r}}_{\mathcal{L}_{u}})_{\mathcal{L}_{s},\mathcal{L}_{t}}\cdot{\left|{\mu^{pq}_{s}}\right>}\otimes{\left|{\mu^{sr}_{u}}\right>}={\left|{\mu^{pt}_{u}}\right>}\otimes{\left|{\mu^{qr}_{t}}\right>}$
(3.15)

Since the $F$-matrices of $D(\mathbb{Z}_{2})$ is also completely trivial, for
a fusion vector ${\left|{\widetilde{\mu}}\right>}\in
V^{\widetilde{\mathcal{L}}\widetilde{\mathcal{L}}}_{\widetilde{\mathcal{L}}}$
we immediately have (2.3)

$\displaystyle(F^{\widetilde{\mathcal{L}}\widetilde{\mathcal{L}}\widetilde{\mathcal{L}}}_{\widetilde{\mathcal{L}}})_{\widetilde{\mathcal{L}},\widetilde{\mathcal{L}}}\cdot{\left|{\widetilde{\mu}}\right>}\otimes{\left|{\widetilde{\mu}}\right>}={\left|{\widetilde{\mu}}\right>}\otimes{\left|{\widetilde{\mu}}\right>}$
(3.16)

Figure 9: We carve out the junction of tubes, and then introduce gapped
boundary. The $F$-move of tubes is obviously trivial since it can be realized
by diffeomorphism.

The next constraint has to do with $R$-move of anyons, which is a bit more
involved. This constraint is obtained by considering a “half-twist” of the
junction of gapped boundaries as shown in Fig. 10. This process amounts to
twisting each tube of gapped boundary by $\pi$ along the meridian. We write
the $\pi$ twist as $T^{\frac{1}{2}}$ since performing it twice gives a single
Dehn twist $T$. $T^{\frac{1}{2}}$ acts diagonally on anyons, and has the form
of

$\displaystyle\begin{split}(T^{\frac{1}{2}})_{a,a}&=1\quad\text{for
$a\in\mathcal{A}_{\mathrm{NS}}^{1},\mathcal{A}_{\mathrm{R}}^{e},\mathcal{A}_{\mathrm{R}}^{m}$}\\\
(T^{\frac{1}{2}})_{a,a}&=i\quad\text{for
$a\in\mathcal{A}_{\mathrm{NS}}^{\psi}$}\\\ \end{split}$ (3.17)

Figure 10: $R$-move for a junction of tubes. One can realize the $R$-move by a
diffeomorphism up to a “half-twist” on each tube, represented by $\pi$
rotation of a red line along its meridian.

By writing the objects for gapped boundary as
$\mathcal{L}_{\mathrm{NS}}=\mathcal{L}_{\mathrm{NS}}^{1}+\mathcal{L}_{\mathrm{NS}}^{\psi}$
and
$\mathcal{L}_{\mathrm{R}}=\mathcal{L}_{\mathrm{R}}^{e}+\mathcal{L}_{\mathrm{R}}^{m}$
with

$\displaystyle\mathcal{L}^{1}_{\mathrm{NS}}=\bigoplus_{a\in\mathcal{A}^{1}_{\mathrm{NS}}}Z_{0a}^{\mathrm{NS}}a,\quad\mathcal{L}^{\psi}_{\mathrm{NS}}=\bigoplus_{a\in\mathcal{A}_{\mathrm{NS}}^{\psi}}Z_{0a}^{\mathrm{NS}}a$
(3.18)
$\displaystyle\mathcal{L}^{e}_{\mathrm{R}}=\bigoplus_{a\in\mathcal{A}^{e}_{\mathrm{R}}}Z_{0a}^{\mathrm{R}}a,\quad\mathcal{L}^{m}_{\mathrm{R}}=\bigoplus_{a\in\mathcal{A}_{\mathrm{R}}^{m}}Z_{0a}^{\mathrm{R}}a$
(3.19)

the action of $R$-move on the fusion vector ${\left|{\mu^{xy}_{z}}\right>}$
for $x,y,z\in D(\mathbb{Z}_{2})$ is then expressed as

$\displaystyle
R^{\mathcal{L}^{x}\mathcal{L}^{y}}_{\mathcal{L}^{z}}{\left|{\mu^{xy}_{z}}\right>}=(T^{-\frac{1}{2}})_{x,x}(T^{-\frac{1}{2}})_{y,y}(T^{\frac{1}{2}})_{z,z}{\left|{\mu^{xy}_{z}}\right>}$
(3.20)

One can see that this gives the same $R$ matrix as that of
$\overline{D(\mathbb{Z}_{2})}$,
$R^{\mathcal{L}^{x}\mathcal{L}^{y}}_{\mathcal{L}^{z}}=(R^{xy}_{z})^{*}$ by
picking a proper $R^{xy}_{z}$ of $D(\mathbb{Z}_{2})$ by suitable vertex basis
transformation (A.1). Hence, one can see that the $R$-matrix element of
$\widetilde{\mathcal{L}}$ for $\breve{\mathcal{C}}\boxtimes D(\mathbb{Z}_{2})$
is trivial and (2.4) follows,

$\displaystyle
R^{\widetilde{\mathcal{L}}^{x}\widetilde{\mathcal{L}}^{y}}_{\widetilde{\mathcal{L}}^{z}}{\left|{\widetilde{\mu}^{xy}_{z}}\right>}=R^{\mathcal{L}^{x}\mathcal{L}^{y}}_{\mathcal{L}^{z}}R^{xy}_{z}{\left|{\widetilde{\mu}^{xy}_{z}}\right>}={\left|{\widetilde{\mu}^{xy}_{z}}\right>}$
(3.21)

###  3.5 U(1)f symmetry-preserving gapped boundary

In this section, we study symmetry-preserving gapped boundary of (2+1)D spin
TQFT. For simplicity, we focus on global symmetry $G_{f}=\mathrm{U}(1)^{f}$
given by the symmetry extension

$\displaystyle\mathbb{Z}_{2}^{f}\to G_{f}\to G_{b},$ (3.22)

where $G_{b}=\mathrm{U}(1)$ is the bosonic symmetry group. Note that
$G_{b}=\mathbb{R}/(\mathbb{Z}/2)$ has periodicity 1/2, since we want the
fermionic group $G_{f}=\mathbb{R}/\mathbb{Z}$ to have periodicity 1.

As reviewed in Appendix B.3, in order to describe a spin TQFT with $G_{f}$
symmetry, we first take $G_{b}$ symmetry on the bosonic theory given by a
minimal modular extension
$\breve{\mathcal{C}}=\breve{\mathcal{C}}_{\mathrm{NS}}\oplus\breve{\mathcal{C}}_{\mathrm{R}}$
as an input. In general, a spin TQFT is constructed by performing a process of
“fermion condensation” for a given bosonic TQFT $\breve{\mathcal{C}}$, see
Appendix C for a review of spin TQFT. This process corresponds to gauging
1-form symmetry generated by a Wilson line of a transparent fermion $\psi$,
and the resulting theory has a dual $\mathbb{Z}_{2}^{f}$ fermion parity
symmetry. So, after fermion condensation starting with the bosonic theory
$\breve{\mathcal{C}}$, we obtain a spin TQFT with $G_{f}$ symmetry.

We demand that [31, 37] (see Appendix B.3 for an explanation)

$\displaystyle U_{\bf g}(\psi,\psi;1)$ $\displaystyle=1$ (3.23)
$\displaystyle\eta_{\psi}({\bf g},{\bf h})$ $\displaystyle=\omega_{2}({\bf
g},{\bf h})$ (3.24)

where $[\omega_{2}]\in H^{2}(BG_{b},\mathbb{Z}_{2})$ is the cohomology class
specifying $G_{f}$ as a group extension of $G_{b}$ by $\mathbb{Z}_{2}$.

The $G_{b}$ action of $\breve{\mathcal{C}}$ is realized as a topological
symmetry of BTC which preserves the $\mathbb{Z}_{2}$ grading
$\breve{\mathcal{C}}=\breve{\mathcal{C}}_{\mathrm{NS}}\oplus\breve{\mathcal{C}}_{\mathrm{R}}$.
So, it induces a topological symmetry of the NS sector
$\breve{\mathcal{C}}_{\mathrm{NS}}=\mathcal{C}$. We can define fractional
charges of anyons in $\mathcal{C}$ analogously to the bosonic case. For a
fixed anyon $a$, let $n$ be the smallest integer such that $a^{n}$ contains
the identity as a fusion product. Choose a sequence of anyons $a,a^{2},\dots
a^{n}=1$ such that $a\times a^{k}$ contains $a^{k+1}$ as a fusion product.
Then define a fractional charge $Q_{a}\in\mathbb{R}/(2\mathbb{Z})$ as [31]

$\displaystyle e^{\pi
iQ_{a}}:=\prod_{m=1}^{n-1}\eta_{a}\left(\frac{1}{2n},\frac{m}{2n}\right)U_{\frac{1}{2n}}(a,a^{m};a^{m+1}),$
(3.25)

where the elements of $G_{b}$ is labeled by numbers in $[0,1/2)$. Since
$[\rho]$ is the identity map, there is a gauge in which we can set $U=1$, so
that $\eta$ satisfies ${\eta_{c}({\bf g},{\bf h})}=\eta_{a}({\bf g},{\bf
h})\eta_{b}({\bf g},{\bf h})$ when $N^{c}_{ab}>0$. Then one can see that
$e^{\pi iQ_{a}}$ satisfies

$\displaystyle e^{\pi iQ_{a}}e^{\pi iQ_{b}}=e^{\pi iQ_{c}}\quad\text{when
$N_{ab}^{c}\neq 0$}.$ (3.26)

Since $\omega_{2}({\bf g},{\bf h})=e^{2\pi
i(\mathbf{g}+\mathbf{h}-[\mathbf{g}+\mathbf{h}]_{1/2})}$, the fractional
charge of a transparent fermion $\psi$ satisfies

$\displaystyle Q_{\psi}=1\mod 2.$ (3.27)

Since $\mathrm{U}(1)$ symmetry does not permute anyons we can set the gauge
$U=1$, then we can write the phases $\eta_{a}({\bf g},{\bf h})$ as

$\displaystyle\eta_{a}({\bf g},{\bf h})=M_{a,\mathfrak{t}({\bf g},{\bf h})},$
(3.28)

for an Abelian anyon $\mathfrak{t}({\bf g},{\bf h})\in
Z^{2}(B\mathrm{U}(1),\mathcal{B})$ where $\mathcal{B}$ is the set of Abelian
anyons in $\breve{\mathcal{C}}$. A representative 2-cocycle $\mathfrak{t}$ is
given by $\mathfrak{t}({\bf g},{\bf h})=v^{2({\bf g}+{\bf h}-[{\bf g}+{\bf
h}]_{1/2})},$ where $v\in\mathcal{B}$ is referred to as the vison,
$\mathbf{g},\mathbf{h}\in\mathbb{R}/(\mathbb{Z}/2)$ takes the values in
$[0,1/2)$, and $[\mathbf{g}+\mathbf{h}]_{1/2}$ means the sum mod 1/2. In
particular, since we have $M_{\psi,v}=-1$ in order to satisfy (3.24), the
vison must be in the R sector $v\in\breve{\mathcal{C}}_{\mathrm{R}}$. Using
the vison, the fractional charge (3.25) can be expressed as

$\displaystyle e^{\pi iQ_{a}}=M_{a,v}.$ (3.29)

Now we are prepared to discuss the symmetry-preserving gapped boundary of spin
TQFT with U(1)f symmetry. Analogously to the bosonic case in Sec. 2.2, we
expect that the partition function of the bulk-boundary system without
operator insertions is invariant under background gauge transformations, since
it is expected that both the bulk and boundary cannot have ’t Hooft anomaly in
order to admit a symmetry-preserving gapped boundary [29]. Though we are not
aware of a rigorous argument for this in fermionic case due to the presence of
subtle anomaly beyond group cohomology intrinsic in fermionic systems [38, 39,
40, 41, 42], here we assume that the partition function is invariant under
background $G_{f}$ gauge transformations.

Then, similar to what we have done for bosonic case in Sec. 2.2, we think of
carving out a solid torus from a spacetime 3-manifold, and introduce a gapped
boundary condition on the boundary of the solid torus. We consider a junction
of symmetry defects $\mathbf{g},\mathbf{h},\mathbf{g}+\mathbf{h}$ piercing the
solid torus, see Fig. 11 When the symmetry defect ends on boundary, it
realizes a symmetry defect of a gapped boundary theory.

By performing background gauge transformations, one can move the symmetry
defects away from the tube of gapped boundary without producing a phase. By
comparing the configuration before and after the gauge transformation, we get
a constraint on the symmetry fractionalization data of the Lagrangian algebra
anyon. In particular, consider a flat U(1)f background gauge field realized by
a junction of three defects, where
$\mathbf{g}+\mathbf{h}-[\mathbf{g}+\mathbf{h}]=1/2$. In that case, due to the
symmetry extension $\mathbb{Z}_{2}^{f}\to\mathrm{U}(1)^{f}\to\mathrm{U}(1)$,
the junction bounds a vortex for fermion parity $\mathbb{Z}_{2}^{f}$ symmetry,
hence the tube of gapped boundary before and after gauge transformation carry
a different spin structure along the longitude. That is, suppose that the
initial tube has $(\mu,\lambda)$=(NS,R), then the final tube has (NS,NS).

Figure 11: Insertion of a tube of gapped boundary through symmetry defects
with a junction. By background gauge transformation, one can move the junction
away from the tube, which shifts spin structure of the tube along its
longitude.

By shrinking the tube of gapped boundary, it follows that for the Lagrangian
algebra anyon in NS sector
$\mathcal{L}_{\mathrm{NS}}^{1}+\mathcal{L}_{\mathrm{NS}}^{\psi}$, moving the
tube across the junction of defects has the effect of acting on
$\mathcal{L}_{\mathrm{NS}}^{\psi}$ by $-1$ phase, while acting on
$\mathcal{L}_{\mathrm{NS}}^{1}$ trivially. Therefore, we conclude that
$\eta_{a}(\mathbf{g},\mathbf{h})$ has the form of

$\displaystyle\eta_{\mathcal{L}_{\mathrm{NS}}^{1}}(\mathbf{g},\mathbf{h})=1,\quad\eta_{\mathcal{L}_{\mathrm{NS}}^{\psi}}(\mathbf{g},\mathbf{h})=\omega_{2}(\mathbf{g},\mathbf{h}).$
(3.30)

In the gauge where $U=1$, it implies that the fractional charge is trivial for
anyons in $\mathcal{A}_{\mathrm{NS}}^{1}$,

$\displaystyle e^{\pi iQ_{a}}=M_{a,v}=1\quad\text{when
$a\in\mathcal{A}_{\mathrm{NS}}^{1}$}.$ (3.31)

We can then show that the vison is condensable $v\in\mathcal{A}_{\mathrm{R}}$,
by using the modular property of the Lagrangian algebra anyon (3.6) and
$S_{v,b}=\frac{d_{b}M^{*}_{v,b}}{\mathcal{D}}$,

$\displaystyle
Z_{0v}^{\mathrm{R}}=\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{1}}2S_{v,b}Z_{0b}^{\mathrm{NS}}=\frac{1}{\mathcal{D}}\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{1}}2d_{b}Z_{0b}^{\mathrm{NS}}=Z_{00}^{\mathrm{NS}}>0.$
(3.32)

Here, $Z_{00}^{\mathrm{NS}}>0$ follows from the observation that the gapped
boundary on a $S^{2}$ is topologically deformed to a thin tube with NS spin
structure ending on the north and south pole, which means that
$\mathrm{Hom}(\mathcal{L}_{\mathrm{NS}},1)$ is non-vanishing. Since the anyons
in $\mathcal{A}_{\mathrm{R}}$ are all bosons as shown in Sec. 3.2, the vison
must be a boson, $\theta_{v}=1$.

##  4 Higher version of $c_{-}$ and Hall conductivity

In this section, we focus on fermionic topological phases with U(1)f symmetry.
We prove the following statement:

######  Theorem 3.

Fermionic topological phase with U(1)f symmetry described by a super-modular
category $\mathcal{C}$ has a U(1)f symmetric gapped boundary only if the
quantity $\zeta_{n}$ defined as

$\displaystyle\zeta_{n}:=\frac{\sum_{a\in\mathcal{C}}e^{i\pi
Q_{a}}d_{a}^{2}\theta_{a}^{n}}{|\sum_{a\in\mathcal{C}}e^{i\pi
Q_{a}}d_{a}^{2}\theta_{a}^{n}|}$ (4.1)

becomes 1 for all $n$ such that $\gcd(n,N_{\mathrm{FS}})=1$.

Here, $N_{\mathrm{FS}}$ is the Frobenius-Schur exponent of super-modular
tensor category $\mathcal{C}$.

###### Proof.

For a given super-modular category $\mathcal{C}$, we pick a minimal modular
extension $\breve{\mathcal{C}}$ where symmetry fractionalization data of
$\mathcal{C}$ lifts to $\mathrm{U}(1)$ symmetry fractionalization of
$\breve{\mathcal{C}}$, which is guaranteed to exist as discussed in Sec. 3.5.

Let $\mathcal{B}$ be the group of Abelian anyons in $\mathcal{C}$. We firstly
show that for any $a\in\mathcal{B}$, $N_{\mathrm{FS}}$ is integer multiple of
$N_{a}$, where we define $N_{a}$ as the smallest integer that $a^{N_{a}}=1$.
We prove this statement by contradiction. Suppose that there exists an Abelian
anyon $a\in\mathcal{B}$ such that $N_{a}$ does not divide $N_{\mathrm{FS}}$.
Note that $\theta_{a\times x}=\theta_{a}\theta_{x}M_{a,x}$ for any
$x\in\mathcal{C}$, which means that $M_{a,x}$ is $N_{\mathrm{FS}}$-th root of
unity. At the same time, $M_{a,x}$ is $N_{a}$-th root of unity since
$M_{a,x}^{N_{a}}=M_{a^{N_{a}},x}=1$, so
$M_{a,x}^{\gcd(N_{\mathrm{FS}},N_{a})}=M_{a^{\gcd(N_{\mathrm{FS}},N_{a})},x}=1$
for all $x\in\mathcal{C}$.

Since $N_{a}$ does not divide $N_{\mathrm{FS}}$,
$\gcd(N_{\mathrm{FS}},N_{a})<N_{a}$ and $a^{\gcd(N_{\mathrm{FS}},N_{a})}\neq
1$. Hence, due to super-modularity, we must have
$a^{\gcd(N_{\mathrm{FS}},N_{a})}=\psi$. This means that
$\gcd(N_{\mathrm{FS}},N_{a})=N_{a}/2$, so $N_{\mathrm{FS}}$ is odd multiple of
$N_{a}/2$, and hence $\gcd(N_{\mathrm{FS}},2N_{a})=N_{a}/2$. Since
$\theta_{a}^{2N_{a}}=M_{a,a}^{N_{a}}=1$, $\theta_{a}$ is
$\gcd(N_{\mathrm{FS}},2N_{a})=N_{a}/2$-th root of unity, so
$\theta_{a^{N_{a}/2}}=(\theta_{a})^{(\frac{N_{a}}{2})^{2}}=1$. This
contradicts with $a^{\gcd(N_{\mathrm{FS}},N_{a})}=\psi$, so $N_{\mathrm{FS}}$
must be an integer multiple of $N_{a}$.

For a vison $v\in\breve{\mathcal{C}}_{\mathrm{R}}$ in (3.29), let $N_{v}$ be
the smallest integer satisfying $v^{N_{v}}=1$. Since $v^{2}\in\mathcal{B}$,
$N_{v}$ is expressed as $N_{v}=2N_{v^{2}}$ where $N_{v^{2}}$ divides
$N_{\mathrm{FS}}$. When $n$ satisfies $\gcd(n,N_{\mathrm{FS}})=1$, we also
have $\gcd(n,N_{v})=1$, because $n$ must be odd since $N_{\mathrm{FS}}$ is
even due to the presence of the fermion $\psi$, and $\gcd(n,N_{v^{2}})=1$.
Then, there exists an Abelian anyon $\tilde{v}$ that satisfies
$v=\tilde{v}^{n}$, by using an integer $m$ such that $mn=1\mod N_{v}$ and
setting $\tilde{v}=v^{m}$. Then, $\zeta_{n}$ is expressed by

$\displaystyle\begin{split}{\sum_{a\in\mathcal{C}}e^{i\pi
Q_{a}}d_{a}^{2}\theta_{a}^{n}}&=\sum_{a\in\breve{\mathcal{C}}}\frac{(M_{v,a}+M_{v\times\psi,a})}{2}d_{a}^{2}\theta_{a}^{n}\\\
&=\sum_{a\in\breve{\mathcal{C}}}\frac{(M_{\tilde{v}^{n},a}+M_{(\tilde{v}\times\psi)^{n},a})}{2}d_{a}^{2}\theta_{a}^{n}\\\
&=\frac{(\theta_{\tilde{v}})^{-n}+(\theta_{\tilde{v}\times\psi})^{-n}}{2}\sum_{a\in\breve{\mathcal{C}}}d_{a}^{2}\theta_{a}^{n}\\\
&=(\theta_{\tilde{v}})^{-n}\sum_{a\in\breve{\mathcal{C}}}d_{a}^{2}\theta_{a}^{n}\\\
&=(\theta_{v})^{-nm^{2}}\sum_{a\in\breve{\mathcal{C}}}d_{a}^{2}\theta_{a}^{n}\end{split}$
(4.2)

When a spin TQFT admits a U(1) symmetric gapped boundary, we must have
$\theta_{v}=1$ as shown in Sec. 3.5. Also, the presence of gapped boundary in
spin TQFT implies that a bosonic theory $\breve{\mathcal{C}}\boxtimes
D(\mathbb{Z}_{2})$ has a Lagrangian algebra anyon, as shown in Sec. 3.3. Now,
$\sum_{a\in\breve{\mathcal{C}}}d_{a}^{2}\theta_{a}^{n}$ is higher central
charge of a bosonic modular theory $\breve{\mathcal{C}}$, and must have a
trivial phase for all $n$ with
$\gcd(n,N_{\mathrm{FS}}(\breve{\mathcal{C}}))=1$, when
$\breve{\mathcal{C}}\boxtimes D(\mathbb{Z}_{2})$ has a Lagrangian algebra
anyon [14]. Here, $N_{\mathrm{FS}}(\breve{\mathcal{C}})$ is Frobenius-Schur
exponent of $\breve{\mathcal{C}}$. So, we have shown that $\zeta_{n}=1$ for
all $n$ with $\gcd(n,N_{\mathrm{FS}}(\breve{\mathcal{C}}))=1$.

Finally, we show that $\gcd(n,N_{\mathrm{FS}}(\breve{\mathcal{C}}))=1$ is
equivalent to $\gcd(n,N_{\mathrm{FS}}({\mathcal{C}}))=1$. This can be checked
as follows. Since $v^{2}\in\mathcal{B}$, $N_{v}$ is expressed in the form of
$N_{v}=2^{r}\times q$ with $r\geq 1$ and odd number $q$. Consider
$x:=v^{2^{r}}\in\mathcal{B}$ which has order $q$. Then, there exists an anyon
$y\in\mathcal{B}$ such that $y^{2^{r}}=x$, given by preparing an integer $m$
such that $m\cdot 2^{r}=1$ mod $q$ and setting $y:=x^{m}\in\mathcal{B}$. Then
we can define $v_{0}:=v\times y^{*}\in\breve{\mathcal{C}}_{\mathrm{R}}$, which
satisfies $v_{0}^{2^{r}}=1$. Then, all anyons
$x\in\breve{\mathcal{C}}_{\mathrm{R}}$ is expressed as $x=a\times v_{0}$ with
some $a\in\mathcal{C}$, so $\theta_{x}=\theta_{a}\theta_{v_{0}}M_{a,v_{0}}$.
$N_{\mathrm{FS}}(\breve{\mathcal{C}})$ is then expressed as
$N_{\mathrm{FS}}(\breve{\mathcal{C}})=N_{\mathrm{FS}}(\mathcal{C})\cdot 2^{t}$
for some integer $t\geq 0$. Since $N_{\mathrm{FS}}(\mathcal{C})$ is even, the
condition $\gcd(n,N_{\mathrm{FS}}(\breve{\mathcal{C}}))=1$ is equivalent to
$\gcd(n,N_{\mathrm{FS}}({\mathcal{C}}))=1$. ∎

Before closing this section, let us explicitly check with an example that the
quantity $\zeta_{n}$ gives an obstruction beyond the chiral central charge
$c_{-}$ and Hall conductivity $\sigma_{H}$ given by Eq. (1.5) and Eq. (1.6).
The simplest example that demonstrates this is a spin TQFT described by a
super-modular category
$\mathcal{C}=\mathrm{U}(1)_{2}\times\mathrm{U}(1)_{-4}\times\\{1,\psi\\}$,
where $\psi$ is a transparent fermion. Let us define $\mathrm{U}(1)^{f}$
symmetry fractionalization by picking up a vison $v$ in R sector, such that
$v$ is a boson. For example, we take the R sector as
$\breve{\mathcal{C}}_{\mathrm{R}}=\mathrm{U}(1)_{2}\times\mathrm{U}(1)_{-4}\times\\{e,m\\}$
with $e,m$ bosons in the $\mathbb{Z}_{2}$ toric code
$D(\mathbb{Z}_{2})=\\{1,\psi,e,m\\}$, and then define $v=e$. Then, one can see
that both $c_{-}$ and $\sigma_{H}$ are trivial: $e^{2\pi ic_{-}}=e^{2\pi
i\sigma_{H}}=1$, while the higher version $\zeta_{n}$ is nontrivial, e.g.,
$\zeta_{3}=-1$, reflecting that higher central charge $\xi_{3}$ of a bosonic
modular extension $\breve{\mathcal{C}}=U(1)_{2}\times U(1)_{-4}\times
D(\mathbb{Z}_{2})$ is a nontrivial phase.

##  5 Discussions

In this work, we present a general framework for gapped boundary of (2+1)D
topological phases with global symmetry, both in the bosonic or fermionic
cases. We then derived obstructions to symmetry-preserving gapped boundary,
mainly focusing on U(1)-symmetric bosonic and U(1)f-symmetric fermionic
phases. Let us comment on several generalizations and future questions. Some
of them are investigated in future work.

First, for a (2+1)D bosonic topological phase, we only considered a bosonic
gapped boundary obtained by condensing bosons. However, it is also possible to
obtain a gapped boundary of a bosonic phase by fermion condensation, which
gives a fermionic gapped boundary condition [43, 44]. For example, [43]
constructed a gapped boundary of a (2+1)D toric code $D(\mathbb{Z}_{2})$ by
condensing a fermion $\psi$, by introducing fermionic degrees of freedom on
the boundary. It would be interesting to make a generalization of the existing
algebraic formalism of anyon condensation to the case for fermionic gapped
boundary of a bosonic phase.

Also, it is a natural question to consider symmetry enrichment of such a
fermionic gapped boundary. For example, consider a $\mathbb{Z}_{2}$ global
symmetry of $D(\mathbb{Z}_{2})$ that permutes anyons as $e\leftrightarrow m$.
Then, we immediately find that a bosonic gapped boundary cannot preserve the
$\mathbb{Z}_{2}$ global symmetry, since condensing $e$ or $m$ explicitly
violates the symmetry. Meanwhile, condensing $\psi$ is left invariant under
this $\mathbb{Z}_{2}$ symmetry action. So, it is expected that fermion
condensation makes possible to obtain richer class of symmetry-preserving
gapped boundary for bosonic phases.

In addition, in the study of gapped boundary with $G$ global symmetry for
bosonic phases, we only considered the Lagrangian algebra anyon that
corresponds to a tube of gapped boundary without $G$ holonomy along its
meridian. We can also consider the object $\mathcal{L}_{\mathbf{g}}$ that is
given by shrinking the tube of gapped boundary with holonomy labeled by
$\mathbf{g}\in G$, which should be an anyon in the twisted sector
$\mathcal{C}_{\mathbf{g}}$ of $G$-crossed modular tensor category
$\mathcal{C}_{G}^{\times}$ [24]. This should define the $G$-crossed version of
the Lagrangian algebra anyon of $\mathcal{C}_{G}^{\times}$ in the form of

$\displaystyle\mathcal{L}_{G}^{\times}:=\bigoplus_{\mathbf{g}\in
G}\mathcal{L}_{\mathbf{g}}.$ (5.1)

It would be interesting to study this $G$-crossed Lagrangian algebra anyon in
$\mathcal{C}_{G}^{\times}$, and obtain a further constraint on the data of
$G$-crossed category required for symmetry-preserving gapped boundary.

As a related work, Ref. [45] comments on an expectation about the algebraic
criteria for symmetry-preserving gapped boundary of a (2+1)D symmetry-enriched
topological phase, based on the Lagrangian algebra of the theory obtained by
gauging the global symmetry of the topological phase. Concretely, they
conjecture that the existence of the symmetry-preserving gapped boundary of
$G$-symmetric topological phase $\mathcal{C}$ is equivalent to the existence
of the Lagrangian algebra anyon $(\mathcal{L}_{G}^{\times})^{G}$ in a
$G$-gauged theory $(\mathcal{C}_{G}^{\times})^{G}$, such that for each
$\mathbf{g}\in G$ there is at least one condensed anyon $a_{\mathbf{g}}$
satisfying $\mathrm{Hom}(a_{\mathbf{g}},(\mathcal{L}_{G}^{\times})^{G})\neq 0$
that carries the $[\mathbf{g}]$ magnetic flux, where $[\mathbf{g}]$ is the
conjugacy class that contains $\mathbf{g}\in G$. Though we do not figure out
how to show that the above Lagrangian algebra is sufficient for the existence
of the symmetry-preserving gapped boundary, we believe that
$(\mathcal{L}_{G}^{\times})^{G}$ containing the magnetic flux for each
conjugacy class $[\mathbf{g}]$ is at least necessary for having a symmetry-
preserving gapped boundary. This is because one can construct the $G$-crossed
Lagrangian algebra anyon $\mathcal{L}_{G}^{\times}$ of the $G$-crossed theory
$\mathcal{C}_{G}^{\times}$ as explained above, which is expected to contain
all the nontrivial twisted sector $\mathbf{g}\in G$. Then, we expect that
there should be a way to obtain the Lagrangian algebra
$(\mathcal{L}_{G}^{\times})^{G}$ of the gauged theory, based on the object
$\mathcal{L}_{G}^{\times}$ of the $G$-crossed theory, which should be phrased
as $G$-equivariantization of the Lagrangian algebra [24]. It would be
interesting to give an explicit proof to these expectations.

Finally, though we mainly focused on application to U(1) global symmetry, it
would be interesting to construct the formula for obstructions to gapped
boundary for other symmetries. In particular, for fermionic phases without any
global symmetry, one should be able to compute $c_{-}$ mod 1/2 for a given
super-modular category. However, even a formula of $c_{-}$ mod 1/2 for a
super-modular category is not known. So, it would be nice to express any
obstruction to gapped boundary of fermionic phases without symmetry in terms
of the data of super-modular category.

## Acknowledgement

We thank Maissam Barkeshli, Kantaro Ohmori and Sahand Seifnashri for useful
discussions. We thank Kansei Inamura, Kantaro Ohmori, Yuji Tachikawa and
Yunqin Zheng for comments on a draft. RK is supported by the JQI postdoctoral
fellowship at the University of Maryland.

##  Appendix A Review of (2+1)D anyon systems

###  A.1 Notations of BTC

In this appendix, we briefly review the notation that we use to describe
braided tensor category (BTC). For a more comprehensive review of the notation
that we use, see, e.g., Ref. [24, 22]. The topologically non-trivial
quasiparticles of a (2+1)D topologically ordered state are referred to as
anyons. In the category theory terminology, they correspond to isomorphism
classes of simple objects of the BTC.

A BTC $\mathcal{C}$ contains splitting spaces $V_{c}^{ab}$, and their dual
fusion spaces, $V_{ab}^{c}$, where $a,b,c\in\mathcal{C}$ are anyons. These
spaces have dimension $\text{dim }V_{c}^{ab}=\text{dim
}V_{ab}^{c}=N_{ab}^{c}$, where the fusion coefficients $N_{ab}^{c}$ determine
the fusion rules. In particular, the fusion rules of the anyons are written as
$a\times b=\sum_{c}N_{ab}^{c}c$, so that fusion from $a\times b\to c$ is
possible if and only if $N_{ab}^{c}\geq 1$. If $N_{ab}^{c}>1$, then each
fusion corresponds to a higher dimensional vector space with more possible
“fusion outcomes”.

The fusion spaces are depicted graphically as:

$\left(d_{c}/d_{a}d_{b}\right)^{1/4}\begin{pspicture}[missing](-2.84526pt,-5.69054pt)(42.67911pt,34.14328pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}{\psline{-<}(19.91692pt,0.0pt)(19.91692pt,-9.95845pt)}{\psline(19.91692pt,0.0pt)(19.91692pt,-15.649pt)}{\psline(19.91692pt,-15.649pt)(7.11317pt,-28.45274pt)}{\psline{-<}(19.91692pt,-15.649pt)(9.95845pt,-25.60747pt)}{\psline(19.91692pt,-15.649pt)(32.72064pt,-28.45274pt)}{\psline{-<}(19.91692pt,-15.649pt)(29.87537pt,-25.60747pt)}\rput[tl]{0.0}(11.38109pt,0.0pt){$c$}\rput[br]{0.0}(39.83383pt,-27.0301pt){$b$}\rput[bl]{0.0}(0.0pt,-27.0301pt){$a$}\scriptsize\rput[bl]{0.0}(24.18483pt,-14.22636pt){$\mu$}\end{pspicture}=\left\langle
a,b;c,\mu\right|\in V_{ab}^{c},$ (A.1)

$\left(d_{c}/d_{a}d_{b}\right)^{1/4}\begin{pspicture}[missing](-2.84526pt,-5.69054pt)(42.67911pt,34.14328pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}{\psline{->}(19.91692pt,0.0pt)(19.91692pt,12.80373pt)}{\psline(19.91692pt,0.0pt)(19.91692pt,15.649pt)}{\psline(19.91692pt,15.649pt)(7.11317pt,28.45274pt)}{\psline{->}(19.91692pt,15.649pt)(8.53581pt,27.0301pt)}{\psline(19.91692pt,15.649pt)(32.72064pt,28.45274pt)}{\psline{->}(19.91692pt,15.649pt)(31.298pt,27.0301pt)}\rput[bl]{0.0}(11.38109pt,0.0pt){$c$}\rput[br]{0.0}(39.83383pt,22.76219pt){$b$}\rput[bl]{0.0}(0.0pt,22.76219pt){$a$}\scriptsize\rput[bl]{0.0}(24.18483pt,9.95845pt){$\mu$}\end{pspicture}=\left|a,b;c,\mu\right\rangle\in
V_{c}^{ab},$ (A.2)

where $\mu=1,\ldots,N_{ab}^{c}$, $d_{a}$ is the quantum dimension of $a$, and
the factors $\left(\frac{d_{c}}{d_{a}d_{b}}\right)^{1/4}$ are a normalization
convention for the diagrams.

Diagrammatically, inner products come from connecting the fusion/splitting
spaces’ lines as:

$\begin{pspicture}[missing](-5.69054pt,-9.95845pt)(34.14328pt,49.7923pt)\small{\psset{linewidth=0.9pt,linecolor=black,border=0pt}\psarc(22.76219pt,19.91692pt){11.38109pt}{120.0}{240.0}}{\psset{linewidth=0.9pt,linecolor=black,arrows=<-,arrowscale=1.4,arrowinset=0.15}\psarc{<-}(22.76219pt,19.91692pt){11.38109pt}{165.0}{240.0}}{\psset{linewidth=0.9pt,linecolor=black,border=0pt}\psarc(11.38109pt,19.91692pt){11.38109pt}{-60.0}{60.0}}{\psset{linewidth=0.9pt,linecolor=black,arrows=->,arrowscale=1.4,arrowinset=0.15}\psarc{->}(11.38109pt,19.91692pt){11.38109pt}{-60.0}{15.0}}\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}{\psline(17.07164pt,29.87537pt)(17.07164pt,44.10175pt)}{\psline{->}(17.07164pt,29.87537pt)(17.07164pt,41.25647pt)}{\psline(17.07164pt,-4.2679pt)(17.07164pt,9.95845pt)}{\psline{->}(17.07164pt,-4.2679pt)(17.07164pt,7.11317pt)}\rput[bl]{0.0}(1.99168pt,15.649pt){$a$}\rput[bl]{0.0}(26.74557pt,15.649pt){$b$}\rput[bl]{0.0}(7.3977pt,35.56592pt){$c$}\rput[bl]{0.0}(6.82864pt,-1.42262pt){$c^{\prime}$}\scriptsize\rput[bl]{0.0}(19.91692pt,29.87537pt){$\mu$}\rput[bl]{0.0}(19.91692pt,4.2679pt){$\mu^{\prime}$}\endpspicture=\delta_{cc^{\prime}}\delta_{\mu\mu^{\prime}}\sqrt{\frac{d_{a}d_{b}}{d_{c}}}\pspicture[missing](4.2679pt,-9.95845pt)(22.76219pt,49.7923pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}{\psline(17.07164pt,-4.2679pt)(17.07164pt,44.10175pt)}{\psline{->}(17.07164pt,-4.2679pt)(17.07164pt,24.18483pt)}\rput[bl]{0.0}(21.33955pt,35.56592pt){$c$}\end{pspicture},$
(A.3)

This is a way of phrasing topological charge conservation. In addition, we
have the usual ‘resolution of the identity’ in a UMTC, phrased
diagrammatically:

$\begin{pspicture}[missing](-2.84526pt,-5.69054pt)(28.45274pt,34.14328pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}{\psline{->}(7.11317pt,0.0pt)(7.11317pt,17.07164pt)}{\psline(7.11317pt,0.0pt)(7.11317pt,28.45274pt)}{\psline{->}(19.91692pt,0.0pt)(19.91692pt,17.07164pt)}{\psline(19.91692pt,0.0pt)(19.91692pt,28.45274pt)}\rput[br]{0.0}(4.2679pt,14.22636pt){$a$}\rput[bl]{0.0}(22.76219pt,14.22636pt){$b$}\end{pspicture}=\sum_{c}\sqrt{\frac{d_{c}}{d_{a}d_{b}}}\begin{pspicture}[missing](-11.38109pt,-5.69054pt)(42.67911pt,36.98856pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}{\psline{->}(19.91692pt,7.11317pt)(19.91692pt,19.91692pt)}{\psline(19.91692pt,7.11317pt)(19.91692pt,22.76219pt)}{\psline(19.91692pt,22.76219pt)(7.11317pt,35.56592pt)}{\psline{->}(19.91692pt,22.76219pt)(8.53581pt,34.14328pt)}{\psline(19.91692pt,22.76219pt)(32.72064pt,35.56592pt)}{\psline{->}(19.91692pt,22.76219pt)(31.298pt,34.14328pt)}{\psline{->}(7.11317pt,-8.53581pt)(17.07164pt,4.2679pt)}{\psline(7.11317pt,-8.53581pt)(19.91692pt,7.11317pt)}{\psline{->}(32.72064pt,-8.53581pt)(22.76219pt,4.2679pt)}{\psline(32.72064pt,-8.53581pt)(19.91692pt,7.11317pt)}\rput[bl]{0.0}(11.38109pt,14.22636pt){$c$}\rput[br]{0.0}(39.83383pt,29.87537pt){$b$}\rput[bl]{0.0}(0.0pt,29.87537pt){$a$}\rput[bl]{0.0}(0.0pt,-5.69054pt){$a$}\rput[br]{0.0}(39.83383pt,-5.69054pt){$b$}\end{pspicture},$
(A.4)

implicitly assuming $N_{ab}^{c}\leq 1$ for all $a,b,c$.

We denote $\bar{a}$ as the topological charge conjugate of $a$, for which
$N_{a\bar{a}}^{1}=1$, i.e.

$\displaystyle a\times\bar{a}=1+\cdots$ (A.5)

Here $1$ refers to the identity particle, i.e. the vacuum topological sector,
which physically describes all local, topologically trivial bosonic
excitations.

The $F$-symbols are defined as the following basis transformation between the
splitting spaces of $4$ anyons:

$\begin{pspicture}[missing](0.0pt,-12.80373pt)(51.21494pt,51.21494pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}{\psline(5.69054pt,42.67911pt)(28.45274pt,14.22636pt)}{\psline(28.45274pt,14.22636pt)(28.45274pt,0.0pt)}{\psline(51.21494pt,42.67911pt)(28.45274pt,14.22636pt)}{\psline(17.07164pt,28.45274pt)(28.45274pt,42.67911pt)}{\psline{->}(17.07164pt,28.45274pt)(8.53581pt,39.12251pt)}{\psline{->}(17.07164pt,28.45274pt)(25.60747pt,39.12251pt)}{\psline{->}(28.45274pt,14.22636pt)(48.36966pt,39.12251pt)}{\psline{->}(28.45274pt,14.22636pt)(19.91692pt,24.89615pt)}{\psline{->}(28.45274pt,0.0pt)(28.45274pt,10.66977pt)}\rput[bl]{0.0}(1.42262pt,45.52438pt){$a$}\rput[bl]{0.0}(27.0301pt,45.52438pt){$b$}\rput[bl]{0.0}(49.7923pt,45.52438pt){${c}$}\rput[bl]{0.0}(14.22636pt,14.22636pt){$e$}\rput[bl]{0.0}(25.60747pt,-8.53581pt){$d$}\scriptsize\rput[bl]{0.0}(8.53581pt,22.76219pt){$\alpha$}\rput[bl]{0.0}(19.91692pt,7.11317pt){$\beta$}\end{pspicture}=\sum_{f,\mu,\nu}\left[F_{d}^{abc}\right]_{(e,\alpha,\beta)(f,\mu,\nu)}\begin{pspicture}[missing](0.0pt,-12.80373pt)(51.21494pt,51.21494pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}{\psline(5.69054pt,42.67911pt)(28.45274pt,14.22636pt)}{\psline(28.45274pt,14.22636pt)(28.45274pt,0.0pt)}{\psline(51.21494pt,42.67911pt)(28.45274pt,14.22636pt)}{\psline(39.83383pt,28.45274pt)(28.45274pt,42.67911pt)}{\psline{->}(17.07164pt,28.45274pt)(8.53581pt,39.12251pt)}{\psline{->}(39.83383pt,28.45274pt)(31.298pt,39.12251pt)}{\psline{->}(28.45274pt,14.22636pt)(48.36966pt,39.12251pt)}{\psline{->}(28.45274pt,14.22636pt)(36.98856pt,24.89615pt)}{\psline{->}(28.45274pt,0.0pt)(28.45274pt,10.66977pt)}\rput[bl]{0.0}(1.42262pt,45.52438pt){$a$}\rput[bl]{0.0}(27.0301pt,45.52438pt){$b$}\rput[bl]{0.0}(49.7923pt,45.52438pt){${c}$}\rput[bl]{0.0}(35.56592pt,12.80373pt){$f$}\rput[bl]{0.0}(25.60747pt,-8.53581pt){$d$}\scriptsize\rput[bl]{0.0}(42.67911pt,22.76219pt){$\mu$}\rput[bl]{0.0}(19.91692pt,7.11317pt){$\nu$}\end{pspicture}.$
(A.6)

To describe topological phases, these are required to be unitary
transformations, i.e.

$\displaystyle\left[\left(F_{d}^{abc}\right)^{-1}\right]_{\left(f,\mu,\nu\right)\left(e,\alpha,\beta\right)}$
$\displaystyle=\left[\left(F_{d}^{abc}\right)^{\dagger}\right]_{\left(f,\mu,\nu\right)\left(e,\alpha,\beta\right)}$
(A.7)
$\displaystyle=\left[F_{d}^{abc}\right]_{\left(e,\alpha,\beta\right)\left(f,\mu,\nu\right)}^{\ast}.$

The $R$-symbols define the braiding properties of the anyons, and are defined
via the the following diagram:

$\begin{pspicture}[missing](-2.84526pt,-5.69054pt)(42.67911pt,34.14328pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}{\psline{->}(19.91692pt,0.0pt)(19.91692pt,12.23466pt)}{\psline(19.91692pt,0.0pt)(19.91692pt,14.22636pt)}{\psarc(22.76219pt,19.15453pt){5.69054pt}{120.0}{240.0}}{\psarc(17.07164pt,19.15453pt){5.69054pt}{-60.0}{35.0}}{\psline(17.45291pt,25.50531pt)(7.59688pt,31.19586pt)}{\psline{->}(17.45291pt,25.50531pt)(10.0606pt,29.77322pt)}{\psline(19.91692pt,24.08267pt)(32.23695pt,31.19586pt)}{\psline{->}(19.91692pt,24.08267pt)(29.77322pt,29.77322pt)}\rput[bl]{0.0}(11.38109pt,0.0pt){$c$}\rput[br]{0.0}(38.4112pt,24.18483pt){$b$}\rput[bl]{0.0}(1.42262pt,24.18483pt){$a$}\scriptsize\rput[bl]{0.0}(23.33124pt,9.95845pt){$\mu$}\end{pspicture}=\sum\limits_{\nu}\left[R_{c}^{ab}\right]_{\mu\nu}\begin{pspicture}[missing](-2.84526pt,-5.69054pt)(42.67911pt,34.14328pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}{\psline{->}(19.91692pt,0.0pt)(19.91692pt,12.80373pt)}{\psline(19.91692pt,0.0pt)(19.91692pt,15.649pt)}{\psline(19.91692pt,15.649pt)(7.11317pt,28.45274pt)}{\psline{->}(19.91692pt,15.649pt)(8.53581pt,27.0301pt)}{\psline(19.91692pt,15.649pt)(32.72064pt,28.45274pt)}{\psline{->}(19.91692pt,15.649pt)(31.298pt,27.0301pt)}\rput[bl]{0.0}(11.38109pt,0.0pt){$c$}\rput[br]{0.0}(39.83383pt,22.76219pt){$b$}\rput[bl]{0.0}(0.0pt,22.76219pt){$a$}\scriptsize\rput[bl]{0.0}(23.33124pt,10.52751pt){$\nu$}\end{pspicture}.$
(A.8)

Under a basis transformation, $\Gamma^{ab}_{c}:V^{ab}_{c}\rightarrow
V^{ab}_{c}$, the $F$ and $R$ symbols change:

$\displaystyle F^{abc}_{def}$
$\displaystyle\rightarrow\check{F}^{abc}_{d}=\Gamma^{ab}_{e}\Gamma^{ec}_{d}F^{abc}_{def}[\Gamma^{bc}_{f}]^{\dagger}[\Gamma^{af}_{d}]^{\dagger}$
$\displaystyle R^{ab}_{c}$
$\displaystyle\rightarrow\check{R}^{ab}_{c}=\Gamma^{ba}_{c}R^{ab}_{c}[\Gamma^{ab}_{c}]^{\dagger}.$
(A.9)

where we have suppressed splitting space indices and dropped brackets on the
$F$-symbol for shorthand. These basis transformations are referred to as
vertex basis gauge transformations. Physical quantities correspond to gauge-
invariant combinations of the data.

The topological twist $\theta_{a}$ is defined via the diagram:

$\theta_{a}=\theta_{\bar{a}}=\sum\limits_{c,\mu}\frac{d_{c}}{d_{a}}\left[R_{c}^{aa}\right]_{\mu\mu}=\frac{1}{d_{a}}\begin{pspicture}[missing](-36.98856pt,-17.07164pt)(36.98856pt,17.07164pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}{\psset{linewidth=0.9pt,linecolor=black}\psarc(20.11893pt,0.0pt){14.22636pt}{-135.0}{135.0}}{\psset{linewidth=0.9pt,linecolor=black}\psarc(-20.11893pt,0.0pt){14.22636pt}{45.0}{315.0}}{\psline(-10.06088pt,10.06088pt)(10.06088pt,-10.06088pt)}{\psset{border=2.3pt}\psline(-10.06088pt,-10.06088pt)(10.06088pt,10.06088pt)}{\psset{border=2.3pt}\psline{->}(-10.06088pt,-10.06088pt)(0.0pt,0.0pt)}\rput[bl]{0.0}(-5.69054pt,-14.22636pt){$a$}\end{pspicture}.$
(A.10)

Finally, the modular, or topological, $S$-matrix, is defined as

$S_{ab}=\mathcal{D}^{-1}\sum\limits_{c}N_{\bar{a}b}^{c}\frac{\theta_{c}}{\theta_{{a}}\theta_{b}}d_{c}=\frac{1}{\mathcal{D}}\begin{pspicture}[missing](0.0pt,5.69054pt)(73.97713pt,36.98856pt)\small{\psset{linewidth=0.9pt,linecolor=black,arrows=<-,arrowscale=1.5,arrowinset=0.15}\psarc{<-}(45.52438pt,19.91692pt){14.22636pt}{167.0}{373.0}}{\psset{linewidth=0.9pt,linecolor=black,border=3pt,arrows=<-,arrowscale=1.5,arrowinset=0.15}\psarc{<-}(25.60747pt,19.91692pt){14.22636pt}{167.0}{373.0}}{\psset{linewidth=0.9pt,linecolor=black}\psarc(25.60747pt,19.91692pt){14.22636pt}{0.0}{180.0}}{\psset{linewidth=0.9pt,linecolor=black,border=3pt}\psarc(45.52438pt,19.91692pt){14.22636pt}{45.0}{150.0}}{\psset{linewidth=0.9pt,linecolor=black}\psarc(45.52438pt,19.91692pt){14.22636pt}{0.0}{50.0}}{\psset{linewidth=0.9pt,linecolor=black}\psarc(45.52438pt,19.91692pt){14.22636pt}{145.0}{180.0}}\rput[bl]{0.0}(2.84526pt,12.80373pt){$a$}\rput[bl]{0.0}(22.76219pt,12.80373pt){$b$}\end{pspicture},$
(A.11)

where $\mathcal{D}=\sqrt{\sum_{a}d_{a}^{2}}$.

We also denote by $\mathcal{A}$ the Abelian group corresponding to fusion of
Abelian anyons, for which each $a\in\mathcal{A}$ satisfies $d_{a}=1$ and
$a\times b$ has a unique fusion product for any $b\in\mathcal{C}$.

The double braid, or mutual statistics, of anyons $a$ and $b$ is defined as

$\displaystyle M_{ab}=\frac{S^{*}_{ab}S_{00}}{S_{0a}S_{0b}}$ (A.12)

and is a phase if either $a$ or $b$ is an Abelian anyon.

###  A.2  Modular and super-modular tensor categories

Physically realizable bosonic topological orders are described by modular
tensor categories, which have the property that the $S$-matrix is unitary.
This means that braiding is non-degenerate, that is, every anyon $a$ can be
detected by its non-trivial mutual statistics with some other anyon $b$.

Fermionic topological phases have local fermions, and locality requires that
these fermions have trivial mutual statistics with all other excitations. One
way to keep track of the anyon fusion and braiding properties in a fermionic
topological phase is to use a super-modular tensor category $\mathcal{C}$. A
super-modular tensor category is a unitary braided fusion category where there
are exactly two transparent anyons: the identity $1$ and a fermion $\psi$.
That is, $\psi$ has $\theta_{\psi}=-1$ and trivial mutual braiding with all
other particles. As such, the braiding is degenerate.

In a super-modular tensor category, the anyons (simple objects) as a set form
the structure $\\{1,a,b,...\\}\times\\{1,\psi\\}$. The $S$ matrix factorizes
as

$\displaystyle S=\tilde{S}\otimes\frac{1}{\sqrt{2}}\left(\begin{matrix}1&1\\\
1&1\end{matrix}\right),$ (A.13)

where $\tilde{S}$ is unitary.

###  A.3 Modular extension of super-modular category

For a given super-modular category $\mathcal{C}$, its modular extension is a
modular tensor category $\breve{\mathcal{C}}$ that contains $\mathcal{C}$ as a
subcategory. In particular, a modular extension $\breve{\mathcal{C}}$ is
called a minimal modular extension if its quantum dimension satisfies
$\mathcal{D}_{\breve{\mathcal{C}}}^{2}=2\mathcal{D}_{\mathcal{C}}^{2}$. It was
shown in [46] that every super-modular category admits a minimal modular
extension. In this paper, we write the minimal modular extension as
$\breve{\mathcal{C}}=\breve{\mathcal{C}}_{\mathrm{NS}}\oplus\breve{\mathcal{C}}_{\mathrm{R}}$,
with $\breve{\mathcal{C}}_{\mathrm{NS}}=\mathcal{C}$.

We have $M_{a,\psi}=1$ for $a\in\breve{\mathcal{C}}_{\mathrm{NS}}$ since
$\breve{\mathcal{C}}_{\mathrm{NS}}=\mathcal{C}$ is super-modular. Meanwhile,
using modularity of $\breve{\mathcal{C}}$, it follows that $M_{a,\psi}=-1$ for
$a\in\breve{\mathcal{C}}_{\mathrm{R}}$, which can be checked by computing
$\sum_{a\in\breve{\mathcal{C}}}S^{*}_{a,\psi}S_{a,1}$ which must be zero since
$S$ is unitary,

$\displaystyle\begin{split}\sum_{a\in\breve{\mathcal{C}}}S^{*}_{a,\psi}S_{a,1}&=\frac{1}{\mathcal{D}_{\breve{\mathcal{C}}}^{2}}\left(\sum_{a\in\breve{\mathcal{C}}_{\mathrm{NS}}}d_{a}^{2}M_{a,\psi}+\sum_{a\in\breve{\mathcal{C}}_{\mathrm{R}}}d_{a}^{2}M_{a,\psi}\right)\\\
&=\frac{1}{2}+\frac{1}{\mathcal{D}_{\breve{\mathcal{C}}}^{2}}\sum_{a\in\breve{\mathcal{C}}_{\mathrm{R}}}d_{a}^{2}M_{a,\psi}\\\
&\geq\frac{1}{2}+\frac{1}{\mathcal{D}_{\breve{\mathcal{C}}}^{2}}\sum_{a\in\breve{\mathcal{C}}_{\mathrm{R}}}d_{a}^{2}\cdot(-1)=0,\end{split}$
(A.14)

where we used $M_{a,\psi}=\pm 1$ in the inequality. So,
$\sum_{a\in\breve{\mathcal{C}}}S^{*}_{a,\psi}S_{a,1}=0$ is satisfied only when
$M_{a,\psi}=-1$ for all $a\in\breve{\mathcal{C}}_{\mathrm{R}}$.

For a given super-modular category $\mathcal{C}$, the choice of a minimal
modular extension $\breve{\mathcal{C}}$ is not unique. In particular, for the
most trivial super-modular category $\mathcal{C}=\\{1,\psi\\}$, there exists
sixteen minimal modular extensions which we denote as
$\breve{\mathcal{C}}^{\nu}$ for $\nu\in\mathbb{Z}_{16}$ [22]. Then, for a
generic super-modular category $\mathcal{C}$ and its minimal modular extension
$\breve{\mathcal{C}}$, we can construct sixteen distinct minimal modular
extensions of $\mathcal{C}$ by considering
$\breve{\mathcal{C}}\boxtimes\breve{\mathcal{C}}^{\nu}$, and then condensing a
pair of fermions $(\psi,\psi)$. This property for sixteen possible choices of
minimal modular extensions is called “sixteen-fold way” [47].

##  Appendix B Review on global symmetry of (2+1)D topological phases

In this appendix, we provide a review of symmetry fractionalization of braided
tensor categories (BTC), summarizing results from [24, 31, 37]. We first
recall the topological symmetries of BTC, and then describe the symmetry
fractionalization of bosonic and fermionic topological phases. We restrict
ourselves to unitary global symmetry for simplicity.

###  B.1  Topological symmetry and braided auto-equivalence

An important property of a BTC $\mathcal{C}$ is the group of “topological
symmetries,” which are related to “braided auto-equivalences” in the
mathematical literature. The topological symmetries consist of the invertible
maps

$\displaystyle\varphi:\mathcal{C}\rightarrow\mathcal{C}.$ (B.1)

We consider the different $\varphi$ modulo equivalences known as natural
isomorphisms of the form

$\displaystyle\Upsilon({\left|{a,b;c}\right>})=\frac{\gamma_{a}\gamma_{b}}{\gamma_{c}}{\left|{a,b;c}\right>}$
(B.2)

The equivalence classes $[\varphi]$ then constitute a group, which we denote
as Aut$(\mathcal{C})$ [24]. The maps $\varphi$ may permute the anyons:

$\displaystyle\varphi(a)=a^{\prime}\in\mathcal{C},$ (B.3)

subject to the constraint that

$\displaystyle N_{a^{\prime}b^{\prime}}^{c^{\prime}}=N_{ab}^{c},\quad
S_{a^{\prime}b^{\prime}}=S_{ab},\quad\theta_{a^{\prime}}$
$\displaystyle=\theta_{a},$ (B.4)

The maps $\varphi$ have a corresponding action on the $F$\- and $R$\- symbols
of the theory, as well as on the fusion and splitting spaces, reviewed below.

###  B.2  Global symmetry and symmetry fractionalization: bosonic case

We now consider a bosonic system which has a unitary global symmetry group
$G$. In the bosonic case, the global symmetry acts on the anyons and the
topological state space through the action of a group homomorphism

$\displaystyle[\rho]:G\rightarrow\text{Aut}(\mathcal{C}).$ (B.5)

We use the notation $[\rho_{\bf g}]\in\text{Aut}(\mathcal{C})$ for a specific
element ${\bf g}\in G$. The square brackets indicate the equivalence class of
symmetry maps related by natural isomorphisms, which we define below.
$\rho_{\bf g}$ is thus a representative symmetry map of the equivalence class
$[\rho_{\bf g}]$. We use the notation

$\,{}^{\bf g}a\equiv\rho_{\bf g}(a).$ (B.6)

Each $\rho_{\bf g}$ has a unitary action on the fusion/splitting spaces:

$\displaystyle\rho_{\bf g}:V_{ab}^{c}\rightarrow V_{\,{}^{\bf g}a\,^{\bf
g}b}^{\,{}^{\bf g}c}.$ (B.7)

We choose a basis ${\left|{a,b;c,\mu}\right>}$ for $V_{ab}^{c}$ and write the
action of $\rho_{\bf g}$ on the basis states as

$\displaystyle\rho_{\bf g}|a,b;c,\mu\rangle=\sum_{\nu}[U_{\bf g}(\,^{\bf
g}a,\,^{\bf g}b;\,^{\bf g}c)]_{\mu\nu}{\left|{\,{}^{\bf g}a,\,^{\bf
g}b;\,^{\bf g}c,\nu}\right>},$ (B.8)

and the action of $\rho_{\bf g}$ defined on the rest of the fusion/splitting
spaces by linearity. Here $U_{\bf g}(\,^{\bf g}a,\,^{\bf g}b;\,^{\bf g}c)$ is
a $N_{ab}^{c}\times N_{ab}^{c}$ matrix.

In the presence of $G$ symmetry defects, a graphical calculus can be developed
for the action of symmetry defects on anyon data. The basic pictures defining
the graphical calculus are given in Fig. 12. Symmetry fractionalization is
specified by a set of phases $\eta_{a}({\bf g,h})$, which satisfy certain
consistency relations which we will discuss shortly. The data $\\{U,\eta\\}$
characterize a symmetry fractionalization class and give us information about
how the group symmetries fractionalize onto the different anyons.

Figure 12: Anyon lines (black) passing through branch sheets (orange) and
graphical definitions of the $U$ and $\eta$ symbols.

There are several consistency conditions that need to be imposed on the
$U,\eta,F,R$ symbols in order for diagrammatic evaluations to be consistent
under different orders of moves. In the case of $N_{ab}^{c}\leq 1$ they can be
written as

$\displaystyle
F^{fcd}_{egl}{F}^{abl}_{efk}=\sum_{h}{F}^{abc}_{gfh}{F}^{ahd}_{egk}{F}^{bcd}_{khl}$
(B.9)
$\displaystyle{R}^{ca}_{e}{F}^{acb}_{deg}{R}^{cb}_{g}=\sum_{f}{F}^{cab}_{def}{R}^{cf}_{d}{F}^{abc}_{dfg}$
(B.10)
$\displaystyle({R}^{ac}_{e})^{-1}{F}^{acb}_{deg}({R}^{bc}_{g})^{-1}=\sum_{f}{F}^{cab}_{def}({R}^{fc}_{d})^{-1}{F}^{abc}_{dfg}$
(B.11)
$\displaystyle{U}_{\mathbf{g}}(^{\mathbf{g}}{a},^{\mathbf{g}}{b};^{\mathbf{g}}{e}){U}_{\mathbf{g}}(^{\mathbf{g}}{e},^{\mathbf{g}}{c};^{\mathbf{g}}{d}){F}^{{}^{\mathbf{g}}{a}^{\mathbf{g}}{b}^{\mathbf{g}}{c}}_{{}^{\mathbf{g}}{d}^{\mathbf{g}}{e}^{\mathbf{g}}{f}}{U}_{\mathbf{g}}^{-1}(^{\mathbf{g}}{b},^{\mathbf{g}}{c};^{\mathbf{g}}{f}){U}_{\mathbf{g}}^{-1}(^{\mathbf{g}}{a},^{\mathbf{g}}{f};^{\mathbf{g}}{d})=F^{abc}_{def}$
(B.12)
$\displaystyle{U}_{\mathbf{g}}(^{\mathbf{g}}{a},^{\mathbf{g}}{b};^{\mathbf{g}}{c})R^{{}^{\mathbf{g}}{a}^{\mathbf{g}}{b}}_{{}^{\mathbf{g}}{c}}{U}_{\mathbf{g}}(^{\mathbf{g}}{a},^{\mathbf{g}}{b};^{\mathbf{g}}{c})^{-1}=R^{ab}_{c}$
(B.13)
$\displaystyle{U}_{\mathbf{g}}({a},{b};{c}){U}_{\mathbf{h}}({a},{b};{c})={U}_{\mathbf{gh}}({a},{b};{c})\frac{\eta_{c}({\bf
g},{\bf h})}{\eta_{a}({\bf g},{\bf h})\eta_{b}({\bf g},{\bf h})}$ (B.14)
$\displaystyle{\eta}_{a}({\bf h},{\bf k}){\eta}_{a}({\bf gh},{\bf
k})={\eta}_{a}({\bf g},{\bf h}){\eta}_{a}({\bf g},{\bf hk})$ (B.15)

The top three are just the standard pentagon and hexagon equations from BTCs
without symmetry. The next two ensure the symmetry action is compatible with
the $F$\- and $R$-symbols. The next ensures that the symmetry action and
symmetry fractionalization are consistent with each other, and the last one is
a generalized associativity condition for the $\eta$ symbols.

These data are subject to an additional class of gauge transformations, which
arise by changing $\rho$ by a natural isomorphism: [24]

$\displaystyle U_{\bf g}(a,b;c)$
$\displaystyle\rightarrow\frac{\gamma_{a}({\bf g})\gamma_{b}({\bf
g})}{\gamma_{c}({\bf g})}U_{\bf g}(a,b;c)$ $\displaystyle\eta_{a}({\bf g},{\bf
h})$ $\displaystyle\rightarrow\frac{\gamma_{a}({\bf gh})}{\gamma_{\,{}^{\bf
g}a}({\bf h})\gamma_{a}({\bf g})}\eta_{a}({\bf g},{\bf h})$ (B.16)

In this paper we will always fix the gauge

$\displaystyle\eta_{1}({\bf g},{\bf h})=\eta_{a}({\bf 1},{\bf
g})=\eta_{a}({\bf g},{\bf 1})$ $\displaystyle=1$ $\displaystyle U_{\bf
g}(1,b;c)=U_{\bf g}(a,1;c)$ $\displaystyle=1.$ (B.17)

One can show that symmetry fractionalization forms a torsor over
$H^{2}_{\rho}(BG,\mathcal{B})$ in the bosonic case, where $\mathcal{B}$ is a
group of Abelian anyons in $\mathcal{C}$. That is, different possible patterns
of symmetry fractionalization can be related to each other by elements of
${H}^{2}_{\rho}(BG,\mathcal{B})$. In particular, given an element
$[\mathfrak{t}]\in{H}^{2}_{\rho}(BG,\mathcal{B})$, we can change the symmetry
fractionalization class as

$\displaystyle\eta_{a}({\bf g},{\bf h})\rightarrow\eta_{a}({\bf g},{\bf
h})M_{a\mathfrak{t}({\bf g},{\bf h})},$ (B.18)

where $\mathfrak{t}({\bf g},{\bf h})\in\mathcal{B}$ is a representative
2-cocycle for the cohomology class $[\mathfrak{t}]$ and $M_{ab}$ is the mutual
braiding in Eq. (A.12).

###  B.3  Fermionic symmetries

We sketch a description of fermionic symmetry in fermionic topological phases
where $\mathcal{C}$ is super-modular, summarizing results from [31, 37]. The
anyons in a super-modular category $\mathcal{C}$ corresponds to the states in
the NS sector of the Hilbert space in spin TQFT, so symmetry fractionalization
for $\mathcal{C}$ is understood as specifying the symmetry action on the NS
sector [36]. To describe the symmetry action on the whole Hilbert space
including the R sector, we need to define symmetry fractionalization on the
modular extension
$\breve{\mathcal{C}}=\breve{\mathcal{C}}_{\mathrm{NS}}\oplus\breve{\mathcal{C}}_{\mathrm{R}}$
of $\mathcal{C}=\breve{\mathcal{C}}_{\mathrm{NS}}$.

As we review in Appendix C, a spin TQFT is constructed by performing a process
of “fermion condensation” for a given bosonic TQFT. This process corresponds
to gauging 1-form symmetry generated by a Wilson line of a transparent fermion
$\psi$, and the resulting theory has a dual $\mathbb{Z}_{2}^{f}$ fermion
parity symmetry.

Suppose we want a fermionic symmetry with a fermionic group $G_{f}$ after
fermion condensation, where $G_{f}$ is defined by a symmetry extension
$\mathbb{Z}_{2}^{f}\to G_{f}\to G_{b}$ with the extension characterized by
$\omega_{2}\in Z^{2}(BG_{b},\mathbb{Z}_{2})$. Then, the initial bosonic theory
(sometimes called a “bosonic shadow theory”) must have $G_{b}$ symmetry, where
the $G_{b}$ symmetry has a certain mixed ’t Hooft anomaly between the
$\mathbb{Z}_{2}$ 1-form symmetry generated by $\psi$ Wilson line [48, 49]. So,
we start with constructing a symmetry action of $G_{b}$ for the bosonic shadow
theory described by a modular category
$\breve{\mathcal{C}}=\breve{\mathcal{C}}_{\mathrm{NS}}\oplus\breve{\mathcal{C}}_{\mathrm{R}}$,
taking into account certain additional constraints which arise from the mixed
’t Hooft anomaly.

The $G_{b}$ action of $\breve{\mathcal{C}}$ is realized as a topological
symmetry of BTC which preserves the $\mathbb{Z}_{2}$ grading
$\breve{\mathcal{C}}=\breve{\mathcal{C}}_{\mathrm{NS}}\oplus\breve{\mathcal{C}}_{\mathrm{R}}$.
So, it induces a topological symmetry of the NS sector
$\breve{\mathcal{C}}_{\mathrm{NS}}=\mathcal{C}$, which was formulated by [31,
37]. One can immediately see that $G_{b}$ cannot permute $\psi$; this follows
from the super-modularity of $\mathcal{C}$, where the topological symmetry has
to preserve the transparent fermion $\psi$.

Then, we need to account for constraints on symmetry fractionalization data
which comes from mixed ’t Hooft anomaly between $G_{b}$ symmetry and a
$\mathbb{Z}_{2}$ 1-form symmetry generated by $\psi$ Wilson line. The ’t Hooft
anomaly is given by a (3+1)D response action $(-1)^{\int A^{*}\omega\cup B}$,
where $A:M\to BG_{b}$ is a $G_{b}$ background gauge field and $B\in
Z^{2}(M,\mathbb{Z}_{2})$ is a background gauge field of $\mathbb{Z}_{2}$
1-form symmetry. This mixed ’t Hooft anomaly implies that a fermion $\psi$
carries symmetry fractionalization characterized by $\omega\in
Z^{2}(BG_{b},\mathbb{Z}_{2})$ [26, 49, 50].

To realize the desired mixed ’t Hooft anomaly, we impose constraints

$U_{\bf g}(\psi,\psi;1)=1$ (B.19) $\displaystyle\eta_{\psi}({\bf
g,h})=\omega_{2}({\bf g,h})$ (B.20)

These equations together with (B.17) guarantees that the phase shifted by
moving the $\psi$ Wilson line across $G_{b}$ symmetry defects is entirely
characterized by (B.20), which correctly realizes the effect of mixed ’t Hooft
anomaly under gauge transformation of 1-form symmetry.

Physically, (B.19) arises from the idea that the symmetry transformation rules
of the local fermion operators in a fermionic theory are set entirely by the
local Hilbert space, that is, the action of the symmetry on states containing
only fermions must be determined entirely by the local action of the symmetry.
Thus the action on the topological state space given by $U_{\bf
g}(\psi,\psi;1)$ must be trivial. In order to preserve above two constraints,
we have a constraint on gauge transformations by a natural isomorphism

$\gamma_{\psi}({\bf g})=1.$ (B.21)

##  Appendix C Review of spin TQFT

In this appendix, we review the construction of spin TQFT based on a method of
fermion condensation, and describe its Hilbert space. The content of this
review is mainly based on [36]. To obtain a spin TQFT, we start with a bosonic
TQFT described by a minimal modular extension $\breve{\mathcal{C}}$ of a
super-modular category $\mathcal{C}$, which is called a “bosonic shadow
theory”. We then gauge $\mathbb{Z}_{2}$ 1-form symmetry generated by a Wilson
line of a transparent fermion $\psi$ in $\mathcal{C}$. The resulting theory
has a dual $\mathbb{Z}_{2}^{f}$ symmetry which is fermion parity symmetry, and
gives a fermionic spin TQFT. The process of gauging the 1-form
$\mathbb{Z}_{2}$ symmetry for $\psi$ is called fermion condensation [51, 52,
43].

The Hilbert space of the spin TQFT after fermion condensation is described by
using the anyons of the initial bosonic shadow theory $\breve{\mathcal{C}}$.
Let us consider the Hilbert space on a torus equipped with spin structure
$T^{2}_{\mu,\lambda}$, where $\mu$ is spin structure along the meridian and
$\lambda$ is that on the longitude. For simplicity, we assume that all anyons
$x\in\breve{\mathcal{C}}_{\mathrm{R}}$ satisfy $x\times\psi\neq x$. For the
case in the presence of anyons $\sigma\times\psi=\sigma$, see [36]. When all
anyons $x\in\breve{\mathcal{C}}_{\mathrm{R}}$ satisfy $x\times\psi\neq x$, the
Hilbert space of the spin TQFT is spanned by states described as follows:

$\displaystyle\begin{split}\mathcal{H}(T^{2}_{\mathrm{NS},\mathrm{NS}}):&\qquad{\left|{a}\right>}+{\left|{a\times\psi}\right>},\quad
a\in\breve{\mathcal{C}}_{\mathrm{NS}}/\\{1,\psi\\}\\\
\mathcal{H}(T^{2}_{\mathrm{NS},\mathrm{R}}):&\qquad{\left|{a}\right>}-{\left|{a\times\psi}\right>},\quad
a\in\breve{\mathcal{C}}_{\mathrm{NS}}/\\{1,\psi\\}\\\
\mathcal{H}(T^{2}_{\mathrm{R},\mathrm{NS}}):&\qquad{\left|{x}\right>}+{\left|{x\times\psi}\right>},\quad
x\in\breve{\mathcal{C}}_{\mathrm{R}}/\\{1,\psi\\}\\\
\mathcal{H}(T^{2}_{\mathrm{R},\mathrm{R}}):&\qquad{\left|{x}\right>}-{\left|{x\times\psi}\right>},\quad
x\in\breve{\mathcal{C}}_{\mathrm{R}}/\\{1,\psi\\}\\\ \end{split}$ (C.1)

where ${\left|{a}\right>}$ denotes a state given by preparing a solid torus
$D^{2}\times S^{1}$ with insertion of a Wilson line for the anyon $a$ along
$S^{1}$. The above expression of the state is understood in a following way.
First, since we are gauging the 1-form symmetry generated by a Wilson line for
$\psi$, each state realized in the gauged theory is averaged over all possible
insertions of the $\psi$ Wilson line in the spacetime 3-manifold, so that the
state is invariant under insertion of $\psi$ Wilson line along both the
meridian and longitude. That explains why the state takes the form of
${\left|{a}\right>}\pm{\left|{a\times\psi}\right>}$ for each spin structure.

Second, we note that the action of a $\psi$ Wilson line with trivial framing
along a curve $C$ on the state on $T^{2}$ characterizes spin structure along
the curve $C$. For example, the action of the $\psi$ Wilson line along the
meridian gives a phase $+1$ (resp. $-1$), if spin structure along the meridian
is $\mu=\mathrm{NS}$ (resp. $\mu=\mathrm{R}$). Due to $M_{x,\psi}=-1$ when
$x\in\breve{\mathcal{C}}_{\mathrm{R}}$, we can see that each state in
(LABEL:eq:spinTQFTH) corresponds to the desired spin structure on $T^{2}$.

Let us explain more precisely how an insertion of the $\psi$ Wilson line
amounts to measuring spin structure, by describing fermion condensation in
detail. In general, fermion condensation in (2+1)D starts with a bosonic
shadow theory $Z_{b}$ with a specific $\mathbb{Z}_{2}$ 1-form symmetry
generated by a fermionic Wilson line $\psi$. Its background gauge field is
written as $B\in Z^{2}(M,\mathbb{Z}_{2})$. We want to gauge the 1-form
symmetry by making $B$ dynamical. However, reflecting the fermionic statistics
of $\psi$, it has a ’t Hooft anomaly whose response action is characterized by

$\displaystyle(-1)^{\int B\cup B}$ (C.2)

This is understood as a “framing anomaly”, which is a phase ambiguity of a
correlation function with $\psi$ Wilson line inserted, under the change of the
framing along the Wilson line [49].

So the bosonic shadow theory has a nontrivial ’t Hooft anomaly, which prevents
us from gauging the 1-form symmetry generated by $\psi$. However, when a
spacetime 3-manifold is equipped with a spin structure, the anomaly (C.2) can
be trivialized. This can be seen by using the Wu relation $B\cup B=w_{2}\cup
B$ mod 2 in cohomology with the second Stiefel-Whitney class $[w_{2}]\in
H^{2}(M,\mathbb{Z}_{2})$ [53]. Then, the anomaly $w_{2}\cup B$ can be
trivialized by introducing spin structure $\xi\in C^{1}(M,\mathbb{Z}_{2})$ on
a 3-manifold, where $\delta\xi=w_{2}$. This means that one can cancel the
anomaly of the bosonic shadow theory (C.2) by introducing a counter-term that
depends on spin structure. So, we write this counter-term as $z(\xi,B)$, which
becomes a phase and depends on spin structure $\xi$. Then, we can gauge the
1-form symmetry of $Z_{b}$ coupled with the spin theory $z(\xi,B)$,

$\displaystyle Z_{f}(\xi)\propto\sum_{B\in
Z^{2}(M,\mathbb{Z}_{2})}Z_{b}(B)z(\xi,B),$ (C.3)

then we obtain a spin theory $Z_{f}(\xi)$. This process is called fermion
condensation. Then, one can see that the fermion condensation is a sum over
the configuration of $\psi$ Wilson lines, weighted by a phase $z(\xi,B)$.

Basically, the phase $z(\xi,B)$ measures a spin structure along the curve
where $\psi$ Wilson line is inserted. Concretely, when we consider $z(\xi,B)$
on a 3-manifold $T^{2}_{\mu,\lambda}\times\mathbb{R}$, then an insertion of a
$\psi$ line along the meridian gives a phase $+1$ (resp. $-1$) when
$\mu=\mathrm{NS}$ (resp. $\mu=\mathrm{R}$), same for the longitude. 333Here,
we assume that the tangent bundle of $T^{2}_{\mu,\lambda}\times\mathbb{R}$ is
framed in the obvious way in $x,y,z$ directions, and the $\psi$ line in $x$ or
$y$ direction carries a trivial framing compared with the background framing
on $T^{2}_{\mu,\lambda}\times\mathbb{R}$. Since the theory $z(\xi,B)$ also
carries a framing anomaly, we need to keep track of framing of the $\psi$
Wilson line when we talk about its eigenvalue. See [51, 54, 49] for detailed
explanation of the theory $z(\xi,B)$.  Hence, (C.3) is understood as sum over
$\psi$ Wilson lines weighted by spin structure along the curve. That explains
why the insertion of a $\psi$ Wilson line measures spin structure.

##  Appendix D $S$-matrix action on boundary states of spin TQFT

In this appendix, we describe a detailed discussion of $S$-matrix action on
boundary states of spin TQFT.

Since the boundary state ${\left|{\mathcal{L}}\right>}$ depends on the spin
structure of the torus, the evaluation of the Hopf link is done by cases. We
write the spin structure for ${\left|{\mathcal{L}}\right>}$ as
$(\mu,\lambda)$, which denotes the meridian and longitude of $T^{2}$
respectively. Similarly, the spin structure around the anyon line $x$ linked
with $\mathcal{L}$ is written as $(\mu^{\prime},\lambda^{\prime})$, which
satisfies $(\mu^{\prime},\lambda^{\prime})=(\lambda,\mu)$.

  * •

When $(\mu,\lambda)=(\mathrm{NS},\mathrm{NS})$, the boundary state becomes
${\left|{\mathcal{L}_{\mathrm{NS},\mathrm{NS}}}\right>}$. We have
$(\mu^{\prime},\lambda^{\prime})=(\mathrm{NS},\mathrm{NS})$, and the state on
$T^{2}_{\mu^{\prime},\lambda^{\prime}}$ is identified as the line operator $x$
in the form of $x=a+a\times\psi$ with $a\in\breve{\mathcal{C}}_{\mathrm{NS}}$.
The Hopf link amplitude is given by the $S$ matrix between
${\left|{\mathcal{L}_{\mathrm{NS},\mathrm{NS}}}\right>}$ and
${\left|{x}\right>}={\left|{a}\right>}+{\left|{a\times\psi}\right>}$ as

$\displaystyle\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{1}}(S_{ab}+S_{a,b\times\psi}+S_{a\times\psi,b}+S_{a\times\psi,b\times\psi})Z_{0b}^{\mathrm{NS}}=\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{1}}4S_{ab}Z_{0b}^{\mathrm{NS}}$
(D.1)

Meanwhile, the Hopf link is regarded as a partition function on $D^{2}\times
S^{1}_{\mathrm{NS}}$ with a gapped boundary and insertion of $x$ along
$S^{1}$. This further reduces to a partition function on $S^{2}\times
S^{1}_{\mathrm{NS}}$ by shrinking boundary of $D^{2}$ into a point, with
insertion of $x\times\overline{\mathcal{L}}_{\mathrm{NS},\mathrm{NS}}$ (see
Fig. 13). This is evaluated as

$\displaystyle{\left<{\mathcal{L}_{\mathrm{NS},\mathrm{NS}}}\right|}({\left|{a}\right>}+{\left|{a\times\psi}\right>})=2Z_{0a}^{\mathrm{NS}}$
(D.2)

By comparing the above two expressions, we obtain

$\displaystyle\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{1}}2S_{ab}Z_{0b}^{\mathrm{NS}}=Z_{0a}^{\mathrm{NS}}\quad\text{for
$a\in\breve{\mathcal{C}}_{\mathrm{NS}}$}.$ (D.3)

Since $S_{ab}=S_{a,b\times\psi}$ for $a\in\breve{\mathcal{C}}_{\mathrm{NS}}$,
we also have

$\displaystyle\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{\psi}}2S_{ab}Z_{0b}^{\mathrm{NS}}=Z_{0a}^{\mathrm{NS}}\quad\text{for
$a\in\breve{\mathcal{C}}_{\mathrm{NS}}$}.$ (D.4)

Figure 13: A space $D^{2}$ with a gapped boundary is topologically equivalent
to $S^{2}$ by shrinking a boundary circle into a point.

  * •

When $(\mu,\lambda)=(\mathrm{NS},\mathrm{R})$, the boundary state becomes
${\left|{\mathcal{L}_{\mathrm{NS},\mathrm{R}}}\right>}$. We have
$(\mu^{\prime},\lambda^{\prime})=(\mathrm{R},\mathrm{NS})$, and the state on
$T^{2}_{\mu^{\prime},\lambda^{\prime}}$ is given in the form of
${\left|{x}\right>}={\left|{a}\right>}+{\left|{a\times\psi}\right>}$ with
$a\in\breve{\mathcal{C}}_{\mathrm{R}}$. The Hopf link amplitude between them
is given by the $S$ matrix as

$\displaystyle\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{1}}(S_{ab}-S_{a,b\times\psi}+S_{a\times\psi,b}-S_{a\times\psi,b\times\psi})Z_{0b}^{\mathrm{NS}}=\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{1}}4S_{ab}Z_{0b}^{\mathrm{NS}}$
(D.5)

Meanwhile, the Hopf link is regarded as a partition function on $D^{2}\times
S^{1}_{\mathrm{NS}}$ with a gapped boundary and insertion of $x$ along
$S^{1}$. This further reduces to a partition function on $S^{2}\times
S^{1}_{\mathrm{NS}}$ by shrinking boundary of $D^{2}$ into a point, with
insertion of $x\times\overline{\mathcal{L}}_{\mathrm{R},\mathrm{NS}}$. This is
evaluated as

$\displaystyle{\left<{\mathcal{L}_{\mathrm{R},\mathrm{NS}}}\right|}({\left|{a}\right>}+{\left|{a\times\psi}\right>})=2Z_{0a}^{\mathrm{R}}$
(D.6)

By comparing the above two expressions, we obtain

$\displaystyle\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{1}}2S_{ab}Z_{0b}^{\mathrm{NS}}=Z_{0a}^{\mathrm{R}}\quad\text{for
$a\in\breve{\mathcal{C}}_{\mathrm{R}}$}.$ (D.7)

Since $S_{ab}=-S_{a,b\times\psi}$ for $a\in\breve{\mathcal{C}}_{\mathrm{R}}$,
we also have

$\displaystyle\sum_{b\in\mathcal{A}_{\mathrm{NS}}^{\psi}}2S_{ab}Z_{0b}^{\mathrm{NS}}=-Z_{0a}^{\mathrm{R}}\quad\text{for
$a\in\breve{\mathcal{C}}_{\mathrm{R}}$}.$ (D.8)

  * •

When $(\mu,\lambda)=(\mathrm{R},\mathrm{NS})$, the boundary state becomes
${\left|{\mathcal{L}_{\mathrm{R},\mathrm{NS}}}\right>}$. We have
$(\mu^{\prime},\lambda^{\prime})=(\mathrm{NS},\mathrm{R})$, and the state on
$T^{2}_{\mu^{\prime},\lambda^{\prime}}$ is given in the form of
${\left|{x}\right>}={\left|{a}\right>}-{\left|{a\times\psi}\right>}$ with
$a\in\breve{\mathcal{C}}_{\mathrm{NS}}$. The Hopf link amplitude between them
is given by the $S$ matrix as

$\displaystyle\sum_{b\in\mathcal{A}_{\mathrm{R}}^{e}}(S_{ab}+S_{a,b\times\psi}-S_{a\times\psi,b}-S_{a\times\psi,b\times\psi})Z_{0b}^{\mathrm{R}}=\sum_{b\in\mathcal{A}_{\mathrm{R}}^{e}}4S_{ab}Z_{0b}^{\mathrm{R}}$
(D.9)

Meanwhile, the Hopf link is regarded as a partition function on $D^{2}\times
S^{1}_{\mathrm{R}}$ with a gapped boundary and insertion of $x$ along $S^{1}$.
This further reduces to a partition function on $S^{2}\times
S^{1}_{\mathrm{R}}$ by shrinking boundary of $D^{2}$ into a point, with
insertion of $x\times\overline{\mathcal{L}}_{\mathrm{NS},\mathrm{R}}$. This is
evaluated as

$\displaystyle\begin{split}{\left<{\mathcal{L}_{\mathrm{NS},\mathrm{R}}}\right|}({\left|{a}\right>}-{\left|{a\times\psi}\right>})&=2Z_{0a}^{\mathrm{NS}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{NS}}^{1}$}\\\
{\left<{\mathcal{L}_{\mathrm{NS},\mathrm{R}}}\right|}({\left|{a}\right>}-{\left|{a\times\psi}\right>})&=-2Z_{0a}^{\mathrm{NS}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{NS}}^{\psi}$}\\\
{\left<{\mathcal{L}_{\mathrm{NS},\mathrm{R}}}\right|}({\left|{a}\right>}-{\left|{a\times\psi}\right>})&=0\quad\text{if
$a\in\breve{\mathcal{C}}_{\mathrm{NS}}$ is not an element of
$\mathcal{A}_{\mathrm{NS}}$}\end{split}$ (D.10)

By comparing the above two expressions, we obtain the nonzero equations as

$\displaystyle\begin{split}\sum_{b\in\mathcal{A}_{\mathrm{R}}^{e}}2S_{ab}Z_{0b}^{\mathrm{R}}&=Z_{0a}^{\mathrm{NS}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{NS}}^{1}$}\\\
\sum_{b\in\mathcal{A}_{\mathrm{R}}^{e}}2S_{ab}Z_{0b}^{\mathrm{R}}&=-Z_{0a}^{\mathrm{NS}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{NS}}^{\psi}$}\\\ \end{split}$ (D.11)

We also have

$\displaystyle\begin{split}\sum_{b\in\mathcal{A}_{\mathrm{R}}^{m}}2S_{ab}Z_{0b}^{\mathrm{R}}&=Z_{0a}^{\mathrm{NS}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{NS}}^{1}$}\\\
\sum_{b\in\mathcal{A}_{\mathrm{R}}^{m}}2S_{ab}Z_{0b}^{\mathrm{R}}&=-Z_{0a}^{\mathrm{NS}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{NS}}^{\psi}$}\\\ \end{split}$ (D.12)

  * •

When $(\mu,\lambda)=(\mathrm{R},\mathrm{R})$, the boundary state becomes
${\left|{\mathcal{L}_{\mathrm{R},\mathrm{R}}}\right>}$. We have
$(\mu^{\prime},\lambda^{\prime})=(\mathrm{R},\mathrm{R})$, and the state on
$T^{2}_{\mu^{\prime},\lambda^{\prime}}$ is given in the form of
$x=a-a\times\psi$ with $a\in\breve{\mathcal{C}}_{\mathrm{R}}$. The Hopf link
amplitude between them is given by the $S$ matrix as

$\displaystyle\sum_{b\in\mathcal{A}_{\mathrm{R}}^{e}}(S_{ab}-S_{a,b\times\psi}-S_{a\times\psi,b}+S_{a\times\psi,b\times\psi})Z_{0b}^{\mathrm{R}}=\sum_{b\in\mathcal{A}_{\mathrm{R}}^{e}}4S_{ab}Z_{0b}^{\mathrm{R}}$
(D.13)

Meanwhile, the Hopf link is regarded as a partition function on $D^{2}\times
S^{1}_{\mathrm{R}}$ with a gapped boundary and insertion of $x$ along $S^{1}$.
This further reduces to a partition function on $S^{2}\times
S^{1}_{\mathrm{R}}$ by shrinking boundary of $D^{2}$ into a point, with
insertion of $x\times\overline{\mathcal{L}}_{\mathrm{R},\mathrm{R}}$. This is
evaluated as

$\displaystyle\begin{split}{\left<{\mathcal{L}_{\mathrm{R},\mathrm{R}}}\right|}({\left|{a}\right>}-{\left|{a\times\psi}\right>})&=2Z_{0a}^{\mathrm{R}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{R}}^{e}$}\\\
{\left<{\mathcal{L}_{\mathrm{R},\mathrm{R}}}\right|}({\left|{a}\right>}-{\left|{a\times\psi}\right>})&=-2Z_{0a}^{\mathrm{R}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{R}}^{m}$}\\\
{\left<{\mathcal{L}_{\mathrm{R},\mathrm{R}}}\right|}({\left|{a}\right>}-{\left|{a\times\psi}\right>})&=0\quad\text{if
$a\in\breve{\mathcal{C}}_{\mathrm{R}}$ is not an element of
$\mathcal{A}_{\mathrm{R}}$}\end{split}$ (D.14)

By comparing the above two expressions, we obtain the nonzero equations as

$\displaystyle\begin{split}\sum_{b\in\mathcal{A}_{\mathrm{R}}^{e}}2S_{ab}Z_{0b}^{\mathrm{R}}&=Z_{0a}^{\mathrm{R}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{R}}^{e}$}\\\
\sum_{b\in\mathcal{A}_{\mathrm{R}}^{e}}2S_{ab}Z_{0b}^{\mathrm{R}}&=-Z_{0a}^{\mathrm{R}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{R}}^{m}$}\\\ \end{split}$ (D.15)

We also have

$\displaystyle\begin{split}\sum_{b\in\mathcal{A}_{\mathrm{R}}^{m}}2S_{ab}Z_{0b}^{\mathrm{R}}&=-Z_{0a}^{\mathrm{R}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{R}}^{e}$}\\\
\sum_{b\in\mathcal{A}_{\mathrm{R}}^{m}}2S_{ab}Z_{0b}^{\mathrm{R}}&=Z_{0a}^{\mathrm{R}}\quad\text{if
$a\in\mathcal{A}_{\mathrm{R}}^{m}$}\end{split}$ (D.16)

## References

  * [1] D. C. Tsui, H. L. Stormer, and A. C. Gossard, “Two-dimensional magnetotransport in the extreme quantum limit,” Phys. Rev. Lett. 48 (1982) 1559.
  * [2] R. B. Laughlin, “Anomalous quantum hall effect: An incompressible quantum fluid with fractionally charged excitations,” Phys. Rev. Lett. 50 (1983) 1395.
  * [3] F. Wilczek, “Magnetic flux, angular momentum, and statistics,” Phys. Rev. Lett. 48 (Apr, 1982) 1144–1146. https://link.aps.org/doi/10.1103/PhysRevLett.48.1144.
  * [4] G. Moore and N. Read, “Nonabelions in the fractional quantum hall effect,” Nuclear Physics B 360 no. 2-3, (1991) 362 – 396.
  * [5] M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman, Y. Oreg, and A. Stern, “Observation of half-integer thermal hall conductance,” Nature 559 no. 7713, (Jun, 2018) 205–210.
  * [6] A. Kitaev and L. Kong, “Models for Gapped Boundaries and Domain Walls,” Commun. Math. Phys. 313 no. 2, (2012) 351–373, arXiv:1104.5047 [cond-mat.str-el].
  * [7] A. Kapustin and N. Saulina, “Topological boundary conditions in abelian Chern-Simons theory,” Nucl. Phys. B845 (2011) 393–435, arXiv:1008.0654 [hep-th].
  * [8] A. Kapustin, “Ground-state degeneracy for abelian anyons in the presence of gapped boundaries,” Phys. Rev. B 89 no. 12, (2014) 125307, arXiv:1306.4254 [cond-mat.str-el].
  * [9] M. Barkeshli, C.-M. Jian, and X.-L. Qi, “Classification of Topological Defects in Abelian Topological States,” Phys. Rev. B 88 (2013) 241103, arXiv:1304.7579 [cond-mat.str-el].
  * [10] L.-Y. Hung and Y. Wan, “Ground State Degeneracy of Topological Phases on Open Surfaces,” Phys. Rev. Lett. 114 no. 7, (2015) 076401, arXiv:1408.0014 [cond-mat.str-el].
  * [11] J. Wang and X.-G. Wen, “Boundary Degeneracy of Topological Order,” Phys. Rev. B 91 no. 12, (2015) 125124, arXiv:1212.4863 [cond-mat.str-el].
  * [12] T. Lan, J. C. Wang, and X.-G. Wen, “Gapped Domain Walls, Gapped Boundaries and Topological Degeneracy,” Phys. Rev. Lett. 114 no. 7, (2015) 076402, arXiv:1408.6514 [cond-mat.str-el].
  * [13] L. Kong and H. Zheng, “A mathematical theory of gapless edges of 2d topological orders. Part I,” JHEP 02 (2020) 150, arXiv:1905.04924 [cond-mat.str-el].
  * [14] J. Kaidi, Z. Komargodski, K. Ohmori, S. Seifnashri, and S.-H. Shao, “Higher central charges and topological boundaries in 2+1-dimensional tqfts,” 2021.  https://arxiv.org/abs/2107.13091.
  * [15] A. Davydov, M. Müger, D. Nikshych, and V. Ostrik, “The witt group of non-degenerate braided fusion categories,” Journal für die reine und angewandte Mathematik (Crelles Journal) 2013 no. 677, (2013) 135–177.
  * [16] J. Fuchs, C. Schweigert, and A. Valentino, “Bicategories for boundary conditions and for surface defects in 3-d TFT,” Commun. Math. Phys. 321 (2013) 543–575, arXiv:1203.4568 [hep-th].
  * [17] A. Davydov, D. Nikshych, and V. Ostrik, “On the structure of the witt group of braided fusion categories,” Selecta Mathematica 19 no. 1, (2013) 237–269, arXiv:1109.5558.
  * [18] F. A. Bais and J. K. Slingerland, “Condensate induced transitions between topologically ordered phases,” Phys. Rev. B 79 (2009) 045316, arXiv:0808.0627 [cond-mat.mes-hall].
  * [19] F. J. Burnell, “Anyon condensation and its applications,” Ann. Rev. Condensed Matter Phys. 9 (2018) 307–327, arXiv:1706.04940 [cond-mat.str-el].
  * [20] S.-H. Ng, A. Schopieray, and Y. Wang, “Higher gauss sums of modular categories,” Selecta Mathematica 25 no. 4, (Aug, 2019) , arXiv:1812.11234 [math.QA]. http://dx.doi.org/10.1007/s00029-019-0499-2.
  * [21] S.-H. Ng, E. C. Rowell, Y. Wang, and Q. Zhang, “Higher central charges and witt groups,” 2020.  https://arxiv.org/abs/2002.03570.
  * [22] A. Kitaev, “Anyons in an exactly solved model and beyond,” Annals of Physics 321 (2006) 2–111, arXiv:cond-mat/0506438.
  * [23] A. Kitaev, “Toward topological classification of phases with short-range entanglement,” 2011.  https://online.kitp.ucsb.edu/online/topomat11/kitaev/.
  * [24] M. Barkeshli, P. Bonderson, M. Cheng, and Z. Wang, “Symmetry fractionalization, defects, and gauging of topological phases,” Phys. Rev. B 100 (Sep, 2019) 115147, arXiv:1410.4540.
  * [25] M. Lapa and M. Levin, “Anomaly indicators for topological orders with u(1) and time-reversal symmetry,” Physical Review B 100 no. 16, (Oct, 2019) , arXiv:1905.00435. http://dx.doi.org/10.1103/PhysRevB.100.165129.
  * [26] F. Benini, C. Córdova, and P.-S. Hsin, “On 2-group global symmetries and their anomalies,” Journal of High Energy Physics 2019 no. 3, (Mar, 2019) 118, arXiv:1803.09336.
  * [27] R. Kobayashi and M. Barkeshli, “(3+1)d path integral state sums on curved u(1) bundles and u(1) anomalies of (2+1)d topological phases,” 2021.  https://arxiv.org/abs/2111.14827.
  * [28] N. Seiberg and E. Witten, “Gapped boundary phases of topological insulators via weak coupling,” Progress of Theoretical and Experimental Physics 2016 no. 12, (2016) 1–86, arXiv:1602.04251v3.
  * [29] R. Thorngren and Y. Wang, “Anomalous symmetries end at the boundary,” Journal of High Energy Physics no. 9, (2021) 17, arXiv:2012.15861.
  * [30] X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, “Symmetry-protected topological orders in interacting bosonic systems,” Science 338 no. 6114, (2012) 1604–1606, http://www.sciencemag.org/content/338/6114/1604.full.pdf.
  * [31] D. Bulmash and M. Barkeshli, “Fermionic symmetry fractionalization in (2+1)d,” 2021.  https://arxiv.org/abs/2109.10913.
  * [32] M. Cheng, M. Zaletel, M. Barkeshli, A. Vishwanath, and P. Bonderson, “Translational symmetry and microscopic constraints on symmetry-enriched topological phases: A view from the surface,” Phys. Rev. X 6 (Dec, 2016) 041068, arXiv:1511.02263. https://link.aps.org/doi/10.1103/PhysRevX.6.041068.
  * [33] N. Reshetikhin and V. G. Turaev, “Invariants of 3-manifolds via link polynomials and quantum groups,” Inventiones mathematicae 103 no. 1, (1991) 547–597. https://doi.org/10.1007/BF01239527.
  * [34] V. G. Turaev, Quantum Invariants of Knots and 3-Manifolds.  De Gruyter, 2010.  https://doi.org/10.1515/9783110221848.
  * [35] V. Drinfeld, S. Gelaki, D. Nikshych, and V. Ostrik, “On braided fusion categories i,” 2010.  https://arxiv.org/abs/0906.0620.
  * [36] D. Delmastro, D. Gaiotto, and J. Gomis, “Global anomalies on the hilbert space,” 2021.  http://arxiv.org/abs/2101.02218.
  * [37] D. Aasen, P. Bonderson, and C. Knapp, “Characterization and classification of fermionic symmetry enriched topological phases,” 2021.  https://arxiv.org/abs/2109.10911.
  * [38] Z.-C. Gu and X.-G. Wen, “Symmetry-protected topological orders for interacting fermions: Fermionic topological nonlinear $\sigma$ models and a special group supercohomology theory,” Phys. Rev. B 90 (Sep, 2014) 115141, arXiv:1201.2648. https://link.aps.org/doi/10.1103/PhysRevB.90.115141.
  * [39] A. Kapustin, R. Thorngren, A. Turzillo, and Z. Wang, “Fermionic Symmetry Protected Topological Phases and Cobordisms,” JHEP 12 (2015) 052, arXiv:1406.7329 [cond-mat.str-el].
  * [40] Q.-R. Wang and Z.-C. Gu, “Construction and classification of symmetry protected topological phases in interacting fermion systems,” Physical Review X 10 (2018) 031055, arXiv:1811.00536.