# Topological and nontopological degeneracies in generalized string-net models

Anna Ritz-Zwilling<EMAIL_ADDRESS>Sorbonne
Université, CNRS, Laboratoire de Physique Théorique de la Matière Condensée,
LPTMC, F-75005 Paris, France  Jean-Noël Fuchs  jean-noel.fuchs@sorbonne-
universite.fr Sorbonne Université, CNRS, Laboratoire de Physique Théorique de
la Matière Condensée, LPTMC, F-75005 Paris, France  Steven H. Simon
<EMAIL_ADDRESS>Rudolf Peierls Centre for Theoretical Physics,
Clarendon Laboratory, Oxford, OX1 3PU, United Kingdom  Julien Vidal
<EMAIL_ADDRESS>Sorbonne Université, CNRS, Laboratoire de
Physique Théorique de la Matière Condensée, LPTMC, F-75005 Paris, France

###### Abstract

Generalized string-net models have been recently proposed in order to enlarge
the set of possible topological quantum phases emerging from the original
string-net construction. In the present work, we do not consider vertex
excitations and restrict to plaquette excitations, or fluxons, that satisfy
important identities. We explain how to compute the energy-level degeneracies
of the generalized string-net Hamiltonian associated to an arbitrary unitary
fusion category. In contrast to the degeneracy of the ground state, which is
purely topological, that of excited energy levels depends not only on the
Drinfeld center of the category, but also on internal multiplicities obtained
from the tube algebra defined from the category. For a noncommutative
category, these internal multiplicities result in extra nontopological
degeneracies. Our results are valid for any trivalent graph and any orientable
surface. We illustrate our findings with nontrivial examples.

###### Contents

  1. I Introduction
  2. II Background information on fusion categories
  3. III Generalized string-net models
    1. III.1 The Hamiltonian
    2. III.2 The output category
    3. III.3 Fluxon model
    4. III.4 Identifying fluxons
  4. IV Spectral degeneracies
    1. IV.1 Moore-Seiberg-Banks formula
    2. IV.2 Application to string-net fluxon model
      1. IV.2.1 Manifold with boundaries
      2. IV.2.2 Total Hilbert-space dimension
  5. V Special cases
    1. V.1 Modular categories
    2. V.2 Commutative categories
  6. VI Examples
    1. VI.1 Rep($S_{3}$) and Vec($S_{3}$) categories
    2. VI.2 Tambara-Yamagami category TY3
    3. VI.3 Haagerup category $\mathcal{H}_{3}$
    4. VI.4 Hagge-Hong category $\mathcal{E}$
  7. VII Conclusion and perspectives
  8. A Summary of the tube algebra
    1. A.1 Tubes $Q$ and their algebra $\mathcal{TA}$
    2. A.2 Decomposition of the tube algebra
    3. A.3 Simple idempotents and nilpotents $p$
    4. A.4 Minimal central idempotents $P_{A}$
    5. A.5 Tube algebra sectors
    6. A.6 The $11$ sector and fluxons
    7. A.7 From tubes to idemnils: $M_{A}$
    8. A.8 Half-braidings $\Omega_{A}$
  9. B Half-braidings for Rep($S_{3}$) and Vec($S_{3}$)
  10. C Fluxon identities
    1. C.1 Proof from the tube algebra
      1. C.1.1 Commutative input category
      2. C.1.2 Noncommutative input category
    2. C.2 Stability inequality
  11. D Generalized Hamiltonian
  12. E Hilbert-space dimension
    1. E.1 Counting edge labelings
    2. E.2 Counting fluxon fusion channels

##  I Introduction

The idea that two-dimensional quantum systems could contain quasiparticles
with exchange statistics different from bosons and fermions emerged about
fifty years ago [1, 2]. These point-like particles, called anyons, could,
under mutual exchange, accumulate “any”, possibly fractional, phase (Abelian
anyons), or even change into a new state orthogonal to the initial one (non-
Abelian anyons). Yet, only recent experiments provided evidence for the
presence of Abelian anyons in the fractional quantum Hall effect [3, 4], and
investigated non-Abelian anyons in quantum processors made of superconducting
qubits [5, 6] or trapped ions [7]. Anyons are a fundamental characteristic of
topologically-ordered phases: gapped phases that appear in strongly
interacting or frustrated systems and that cannot be identified by local
symmetries. In particular, anyons are connected to one of the defining
properties of topological order: a ground-state degeneracy which depends on
the surface topology on which the system is defined, and which is robust
against small local perturbations (a fact which motivated the idea of using
topological order to implement fault tolerant quantum computation [8, 9]). The
excited states of topologically-ordered phases also show non-trivial
degeneracies related to the fusion of anyons.

Some of the most studied models for topologically-ordered phases are the
string-net models introduced by Levin and Wen [10], and their generalizations
[11, 12, 13, 14]. These models, while not being able to produce all possible
topological orders, can produce exactly those topological orders that have
gappable edges and are achiral. The string-net models are exactly solvable.
While the energy levels are easy to obtain, the corresponding spectrum
degeneracies are non-trivial and are the main focus of the present paper.

The string-net models are built from planar diagram algebras — so-called
(spherical) unitary fusion categories (UFCs). A UFC is equipped with a set of
quantum numbers (“particles” or “simple objects” or “labels” or “strings”)
obeying certain fusion rules. The string-net model is then built on some
trivalent graph [dual to a triangulation of an orientable two-dimensional (2D)
manifold] as shown in Fig. 1. The full Hilbert space is given by assigning a
label (a simple object) to each directed edge of the graph.

Figure 1: A portion of a trivalent graph with oriented edges.

Given a UFC $\mathcal{C}$, the string-net model built from $\mathcal{C}$
realizes a topological order [15] corresponding to the Drinfeld center
$\mathcal{Z}(\mathcal{C})$ [16] which is a well-defined (2+1)D topological
quantum field theory (TQFT), or, in mathematical terms, a unitary modular
tensor category (UMTC). For example, the vector space dimension assigned to a
2D compact manifold of genus $g$ by $\mathcal{Z}(\mathcal{C})$ matches the
ground-state degeneracy of the string-net model built from $\mathcal{C}$ on
the same manifold. However, what is often not appreciated is that the
agreement between the vector space dimension assigned by
$\mathcal{Z}({\mathcal{C}})$ and the degeneracy of states in the string-net
model may not be perfect when one considers string-nets on manifolds with
boundary, or the case where quasiparticles are present, i.e., excitations
above the ground state. In particular, for cases other than the ground state
on a closed manifold, the string-net model displays both topological and
nontopological degeneracies.

By “topological degeneracies,” we mean degeneracies that arise from multiple
topological fusion channels in the ${\cal Z}({\cal C})$ topological quantum
field theory. As we will discuss below in section IV, the degeneracy of the
ground state on a surface of nonzero genus is such a topologcial degeneracy.
This type of degeneracy cannot be split by local perturbations so long as the
excitation gap does not close. Here, the notion of locality is related to
operators acting on a typical scale smaller than the systole (shortest
noncontractible loops). Degeneracies that arise from multiple fusion channels
of quasiparticle excitations, if present, will also be called topological.
These can be split by perturbations only if the perturbing operators connect
the positions of the multiple excitations. So for example, if two excitations
are very close to each other, a fairly local perturbation operator could in
principle split such degeneracy. Topological degeneracies, independent of how
easy they are to split, are encoded in the fusion rules of
$\mathcal{Z}(\mathcal{C})$.

However, in the case of a string-net model built from a noncommutative UFC
there are additional degeneracies associated with quasiparticles that are not
topological, and counting them requires the knowledge of other quantities that
will be discussed below. Indeed, in a given topological sector, one may still
have some additional degeneracies which can be lifted by completely local
perturbations.

For a string-net model built from a category $\cal C$, certain quasiparticle
types of the emergent theory $\mathcal{Z}(\mathcal{C})$ play a special role
and are denoted as “fluxons” or plaquette excitations. These elementary
excitations are always bosons that can be excited in the string-net model
without violating the fusion condition of the model at every trivalent vertex
(see Sec. III for more details). In the current paper, we will focus on this
type of excitations which only exists for smooth boundaries.

Excitations that are not fluxons and boundaries that are not smooth generally
require violations of the fusion condition. This is more complicated since
there may be inequivalent ways to define the model once the fusion conditions
are violated. Note that in Ref. [17] a particular extension of the string-net
model (known as “extended string-net model”) is considered that contains all
topological quasiparticle types but never violates any vertex constraints at
the price of some other complications. We leave for future work analysis of
that extended model.

It is worth noting that being a fluxon for some particular particle types of a
UMTC is dependent on the microscopic string-net model and is not a property of
the TQFT. In particular it may happen that the same TQFT may be constructed
either as the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of a UFC
$\mathcal{C}$ or as the Drinfeld center $\mathcal{Z}(\mathcal{C}^{\prime})$ of
another UFC $\mathcal{C}^{\prime}$. When
$\mathcal{Z}(\mathcal{C}^{\prime})\simeq\mathcal{Z}(\mathcal{C})$ we say that
$\mathcal{C}$ and $\mathcal{C}^{\prime}$ are Morita equivalent [18]. However,
the fluxons of the string-net model built from $\mathcal{C}$ are generally
different from the fluxons of the string-net model built from
$\mathcal{C}^{\prime}$. This is quite natural since a fluxon is defined as
satisfying the fusion rules of the underlying UFC that the string-net model is
built from.

In the case where the fusion rules of the category $\mathcal{C}$ are
commutative, i.e., when $a\times b=b\times a$ for all objects $a$ and $b$ in
the category, there is a substantial simplification since there are no
internal multiplicities. One can then find the topological degeneracy of a 2D
orientable manifold with any number of quasiparticles on it using classic
formulas found by Moore, Seiberg and Banks [19] and Verlinde [20]. However, in
the case where the fusion rules of the category $\mathcal{C}$ are
noncommutative, i.e., $a\times b\neq b\times a$ for some $a$ and $b$ in the
category, the situation is more complicated. Here, for some of the fluxons,
there is an additional internal degeneracy factor on top of the TQFT/conformal
field theory prediction of Moore, Seiberg, Banks, and Verlinde [19, 20]. We
will be particularly interested in this case.

It may seem to be quite an unusual situation to have a fusion category with
noncommutative fusion rules. However, this is precisely the case for the
famous Kitaev quantum double model [8] for a non-Abelian group, which is the
first (and potentially simplest) anyon model known to be universal for quantum
computation. Here, we are thinking of the Kitaev model on the dual lattice
compared to Kitaev’s original construction, see [21, 22]. In this model, the
category is based on a group $G$ and the fusion rules simply follow the group
multiplication rules. In cases where $G$ is a noncommutative group we have a
string-net model with noncommutative fusion rules. The Kitaev quantum double
model is thus a string-net model based on the trivial categorification of $G$,
which is known as the category ${\cal C}=$ Vec($G$). This is an interesting
case because there is a Morita equivalence to the commutative fusion category
based on the representations of the group ${\cal C^{\prime}}=$ Rep($G$) where
the fusion rules follow the multiplication rules of the group irreducible
representations. Although Vec($G$) and Rep($G$) have the same Drinfeld center,
and hence string-net models built from the two are described by the same TQFT,
we will see that their nontopological degeneracies are quite different.

In addition to UFCs based on noncommutative groups, there are other
noncommutative fusion categories, such as the Haagerup category [23, 24],
which we will discuss in detail below. Construction of string-net models from
these categories (and more generally from categories not possessing the
tetrahedral symmetry, see e.g. [22]) requires some care and is discussed in
details, for example, in Refs. [11, 12, 13, 14]. This is known as “generalized
string-net models” and should not be confused with the “extended string-net
models” of Ref. [17].

Our goal is to compute the spectral degeneracies of the generalized string-net
models, hence extending the recent results obtained in Ref. [25] to any UFC.
Let us stress that the calculation of degeneracies for some specific UFCs has
been the subject of several works in the last decade [26, 27, 28, 29, 30]. Our
analysis will begin by constructing the Drinfeld center $\cal Z({\cal C})$ for
a string-net model built from any input UFC $\cal C$. Building the Drinfeld
center can in general be quite complicated. Our approach will rely on the so-
called Ocneanu tube algebra [31, 32], which gives us all the objects in $\cal
Z({\cal C})$ together with their internal degeneracies. Then, using the Moore-
Seiberg-Banks formula [19] supplemented by degeneracy factors we will
calculate the degeneracies of each energy levels. This final step requires
several nontrivial manipulations which we will emphasize. At the end, we will
obtain expressions for the spectral degeneracies of generalized string-net
models (restricted to the fluxon sector). These will be used in an analysis of
the thermal properties of these models in a forthcoming paper.

The paper is structured as follows. In Sec. II, we provide a few basic notions
on fusion categories. In Sec. III, we give a short introduction to generalized
string-net models, with a particular emphasis on the construction and
description of the emergent topological phase. Section IV contains our main
result, namely the formula for the spectrum degeneracies in the most general
case. We make the link with previously obtained results in some particular
cases. Special cases such as modular or commutative categories are discussed
in Sec. V. We further illustrate our results with examples in Sec. VI. In
particular, we study the case of the Morita-equivalent categories Rep($S_{3}$)
and Vec($S_{3}$), corresponding to the same Drinfeld center
$\mathcal{Z}(S_{3})$ but having different energy-level degeneracies. We also
study more involved cases such as the Haagerup, the Tambara-Yamagami and the
Haage-Hong categories. Eventually, we conclude and discuss perspectives in
Sec. VII. In appendices, we give details on the tube algebra (see Appendices A
and B), on fluxon identities (see Appendix C), on a generalized Hamiltonian
(see Appendix D) and on the Hilbert-space dimension (see Appendix E).

##  II Background information on fusion categories

Before starting our discussion, we will need to introduce a few concepts from
the theory of TQFTs and fusion categories. Those familiar with the field can
skip this section. A UFC $\mathcal{C}$ is essentially defined by a set of
simple objects, a set of fusion rules, and a set of $F$-matrices. We will now
describe each of these in turn.

The first property of a UFC is the set of simple objects (also called
particles, labels, quantum numbers, strings, superselection sectors, …). The
identity $1$ (also called the vacuum or trivial object) is one of these
objects. Below, we label the (non-identity) simple objects of the UFC with
Roman letters $a\in\mathcal{C}$.

The second property of the UFC is the set of fusion rules. This defines how
objects “multiply” together, such as $a\times b=2c+f+h$. These fusion rules
are encoded in coefficients $N_{ab}^{c}$. In particular, we write

$a\times b=\sum_{c}N_{ab}^{c}\>c,$ (1)

where the elements $N_{ab}^{c}$ are just the nonnegative integer coefficients
of the object $c$ on the right-hand side of the equation. With only these two
properties (simple objects and fusion rules), one has a fusion ring. For a
list of small multiplicity-free fusion rings, see Ref. [33].

$a$$b$$c$$\mu$ Figure 2: Oriented fusion vertex corresponding to Eq. (1). A
fusion is “allowed” if $N_{ab}^{c}>0$ where $a$, $b$ and $c$ are in
counterclockwise order. In this case the index at the vertex takes values
$\mu=1,\ldots,N_{ab}^{c}=N_{ab\bar{c}}$. In cases where $N_{ab}^{c}=1$ we do
not write $\mu$.

If there are $N_{\mathcal{C}}$ simple objects in the category, we can think
$N_{ab}^{c}$ as being a set of $N_{\mathcal{C}}$ square matrices $N_{a}$ which
are each $N_{\mathcal{C}}$-dimensional, with rows and columns indexed by
$b,c$. The quantum dimension $d_{a}$ is defined to be the largest eigenvalue
of $N_{a}$, which is guaranteed to be positive. The so-called total quantum
dimension is

$\mathcal{D}=\mbox{$\sqrt{\sum\limits_{a}d_{a}^{2}}$}.$ (2)

The fusion rules are always assumed to be associative

$a\times(b\times c)=(a\times b)\times c,\>\text{for all }(a,b,c)\in\cal C.$
(3)

If $a\times b=b\times a,\>\text{for all }(a,b)\in\cal C$, or equivalently
$N_{ab}^{c}=N_{ba}^{c}$, we say that the fusion rules are commutative.
Conversely if there exists $(a,b)\in\cal C$ such that $a\times b\neq b\times
a$, then we say that the fusion rules are noncommutative. Most studies of
string-net models have focused on the simple case of commutative fusion rules
with $N_{ab}^{c}\leqslant 1$ (no fusion multiplicities), but we will be more
general.

Fusion with the identity object $1$ is trivial, meaning that $1\times
a=a\times 1=a,\>\text{for all }a\in{\cal C}$. For each simple object $a$ there
is a unique dual simple object $\bar{a}$ such that
$N_{a\bar{a}}^{1}=N_{\bar{a}a}^{1}=1$. It is possible that $a=\bar{a}$. It is
always the case that $N_{ab}^{c}=N_{\bar{b}\bar{a}}^{\bar{c}}$.

It is also sometimes useful to define

$N_{abc}=N_{ab}^{\bar{c}}.$ (4)

This notation is convenient because $N_{abc}=N_{bca}=N_{cab}$ even for
noncommutative fusion.

The fusion of objects in the category is represented by diagrams with labeled
and directed edges. For example, fusion of $a$ and $b$ to $c$ is represented
by the diagram shown in Fig. 2. Note that we have adopted the orientation
convention given in Refs. [14, 13] (one reads $a,b,c$ counterclockwise for the
diagram of the vertex $N_{ab}^{c}$). Reversing the arrow on a labeled edge
changes the edge label to its dual.

In a UFC, a fusion vertex, such as the one in Fig. 2, represents a Hilbert
space of dimension $N_{ab}^{c}$ with basis vectors labeled with greek letter
$\mu=1\ldots N_{ab}^{c}$. The diagram on the left of Fig. 3 describes a
Hilbert space of dimension $\sum_{d}N_{d\bar{b}\bar{a}}N_{e\bar{c}\bar{d}}$
and this space is spanned by the possible values of $d,\mu,\nu$. On the other
hand, the diagram on the right describes a Hilbert space of dimension
$\sum_{f}N_{f\bar{c}\bar{b}}N_{e\bar{f}\bar{a}}$ and this space is spanned by
the possible values of $f,\alpha,\beta$. Due to the associativity condition on
fusion, the dimension of the space in the two descriptions are the same, and
the unitary $F$-matrix shown in Fig. 3 relates the two bases to each other.
There is a consistency condition on the $F$-matrices, known as the pentagon
equation [34, 35, 16, 22], that assures that multiple changes of basis in
complicated diagrams will give a consistent result.

The essence of the UFC is that it gives us a consistent planar diagram
algebra. The values of labeled diagrams can be related to each other via
$F$-moves, and any “closed” diagram (diagrams where all edges end on trivalent
vertices at both of their ends) can be evaluated to be a complex number [22].

As mentioned above, the fusion rules of a UFC may be commutative or
noncommutative. A simple example of the noncommutative case is when one
considers a UFC built from noncommutative group elements where fusion rules
are simply given by the multiplication table of the group considered. A more
complicated example of a UFC with noncommutative fusion rules is the Haagerup
category $\mathcal{H}_{3}$ [23, 24] which we will discuss further below.

$a$$b$$c$$d$$\mu$$\nu$$e$$=\displaystyle{\sum_{f,\alpha,\beta}}\left[F^{abc}_{e}\right]_{(d\mu\nu)(f\alpha\beta)}$$a$$b$$c$$f$$e$$\alpha$$\beta$
Figure 3: The space described by the set of diagrams on the left, spanned by
different values of $d,\mu,\nu$, is the same as the space described by the
diagrams on the right, spanned by different values of $f,\beta,\alpha$. The
$F$-matrix describes the unitary basis transformation between the two
descriptions.

When fusion rules are commutative, several types of UFC can further be
distinguished as depicted in Fig. 4. First, one may ask whether a commutative
UFC is braided or not. A unitary braided fusion category is basically an anyon
theory. The planar diagram rules that one considers for UFCs are extended to
apply to diagrams with over- and under-crossings. Thus (2+1)-dimensional
diagrams, corresponding essentially to space-time diagrams for world-lines of
anyons, can be given values. The rules for handling over- and under-crossings
(known as $R$-matrix) must also satisfy a further consistency condition known
as the hexagon equations [34, 35, 16, 22].
It may be the case that for a given UFC, there is no consistent solution to
these equations, and the UFC is not braided. For instance, the Tambara-
Yamagami category TY3 [36], which we will discuss further below, is
commutative but non braided.

Unitary Fusion Category
---
(UFC)
Non-commutative
---
fusion rules
Commutative
---
fusion rules
Non-Braided
---
Braided
---
(UBFC)
Non-Modular
---
Modular
---
(UMTC)
Figure 4: A possible classification of unitary fusion categories.

For unitary braided tensor categories (UBFC), important quantities to consider
are the so-called $T$-matrix and $S$-matrix. The $T$-matrix,
$T_{a,b}=\delta_{a,b}\theta_{a}$, is simply a diagonal matrix of the twist
factors $\theta_{a}=\mathrm{e}^{2\mathrm{i}\pi s_{a}}$, as shown in Fig. 5,
where $s_{a}$ is the topological spin.

$a$$=\theta_{a}$$a$ Figure 5: Twist factors $\theta$ appearing in the
$T$-matrix.

The $S$-matrix is a symmetric matrix of elements $S_{a,b}=S_{b,a}$ which
evaluate a loop labeled $a$ linked with a loop labeled $b$ and then normalized
with the total quantum dimension, as shown in Fig. 6. It gives the exchange
statistics of the anyons $a$ and $b$. In the particular case where one of the
anyons is the vacuum, the $S$-matrix element takes a simple value in terms of
the quantum dimensions:

$S_{1,a}=\mbox{
{\begin{pspicture}(0.0pt,25.60747pt)(56.90549pt,56.90549pt)\psset{linewidth=1.1381pt,linecolor=black,arrowsize=7.11317pt
0.0}
{\psset{border=4pt,bordercolor=white}\psarc(28.45274pt,28.45274pt){21.33955pt}{0.0}{370.0}}
{\psarc{->}(28.45274pt,28.45274pt){21.33955pt}{180.0}{365.0}}
\rput(56.90549pt,28.45274pt){{\@@@scalebox{1.3}{\color[rgb]{0,0,0}$a$}}}
\end{pspicture}}}~{}~{}\times~{}~{}\frac{1}{\mathcal{D}}=\frac{d_{a}}{\mathcal{D}}.$
(5)

A word of caution is in order. Diagrams in Figs. 5 and 6 as well as in Eq. (5)
are evaluated meaning that they give complex numbers corresponding to quantum
amplitudes. Most other diagrams that will appear later in this paper, e.g., in
Figs. 8 or 9 and in Appendix E, are used differently and as a way to compute
the dimension of Hilbert spaces, similarly to what is done in Figs. 2 and 3.
The two ways of considering a diagram, i.e., either as a quantum amplitude or
as a way of counting possible labelings, should be clearly distinguished. For
example, in the second case, loops or bubbles can never be contracted, whereas
in the first case, an isolated loop of $a$ is evaluated as the quantum
dimension $d_{a}$ [see Eq. (5)].

If for all $a$ and $b$ there is a single $c$ such that $N_{ab}^{c}>0$, then
the category is said to be Abelian (in the sense that the corresponding anyons
will all be Abelian, i.e., exchange statistics is just a phase) and non-
Abelian otherwise. Note that for a fusion category, and in contrast to a
group, the notions of commutativity and of Abelianity are distinct. For
instance, the Fibonacci UFC is commutative but non-Abelian.

UBFCs can still be split into two families according to whether their
$S$-matrix is unitary, in which case we say the theory is modular, or not. A
UBFC is modular if it has no transparent particles besides the identity —
where “transparent” means that it accumulates no phase when braided all the
way around any other particle. If a UBFC is modular, it is known as a unitary
modular tensor category, or UMTC. A crucial example of a UMTC is the Drinfeld
center $\cal{Z}({\cal C})$, which is a theory constructed from any UFC $\cal
C$. We will discuss this example in detail below. Collectively, $T$ and $S$
are then known as the modular matrices.

The $S$-matrix of a UMTC has the crucial property that it simultaneously
diagonalizes all of the fusion matrices. The famed Verlinde formula [37, 20]
states that

$N_{a}=S\Lambda_{a}S^{\dagger},\text{ where
}(\Lambda_{{a}})_{bc}=\delta_{b,c}\,\frac{S_{a,b}}{S_{1,b}}.$ (6)

In fact, a similar formula holds true for any commutative UFC (even if non
modular or non braided). One can always define a matrix $\tilde{s}$, sometimes
known as the mock $S$-matrix, that simultaneously diagonalizes all of the
fusion matrices as in Eq. (6).

A quantity known as the topological central charge $c$ may be defined via

$\mathrm{e}^{2\mathrm{i}\pi
c/8}=\frac{1}{\mathcal{D}}\sum_{a}d_{a}^{2}\,\theta_{a}.$ (7)

We say a theory is achiral if $c=0$, even if the theory breaks time-reversal
invariance (we will see an example of this below in Sec. VI.2). Strictly
speaking, Eq. (7) only defines the topological central charge modulo 8 and one
should distinguish it from the chiral central charge (defined without the
modulo 8), which determines the thermal Hall conductance [38, 34]. However,
this distinction will not be important for us.

$S_{a,b}=\mbox{
{\begin{pspicture}(0.0pt,25.60747pt)(85.35823pt,56.90549pt)\psset{linewidth=1.1381pt,linecolor=black,arrowsize=7.11317pt
0.0} {\psarc(28.45274pt,28.45274pt){21.33955pt}{0.0}{180.0}}
{\psset{border=4pt,bordercolor=white}\psarc(56.90549pt,28.45274pt){21.33955pt}{0.0}{180.0}}
{\psarc{->}(56.90549pt,28.45274pt){21.33955pt}{180.0}{365.0}}
{\psset{border=4pt,bordercolor=white}\psarc(28.45274pt,28.45274pt){21.33955pt}{180.0}{370.0}}
{\psarc{->}(28.45274pt,28.45274pt){21.33955pt}{180.0}{365.0}}
\rput(56.90549pt,28.45274pt){{\@@@scalebox{1.3}{\color[rgb]{0,0,0}$a$}}}
\rput(85.35823pt,28.45274pt){{\@@@scalebox{1.3}{\color[rgb]{0,0,0}$b$}}}
\end{pspicture}}}~{}~{}\times~{}~{}\frac{1}{\mathcal{D}}$ Figure 6: The
$S$-matrix.

##  III Generalized string-net models

Inspired by the original construction proposed by Levin and Wen [10], but
without assuming tetrahedral symmetry, the generalized string-net models [12,
11, 14, 13] allow us to generate all topological phases described by the
Drinfeld center $\mathcal{Z(C)}$ of a UFC $\mathcal{C}$. To compute the
spectral degeneracies of generalized string-net models, we do not have to
explain the detailed construction of the model which can be found in Refs.
[14, 13]. We will nonetheless give a rough picture of it here.

The generalized string-net model can be defined on any trivalent graph (see
Fig. 1) and its construction only requires a (input) UFC $\mathcal{C}$. The
Hilbert space $\mathcal{H}$ is defined by the set of all possible allowed edge
and vertex labelings. The trivalent graph is intended to be dual to a
triangulation of a genus $g$ two-dimensional orientable manifold.

###  III.1 The Hamiltonian

The Hamiltonian for a string-net model is written in terms of a set of
operators $A_{v}$, defined at the position of each vertex $v$, and operators
$B_{p}$ defined at the position of each plaquette $p$. These operators are
projectors, meaning that $A_{v}^{2}=A_{v}$ and $B_{p}^{2}=B_{p}$ or
equivalently that their eigenvalues are 0 and 1 only. Further these operators
all commute with each other:
$[A_{v},A_{v^{\prime}}]=[A_{v},B_{p}]=[B_{p},B_{p^{\prime}}]=0$, for all
$v,v^{\prime},p,p^{\prime}$.

The exactly solvable string-net Hamiltonian is

$H=-J_{v}\\!\\!\\!\sum_{{\rm vertices}\,v}A_{v}\,\,-J_{p}\\!\\!\\!\sum_{{\rm
plaquettes}\,p}B_{p},$ (8)

with $(J_{v},J_{p})>0$. The $A_{v}$ operators yield unity if the fusion rules
of the input category $\mathcal{C}$ are obeyed at vertex $v$, and give zero
otherwise. In a lattice gauge theory analogy, $A_{v}=1$ corresponds to
satisfying a Gauss law at vertex $v$, while $A_{v}=0$ corresponds to having a
“charge” at this vertex, i.e., a vertex excitation. The $B_{p}$ operators can
be thought of as assuring that no “flux” penetrates the plaquette $p$ (the
notion of flux being dual to that of charge). The eigenvalue $B_{p}=1$
corresponds to having no flux in the plaquette $p$ and $B_{p}=0$ to having a
flux, i.e., a plaquette excitation. The detailed form of the $B_{p}$ operator
is a bit complicated, and is given in terms of $F$-symbols of the input UFC.
Since we do not need these for our purpose, we refer the interested reader to
Refs. [13, 14] for more details (see Refs. [10, 22] for an introduction to the
simple case with only commutative fusion rules).

The ground-state space of the model is spanned by all states
$|\psi_{0}\rangle$ such that
$A_{v}|\psi_{0}\rangle=B_{p}|\psi_{0}\rangle=|\psi_{0}\rangle$. Excited states
$|\psi\rangle$ violate these constraints, i.e., either $A_{v}|\psi\rangle=0$
and/or $B_{p}|\psi\rangle=0$ for some $(v,p)$.

###  III.2 The output category

The excitations of the system correspond to nontrivial objects of the output
category $\mathcal{Z(C)}$, known as the Drinfeld center of $\cal C$ [12].
Here, we will discuss how to infer their properties from the input category
$\mathcal{C}$.

By construction, the Drinfeld center $\mathcal{Z(C)}$ of a UFC $\mathcal{C}$
is a UMTC whose objects are built from $\mathcal{C}$. As a UMTC, the output
category $\mathcal{Z(C)}$ describes an achiral $(c\mod 8=0)$ anyon theory
(see, e.g., Refs. [34, 35, 16, 22] for more informations). As mentioned in
Ref. [14], topological order associated with Drinfeld centers are believed to
be the most general class of bosonic topological order compatible with gapped
boundaries [39, 11, 40]. One way to build the Drinfeld center of a UFC
consists in deriving the Ocneanu tube algebra [31, 32] and finding its
irreducible representations. The main lines of this construction are
summarized in Appendix A and some examples are given in Appendix B. In the
following, we will denote by Roman letters simple objects of $\mathcal{C}$ and
by capital Roman letters simple objects of $\mathcal{Z(C)}$, except for the
vacua which are denoted $1$ and ${\bf 1}$, respectively.

Any simple object $A$ of the Drinfeld center is given by

$A=\big{(}\mathop{\oplus}_{s}n_{A,s}\,s,\Omega_{A}\big{)},$ (9)

where $n_{A,s}$ is a nonnegative integer (internal multiplicity) counting the
number of times the simple object $s\in\mathcal{C}$ appears in the simple
object $A\in\mathcal{Z}(\mathcal{C})$, and where the half-braiding tensor
$\Omega_{A}$ gives all braiding properties of $A$ (see Appendices A and B for
more details). Practically, one may see $A$ as a rope made of different types
of strands (with a weight $n_{A,s}$ for the strand $s$) endowed with braiding
properties defined by $\Omega_{A}$. If the input category $\mathcal{C}$ is
commutative, one has $n_{A,s}\in\\{0,1\\}$. By contrast, for noncommutative
$\mathcal{C}$, $n_{A,s}$ can take integer values larger than 1. Actually, the
$\Omega_{A}$’s entirely define $\mathcal{Z}(\mathcal{C})$ but they also
contain more information since they keep track of the original UFC
($n_{A,s}$’s can be expressed in terms of these fundamental quantities). Let
us also stress that the internal multiplicities $n_{A,s}$’s have nothing to do
with the fusion multiplicities discussed in Sec. II.

Finally, let us mention that an alternative route to the description of the
excitations consists in finding operators associated to closed strings which
commute with the Hamiltonian (10). Such an approach based on string operators
is detailed in Refs. [10, 14] and the relation between the two approaches is
discussed in [12].

###  III.3 Fluxon model

In this work, we will focus on a somewhat simpler Hamiltonian than the full
string-net Hamiltonian of Eq. (8), since we shall strictly enforce the vertex
constraint. This is essentially equivalent to taking $J_{v}\rightarrow\infty$
in Eq. (8) and considering only low-energy physics. Even more simply we can
think of this constraint as restricting the Hilbert space to include only the
states that are +1 eigenstates of $A_{v}$ for all vertices. Within this
restricted Hilbert space and setting $J_{p}=1$, the Hamiltonian takes the
simpler form

$H=-\sum_{{\rm plaquettes\,}p}B_{p}.$ (10)

The ground-state space is entirely contained in this restricted Hilbert space.
All excitations of the model that remain within this space are plaquette
excitations, and known as fluxons. These are the excitations we will focus on
in this paper. In Eq. (10), $B_{p}$ is the projector onto the vacuum of
$\mathcal{Z(C)}$ in the plaquette $p$. Consequently, the ground-state energy
of the system is given by $E_{0}=-N_{\rm p}$ where $N_{\rm p}$ is the total
number of plaquette and a state with $q$ fluxons has an energy $E_{q}=-N_{\rm
p}+q=E_{0}+q$.

The Hamiltonian (10) assigns a constant energy penalty of one unit to each
plaquette which is not in the vacuum state. It is easy to generalize Eq. (10)
to a form that instead assigns a different energy to each different type of
fluxon excitation. Similarly, we can also study the case where the energy
penalties depend on which particular plaquette of the lattice we are
considering. We briefly address these more general cases in Appendix D.

###  III.4 Identifying fluxons

We will need to determine which objects from $\cal{Z}(\cal{C})$ are actually
fluxons. Using the tube algebra construction, it is straightforward to see
that $A$ is a fluxon iff $n_{A,1}\neq 0$ (see Appendix A). This definition
also includes the trivial fluxon $A=\mathbf{1}$ (vacuum) although, strictly
speaking, it is not an excitation. Other fluxons will be called nontrivial in
the following. The set of all fluxons in $\cal{Z}(\cal{C})$ will be denoted
$\mathcal{F}$, while that of nontrivial fluxons will be denoted
$\mathcal{F}^{*}$. As shown in Appendix C, the vector ${\bf n}_{1}$, the
components of which are $n_{A,1}$, obeys

$\displaystyle S{\bf n}_{1}$ $\displaystyle=$ $\displaystyle{\bf n}_{1},$ (11)
$\displaystyle T{\bf n}_{1}$ $\displaystyle=$ $\displaystyle{\bf n}_{1},$ (12)

where $S$ and $T$ are the modular matrices of $\mathcal{Z(C)}$. In the
following, we will make an extensive use of these important relations. Similar
equations have appeared in other contexts such as anyon condensation [41] or
gapped boundaries [42, 43]. Here they are obtained as properties of fluxons
(i.e. plaquette excitations).

In addition, if the input category $\mathcal{C}$ is commutative
(noncommutative), the number $N_{\mathcal{F}}$ of fluxons equals (is strictly
smaller than) the number $N_{\mathcal{C}}$ of simple objects in $\mathcal{C}$.
Other simple objects (non fluxons) in $\mathcal{Z(C)}$ are associated to
violations of the vertex constraint (see Sec. III.1). At first glance, it may
seem strange to worry about these latter excitations since they do not belong
to the Hilbert space $\mathcal{H}$. However, we will see that they do play a
role in the degeneracies of the energy spectrum.

Finally, let us make a short comment on Morita-equivalent categories. As
mentioned in the introduction, two different input categories may have the
same Drinfeld center. Thus, one could naively think that two string-net models
built from these two categories have the same plaquette excitations but, in
general, this is not true. Indeed, as discussed above, multiplicities
$n_{A,s}$ do not depend only on $\mathcal{Z}(\mathcal{C})$, but also on
$\mathcal{C}$. We will illustrate such a situation in Sec. VI.1.

##  IV Spectral degeneracies

The goal of the present paper is to explain how to compute the degeneracy of
the $q$-fluxon energy level in a string-net model. As can be anticipated, this
degeneracy depends on the surface topology and on the input UFC which, as
discussed above, determines the nature of the excitations. Before proceeding
further, let us stress that this degeneracy has recently been computed in Ref.
[25] for any modular input UFC. Here, we aim at giving a general expression
valid for any input UFC (see Fig. 4 for an overview).

###  IV.1 Moore-Seiberg-Banks formula

We start by considering a TQFT (or UMTC) generically called $\mathcal{U}$ on a
2D orientable manifold of genus $g$. Here, we have in mind that $\mathcal{U}$
will be a Drinfled center so that we label its simple objects by capital Roman
letters. We can generically decompose this manifold into pants diagrams that
are sewn together as in Fig. 7 (assume for now $g>1$ although in the end we do
not need this). Each hole of the pants is labeled with a simple object of
$\mathcal{U}$. The ground-state degeneracy of the resulting manifold is given
by all ways consistent with the fusion rules of $\mathcal{U}$ in which these
holes can be labeled and then sewn together. This problem of counting the
dimension of the ground-state subspace is then mapped to a problem of counting
labelings of a planar diagram. For example, a two-handled torus can be
assembled from two pants so that the ground-state degeneracy is given as in
Fig. 8. Note that by using associativity of fusion, one can restructure the
fusion diagram in multiple equivalent ways, an example of which is shown in
Fig. 8 (see the discussion in Ref. [44]).

$C$$A$$B$(274.8502655,749.5456764)
(274.8502655,744.02282891)(250.22592673,739.5456764)(219.8502655,739.5456764)
(189.47460427,739.5456764)(164.8502655,744.02282891)(164.8502655,749.5456764)
(164.8502655,755.0685239)(189.47460427,759.5456764)(219.8502655,759.5456764)
(250.22592673,759.5456764)(274.8502655,755.0685239)(274.8502655,749.5456764)
(274.8502655,749.5456764)
(274.8502655,744.02282891)(250.22592673,739.5456764)(219.8502655,739.5456764)
(189.47460427,739.5456764)(164.8502655,744.02282891)(164.8502655,749.5456764)
(164.8502655,755.0685239)(189.47460427,759.5456764)(219.8502655,759.5456764)
(250.22592673,759.5456764)(274.8502655,755.0685239)(274.8502655,749.5456764)
(164.89848328,604.44417494)
(164.89848328,598.92132744)(145.87058514,594.44417494)(122.39848328,594.44417494)
(98.92638142,594.44417494)(79.89848328,598.92132744)(79.89848328,604.44417494)
(79.89848328,609.96702244)(98.92638142,614.44417494)(122.39848328,614.44417494)
(145.87058514,614.44417494)(164.89848328,609.96702244)(164.89848328,604.44417494)
(164.89848328,604.44417494)
(164.89848328,598.92132744)(145.87058514,594.44417494)(122.39848328,594.44417494)
(98.92638142,594.44417494)(79.89848328,598.92132744)(79.89848328,604.44417494)
(79.89848328,609.96702244)(98.92638142,614.44417494)(122.39848328,614.44417494)
(145.87058514,614.44417494)(164.89848328,609.96702244)(164.89848328,604.44417494)
(339.80459595,604.49492567)
(339.80459595,598.97207818)(320.77669781,594.49492567)(297.30459595,594.49492567)
(273.83249409,594.49492567)(254.80459595,598.97207818)(254.80459595,604.49492567)
(254.80459595,610.01777317)(273.83249409,614.49492567)(297.30459595,614.49492567)
(320.77669781,614.49492567)(339.80459595,610.01777317)(339.80459595,604.49492567)
(339.80459595,604.49492567)
(339.80459595,598.97207818)(320.77669781,594.49492567)(297.30459595,594.49492567)
(273.83249409,594.49492567)(254.80459595,598.97207818)(254.80459595,604.49492567)
(254.80459595,610.01777317)(273.83249409,614.49492567)(297.30459595,614.49492567)
(320.77669781,614.49492567)(339.80459595,610.01777317)(339.80459595,604.49492567)
(165,605.00000472)
(191.07355,665.88209472)(216.96954,664.44416472)(233.5863,641.01015472)
(255,605.00000472) (80,605.00000472)
(100,630.00000472)(105,640.00000472)(130.55584,689.94924472)
(156.52229,740.70102472)(165,750.00000472)(165,750.00000472)
(274.49492,750.50507472)
(300,690.00000472)(305,690.00000472)(340,605.00000472)

$A$$B$$C$$\longleftrightarrow$

Figure 7: A pants diagram. Any two-dimensional orientable manifold with $g>1$
can be decomposed into pants diagrams that are then sewed together. This can
be mapped to sewing together trivalent vertices to build a planar fusion
diagram. Creating a sphere or a torus in a similar way would require addition
of hemispheres. However, computing the degeneracies is equivalent to assigning
the identity or vacuum to one (torus) or all (sphere) holes of a pant.
$A$$B$$C$$A$$C$$D$$\displaystyle{\sum_{ABC}}$$\displaystyle{=\sum_{ABC}N_{ACB}N_{\bar{A}\bar{B}\bar{C}}}=$$\displaystyle\sum_{ACD}N_{AD\bar{A}}N_{C\bar{C}\bar{D}}=\sum_{ACD}$
Figure 8: The ground-state degeneracy of a two-handled torus. Associativity
allows restructuring of the shape of the diagram. The restructuring of the
diagram is basically an $F$-move (see Fig. 3) except that here we are only
interested in the total dimension of the space described by the diagram.

A puncture in the manifold surface may be assigned a simple object type –
essentially any excitation (including the trivial vacuum, which is not really
an excitation). This corresponds to leaving an open, labeled, pant-hole when
we sew together our sets of pants. The most general case we can consider is
then a genus $g$ surface with $m$ labeled punctures, and the corresponding
fusion diagram is shown in Fig. 9. Thus the Hilbert-space dimension
$\dim_{\mathcal{U}}$ associated with this labeled punctured surface just
amounts to summing the product of $N$-matrices (one for each trivalent vertex)
over all possible internal indices of this diagram.

$\dim_{\mathcal{U}}(g;A_{1},\ldots,A_{m})=$$\displaystyle g\mbox{ loops
}\left\\{\rule{0.0pt}{65.0pt}\right.$$A_{1}$$A_{2}$$A_{3}$$\ldots$$A_{m}$

$\displaystyle\left\\{\rule{0.0pt}{65.0pt}\right.$

$m$labeled punctures

Figure 9: The fusion diagram for a genus $g$ surface with $m$ labeled
punctures. Arrows are not drawn on edges for simplicity of presentation. Each
vertex in this diagram is a fusion matrix $N$ and each edge is a sum over
simple objects in $\mathcal{U}$. See the example in Fig. 8. For brevity, the
sums over unlabeled edges are not written explicitly here. The sum implied by
this diagram can be reduced to Eq. (13).

While it may look like the sum over all of these internal indices is
complicated, using the Verlinde formula and the $S$-matrix of $\mathcal{U}$
[see Eq. (6)], the sum from Fig. 9 can be simplified to the form

$\mbox{$\dim_{\mathcal{U}}(g;A_{1},\ldots,A_{m}$)}=\sum_{C}\left[\prod_{j=1}^{m}S_{A_{j},C}\right]S_{\mathbf{1},C}^{2-2g-m},$
(13)

where the sum is performed over all $C\in\mathcal{U}$. This formula, although
it appeared first in the work of Moore and Seiberg [19], is credited to Moore,
Seiberg and Banks in a footnote. A special case of this formula was obtained
earlier by Verlinde [20]: when there are no punctures in the orientable
surface of genus $g$, the Hilbert-space dimension associated with the manifold
is just

$\dim_{\mathcal{U}}(g)=\sum_{C}S_{\mathbf{1},C}^{2-2g}.$ (14)

Another special limit of Eq. (13) is the case where there are three punctures
in a $g=0$ spherical surface (see again Fig. 7). In this case, one recovers
precisely the Verlinde formula (6). Note that Eq. (13) has a useful property
that if one of the punctures $j$ is labeled by the vaccum, i.e., if
$A_{j}=\mathbf{1}$, it can simply be dropped from the list. So, for example,
one has

$\dim_{\mathcal{U}}(g;A_{1},\dots,A_{m-1},\mathbf{1})=\dim_{\mathcal{U}}(g;A_{1},\dots,A_{m-1}).$
(15)

This makes sense since the vacuum quantum number is equivalent to the absence
of an excitation.

###  IV.2 Application to string-net fluxon model

We now apply the Moore-Seiberg-Banks formula to the string-net model where we
only allow fluxon excitations, i.e., we assume all vertex conditions are
always satisfied. Recall that the string-net model is built from a category
$\cal C$ and the emergent TQFT is the Drinfeld center ${\cal Z}({\cal C})$. As
discussed in Sec. III.3, fluxons are certain simple objects of ${\cal Z}({\cal
C})$.

Let us consider a string-net model on a graph with $N_{\text{p}}$ plaquettes
embeded on a surface of genus $g$ and a situation in which we have excited $q$
plaquettes by setting $A_{1}$, $A_{2}$… $A_{q}$ (nontrivial) fluxons labels in
these plaquettes. There will still be a degeneracy which comes from the
different possibilities how these fluxons can fuse together.

From the previous Sec. IV.1, one might guess that the corresponding degeneracy
is given by $\dim_{{\cal Z}({\cal C})}(g;A_{1},\ldots,A_{q})$ defined in Eq.
(13), where the UMTC in question here is ${\cal Z}({\cal C})$. In the case
where $\cal C$ has commutative fusion rules, this guess is in fact correct.
However, more generally, the number of states built from the category $\cal C$
on a surface of genus $g$ with fluxons $A_{i}$ through plaquettes $i=1\ldots
q$ is given by

$\widetilde{\dim}_{\mathcal{C}}(g;A_{1},\ldots,A_{q})=\dim_{{\cal Z}({\cal
C})}(g;A_{1},\ldots,A_{q})\prod_{i=1}^{q}n_{A_{i},1},$ (16)

where the first term on the right-hand side is given by Eq. (13) and where we
have used the property of Eq. (15) that plaquettes in the vacuum state do not
contribute to the degeneracy. The second term is made from the $n_{A_{i},1}$
degeneracy factors assigned to fluxons in Sec. III.4 above. These internal
multiplicities depend on the fluxons and, as explained in Sec. III.2, they may
be larger than 1 for noncommutative $\cal C$. Equation (16) is one of the key
statements made in this work.

The intuition behind Eq. (16) comes from an understanding that the string
operators for quasiparticles are actually built from the tube algebra (see
Appendix A). For each object type in ${\cal Z}({\cal C})$ there are multiple
tube types, and this provides the additional degeneracy factor $n_{A,1}$. The
reason that the second index is 1 is that the end of the string operator
should not have a vertex violation in our model (we consider plaquette
violations only) which requires the tube end to be labeled with $1\in{\cal C}$
only.

From Eq. (16), it is then easy to compute the total degeneracy of the
$q$-fluxon level with energy $E_{q}=-N_{\text{p}}+q$, since it is enough to
sum over all possibilities for the labels $A_{1}\ldots A_{q}$. Up to a
combinatorial prefactor given by the binomial coefficient
$\left(\begin{array}[]{c}N_{\mathrm{p}}\\\ q\end{array}\right)$ (which
accounts for the number of ways to choose $q$ excited plaquettes among
$N_{\text{p}}$ plaquettes), the degeneracy on a genus $g$ surface with $b=0$
holes is then given by

$\displaystyle D_{{\cal C}}(g,0,q)$ $\displaystyle=$
$\displaystyle\sum_{A_{1},\ldots,A_{q}\in\mathcal{F}^{*}}\widetilde{\dim}_{\mathcal{C}}(g;A_{1},\ldots,A_{q}),$
(17) $\displaystyle=$ $\displaystyle\quad\sum_{A\in{\cal Z}({\cal
C})}S_{\mathbf{1},A}^{2-2g-q}(n_{A,1}-S_{\mathbf{1},A})^{q},$

where the final form is obtained by using Eqs. (16)-(13), and (11). By adding
the proper local operator to the Hamiltonian, it is possible to identify the
fluxon labels present in the different plaquettes. Therefore, this total
degeneracy carries some additional nontopological degeneracy with respect to
Eq. (16). Thus in the presence of small amounts of “disorder”, the degeneracy
calculated in Eq. (17) is genericallly lifted. The above equation for the
degeneracy of a level with $q$-fluxons and involving the internal
multiplicities $n_{A,1}$ is the main result of the present work.

####  IV.2.1 Manifold with boundaries

We now extend the previous results to string-net models on surfaces with
boundaries. For a given topological order, there are several types of possible
gapped boundaries [39, 43]. Here, we only consider smooth boundaries
corresponding to the condensation of the fluxons. Consider starting with an
orientable manifold without boundary of genus $g$. Then we can imagine poking
$b$ (potentially large) holes in this manifold to obtain a manifold with
boundary. These holes are essentially (potentially large) plaquettes
themselves — the only distinction between a boundary hole and a regular
plaquette is that in Eq. (10) we sum over only plaquettes that are not these
boundary holes, i.e., the energy cost of a boundary hole is zero.

To understand this additional degeneracy, we can start by looking at its
contribution in the ground state. The ground-state degeneracy of the string-
net model is given by summing over all possible fluxons (including the vacuum
fluxon) that can be present in each hole. All other plaquettes are assumed to
be in the ground state, i.e, have the vacuum fluxon quantum number
$\mathbf{1}$. In other words, the number $q$ of excited fluxons is $0$. As
mentioned above, if a plaquette is known to be in the vacuum state, it does
not contribute to any degeneracy. The ground-state degeneracy is

$\displaystyle D_{{\cal C}}(g,b,0)$ $\displaystyle=$
$\displaystyle\sum_{A_{1},\ldots,A_{b}\in\mathcal{F}}\widetilde{\dim}_{\mathcal{C}}(g;A_{1},\ldots,A_{b}),$
(18) $\displaystyle=$ $\displaystyle\>\>\>\sum_{A\in{\cal Z}({\cal
C})}S_{\mathbf{1},A}^{2-2g-b}n_{A,1}^{b}.$

Note that, apart from the substitution of $b$ with $q$, the only difference
between Eq. (18) and Eq. (17) comes from the fact that we sum either on
$\mathcal{F}$ or on $\mathcal{F^{*}}$. Equation (18) should be considered as a
generalization of Eq. (14) to the case with boundaries and with internal
multiplicities. In the absence of boundary, i.e., for $b=0$, one recovers Eq.
(14):

$\displaystyle D_{{\cal C}}(g,0,0)$ $\displaystyle=$
$\displaystyle\dim_{\mathcal{Z}(\mathcal{C})}(g)=\sum_{A\in\mathcal{Z}(\mathcal{C})}S_{\mathbf{1},A}^{2-2g},$
(19)

where, by convention, we set $0^{0}=1$.

We are now ready to compute the degeneracies of the excited states in the most
general situation. This essentially merges the calculations of the previous
two sections. We choose $q$ plaquettes that can have any non-vacuum fluxon
(punctures labeled from $1$ to $q$), whereas the $b$ boundaries (punctures
labeled from $q+1$ to $b$) can have any fluxon including the vacuum.

Up to the binomial factor $\left(\begin{array}[]{c}N_{\mathrm{p}}\\\
q\end{array}\right)$ discussed before Eq. (17), the degeneracy of the
excitation level, $E_{q}=-N_{\text{p}}+q$, is then

$\displaystyle D_{{\cal C}}(g,b,q)$ $\displaystyle=$
$\displaystyle\sum_{\scriptsize\begin{array}[]{c}A_{1},\ldots,A_{q}\in\mathcal{F}^{*}\\\
A_{q+1},\ldots,A_{q+b}\in\mathcal{F}\end{array}}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\widetilde{\dim}_{\mathcal{C}}(g;A_{1},\ldots,A_{q+b}),$
(22) $\displaystyle=$ $\displaystyle\sum_{A\in{\cal Z}({\cal
C})}S_{\mathbf{1},A}^{2-2g-q-b}(n_{A,1}-S_{\mathbf{1},A})^{q}\,\,n_{A,1}^{b},\quad\quad$
(23)

which coincides with Eq. (17) in the compact case ($b=0$), as it should.

####  IV.2.2 Total Hilbert-space dimension

To end this section, let us calculate the total dimension of the Hilbert space
$\mathcal{H}$ of a string-net model built from a category $\cal C$ on an
orientable surface of genus $g$ with $N_{\text{p}}$ plaquettes, $b$
boundaries, and without vertex defects. This is the total Hilbert-space
dimension remaining at finite energy once we have taken $J_{v}$ to infinity in
the Hamiltonian Eq. (8). One approach to this is to sum over the degeneracy of
the string-net model for all possible fluxon states (including the vacuum
$\mathbf{1}$) being present in all ($N_{\text{p}}+b$) “punctures”. Thus, we
have

$\displaystyle\dim\mathcal{H}$ $\displaystyle=$
$\displaystyle\sum_{A_{1},\ldots,A_{N_{\text{p}}+b}\in{\cal Z}({\cal
C})}\widetilde{\dim}_{\mathcal{C}}(g;A_{1},\ldots,A_{N_{\text{p}}+b}),\quad$
(24) $\displaystyle=$ $\displaystyle\sum_{A\in{\cal Z}({\cal
C})}S_{\mathbf{1},A}^{2-2g-(N_{\text{p}}+b)}n_{A,1}^{N_{\text{p}}+b},$ (25)
$\displaystyle=$ $\displaystyle D_{{\cal C}}(g,N_{\text{p}}+b,0),$ (26)

where we have used Eq. (16) and the fact that $n_{A,1}$ is nonzero only if $A$
is a fluxon (including the identity), i.e., $A\in\mathcal{F}$.

For a trivalent graph, one further has $N_{\text{e}}=\frac{3}{2}N_{\text{v}}$,
where $N_{\text{e}}$ and $N_{\text{v}}$ denote the number of edges and the
number of vertices, respectively. The Euler-Poincaré characteristic on a
genus-$g$ surface with $b$ boundaries, is then given by

$\chi=2-2g-b=N_{\text{p}}-N_{\text{e}}+N_{\text{v}}=N_{\text{p}}-\frac{1}{2}N_{\text{v}}.$
(27)

Thus, we obtain the delightfully simple result for the Hilbert-space dimension
of any generalized string-net model:

$\dim\mathcal{H}=\sum_{A\in{\cal Z}({\cal
C})}S_{\mathbf{1},A}^{-N_{\text{v}}/2}n_{A,1}^{N_{\text{p}}+b}.$ (28)

One straightforwardly sees that for commutative UFCs ($n_{A,1}=0$ or $1$),
this dimension only depends on the number of vertices whereas, in the
noncommutative case for which $n_{A,1}$ can be larger than 1,
$\dim\mathcal{H}$ also depends on $N_{\text{p}}$. Hence, in this latter case,
the Hilbert-space dimension is sensitive to the surface topology.

Replacing $N_{\text{p}}+b$ by $2-2g+N_{\text{v}}/2$ and keeping in mind that
$S_{\mathbf{1},A}=d_{A}/\mathcal{D}$ and $n_{A,1}\leq d_{A}$, one gets, in the
thermodynamic limit:

$\dim\mathcal{H}\simeq\mathcal{D}^{N_{\text{v}}/2}\sum_{{\rm
pure}\>A}n_{A,1}^{2-2g},$ (29)

where “pure” fluxons are fluxons for which $n_{A,1}=d_{A}$.

Another way of obtaining this result is to return to the original string-net
graph (as shown in Fig. 1) and label all possible edges with simple objects of
$\cal C$ in all possible ways such that all the vertex conditions are
satisfied (and counting fusion multiplicities $N_{abc}$ at each satisfied
vertex). The general proof that this counting gives the same result as Eq.
(28) is given in Appendix E for the case $b=0$.

##  V Special cases

In the previous sections, all results were given in terms of data from the
Drinfeld center and the tube algebra. Yet, for some special cases, we can also
write Eq. (23) completely or partially in terms of data from the input
category. We will discuss these cases here, namely when the input category is
a UMTC and when the input category is commutative.

###  V.1 Modular categories

When the input category of the string-net model is modular, it is possible to
compute the degeneracies of the excited levels by using only the fusion
properties of the input category. The derivation of this equation is presented
in Ref. [25], where the partition function is computed from a microscopic
point of view for string-net models built from an input category $\mathcal{C}$
which is a UMTC. We aim here at making the link between the results of the
previous sections and the ones of Ref. [25]. In the case where the input
category $\mathcal{C}$ is a UMTC, the Drinfeld center is the product of two
copies of $\mathcal{C}$ of opposite chiralities [45]:

$\mathcal{Z}(\mathcal{C})=\mathcal{C}\times\bar{\mathcal{C}}.$ (30)

In this case, any object $A$ of the center can be represented by a couple
$(i,j)$ of objects $i\in\mathcal{C}$ and $j\in\bar{\mathcal{C}}$. The
$S$-matrix of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ has a simple
expression in terms of the $S$-matrix of the input category $\mathcal{C}$ (for
clarity, we will denote them respectively $S$ and $s$):

$S_{A,B}=S_{(i,j),(p,q)}=s_{i,p}s^{*}_{j,q}.$ (31)

A particle $A=(i,j)$ of $\mathcal{Z}(\mathcal{C})$ is a fluxon iff $i=j$ and
$n_{A,1}=1$, so that $S_{\mathbf{1},A}=s_{1,i}^{2}$. In all other cases,
$n_{A,1}=0$. Thus, when $b=0$, we can rewrite (22) as

$\displaystyle D_{{\cal C}}(g,0,q)=$
$\displaystyle\sum_{i,j\in\mathcal{C}}(s_{1,i}s_{1,j})^{\chi}\left(\frac{\delta_{i,j}}{s_{1,i}s_{1j}}-1\right)^{q},$
$\displaystyle=$
$\displaystyle(-1)^{q}\Big{\\{}\sum_{i\in\mathcal{C}}s_{1,i}^{2\chi}[(1-s_{1,i}^{-2})^{q}-1]+D_{{\cal
C}}(g,0,0)\Big{\\}},$

where $D_{{\cal
C}}(g,0,0)=\bigg{(}\sum_{i\in\mathcal{C}}s_{1,i}^{\chi}\bigg{)}^{2}$ is the
ground-state degeneracy. Up to the usual binomial factor, Eq. (LABEL:julien)
is exactly Eq. (10) in Ref. [25].

###  V.2 Commutative categories

The previous equations do not hold in the case where the input category is
commutative but not modular, since we cannot define a modular $S$-matrix.
However, for any commutative input category, we can always find a unitary
matrix $\tilde{s}$, called the mock $S$-matrix, that simultaneously
diagonalizes its fusion matrices $N_{i}$. A Verlinde-like formula (6) holds
between the $N_{i}$ and $\tilde{s}$, see Eq. (86). The columns of the matrix
$\tilde{s}$ can be indexed by the fluxons $A$ in $\mathcal{Z}(\mathcal{C})$
(for details on the mock $S$-matrix and the “mixed” notation for the matrix
elements $\tilde{s}_{i,A}$ with $i\in\mathcal{C}$ and $A\in\mathcal{F}$, see
Appendix C.1.1). In particular, if $A$ is a fluxon,
$S_{\mathbf{1},A}=d_{A}/\mathcal{D}=\tilde{s}_{1,A}^{2}$. Also, for any
commutative UFC, the multiplicities $n_{A,1}$ only take values 0 or 1, and the
number of fluxons is equal to the number of simple objects in the input
category. With these insights at hand, we can rewrite Eq. (23) as

$D_{{\cal
C}}(g,0,q)=(-1)^{q}\Big{\\{}\sum_{A\in\mathcal{F}}\tilde{s}_{1,A}^{2\chi}[(1-\tilde{s}_{1,A}^{-2})^{q}-1]+D_{{\cal
C}}(g,0,0)\Big{\\}},$ (33)

which is very similar to Eq. (LABEL:julien) with $s_{1,i}$ replaced by
$\tilde{s}_{1,A}$ and fluxons labeled by $A\in\mathcal{F}$ instead of
$i\in\mathcal{C}$. However, in the present case, there is no general formula
for the ground-state degeneracy $D_{{\cal C}}(g,0,0)$ in terms of $\tilde{s}$.
One could think of using Eq. (19) to compute the ground-state degeneracy with
$S_{\mathbf{1},A}=\tilde{s}_{1,A}^{2}$, but this relation only holds for
fluxons and not for all $A\in\mathcal{Z(C)}$.

Generally speaking, it is not possible to compute the ground-state degeneracy
$D_{{\cal C}}(g,0,0)$ simply from the knowledge of the fusion rules of the
input category. In fact, two input categories that have the same fusion rules
but different $F$-symbols generally lead to two different Drinfeld centers. In
particular, the number of simple objects of the Drinfeld centers need not be
the same [46]. And even if the numbers match, the corresponding quantum
dimensions $d_{A}$ need not be equal and therefore the level degeneracies can
differ. However, there still is a way to write $D_{{\cal C}}(g,0,0)$ only in
terms of the input data. This way, which is a bit cumbersome, requires the use
of the $F$-symbols of the category (see Ref. [29]).

##  VI Examples

This section is devoted to non-trivial examples that illustrate the formulas
obtained in the previous sections. Before discussing some specific examples,
let us give some general results that are valid for any input category ${\cal
C}$. In Table 1, we give the degeneracies $D_{{\cal C}}(g,b,q)$ [see Eq. (23)]
of the first three energy levels for various simple surface topologies.

$D_{{\mathcal{C}}}$ | $(g,b)=(0,0)$ | $(0,1)$ | $(0,2)$ | $(1,0)$
---|---|---|---|---
$q=0$ | 1 | 1 | $N_{\mathcal{C}}$ | $N_{\mathcal{Z}}$
1 | 0 | $N_{\mathcal{C}}-1$ |  |
2 | $N_{\mathcal{C}}-1$ |  |  |
Table 1: Degeneracies $D_{{\cal C}}(g,b,q)$ of the $q$-th excited level (row
index: $q=0,1,2$) for various surface topologies characterized by their genus
$g$ and their $b$ boundaries [column index $(g,b)$]. The number of simple
objects in the input category $\mathcal{C}$ and in its Drinfeld center
$\mathcal{Z}(\mathcal{C})$ are denoted by $N_{\mathcal{C}}$ and
$N_{\mathcal{Z}}$, respectively. Empty entries depend on the input category
details.

To obtain these results, we used Eq. (11) which leads to

$\sum_{A\in{\cal Z}({\cal C})}S_{\mathbf{1},A}n_{A,1}=n_{\mathbf{1},1}=1,$
(34)

and the following identity:

$\sum_{A}n_{A,1}^{2}=N_{\mathcal{C}},$ (35)

which is shown in Appendix A.

Whenever possible, i.e., if the Hilbert space is not too large, we checked
these degeneracies by exact diagonalizations of the Hamiltonian (10) on
trivalent graphs with the corresponding topology. For instance, starting from
a cube ($g=b=0$, and $N_{\text{v}}=8$), we can build a “disk” ($g=0$, $b=1$)
by removing one plaquette operator in the Hamiltonian or a “cylinder” ($g=0$,
$b=2$) by removing two opposite plaquette operators.

###  VI.1 Rep($S_{3}$) and Vec($S_{3}$) categories

We present here two string-net models built from two Morita-equivalent
categories: Rep($S_{3}$) and Vec($S_{3}$), where $S_{3}$ is the symmetric
group on a set of 3 elements (symmetry group of the oriented equilateral
triangle). They correspond respectively to the category of irreducible
representations of $S_{3}$ and the category of the elements of the group
$S_{3}$.

The category $\mathcal{C}$ = Rep($S_{3}$) contains $N_{\mathcal{C}}=3$ simple
objects $\\{1,2,3\\}$ with quantum dimensions $\\{1,1,2\\}$, which correspond
to the irreducible representations of $S_{3}$. Fusion rules and $F$-symbols
can be found, e.g., in Ref. [10]. The fusion rules are commutative but the
category is non-Abelian, braided, and non modular.

The category $\mathcal{C}$ = Vec($S_{3}$) contains $N_{\mathcal{C}}=6$ simple
objects $\\{e,\zeta,\zeta^{2},\tau,\tau\zeta,\tau\zeta^{2}\\}$, which are the
elements of the group $S_{3}$, ($e$ is the identity element, $\zeta$ is a
$2\pi/3$ rotation and $\tau$ is a mirror). The fusion rules are simply the
multiplication rules of the group. All elements have quantum dimension
$d_{s}=1$. The fusion rules are noncommutative but the category is Abelian.

The two categories are Morita-equivalent and lead to the same Drinfeld center
$\mathcal{Z}(S_{3})$. The latter contains $N_{\mathcal{Z}}=8$ simple objects,
denoted by $\\{A,B,C,D,E,F,G,H\\}$, see Ref. [47]. The quantum dimensions are
$\\{1,1,2,3,3,2,2,2\\}$, and the total quantum dimension is $\mathcal{D}=6$.

While the two categories share the same center $\mathcal{Z}(S_{3})$, the tube
algebras are different (see Appendix B). In particular, the internal
multiplicities of the quasiparticles, $n_{J,s}$ differ. We give them as
rectangular matrices $n_{\mathcal{C}}$ with $N_{\mathcal{C}}$ columns and
$N_{\mathcal{Z}}$ rows:

$n_{\text{Rep}(S_{3})}=\begin{pmatrix}1&0&0\\\ 0&1&0\\\ 0&0&1\\\ 1&0&1\\\
0&1&1\\\ 1&1&0\\\ 0&0&1\\\ 0&0&1\\\ \end{pmatrix},$ (36)
$n_{\text{Vec}(S_{3})}=\begin{pmatrix}1&0&0&0&0&0\\\ 1&0&0&0&0&0\\\
2&0&0&0&0&0\\\ 0&0&0&1&1&1\\\ 0&0&0&1&1&1\\\ 0&1&1&0&0&0\\\ 0&1&1&0&0&0\\\
0&1&1&0&0&0\\\ \end{pmatrix}.$ (37)

Quantum dimensions of the quasiparticles in the center and the multiplicities
are the only quantities we need to compute the degeneracies of any energy
level with Eq. (23). In both cases there are three types of fluxons
(quasiparticles $J$ with $n_{J,1}>0$) but they correspond to two different
subsets of the simple objects of $\mathcal{Z}(S_{3})$: $A,D,F$ for
Rep($S_{3}$), and $A,B,C$ for Vec($S_{3}$) (note that $n_{C,1}=2$). As a
consequence, the spectral degeneracies of the two models are different (see
Tables 2 and 3).

$D_{{\rm Rep}(S_{3})}$ | $(g,b)=(0,0)$ | $(0,1)$ | $(0,2)$ | $(1,0)$
---|---|---|---|---
$q=0$ | 1 | 1 | 3 | 8
1 | 0 | 2 | 8 | 3
2 | 2 | 6 | 30 | 35
3 | 4 | 24 | 134 | 129
4 | 20 | 110 | 642 | 647
Table 2: Degeneracies of the $q^{th}$ excited state of a string net built from Rep($S_{3}$) for various surface topologies up to $q=4$. Here $g$ is the genus and $b$ is the number of boundaries.  $D_{{\rm Vec}(S_{3})}$ | $(g,b)=(0,0)$ | $(0,1)$ | $(0,2)$ | $(1,0)$
---|---|---|---|---
$q=0$ | 1 | 1 | 6 | 8
1 | 0 | 5 | 30 | 10
2 | 5 | 25 | 150 | 80
3 | 20 | 125 | 750 | 370
4 | 105 | 625 | 3750 | 1880
Table 3: Degeneracies of the $q^{th}$ excited state of a string net built from
Vec($S_{3}$) for various surface topologies up to $q=4$. Here $g$ is the genus
and $b$ is the number of boundaries.

The dimension of the Hilbert space [see Eq. (28)] is

$\dim\mathcal{H}=\left(1+3^{-N_{\text{v}}/2}+2^{-N_{\text{v}}/2}\right)\sqrt{6}^{N_{\text{v}}},$
(38)

for Rep($S_{3}$) and

$\dim\mathcal{H}=(1+1+2^{2-2g})\sqrt{6}^{N_{\text{v}}},$ (39)

for Vec($S_{3}$). We provide a few values of $\dim\mathcal{H}$ in Table 4 for
some simple trivalent graphs.

$\dim\mathcal{H}$ | $(g,N_{\text{v}})=(0,2)$ | (1,2) | (0,4) | (1,4) | (0,6) | (1,6)
---|---|---|---|---|---|---
Rep($S_{3}$) | 11 | 11 | 49 | 49 | 251 | 251
Vec($S_{3}$) | 36 | 18 | 216 | 108 | 1296 | 648
TY3 | 18 | 18 | 90 | 90 | 486 | 486
$\mathcal{H}_{3}$ | 63 | 45 | 1431 | 1323 | 46494 | 45846
Table 4: Hilbert-space dimension $\dim\mathcal{H}$ for some simple compact
surfaces ($b=0$) and for some input categories. Each row corresponds to a
given category and each column is indexed by $(g,N_{\text{v}})$ with $g$ the
genus and $N_{\text{v}}$ the number of vertices.

The fusion matrices of the commutative category Rep$(S_{3})$ are
simultaneously diagonalized by the following mock $S$-matrix:

$\tilde{s}=\frac{1}{\sqrt{6}}\left(\begin{array}[]{ccc}1&\sqrt{3}&\sqrt{2}\\\
1&-\sqrt{3}&\sqrt{2}\\\ 2&0&-\sqrt{2}\end{array}\right).$ (40)

This matrix contains the transformation $\tilde{s}_{i,J}$ between the simple
objects $i\in\\{1,2,3\\}=\mathcal{C}$ of the input category and the fluxons
$J\in\\{A,D,F\\}=\mathcal{F}$ of the output category $\mathcal{Z(C)}$. Here,
it was chosen such that the first row only contains strictly positive
elements, in which case, one has $\tilde{s}_{1,J}=\sqrt{d_{J}/\mathcal{D}}$
and $\tilde{s}_{i,\mathbf{1}}=d_{i}/\sqrt{\mathcal{D}}$ (see Appendix C.1.1).

In conclusion, Rep($S_{3}$) and Vec($S_{3}$) are two Morita-equivalent UFCs
that correspond to the same Drinfeld center but give different Hilbert spaces
and spectral degeneracies (and hence have different partition functions).

Note that, in contrast to the case presented here, there also exist UFCs
corresponding to different Drinfeld centers that can give rise to the same
energy spectrum and level degeneracies. This is the case, e.g., for
Vec($\mathbb{Z}_{2}$) and the semion theory Vec${}^{\omega}(\mathbb{Z}_{2})$
(nontrivial cocycle of $\mathbb{Z}_{2}$).

###  VI.2 Tambara-Yamagami category TY3

The Tambara-Yamagami category for $\mathds{Z}_{3}$ (notated TY3) is
interesting as it breaks the tetrahedral symmetry and therefore can only be
realized by the generalized string-net model (see for example [14]) and not by
the original Levin-Wen model [10]. It is commutative (but non-Abelian and non-
braided), so that $n_{A,1}=1$ for all fluxons. It has $N_{\mathcal{C}}=4$
simple objects $\\{1,2,3,\sigma\\}$, with quantum dimensions
$\\{1,1,1,\sqrt{3}\\}$. There are two different solutions to the pentagonal
equations indexed by $p=\pm 1$ (see [14] for more details on the $F$-symbols).
The Drinfeld center $\mathcal{Z}({\rm TY}_{3})$ contains $N_{\mathcal{Z}}=15$
elements $\\{1,...,15\\}$ with dimensions
$\\{1,1,1,1,1,1,2,2,2,\sqrt{3},\sqrt{3},\sqrt{3},\sqrt{3},\sqrt{3},\sqrt{3}\\}$
and the total quantum dimension is $\mathcal{D}=6$. For the $p=1$ model, the
internal multiplicities are:

$n_{{\rm TY}_{3}}=\begin{pmatrix}1&0&0&0\\\ 1&0&0&0\\\ 0&1&0&0\\\ 0&1&0&0\\\
0&0&1&0\\\ 0&0&1&0\\\ 1&1&0&0\\\ 1&0&1&0\\\ 0&1&1&0\\\ 0&0&0&1\\\ 0&0&0&1\\\
0&0&0&1\\\ 0&0&0&1\\\ 0&0&0&1\\\ 0&0&0&1\\\ \end{pmatrix}.$ (41)

The fluxons are therefore $\mathbf{1},2,7$ and $8$ and, using Eq. (23), we can
easily compute the degeneracies. Some examples are given in Table 5.

$D_{{\rm TY}_{3}}$ | $(g,b)=(0,0)$ | $(0,1)$ | $(0,2)$ | $(1,0)$
---|---|---|---|---
$q=0$ | 1 | 1 | 4 | 15
1 | 0 | 3 | 14 | 3
2 | 3 | 11 | 58 | 69
3 | 8 | 47 | 266 | 255
4 | 39 | 219 | 1282 | 1293
Table 5: Degeneracies of the $q^{th}$ excited state of a string net built from
TY3 for various surface topologies up to $q=4$. Here $g$ is the genus and $b$
is the number of boundaries.

The Hilbert-space dimension [see Eq. (28)] is

$\dim\mathcal{H}=2\left(1+2^{-N_{\text{v}}/2}\right)\sqrt{6}^{N_{\text{v}}},$
(42)

and a few examples are given in Table 4. Interestingly, this topological order
breaks time-reversal symmetry [14] but still has a vanishing topological
central charge ($c\mod 8=0$).

The fusion matrices of TY3 are simultaneously diagonalized by the following
mock $S$-matrix:

$\tilde{s}=\frac{1}{\sqrt{6}}\left(\begin{array}[]{cccc}1&1&\sqrt{2}&\sqrt{2}\\\
1&1&\sqrt{2}\mathrm{e}^{\mathrm{i}2\pi/3}&\sqrt{2}\mathrm{e}^{-\mathrm{i}2\pi/3}\\\
1&1&\sqrt{2}\mathrm{e}^{-\mathrm{i}2\pi/3}&\sqrt{2}\mathrm{e}^{\mathrm{i}2\pi/3}\\\
\sqrt{3}&-\sqrt{3}&0&0\end{array}\right).$ (43)

It contains the transformation $\tilde{s}_{i,A}$ between the simple objects
$i\in\\{1,2,3,\sigma\\}=\mathcal{C}$ of the input category and the fluxons
$A\in\\{\mathbf{1},2,7,8\\}=\mathcal{F}$ of the output category
$\mathcal{Z(C)}$.

###  VI.3 Haagerup category $\mathcal{H}_{3}$

The Haagerup category $\mathcal{H}_{3}$ is a good example of the universality
of our formula: it is neither commutative, nor braided, nor Abelian, nor does
it respect tetrahedral symmetry, see e.g. [24]. It has $N_{\mathcal{C}}=6$
simple objects $\\{1,\alpha,\alpha^{*},\rho,\alpha\rho,\alpha^{*}\rho\\}$ with
quantum dimensions $\\{1,1,1,d_{\rho},d_{\rho},d_{\rho}\\}$ where
$d_{\rho}=\frac{3+\sqrt{13}}{2}$. The Drinfeld center $\mathcal{H}_{3}$
contains $N_{\mathcal{Z}}=12$ simple objects
$\\{\mathbf{1},\mu^{1},\mu^{2},\mu^{3},\mu^{4},\mu^{5},\mu^{6},\pi_{1},\pi_{2},\sigma^{1},\sigma^{2},\sigma^{3}\\}$
with quantum dimensions
$\\{1,3d_{\rho},3d_{\rho},3d_{\rho},3d_{\rho},3d_{\rho},3d_{\rho},3d_{\rho}+1,3d_{\rho}+2,3d_{\rho}+2,3d_{\rho}+2,3d_{\rho}+2\\}$
so that the total quantum dimension is $\mathcal{D}=3(1+d_{\rho}^{2})$ [24].
The internal multiplicities are given by [48]

$n_{\mathcal{H}_{3}}=\left(\begin{tabular}[]{cccccc}1&0&0&0&0&0\\\
0&0&0&1&1&1\\\ 0&0&0&1&1&1\\\ 0&0&0&1&1&1\\\ 0&0&0&1&1&1\\\ 0&0&0&1&1&1\\\
0&0&0&1&1&1\\\ 1&0&0&1&1&1\\\ 2&0&0&1&1&1\\\ 0&1&1&1&1&1\\\ 0&1&1&1&1&1\\\
0&1&1&1&1&1\end{tabular}\right),$ (44)

so that the three fluxons are $\mathbf{1}$, $\pi_{1}$, and $\pi_{2}$. Some
degeneracies computed from Eq. (23) are given in Table 6.

$D_{\mathcal{H}_{3}}$ | $(g,b)=(0,0)$ | $(0,1)$ | $(0,2)$ | $(1,0)$
---|---|---|---|---
$q=0$ | 1 | 1 | 6 | 12
1 | 0 | 5 | 57 | 33
2 | 5 | 52 | 1311 | 1245
3 | 47 | 1259 | 42384 | 42000
4 | 1212 | 41125 | 1456539 | 1454673
Table 6: Degeneracies of the $q^{th}$ excited state of a string net built from
$\mathcal{H}_{3}$ for various surface topologies up to $q=4$. Here $g$ is the
genus and $b$ is the number of boundaries.

The Hilbert-space dimension [see Eq. (28)] is given by

$\dim\mathcal{H}=\left[1+\frac{1}{(3d_{\rho}+1)^{\frac{N_{\text{v}}}{2}}}+\frac{2^{2-2g+\frac{N_{\text{v}}}{2}}}{(3d_{\rho}+2)^{\frac{N_{\text{v}}}{2}}}\right]\mathcal{D}^{\frac{N_{\text{v}}}{2}},$
(45)

(see also Table 4).

###  VI.4 Hagge-Hong category $\mathcal{E}$

This is a simple example of an input category $\mathcal{C}=\mathcal{E}$ with
fusion multiplicities [24]. It is commutative, not braided and it breaks the
tetrahedral symmetry. It contains $N_{\mathcal{C}}=3$ simple objets
$\\{1,x,y\\}$ with quantum dimensions $\\{1,d_{x},1\\}$, where
$d_{x}=\sqrt{3}+1$ (see Ref. [24] for more details). The Drinfeld center
$\mathcal{Z}(\mathcal{E})$ contains $N_{\mathcal{Z}}=10$ simple objects
$\\{\mathbf{1},Y,X_{1},X_{2},X_{3},X_{4},X_{5},U,V,W\\}$ with quantum
dimensions $\\{1,1,d_{x},d_{x},d_{x},d_{x},d_{x},d_{x}+1,d_{x}+1,d_{x}+2\\}$
so that $\mathcal{D}=2d_{x}+4$. Internal multiplicities are given by [24]

$n_{\mathcal{E}}=\left(\begin{tabular}[]{ccc}1&0&0\\\ 0&0&1\\\ 0&1&0\\\
0&1&0\\\ 0&1&0\\\ 0&1&0\\\ 0&1&0\\\ 1&1&0\\\ 0&1&1\\\
1&1&1\end{tabular}\right),$ (46)

so that the three fluxons are $\mathbf{1}$, $U$, and $W$. This allows us to
compute the degeneracies for a few systems (see Table 7).

$D_{\mathcal{E}}$ | $(g,b)=(0,0)$ | $(0,1)$ | $(0,2)$ | $(1,0)$
---|---|---|---|---
$q=0$ | 1 | 1 | 3 | 10
1 | 0 | 2 | 11 | 4
2 | 2 | 9 | 75 | 82
3 | 7 | 66 | 611 | 604
4 | 59 | 545 | 5139 | 5146
Table 7: Degeneracies of the $q^{th}$ excited state of a string net built from
$\mathcal{E}$ for various surface topologies up to $q=4$. Here $g$ is the
genus and $b$ is the number of boundaries.

The Hilbert-space dimension [see Eq. (28] is therefore

$\dim\mathcal{H}=\left[1+(\sqrt{3}+2)^{\frac{-N_{\text{v}}}{2}}+(\sqrt{3}+3)^{\frac{-N_{\text{v}}}{2}}\right]\mathcal{D}^{\frac{N_{\text{v}}}{2}},$
(47)

(see Table 4).

The mock $S$-matrix is

$\tilde{s}=\frac{1}{\sqrt{2d_{x}+4}}\left(\begin{array}[]{ccc}1&\sqrt{d_{x}+1}&\sqrt{d_{x}+2}\\\
d_{x}&-\sqrt{2}&0\\\ 1&\sqrt{d_{x}+1}&-\sqrt{d_{x}+2}\end{array}\right),$ (48)

where $\tilde{s}_{i,A}$ with $i\in\\{1,x,y\\}$ and
$A\in\\{\mathbf{1},U,W\\}=\mathcal{F}$.

##  VII Conclusion and perspectives

In this work, we have studied the generalized string-net model with arbitrary
input category and restricted to the charge-free sector (all vertex
constraints are satisfied) so that only fluxons (plaquette excitations) are
present as real excitations at the single plaquette level. This does not
exclude the presence of other excitations resulting from the fusion of several
fluxons. The main results are analytical expressions for the level
degeneracies. In particular, in the case of a noncommutative input category,
we show that part of the degeneracy is contained in nontrivial factors
$n_{A,1}$ obtained from the tube algebra and not only from the output category
(Drinfeld center). An important result is the generalization of the Moore-
Seiberg-Banks formula, Eq. (16), that includes this nontopological degeneracy.
For example, two Morita-equivalent categories such as Rep$(S_{3})$ and
Vec$(S_{3})$ have the same Drinfeld center $\mathcal{Z}(S_{3})$ but the
corresponding string-net models have different fluxons and different Hilbert
spaces. In a forthcoming publication [49], we will use these degeneracies to
compute the partition function of the generalized string-net model and to
study its finite temperature properties.

Finally, we wish to use a similar approach to extend the computation of level
degeneracies to models (such as Kitaev quantum double [8] or extended string-
nets [17]) that have all elementary excitations of the Drinfeld center and not
only fluxons.

###### Acknowledgements.

We thank N. Bultinck, B. Douçot, S. Dusuel, L. Lootens, and J. Slingerland for
illuminating discussions. A.R.-Z. and J.-N.F. acknowledge financial support
from iMAT Sorbonne Université Emergence through a “bourse exploratoire”.
J.-N.F. acknowledges the support of the quantum information center at Sorbonne
Université (QICS) via an Emergence grant for the TQCNAA project. S.H.S.
acknowledges support from EPSRC Grant EP/S020527/1. Statement of compliance
with EPSRC policy framework on research data: This publication is theoretical
work that does not require supporting research data.

##  Appendix A Summary of the tube algebra

This appendix contains an introduction to the tube algebra $\mathcal{TA}$ of a
given UFC $\mathcal{C}$ and to its decomposition, as a way of finding the
topological quasiparticles, i.e., the simple objects of the Drinfeld center
$\mathcal{Z}(\mathcal{C})$. The tube algebra was introduced by Ocneanu [31,
32]. A detailed account is given by Lan and Wen [12], who call it the
$Q$-algebra, which we mainly follow. For a pedagogical introduction, see Ref.
[22].

Figure 10: Decomposing the tube algebra $\mathcal{TA}$ gives the half-
braidings $\Omega_{A}$ (or, almost equivalently, the minimal central
projectors $P_{A}$) from the input category $\mathcal{C}$. From there, the
Drinfeld center $\mathcal{Z}(\mathcal{C})$ follows, as well as the internal
multiplicities $n_{A,1}$ and the quantum dimensions $d_{A}$, that give access
to the energy-level degeneracies $D_{\mathcal{C}}$ of the string-net model.

Objects in $\mathcal{TA}$ are called tubes and denoted $Q$, following Ref.
[12]. The composition rules of these objects constitute the essence of the
tube algebra. From these composition rules, one builds a canonical
representation of $\mathcal{TA}$. This representation can be block-
diagonalized and the irreducible blocks correspond to the topological sectors.
Decomposing the tube algebra means finding its center
$\mathcal{Z}(\mathcal{TA})$, containing objects in $\mathcal{TA}$ that commute
with every tube. In other words, it means obtaining the projectors $P_{A}$
onto the topological quasiparticles $A\in\mathcal{Z}(\mathcal{C})$ as linear
combinations of the tubes $Q$. Almost equivalently, it means obtaining the
half-braidings $\Omega_{A}$, that appear in the standard definition of the
Drinfeld center $\mathcal{Z}(\mathcal{C})$.

From the half-braidings, it is easy to obtain all of the properties of the
Drinfeld center, e.g., the $S$ and $T$ matrices, and therefore the fusion
matrices, the quantum dimensions of the simple objects [10, 12]), and more
informations such as the internal multiplicities. The multiplicities $n_{A,1}$
and the quantum dimensions $d_{A}$ of the quasiparticles
$A\in\mathcal{Z}(\mathcal{C})$ are all that is needed to compute the
degeneracies of the energy levels of a string-net model [see Eq. (17) and Fig.
10].

In the following, we first define the tubes and obtain their algebra. Then, we
explain how to decompose the tube algebra in order to find the topological
quasiparticles and their internal multiplicities.

###  A.1 Tubes $Q$ and their algebra $\mathcal{TA}$

The main objects of $\mathcal{TA}$ are the tubes $Q_{rsj}^{i}\equiv irsj$ [see
Fig. 11(a)], where $i,r,s,j$ are simple objects of $\mathcal{C}$. A tube has
two open string ends, labeled by $r$ and $s$, and two strings without open
ends called $i$ and $j$. Here the vertex indices are dropped.

Figure 11: (a) Graphical representation of a tube $Q_{rsj}^{i}$. The vertical
dashed lines indicate periodic boundary conditions so that the rectangle is
topologically equivalent to a tube or cylinder. The crosses indicate the open
ends of the vertical strings. (b) Vertical stacking of two tubes
$Q^{i}_{rsj}Q^{k}_{s^{\prime}tl}$ The horizontal dashed line indicates the
stacking.

The only allowed tubes are those that respect the fusion rules of
$\mathcal{C}$, i.e., $i\times j=\sum_{k}N_{ij}^{k}\,k$ at their two inner
vertices. Therefore, the number of tubes $N_{\mathcal{T}}$, i.e. the dimension
of the tube algebra, is given by a product of two fusion matrices (applying to
the two vertices):

$N_{\mathcal{T}}=\sum_{i,j,r,s}N_{si}^{j}N_{j\bar{r}}^{i}.$ (49)

The tubes can be composed (or multiplied): $Q^{i}_{rsj}Q^{k}_{s^{\prime}tl}$
means that $Q^{k}_{s^{\prime}tl}$ is stacked on top of $Q^{i}_{rsj}$ [see Fig.
11(b)]. The $\mathcal{TA}$ has the following structure [12]

$Q^{i}_{rsj}Q^{k}_{s^{\prime}tl}=\sum_{m,n}\delta_{ss^{\prime}}\sqrt{\frac{d_{i}d_{k}}{d_{m}}}F^{i\overline{j}s}_{\overline{k}l\overline{n}}F^{\overline{r}\overline{i}j}_{\overline{k}n\overline{m}}F^{tk\overline{l}}_{i\overline{n}m}\,\,Q^{m}_{rtn},$
(50)

and is noncommutative in general. The coefficients in front of $Q^{m}_{rtn}$
in the sum on the right-hand side are the structure constants of the tube
algebra. One can observe that the stacking is determined by the labels $r$ and
$s$ of the open strings. Therefore, the tube algebra splits in different
sectors labeled by $rs$. This will play a role later in the decomposition of
the tube algebra.

Two important sets of tubes deserve a particular attention:

Figure 12: Two particular sets of tubes: (a) horizontal closed string
$Q_{11i}^{i}$; (b) vertical open string $Q^{1}_{rrr}$.

(1) the $N_{\mathcal{C}}$ tubes $Q_{11i}^{i}$ which have no open ends and are
formed by “horizontal” closed strings [see Fig. 12(a)]. Each such tube
corresponds to a simple object $i$ of $\mathcal{C}$. The $11$ sector consists
only of these tubes and the tube algebra restricted to the $11$ sector
reflects directly the algebra of the input category:

$Q_{11i}^{i}Q_{11j}^{j}=\sum_{k}N_{ij}^{k}Q_{11k}^{k}.$ (51)

(2) The $N_{\mathcal{C}}$ tubes $Q^{1}_{rrr}$ formed by a “vertical” open
string [see Fig. 12(b)] which correspond to the identity operator
$\mathds{1}_{rr}$ in the $rr$ sector of $\mathcal{TA}$:

$Q^{1}_{rrr}Q^{k}_{rrl}=Q^{k}_{rrl}Q^{1}_{rrr}=Q^{k}_{rrl}.$ (52)

Graphically, it is easy to see that stacking $Q^{1}_{rrr}$ on top or below of
any tube of the same sector returns that second tube. The identity over
$\mathcal{TA}$ is the sum over all of the $\mathds{1}_{rr}$:

$\mathds{1}=\sum_{r}\mathds{1}_{rr}=\sum_{r}Q^{1}_{rrr}.$ (53)

###  A.2 Decomposition of the tube algebra

The Wedderburn-Artin theorem states that a semi-simple algebra (such as
$\mathcal{TA}$) is isomorphic to a direct sum of simple matrix algebras (see,
e.g., Ref. [50]). It means that the tube algebra splits into $N_{\mathcal{Z}}$
matrix algebras $\mathcal{M}_{n}$ of dimension $n$:

$\mathcal{TA}\simeq\oplus_{n_{A}}\mathcal{M}_{n_{A}},$ (54)

with $A=1,...,N_{\mathcal{Z}}$. Each block corresponds to an irreducible
representation (or simple module) of the algebra and therefore to a simple
object (a topological quasiparticle) $A$ in the Drinfeld center
$\mathcal{Z}(\mathcal{C})$.

In the following, we represent the tube algebra in a vector space and search
for its block structure in order to identify the irreducible representations
$A$ and their dimensions $n_{A}$.

###  A.3 Simple idempotents and nilpotents $p$

A first step is to construct a vector space of dimension $V=\sum_{A}n_{A}$ in
which to represent the tube algebra by finding an orthonormal basis of vectors
$\\{|\alpha\rangle\\}$. When decomposing the tube algebra, one does not know a
priori the dimension $V$ of the vector space, the labels $A$ of the
irreducible representations, their dimensions $n_{A}$ or their total number
$N_{\mathcal{Z}}$. These are the output of the decomposition. Therefore, for
the moment, we do not give the final labels of these states and just use greek
letters such as $\alpha$ as a generic label (later, we will see that $\alpha$
can actually correspond to several labels).

The basis is found by building $V$ simple orthogonal idempotents (or
projectors) $p$ as linear combinations of the tubes $Q$. Simple means that
they cannot be further decomposed or, in other words, that they are projectors
onto one-dimensional subspaces. We denote simple idempotents by
$p^{\alpha\alpha}=|\alpha\rangle\langle\alpha|$ (known as “diagonal elements”,
as later we will also introduce “off-diagonal elements” like $p^{\alpha\beta}$
with $\beta\neq\alpha$) and they satisfy the orthogonality relation

$p^{\alpha\alpha}p^{\beta\beta}=\delta_{\alpha,\beta}\,p^{\alpha\alpha},$ (55)

and $\mathrm{Tr}\,p^{\alpha\alpha}=1$, meaning that the projected subspace is
one dimensional. The identity in the vector space is given by the sum over all
these simple idempotents:

$\mathds{1}=\sum_{\alpha}p^{\alpha\alpha}.$ (56)

The next step is to find which simple idempotents belong to the same
irreducible block or, in other words, which simple idempotents correspond to
the same topological quasiparticle $A$. In order to do so, we need to find
$(N_{\mathcal{T}}-V)$ nilpotents $p^{\alpha\beta}$ with $\alpha\neq\beta$,
which are also linear combinations of the tubes that satisfy
$(p^{\alpha\beta})^{2}=0$. They are traceless and connect one-dimensional
subspaces that belong to the same block. For example, if

$p^{\alpha\beta}p^{\beta\alpha}=p^{\alpha\alpha},\text{ and
}p^{\beta\alpha}p^{\alpha\beta}=p^{\beta\beta},$ (57)

it means that $|\alpha\rangle$ and $|\beta\rangle$ belong to the same
irreducible block (say $A$) and that
$p^{\alpha\beta}=|\alpha\rangle\langle\beta|=(p^{\beta\alpha})^{\dagger}$
(“off-diagonal elements”). Therefore, we denote $p_{A}^{\alpha\alpha}$,
$p_{A}^{\beta\beta}$, $p_{A}^{\alpha\beta}$ and $p_{A}^{\beta\alpha}$ the
simple idempotents and nilpotents that belong to the same irreducible block.

More generally, simple idempotents and nilpotents satisfy the following key
equation:

$p_{A}^{\alpha\beta}p_{B}^{\gamma\delta}=\delta_{A,B}\delta_{\beta,\gamma}p_{A}^{\alpha\delta}.$
(58)

In total, there are as many simple idempotents and nilpotents as tubes. When
we want to refer to them collectively, we will call them “idemnils” and use a
small $p$.

###  A.4 Minimal central idempotents $P_{A}$

Once we have found the $N_{\mathcal{T}}$ idemnils and we have identified the
ones that belong to the same irreducible block, we construct the minimal (or
simple) central idempotents $P_{A}$. Similarly to the simple idempotents, they
are orthogonal idempotents

$P_{A}P_{B}=\delta_{A,B}P_{A},$ (59)

but they also commute with any object of $\mathcal{TA}$, i.e., they belong to
the center $\mathcal{Z}(\mathcal{TA})$ of the tube algebra. In addition, they
are minimal, i.e., they cannot be decomposed further into a sum of other
central idempotents. By summing over all simple idempotents that belong to the
same irreducible block, we get

$P_{A}=\sum_{\alpha}p_{A}^{\alpha\alpha},$ (60)

which is the projector onto a topological quasiparticle $A$. The dimension of
the irreducible representation $A$ (or the multiplicity of $A$) is

$n_{A}=\mathrm{Tr}\,P_{A}.$ (61)

We can also define partial multiplicities such as

$n_{A,\alpha}=\mathrm{Tr}\,p_{A}^{\alpha\alpha}=0\text{ or }1,$ (62)

for the topological quasiparticle $A$ in the $\alpha$ sector. Here it is
important to realize that $\alpha$ may denote either a single label (this will
be the case for a commutative input category) or a pair of labels (this will
be the case for a noncommutative input category).

The number of minimal central idempotents (topological sectors),
$N_{\mathcal{Z}}$ is such that:

$N_{\mathcal{Z}}=\sum_{A}1\leqslant V=\sum_{A}n_{A}\leqslant
N_{\mathcal{T}}=\sum_{A}n_{A}^{2}.$ (63)

###  A.5 Tube algebra sectors

When decomposing the tube algebra, a substantial simplification comes from the
fact that, as we have seen, the tube algebra separates into independent
sectors labeled by the open ends $r$ and $s$ of a tube $Q^{i}_{rsj}$.
Therefore, the search for idemnils can be done sector by sector (indexed by
$rs$). A simple idempotent necessarily belongs to a diagonal sector $rr$,
whereas a nilpotent belongs to an off-diagonal sector $rs$ with $s\neq r$.

If the input category $\mathcal{C}$ is commutative, the tube algebra in a
given sector $rs$ is commutative so that the corresponding vector space has a
dimension $V^{rs}$ equal to the number of tubes

$N_{\mathcal{T}}^{rs}=\sum_{i,j}N_{si}^{j}N_{j\bar{r}}^{i}.$ (64)

By summing over all sectors $rs$, one recovers Eq. (49).

A simple idempotent is labeled as $p_{A}^{rr}$ and a nilpotent as $p_{A}^{rs}$
with $s\neq r$ (meaning that $\alpha$ is actually $r\in\mathcal{C}$ here).
Also, one has that

$n_{A,r}=\mathrm{Tr}\,p_{A}^{rr}=0\text{ or }1.$ (65)

If the input category $\mathcal{C}$ is noncommutative, the tube algebra in a
given $rs$ sector need not be commutative. If it is noncommutative, it means
that the corresponding vector space’s dimension $V^{rs}<N_{\mathcal{T}}^{rs}$
and that one needs to introduce an extra index $a$ to label orthonormal basis
vectors. More generally, a simple idempotent is labeled as $p_{A}^{rr,aa}$ and
a nilpotent as $p_{A}^{rs,ab}$ with $s\neq r$ or $b\neq a$ (meaning that
$\alpha$ is actually $r,a$ with $r\in\mathcal{C}$ and $a=1,...,n_{A,r}$ here).
We call $P_{A}^{rr}=\sum_{a}p_{A}^{rr,aa}$ the projector onto the
quasiparticle $A$ in sector $rr$ (our convention is that an idempotent denoted
by a small $p$ is a simple idempotent that has a trace of $1$, whereas a
capital $P$ is a projector with a trace that can be larger than $1$. Also note
that $P_{A}^{rr}$, in contrast to $P_{A}$, does generally not belong to the
center of the tube algebra). It projects onto a subspace of dimension

$n_{A,r}=\mathrm{Tr}\,P_{A}^{rr},$ (66)

which can also be interpreted as a multiplicity for the topological
quasiparticle $A$ in a given sector. An important fact is that, for a
noncommutative input category, $n_{A,r}$ is no longer restricted to $0$ or $1$
but can be larger. Examples are given in Sec. VI.1 and in Sec. VI.3.

The eigenvectors $\\{|A,r,a\rangle\\}$ (of eigenvalue $1$) of the simple
idempotents form an orthonormal basis of the total vector space. The idemnils
can be written as

$p_{A}^{rs,ab}=|A,r,a\rangle\langle A,s,b|,$ (67)

and Eq. (58) becomes

$p_{A}^{rs,aa^{\prime}}p_{B}^{s^{\prime}t,b^{\prime}b}=\delta_{A,B}\delta_{s,s^{\prime}}\delta_{b^{\prime},a^{\prime}}p_{A}^{rt,ab}.$
(68)

Equation (62) becomes $n_{A,r,a}=\mathrm{Tr}\,p_{A}^{rr,aa}=0$ or $1$ even if
$n_{A,r}$ can be strictly greater than $1$.

Next, each $P_{A}$ is constructed as a sum of the corresponding simple
idempotents $p_{A}$, so that Eq. (60) becomes:

$P_{A}=\sum_{r\in\mathcal{C}}P_{A}^{rr}=\sum_{r\in\mathcal{C}}\sum_{a=1}^{n_{A,r}}p_{A}^{rr,aa}.$
(69)

The multiplicity of $A$ (or the dimension of the irreducible representation
$A$) is given by:

$n_{A}=\sum_{r}n_{A,r}.$ (70)

###  A.6 The $11$ sector and fluxons

When considering generalized string-net models in the charge-free sector, one
particular type of tubes plays a major role: the tubes in the $11$ sector.
These $N_{\mathcal{C}}$ tubes are formed by closed horizontal strings [see
Fig. 12(a)] and have no open ends. Because of this latter characteristic, they
correspond to pure plaquette excitations, i.e., fluxons. Among the objects of
$\mathcal{Z}(\mathcal{C})$, fluxons are identified through the fact that the
corresponding $P_{A}$ has non-zero weight on the $11$ sector, i.e.,

$n_{A,1}\geqslant 1.$ (71)

In the $11$ sector, the number of tubes $N_{\mathcal{T}}^{11}=N_{\mathcal{C}}$
is larger or equal than the vector space dimension $V^{11}$ which is larger or
equal than the number of fluxons $N_{\mathcal{F}}$:

$N_{\mathcal{C}}=\sum_{A}n_{A,1}^{2}\geqslant V_{11}=\sum_{A}n_{A,1}\geqslant
N_{\mathcal{F}}=\sum_{A}\text{sgn }n_{A,1}.$ (72)

In the above equation, the equalities occur when $\mathcal{C}$ is commutative,
i.e. when $n_{A,1}$ is either $0$ or $1$. Moreover, the projector on the
vacuum particle $A=\mathbf{1}$ of $\mathcal{Z}(\mathcal{C})$ is a weighted sum
of all tubes in the $11$ sector:

$P_{\mathbf{1}}=p_{\mathbf{1}}^{11}=\sum_{i}\frac{d_{i}}{\mathcal{D}}Q^{i}_{11i}.$
(73)

This projector is equal to the “Kirby strand” [22]. When the contour over
which it acts is a single plaquette, $P_{\mathbf{1}}$ is the same as $B_{p}$
defined in Sec. III.1.

###  A.7 From tubes to idemnils: $M_{A}$

Finally, we express the tubes as linear combinations of the idemnils (this
corresponds to writing the irreducible representations of the tubes in the
basis spanned by the vectors $|A,r,a\rangle$) :

$Q_{rsj}^{i}=\sum_{A,a,b}(M^{i}_{A,rsj})_{a,b}\,p_{A}^{rs,ab},$ (74)

where

$(M^{i}_{A,rsj})_{a,b}=\langle A,r,a|Q_{rsj}^{i}|A,s,b\rangle.$ (75)

The coefficients $M_{A}$ in this decomposition are sometimes called modules
(see Ref. [12]).

The quantity $M^{i}_{A,rsj}$ is often written as an $N_{\mathcal{Z}}\times
N_{\mathcal{T}}$ matrix with row index $A$ and column index $irsj$. Each
element $(M^{i}_{A,rsj})_{a,b}$ is itself a small matrix with row index
$a=1,\dots,n_{A,r}$ and column index $b=1,\dots,n_{A,s}$.

The internal multiplicities (and in particular the important $n_{A,1}$) are
given by:

$n_{A,s}=\mathrm{Tr}\,M_{A,sss}^{1}.$ (76)

From Eq. (76) and the knowledge of $d_{s}$, one also obtains the quantum
dimensions of the simple objects in $\mathcal{Z}(\mathcal{C})$ as

$d_{A}=\sum_{s}n_{A,s}d_{s}\geqslant n_{A}\geqslant n_{A,1},$ (77)

where, for the inequalities, we used that $d_{s}\geqslant 1$ and
$n_{A,s}\geqslant 0$. The multiplicity $n_{A,1}$ and the quantum dimensions
$d_{A}$ are all that is needed to compute the spectrum degeneracies [see Eq.
(22)]. As a consequence of the above inequality and the fact that
$d_{\mathbf{1}}=1$, when $A=\mathbf{1}$, one has
$n_{\mathbf{1}}=n_{\mathbf{1},1}=1$.

###  A.8 Half-braidings $\Omega_{A}$

Levin and Wen [10] have developed an alternative way of finding the
topological quasiparticles from the input category. Instead of the projectors
$P_{A}$ on the topological quasiparticles, they find the (closed) string
operators $W_{A}$ that also belong to the tube algebra center but obey the
fusion algebra of $\mathcal{Z}(\mathcal{C})$:

$W_{A}W_{B}=\sum_{C}N_{A,B}^{C}W_{C}.$ (78)

It is a simple matter to go from the projectors $P_{A}$’s to the string
operators $W_{A}$’s or vice-versa. For example

$P_{A}=S_{1,A}\sum_{B}S_{A,B}^{*}W_{B}.$ (79)

The output of their approach is not $M_{A}$ but is called $\Omega_{A}$, which
is a half-braiding in the standard construction of the Drinfeld center (see,
e.g., Ref. [24, 16]). Actually the two quantities $M_{A}$ and $\Omega_{A}$ are
closely related [see Eq. (62) in Ref. [12]]:

$M_{A,rsj}^{i}=\Omega_{A,rsj}^{i}d_{i}\sqrt{\frac{d_{s}}{d_{r}}}.$ (80)

Eventually, from $\Omega_{A}$ (or $M_{A}$), all properties of the Drinfeld
center $\mathcal{Z}(\mathcal{C})$, such as the $S$, $T$ and fusion matrices,
can also be easily obtained (see Eqs. (64), (65) and (60) in Ref. [12]).

In order to obtain the $S-$ and $T-$ matrices of the Drinfeld center as well
as the internal multiplicities $n_{A,1}$, it is enough to know the
coefficients of the decomposition of the minimal central idempotents $P_{A}$
on the tubes $Q^{i}_{rsj}$. Note that the half-braidings $\Omega_{A}$ contain
a little more information as they give the full decomposition of the idemnils
$p_{A}^{rs}$ on the tubes. This extra information is needed for certain
observables such as Wegner-Wilson loops (see Ref. [49]).

In practice, finding the block structure of the tube algebra is not always an
easy task. Lan and Wen give a procedure that works well in simple cases and
which they call idempotent decomposition [12]. Another, more
systematic/algorithmic approach to obtain the minimal central idempotents from
the tube algebra is described in Appendix C of Ref. [51].

In the following appendix, we give the result of the decomposition of the tube
algebra for two Morita-equivalent categories.

##  Appendix B Half-braidings for Rep($S_{3}$) and Vec($S_{3}$)

The Drinfeld center for Rep($S_{3}$) is given, e.g., in Ref. [47]. Some
details about its tube algebra can be found in an Appendix in Ref. [51] and
the half-braidings are derived in Ref. [52]. Since all $\Omega^{i}_{J,rsj}$
are one-dimensional, we write them under the form of a matrix $\Omega$ with
$8$ rows indexed by particles $J=\\{A,B,C,D,E,F,G,H\\}$ and $17$ columns
indexed by tubes
$irsj=\\{1111,2112,3113,1222,2221,3223,1333,2333,3331,3313,3133,3213,3123,3233,3323\\}:$

$\Omega=\left(\begin{tabular}[]{ccccccccccccccccc}1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&1&1&-1&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&1&-1&$\frac{1}{2}$&-$\frac{1}{2}$&$\frac{1}{\sqrt{2}}$&0&0&0&0&0&0\\\
1&-1&0&0&0&0&1&1&$\frac{1}{2}$&$\frac{1}{2}$&0&$2^{\frac{1}{4}}$&$\frac{1}{2^{\frac{3}{4}}}$&0&0&0&0\\\
0&0&0&1&-1&0&1&1&-$\frac{1}{2}$&-$\frac{1}{2}$&0&0&0&0&0&-$\frac{i}{2^{\frac{3}{4}}}$&-$i2^{\frac{1}{4}}$\\\
1&1&-$\frac{1}{2}$&1&1&$\frac{1}{2}$&0&0&0&0&0&0&0&$\frac{\sqrt{3}}{2}$&$\frac{\sqrt{3}}{2}$&0&0\\\
0&0&0&0&0&0&1&-1&$-\frac{1+i\sqrt{3}}{4}$&$\frac{1+i\sqrt{3}}{4}$&$-\frac{1-i\sqrt{3}}{2^{3/2}}$&0&0&0&0&0&0\\\
0&0&0&0&0&0&1&-1&$-\frac{1-i\sqrt{3}}{4}$&$\frac{1-i\sqrt{3}}{4}$&$-\frac{1+i\sqrt{3}}{2^{3/2}}$&0&0&0&0&0&0\end{tabular}\right).$
(81)

For Vec($S_{3}$), we have decomposed the tube algebra and give here the
resulting half-braidings $\Omega_{J,rsj}^{i}$ as a matrix with $8$ rows
indexed by $J$ as before and $36$ columns indexed by the tubes

$\displaystyle irsj$ $\displaystyle=$
$\displaystyle\\{1111,2112,3113,4114,5115,6116,1222,2223,3221,1333,2331,3332,1444,4441,1555,5551,1666,6661,$
$\displaystyle
4235,5236,6234,4326,5324,6325,2456,6452,3546,6543,3465,5463,2645,5642,2564,4562,3654,4653\\},$

where
$s=\\{1,2,3,4,5,6\\}=\\{e,\zeta,\zeta^{2},\tau,\tau\zeta,\tau\zeta^{2}\\}$.
Defining $\omega={\rm e}^{2{\rm i}\pi/3}$ and $\omega^{*}={\rm e}^{-2{\rm
i}\pi/3}$, the half-braiding matrix reads

$\Omega=\left(\begin{tabular}[]{cccccccccccccccccccccccccccccccccccc}1&1&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
1&1&1&-1&-1&-1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
2&-1&-1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&1&1&1&1&1&0&0&0&0&0&0&1&1&1&1&1&1&1&1&1&1&1&1\\\
0&0&0&0&0&0&0&0&0&0&0&0&1&-1&1&-1&1&-1&0&0&0&0&0&0&1&-1&1&-1&1&-1&1&-1&1&-1&1&-1\\\
0&0&0&0&0&0&1&1&1&1&1&1&0&0&0&0&0&0&1&1&1&1&1&1&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&1&$\omega^{*}$&$\omega$&1&$\omega$&$\omega^{*}$&0&0&0&0&0&0&1&$\omega^{*}$&$\omega$&1&$\omega$&$\omega^{*}$&0&0&0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&1&$\omega$&$\omega^{*}$&1&$\omega^{*}$&$\omega$&0&0&0&0&0&0&1&$\omega$&$\omega^{*}$&1&$\omega^{*}$&$\omega$&0&0&0&0&0&0&0&0&0&0&0&0\end{tabular}\right).$
(82)

As $n_{C,1}=2$ [see Eq. (37)], beware that $\Omega_{C,11j}^{i}$ is actually a
$2\times 2$ matrix of which Eq. (82) only gives the trace.

##  Appendix C Fluxon identities

In Appendix A on the decomposition of the tube algebra, fluxons are identified
as topological quasiparticles $A$ that have $n_{A,1}>0$. One can define a
vector $\mathbf{n}_{1}$ with components $n_{A,1}$ with
$A=1,...,N_{\mathcal{Z}}$ such that it has non-zero entries only when $A$ is a
fluxon. Equations (11-12) are extra relations satisfied by the vector
$\mathbf{n}_{1}$ that we call fluxon identities. Similar equations first
appeared in the context of anyon condensation [41] and gapped boundaries [42,
43]. Equation (12) involving the $T$-matrix means that fluxons are necessarily
bosons, i.e., they have a trivial twist $\theta_{A}=1$. When
$\mathcal{Z}(\mathcal{C})$ is Abelian, Eq. (11) involving the $S$-matrix means
that fluxons also have trivial mutual statistics [42]. When
$\mathcal{Z}(\mathcal{C})$ is non-Abelian, the interpretation is less obvious.
In the following, we provide a proof of the fluxon identites from the tube
algebra, and comment on the relation with anyon condensation.

###  C.1 Proof from the tube algebra

The tubes in the $11$ sector are the horizontal closed (input) strings
$Q^{i}_{11i}$. Therefore, the tube algebra restricted to the $11$ sector is
just the fusion algebra of the input category and it decouples from the rest
of the tube algebra [see Eq. (51)]. In particular

$Q^{1}_{111}Q^{1}_{111}=\sum_{k}N_{11}^{k}Q^{k}_{11k}=Q^{1}_{111},$ (83)

is the projector onto the $11$ sector (it is the empty tube). The vertical
tubes $Q^{1}_{rrr}$ are the projectors onto the $rr$ sectors [see Eq. (52)].

####  C.1.1 Commutative input category

In the commutative case, we can use the mock $S$-matrix $\tilde{s}$ to
diagonalize simultaneously all the fusion matrices $N_{i}$. It is a unitary
matrix but, unlike the $S$-matrix, it is not symmetric in general (a special
case is when the input category is modular, in which case $\tilde{s}$ is a
genuine $S$-matrix). Naturally, one would label the matrix elements as
$\tilde{s}_{i,j}$ with $i$ and $j\in\mathcal{C}$. However, $\tilde{s}$ is also
a unitary transformation between input strings and fluxons (see Eqs. (84) and
(85) below). As such, it makes physically more sense to write a matrix element
of $\tilde{s}$ as $\tilde{s}_{i,A}$ with rows indexed by input labels
$i=1,..,N_{\mathcal{C}}$ and columns by fluxons $A\in\mathcal{F}$ with
$N_{\mathcal{F}}=N_{\mathcal{C}}$. The matrix $\tilde{s}$ can be chosen such
that its first row only contains strictly positive elements, in which case,
$\tilde{s}_{1,A}=\sqrt{d_{A}/\mathcal{D}}=\sqrt{S_{\mathbf{1},A}}$. The mock
$S$-matrix is not yet unique as its columns can always be permuted. By
convention, we choose that the first column corresponds to the vacuum
$A=\mathbf{1}$ of the output category so that
$\tilde{s}_{i,\mathbf{1}}=d_{i}/\sqrt{\mathcal{D}}$. Examples of mock
$S$-matrix are given in Secs. VI.1, VI.2 and VI.4.

In all of Sec. C.1, $A$ is taken to be a fluxon. The simple idempotents
$p_{A}^{11}$ are easily found as linear combination [53] of the horizontal
tubes $Q^{i}_{11i}$

$p_{A}^{11}=\tilde{s}_{1,A}\sum_{i}\tilde{s}_{i,A}^{*}Q^{i}_{11i},$ (84)

and vice-versa:

$Q^{i}_{11i}=\sum_{A}\frac{\tilde{s}_{i,A}}{\tilde{s}_{1,A}}p_{A}^{11}.$ (85)

Using the Verlinde-like equation

$N_{ij}^{k}=\sum_{A}\frac{\tilde{s}_{i,A}\tilde{s}_{j,A}\tilde{s}_{k,A}^{*}}{\tilde{s}_{1,A}},$
(86)

and the unitarity of the mock $S$-matrix, one can check that indeed

$p_{A}^{11}p_{B}^{11}=p_{A}^{11}\delta_{A,B}.$ (87)

We now introduce the projector $Q_{1}$ onto the fluxons in the $11$ sector of
the tube algebra. By definition

$Q_{1}=\sum_{A}p_{A}^{11},$ (88)

where $p_{A}^{11}$ only exists if $A$ is a fluxon, i.e., if $n_{A,1}=1$. The
above sum puts a weight $1$ on fluxons and $0$ on non-fluxons, so that

$\mathrm{Tr}\,Q_{1}=\sum_{A}\mathrm{Tr}\,p_{A}^{11}=\sum_{A}n_{A,1}=N_{\mathcal{C}},$
(89)

which shows that $Q_{1}$ is closely related to the vector $\mathbf{n}_{1}$.

From (85) with $i=1$, one gets

$\sum_{A}p_{A}^{11}=Q^{1}_{111},$ (90)

which is simply the empty tube. Therefore, the projector $Q_{1}$ onto the
fluxons in the $11$ sector is also the projector $Q^{1}_{111}$ onto the $11$
sector.

The action of the modular matrices $S$ and $T$ on the tubes is well-known:
they act trivially on the empty tube (see Sec. 28.3 in [22]). So that

$SQ_{1}=Q_{1}\text{ and }TQ_{1}=Q_{1},$ (91)

which are the fluxon identities, Eqs. (11) and (12).

####  C.1.2 Noncommutative input category

In the noncommutative case, we also have that the tube algebra restricted to
the $11$ sector is the fusion algebra of the input category and that it
decouples from the rest of the tube algebra. However, it is noncommutative and
there is no mock-$S$ matrix. A consequence is that the number of tubes
$N_{\mathcal{C}}$ is strictly larger than the vector space dimension $V_{11}$
which is strictly larger than the number of fluxons $N_{\mathcal{F}}$.

From the $N_{\mathcal{C}}$ tubes $Q^{i}_{11i}$, we build $N_{\mathcal{C}}$
idemnils $p_{A}^{11,ab}$, corresponding to only $N_{\mathcal{F}}$ (non-simple)
idempotents $P_{A}^{11}=\sum_{a=1}^{n_{A,1}}p_{A}^{11,aa}$ in the $11$ sector
(the central idempotents are $P_{A}=\sum_{s}P_{A}^{ss}$). Now
$\mathrm{Tr}\,P_{A}^{11}=n_{A,1}$ can be larger than 1. We can again define
the projector $Q_{1}$ onto the fluxons in the $11$ sector by:

$Q_{1}=\sum_{A}P_{A}^{11}=\sum_{A,a}p_{A}^{11,aa}.$ (92)

From the transformation (74) between tubes and idemnils, one has

$Q^{1}_{111}=\sum_{A,a,b}(M^{1}_{A,111})_{a,b}\,p_{A}^{11,ab}=\sum_{A,a}\,p_{A}^{11,aa},$
(93)

as $(M^{1}_{A,111})_{a,b}=(\Omega^{1}_{A,111})_{a,b}=\delta_{a,b}$ (see above
Eq. (40) in Ref. [14] or below Eq. (52) and Eq. (53) in Ref. [12]). Therefore
$Q_{1}$ is also the projector onto the $11$ sector, i.e. the empty tube
$Q^{1}_{111}$, from which the fluxon identities (91) follow as in the previous
subsection.

###  C.2 Stability inequality

In order to prove an important stability inequality that $\mathbf{n}_{1}$
needs to satisfy, we make a detour into anyon condensation. Anyon condensation
is a general mechanism that allows one to describe a phase transition from a
topological order described by a UMTC $\mathcal{A}$ to another described by a
UMTC $\mathcal{U}$ [54]. We therefore reverse the perspective and imagine
that, instead of building the Drinfeld center from an input category, we
condense some bosons of the Drinfeld center to recover the input category. We
will use the general formalism of anyon condensation [54, 41] and apply it to
the particular case in which we start from the Drinfeld center
$\mathcal{Z}(\mathcal{C})$ and condense it towards a trivial order. This is
known as anyon condensation to the vacuum, which is intimately related to
finding how many types of gapped boundaries are possible for a given
topological order [39, 43, 55].

Condensation to the vacuum relates the UMTC
$\mathcal{A}=\mathcal{Z}(\mathcal{C})$ to the trivial UMTC $\mathcal{U}$
(total quantum dimension $\mathcal{D}_{\mathcal{U}}=1$), via a UFC
$\mathcal{T}=\mathcal{C}$, where $\mathcal{U}$ is included in $\mathcal{T}$.
Generally speaking, anyon condensation

$\mathcal{A}\to\mathcal{T}\to\mathcal{U},$ (94)

is described by a rectangular matrix $n_{A,s}$, called restriction or lifting
matrix, with $A\in\mathcal{Z}(\mathcal{C})$ and $s\in\mathcal{U}$ (in our
case, $s=1$ only). Here, this matrix is the vector $\mathbf{n}_{1}$ and the
condensing bosons are the fluxons. The condensation equations (see, e.g., Eqs.
(20a) and (20b) in Ref. [41]) relate the $S$\- and $T$\- matrices of the two
UMTCs $\mathcal{A}$ and $\mathcal{U}$ as

$T_{\mathcal{A}}\mathbf{n}_{1}=\mathbf{n}_{1}T_{\mathcal{U}}\textnormal{ and
}S_{\mathcal{A}}\mathbf{n}_{1}=\mathbf{n}_{1}S_{\mathcal{U}}.$ (95)

As $T_{\mathcal{U}}=1$ and $S_{\mathcal{U}}=1$ are trivial $1\times 1$
matrices, and $T_{\mathcal{A}}=T$ and $S_{\mathcal{A}}=S$ are the $T$\- and
$S$\- matrices of $\mathcal{Z}(\mathcal{C})$, we find Eq. (11-12). Anyon
condensation should also hold in the case where the UFC $\mathcal{T}$ has
noncommutative fusion rules, as briefly discussed by Bais and Slingerland
[54].

In the context of anyon condensation, one requires the commutation of fusion
and restriction [41, 54], i.e.,

$\sum_{C}N_{AB}^{C}n_{C,t}=\sum_{r,s}n_{A,r}n_{B,s}\tilde{N}_{rs}^{t}.$ (96)

In this appendix (and in the following), in order to avoid confusion,
$\tilde{N}$ denotes the fusion matrices for $\mathcal{C}$, whereas $N$ denotes
the fusion matrices for $\mathcal{Z}(\mathcal{C})$. The stability inequality
follows by taking $t=1$ so that

$\sum_{C}N_{AB}^{C}n_{C,1}=n_{A,1}n_{B,1}+\sum_{r\neq
1}n_{A,r}n_{B,\bar{r}}\geqslant n_{A,1}n_{B,1},$ (97)

An advantage of this inequality is that it can serve as a test of the
coefficients $n_{A,1}$ without the knowledge of the complete
lifting/restriction matrix $n_{A,s}$ and the fusion matrix $\tilde{N}$.

An interesting question to ask is: given an achiral topological order
characterized by a Drinfeld center $\mathcal{Z}(\mathcal{C})$ with modular
matrices $S$ and $T$, what are the possible gapped boundaries or condensations
to the vacuum [41, 43, 55] ?

A partial answer to that question is known (see, e.g., Refs [43, 56, 55]): a
necessary (but not sufficient) condition for a gapped boundary is to have a
vector $\mathbf{n}_{1}$ that satisfies the fluxon identities Eqs. (11)-(12)
and the stability condition (97), i.e., to have a stable fluxon set. This is
equivalent to the notion of Lagrangian algebras in $\mathcal{Z}(\mathcal{C})$
[57]. For each such stable $\mathbf{n}_{1}$, there is a corresponding boundary
theory described by a UFC $\mathcal{C}_{b}$. Obviously, $\mathcal{C}$ is one
of the possible boundary theories $\mathcal{C}_{b}$. Also, all the boundary
theories are Morita-equivalent, i.e.
$\mathcal{Z}(\mathcal{C}_{b})\simeq\mathcal{Z}(\mathcal{C})$. This is the
bulk-boundary correspondence for achiral topologically-ordered phases.

For example, given $\mathcal{Z}(S_{3})$ (see Sec. VI.1 for the notations), one
finds five possible $\mathbf{n}_{1}$’s satisfying (11) and (12), one of which
($\mathbf{n}_{1}=\\{1,1,1,0,0,1,0,0\\}^{T}$ which for short we note
$\mathbf{n}_{1}=(A,B,C,F)$ according to its non-zero elements) does not
satisfy the stability condition (97) [55]. Among the four stable solutions,
two are related by symmetry and correspond to Vec($S_{3}$) [i.e.
$\mathbf{n}_{1}=(A,B,2\times C)$ and $\mathbf{n}_{1}=(A,B,2\times F)$], and
two are related by symmetry and correspond to Rep($S_{3}$) [i.e.
$\mathbf{n}_{1}=(A,D,F)$ and $\mathbf{n}_{1}=(A,C,D)$], accounting for the
symmetry between the $C$ and $F$ quasiparticles [47].

Another example is that of $\mathcal{Z}(\mathcal{H}_{3})$ (see Sec. VI.3 for
the notations). In this case, we find three stable solutions. Two solutions
related by symmetry [i.e.
$\mathbf{n}_{1}=\\{1,0,0,0,0,0,0,1,2,0,0,0\\}^{T}=(\mathbf{1},\pi_{1},2\times\pi_{2})$
and $\mathbf{n}_{1}=(\mathbf{1},\pi_{1},2\times\sigma_{1})$], due to the
symmetry between the $\pi_{2}$ and $\sigma_{1}$ quasiparticles, point to the
fusion ring $H_{6}$ and one [i.e.
$\mathbf{n}_{1}=(\mathbf{1},\pi_{1},\pi_{2},\sigma_{1}$)] to the fusion ring
$H_{4}$ [58]. The fusion ring $H_{6}$ corresponds to the UFCs
$\mathcal{H}_{3}$ and $\mathcal{H}_{2}$, whereas $H_{4}$ refers to the UFC
$\mathcal{H}_{1}$. The two categories $\mathcal{H}_{3}$ and $\mathcal{H}_{2}$
have the same fusion rules but different $F$ symbols.

##  Appendix D Generalized Hamiltonian

The Hamiltonian that we consider in the main text, Eq. (10), assigns the same
energy penalty +1 to each fluxon which is not the vacuum. However, we can
assign different energy penalties to each fluxon type. Further, we can also
assign a plaquette-dependent energy penalties. More generally, let us define a
projector

$B_{p}^{A}=\left\\{\begin{array}[]{ll}1&\mbox{if plaquette $p$ has fluxon
excitation $A$,}\\\ 0&\mbox{otherwise.}\end{array}\right.$

We can then write a more general Hamiltonian

$H_{\text{gen}}=\sum_{{\text{plaquettes }}\,p}\,\,\sum_{A\in{\cal
F}}\,\,E^{A}_{p}\,\,B_{p}^{A},$ (98)

where $E^{A}_{p}$ is the energy cost of having fluxon $A$ on plaquette $p$ and
$\cal F$ is the set of fluxons. So long as $E^{\bf 1}_{p}<E^{A}_{p}$ for all
$A\neq{\bf 1}$ and all $p$, then the ground state is still the state where all
plaquettes are in the vacuum state. The Hamiltonian we consider in the main
text corresponds to $E_{p}^{A}=0$ for all $A\neq{\bf 1}$ and $E_{p}^{\bf
1}=-1$, for all plaquettes $p$.

It is not hard to construct the projectors $B_{p}^{A}$ from the structure of
the tube algebra by inserting a projector $P_{A}^{11}$ inside the plaquette
$p$ and then fusing it into the edges of the plaquette. For the case where
$n_{A,1}>1$, in fact there is freedom to generalize Hamiltonian Eq. (98)
further. Since the space of states where a plaquette $p$ has fluxon $A$ is
$n_{A,1}$-dimensional, one can add terms to split this added degeneracy. Thus
we can even more generally write

$H_{\text{gen2}}=\sum_{{\rm{plaquettes}}\,p}\,\,\sum_{A\in{\cal
F}}\,\,\sum_{a=1}^{n_{A,1}}\,\,E^{A,a}_{p}\,\,B_{p}^{A,a},$ (99)

where $E^{A,a}_{p}$ are a set of coefficients and the operators $B_{p}^{A,a}$
are obtained by inserting the simple idempotent $p_{A}^{11,aa}$ inside
plaquette $p$ and fusing it into the edges. Again we assure that the vacuum is
the ground state when $E^{\mathbf{1}}_{p}$ is smaller than any $E^{A,a}_{p}$
for all $A\neq{\bf 1}$ and $a=1,...,n_{A,1}$.

Calculation of spectral degeneracy can in principle be done analytically for
any of these Hamiltonians. One always starts with the Moore-Seiberg-Banks
formula, and sums over all possible fluxon labellings of each plaquette.

Let us take an example of the category Vec($S_{3}$) as discussed in Sec. VI.1.
There are three fluxon types labeled $A,B,C$. Here $A$ is the vacuum, which we
give energy $E^{A}=0$. Then $B$ we give energy $E^{B}=x$. Finally $C$ is two-
dimensional, i.e. $n_{C,1}=2$, and we call $E^{C,1}=y$ and $E^{C,2}=z$ the two
values of $E^{C,a}$ with $a=1,2$. For simplicity here, we assume all
plaquettes are identical to each other, although this is not necessary.

Assume that $x,y,z$ are incommensurate, i.e., all ratios of these values are
irrational. If we consider a total energy

$E=n_{x}x+n_{y}y+n_{z}z,$ (100)

with integers $n_{x},n_{y},n_{z},$ then the number of eigenstates of the
system having exactly this energy is given by the number of ways we can have
$n_{x}$ plaquettes having $B$ fluxons, and $n_{y}+n_{z}$ plaquettes having $C$
fluxons with $n_{y}$ of these in the $y$ eigenstate and $n_{z}$ of these in
the $z$ eigenstate, and all remaining plaquettes in the vacuum eigenstate.
This number is given by

$\displaystyle\frac{N_{\text{p}}!}{n_{x}!n_{y}!n_{z}!(N_{\text{p}}-n_{x}-n_{y}-n_{z})!}\times$
$\displaystyle{\rm dim}_{{\cal Z}(S_{3})}(g,n_{x}\mbox{ fluxons
}B,(n_{y}+n_{z})\mbox{ fluxons }C).\quad\quad\quad$ (101)

##  Appendix E Hilbert-space dimension

In this appendix, we use two different methods to count the total Hilbert-
space dimension of string-net model having no vertex defects, but having any
possible fluxon excitations. In other words, we are restricting the Hilbert
space as discussed in Sec. III.3 and counting the size of this restricted
Hilbert space,i.e., all possible excitations of the Hamiltonian Eq. (10).

The first method we use consists in counting all possible edge and vertex
labellings that do not violate the vertex constraints (see Sec. E.1). The
second method, in Sec. E.2, is to count all possible labelings of the fluxon
states of the plaquettes.

Both calculations are actually sums over fusion networks. In Sec. E.1, we are
summing over fusion trees of $\cal C$ whereas, in Sec. E.2, we are summing
over fusion trees of $\cal Z(\cal C)$ (see the discussion on the derivation of
the Moore-Seiberg-Banks formula in Sec. IV.1). In both cases, the fusion trees
are sums over fusion multiplicity matrices. In Sec. E.1 we are summing fusion
matrices of $\cal C$ whereas in Sec. E.2 we are summing fusion matrices of
${\cal Z}({\cal C})$. These two different fusion multiplicity matrices are
related to each other via the commutativity of fusion and restriction [54,
41]. In particular, we have Eq. (96), where $r,s,t\in\cal C$ and $A,B,C\in\cal
Z(C)$ and $n_{A,r}$ are lifting coefficients. To clearly distinguish the two
sets of fusion matrices we call $\tilde{N}$ that in the input category
$\mathcal{C}$ and $N$ that in the output category $\mathcal{Z}(\mathcal{C})$.

For simplicity, we consider an (orientable) surface of genus $g$ without
boundaries ($b=0$) for which the Euler-Poincaré characteristic tells us that

$2-2g=N_{\text{p}}-N_{\text{e}}+N_{\text{v}}.$ (102)

For a trivalent graph, we also have $N_{\text{e}}=3N_{\text{v}}/2$, so that

$N_{\text{p}}=2-2g+N_{\text{v}}/2.$ (103)

###  E.1 Counting edge labelings

Here we want to count up all allowed labellings of the trivalent-graph edges.
Each oriented edge is labeled with a simple object in $\cal C$ and a vertex
with incoming arrows $a$ and $b$ and $c$ reading counterclockwise around the
vertex ($a,b,c\in\cal C$) gets a value
$\tilde{N}_{abc}=\tilde{N}_{ab}^{\bar{c}}$ (see Fig. 2). We then sum over all
edge labels (summing over $\tilde{N}$’s is equivalent to counting all vertex
labels $\mu\in 1\ldots\tilde{N}_{abc}$). An example of summing over a simple
graph is shown in Fig. 13.

$g$$c$$b$$a$$d$$f$$\displaystyle=\sum_{a,b,c,d,f,g}\tilde{N}_{adb}\tilde{N}_{fc\bar{d}}\tilde{N}_{\bar{b}\bar{c}\bar{g}}\tilde{N}_{\bar{f}\bar{a}g}$
Figure 13: Finding the total string-net model Hilbert-space dimension (with no
vertex defects) by summing $\tilde{N}$’s at the vertices over all edge
labellings.

For more complex graphs, one may use restructuring moves (or “$F$-moves”,
which here are just the associativity of fusion) to change the connectivity of
the graph without altering the Hilbert-space dimension, as shown in Fig. 14.

$\displaystyle\sum_{c}$$b$$a$$c$$d$$f$$\displaystyle=\sum_{g}$$b$$a$$g$$d$$f$,
$\displaystyle\sum_{c}\tilde{N}_{abc}\tilde{N}_{df\bar{c}}=\sum_{g}\tilde{N}_{afg}\tilde{N}_{bd\bar{g}}$.
Figure 14: The associativity constraint allows restructuring the graph without
changing the Hilbert-space dimension.

As discussed in Ref. [44], if the category has commutative fusion rules, or if
the graph is on a surface of genus $g=0$ (a sphere), then by using these
restructuring moves, any graph can be reduced to a chain of bubbles

$L_{ij}=\sum_{ab}\raisebox{-19.91684pt}{{
\begin{pspicture}(0.0pt,-28.45274pt)(85.35823pt,28.45274pt){\psset{ArrowInside=->,arrowsize=6.0pt
0.0}\psline(7.11317pt,0.0pt)(28.45274pt,0.0pt)}\rput(14.22636pt,7.11317pt){$i$}
\rput(71.13185pt,7.11317pt){$j$} \rput(42.67911pt,-21.33955pt){$a$}
\rput(42.67911pt,21.33955pt){$b$} {\pscircle(42.67911pt,0.0pt){14.22636pt}}
{\psset{arrowsize=6.0pt
0.0}\psarc{->}(42.67911pt,0.0pt){14.22636pt}{0.0}{90.0}}
{\psset{arrowsize=6.0pt
0.0}\psarc{<-}(42.67911pt,0.0pt){14.22636pt}{270.0}{360.0}}
{\psset{ArrowInside=->,arrowsize=6.0pt
0.0}\psline(56.90549pt,0.0pt)(78.24504pt,0.0pt)}\end{pspicture}}}=\sum_{ab}\tilde{N}_{abi}\tilde{N}_{\bar{a}\bar{j}\bar{b}},$
(104)

so that, in these cases, the total Hilbert-space dimension is given by

${\dim\mathcal{H}}={\rm Tr}[L^{N_{\text{v}}/2}].$ (105)

Thus, for commutative fusion rules, the Hilbert-space dimension is independent
of the genus of the surface and only depends on the number of vertices of the
graph.

However, in the case where fusion rules are noncommutative, the situation is
different, and on a surface of nonzero genus, one cannot reduce the fusion
diagram to a chain of bubbles of the form $L$. To understand this, let us
first consider the case $g=1$ (torus). We draw the torus as a rectangle with
opposite edges identified. Using the above restructuring moves (Fig. 14), we
can reduce any trivalent graph to the following (where the graph edges are
black, and we do not draw arrows or labels on these edges for simplicity).

(106)

If there are multiple lines running around the handles, Fig. 106 can be
reduced to single lines running around the handle by using restructuring moves
(Fig. 14). However if there is a single line going around a handle (as shown
in the above figure), it cannot be removed by restructuring moves.

Relaxing the restriction that we need to draw all of our diagrams in the
plane, the diagram (106) can also be expressed equivalently as

(107)

where the over-crossing could equally well be drawn as an undercrossing, since
it is only the connectivity that matters. But note that due to
noncommutativity of the fusion rules, it is not possible to remove the under-
crossing and flatten the diagram.

We thus define a twisted bubble

$\tilde{L}_{ij}=\sum_{ab}\raisebox{-19.91684pt}{{
\begin{pspicture}(-14.22636pt,-28.45274pt)(85.35823pt,28.45274pt){\psset{ArrowInside=->,arrowsize=6.0pt
0.0}\psline(-7.11317pt,0.0pt)(7.11317pt,0.0pt)}\rput(0.0pt,7.11317pt){$i$}
\rput(71.13185pt,7.11317pt){$j$} \rput(42.67911pt,-15.649pt){$a$}
\rput(42.67911pt,15.649pt){$b$}
{\psset{ArrowInside=->,ArrowInsidePos=.375,arrowsize=6.0pt
0.0}\psbezier(56.90549pt,0.0pt)(42.67911pt,34.14328pt)(21.33955pt,-34.14328pt)(7.11317pt,0.0pt)}{\psset{border=2pt,bordercolor=white}\psbezier(56.90549pt,0.0pt)(42.67911pt,-34.14328pt)(21.33955pt,34.14328pt)(7.11317pt,0.0pt)}{\psset{ArrowInside=->,ArrowInsidePos=.375,arrowsize=6.0pt
0.0}\psbezier(56.90549pt,0.0pt)(42.67911pt,-34.14328pt)(21.33955pt,34.14328pt)(7.11317pt,0.0pt)}{\psset{ArrowInside=->,arrowsize=6.0pt
0.0}\psline(56.90549pt,0.0pt)(78.24504pt,0.0pt)}\end{pspicture}}}=\sum_{ab}\tilde{N}_{bai}\tilde{N}_{\bar{a}\bar{j}\bar{b}},$
(108)

where again it does not matter if we drew an over- or under-crossing. It is
also worth noting that $\tilde{L}_{1j}=L_{1j}$ since fusion with $1$ is
commutative.

In this notation, the Hilbert-space dimension on a $g=1$-surface (torus) is
given by

${\dim\mathcal{H}}={\rm Tr}[\tilde{L}\,\,L^{N_{\text{v}}/2-1}],$ (109)

since two vertices occur within the twisted bubble and the remaining vertices
are within the planar graph which can be reduced to a bubble chain of $L$’s.
Note that for the case of commutative fusion we have $\tilde{L}=L$ and we
recover the previous result (105).

To extend this to a genus-$g$ surface, we start with the planar representation
of this surface as a $4g$-sided polygon with certain edges identified as in
the following picture

(110)

As with the simple torus, a trivalent graph within this polygon can always be
reduced, but a single line around a handle cannot be removed. We can thus
reduce any graph to the following

(111)

Again, analogous to what we did with the torus, we can remove the periodic
boundary conditions and achieve the same connectivity if we allow over- or
under-crossings of the graph. In particular this is entirely equivalent to

(112)

So we have $g$ copies of what we called $\tilde{L}$, each accounting for two
vertices, and all the remaining vertices are in some trivalent planar graph.

Unfortunately, this structure is still somewhat inconvenient, so we use
restructuring moves to connect together the legs of each twisted bubble,
getting

(113)

We then define

$\displaystyle K_{cd}$ $\displaystyle=$
$\displaystyle\sum_{abijk}\raisebox{-42.67912pt}{{
\begin{pspicture}(-14.22636pt,-28.45274pt)(85.35823pt,28.45274pt)\rput(-7.11317pt,7.11317pt){$i$}
\rput(71.13185pt,7.11317pt){$j$} \rput(42.67911pt,-15.649pt){$a$}
\rput(42.67911pt,15.649pt){$b$}
{\psset{ArrowInside=->,ArrowInsidePos=.375,arrowsize=6.0pt
0.0}\psbezier(56.90549pt,0.0pt)(42.67911pt,34.14328pt)(21.33955pt,-34.14328pt)(7.11317pt,0.0pt)}{\psset{border=2pt,bordercolor=white}\psbezier(56.90549pt,0.0pt)(42.67911pt,-34.14328pt)(21.33955pt,34.14328pt)(7.11317pt,0.0pt)}{\psset{ArrowInside=->,ArrowInsidePos=.375,arrowsize=6.0pt
0.0}\psbezier(56.90549pt,0.0pt)(42.67911pt,-34.14328pt)(21.33955pt,34.14328pt)(7.11317pt,0.0pt)}{\psset{ArrowInside=->,arrowsize=6.0pt
0.0}\psbezier(56.90549pt,0.0pt)(78.24504pt,0.0pt)(56.90549pt,28.45274pt)(35.56592pt,28.45274pt)}{\psset{ArrowInside=->,ArrowInsidePos=.7,arrowsize=6.0pt
0.0}\psbezier(35.56592pt,28.45274pt)(14.22636pt,28.45274pt)(-14.22636pt,0.0pt)(7.11317pt,0.0pt)}{\psset{ArrowInside=->,arrowsize=6.0pt
0.0}\psline(35.56592pt,28.45274pt)(35.56592pt,42.67911pt)}{\psset{ArrowInside=->,arrowsize=6.0pt
0.0}\psline(7.11317pt,42.67911pt)(35.56592pt,42.67911pt)}{\psset{ArrowInside=->,arrowsize=6.0pt
0.0}\psline(35.56592pt,42.67911pt)(64.01866pt,42.67911pt)}\rput(28.45274pt,35.56592pt){$k$}
\rput(0.0pt,42.67911pt){$c$} \rput(71.13185pt,42.67911pt){$d$}
\end{pspicture}}},$ (114) $\displaystyle=$
$\displaystyle\sum_{ijk}\tilde{L}_{ij}N_{\bar{i}j\bar{k}}N_{ck\bar{d}},$ (115)

Note that $K$ commutes with $L$ (by associativity of fusion).

Now the total Hilbert-space dimension is then

${\dim\mathcal{H}}={\rm Tr}[K^{g}L^{N_{\text{v}}/2-2g}],$

i.e., there are $g$ twisted loops, each factor of $K$ includes four vertices,
and the remaining vertices are in the trivalent planar graph.

The diagram that gives the twisted bubble $\tilde{L}$ [see Eq. (108)] is
identical to the diagram we used to define a tube $Q$ in Fig. 11, the
difference between the two being only that we have written the tube algebra on
a rectangle with sides identified. $\tilde{L}_{ij}$ is thus equal to the
number of all the tube diagrams $Q^{b}_{ija}$ summed over $a$ and $b$. In
other words, it is the number of tubes $N_{\mathcal{T}}^{ij}$ in a given $ij$
sector of the tube algebra [see Eq. (64)]. Since the tube algebra is also
spanned by the idemnils, we can also calculate the $\tilde{L}_{ij}$ by
counting idemnils $p^{ij,xy}_{A}$ for all $x,y$ and then summing over all
simple objects $A\in\mathcal{Z}(\mathcal{C})$. Since here $x\in 1\ldots
n_{A,x}$ and $y\in 1\ldots n_{A,y}$ we have

$\tilde{L}_{ij}=\sum_{A}n_{A,i}n_{A,j}.$ (116)

###  E.2 Counting fluxon fusion channels

We now calculate the same Hilbert-space dimension of a string-net model by
summing up all possible fluxon fusion channels assigning all possible fluxons
to all plaquettes.

We are going to use a graphical notation that a quantum number in $\cal Z(C)$
is a black edge, but a quantum number in $\cal C$ is red. A lifting
coefficient is a green dot. In particular this means that Eq. (96) can be
expressed graphically as

$\displaystyle\sum_{C}N_{AB}^{C}n_{C,t}=\sum_{r,s}n_{A,r}n_{B,s}\tilde{N}_{rs}^{t},$
(117) (118)

For simplicity of notation we do not draw arrows on edges.

To determine the full Hilbert-space dimension, we start with the Moore-
Seiberg-Banks formula, which graphically looks like this (see Eq. 9):

$\displaystyle g\mbox{ loops
}\left\\{\rule{0.0pt}{65.0pt}\right.$$A_{1}$$A_{2}$$A_{3}$$\ldots$$A_{N_{\text{p}}}$

$\displaystyle\left\\{\rule{0.0pt}{65.0pt}\right.$

$N_{\text{p}}$labeled punctures

Now we want to sum the value of this diagram over the product of $n_{A,1}$
which selects fluxons. This is graphically depicted as

$\displaystyle g\mbox{ loops
}\left\\{\rule{0.0pt}{65.0pt}\right.$$\ldots$$1$$1$$1$$1$

Now using the graphical identity Eq. (118) with $t=1$, we can push all the
green dots through the vertices to obtain the following: $\displaystyle
g\mbox{ loops }\left\\{\rule{0.0pt}{65.0pt}\right.$$\ldots$$1$$1$$1$$1$

Here the double green dot is meant to be connected by a small black line
(which is not drawn). Given our graphical notation, the double green dot has
the following value

$\sum_{A}n_{A,i}n_{A,j}=\tilde{L}_{ij},$ (119)

where the equivalence to $\tilde{L}$, the twisted bubble, is from Eq. (119)
above. Note that as mentioned just below Eq. (108), we have
$\tilde{L}_{1j}=L_{1j}$ so that the double green dots at the top of the
diagram can be replaced by $L$ rather than $\tilde{L}$. Thus the Hilbert-space
dimension is given by the diagram

$\displaystyle g\mbox{ loops }\left\\{\rule{0.0pt}{65.0pt}\right.$$\ldots$

This is now a trivalent diagram with edges living in $\cal C$, of the form of
Eq. (112). That is, $g$ twisted bubbles connected to a trivalent planar
diagram.

The final step is to check that the number of vertices in this diagram is the
same as the number of vertices in the original graph for the string net. We
started with $N_{\text{p}}$ labeled punctures (at the top of the diagram) and
$g$ loops. The final diagram has $2(N_{\text{p}}-1)-1$ vertices in the top
half of the diagram and $4g-1$ vertices in the bottom half of the diagram,
giving a total of $2N_{\text{p}}+4g-4$ vertices. Using Eq. (103), this is
precisely the number of vertices in the original string-net model.

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