SciPost Phys. 16, 143 (2024)

Fusion surface models: 2+1d latticemodels from fusion 2-categories
Kansei Inamura1 and Kantaro Ohmori2
1 Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
2 Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan

Abstract
We construct (2+1)-dimensional lattice systems, which we call fusion surface models. These models have finite non-invertible symmetries described by general fusion
2-categories. Our method can be applied to build microscopic models with, for example, anomalous or non-anomalous one-form symmetries, 2-group symmetries, or noninvertible one-form symmetries that capture non-abelian anyon statistics. The construction of these models generalizes the construction of the 1+1d anyon chains formalizedby Aasen, Fendley, and Mong. Along with the fusion surface models, we also obtain thecorresponding three-dimensional classical statistical models, which are 3d analogues ofthe 2d Aasen-Fendley-Mong height models. In the construction, the “symmetry TFTs” forfusion 2-category symmetries play an important role.
Copyright K. Inamura and K. Ohmori.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 30-06-2023
Accepted 07-05-2024
Check forupdatesPublished 03-06-2024
doi:10.21468/SciPostPhys.16.6.143

Contents
1 Introduction and summary
1.1 Motivation
1.2 Review of the Aasen-Fendley-Mong model
1.3 Generalization to 2+1 dimensions
1.3.1 Fusion surface models
1.4 Structure of the paper

2
2
4
11
13
17

2 Preliminaries
2.1 Fusion 2-categories
2.2 Douglas-Reutter TFT

17
18
21

3 3d height models
3.1 3d classical statistical models from 4d Douglas-Reutter TFT
3.2 3d height models on a triangulated cubic lattice

23
23
24

4 2+1d fusion surface models
4.1 2+1d fusion surface models from 3d height models
4.2 Unitarity of the model
4.3 Graphical representation

27
27
31
33

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SciPost Phys. 16, 143 (2024)

4.4

Fusion 2-category symmetry
4.4.1 General case
4.4.2 Non-invertible 1-form symmetry
4.4.3 Anomalous finite group symmetry

38
38
38
40

5 Examples
5.1 Lattice models with anomalous invertible 1-form symmetries
5.2 Kitaev honeycomb model without a magnetic field
5.3 Non-chiral topological phases with fusion 2-category symmetries

41
42
44
46

References

49

1

Introduction and summary

1.1

Motivation

Symmetry plays a fundamental role in both constructing and analyzing models of physical systems. Recently, various generalizations of symmetry have been introduced, including higherform symmetry [1], higher-group symmetry [2–5], and even more general non-invertible
symmetry [6, 7]. These generalized symmetries greatly extend the applicability of various
symmetry-based techniques in theoretical physics, and have therefore been one of the main
topics in the field.
The core principle of the generalizations is the correspondence between symmetry operations and topological defects/operators [1], see Figure 1 for an illustration of this correspondence. In particular, symmetry operations for a conventional symmetry correspond to
invertible topological defects with codimension one, which form a group under the fusion.
The generalizations of a conventional symmetry are achieved by relaxing the requirements
for the dimensionality and invertibility of topological defects: higher-form symmetries are
generated by topological defects with higher codimensions and non-invertible symmetries are
generated by topological defects that do not have their inverses.1 We note that topological defects associated with a non-invertible symmetry can have arbitrary codimensions, andtherefore non-invertible symmetries include higher-form symmetries as special cases. Whilehigher-form symmetries are still described by groups, non-invertible symmetries are no longer
described by groups in general because the fusion rules of the associated topological defects
are not necessarily group-like.
In 1+1 dimensions, finite non-invertible symmetries are generally described by fusion categories [6, 7, 9] (see also [10] for an earlier reference), which are natural generalizations offinite groups.2 For this reason, finite non-invertible symmetries in 1+1 dimensions are calledfusion category symmetries [19]. Fusion category symmetries are particularly well studied
in the context of rational conformal field theories [7, 20–26] and topological field theories(TFTs) [6, 10, 19, 27–30]. See also, e.g., [31–52] for recent developments. Although thesesymmetries were originally discussed in the context of quantum field theories (QFTs), they1

One can further generalize the notion of symmetry by allowing the defects to be non-topological along somespatial directions. Symmetries generated by such defects are called subsystem symmetries, which are typicallyexhibited by fractonic systems [8]. We do not investigate this direction in this paper.
2
Precisely, while finite non-invertible symmetries of 1+1d bosonic systems are described by fusion categories,
finite non-invertible symmetries of 1+1d fermionic systems are described by superfusion categories [11–18]. In
this paper, we will only consider bosonic systems. Technically, all fusion categories and fusion 2-categories that
we will discuss in this paper are supposed to be spherical. We also emphasize that we do not consider non-unitary
theories where symmetry categories can be non-semisimple.

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SciPost Phys. 16, 143 (2024)

Ug

Ug

Twisted boundary condition

Symmetry action on the state space

Figure 1: A timelike insertion of a symmetry defect realizes a twisted boundary condition, while a spacelike insertion does a symmetry operation on a state.
also exist on the lattice. In particular, we can systematically construct 1+1d lattice models
with general fusion category symmetries, which are known as anyon chain models [33,53,54].
In higher dimensions, finite non-invertible symmetries in unitary theories are expected to
be described by fusion higher categories [55–58].3 Since the discovery of concrete realizationsof such symmetries in lattice models [59] and QFTs [60–64], non-invertible symmetries inhigher dimensions have been studied intensively in various contexts, see, e.g., [59–121] forrecent advances and also [122, 123] for earlier discussions. However, systematic constructionof physical systems with general fusion higher category symmetries is still lacking.
Given the generalization of symmetry, one may wonder whether we can utilize it to build
physical models with a given generalized symmetry. This question was answered affirmatively
by Aasen, Fendley, and Mong for fusion category symmetries in 1+1 dimensions [33]:4 they
constructed explicit two-dimensional classical statistical models acted upon by a given fusioncategory. The corresponding 1+1d quantum lattice models turn out to be the anyon chain
models, which have the given fusion category symmetries. In this paper, we generalize theirconstruction to 2+1 dimensions. Namely, we construct three-dimensional classical statisticalmodels and the corresponding (2+1)-dimensional quantum lattice systems that are acted uponby a given fusion 2-category. We call our 2+1d quantum lattice models the fusion surface
models, which are (2+1)-dimensional analogues of the 1+1d anyon chains. By construction,
the fusion surface models have finite non-invertible symmetries described by general fusion
2-categories, i.e., fusion 2-category symmetries.
The fusion 2-category symmetry in 2+1 dimensions has a particular significance: it includes the symmetry of anyons in topological orders as a special example. Within a generalized
Landau paradigm, the existence of anyons in topologically ordered phases can be regarded asa consequence of a spontaneously broken higher (potentially non-invertible) symmetry [69].
Therefore, the fusion surface models with non-invertible higher symmetry provide candidates
that might realize a given topological order. In other words, if the model has a gapped point,
it is guaranteed that the IR phase contains the anyons we used as an input to the model. While
our models include the Levin-Wen string-net models [125] that realize non-chiral topologicalorders, it probably requires a numerical study to see whether our model can realize chiral
topological orders.
Another example of a fusion 2-category symmetry is a finite 2-group (a.k.a. invertible)
symmetry with and without an ’t Hooft anomaly [2, 3, 5]. A 2-group is a symmetry structure
where a conventional symmetry is non-trivially intertwined with an invertible higher symme3
The definition of fusion n-category for n = 2 is given in [55] in detail, and it will also be reviewed in Section 2.1.
For n ≥ 3, the definition is proposed in [57, Definition II.9], upon the technical assumption mentioned in Remark I.2of the reference.
4
A statistical-mechanical model, which was defined following a similar conceptual framework, has been previously presented in [124].

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SciPost Phys. 16, 143 (2024)

b
ϕ

a

a

a

⊗

b

=

X

c
Na,b

c
c
(va,b
)i
b

c

c

a
a ∈ Simp C

ϕ ∈ HomC (a, b)

(a)

(b)

(d)

(c)

Figure 2: Data defining a fusion (1-)category C . (a) Topological lines in 1+1 dimensions define objects of C . In particular, simple (i.e., indecomposable) topologicallines correspond to simple objects, whereas superpositions of them correspond tonon-simple objects. (b) Topological line-changing operators define morphisms between topological lines. (c) The stacking of two topological lines a and b is denotedby a ⊗ b and it is in general a superposition of simple topological lines. (d) By locallycfusing two lines a and b, we get topological junction operators (va,b)i connecting
three lines a, b and c. The number of independent junction operators is equal to thefusion coefficient Nac b .
d

v2

=

x
a

v1
b

c

X

d
v4

(Fdabc )(x;v1 ,v2 ),(y;v3 ,v4 )ay,v3 ,v4

y
v3
b

c

Figure 3: The F -move in a fusion category.
try. Our method naturally works for constructing lattice models that possess such a symmetry
structure.
In the rest of the introduction, we briefly review the fusion category symmetry in 1+1
dimensions and the Freed-Teleman-Aasen-Fendley-Mong (FT-AFM) construction [33, 124].5
This would serve as a stepping stone to the (2+1)-dimensional case, which is a straightforward generalization of the (1+1)-dimensional case but is apparently more complicated. After
reviewing the FT-AFM construction in 1+1d, we will outline the construction of the 2+1d
fusion surface models.

1.2

Review of the Aasen-Fendley-Mong model

Fusion category. In 1+1 dimensions, a finite generalized symmetry is described by a fusioncategory, which is a generalization of a finite group. It describes the algebraic structure of
topological defects of codimension one, or equivalently topological lines, in 1+1 dimensions.6
More explicitly, a fusion category C contains the following data (see, e.g., [6, 7] for a detailed
explanation and more examples for physicists):• Simp C : The finite set of (isomorphism classes of) “simple objects”. A simple objecta ∈ Simp C represents an oriented indecomposable topological defect line in 1+1d.
There is a special object I ∈ Simp C representing the trivial defect. See Figure 2a.
5
We slightly generalize the presentation in [33] in that we allow the multiplicity of fusion coefficients. While
such symmetries are not so common in 1+1 dimensions, in 2+1 dimensions there are multiple inequivalent junctions as long as there is a bulk topological line. Thus, we consider the multiplicity in 1+1 dimensions as a warm-up.
6
When finite one-form symmetries are also present, the whole symmetry is described by a multifusion category,
and it is related to the concept of decomposition or “universes” in a (1+1)-dimensional system. See, for example,[32, 126, 127] for discussions on this point.

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SciPost Phys. 16, 143 (2024)

F

F

F

F
F
Figure 4: The pentagon identity.

L• Obj C : the set of objects. Any object b ∈ Obj C takes the form of b =a∈Simp C na awhere a runs over simple objects and na ’s are non-negative integers. An object a1 ⊕ a2
represents a superposition of defects a1 and a2 : The correlation function containing
a1 ⊕ a2 is the sum of the correlation function containing a1 and the one containing a2 .
• HomC (a, b): the “hom space” between two objects a and b. A morphism φ ∈ HomC (a, b)
from a to b represents a topological line-changing operator connecting the lines aand b. See Figure 2b. Such operators form a (finite-dimensional) C-vector spacebecause they can be added and multiplied by complex numbers. In addition, in
HomC (a, a) =: EndC (a), there is the identity operator/morphism ida . Two line-changing
operators φ1 ∈ HomC (a, b) and φ2 ∈ HomC (b, c) can be composed, and the composition
defines an element φ2 ◦ φ1 ∈ HomC (a, c). The hom space between two simple objects
a, b ∈ Simp C is one dimensional when a = b and zero-dimensional otherwise. In particular, for a simple object a ∈ Simp C , there is a canonical isomorphism EndC (a) ∼= C,which maps λ ida ∈ EndC (a) to λ ∈ C.
• a ⊗ b ∈ Obj C : The tensor product of objects a and b. This corresponds to the fusion of topological lines a and b. See Figures 2c and 2d. We can expand the tensor
Lcc
product as a ⊗ b = c∈Simp C Nabc, where the non-negative integers Nab
are called fusion coefficients. As a physicists’ convention, we fix a particular (non-canonical) basiscBasis(a ⊗ b, c) := {(va,b)i }i=1,··· ,N c of the hom space Hom(a ⊗ b, c) for a, b, c ∈ Simp C .
ab•

F -symbols (Fda bc )(x;v1 ,v2 ),( y;v3 ,v4 ) ∈ C:

complex numbers that govern the “F -move” depicted in Figure 3. Specifically, the F -symbols encode the relationship between two
different ways of composing basis morphisms via the following equation:7
X
v2 ◦ (v1 ⊗ idc ) =(Fdabc )(x;v1 ,v2 ),( y;v3 ,v4 ) v4 ◦ (ida ⊗v3 ) ∈ Hom(a ⊗ b ⊗ c, d) . (1)y∈Simp C ,v3 ∈Basis(b⊗c, y),v4 ∈Basis(a⊗ y,d)

The F -symbols should satisfy the pentagon identity depicted in Figure 4 [129].
In addition, a fusion category has the following data regarding “dual”, which is a relaxednotion of the inverse:
Precisely, the left- and right-hand sides of eq. (1) differ by an isomorphism αa,b,c : (a ⊗ b) ⊗ c → a ⊗ (b ⊗ c)called an associator, which is assumed to be the identity here. This assumption is always possible due to Mac Lane
strictness theorem [128].
7

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SciPost Phys. 16, 143 (2024)

a
a

a∗

=

=

a∗

ev
a

(b)

(a)

ev
=
a

a∗

=

dim a =

a

a∗

a

coev
a

(c)

coev
(d)

Figure 5: The duality structure in a fusion (1-)category C . (a) The dual a∗ of a
topological line a is its orientation reversal. (b) There is a morphism eva : a∗ ⊗ a → I
called evaluation defined by the figure. (c) There is a morphism coeva : I → a∗ ⊗ acalled coevaluation defined by the figure. (d) The expectation value of a loop of a
topological line a is called the “quantum dimension” and is denoted by dim a. In themathematical terms it is the composition eva ◦ coeva ∈ Hom(I, I) ∼= C.
• a∗ ∈ Obj C : The dual of an object a. This represents the orientation reversal of a, see
Figure 5a.
• eva ∈ HomC (a∗ ⊗ a, I): The evaluation morphism. This represents the pair-annihilationof topological lines a and a∗ , see Figure 5b.
• coeva ∈ HomC (I, a∗ ⊗ a): The coevaluation morphism. This represents the pair-creationof topological lines a and a∗ , see Figure 5c.8
• dim(a) ∈ C: The quantum dimension of an object a. This quantity is defined by theequality eva ◦ coeva = dim(a) id I and thus corresponds to the vacuum expectation valueof a loop of a, see Figure 5d. If the quantum dimensions of a and a∗ agree with eachother for every object a ∈ Obj C , the fusion category C is said to be spherical.
The above data that satisfy appropriate consistency conditions define a fusion category [6,7,9].
Examples. Let us see a few basic examples of fusion categories that naturally appear inphysical systems.
• Finite group. Topological defects for a finite group symmetry G form the fusion category VecG of G-graded vector spaces. The category VecG consists of simple objects L glabeled by group elements g ∈ G. These simple objects obey the group-like fusion rules
L g1 ⊗ L g2 = L g1 g2 and have trivial F -symbols.9 The dual of an object L g is its inverse,
i.e., we have L ∗g = L g −1 . In particular, when G = {id} is the trivial group, VecG reduces
to the category Vec of finite-dimensional vector spaces, which corresponds to the trivial(i.e., no) symmetry. We note that all simple objects of VecG are invertible.
• Ising category. A basic example of a non-invertible fusion category arises in the critical
Ising model [7, 23, 130]. The category contains simple objects η for the Z2 spin-flip
symmetry and N for the Kramers-Wannier self-duality, forming the fusion category Ising
There are left and right evaluation/coevaluation morphisms depending on whether we consider a ⊗ a∗ ora ⊗ a. For a unitary fusion category, any two of them are automatically determined by the other two because of
the unitary structure. See, e.g., [6, 9] for more details.
9
The F -symbols become non-trivial when the finite group symmetry G is anomalous.
8∗

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SciPost Phys. 16, 143 (2024)

called the “Ising category”. The latter object does not constitute a conventional symmetry
but does a non-invertible symmetry. The fusion rules of the simple objects are given by
η⊗η∼
=I,

η⊗N ∼
=N ⊗η∼=N,

N ⊗N ∼= I ⊕ η.

(2)

As we can see from the above equation, the simple object N is indeed non-invertible.
• Representation category. Another significant example of a fusion category is the representation category Rep(G) for a finite group G. In this category, simple objects are
irreducible representations of G, general objects are general finite dimensional representations, morphisms are intertwiners, the tensor product of objects is the ordinary
tensor product of representations, and the dual is the complex conjugation. When G is
non-abelian, Rep(G) contains irreducible representations of dimension greater than 1,
which are non-invertible.
Symmetry TFT construction. In [33], Aasen, Fendley, and Mong constructed both twodimensional classical statistical mechanical models and (1+1)-dimensional quantum chain
models based on fusion categories.
As noted in their paper [33], these models can be naturally understood in terms of
three-dimensional topological field theory known as the Turaev-Viro-Barrett-Westbury (TVBW)
model [131, 132]. Here, the TVBW model plays the role of what is called “symmetry topological field theory (SymTFT)” in the QFT literature [19, 91, 124, 133–135] and “categoricalsymmetry” in the condensed matter literature [58, 136–141].10
The TVBW model is a state sum model on a (2+1)-dimensional (oriented) spacetime lattice. The input datum of the state sum is a (spherical) fusion category C , and the TVBW model
constructed from a fusion category C is denoted by TVBW(C ). The model describes the topological order whose anyon data are described by the Drinfeld center Z(C ) of C , which is a
modular tensor category made out of a fusion category C .
The two-dimensional statistical mechanical model in [33], which we call the AFM heightmodel, can be constructed by placing the TVBW model on a slab, that is, the direct product of aninterval I and a two-dimensional oriented closed surface Σ, see Figure 6. On the left and rightboundaries of the slab I × Σ, we impose topological and non-topological boundary conditions
respectively. The non-topological boundary condition is defined by decorating the “Dirichlet”
boundary of TVBW(C ) with a network of defects as depicted in Figure 7. Here, the Dirichletboundary condition is a topological boundary condition such that the category of topologicallines on the boundary is the input fusion category C . For simplicity, just as in [33, 124], wechoose the Dirichlet boundary condition as the topological boundary condition on the left
boundary.11
To see the symmetry of the AFM height model, we consider the topological lines (or anyons)in TVBW(C ). Although we can insert any anyons labeled by objects of Z(C ) in the 3d bulk,
some of them can be absorbed by the topological boundary on the left. Thus, the nontrivial
topological lines in the AFM height model are identified with the lines on the topologicalboundary, which form the fusion category C .12
10

The idea of using the Turaev-Viro model to construct and study 2d statistical-mechanical systems had alreadyappeared in [124].
11
In general, topological boundary conditions of TVBW(C ) are in one-to-one correspondence with (finite
semisimple) module categories over C (equipped with a module trace) [122, 123, 142, 143]. In particular, theDirichlet boundary condition corresponds to the regular C -module category C .
12
If the topological boundary condition on the left boundary is the one labeled by a C -module category M,
topological lines on the boundary form a fusion category FunC (M, M), which is the category of C -module endofunctors of M [122, 123, 142]. Choosing a different topological boundary condition corresponds to gauging (apart of) the fusion category symmetry C [44, 48, 91, 133].

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SciPost Phys. 16, 143 (2024)

topological boundary

decorated boundary

2+1d TFT
TVBW(C)

Figure 6: The AFM height model realized as the TVBW model on a slab with one topological boundary and one non-topological boundary. The non-topological boundary
is obtained by decorating the Dirichlet boundary with a defect network as shown in
Figure 7.

F

ρ

T

ρ
F

Feρ

ρ

ρ
Fρ

Feρ

ρ

ρ
Fρ

F

Fρ

Feρ

ρ
F

Feρ

ρ
F

Fρ

Feρ

F

Fρ

Feρ

ρ
F

Fρ

F

Fρ

Feρ

ρ
F

ρ ∈ Obj C
F ∈ Obj CFρ ∈ HomC (ρ, F ⊗ ρ)Feρ ∈ HomC (ρ ⊗ F, ρ)Feρ

Figure 7: The decorated boundary in Figure 6. The transfer matrix T of the AFM
height model is defined by the region indicated in the figure.
AFM height model. Let us unpack the above abstract construction to obtain explicit 2d classical statistical models. The input data of the AFM height model are listed as follows:• A (spherical) fusion category C ,• an object ρ ∈ Obj C ,• an object F ∈ Obj C ,• morphisms Fρ ∈ HomC (ρ, F ⊗ ρ) and Feρ ∈ HomC (ρ ⊗ F , ρ).
Note that both ρ and F are not necessarily simple. Based on the above data,13 we explicitly
describe the AFM height model on a two-dimensional torus Σ = T 2 .
In order to define the model based on the above data, we first draw a defect network on
the torus T 2 as shown in Figure 7. This defect network plays the role of the spacetime lattice
on which dynamical variables reside. Specifically, we assign a dynamical variable Γi ∈ Simp C
to each plaquette i, and also assign a dynamical variable Γi j ∈ Basis(Γi ⊗ ρ ∗ , Γ j ) to each verticalsegment (i.e., a black edge) separating plaquettes i and j, see Figure 8 for the assignment ofthese dynamical variables. The dynamical variables Γi j on edges are trivial when all the fusionccoefficients Nab
are either 0 or 1, which is often assumed in the literature for simplicity. The
These data are redundant. In particular, for an invertible element φ ∈ HomC (F , F ), modifying (Fρ , Feρ ) into((φ ⊗ idρ ) ◦ Fρ , Feρ ◦ (idρ ⊗φ −1 )) does not change the model. In addition, it turns out that the choice F = ρ ∗ ⊗ ρ
can reproduce the most general model (for a fixed ρ). In this case, we can fix the above ambiguity by the conditionof, for example, Feρ = evρ∗ ⊗ idρ . With this gauge fixing, the pair (ρ, Fρ ) parametrizes the model without obvious
redundancies except for the overall scaling.
13

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SciPost Phys. 16, 143 (2024)

Γℓ
Γiℓ

Γℓk

Γi

Γk

Γi ∈ Simp CΓij ∈ Basis(Γi ⊗ ρ∗ , Γj )

Γij

Γjk
Γj

Figure 8: The dynamical variables in the AFM height model.
statistical mechanical partition function is defined by the sum of the Boltzmann weights for all
possible configurations of dynamical variables:
X YZ=
Wh (Γ~ ) .
(3){Γi },{Γi j } hHere, h runs over the horizontal segments (i.e., the blue and green edges) in Figure 7, andWh (Γ~ ) ∈ C is the local Boltzmann weight depending on the dynamical variables around h.
More specifically, when h is the green edge in Figure 8, the local Boltzmann weight Wh (Γ~ )depends only on four objects {Γi , Γ j , Γk , Γ` } and four basis morphisms {Γi j , Γ jk , Γi` , Γ`k } appearing
in the same figure. The explicit form of the Boltzmann weight Wh (Γ~ ) is given by the following
diagrammatic equation:
Γℓ

Γiℓ
ρ

Wh (Γ~ ) =qρFdim Γ j dim Γ` × Γi Fρ
ρ
Γij

Γℓk

Feρ Γk ,

(4)

ρ
Γj

Γjk

c
where (vac b )i ∈ HomC (c, a ⊗ b) is the dual junction of (vab)i ∈ Basis(a ⊗ b, c) that satisfiesccevc ◦(idc ∗ ⊗(vab)i ) ◦ (idc ∗ ⊗(vab) j ) ◦ coevc = δi j .

(5)c
We write the basis of Hom(c, a ⊗ b) as Basis(c, a ⊗ b) := {(vab)i }i=1,··· ,N c . The weight (4)abcan also be written explicitly in terms of F -symbols. If we define the transfer matrix T by
the Boltzmann weight on the region indicated in Figure 7, we can write the partition functionof the AFM height model as Z = Tr T N , where N is the number of plaquettes in the verticaldirection.

Quantum anyon chain. We can obtain a (1+1)-dimensional quantum chain model known as
the anyon chain [53,54] by taking the anisotropic limit of the above two-dimensional statisticalmechanical model [33]. The Hilbert space H of the model is spanned by the fusion trees
depicted in Figure 9. Here, we assign a simple object Γi ∈ C to each segment i connecting thevertical ρ lines, and assign a morphism Γi,i+1 ∈ Basis(Γi ⊗ ρ ∗ , Γi+1 ) to each vertex connecting
the segments i, i + 1 and the vertical ρ line. These simple objects and basis morphisms are the9SciPost Phys. 16, 143 (2024)

Γi−2,i−1
Γi−2

Γi−1,i

Γi−1

Γi,i+1ΓiρΓi+1,i+2

Γi+1

ρ

Γi+2

ρ

ρ

Figure 9: The fusion tree defining the Hilbert space of the quantum anyon chain.
dynamical variables of the model.14 An assignment of simple objects Γi and Γi+1 is prohibitedΓ
if the fusion coefficient NΓ i+1∗ is zero.
iρThe Hamiltonian H of the model is derived by expanding the transfer matrix of the AFM
height model as T = idH − εH + O(ε2 ) in the anisotropic limit, P
where ε is a small parame15
ter. The Hamiltonian obtained in this way is of the form H = − i ĥi−1,i,i+1 , where the local
interaction ĥi−1,i,i+1 can be expressed diagrammatically as16
Γi−1,i
Γi−1

Γi−1,iΓi,i+1
Γi

Γi+1

ρ

Γi−1

=

ĥi−1,i,i+1

Γi,i+1
Γi

Fρ

ρ

Γi+1
Feρ

F
ρ

Γi−1,i=X

A(Fint )

Γi−1

Γi
v

Γi+1
w

f

Fint

ρ

Γi,i+1

ρ

.

ρ

(6)
Here, Fint = { f , v, w} denotes the set consisting of a simple object f ∈ Simp C and morphismsv ∈ Basis(ρ, f ⊗ ρ) and w ∈ Basis(ρ ⊗ f , ρ) that appear in the diagram on the right-hand side.
The weight A(Fint ) is a complex number determined by (F , Fρ , Feρ ).
The above 1+1d model has a fusion category symmetry C . The symmetry acts on thesystem “from above” as shown in Figure 10. That is, we define the action of a topological linea ∈ Obj C by placing it above the fusion tree and fusing it into the tree using the F -move. Thissymmetry action commutes with the Hamiltonian (6) because it acts on the fusion tree “frombelow”:


a









Γi−1,i
Γi−1

Γi−1,iΓi,i+1

Γi

ρ

Γi−1

Γi+1

ρ

Γi

Fρ

,

Γi,i+1
Γi+1
Feρ

F
ρ

ρ




 = 0.


(7)

If we write both the Hamiltonian and the symmetry action in terms of the F -symbols, thecommutation relation (7) follows from the pentagon identity shown in Figure 4.
Examples.

Let us consider several examples of the anyon chain models.

L• Spin chains. When C = VecZN and ρ = g∈ZN L g , the state space of the anyon chain
model becomes the tensor product of N -dimensional on-site Hilbert spaces. Namely,
14
The dimension of the Hilbert space asymptotically grows as dim H ∼ (dim ρ) L where L  1 is the number ofvertical ρ lines. Thus, we can regard ρ as the degree of freedom at each site. Note that dim ρ is not necessarily
an integer.
15
As we can see from eq.
(4), the parameter of the AFM height model is a morphism
ϕ := (Feρ ⊗ idρ ) ◦ (idρ ⊗ Fρ ) ∈ Hom(ρ ⊗ ρ, ρ ⊗ ρ). The anisotropic limit is defined by choosing ϕ = idρ⊗ρ + εϕ 0
and taking the limit of ε  1. By abuse of notation, ϕ 0 is also written as (Feρ ⊗ idρ ) ◦ (idρ ⊗ Fρ ) in eq. (6).
16
In the original paper by Aasen, Fendley, and Mong [33], they use a different basis for the local Hamiltonianĥi−1,i,i+1 . However, the choice of a basis does not affect the family of Hamiltonians obtained in this way, up to a
reparametrization, because the two bases are related by F -moves.

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SciPost Phys. 16, 143 (2024)

a
Γi−2

Γi−1

Γi+1

Γi

Γi+2

Γi−2

Γi−1

Γi+1

a ⊗ Γi

Γi+2

F
ρ

ρ

ρ

ρ

ρ
Γi−2

ρ

ρ

ρ

a ⊗ Γi a ⊗ Γi+1

Γi−1

Γi+2

F
ρ

ρ

ρ

ρ

Figure 10: The action of a line a ∈ Obj C on the Hilbert space H. This action changes
the dynamical variable on each edge from Γi to a ⊗ Γi . The dynamical variables on
the vertices are also affected accordingly. The resulting fusion tree can be written asa linear combination of fusion trees whose edges are labeled by fusion channels ofa ⊗ Γi .
we have a ZN -valued spin on each site. The Hamiltonian of the model preserves theon-site ZN symmetry that rotates these spins. Thus, the anyon chain model in this casereduces to an ordinary ZN -symmetric spin chain. More generally, if we choose C = VecG
Land ρ = g∈G L g , we obtain a G-symmetric spin chain whose on-site Hilbert space isthe regular representation of G. Furthermore, we can also consider spin chains withanomalous finite group symmetries by choosing C = VecωG , the category of G-graded
vector spaces with a twist ω ∈ H 3 (G, U(1)).
• Gauged spin chains. If we choose C = Rep(G), we obtain the G-gauged version of
the spin chains, where the choice of ρ ∈ Rep(G) determines the on-site Hilbert spaceof the ungauged G-symmetric spin chain. More specifically, the Rep(G) symmetry canbe ungauged by replacing the Dirichlet boundary condition on the left boundary of the
SymTFT with another topological boundary condition labeled by a Rep(G)-module category Vec [44, 48, 133]. This ungauging procedure results in a G-symmetric spin chain
whose on-site Hilbert space is ρ ∈ Rep(G).
• Critical Ising model. When C is the Ising category and ρ is the Kramers-Wannier duality
line, the anyon chain model reproduces the critical Ising model [33].
• Golden chain. When C is the Fibonacci category and ρ is the unique non-invertibleline,17 we obtain the golden chain [53].
• Haagerup model. The Haagerup category H3 is a fusion category that is directly related to neither finite groups nor affine Lie algebras [144–146]. It is generated by a Z3
invertible line η and a self-dual non-invertible line ρ that satisfy the following fusion
rules:
ρ⊗η∼(8)= η2 ⊗ ρ , ρ ⊗ ρ ∼= I ⊕ (I ⊕ η ⊕ η2 ) ⊗ ρ .
Numerical studies in [40,41] suggest that the anyon chain model and the corresponding
statistical mechanical model with the Haagerup symmetry H3 contain a critical pointwith central charge c ∼ 2, but the conclusive identification of the phase is elusive so far.

1.3

Generalization to 2+1 dimensions

Our strategy for constructing (2+1)-dimensional models, which we call the fusion surface
models, is to directly generalize the story reviewed above.
17

The Fibonacci category consists of two simple objects I and τ that satisfy the fusion rule τ ⊗ τ = I ⊕ τ.

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SciPost Phys. 16, 143 (2024)

Figure 11: The diagrammatic representations of objects, 1-morphisms, and 2morphisms in a fusion 2-category. An object A is represented by an oriented surface,
whose coorientation is specified by the red arrow perpendicular to the surface. A
1-morphism a : A → B is represented by an oriented line at the interface between
surfaces A and B. A 2-morphism α : a ⇒ b is represented by a junction of lines aand b. In our convention for surface diagrams, a 1-morphism is read from right toleft, and a 2-morphism is read from bottom to top. The red arrow specifying thecoorientation of a surface will be omitted when the coorientation is clear from thecontext.
Fusion 2-category. In higher dimensions, finite generalized symmetries are expected to be
described by higher categories [55–58]. In particular, we can naturally expect that finite generalized symmetries of (2+1)-dimensional unitary bosonic systems are generally describedby fusion 2-categories. The precise definition of a (spherical) fusion 2-category can be foundin [55]. Here we review the concept of fusion 2-category very briefly. A longer review of fusion
2-categories will be provided in Section 2.1.
In 2+1 dimensions, a defect can have 2-, 1-, or 0-dimensional volume. Correspondingly, afusion 2-category C consists of• Obj C : The set of objects,• HomC (A, B): The 1-category of 1-morphisms between objects A, B ∈ Obj C , and• HomA→B (a, b):Thea, b ∈HomC (A, B).18

vectorspaceof

2-morphisms

between

1-morphisms

As depicted in Figure 11, each element of Obj C corresponds to a two-dimensional topologicalsurface, each object of HomC (A, B) corresponds to a topological interface between two surfaces
A and B, and each element of HomA→B (a, b) corresponds to a topological interface between
topological interfaces a and b. Note that both 1- and 2-morphisms can be composed, e.g., fora ∈ HomC (A, B) and b ∈ HomC (B, C), there exists b ◦ a ∈ HomC (A, C). Similarly, two objects A
and B can be stacked on top of each other, which defines the tensor product AB ∈ Obj C , see
Figure 12. Furthermore, a fusion 2-category is also equipped with the duality data such as thedual A# of an object A, the dual a∗ of a 1-morphism a, and the evaluation and coevaluation
morphisms associated with them.
Symmetry TFT construction. In [55], a state sum model on a four-dimensional (oriented)
spacetime lattice is defined based on a (spherical) fusion 2-category C . We call this state sum
model the Douglas-Reutter (DR) model and denote it as DR(C ). The DR model is a fourdimensional version of the TVBW model, and thus we can utilize it to generalize the FT-AFM
construction to one higher dimension.
In order to generalize the FT-AFM construction, we consider the DR model DR(C ) on afour-dimensional slab I × T 3 where I is an interval and T 3 is a three-dimensional torus, see
the left panel of Figure 13. On the left boundary of the slab, we impose the Dirichlet boundary
18

The vector space HomA→B (a, b) is a hom space of the 1-category HomC (A, B).

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SciPost Phys. 16, 143 (2024)

Figure 12: The diagrammatic representation of the tensor product AB. A surface
labeled by A appears in front of a surface labeled by B.

topological boundary

decorated boundary

3+1d TFT
DR(C)

g
f

Figure 13: Schematic description of the fusion surface model.
condition of DR(C ).19 In particular, this means that topological defects on the left boundary
are described by the fusion 2-category C that we started with. On the other hand, on the rightboundary, we impose a non-topological boundary condition that is obtained by decorating theDirichlet boundary with the defect network shown in the right panel of Figure 13.
Since the bulk of DR(C ) is topological, the configuration depicted in the left panel of Figure
13 defines a purely 3d classical statistical model, which we call the 3d height model. Furthermore, by taking the anisotropic limit of the 3d height model, we can define the corresponding
2+1d quantum lattice model, which we call the fusion surface model. The derivation of the
3d height models and 2+1d fusion surface models will be explained in detail in sections 3 and4 respectively. In the rest of this subsection, we will briefly summarize the definition of the
2+1d fusion surface model and describe its fusion 2-category symmetry.
1.3.1

Fusion surface models

Input data. The fusion surface model is a 2+1d quantum model on a honeycomb lattice.20
In order to define the state space of this model, we fix the following data, see Figure 14:• A (spherical) fusion 2-category C ,• objects ρ, σ, λ ∈ Obj C ,• 1-morphisms f ∈ HomC (ρσ, λ) and g ∈ HomC (σρ, λ).
19More generally, we can also use a different topological boundary condition on the left boundary, which should
be labeled by a module 2-category over C . A different choice of a module 2-category would correspond to adifferent way of gauging the fusion 2-category symmetry C . In the case of a non-anomalous finite group symmetry
2VecG , the relation between the choice of a module 2-category and the (twisted) gauging is studied in [113]. We
do not explore this generalization in this paper.
20
Although a honeycomb lattice is convenient for our purpose, e.g. because at each vertex the minimal number(three) of edges meet, it should be straightforward to generalize the model to another lattice.

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SciPost Phys. 16, 143 (2024)

σ

ρ

λ
g

f
λ

ρ

σ

g

f

ρ, λ, σ ∈ Obj C
f ∈ HomC (ρ□σ, λ)λg ∈ HomC (σ□ρ, λ)

ρ

σ
f

g

λ

ρ

σ

Figure 14: Data defining the state space.

Fλ
g

f
Fλσ
Fρλ

Fρ
g
Fσρ

Feσ

Fσ
f
Fσρ

F
g

f
Fρλ
Fλσ

Feρ

Feλ

Figure 15: Data defining the Hamiltonian. The domains/codomains of the symbols
are summarized in eqs. (9) and (10).
Moreover, to define the Hamiltonian, we fix the data listed below, see also Figure 15:• An object F ∈ Obj C ,• 1-morphisms

Fλ ∈ HomC (F λ, λ) ,Feλ ∈ HomC (λF , λ) ,Fρ ∈ HomC (F ρ, ρ) ,Feρ ∈ HomC (ρF , ρ) ,Fσ ∈ HomC (F σ, σ) ,Feσ ∈ HomC (σF , σ) ,(9)

• 2-morphismsfFλσ ∈ HomρσF →λ (Feλ ◦ ( f 1F ), f ◦ (1ρ Feσ )) ,F g ∈ HomσF ρ→λ (g ◦ (Feσ 1ρ ), g ◦ (1σ Fρ )) ,σρfFρλ ∈ HomF ρσ→λ ( f ◦ (Fρ 1σ ), Fλ ◦ (1F  f )) ,g
Fλσ ∈ HomF σρ→λ (Fλ ◦ (1F g), g ◦ (Fσ 1ρ )) ,fFσρ
∈ HomρF σ→λ ( f ◦ (1ρ Fσ ), f ◦ (Feρ 1σ )) ,(10)Fρλ ∈ HomσρF →λ (g ◦ (1σ Feρ ), Feλ ◦ (g1F )) .
g

State space. The state space H0 of the fusion surface model on a honeycomb lattice is a
specific subspace of a larger state space H that is spanned by fusion diagrams of the following

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SciPost Phys. 16, 143 (2024)

form:

.

(11)

Dynamical variables living on plaquettes (written in white), edges (written in black), andvertices (written in black) of the honeycomb lattice are labeled by simple objects, simple 1morphisms, and basis 2-morphisms of C respectively. More precisely, the plaquette variables
take values in the set of representatives of connected components of simple objects of C , andthe edge variables take values in the set of representatives of isomorphism classes of simple
1-morphisms of C , see Section 2.1 for the terminology. The dynamical variables on plaquettes
are denoted by Γi in the above equation, whereas the dynamical variables on edges and vertices
are not specified in order to avoid cluttering the diagram. The colored surfaces in eq. (11)are labeled by objects ρ, σ, and λ, which are not dynamical variables of the model. Similarly,
the colored edges in eq. (11) are labeled by 1-morphisms f and g, which are not dynamicalvariables as well. The state space H0 is the subspace of H on which the eigenvalue of theplaquette operator B̂ p defined by the following equation is 1 for every plaquette p:

=

B̂ p

X
Γ45 ∈End(Γ4 )dim(Γ45 )Dim(Γ4 ).

(12)Here, dim(Γ45 ) is the quantum dimension of a simple 1-morphism Γ45 , andDim(Γ4 ) := dim(Γ4 ) dim(EndC (Γ4 )) is the product of the quantum dimension of a simple object
Γ4 and the total dimension of a fusion 1-category EndC (Γ4 ) := HomC (Γ4 , Γ4 ), see Section 2 forthe definitions of these quantities. The diagram on the right-hand side of eq. (12) is evaluatedby fusing the loop labeled by Γ45 to the edges of the honeycomb lattice. We note that B̂ p is alocal commuting projector just like the plaquette operator of the Levin-Wen model [125]. The
projector to the subspace H0 is given by the product of B̂ p ’s on all plaquettes, namely, we have!
Y
H0 =B̂ p H .
(13)
p: plaquettes

Hamiltonian.

The Hamiltonian of the model is given by H = −P

p: plaquettes T̂p , where eachterm T̂p is defined by the following diagrammatic equation:

T̂p

=

.

(14)

The diagram on the right-hand side is evaluated by fusing the yellow surface, yellow edges,
and yellow vertices into the honeycomb lattice. Here, they are labeled by an object F ∈ Obj C ,15

SciPost Phys. 16, 143 (2024)

Figure 16: The action of a fusion 2-category symmetry is defined by the fusion of
surface and line defects to the spatial lattice.
1-morphisms (9), and 2-morphisms (10) as shown in Figure 15. If we expand them in termsof simple objects, simple 1-morphisms, and basis 2-morphisms, the Hamiltonian (14) can also
be written as

T̂p

=

X

A(Fint )

,

(15)

Fint

where the weight A(Fint ) is a complex number, and the summation on the right-hand side istaken over all possible simple objects, simple 1-morphisms, and basis 2-morphisms labelingthe yellow surface, yellow edges, and yellow vertices in the diagram. The labels summed overare collectively denoted by Fint in the above equation. Specifically, Fint consists of one simple
object, six simple 1-morphisms, and six basis 2-morphisms. Under several assumptions thatwe spell out in Section 4.2, we can show that the Hamiltonian (15) becomes Hermitian if theweight satisfies A(Fint ) = A(F int )∗ , where F int basically means the dual of Fint , see Section
4.2 for more details.21
Symmetry. The fusion surface model defined above has a fusion 2-category symmetry described by the input fusion 2-category C . The action of the symmetry is defined by the operationof fusing topological surfaces and topological lines into the honeycomb lattice from above asshown in Figure 16. This symmetry action can be written in terms of the 10-j symbols of thefusion 2-category. The commutativity of the symmetry action and the Hamiltonian (15) isguaranteed by the coherence conditions on the 10-j symbols. This is because the symmetry
operator acts on a state from above, while the Hamiltonian acts on a state from below, see eq.
(7) for an analogous statement in 1+1d.
Examples.
this paper.

Let us see several examples of the fusion surface model that we will discuss in• Spin models with anomalous finite group symmetries. When the input fusion 24
category is the 2-category 2Vecω
G of G-graded 2-vector spaces with a twist ω∈ H (G,U(1))[55], the fusion surface model has a finite group symmetry G with an anomaly ω. In
21

We can always make the Hamiltonian Hermitian by adding the Hermitian conjugate, which would not violate
the fusion 2-category symmetry of the model.

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SciPost Phys. 16, 143 (2024)

particular, when ρ, σ, and λ are the sum of all simple objects ρ = σ = λ =  g∈G g,
the dynamical variables of the model are G-valued spins on all plaquettes. Thus, thefusion surface model in this case reduces to an ordinary spin model with an anomalous
finite group symmetry. We will study this example in Section 4.4.3. As a special case,the fusion surface model includes the anomaly free G-symmetric spin model discussed
in [113].
• Lattice models with non-invertible and invertible 1-form symmetries. When the
input fusion 2-category is (the condensation completion of) a ribbon category B ,22 thefusion surface model has a non-invertible 1-form symmetry described by B . We will
discuss this example briefly in Section 4.4.2. In particular, when the fusion rules of
B are group-like, the fusion surface model reduces to an ordinary spin model with ananomalous invertible 1-form symmetry. This example will be discussed in more detail in
Section 5.1.
• Kitaev honeycomb model without a magnetic field. When the input fusion 2-categoryis (the condensation completion of) the Ising category, we can obtain the Kitaev honeycomb model without a magnetic field [149] as a variant of the fusion surface model. We
will consider this example in Section 5.2.
• Non-chiral topological phases with fusion 2-category symmetries. For any fusion
2-category C , we can construct a commuting projector Hamiltonian with C symmetry bydefining the input data of the fusion surface model using a separable algebra in C . Since
the Hamiltonian is the sum of local commuting projectors, this model would realize a
non-chiral topological phase with C symmetry. We expect that all non-chiral topological phases with arbitrary fusion 2-category symmetries can be realized in this way bychoosing a separable algebra appropriately. This example will be discussed in Section
5.3.

1.4

Structure of the paper

This paper is organized as follows. In Section 2, we review fusion 2-categories and the state
sum construction of the 4d Douglas-Reutter TFT. In Section 3, we define the 3d height models
on a cubic lattice, which are three-dimensional analogues of the 2d AFM height models. In
Section 4, we derive the 2+1d fusion surface models on a honeycomb lattice by taking anappropriate limit of the 3d height models. In particular, we see that the fusion surface models
are (2+1)-dimensional analogues of the 1+1d anyon chain models. We also investigate theunitarity and fusion 2-category symmetries of these models. Finally, in Section 5, we study
several examples of the fusion surface models, including those that would realize general
non-chiral topological phases with fusion 2-category symmetries.

2

Preliminaries

Throughout the paper, we suppose that the base field of a fusion 2-category is C.
22
The condensation completion physically means that we add in topological surfaces consisting of the condensates of topological lines. Mathematically, the condensation completion of a ribbon category B is described by
the 2-category Mod(B ) of module categories over B . Practically, the construction of our lattice model only uses
B rather than its condensation completion. We note that there is a closely related notion called orbifold completion [27,147], which might agree with the condensation completion in the case of our interest. We refer the reader
to [56, Remark 1.4.7] and [148] for the potential difference between these notions.

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SciPost Phys. 16, 143 (2024)

2.1

Fusion 2-categories

Finite symmetries in 2+1 dimensions are characterized by the algebraic structure of topologicalsurfaces, topological lines, and topological point defects. In unitary theories, these defects are
expected to form a spherical fusion 2-category. In this section, we briefly review the basics offusion 2-categories. We refer the reader to [55] for more details, see also [150].
A fusion 2-category C consists of objects, 1-morphisms between objects, and 2-morphisms
between 1-morphisms. The 1-morphisms between objects A and B form a finite semisimple
1-category, which is denoted by HomC (A, B). The subscript C of HomC (A, B) will often be
omitted in what follows. The 2-morphisms between 1-morphisms a and b form a finite dimensional vector space, which is denoted by Hom(a, b). Physically, objects, 1-morphisms, and2-morphisms correspond to surface defects, line defects, and point defects respectively. The
diagrammatic representations of these data are shown in Figure 11.
The fusion of topological surfaces labeled by A and B corresponds to taking the tensor
product of objects A and B. The tensor product of A and B, which is denoted by AB, isrepresented by the diagram of layered two surfaces, where the surface labeled by A is put infront of the surface labeled by B, see Figure 12. The unit of the tensor product is called a unit
object and is denoted by I. The unit object I corresponds to a trivial surface defect, which isrepresented by an invisible diagram.
The fusion of topological lines labeled by a and b corresponds to the composite of 1morphisms a and b, which is denoted by a ◦ b. In the diagrammatic representation of thecomposite a ◦ b, the line labeled by a is on the left of the line labeled by b. There is also asimilar correspondence between the fusion of topological point defects and the compositionof 2-morphisms.
Every object and every 1-morphism of a fusion 2-category C have their duals. The dual#
A of an object A is represented by the orientation reversal of a surface diagram labeled by A.
Similarly, the dual a∗ of a 1-morphism a is represented by the orientation reversal of a line
labeled by a.23 We note that a∗ is a 1-morphism from B to A when a is a 1-morphism from
A to B. Taking the duals of objects and 1-morphisms is involutive, i.e., we have A## = A anda∗∗ = a.
An object A ∈ C is called a simple object if the vector space of 2-endomorphisms of theidentity 1-morphism 1A is one-dimensional, i.e., End(1A) := Hom(1A, 1A) ∼= C. The unit object
I of a fusion 2-category is required to be simple. Similarly, a 1-morphism a ∈ Hom(A, B) is
called a simple 1-morphism if its endomorphism space End(a) is a one-dimensional vector
space. We note that the identity 1-morphism of a simple object is simple. A fusion 2-category
C has only finitely many (isomorphism classes of) simple objects and simple 1-morphisms
between simple objects.
Any objects and 1-morphisms in C can be decomposed into finite direct sums of simple
objects and simple 1-morphisms respectively. The direct sum of objects A and B is denoted byA B, whereas the direct sum of 1-morphisms a and b is denoted by a ⊕ b. Simple objects andsimple 1-morphisms are indecomposable, which means that they cannot be decomposed into
direct sums any further.
Since the unit object I is simple, the identity 1-morphism 1 I has a one-dimensional vector
space of 2-endomorphisms End(1 I ) ∼= C. This implies that any 2-endomorphism of 1 I is proportional to the identity 2-morphism and hence can be identified with a number. In particular,a closed surface diagram, when viewed as a 2-endomorphism of 1 I , gives rise to a complex
number. This enables us to define complex numbers dimR (a) and dim L (a) for a 1-morphism
23

In the mathematical literature, the dual of a 1-morphism is often called the adjoint.

18

SciPost Phys. 16, 143 (2024)a ∈ Hom(A, B) by the following sphere diagrams:

dimR (a) =dim L (a) =

,

.

(16)

We call these quantities the right and left dimensions of a. Here, we implicitly used the(co)evaluation 1- and 2-morphisms to define the cups and caps as in the definition of thequantum dimension in a fusion 1-category, cf. Figure 5d.
The left and right dimensions agree with each other when a fusion 2-category C is equippedwith an additional structure called a pivotal structure. In a pivotal fusion 2-category, the right(or equivalently left) dimension of a 1-morphism a is simply denoted by dim(a) and is called
the quantum dimension of a. In particular, the quantum dimension of the identity 1-morphism
1A ∈ End(A) is denoted by dim(A) := dim(1A) and is called the quantum dimension of A.24 A
pivotal fusion 2-category C is said to be spherical if the quantum dimension of every object
agrees with the quantum dimension of its dual. In the rest of this paper, a fusion 2-category C
always means a spherical fusion 2-category.
The quantum dimension of a 1-morphism a ∈ Hom(A, B) can be understood as the trace of
the identity 2-morphism of a. More generally, the trace of a 2-morphism α ∈ End(a) is definedas the value of the following sphere diagram:

Tr(α) =.

(17)

The trace defines a non-degenerate pairing 〈α, β〉 := Tr(αβ) between 2-morphismsα ∈ Hom(a, b) and β ∈ Hom(b, a). The dual bases of the vector spaces Hom(a, b) andHom(b, a) with respect to the above non-degenerate pairing are denoted by {αi } and {αi },which satisfy 〈αi α j 〉 = δi j . We call αi and αi basis 2-morphisms or normalized 2-morphisms.
In a fusion 1-category, the associativity of the tensor product is captured by the F -symbols.
In a fusion 2-category, the tensor product among objects satisfies a higher associativity, whichis captured by the data called the 10-j symbols. Specifically, the 10-j symbols z+ (Γ ; [01234])
24
Simple objects A and B can have different quantum dimensions even if they are isomorphic to each other.
Physically, isomorphic objects with different quantum dimensions differ by an invertible 2d TFT.

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SciPost Phys. 16, 143 (2024)

and z− (Γ ; [01234]) are defined by the following diagrammatic equations:25

=

X

X

dim(Γ024 )z+ (Γ ; [01234]),

Γ024 Γ0124 ,Γ0234

(18)

=

X

X

dim(Γ024 )z− (Γ ; [01234]).

Γ024 Γ0124 ,Γ0234(19)
Here, the diagram in the above equation consists of ten surfaces [i j] labeled by simple objects Γi j where 0 ≤ i < j ≤ 4, ten lines [i jk] labeled by simple 1-morphisms
Γi jk ∈ HomC (Γi j Γ jk , Γik ) where 0 ≤ i < j < k ≤ 4, and five points [i jkl] labeled by basis
2-morphisms Γi jkl ∈ Hom(Γikl ◦ (Γi jk 1Γkl ), Γi jl ◦ (1Γi j Γ jkl )) where 0 ≤ i < j < k < l ≤ 4. The
summation on the right-hand side is taken over (isomorphism classes of) simple 1-morphisms
Γ024 and basis 2-morphisms Γ0124 and Γ0234 . The braiding of two lines Γ012 and Γ234 on theright-hand side represents the interchanger 2-isomorphism [55], which reduces to the ordinary braiding isomorphism when Γ012 and Γ234 are 1-endomorphisms of the unit object I.
For later convenience, we define the notion of connected components of simple objects
and simple 1-morphisms in a fusion 2-category C . Simple objects A and B are connected if andonly if there exists a non-zero 1-morphism between them, and we say they are in the sameconnected component. Similarly, simple 1-morphisms a and b are connected if and only ifthere exists a non-zero 2-morphism between them. We note that the connected component of asimple object is bigger than the isomorphism class of the simple object.26 This is because there
can be non-zero 1-morphisms between simple objects A and B even if they are not isomorphic
to each other. The set of connected components of simple objects in C is denoted by π0 C . By aslight abuse of notation, we will also write the set of representatives of connected components
as π0 C .
25
The 10-j symbols defined here are the same as those defined in [55]. Indeed, eq. (18) and (19) reduce to theoriginal definition of the 10-j symbols given in [55] if we take the trace of these equations after post-composing a
2-morphism that is the dual of a summand on the right-hand side.
26
On the other hand, the connected component of a simple 1-morphism agrees with the isomorphism class. Thisis because a simple 1-morphism a : A → B in a fusion 2-category is a simple object in a finite semisimple 1-category
Hom(A, B) and every non-zero morphism between simple objects in such a 1-category is an isomorphism due to
Schur’s lemma.

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SciPost Phys. 16, 143 (2024)

Table 1: The relation between the Douglas-Reutter TFT and other 4d TFTs [55].
Fusion 2-category
Finite group (with a twist)
Finite 2-group (with a twist)
Ribbon category
G-crossed braided fusion category

Douglas-Reutter TFT(twisted) Dijkgraaf-Witten TFT [155](twisted) Yetter TFT [156, 157]
Crane-Yetter TFT [158–160]
Cui TFT [161]Figure 17: The compatibility conditions on a configuration of dynamical variables.
The left figure represents the compatibility between the dynamical variable on a 2simplex [i jk] and those on the boundary of [i jk]. The right figure represents thecompatibility between the dynamical variable on a 3-simplex [i jkl] and those on theboundary of [i jkl].

2.2

Douglas-Reutter TFT

The Douglas-Reutter TFT is a four-dimensional oriented topological field theory obtained froma spherical fusion 2-category C [55]. This TFT generalizes various 4d TFTs known in theliterature, see table 1.27 In this section, we review the Douglas-Reutter TFT, which is denoted
by DR(C ), following Walker’s universal state sum [162].
Let M be a closed oriented 4-manifold. In order to define the partition function of the
Douglas-Reutter TFT DR(C ) on M , we first choose a triangulation of M . We also give a branching structure on the triangulated 4-manifold M by choosing a global order o of 0-simplices.
The dynamical variables of the Douglas-Reutter TFT DR(C ) are simple objects Γi j living on
1-simplices [i j], simple 1-morphisms Γi jk living on 2-simplices [i jk], and basis 2-morphisms
Γi jkl living on 3-simplices [i jkl], where i < j < k < l with respect to the global order o of 0simplices. Here, Γi j and Γi jk are taken from the set of representatives of connected components
of simple objects and simple 1-morphisms. A configuration of the above dynamical variables
must be compatible with the monoidal structure of a fusion 2-category C in the following
sense: a simple 1-morphism Γi jk is a 1-morphism from Γi j Γ jk to Γik and a basis 2-morphism
Γi jkl is a 2-morphism from Γikl ◦ (Γi jk 1Γkl ) to Γi jl ◦ (1Γi j Γ jkl ). These compatibility conditions
can be expressed by the fusion diagrams shown in Figure 17. A configuration Γ of dynamicalvariables is referred to as a C -state.
The partition function of DR(C ) is given by the sum of appropriate weights over all possible
C -states. More specifically, the partition function ZDR(M ) on a closed oriented 4-manifold M
is defined by the following formula: [55, 162]
ZDR(M ) =XYΓ

0-simplices [i]×Y
2-simplices [i jk]

1
dim(C )

Y
1-simplices [i j]1Dim(Γi j )Ydim(Γi jk )zεo ([i jklm]) (Γ ; [i jkl m]) .

(20)

4-simplices [i jklm]

There are also other 4d TFTs such as unoriented TFTs [151,152], spin and pin TFTs [153], and TFTs with U(1)symmetry [154], which are not examples of the Douglas-Reutter TFT. We do not consider these TFTs in this paper.
It would be interesting to generalize our analyses to these TFTs.
27

21

SciPost Phys. 16, 143 (2024)

P
Here, dim(C ) = X ∈π0 C (dim(End(X )))−1 is the total dimension of a fusion 2-category C ,28
Dim(Γi j ) := dim(Γi j )dim(End(Γi j )) is the product of the quantum dimension dim(Γi j ) of asimple object Γi j and the global dimension dim(End(Γi j )) of the endomorphism 1-category
End(Γi j ), and dim(Γi jk ) is the quantum dimension of a simple 1-morphism Γi jk . The weightzεo ([i jklm]) (Γ ; [i jkl m]) on a 4-simplex [i jkl m] is the 10-j symbol defined by eqs. (18) and (19).
The subscript εo ([i jkl m]) is a sign determined by the orientation of a 4-simplex [i jkl m] inthe following way: εo ([i jkl m]) is + if the orientation of [i jkl m] induced by the orientation of
the underlying manifold M agrees with the one induced by the global order o of 0-simplices,and εo ([i jkl m]) is − otherwise.
The formula (20) is based on Walker’s universal state sum construction [162], which isslightly different from the original formulation by Douglas and Reutter [55]. In the original
paper by Douglas and Reutter, dynamical variables Γi j on 1-simplices are taken from the set of
isomorphism classes of simple objects rather than the set of connected components of them.
Accordingly, the scalar factor on each 1-simplex [i j] is further divided by the number of (isomorphism classes of) simple objects in the connected component of Γi j . In the subsequent
sections, we will use Walker’s universal state sum construction instead of the original formulation by Douglas and Reutter because the former can immediately be applied to manifolds
with general cell decompositions, which is convenient for our purposes.
In order to apply Walker’s universal state sum to a closed 4-manifold M with a general celldecomposition, we begin with turning the cell decomposition into a handle decomposition by
thickening each cell. A thickened j-cell is called a j-handle, which has the shape of B j × B 4− jwhere B n is an n-dimensional ball. We sometimes use the terms j-cell and j-handle interchangeably. The boundary of a j-cell is topologically a 3-sphere S 3 consisting of two regions
S j−1 × B 4− j and B j ×S 3− j . The former region is glued to i-cells for i < j, while the latter region
is glued to i-cells for i > j. Given a handle decomposition as above, we can write down thepartition function of the Douglas-Reutter TFT on M as
X Y
Y
1
1
ZDR(M ) =dim(C ) 3-handles h Dim(Γ (h3 ))Γ 4-handles h4
(21)
Y
Y3
×
ev(Γ (∂ h2 ))ev(Γ (∂ h0 )) ,2-handles h20-handles h0

where a C -state Γ is a compatible assignment of simple objects Γ (h3 ), simple 1-morphisms
Γ (h2 ), and basis 2-morphisms Γ (h1 ) to 3-handles h3 , 2-handles h2 , and 1-handles h1 respectively. In the above equation, Γ (∂ h j ) represents the surface diagram that appears as the intersection of the 3-sphere ∂ h j and the original (i.e., not thickened) cell decomposition labeled
by a C -state Γ . Since the surface diagram Γ (∂ h j ) is closed, it defines a 2-morphism from theunit object I to itself, which is canonically identified with a complex number. This complex
number, which would be expressed in terms of the 10-j symbols and the quantum dimensions,is denoted by ev(Γ (∂ h j )) in the above equation. We note that eq. (21) reduces to eq. (20)when the cell decomposition is the dual of a triangulation.
We can also apply Walker’s universal state sum to manifolds with boundaries. In order tocompute the partition function on an oriented 4-manifold M with boundary ∂ M , we endow
M with a cell decomposition and label the boundary 2-cells, boundary 1-cells, and boundary
0-cells by simple objects, simple 1-morphisms, and basis 2-morphisms in a consistent manner.
The labeling on the boundary cells defines a fusion diagram on ∂ M , which we denote by F .
A fusion diagram F on the boundary is called a coloring. When the boundary ∂ M is coloredby a fusion diagram F , a C -state Γ in the bulk is constrained so that the dynamical variable
28
The global dimension dim(End(X )) of End(X ) for a simple object X ∈ π0 C is given by the sum of the squared
norm kxk2 = (dim(x)/dim(X ))2 for all (isomorphism classes of) simple objects x ∈ End(X ). We note that kxk is
the quantum dimension of x viewed as an object of a fusion 1-category End(X ).

22

SciPost Phys. 16, 143 (2024)

Figure 18: A schematic picture of the 3d classical statistical model obtained fromthe 4d Douglas-Reutter TFT on a slab N × [0, 1]. We impose a topological boundary
condition τ on the left boundary N × {0}. On the other hand, we P
impose a (notnecessarily topological) boundary condition decorated by a coloring F A(F )F on
the right boundary N ×{1}. The above geometry defines a genuine three-dimensional
system because the four-dimensional bulk is topological.
Γ (h j ) on a bulk j-cell intersecting the boundary ∂ M agrees with the label on the boundary( j − 1)-cell h j ∩ ∂ M . For this setup, the partition function on an oriented 4-manifold M witha non-empty boundary can be written as
ZDR(M ; F ) = NF
×Y
1
1
dim(C ) 3-handles h Dim(Γ (h3 ))Γ 4-handles h4
3YYev(Γ (∂ h2 ))ev(Γ (∂ h0 )) ,XY

2-handles h2(22)0-handles h0

where the products on the right-hand side are taken over j-handles that do not intersect theboundary. The numerical factor NF is a complex number that depends only on the coloring
F on the boundary. In particular, NF does not depend on the cell decomposition of the bulk.
The detailed definition of NF does not matter for later applications as we will see shortly in
Section 3.1.

3

3d height models

3.1

3d classical statistical models from 4d Douglas-Reutter TFT

We construct three-dimensional classical statistical models based on the Douglas-Reutter TFT.
Specifically, we put the DR theory on a slab N ×[0, 1], where N is a closed oriented 3-manifold
equipped with a cell decomposition. See Figure 18. The partition function of this 3d classicalstatistical model is given by
X
Z(A; τ) =A(F )ZDR(N × [0, 1]; τ, F ) , A(F ) ∈ C .
(23)
F

Here, ZDR(N × [0, 1]; τ, F ) is the partition function of the Douglas-Reutter TFT on a slab
N × [0, 1], where τ and F specify the boundary conditions on the left and right boundaries
respectively. Concretely, τ is a topological boundary condition on the left boundary N × {0}and F is a coloring on the right boundary N × {1}.29 We note that the numerical factor NF
in eq. (22) is absorbed into the redefinition of the weight A(F ) in eq. (23) and therefore theprecise form of NF does not matter.
29
We expect that a topological boundary τ is obtained by decorating the Dirichlet boundary by a fine mesh of
topological surfaces labeled by an algebra object of the input fusion 2-category. In the subsequent sections, we will
restrict our attention to the case where τ is the Dirichlet boundary.

23

SciPost Phys. 16, 143 (2024)

The above partition function defines a purely three-dimensional system because we cansquash the bulk TFT due to its topological nature. Indeed, we can write the right-hand side of
the above equation without using the four-dimensional bulk at all. To see this, we choose a celldecomposition of N ×[0, 1] so that every j-cell in the bulk is of the form c j−1 ×[0, 1], where c j−1
is a ( j − 1)-cell on the boundary. In particular, there are no 0-cells in the bulk. For this choiceof a cell decomposition, we can think of a C -state, which is originally defined as a collectionof dynamical variables in the bulk, as a collection of dynamical variables on the boundary
N × {1} by projecting the dynamical variables on j-cells in the bulk onto the corresponding( j − 1)-cells on the boundary N × {1}. More specifically, a C -state assigns a simple object, a
simple 1-morphism, and a basis 2-morphism to each 0-cell, 1-cell, and 2-cell on the boundary
N × {1}. We note that the dynamical variables are now living only on the boundary N × {1}.
Since the projection of dynamical variables in the bulk to the boundary preserves the locality,
the partition function (23) can be viewed as a genuine 3d classical statistical model.
The symmetry of the above 3d classical statistical model is determined by the pair of the
4d TFT DR(C ) in the bulk and a topological boundary condition τ on the left boundary. For
this reason, the bulk topological field theory is called a symmetry TFT [19, 91, 124, 133–135]
or categorical symmetry [58, 136–141] in the literature. Specifically, the symmetry of the 3dmodel is generated by topological defects living on the topological boundary τ. Therefore, adifferent choice of a topological boundary condition gives rise to a different symmetry.30
In what follows, we take τ to be the Dirichlet boundary condition, which means that thecoloring on the left boundary N × {0} is a trivial fusion diagram. In this case, eq. (22) implies
that we can shrink the left boundary N × {0} to a point when computing the partition function
because shrinking the Dirichlet boundary to a point does not affect the surface diagrams Γ (∂ h j )in eq. (22). More specifically, the partition function on a slab N ×[0, 1] agrees, up to a constantN0 , with the partition function on a cone pt ∗ N , which is a (singular) manifold obtained byshrinking the Dirichlet boundary to a point. Thus, the partition function of the 3d classicalstatistical model can be written as
X
Z(A) =
A(F )ZDR(pt ∗ N ; F ) .
(24)
F

We note that the scalar factor N0 is absorbed into the redefinition of A(F ).
Before proceeding, we emphasize that the simple objects, simple 1-morphisms, and basis
2-morphisms contained in the coloring F are regarded as dynamical variables of the 3d modelfor the time being. These dynamical variables will be integrated out when we will define the
3d height models in the next subsection. Hence, the dynamical variables of the 3d heightmodels consist only of simple objects, simple 1-morphisms, and basis 2-morphisms containedin the C -state Γ .

3.2

3d height models on a triangulated cubic lattice

Let us now explicitly construct the 3d height model based on the above general idea of constructing 3d classical statistical models. The lattice Λ on which the 3d height model is definedis a cubic lattice endowed with a triangulation and a branching structure as shown in Figure
19. The underlying 3-manifold of Λ is supposed to be a 3-torus T 3 .
A configuration of dynamical variables is specified by a pair (Γ , F ) of a C -state Γ and acoloring F . As we mentioned in the previous subsection, a C -state Γ assigns a simple object Γito each 0-simplex [i], a simple 1-morphism Γi j to each 1-simplex [i j], and a basis 2-morphism
Γi jk to each 2-simplex [i jk]. On the other hand, a coloring F consists of a simple object
30
Given a topological boundary condition, we can obtain another topological boundary condition by condensing
a separable algebra formed by a set of topological defects on the boundary. The condensation of a separable algebraon the topological boundary is regarded as the gauging of the fusion 2-category symmetry of the 3d model.

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SciPost Phys. 16, 143 (2024)

Figure 19: The cubic lattice on which the 3d height model is defined is equipped withthe above triangulation and branching structure.

Fi j on each 1-simplex [i j], a simple 1-morphism Fi jk on each 2-simplex [i jk], and a basis
2-morphism Fi jkl on each 3-simplex [i jkl].31 This difference is because, in the 4d DouglasReutter theory on the cone, while Γ ’s are assigned to the cells connecting the vertex [pt] andthe right boundary, F ’s are assigned to the simplices on the boundary.
The partition function of a general 3d classical statistical model defined in the previous
subsection can be written as the sum of the Boltzmann weights over all possible configurationsof dynamical variables on Λ as follows:
XX
Y
Y
Y
1
Z(A) =A(F )dim(Γi j )z(Γ , F ; c) .
(25)
Dim(Γ )
ΓF

0-simplices [i]i1-simplices [i j]cubes cHere, the weight z(Γ , F ; c) on a cube c is given by the product of the weights on the 3-simplices
contained in c. By construction, the weight on a 3-simplex [i jkl] is the 10-j symbol on thecorresponding 4-simplex pt ∗ [i jkl], whose vertices are ordered as pt < i, j, k, l. Therefore, if
we label the vertices of a cube c by 1, 2, · · · , 8 as shown in Figure 19, we can write the weightz(Γ , F ; c) asz(Γ , F ; c) = zε (Γ , F ; pt ∗ [1245])zε (Γ , F ; pt ∗ [2456])zε (Γ , F ; pt ∗ [4568])
× z−ε (Γ , F ; pt ∗ [1345])z−ε (Γ , F ; pt ∗ [3457])z−ε (Γ , F ; pt ∗ [4578]) ,(26)

where ε = ± is a sign determined by the choice of an orientation of the underlying manifold
T 3 . We note that the relative signs for different 3-simplices [i jkl] are determined solely by
the branching structure on Λ, which is independent of the choice of the orientation of T 3 .
For example, the relative sign for 3-simplices [1245] and [2456] can be computed as follows.
We first suppose that each 4-simplex pt ∗ [i jkl] has an orientation εi jkl .32 In this case, a4-simplex pt ∗ [1245] induces an orientation −ε1245 on a 3-simplex pt ∗ [245], whereas a 4simplex pt ∗ [2456] induces an orientation ε2456 on pt ∗ [245]. These induced orientations
must be opposite to each other because the underlying (singular) manifold pt ∗ T 3 is oriented.
Therefore, we find ε1245 = ε2456 , which shows that the relative sign for [1245] and [2456] ispositive. Similarly, we can compute the relative signs for other 3-simplices. We can also checkthat ε in eq. (26) does not depend on a cube c by computing the relative signs for 3-simplices
contained in adjacent cubes.
A 3d height model is obtained by choosing a weight A(F ) appropriately as we describe
below. In order to define the 3d height model, we first take A(F ) to be the product of local
weights on cubes:
Y
A(F ) =Ac (F c ) .
(27)
cubes cA coloring F gives rise to a fusion diagram on the dual cell decomposition of Λ.
We define the orientation of an n-simplex [i0 · · · in−1 ] to be positive if it is an even permutation of i0 , · · · , in−1 .
Otherwise, the orientation of [i0 · · · in−1 ] is defined to be negative.
31
32

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SciPost Phys. 16, 143 (2024)

Figure 20: We fix the coloring on the boundary of each cube as above, while wekeep the coloring inside each cube dynamical. The dual fusion diagram of the above
coloring is shown in Figure 21.

Figure 21: The coloring (28) on a triangulated cube defines a fusion diagram on the
dual cell decomposition, which looks like a hexagonal prism as shown above. The
yellow surface, lines and points in the above diagram are the duals of the internaledge, triangles and tetrahedra of a triangulated cube.
We note that the function Ac (F c ) can depend on a cube c, meaning that the Boltzmann weight
can be non-uniform on the lattice Λ. The argument F c denotes the set of dynamical variables
contained in the coloring on a cube c. More specifically, F c consists of 19 simple objects Ficj
on 1-simplices [i j] in c, 12 simple 1-morphisms Ficjk on 2-simplices [i jk] in c, and 6 basis
2-morphisms Ficjkl on 3-simplices [i jkl] in c. The superscript c will be omitted when it is clear
from the context. Furthermore, we fix the coloring on the boundary of each cube so that we
can integrate out the coloring F later while preserving the locality of the Boltzmann weight.
In other words, we require that a local weight Ac (F c ) is non-zero only when a coloring F csatisfies the following conditions, see also Figure 20:
c
c
c
c
c
c
F12
= F34
= F56
= F78
= F35
= F46
= ρ,
c
c
c
c
c
c
F13
= F24
= F57
= F68
= F25
= F47
= σ,
c
c
c
c
c
c
F15
= F26
= F37
= F48
= F14
= F58
= λ,

(28)

c
c
c
c
c
c
F124
= F125= F347= F357= F468= F568
=f,
c
c
c
c
c
c
F134 = F135 = F246 = F256 = F478 = F578 = g .

Here, ρ, σ, and λ are simple objects, and f : ρσ → λ and g : σρ → λ are simple 1morphisms, all of which are chosen arbitrarily.33 We emphasize that the choice of these simple
objects and simple 1-morphisms does not depend on a cube c. The above coloring defines afusion diagram on the dual of a triangulated cube as shown in Figure 21. The coloring inside accube c, which is denoted by Fint
, remains dynamical after fixing the coloring on the boundarycc0of c. Since the coloring Fint inside c can be chosen independently of the coloring Fint
insideany other cube c 0 , the summation over all colorings F in eq. (25) can be factorized into the
Although we assume that objects ρ, σ, λ, and 1-morphisms f , g are simple, a similar derivation of the 3d heightmodel can be applied even when they are non-simple.
33

26

SciPost Phys. 16, 143 (2024)csummations over Fint
for all cubes c. Therefore, we can write the partition function (25) as

Z(A) =

XY
Γ

[i]

Y
YX
1
c
dim(Γi j )Ac (Fint)z(Γ , F ; c) .
Dim(Γi )
c
c
[i j]

(29)

Fint

For later convenience, we write the above partition function in a more compact form as
X Y
Z(A) =Wc (Γ ; c) ,(30)
Γ

c: cubes

where the Boltzmann weight Wc (Γ ; c) on a cube c is defined by
v
Q
Q
u
c
c
uY
i=1,2,3
k=6,7,8 dim(Γi j )dim(Γ jk ) Xc tcWc (Γ ; c) = dim(Γ45 )Ac (Fint)z(Γ , F ; c) .
c
Dim(Γ
)
c
j
j=4,5

(31)

Fint

We call the 3d classical statistical model defined by the above partition function a 3d heightmodel because this model is a three-dimensional analogue of the 2d AFM height model in [33].
We note that a coloring F is already integrated out in eq. (30) and hence is no longer regarded
as a dynamical variable of the 3d height model.
Although we do not describe in detail, we can also incorporate topological defects by inserting them on the left (i.e., topological) boundary of the Douglas-Reutter theory before squashing the four-dimensional bulk. These topological defects generate the symmetry of the 3dheight model. In Section 4.4, we will see how this symmetry is realized in the corresponding
2+1d quantum model.

42+1d fusion surface modelscThroughout this section, we suppose that the weight Ac (Fint) does not depend on cube c andcwrite it simply as A(Fint ).

4.1

2+1d fusion surface models from 3d height models

In this subsection, we derive the Hamiltonian of the 2+1d fusion surface model on a honeycomb lattice, which is the quantum counterpart of the 3d height model on the triangulated
cubic lattice Λ. To this end, we first choose a time direction on Λ. The time direction on eachcube is given by the direction from vertex [4] to vertex [5]. We call vertices [4] and [5] theinitial vertex and the final vertex respectively. The above choice of a time direction enables us
to define the initial time slice and the final time slice for each cube as follows: The initial timeslice consists of the faces containing the initial vertex [4], whereas the final time slice consistsof the faces containing the final vertex [5], see Figure 22.34 A global time slice on the wholecubic lattice is illustrated in Figure 23, where the cubes are colored in blue, green, and yellow
for later convenience.
The triangulation of a cubic lattice shown in Figure 19 gives rise to a triangular lattice on asingle time slice, which is the Poincaré dual of a honeycomb lattice. We illustrate the relation
between a triangulated cubic lattice, a triangular lattice, and a honeycomb lattice in Figure
24. The plaquettes of the honeycomb lattice in Figure 24 are colored in accordance with thecolors of the cubes of the cubic lattice. We note that dynamical variables on the honeycomb
34

We can equally choose the time direction in the opposite way, and we will end up with the same (family of)models.

27

SciPost Phys. 16, 143 (2024)

Figure 22: The initial time slice (left) and the final time slice (right) on a single cube.

(a)
(b)

Figure 23: (a) A colored cubic lattice. A blue cube is surrounded by three green
cubes and three yellow cubes. A green cube is surrounded by three yellow cubes andthree blue cubes. A yellow cube is surrounded by three blue cubes and three green
cubes. (b) The initial and final time slices at t = 0, 1 viewed from the z direction.
Each red line represents the intersection of a time slice and a plane perpendicular to
the z-axis.

Figure 24: A time slice on the triangulated cubic lattice (left) becomes a triangularlattice (middle), which is related to a honeycomb lattice (right) by the Poincaré duality. The color of a vertex v of the cubic lattice represents the color of the cube whose
initial vertex is v. A vertex of the triangular lattice has the same color as the colorof the corresponding vertex of the cubic lattice, while a plaquette of the honeycomb
lattice has the same color as the color of its dual vertex of the triangular lattice.

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SciPost Phys. 16, 143 (2024)

lattice are simple objects, simple 1-morphisms, and basis 2-morphisms living on plaquettes,
edges, and vertices.35
Based on the above definition of a time slice, we define the transfer matrix of the 3d heightmodel on Λ. The transfer matrix T̂ is a linear map from the state space on a time slice at t = 0
to the state space on another time slice at t = 1. Physically, this linear map represents theimaginary time evolution from t = 0 to t = 1. The state space on a time slice is spanned bypossible configurations of dynamical variables on the honeycomb lattice. Specifically, the state
space H is given by H = Span{|Γ 〉}, where |Γ 〉 denotes a state corresponding to a configuration
Γ on the honeycomb lattice. Pictorially, we will often write |Γ 〉 as+|Γ 〉 =,(32)

where we omitted labels on the edges and vertices on the right-hand side in order to avoid
cluttering the notation. The inner product of states in H is defined by 〈Γ 0 |Γ 〉 = δΓ ,Γ 0 .
If we choose the initial time slice at t = 0 and the final time slice at t = 1 as shown in
Figure 23, the transfer matrix T̂ is factorized into the product of three linear maps
T̂ = T̂yellow T̂green T̂blue ,(33)

where T̂blue , T̂green , and T̂yellow are the imaginary time evolutions on the blue plaquettes, green
plaquettes, and yellow plaquettes respectively. More specifically, the linear map T̂blue is givenby the product of local transfer matrices on the blue plaquettes. The matrix element of the local
transfer matrix T̂p on a plaquette p is defined by the Boltzmann weight on the corresponding
cube c with the dynamical variables inside c integrated out, namely,
*

+
=

T̂p

X

Wc (Γ ; c) ,

(34)

c
Γint

c
where c is a cube whose initial vertex is dual to the plaquette p and Γint
is the collectionof dynamical variables inside c. We note that the local transfer matrices on blue plaquettes
commute with each other because any two plaquettes of the same color are not adjacent toeach other. Hence, the product of these transfer matrices is defined unambiguously. The othertwo linear maps T̂green and T̂yellow in eq. (33) are also defined by the products of local transfer
matrices on the green plaquettes and the yellow plaquettes respectively. Therefore, we have
Y
Y
Y
T̂ =
T̂p y
T̂p g
T̂p b .
(35)
yellow plaquettes p ygreen plaquettes p g

blue plaquettes p b

The partition function (25) of the 3d height model can be written in terms of the above transfer
matrix as Z(A) = TrH ( T̂ N ), where N is the number of lattice sites in the time direction.
The transfer matrix formalism of the 3d height model enables us to write down the Hamiltonian of the corresponding 2+1d quantum model on a honeycomb lattice. Specifically, wedefine the Hamiltonian of the 2+1d quantum lattice model by
X
H =−
T̂p ,
(36)
plaquettes p
35

Equivalently, simple objects, simple 1-morphisms, and basis 2-morphisms are living on vertices, edges, andplaquettes of the triangular lattice.

29

SciPost Phys. 16, 143 (2024)

where T̂p is the local transfer matrix on a plaquette p defined by eq. (34). However, the above
2+1d model is not precisely the quantum counterpart of the 3d height model. This is because
the imaginary time evolution e−εH on the state space H does not become the transfer matrixof the 3d height model even when ε  1. In other words, the transfer matrix T̂ of the 3dheight model cannot be expanded as T̂ = idH −εH + O(ε2 ) due to the fact that the state space
H contains a lot of states that are redundant in the description of the 3d height model.
The appropriate quantum counterpart of the 3d height model is obtained by restricting thestate space of the above 2+1d model to a specific subspace of H. To see this, we first noticethat the partition function of the 3d height model can also be written as
Z(A) = TrH ( T̂0 T̂ T̂0 )N = TrH0 ( T̂ N ) ,(37)cwhere T̂0 is the transfer matrix for the trivial weight A(Fint) = δF c ,1 and H0 := T̂0 H is theintccimage of T̂0 . Here, the trivial weight means that A(Fint) is one if Fint
is a trivial coloring and2zero otherwise. We note that T̂0 is a projector, that is, it satisfies T̂0 = T̂0 . This is because
the Dirichlet boundary decorated by the trivial coloring is topological in the imaginary time
direction. More explicitly, T̂0 can be written as the product of local commuting projectors as ineq. (52), which makes it clear that T̂0 is a projector. The first equality of eq. (37) follows fromthe relation T̂ = T̂0 T̂ T̂0 , which is an immediate consequence of the fact that T̂0 is the transfer
matrix for the trivial weight. The second equality of eq. (37) follows from the definition of
H0 . Equation (37) motivates us to consider a 2+1d quantum lattice model whose state spaceon the honeycomb lattice is H0 rather than H. The Hamiltonian of this model is given by eq.
(36), where the domain of the Hamiltonian is now restricted to H0 . The restriction of the state
space to H0 makes sense because the Hamiltonian (36) does not mix states in H0 and those in
the kernel ker( T̂0 ) of T̂0 due to the equality T̂ = T̂0 T̂ T̂0 . The 2+1d quantum lattice model on
H0 is precisely the quantum counterpart of the 3d height model. Indeed, if we take the weightccA(Fint) of the 3d height model to be slightly off from the trivial weight, i.e., δF c ,1 +εA(Fint
) for
int

ε  1, the transfer matrix T̂ can be expanded in ε as T̂ = T̂0 − εH T̂0 + O(ε2 ),36 which reduces
to idH0 − εH + O(ε2 ) on H0 . Namely, our 2+1d lattice model whose state space is H0 can beobtained by taking the anisotropic limit of the 3d height model. Here, we emphasize that the
point in passing to the smaller state space H0 is to find the completely anisotropic limit where
the transfer matrix becomes the identity, around which we can expand the transfer matrix.
The introduction of H0 is also motivated by the 3+1d perspective. Specifically, the state
space of the DR theory on a time slice that is perpendicular to the boundaries depicted in
Figure 18, is represented by H0 , not H. This can be understood by noting that T̂0 is thetransfer matrix when the decoration remains invariant under time translation, and such atransfer matrix should be evaluated as the identity operator on the state space.
There is also another way to see the reduction of the state space from a 3+1d point of view.
As we will discuss in Section 4.3, T̂0 is the projection onto the eigenspace of a generalized
Levin-Wen plaquette operator. This insight implies that H corresponds to the state space for
configurations where a bunch of hollow cylinders connects the two boundaries of the 3+1d slab
in Figure 18. Each cylinder terminates at the center of a plaquette on the right boundary andextends horizontally in the figure, and inside it, there is the trivial phase. The projector T̂0 is
responsible for closing these holes, cf. the original discussion by Levin and Wen [125, Appendix
C].
The discrepancy between the naive coloring space H and the space H0 stems from thedifference in the homotopy type of the honeycomb lattice, on which the coloring is defined,
and the homotopy type of the continuum space on which the DR theory is defined. The former
has almost as many generators as plaquettes, while the latter depends only on the global shape36

We note that H T̂0 = T̂0 H T̂0 because T̂ and T̂0 commute with each other due to T̂ = T̂0 T̂ T̂0 .

30

SciPost Phys. 16, 143 (2024)of the space (e.g. either torus or open disk). This discrepancy does not occur in 1+1d open
anyon chain.
We note that the derivation of the above 2+1d quantum lattice model from the 3d heightmodel is parallel to the derivation of the 1+1d anyon chain model from the 2d AFM heightmodel elaborated on in [33]. Thus, we can think of our lattice model as a 2+1d analogue of theanyon chain model. Indeed, as we will see in Section 4.3, our 2+1d model admits a graphicalrepresentation analogous to the anyon chain model. We call these 2+1d lattice models fusion
surface models.
As we will discuss in Section 4.4, the 2+1d fusion surface model has an exact fusion 2category symmetry described by the input fusion 2-category C . Equivalently, the 2+1d quantum lattice model defined by the same form of the Hamiltonian (36) acting on a larger state
space H has a fusion 2-category symmetry C only on its subspace H0 ⊂ H.37 The existence of
this fusion 2-category symmetry is guaranteed by the symmetry TFT construction depicted in
Figure 18.

4.2

Unitarity of the model

In this subsection, we spell out the condition for the Hamiltonian (36) to be Hermitian under
several assumptions on the input fusion 2-category C . Let us first list the assumptions that we
make. The first assumption is that the set of representatives of the connected components ofsimple objects is closed under taking the dual up to isomorphism. Namely, for the representative X of every connected component, there is a connected component whose representative
Y is isomorphic to the dual object X # . This isomorphism is assumed to preserve the quantum
dimension and the 10-j symbol. The precise meaning of this assumption will become clear in alater computation. Similarly, for every representative x of simple 1-morphisms in HomC (X , Y ),there is a representative y of simple 1-morphisms in HomC (Y, X ) that is isomorphic to x ∗ , andwe assume that this isomorphism preserves the 10-j symbol.38 We also make an assumptionthat the quantum dimensions of the representatives of simple objects and simple 1-morphisms
are positive real numbers.39 Finally, we assume that the 10-j symbol has the properties that
we call the reflection positivity and the 4-simplex symmetry. The reflection positivity of the
10-j symbol zε (Γ ; [01234]) is the property that flipping the orientation of a 4-simplex [01234]
amounts to taking the complex conjugation of the 10-j symbol:40
z−ε (Γ ; [01234]) = zε (Γ ; [01234])∗ .

(38)

The 4-simplex symmetry of the 10-j symbol zε (Γ ; [01234]) is the invariance under any permutation σ ∈ S 5 of vertices of a 4-simplex [01234]:zε (Γ ; [01234]) = zε·sgn(σ) (Γ ; [σ(0)σ(1)σ(2)σ(3)σ(4)]) .

(39)

The signature sgn(σ) of a permutation σ is + if σ is an even permutation and it is − if σ is an
odd permutation.41
This kind of symmetry is called exact emergent symmetry in [163].
This assumption particularly implies that the Frobenius-Schur indicator of a self-dual simple 1-morphism istrivial.
39
The quantum dimension of a simple object is multiplied by λ2 if we stack an invertible 2d TFT on top of it,
where λ2 is the partition function of the invertible 2d TFT on a sphere. We note that λ has to be real and hence
λ2 is positive if this invertible 2d TFT is reflection positive [164].
40
The term “reflection positivity” originates from an analogy to the property of a reflection positive quantum
field theory: The partition function of a reflection positive quantum field theory becomes its complex conjugate if
the orientation of the underlying spacetime is reversed.
41
We expect that these conditions, e.g., the triviality of the Frobenius-Schur indicators of 1-morphisms, can berelaxed to the axioms of what we should call a unitary fusion 2-category. In the more general cases, the Hermiticity
condition (43) should be modified to include, e.g., the Frobenius-Schur indicators: see Section 5.1. We do not
explore the most general conditions in this paper.
37
38

31

SciPost Phys. 16, 143 (2024)

When the permutation σ is non-trivial, the right-hand side of eq. (39) involves objects
and morphisms that are dual to those on the left-hand side. Let us illustrate this point byconsidering the simplest example where σ = (01) is the transposition of 0 and 1. In thiscase, the 4-simplex symmetry (39) reduces to zε (Γ ; [01234]) = z−ε (Γ ; [10234]). The righthand side of this equation involves a simple object Γ10 , whereas the left-hand side involves
another simple object Γ01 . These simple objects are supposed to be dual to each other, i.e., we#have Γ10 = Γ01. Similarly, the right-hand side z−ε (Γ ; [10234]) involves a simple 1-morphism
Γ10 j : Γ10 Γ0 j → Γ1 j for 2 ≤ j ≤ 4, whereas the left-hand side zε (Γ ; [01234]) involves anothersimple 1-morphism Γ01 j : Γ01 Γ1 j → Γ0 j . These simple 1-morphisms are related to each otherby an appropriate duality that contains both the object-level duality and the morphism-level
duality. Specifically, the relation between Γ01 j and Γ10 j is expressed as

=

,

(40)

#
#
#
∗
∗ #
where Γ01j : Γ0 j → Γ01 Γ1 j is the morphism-level dual of Γ01 j and (Γ01 j ) : Γ1 j Γ01 → Γ0 j is∗the object-level dual of Γ01
j . The relation between the basis 2-morphisms on the left-hand sidezε (Γ ; [01234]) and those on the right-hand side z−ε (Γ ; [10234]) is also given in a similar way.
The 4-simplex symmetry (39) implies that the 10-j symbol on a 4-simplex does not depend on
the choice of a branching structure on it. This is a natural generalization of the tetrahedral
symmetry of the 6-j symbol of a fusion 1-category [165].
Let us now derive the condition for the Hermiticity of the Hamiltonian (36) based on
the above assumptions. We first write down the matrix element of the local Hamiltonian T̂pexplicitly as follows:

*

+
T̂p

=

X

X

X

X

X

A(F45 ; F145 , · · · , F458 ; F1245 , · · · , F4578 )Γ45 Γ145 ,··· ,Γ458 F45 F145 ,··· ,F458 F1245 ,··· ,F4578(41)v
u Y dim(Γ )dim(Γ )dim(Γ )dim(Γ )dim(Γ )dim(Γ )
1j
2j
3j
j6
j7
j8
t
× dim(Γ45 )Dim(Γ j )j=4,5
× zε (Γ , F ; pt ∗ [1245])zε (Γ , F ; pt ∗ [2456])zε (Γ , F ; pt ∗ [4568])
× z−ε (Γ , F ; pt ∗ [1345])z−ε (Γ , F ; pt ∗ [3457])z−ε (Γ , F ; pt ∗ [4578]) .
The summation on the right-hand side is taken over the representatives of simple objects,
simple 1-morphisms, and basis 2-morphisms. Due to the assumptions, the matrix element of32

SciPost Phys. 16, 143 (2024)

the Hermitian conjugate of T̂p can be computed as
∗

*

+
T̂p†

=

X

X

X

*

+

=
X

X

T̂p

A(F54 ; F154 , · · · , F548 ; F1254 , · · · , F5478 )∗Γ54 Γ154 ,··· ,Γ548 F54 F154 ,··· ,F548 F1254 ,··· ,F5478(42)v
u Y dim(Γ )dim(Γ )dim(Γ )dim(Γ )dim(Γ )dim(Γ )
1j
2j
3j
j6
j7
j8
t
× dim(Γ45 )Dim(Γ j )j=4,5
× zε (Γ , F ; pt ∗ [1245])zε (Γ , F ; pt ∗ [2456])zε (Γ , F ; pt ∗ [4568])
× z−ε (Γ , F ; pt ∗ [1345])z−ε (Γ , F ; pt ∗ [3457])z−ε (Γ , F ; pt ∗ [4578]) .
The Hamiltonian (36) is Hermitian if and only if the above two quantities (41) and (42) agreewith each other. We emphasize that Γ45 , F45 , etc. involved in the 10-j symbols in eq. (42)are not representatives themselves in general but the appropriate duals of the representatives
Γ54 , F54 , etc. Although they are not representatives, they are isomorphic to representatives
because the set of representatives is assumed to be closed under taking the dual up to isomorphism. Since we are assuming that these isomorphisms preserve the quantum dimension and10-j symbol, we can identify the summands on the right-hand side of eq. (42) with those on
the right-hand side of eq. (41). Therefore, the Hermiticity condition on the Hamiltonian (36)reduces toA(F45 ; F145 , · · · , F458 ; F1245 , · · · , F4578 ) = A(F54 ; F154 , · · · , F548 ; F1254 , · · · , F5478 )∗ ,(43)

where the arguments on the right-hand side are the representatives of the connected components of appropriate duals of the arguments on the left-hand side. The above equation can beccwritten simply as A(Fint) = A(F int )∗ , where the bar represents the appropriate dual.42

4.3Graphical representation

In this subsection, we give a graphical representation of the 2+1d fusion surface model that
we obtained from the 3d height model. To begin with, we consider a graphical representationof a state in the larger state space H. As we mentioned in Section 4.1, states in H are in oneto-one correspondence with possible configurations of dynamical variables on a honeycomb
lattice. A configuration of dynamical variables is constrained by the monoidal structure of the
input fusion 2-category C . For example, a simple 1-morphism Γi j on an edge ei j is constrainedby the simple objects Γi and Γ j on the adjacent plaquettes pi and p j . More specifically, Γi j
has to be a simple 1-morphism from Γi Fi j to Γ j , where Fi j is a simple object assigned to a
1-simplex [i j] of the original 3d lattice Λ that is dual to an edge ei j on the honeycomb lattice.
We recall that Fi j is fixed due to eq. (28), meaning that Fi j is not dynamical. Similarly, abasis 2-morphism Γi jk on a vertex vi jk at the junction of three edges ei j , e jk , and eik must be
a 2-morphism between Γ jk ◦ (Γi j 1F jk ) and Γik ◦ (1Γi Fi jk ). The local constraints around allvertices combine dynamical variables on the honeycomb lattice into a single fusion diagram.
42

In Section 5.1, we will see an example where the Hermiticity condition (43) is modified due to the non-trivial
Frobenius-Schur indicator of a simple 1-morphism.

33

SciPost Phys. 16, 143 (2024)

partialfusionbubbleremoval

−−−−→

−−−→

Figure 25: The diagrammatic representation of the Hamiltonian is evaluated by combining the partial fusion around edges and the bubble removal around vertices.
Therefore, a state on the honeycomb lattice can be identified with a fusion diagram as follows:

+
= NΓ

.

(44)

Here, the left-hand side is an orthonormal basis of the state space H on the honeycomb latticeand NΓ is a normalization factor defined by
v
Y
u Y
1
NΓ = tdim(Γi j ) .
(45)
Dim(Γi ) edges
plaquettes
The action of the local Hamiltonian T̂p on a state (44) is graphically expressed as

=

T̂p

XXX
Γ5

A(Fint )Γ45 Fintdim(Γ45 )Dim(Γ5 ). (46)The yellow surface and the small white plaquette on the right-hand side are labeled by simple
objects F45 and Γ5 respectively. The edges and vertices are also labeled by simple 1-morphisms
and basis 2-morphisms, although the labels are omitted in the above equation due to the lackof space. For example, the loop at the junction of three surfaces Γ4 , Γ5 , and F45 is labeled by asimple 1-morphism Γ45 : Γ4 F45 → Γ5 . The other labels can also be deduced from the labels
already specified in the above equation. The right-hand side of eq. (46) is evaluated in two
steps as shown in Figure 25. In what follows, we show that the above graphical representation
gives the correct matrix element (34) of the local Hamiltonian T̂p by explicitly evaluating thefusion diagram step by step.
The first step of the evaluation is the partial fusion, which is represented by the following
diagrammatic equality of 2-morphisms:

=

X X

=

d (πi ,ιi )XXddim(d). (47)

αi

The 2-morphisms πi and ιi in the above equation are the projection and inclusion 2-morphisms,
whereas αi and αi are basis 2-morphisms. The defining property of the projection and inclusion
34

SciPost Phys. 16, 143 (2024)

2-morphisms is that they are dual to each other and satisfy πi · ι j = δi j 1. In particular, 2morphisms πi and ιi are proportional to basis 2-morphisms αi and αi . The proportionality
constant can be figured out by comparing the trace of ιi · πi with that of αi · αi . The trace of
ιi · πi is equal to the quantum dimension of the target 1-morphism d of πi , while the traceof αi · αi is unity because αi is normalized. Therefore, we have ιi · πi = dim(d)αi · αi , whichshows the second equality of eq. (47). We perform this partial fusion for all edges around thecentral plaquette labeled by Γ5 .
The second step of the evaluation is to remove the small bubbles that are localized aroundthe vertices after we perform the partial fusion. As an example, we focus on the bubble at theleft bottom vertex v124 . In order to remove the bubble, we first notice that the configurationof surfaces around a vertex v124 can be identified with the left-hand side of the 10-j move (18)
as follows:43

=

.

(48)

This identification makes it clear that the 10-j move around a vertex v124 deforms the fusion
diagram as

=

XXX
0
Γ25

0
Γ125
Γ245

0
dim(Γ25)z+ (Γ , F ; pt∗[1245]). (49)

We can now remove the bubble on the right-hand side by using the fact that the composition000
of Γ245and Γ245 is non-zero only when Γ25= Γ25 and Γ245
= Γ 245 . When non-zero, the above
composite map is a 2-endomorphism of Γ25 , which is proportional to the identity 2-morphism0−10 δ
because Γ25 is simple. More specifically, we have Γ245
· Γ245 = δΓ25 ,Γ25
idΓ25 ,
0 dim(Γ25 )Γ 245 ,Γ245
which can be verified by computing the trace of both sides. Therefore, eq. (49) reduces to=X

z+ (Γ , F ; pt ∗ [1245])

.

(50)

Γ125

Similar equations also hold for the other vertices.44
Combining eqs. (44), (45), (46), (47), and (50) leads to eq. (41) with ε being +. Thus, wefind that the 2+1d quantum lattice model defined on the larger state space H has a graphicalrepresentation (46). This graphical representation can further be simplified on the subspace
43
44

Equation (48) involves the identification of 2-morphisms related by the duality.
Precisely, the 10-j symbol z+ on the right-hand side is replaced by z− depending on vertices.

35

SciPost Phys. 16, 143 (2024)

H0 ⊂ H as follows:

=

T̂p

X

A(Fint )

.

(51)

Fint

This is the graphical representation of the Hamiltonian of the 2+1d fusion surface model. We
will show the above equation in the rest of this subsection.
Before we derive eq. (51), we first specify the subspace H0 ⊂ H in more detail. As alluded
to in Section 4.1, the subspace H0 is defined as the image of the transfer matrix T̂0 for thetrivial weight. The transfer matrix T̂0 is given by the product of local transfer matrices B̂ p ,namely,YT̂0 =B̂ p ,(52)
plaquettes p

where B̂ p is represented by the following diagrammatic equation:

X

=

B̂ p

Γ45 ∈End(Γ4 )dim(Γ45 )Dim(Γ4 ).

(53)

We note that B̂ p is a local commuting projector, i.e., it satisfies B̂ p B̂ p0 = B̂ p0 B̂ p and B̂ p2 = B̂ p .45
Therefore, the subspace H0 is spanned by the states satisfying B̂ p = 1 for all the plaquettes:

H0 = T̂0 H = Span{|Γ 〉 ∈ H | B̂ p |Γ 〉 = |Γ 〉 , ∀p} .

(54)

On this subspace, a contractible loop of x ∈ End(Γ4 ) on a plaquette acts as a scalar multiplication. This is because the loop operator B̂ px for a contractible loop of x on a plaquette p can beabsorbed by the projector B̂ p as follows:B̂ px B̂ p = B̂ px

XΓ45 ∈End(Γ4 )dim(Γ45 ) Γ45
dim(x)
B̂ p =
B̂ p .
Dim(Γ4 )dim(Γ4 )(55)

This equation implies that a contractible loop of x can be shrunk at the expense of multiplying
a scalar factor dim(x)/dim(Γ4 ), which is the quantum dimension of an object x in a fusion
1-category End(Γ4 ).
On the subspace H0 , we can also define the states whose plaquette variables are not representatives in π0 C . This is achieved by demanding that the contractible loop on a plaquette
can be shrunk at the expense of multiplying a scalar factor. Specifically, such a state is definedby:=dim(Γ40 )dim(x)

45

.

(56)

The local commuting projector B̂ p is nothing but the plaquette term of the Levin-Wen model for the input fusion
1-category End(Γ4 ) [125].

36

SciPost Phys. 16, 143 (2024)

The left-hand side is well-defined on H0 because the right-hand side does not depend on the
choice of x ∈ Hom(Γ40 , Γ4 ) when projected onto H0 . Indeed, the composite of the loop operatorB̂ px on the right-hand side and the projector B̂ p is independent of x:dim(Γ40 )dim(x)

X

B̂ p B̂ px =y∈Hom(Γ40 ,Γ4 )dim( y) yB̂ .
Dim(Γ4 ) p(57)

Let us now show that eq. (46) reduces to eq. (51) on H0 . To this end, we use the following
expression for a simple 1-morphism Γ45 : Γ4 F45 → Γ5 on the right-hand side of eq. (46):
0
Γ45 ∼
◦P.
= Γ45

(58)

0
Here, P is the projection 1-morphism from Γ4 F45 ∼is a=  Γ50 to a fusion channel Γ50 and Γ45
0
simple 1-morphism from Γ5 to Γ5 . We note that the fusion channels are uniquely determined
only up to isomorphism. Physically, isomorphic fusion channels differ by invertible 2d TFTs
stacked to topological surfaces. We can and will always choose the fusion channels properly
so that the dimension of the projection 1-morphism P agrees with the dimension of its target
Γ50 , i.e., we have dim(P) = dim(Γ50 ). This choice of the fusion channels in particular implies
0that simple 1-morphisms Γ45 and Γ45
have the same dimension. By substituting eq. (58) into0
the right-hand side of eq. (46) and shrinking the loop of Γ45, we find

=

T̂p

XXX

A(Fint )0
Fint Γ50 ,P Γ45

0 2
dim(Γ45).

Dim(Γ5 )dim(Γ50 )

(59)
P
0 2
0
Due to the equality dim(End(Γ5 )) =dim(Γ)/dim(Γ
)dim(Γ
),
which
was
0
0
5
Γ45 ∈Hom(Γ5 ,Γ5 )45
5shown in [55], the above equation reduces to

=

T̂p

XX

A(Fint ).

(60)

Fint Γ50 ,P

The right-hand side of the above equation can be written as

X

=

.

(61)

Γ50 ,PTo show this equation, we notice that the fusion of two surfaces Γ4 and F45 satisfies
=

X

,

c(P)

(62)

Γ50 ,Pfor some coefficient c(P) ∈ C. If we compose the both sides with the inclusion 1-morphism
I : Γ50 →  Γ50 ∼= Γ4 F45 , the above equation reduces to= c(P)

37

,

(63)

SciPost Phys. 16, 143 (2024)

where the projection 1-morphism P on the right-hand side is the dual of the inclusion
1-morphism I on the left-hand side. One can extract the coefficient c(P) by taking thetrace of eq. (63) as c(P) = dim(I)/dim(Γ50 ) = 1, where the last equality follows fromdim(I) = dim(P) = dim(Γ50 ). Thus, we find that eq. (61) holds,46 which then implies eq. (51)due to eq. (60).

4.4

Fusion 2-category symmetry

4.4.1

General case

The graphical representation (51) makes it clear that the 2+1d fusion surface model has afusion 2-category symmetry. The action of a fusion 2-category symmetry C is defined by thefusion of surface defects and/or line defects to the fusion diagram representing a state, see Figure 16 for a schematic picture of the symmetry action. The commutativity of the Hamiltonian
and the symmetry action automatically follows from the coherence conditions for a fusion 2category. In the following subsections, we study several simple examples of fusion 2-category
symmetries to demonstrate that the fusion surface models actually have symmetries whose
actions are defined in the above fashion.
Before proceeding, we emphasize that in general, the action of a fusion 2-category symmetry is well-defined only on the projected state space H0 . This is because if we try to define
the symmetry action by fusing a topological surface defect to a fusion diagram representing astate, we generically end up with a fusion diagram whose plaquette variables are not in theset of representatives of simple objects. Such a fusion diagram can be canonically identified
with a state only on H0 as in eq. (56).47
4.4.2

Non-invertible 1-form symmetry

The fusion 2-category that describes a (potentially) non-invertible 1-form symmetry is the 2category Mod(B ) of B -module categories,48 where B is the ribbon 1-category of topologicalline defects.49 The fusion 2-category Mod(B ) has only one connected component of simple
objects, whose representative is chosen to be a unit object I, i.e., the regular B -module. The
endomorphism category of I ∈ Mod(B ) is equivalent to B . In a general fusion 2-category C ,the endomorphism category EndC (I) describes the 1-form part of the whole symmetry. The
action of the 1-form part of a general fusion 2-category symmetry C can be defined in the same
way as the action of Mod(B ) symmetry that we will discuss below. In this subsection, we will
focus on the case where C = Mod(B ) for simplicity.
Since the connected component of simple objects of Mod(B ) is unique, we do not havedynamical variables on the plaquettes of a honeycomb lattice. Therefore, the dynamical variables of the model are living only on the edges and vertices. These dynamical variables arelabeled by simple objects and basis morphisms of a ribbon 1-category B .
The diagrammatic representation (46) of the Hamiltonian acting on the larger state space46
The fact that P is the projection 1-morphism from Γ4 F45 to Γ50 is not sufficient to show eq. (61) because theleft-hand side of eq. (61) depends on the choice of the fusion channels through an isomorphism Γ4 F45 ∼=  Γ50 ,0
whereas the right-hand side does not. Our claim is that eq. (61) is satisfied when each fusion channel Γ5 is chosen
so that dim(P) = dim(Γ50 ) .
47
As we will see in Section 4.4.2, the action of line defects is well-defined also on H because the edge variables
are always in the set of representatives.
48
Here, a 1-form symmetry refers to a symmetry generated by codimension 2 topological defects, which may or
may not be invertible. This symmetry reduces to an ordinary (group-like) 1-form symmetry when the topologicaldefects are invertible. In this paper, non-invertible 1-form symmetries are also simply called 1-form symmetries.
49
The fusion 2-category Mod(B ) is spherical when B is a ribbon 1-category, see Example 2.3.5 of [55].

38

SciPost Phys. 16, 143 (2024)−→−→

Figure 26: A (potentially) non-invertible 1-form symmetry B acts on states by thefusion of a topological line defect, which is written in light orange in the above figure.
We use the F -symbols and the R-symbols of a ribbon 1-category B to fuse the line
defect with the edges of a honeycomb lattice.

H is given by

=

T̂p

XX

A(Fint )Γ45 Fintdim(Γ45 )
D

,

(64)

where D is the total dimension of a ribbon 1-category B . The right-hand side is evaluatedby fusing the loops of Γ45 and Fint = {F145 , · · · , F458 } to the nearby edges. The action of
the 1-form symmetry is defined by the fusion of topological line defects to the honeycomb
lattice as shown in Figure 26. The commutativity of the Hamiltonian and the symmetry action
follows from the coherence conditions (i.e., the pentagon and hexagon equations) of a ribbon
category. It is straightforward to generalize the action of a line defect illustrated in Figure 26
to the action of a general defect network, which also clearly commutes with the Hamiltonian(64). In particular, we can explicitly define the action of condensation defects on the lattice[56, 68, 166].
We emphasize that the action of Mod(B ) symmetry commutes with the Hamiltonian not
only on H0 but also on H. This might seem to imply that our 2+1d lattice model has a noninvertible 1-form symmetry Mod(B ) on the entire state space H. However, the symmetry on H
is not the usual 1-form symmetry because the action of a symmetry operator on a contractible
loop is non-trivial. Such a symmetry on the lattice is called a 1-symmetry rather than 1-form
symmetry in the literature [58, 136]. Therefore, our 2+1d lattice model has a 1-symmetry on
the entire state space H, which reduces to a 1-form symmetry Mod(B ) on H0 .
We note that the Hamiltonian (64) is factorized into the product of two commuting operators asT̂p = B̂ p T̂p0 = T̂p0 B p ,(65)
where B̂ p and T̂p0 are defined by

B̂ p

=

X dim(Γ )
45
D
Γ

,

(66)

45

T̂p0

=

X

A(Fint )

.

(67)

Fint

Eq. (66) can be regarded as a special case of eq. (67). Both of the above operators preserve
P the00non-invertible 1-symmetry on H. In particular, a new Hamiltonian defined by H = − p T̂palso possesses the same non-invertible 1-symmetry.
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SciPost Phys. 16, 143 (2024)

4.4.3

Anomalous finite group symmetry

We consider the case of a finite group symmetry G with an anomaly [ω] ∈ H 4 (G, U(1)).
A fusion 2-category 2 VecωG describing an anomalous finite group symmetry G consists ofsimple objects labeled by group elements. The 10-j symbol is given by a 4-cocycle ω asz+ (Γ ; [i jkl m]) = ω(Γi j , Γ jk , Γkl , Γl m ). Since 2 VecωG does not have non-trivial 1-morphisms and2-morphisms in the sense that Hom2VecωG (g, h) ∼= δ g,h Vec as a 1-category, we do not have dynamical variables on the edges and vertices of a honeycomb lattice. Thus, the dynamicalvariables are living only on the plaquettes. The dynamical variable on each plaquette takes
values in G.
In order to obtain a non-trivial model with an anomalous finite group symmetry G, weneed to choose objects ρ, σ, and λ in eq. (28) to be non-simple.50 In the following, we choose
ρ, σ, and λ to be the sum of all simple objects, i.e., we haveρ=σ=λ=

g.

(68)

g∈G

In this case, there are effectively no constraints on the configuration of dynamical variables on
the plaquettes. More specifically, the state space H of the model is given by the tensor productof local Hilbert spaces C|G| on the plaquettes of the honeycomb lattice. We note that there isno difference between H and H0 for this example because the local commuting projector B̂ p
on each plaquette is the identity operator due to the absence of non-trivial 1-morphisms.
The matrix element (41) of the local Hamiltonian T̂p is given by
*

+
T̂p

= A(g4−1 g5 )ω(g1 , g1−1 g2 , g2−1 g4 , g4−1 g5 )ω(g2 , g2−1 g4 , g4−1 g5 , g5−1 g6 )ω(g4 , g4−1 g5 , g5−1 g6 , g6−1 g8 ).
ω(g1 , g1−1 g3 , g3−1 g4 , g4−1 g5 )ω(g3 , g3−1 g4 , g4−1 g5 , g5−1 g7 )ω(g4 , g4−1 g5 , g5−1 g7 , g7−1 g8 )(69)
The diagrammatic representation (51) of the above Hamiltonian clearly shows that this model
has an anomalous finite group symmetry whose action is defined by fusing topological surface
defects to the honeycomb lattice from above.
Let us explicitly compute the action of a finite group symmetry G with anomaly ω.51 To
this end, we precisely define the process of fusing a surface defect labeled by g ∈ G to thehoneycomb lattice from above. As illustrated in Figure 27a, instead of performing the fusion
at one time, we first fuse a surface defect to the honeycomb lattice only inside each plaquette.
We then slightly deform the surface defect as shown in Figure 27b and perform the partial
fusion of the defect around the vertices so that the defect looks like Figure 27c. Removing
the small bubbles in Figure 27c completes the fusion of a surface defect. As we discussed in
Section 4.3, removing a bubble at a vertex amounts to multiplying the 10-j symbol. More
Objects ρ, σ, and λ were originally supposed to be simple in eq. (28). However, the diagrammatic representation of the model given in Section 4.3 enables us to generalize them to non-simple objects straightforwardly.
51
See [167] for the action of an anomalous finite group symmetry in general dimensions.
50

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SciPost Phys. 16, 143 (2024)

(a)

(b)

(c)

Figure 27: We fuse a surface defect to the honeycomb lattice from above by combining (a) the partial fusion, (b) the deformation of the defect, and (c) the bubble
removal. The orange region in the above figures represents the region where thesurface defect is already fused, while the white region represents the region where
the surface defect is not fused yet.
specifically, we have= ω(g, g i , g i−1 g j , g −1
j gk ),(70)∗= ω(g, g i , g i−1 g j , g −1
j gk ).

Therefore, the action of g ∈ G on a general state |{g i }〉 can be written asYY∗Û g |{g i }〉 =ω(g, g i , g i−1 g j , g −1g)ω(g, g i , g i−1 g j , g −1kjj g k ) |{g g i }〉 ,(71)

where the first and the second products on the right-hand side are taken over vertices shownin the first and the second equalities in eq. (70). A straightforward calculation shows that thesymmetry action (71) commutes with the Hamiltonian (69) due to the cocycle condition on
ω. We note that the anomalous finite group symmetry of the Hamiltonian (69) is preserved
even if the weight A(g4−1 g5 ) also depends on other variables of the form g i−1 g j . In particular,when G is anomaly-free, our model reduces to the G-symmetric models discussed in [113].

5

ExamplesIn this section, we discuss several examples of the fusion surface model and its variants. In
sections 5.1 and 5.2, we consider the 2+1d lattice models only with 1-symmetries. The 1symmetries are present on the larger state space H as we observed in Section 4.4.2. As such,
we consider the 2+1d lattice models defined on H without projecting to the subspace H0 .
On the other hand, in Section 5.3, we consider the 2+1d lattice models with general fusion
2-category symmetries. Since such symmetries are present only on the projected subspace H0 ,we need to consider the fusion surface models whose state space is H0 rather than H. In thefollowing, we use 1-symmetry and 1-form symmetry interchangeably when no confusion canarise. For the 2+1d models with 0-form anomalous finite group symmetries, see Section 4.4.3.

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SciPost Phys. 16, 143 (2024)

5.1

Lattice models with anomalous invertible 1-form symmetries

Let A be a finite abelian group. Anomalies of an invertible 1-form symmetry A are characterized
by the F -symbols and R-symbols defined by the following equations:= F (a, b, c)= R(a, b)

,

.

(72)

The quantum dimensions of invertible lines are all given by one, i.e., dim(a) = 1 for all a ∈ A.
In what follows, we will explicitly write down the Hamiltonian of a lattice model with ananomalous 1-form symmetry A.
To define the model, we choose f and g in eq. (28) to be the sum of all group elementsLa ∈ A, i.e., we have f = g = a∈A a. For this choice of f and g, we can take the dynamicalvariables on different edges independently. Therefore, the state space H of the model is givenNby the tensor product of the local Hilbert spaces on all edges: H = edges C|A| . The HamiltoP 0nian is of the form H = − T̂p , where the local Hamiltonian T̂p0 on a plaquette p is generally
given by eq. (67).
As an example, we consider the Hamiltonian that consists only of the following three terms:
T̂p0 =X

J x (a)+ J y (a)+ Jz (a)

.

a∈A

(73)
Because of the group-like fusion rules, the labels on the orange edges in the above equation are
uniquely determined by the configuration of dynamical variables on the honeycomb lattice.
A more general Hamiltonian can be obtained by adding the terms given by the products ofoperators appearing on the right-hand side. For simplicity, we will focus on the Hamiltonian(73) in the rest of this subsection.
Let us express the above Hamiltonian in terms of F -symbols and R-symbols. To this end,
we first resolve the 4-valent vertices in eq. (73) into trivalent vertices as follows:

=

.

(74)

The labels Γi jk on the dotted edges are uniquely determined by the fusion rule Γi jk = Γ jk Γi j .
The above resolution of 4-valent vertices enables us to compute each term in eq. (73) as
=

=

=

−1
R(Γ12 , a)F (Γ26 , Γ26
Γ246 a−1 , a)F (Γ24 a−1 , Γ12 , a),−1
F (Γ46 , Γ24 a−1 , a)F (Γ24 a−1 , a, Γ12 )F (Γ14 , Γ14
Γ124 a−1 , a)−1
R(Γ24 , a)F (Γ46 , Γ24 , a)F (Γ48 , Γ48
Γ468 , a)−1
F (Γ26 , Γ26
Γ246 , a)F (Γ46 , a, Γ24 )F (Γ68 , Γ46 , a)−1
F (Γ48 , a, a−1 Γ48
Γ478 ).

−1
F (Γ48 , a, a−1 Γ48
Γ468 )

42,

SciPost Phys. 16, 143 (2024)

Figure 28: A honeycomb lattice has three types of edges that we call x-links, y-links,
and z-links. These edges are written in red, green, and blue in the above figure.
If we write the above operators as Ô x (a), Ô y (a), and Ôz (a) respectively, the total Hamiltonianof the model can be written as


X X
X
X

H =−
J x (a)Ô x (a) +J y (a)Ô y (a) +Jz (a)Ôz (a) ,
(75)
a∈A

x-linksy-linksz-links

where x-links, y-links, and z-links are depicted by red, green, and blue edges in Figure 28.
Anomalous Z2 1-form symmetry. As the simplest example, we consider the case of ananomalous Z2 1-form symmetry. There are four possible anomalies of a Z2 1-form symmetry, which are specified by the following F -symbols and R-symbols:(F (η, η, η), R(η, η)) = (1, 1), (1, −1), (−1, i), (−1, −i) .

(76)

Here, η denotes the generator of Z2 . The other components of the F -symbols and R-symbols
are trivial. The Z2 1-form symmetries with the above anomalies are called bosonic, fermionic,
semionic, and anti-semionic Z2 1-form symmetries respectively.52 For each of these Z2 1-form
symmetries, we can write down the operators Ô x (η), Ô y (η), and Ôz (η) in terms of the Pauli
operators.
For a bosonic Z2 1-form symmetry, the operators Ô x (η), Ô y (η), and Ôz (η) are all givenby the Pauli X operator. Thus, the Hamiltonian (75) can be written graphically as
H =−

X

Jx

−

x-linksX

Jy−y-links

X

Jz

,

(77)

z-links

where the parameters J x := J x (η), J y := J y (η), and Jz := Jz (η) are real numbers so thatthe above Hamiltonian satisfies the Hermiticity condition (43). Here and in the rest of thissubsection, we set J x (1), J y (1), and Jz (1) to zero for the unit element 1 ∈ Z2 without loss ofgenerality. The above model realizes a trivial phase with bosonic Z2 1-form symmetry.
For a fermionic Z2 1-form symmetry, the Hamiltonian (75) can be written as
H =−X
x-links

Jx

−

X

Jy

y-links

−

X

Jz

.

(78)

z-links

The parameters J x , J y , and Jz are again chosen to be real due to the Hermiticity condition(43). We note that the qubits on the z-links are decoupled from those on the x-links and ylinks. In particular, the qubits on the z-links have a uniquely gapped ground state given by a
52

The bosonic Z2 1-form symmetry is non-anomalous, whereas the others are anomalous.

43

SciPost Phys. 16, 143 (2024)

trivial product state. Therefore, in the low-energy limit, the above Hamiltonian reduces to thestacking of decoupled 1+1d quantum spin chains consisting of the qubits on the x-links andy-links. These quantum spin chains can be coupled to the qubits on the z-links by adding theterms such as Ô x (η)Ôz (η) and Ô y (η)Ôz (η).
For a semionic Z2 1-form symmetry, the Hamiltonian (75) can be written as

H =−X
x-links

Jx

−

X

X

−

Jy

y-links

Jz

,

(79)

z-links

where C Z denotes the controlled-Z operator that acts on the qubits on the two edges connected by a small arc. The Pauli X operator in each term acts on the middle edge after thesequence of the controlled-Z and the Pauli Z operators. We note that the Hamiltonian (79) is
not Hermitian when the Hermiticity condition (43) is satisfied, namely, when J x , J y , and Jz
are real numbers. This indicates that the Hermiticity condition (43) is invalid for a semionicZ2 1-form symmetry. This is because the generator η of a semionic Z2 1-form symmetry has
a non-trivial Frobenius-Schur indicator, which violates the assumption used in the derivationof the Hermiticity condition (43). We can make the Hamiltonian (79) Hermitian by taking alinear combination of H and its Hermitian conjugate H † . Adding H † to the original Hamiltonian H does not break the semionic Z2 1-form symmetry because H † commutes with thesymmetry action when H does. Similar arguments apply to the case of an anti-semionic Z2
1-form symmetry.

5.2

Kitaev honeycomb model without a magnetic field

The Kitaev honeycomb model without a magnetic field is an exactly solvable model of qubits
on a honeycomb lattice, which exhibits an abelian topological order or a gapless excitation
depending on the parameter of the model [168]. As we will see below, this model can beobtained as a variant of the 2+1d fusion surface model.
The fusion 2-category that we use as an input is the 2-category Mod(Ising) of Ising-module
categories, where Ising denotes the modular tensor category describing the Ising TQFT. The
Ising category consists of three simple objects {1, η, σ}, which are subject to the following
fusion rules:
η⊗η∼
(80)
= 1, η ⊗ σ ∼=σ⊗η∼= σ, σ ⊗σ ∼= 1 ⊕ η.
The non-trivial F -symbols and R-symbols are summarized as follows [168]:1
1(Fησησ )σσ = (Fσηση )σσ = −1 , (Fσσσσ )11 = (Fσσσσ )1η = (Fσσσσ )η1 = p , (Fσσσσ )ηη = − p ,
2
2
ηη
ησ
ση
σσ
−iπ/8
σσ
3iπ/8
R1 = −1 , Rσ = Rσ = −i , R1 = e, Rη = e.
(81)
As discussed in full generality in Section 4.4.2, 1-endomorphisms of the unit object of thefusion 2-category Mod(Ising) form the Ising category. In particular, simple 1-morphisms of
Mod(Ising) are labeled by simple objects 1, η, and σ of Ising. We note that the invertible
object η generates an anomalous Z2 1-form symmetry.
In order to obtain the Kitaev honeycomb model as a variant of the fusion surface model,
we choose both f and g in eq. (28) to be σ. Furthermore, we fix the labels on the edges of
the honeycomb lattice to σ, or in other words, we only consider the sector where all edges arelabeled by σ. We note that the restriction to this sector violates the Ising 1-form symmetry, but
still preserves the anomalous Z2 1-form symmetry generated by η. The dynamical variables inthis sector are living only on the vertices of the honeycomb lattice. The local Hilbert space on
44

SciPost Phys. 16, 143 (2024)

each vertex is given by Hom(σ ⊗ σ, σ ⊗ σ) ∼= C2 , which means that we have a qubit on eachvertex. The total Hilbert space of the model is given by the tensor product of the local Hilbert
spaces on all vertices.
We define the local Hamiltonian on each plaquette asT̂p0 = J x

+ Jy

+ Jz

, (82)

where the green edges are all labeled by σ. Coupling constants J x , J y , and Jz have to be real
due to the Hermiticity condition (43). We note that the above Hamiltonian is an example ofthe local Hamiltonian (67) except that the labels on the edges in eq. (82) are fixed, whereas
those in eq. (67) are dynamical. For computational purposes, we resolve each 4-valent vertex
into two trivalent vertices as follows:

=

.

(83)

Qubits on the left-hand side are living on the 4-valent vertices, whereas qubits on the righthand side are living on the black edges. The qubits on the right-hand side are denoted by thesame letters as the qubits on the left-hand side. After the resolution of the 4-valent vertices,
we can evaluate each term on the right-hand side of eq. (82) as== − i(−1)δΓ124 ,η

== i(−1)δΓ246 ,η

== (−1)δΓ468 ,η +δΓ478 ,η

,

,

.

In terms of the Pauli operators, we can write the above operators as −Y124 X 246 , Y246 X 468 ,and Z468 Z478 respectively, where X i jk , Yi jk , and Zi jk denote the Pauli X , Y , and Z operators
acting on the qubit Γi jk . If we rotate the bases of qubits Γ124 and Γ468 by π/4, the first and thesecond operators become X 124 X 246 and Y246 Y468 respectively. Therefore, the Hamiltonian in
the rotated basis can be written as
X
X
X
H =−
Jx X i X j −J y Yi Y j −Jz Zi Z j ,(84)x-linksy-linksz-links

where x-links, y-links, and z-links are three different types of edges shown in Figure 28. The
above is the Hamiltonian of the Kitaev honeycomb model without a magnetic field [168]. Remarkably, our formulation makes the anomalous Z2 1-form symmetry of the Kitaev honeycomb
model manifest. The symmetry operator on a closed loop indeed agrees with the loop operator
defined in Kitaev’s original paper [168]. This symmetry guarantees that the Kitaev honeycomb
model without a magnetic field realizes non-trivial phases everywhere in the phase diagram.
We note that applying a magnetic field explicitly breaks the anomalous Z2 1-form symmetry.
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SciPost Phys. 16, 143 (2024)

5.3

Non-chiral topological phases with fusion 2-category symmetries

In this subsection, we will sketch out how to obtain the 2+1d fusion surface models thatrealize non-chiral topological phases with fusion 2-category symmetries. As we discussed in
Section 4, the fusion surface model is the quantum counterpart of the 3d height model, which
is obtained by putting the 4d Douglas-Reutter TFT on a slab as shown in Figure 18. The
dynamics of the fusion surface model is determined by the choice of a decorated boundary
condition on the right boundary of the slab. In particular, when the decorated boundary is
topological, the corresponding fusion surface model realizes a topological phase with fusion
2-category symmetry.
Dirichlet boundary and spontaneous symmetry breaking. The simplest example of a topological boundary condition is the Dirichlet boundary condition, which is defined by the trivial
coloring on the decorated boundary. Specifically, for the Dirichlet boundary condition, thesimple objects ρ, σ, λ, and the simple 1-morphisms f , g in eq. (28) are the unit object I and
the identity 1-morphism 1 I respectively. This implies that the graphical representation (44)of a state is given by a planar fusion diagram on a honeycomb lattice. We note that all theplaquettes of the honeycomb lattice are labeled by the same simple object because the simple
objects on the adjacent plaquettes have to be connected. Therefore, the state space H0 splits
into sectors labeled by (representatives of connected components of) simple objects of a fusionL
2-category C , namely, we have H0 = X ∈π0 C H0X . Each sector H0X is the image of the projectorQp B̂ p , where B̂ p is the plaquette operator defined by

B̂ p

=

X
x∈End(X )kxkdim(End(X ))

.

(85)

Here, kxk = dim(x)/dim(X ) is the norm of a simple 1-morphism x, i.e., the quantum dimension of x viewed as a simple object of a fusion 1-category End(X ). The Hamiltonian (51) acting
on the state space H0 is (proportional to) the identity operator because the weight A(Fint ) iszero when the coloring Fint is non-trivial. Thus, the ground state subspace of the model is
H0 itself. Since the plaquette operator (85) is the same as that of the Levin-Wen model for
an input fusion 1-category End(X ) [125], each sector H0X of our model realizes a non-chiral
topological order described by the Drinfeld center Z(End(X )) of End(X ).53 The non-chiral
topological orders realized on different sectors are mixed by the action of a fusion 2-category
symmetry C . Physically, this means that the fusion 2-category symmetry C is spontaneously
broken.
WeP
note that the projection to H0 can also be implemented dynamically by the Hamiltonian
H = − p B̂ p acting on a larger state space H spanned by all possible fusion diagrams on thehoneycomb lattice. However, in this case, the fusion 2-category symmetry is not exact on thelattice but emergent in the low-energy limit.
General topological boundaries and non-chiral topological phases. A more general topological boundary condition gives rise to a more general topological phase with fusion 2category symmetry. In general, topological boundaries of the Douglas-Reutter TFT DR(C ) thatgive rise to non-chiral topological phases with C symmetry are expected to be in one-to-one correspondence with (the equivalence classes of) finite semisimple module 2-categories over C .54
53

The constraints from the vertex terms of the Levin-Wen model are already imposed on the state space.
This is a 4d analogue of the fact that topological boundaries of 3d Turaev-Viro TFT are in one-to-one correspondence with (the equivalence classes of) finite semisimple module categories over the input fusion 1-category
54

46

SciPost Phys. 16, 143 (2024)

Since (the equivalence classes of) finite semisimple module 2-categories over C are in one-toone correspondence with (the Morita equivalence classes of) separable algebras in C [171],55
there should also be a one-to-one correspondence between non-chiral topological boundaries
of DR(C ) and separable algebras in C . In particular, the Dirichlet boundary corresponds to thetrivial algebra I ∈ C . A general non-chiral topological boundary would be realized by condensing a separable algebra A ∈ C on the Dirichlet boundary. In other words, the coloring F on a
non-chiral topological boundary would be given by the condensation of a separable algebra A,
which is a fine mesh of topological surfaces labeled by A [56, 147, 173]. Indeed, as we will see
below, when the coloring F is the condensation of a separable algebra A ∈ C , the corresponding 2+1d fusion surface model has a commuting projector Hamiltonian, which suggests thatthe model realizes a non-chiral topological phase with fusion 2-category symmetry C .
In order to obtain the 2+1d fusion surface models for general non-chiral topological phases,
we first briefly recall the definition of a separable algebra in a fusion 2-category C [174]. An
algebra A in C is an object equipped with a multiplication 1-morphism m : AA → A that is
associative up to coherence 2-isomorphism µ satisfying56

=

.

(86)

An algebra A ∈ C is called a rigid algebra if the multiplication 1-morphism m : AA → A has adual 1-morphism m∗ : A → AA. There are a lot of coherence conditions associated with thisduality, which we assume implicitly in the following, see [174, 175] for the precise definition.
A rigid algebra A ∈ C is called a separable algebra if the multiplication 1-morphism m and its
dual m∗ satisfy the following conditions:

=

=

,

=

(87)

.

(88)

The 2-morphisms in eq. (88) should be regarded as appropriate duals of the associativity 2-isomorphism µ and its inverse µ−1 . By a slight abuse of notation, we write these 2isomorphisms simply as µ and µ−1 in the above equation.57 It is conjectured in [176] and is[19, 122, 123, 142, 143]. For the 4d Dijkgraaf-Witten theory, the correspondence between topological boundaries
and module 2-categories over 2VecG is studied in, e.g., [169, 170].
55
This is a categorified version of Ostrik’s theorem [172].
56
Precisely, an algebra in a fusion 2-category is also equipped with a 1-morphism i : I → A that satisfies the unitality condition up to coherence 2-isomorphism. We will not use this datum explicitly in the following discussions.
57
These 2-isomorphisms are denoted by ψl and ψ r in [174].

47

SciPost Phys. 16, 143 (2024)

proven in [177] that rigid algebras in a fusion 2-category over C are automatically separable.
We note that separable algebras are closely related to the orbifold data of 3d topological field
theories [147, 173].
Based on the above definition of a separable algebra, we now write down the Hamiltoniansof the fusion surface models that realize non-chiral topological phases with fusion 2-category
symmetry C . As we mentioned above, the fusion surface model for a non-chiral topologicalphase is obtained by choosing the coloring F to be the condensation of a separable algebraA ∈ C , i.e., Fi j = A, Fi jk = m, and Fi jkl = µ. The Hamiltonian of this model is given byP
H = − p T̂p , where the local Hamiltonian T̂p is represented by the following fusion diagram:

=

T̂p

.

(89)

Here, we suppose that the state space of the model is H0 rather than H so that the model
has an exact fusion 2-category symmetry. It is obvious that the plaquette term T̂p commute
with another plaquette term T̂p0 when the plaquettes p and p0 are apart from each other. The
commutativity of T̂p and T̂p0 for adjacent plaquettes also follows from eqs. (86) and (88). For
example, we haveT̂p T̂p0 ==== T̂p0 T̂p .

(90)

Furthermore, the plaquette term T̂p is a projector, which means that it satisfies

T̂p2 === T̂p .

(91)

Here, we used eqs. (86) and (87) in the first and the third equalities respectively. The second
equality follows from the identity µ−1 µ = µµ−1 = id. Thus, we find that the 2+1d fusion surface model obtained from a separable algebra A ∈ C has a commuting projector Hamiltonian,which strongly suggests that this model realizes a non-chiral topological phase with fusion
2-category symmetry C .
Let us finally consider some simple examples. When C is the 2-category 2Vec of finite
semisimple 1-categories, separable algebras are given by multifusion 1-categories [174]. In
this case, we expect that eq. (89) reduces to the Hamiltonian of the Levin-Wen model, which
realizes the most general non-chiral topological order without symmetry.58 More generally,
when C is the 2-category 2VecG of G-graded finite semisimple 1-categories, separable algebras
58
The original Levin-Wen model takes a fusion 1-category as an input [125]. The Levin-Wen model whose input
is a multifusion 1-category is investigated in [178].

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SciPost Phys. 16, 143 (2024)

are given by G-graded multifusion 1-categories [174]. In this case, we expect that eq. (89)reduces to the Hamiltonian of the symmetry enriched Levin-Wen model [179, 180], which
realizes the most general non-chiral topological order enriched by a finite group symmetry
G [3].

Acknowledgments
Funding information KI is supported by FoPM, WINGS Program, the University of Tokyo,
and also by JSPS Research Fellowship for Young Scientists. KO is supported by JSPS KAKENHI
Grant-in-Aid No.22K13969 and the Simons Collaboration on Global Categorical Symmetries.

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