G-Crossed Modularity of Symmetry Enriched Topological Phases Arman Babakhani1 2and Parsa Bonderson3 1Department of Chemistry University of California Santa Barbara California 93106 USA

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G-Crossed Modularity of Symmetry Enriched Topological Phases

Arman Babakhani1, 2 and Parsa Bonderson3

1Department of Chemistry, University of California, Santa Barbara, California 93106 USA

2Information Sciences Institute, Marina Del Rey, CA, 90292, USA

3Microsoft Station Q, Santa Barbara, California 93106-6105 USA

(Dated: September 12, 2023)

The universal properties of (2 + 1)D topological phases of matter enriched by a symme-

try group Gare described by G-crossed extensions of unitary modular tensor categories

(UMTCs). While the fusion and braiding properties of quasiparticles associated with the

topological order are described by a UMTC, the G-crossed extensions further capture the

properties of the symmetry action, fractionalization, and defects arising from the interplay

of the symmetry with the topological order. We describe the relation between the G-crossed

UMTC and the topological state spaces on general surfaces that may include symmetry

defect branch lines and boundaries that carry topological charge. We deﬁne operators in

terms of the G-crossed UMTC data that represent the mapping class transformations for

such states on a torus with one boundary, and show that these operators provide projective

representations of the mapping class groups. This allows us to represent the mapping class

group on general surfaces and ensures a consistent description of the corresponding symme-

try enriched topological phases on general surfaces. Our analysis also enables us to prove

that a faithful G-crossed extension of a UMTC is necessarily G-crossed modular.

I. INTRODUCTION

Gapped quantum systems at zero temperature may manifest distinct topologically ordered

phases of matter [1, 2]. Topological phases are characterized by emergent phenomena that cannot

be distinguished by local observables and do not require nor depend on symmetries of the system.

As such, these are universal properties of the phase that are robust to local perturbations, which

can only change when the system undergoes a phase transition. These topological phenomena

include ground state degeneracies that depend only on the topology of the system and quasiparticle

excitations that exhibit exotic exchange statistics. For (2 + 1)D topological phases, the universal

properties of the emergent quasiparticles, such as their fusion and exchange statistics, are captured

by algebraic structures known as unitary modular tensor categories (UMTCs) [3–5]. Moreover, the

UMTC describing a topological phase can be used to deﬁne a (2 + 1)D topological quantum ﬁeld

theory (TQFT), which describes the universal properties of the system on manifolds of arbitrary

arXiv:2210.14943v2 [cond-mat.str-el] 10 Sep 2023

topology. In this way, the UMTC encodes all the universal properties of a (2 + 1)D topological

phase, except for the value of the chiral central charge, which it only encodes mod 8.

When topological order exists in a system with symmetry, the interplay between the two allows

for a richer array of topological properties, giving rise to possibly distinct symmetry enriched topo-

logical (SET) phases. This enriched structure includes that action of the symmetry on topological

charges and states, fractionalization of symmetry quantum numbers carried by topologically non-

trivial objects, and symmetry defects. For (2 + 1)D SET phases with on-site unitary symmetries,

the universal properties of these features are captured by algebraic structures known as G-crossed

UMTCs [6–9]. The symmetry defects in such theories have a similar notion of fusion, with a G-

grading imposed, and a generalized notion of braiding, which incorporates the symmetry action

and fractionalization, and extends them to defects.

A deﬁning property of a UMTC is the notion of modularity, which is a non-degeneracy condition

of braiding, i.e. the vacuum charge is the unique topological charge (quasiparticle type) which has

trivial braiding with all topological charges. Physically, this implies that braiding distinguishes be-

tween all topological charge values. Equivalently, modularity of a unitary braided tensor category

(UBTC) can be succinctly expressed as unitarity of a certain topological invariant known as the

S-matrix, together with the condition that the number of topological charge types (simple objects)

is ﬁnite. Modularity implies that the Sand Tmatrices of the theory provide a projective repre-

sentation of the modular group, i.e. the mapping class group of the torus, yielding well-deﬁned

basis transformations of the state space. More generally, it allows the UMTC to be used to deﬁne

a (2 + 1)D TQFT [4, 5] which constitutes the low-energy eﬀective theory of the topological phase.

In particular, this can be used to describe the topological state spaces and operations on surfaces

of arbitrary topology and topological charge content, e.g. ascribed to boundaries or quasiparticles.

Modularity for G-crossed categories was discussed in Refs. 6–9. 1In Ref. 9, G-crossed UMTCs

were used to describe the topological state spaces of bosonic SET phases on the torus in the presence

of symmetry defect branch lines, and also provide the representation of modular transformations

on these states. In this paper, we extend the analysis of G-crossed modularity to describe the

topological state spaces and representations of the mapping class transformations of SET phases on

the torus with a boundary, which can carry nontrivial topological charge as well as have symmetry

1Refs. 6–9 use three slightly diﬀerent deﬁnitions of G-crossed modularity. In this paper, we introduce yet another

slightly diﬀerent deﬁnition: a G-crossed UMTC is deﬁned to be a G-crossed UBTC B×

Gfor which |Bg|, the number

of topological charges (simple objects) in each g-sector, is ﬁnite for all g∈G, and the operator Sdeﬁned by

Eq. (27) is unitary. We will show that these diﬀerent deﬁnitions of G-crossed modularity are equivalent under the

condition that B×

Gis faithful, that is |Bg|̸= 0 for all g∈G.

FIG. 1: Longitudinal (l) and meridional (m) oriented generating cycles of a 2D torus for a particular

embedding in 3D space.

defect branch lines terminating on it. This allows one to describe the topological state space and

operations of SET phases on surfaces of arbitrary topology and topological charge content in the

presence of symmetry defects and branch lines. Additionally, our analysis allows us to provide a

direct proof that a G-crossed UBTC B×

Gis G-crossed modular if and only if it is a faithful G-crossed

extension of a UMTC B0.

II. STATES AND MODULAR TRANSFORMATIONS ON A TORUS

In this section, we review the description of topological ground states and modular transforma-

tions of SET phases on a 2D torus with no boundary, using a corresponding G-crossed UMTC B×

following Ref. 9. We provide a brief review of the G-crossed UBTC formalism in Appendix A.

A torus can be speciﬁed by an ordered pair of oriented generating cycles with intersection

number +1. We write a particular choice of such an ordered pair as (l, m), which can be thought

of as the longitudinal and meridional cycles for a particular embedding of the torus in 3D space,

as shown in Fig. 1. Every choice of ordered pair of generating cycles with intersection number +1

can be related to each other through linear transformations







l′

m′



=





α β

γ δ









m



=





αl +βm

γl +δm



,(1)

where α, β, γ, δ ∈Zand αδ −βγ = 1. These conditions on the linear transformation ensure

generating pairs are mapped to generating pairs while preserving the intersection number. These

transformations form the modular group SL(2,Z), which is isomorphic to the mapping class group

of the torus, i.e. the automorphisms of the surface modulo the continuous deformations.

Denoting a compact, orientable surface with genus gand nboundary components as Σg,n, we

can write the mapping class group of the torus Σ1,0in the presentation

MCG(Σ1,0)∼

=⟨s,t|(st)3=s2,s4=1⟩(2)

The generators used for this presentation can be chosen to correspond to the following linear

transformations

s∼

=





0−1

1 0 



,t∼

=





1 1

0 1



.(3)

The generator sinterchanges the two generating cycles, with a relative sign that preserves the +1

intersection number. The generator t, which is a “Dehn twist” around the second cycle, shears the

torus.

As with topological phases without symmetry, a basis for the ground state space on a torus

in the presence of symmetry defect branch lines is deﬁned with respect to a choice of ordered

pair of generating cycles. Since there are no boundaries, the symmetry branch lines have no

endpoints. (For these purposes, we treat bulk quasiparticles and defects as boundaries.) As such,

any conﬁguration of branch lines can be deformed to a conﬁguration with one branch loop around

each of the two generating cycles. We say that the system is in the (g,h)-sector with respect

to the pair (l, m), when it can be deformed to a conﬁguration with a g-branch loop around the

lcycle and a h-branch loop around the mcycle. The group elements gand hmust commute,

i.e. gh =hg, for such a conﬁguration, as otherwise it would result in a nontrivial residual defect

(branch line endpoint) that would require a boundary. One can envision processes that transform

the system between diﬀerent (g,h)-sectors by pair-creating defects, transporting them around

nontrivial cycles, and then pair-annihilating the defects. In particular, the (g,h)-sector is obtained

from the (0,0)-sector by creating a h-¯

hpair of defects, transporting the h-defect around the m

cycle (once in the positive direction), and annihilating the pair of defects, and then creating a

g-¯g pair of defects, transporting the g-defect around the lcycle and then annihilating the pair of

defects.

We denote the topological ground state space on the torus in a given (g,h)-sector as H(g,h)

Σ1,0.

We can write basis states for a given (g,h)-sector with respect to (l, m) in a similar manner as for

a topological phase without symmetry. In particular, we can write orthonormal basis states for

the (g,h)-sector as Φag(g,h)

(l,m), where ag∈ Bh

g. Here, Bh

g={ag∈ Bg|hag=ag}is the subset of

g-defect topological charges that are h-invariant. It follows that

dim H(g,h)

Σ1,0=Bh

g.(4)

(a)

(b)

(c)

FIG. 2: Planar representations of the topological ground states of the torus in the presence of symmetry

defect branch lines for diﬀerence choices of pairs of generating cycles. The wavy lines in color represent

defect branch lines, while the grey ribbons correspond to quasiparticle/defect ribbon operators of a speciﬁed

topological charge. (a) Ground state Φag(g,h)

(l,m)in the (g,h) sector with respect to (l, m). (b) Ground state

|Φbh⟩(h,¯

(m,−l)in the (h,¯g) sector with respect to (m, −l). (c) Ground state Φag(gh,h)

(l−m,m)in the (gh,h) sector

with respect to (l−m, m). The displayed conﬁgurations correspond to the same sector of defect branch line

conﬁgurations, represented with respect to diﬀerent pairs of generating cycles. The basis states with respect

to these diﬀerent choices of pairs of generating cycles are related by G-crossed modular transformations S

and T, as speciﬁed in Eqs. (6) and (7).

The state Φag(g,h)

(l,m)corresponds to the conﬁguration shown in Fig. 2(a), where there is a defect

ribbon operator loop of deﬁnite charge agaround the lcycle. For this state, if a topological charge

measurement is performed around the mcycle (which measures the topological charge passing

through the loop m), the measurement outcome will have a deﬁnite outcome of ag. On the other

hand, the topological charge value passing through the loop lwould be in a superposition for

this state, reﬂecting the fact that quasiparticle/defect ribbon operators that loop around the two

diﬀerent cycles are non-commuting operators, i.e. cannot be simultaneously diagonalized. One

can envision obtaining the state Φag(g,h)

(l,m)from |Φ00⟩(0,0)

(l,m)by creating a h-¯

hpair of defects from

vacuum (the topological charges of these are unimportant), transporting the h-defect around the m

cycle, and annihilating the pair of defects, and then creating a ag-agpair of defects from vacuum,

transporting the ag-defect around the lcycle and then annihilating the pair of defects. Annihilation

is not a unitary or deterministic process; rather, one should think of it as a fusion operation,

involving a topological charge measurement of the pair, for which the fusion and measurement

outcome is 0, the vacuum fusion channel. If we obtain an undesired fusion outcome from this

process, we can simply repeat the preparation process until we obtain the desired fusion outcome

0. Since the ag-defect will cross the h-branch line in this process, it must be h-invariant to be able

to pair-annihilate, which is why we required ag∈ Bh

g. Otherwise, the resulting charge ¯

hagcould

not fuse with aginto vacuum, i.e. there would necessarily be a nontrivial quasiparticle charge

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G-CrossedModularityofSymmetryEnrichedTopologicalPhasesArmanBabakhani1,2andParsaBonderson31DepartmentofChemistry,UniversityofCalifornia,SantaBarbara,California93106USA2InformationSciencesInstitute,MarinaDelRey,CA,90292,USA3MicrosoftStationQ,SantaBarbara,California93106-6105USA(Dated:September12,2023)...

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G-Crossed Modularity of Symmetry Enriched Topological Phases Arman Babakhani1 2and Parsa Bonderson3 1Department of Chemistry University of California Santa Barbara California 93106 USA

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