# Duality Defects in $E_{8}$

Ivan M. Burbano 2,3,4 Justin Kulp,444Corresponding author. 5 Jonas Neuser
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Department of
Physics, University of California, 366 Physics North, Berkeley, 94720-7300,
CA, USAPerimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5,
CanadaDepartment of Physics & Astronomy, University of Waterloo, Waterloo, ON
N2L 3G1, CanadaInstitute for Quantum Gravity, University of Erlangen-Nürnberg,
Staudtstraße 7 / B2, 91058 Erlangen, Germany

###### Abstract

We classify all non-invertible Kramers-Wannier duality defects in the $E_{8}$
lattice Vertex Operator Algebra (i.e. the chiral $(E_{8})_{1}$ WZW model)
coming from $\mathbb{Z}_{m}$ symmetries. We illustrate how these defects are
systematically obtainable as $\mathbb{Z}_{2}$ twists of invariant sub-VOAs,
compute defect partition functions for small $m$, and verify our results
against other techniques. Throughout, we focus on taking a physical
perspective and highlight the important moving pieces involved in the
calculations. Kac’s theorem for finite automorphisms of Lie algebras and
contemporary results on holomorphic VOAs play a role. We also provide a
perspective from the point of view of (2+1)d Topological Field Theory and
provide a rigorous proof that all corresponding Tambara-Yamagami actions on
holomorphic VOAs can be obtained in this manner. We include a list of
directions for future studies.

##  1 Introduction

A topological defect line in a 2d QFT corresponds to some oriented (possibly
charged) 1d inhomogeneity in the system, which is invariant under continuous
deformations, so long as the deformation does not move the inhomogeneity past
any other operators. Historically, such topological defect lines have been of
interest in CFT for their connection to BCFT, twisted boundary conditions, and
orbifolds Cardy:1986gw ; Oshikawa:1996 ; Oshikawa:1996_2 ; Petkova:2000 ;
Fuchs:2002cm ; Fuchs:2003id ; Fuchs:2004dz ; Fuchs:2004xi ; Fjelstad:2005ua .
However, more modern applications have exploded in recent years, including:
notions of generalized orbifolds and duality FR_HLICH_2010 ; theoMoonshine ;
Beigi_2011 ; connections to anyon condensation Bais:2008ni ; Kong:2013aya ;
better understanding of defects in statistical physics systems Aasen:2016dop ;
aasen2020topological ; constraints for RG flows Chang_2019 ;
thorngren2019fusion ; gaiotto2020orbifold ; KikuchiRCFT ; constraints for 2d
modular bootstrap Lin:2019kpn ; lin2021mathbbzn ; Pal:2020wwd ;
collier2021bootstrapping ; and even statements about confinement in (1+1)d QCD
komargodski2020symmetries ; Nguyen:2021naa ; Delmastro:2021otj  and quantum
gravity rudelius2020topological . Very broadly, these ideas fit into a series
of research programs studying applications of symmetries in quantum field
theory and their generalizations to “higher-form symmetries” (see e.g.
Gaiotto_2015 ) and “non-invertible symmetries” (see e.g. Bhardwaj_2018  in
2d).

Our goal in this paper is to understand the construction of
$\operatorname{\mathbb{Z}}_{m}$ duality defects in the chiral $(E_{8})_{1}$
WZW model, i.e. lines which separate the theory from its
$\operatorname{\mathbb{Z}}_{m}$ orbifold, and their associated twisted
partition functions. Equivalently, we will focus on
$\operatorname{\mathbb{Z}}_{m}$ Tambara-Yamagami category actions on the
$E_{8}$ lattice Vertex Operator Algebra (VOA). Let us briefly motivate some of
these choices of specialization.

First, duality defects are an interesting object of study in the realm of non-
invertible symmetries. They are non-invertible topological defect lines which
separate a theory from a gauged theory, and help to explain phenomena such as
Kramers-Wannier duality in statistical physics KramWan ; Frohlich_2004 ;
Frohlich:2006 ; Frohlich:2009gb  (see Koide:2021zxj ; Choi:2021kmx ;
Kaidi:2021xfk  for related work in 4d). They also provide “spin-selection
rules” (see e.g. Chang_2019 ) and other constraints on CFTs, which play a role
in e.g. the 2d modular bootstrap. There is also a sense (which we will review)
in which duality defects are the simplest defects after symmetry defects, this
makes them a good tool for understanding the physics of non-invertible defect
lines.

To motivate the choice of chiral $(E_{8})_{1}$, we compare to two other
systems with duality defects: the Ising CFT and the Monster CFT. The Ising CFT
has a $\operatorname{\mathbb{Z}}_{2}$ duality defect, and all of the
properties of this defect can be understood from the fact that it has a (2+1)d
interpretation as one of the three simple anyons in the Ising TFT. By
comparison, the TFT/MTC associated to chiral $(E_{8})_{1}$ is trivial (it has
no irreducible modules besides itself), so we cannot use this approach to
study its duality defects. Moving on, the Monster CFT has
$\operatorname{\mathbb{Z}}_{p}$ duality defects (where $p$ is a prime dividing
the order of the Monster group) and in MonsterCFT  the
$\operatorname{\mathbb{Z}}_{2}$ defects of the Monster CFT were found by
fermionization (see Section 3.5.2). Generalizing such an analysis to a
“parafermionization” could be possible Yao:2020dqx ; thorngren2021fusion , but
our analysis evades the technical difficulties involved with this process.

All in all, our strategy should generalize to theories with duality defects
who do not lift to lines in the representation category of the theory, and
which are more formidable than minimal models, such as the Monster and other
Sporadic group CFTs. Additionally, we conjecture that it will work for higher
$n$-ality defects.

The chiral $(E_{8})_{1}$ theory is also interesting in its own right. For
example, it is the unique holomorphic VOA with central charge $8$, and hence
describes the boundary edge modes of Kitaev’s (2+1)d $E_{8}$ phase kitaevTalk
(see Kaidi:2021gbs  for a recent discussion). Commutant pairs inside the
chiral $(E_{8})_{1}$ theory have also appeared in general CFT contexts and the
study of MLDEs and the conformal bootstrap Mukhi:2020sxt ; Hegde:2021sdm ;
lin2021mathbbzn ; collier2021bootstrapping .

Note: throughout the paper we will always write the total number of spacetime
dimensions.

###  1.1 Outline of the Paper

In this paper we want to construct $\operatorname{\mathbb{Z}}_{m}$ duality
defects of the chiral $E_{8}$ theory, but a side-mission will be to carve a
pathway through the literature in a ground-up way for physicists, focusing on
examples and highlighting various moving pieces to enable explicit
calculations. We also hope this will be of benefit to non-physicists to
understand the physical point of view. We do not review any formal
mathematical definitions (e.g. of fusion categories or vertex algebras) as
such reviews have been done better in other places, although we do provide
references to those materials throughout. When we do introduce some
mathematical formalism, we try to emphasize the physical context so that the
mathematical definitions seem “inevitable.”

In Section 2 we provide background for defects in 2d CFT. We recap how
topological defect lines appear in CFTs in Section 2.1 and how they relate to
gauging symmetries (or orbifolds) in Section 2.2; we also set notation for the
rest of the paper. We briefly recount the story of the critical Ising model
for concreteness in Section 2.3. We conclude in Section 2.4 by providing the
data needed to define the fusion categories $\operatorname{\mathrm{Vec}}_{G}$
and $\operatorname{\mathrm{TY}}(A,\chi,\tau)$ and provide a physical story for
those data.

In Section 3 we build all the tools necessary to compute duality defects in
the chiral $(E_{8})_{1}$ theory. Since we prefer to think of this theory from
the lattice VOA viewpoint, we describe the construction of lattice VOAs in
Section 3.1. Understanding symmetries of VOAs with all the technical details
is important for our construction, so in Section 3.2 we recount the classic
story about lifts from the underlying lattice, Kac’s theorem for finite order
automorphisms of Lie algebras, and a (computer-implementable) way to compute
the relevant twisted characters. In Section 3.3 we discuss orbifolds from the
lattice VOA perspective to make both calculations and the connections to
physics concrete. In Section 3.4 we explain how to compute duality defected
partition functions in the chiral $E_{8}$ theory, and in Section 3.5 we
implement our proposal for $\operatorname{\mathbb{Z}}_{2}$ symmetries and
compare to the result obtained by fermionization.

In Section 4 we show the duality defected partition functions for
$\operatorname{\mathbb{Z}}_{3}$, $\operatorname{\mathbb{Z}}_{4}$, and
$\operatorname{\mathbb{Z}}_{5}$. The purpose of this section is not just the
enumeration of tables of defects, but to show-off potential complications that
may arise in computing defects. For instance, we see the relevance of order-
doubling in Section 4.1, and what happens when the invariant sub-VOA does not
come from a root lattice in Section 4.2 and Section 4.3. In Section 4.2 we
also describe a computer algorithm for systematically computing
($q$-expansions of) defects.

In Section 5 we rephrase our discussion of duality defects from the point of
view of (2+1)d TFTs and gapped boundary conditions. In Section 5.1 we provide
a brief review of the relationship between (2+1)d TFTs, Modular Tensor
Categories (MTCs), and their gapped boundary conditions, then discuss how
orbifolds of 2d theories (and duality defects) can be understood from this
picture in Section 5.2. We conclude with a description of how this applies to
holomorphic VOAs (like the $E_{8}$ lattice VOA) in Section 5.2.1.

We end the paper with a list of open problems in Section 6. Namely those where
explicit computations could give examples of the underlying physics.

In Appendix A we give an alternative way to compute one of our
$\operatorname{\mathbb{Z}}_{3}$ duality defects from the Potts CFT, framing it
from a (2+1)d point of view. In Appendix B we make some technical comments on
canonicality and symmetric non-degenerate bicharacters associated with the
duality defects.

##  2 Topological Defects in 2d

The first example of a topological defect line in 2d occurs in the study of
$0$-form symmetries.111Since we will not discuss higher-form symmetries at all
(until Section 5), from here out we will drop the word $0$-form. Given a
theory with a global $G$ symmetry, each element $g\in G$ corresponds to a
topological line defect $X_{g}$ in the 2d theory: if $X_{g}$ sweeps past an
operator insertion, it acts by $g$ in the appropriate way (see e.g.
Gaiotto_2015 ; yujiCERN ). In the case that $G$ is a continuous group, these
topological lines are simply the (exponentials of the) Noether charges
obtained by integrating the conserved currents associated to $G$ along the
line. Their topological nature can be understood as a consequence of the Ward
identities for such currents.

Continuous or discrete, the topological defect lines associated with
symmetries compose in a natural way: if two symmetry defects labelled by
$g,h\in G$ are brought sufficiently close together, in parallel, and with the
same orientation, they fuse as $X_{g}\otimes X_{h}=X_{gh}$. As a result, we
find that topological defect lines associated with symmetries are invertible,
with $X_{g^{-1}}$ the left and right inverse of the line $X_{g}$.

However, not all topological defect lines in a theory are invertible. Such
non-invertible defect lines in QFTs provide a rich source of information about
a theory, because they are a manifestation of some non-trivial topological
data including dualities, anomalies, and other constraints on RG flows.

The classical example of a non-invertible topological defect line is the
“Kramers-Wannier” duality defect in the Ising model Frohlich_2004 ;
Frohlich:2006 ; FR_HLICH_2010 , which relates correlators in the Ising model
to disorder correlators. We briefly review this in Section 2.3 (see also
Thorngren:2018bhj ; Ji:2019ugf ; MonsterCFT  for recent discussions of the
Kramers-Wannier duality defect).

Another source of topological defect lines are the “Verlinde lines” of
Rational Conformal Field Theories (RCFTs), which are not mutually exclusive
from the two previous examples. Verlinde lines are a natural consequence of
the bulk-boundary relationship between a 2d RCFT and its associated (2+1)d
TFT: they can be understood as 2d topological defects coming from anyons of
the associated bulk (2+1)d TFT, brought to the boundary where the 2d RCFT data
lives Verlinde:1988sn ; Petkova:2000 .

In this paper, we will focus on topological defect lines which are captured by
the mathematical data of a fusion category. Fusion categories describe
theories of topological defect lines which are particularly discrete in that
they are defined to only have a finite number of simple (irreducible) lines.
In addition to being mathematically tractable, fusion categories are also
interesting because they are “rigid” and cannot be “continuously deformed”
(under RG flows, say), which is mathematically captured by the concept of
Ocneanu rigidity ENO . This means that they put constraints on RG flows very
closely related to those implied by ’t Hooft anomalies tHooft:1979rat , see
Section 7 of Chang_2019  for a modern discussion.

By now, fusion categories have appeared in a variety of ways in the physics
literature, broadly in the intersection of fields that study topological
aspects of quantum field theories. This includes more traditional “high-
energy” contexts (e.g. Chang_2019 ; komargodski2020symmetries ; Bhardwaj_2018
; thorngren2019fusion ; gaiotto2020orbifold ) and especially in the study of
condensed matter systems and quantum computation (e.g. Kong:2013aya ;
Lan:2013wia ; Kong:2014qka ; Ji:2019eqo ; Ji:2019jhk ).

Rather than re-define fusion categories, we recommend the enthusiastic physics
reader to check out Bhardwaj_2018  for the definitions of a fusion category
(called a “symmetry category” there) and to Section 3 of Chang_2019 . For the
reader in a hurry, we recommend Section 5 and Appendix A of
komargodski2020symmetries .

###  2.1 CFT and Defect Partition Functions

Since a defect is an inhomogeneity in our system, part of defining a line
defect in 2d involves specifying boundary conditions for fields on either side
of the defect line. We can do this in a 2d CFT by describing how it acts on
states of the Hilbert space.

In a 2d CFT on the plane, any defect line $X$ defines an operator $\hat{X}$ on
the Hilbert space $\mathcal{H}$ of states on a circle, which acts as follows:
place a state $\phi\in\mathcal{H}$ at the origin and the defect on the unit
circle. Then the state that this setup provides is by definition
$\hat{X}\phi\in\mathcal{H}$ FR_HLICH_2010 . Said more cylindrically, prepare
the state $\phi$ in the infinite past and have the defect line waiting “half
way” up the cylinder, then the out state is by definition $\hat{X}\phi$.

From this, we obtain a very algebraic definition of a topological defect line
as one that commutes with all the modes of the stress tensor

$[\hat{L}_{n},\hat{X}]=0=[\hat{\bar{L}}_{n},\hat{X}]\,.$ (1)

The topological defect line $X$ does not need to lie on a closed path.
However, if we do have a defect line along an open oriented path, then we will
also have to specify how the line operator ends. The Hilbert space
$\mathcal{H}_{X}$ of operators upon which $X$ can end is the Hilbert space of
the theory on a circle, except with a single future oriented $X$ defect
piercing the circle.

To obtain the CFT partition functions, we simply periodicitize time in the
cylinder setups. The partition function with the $X$ line inserted along a
spacelike slice (a twist in Euclidean time) is given by

$Z^{X}(\tau,\bar{\tau}):=\Tr_{\mathcal{H}}\hat{X}q^{L_{0}-\frac{c}{24}}\bar{q}^{\bar{L}_{0}-\frac{c}{24}}\,,$
(2)

where $q:=e^{2\pi i\tau}$. Likewise, the partition function with the $X$ line
inserted along a timelike slice (a twist in space) is given by

$Z_{X}(\tau,\bar{\tau}):=\Tr_{\mathcal{H}_{X}}q^{L_{0}-\frac{c}{24}}\bar{q}^{\bar{L}_{0}-\frac{c}{24}}\,.$
(3)

We illustrate these setups in Figure 1. Note that the two are related by the
modular $S$-transformation $\tau\mapsto-\frac{1}{\tau}$ (and similarly for
antiholomorphic), because the $S$-transformation swaps the two torus cycles.

$X$

$X$

Figure 1: On the left, an $X$ defect line (red) is inserted along a spatial
slice, this gives a “twist” by $X$ in the time direction. On the right, the
defect line is inserted along the time direction. Periodicitizing this setup
on the right, we get the trace over the $X$-twisted Hilbert space
$\mathcal{H}_{X}$.

As an almost trivial example, if $X_{-}$ is a topological defect line
generating a $\operatorname{\mathbb{Z}}_{2}$ symmetry, then, basically by
definition, if $\phi_{i}$ is charged under the $\operatorname{\mathbb{Z}}_{2}$
symmetry we have

$\hat{X}_{-}\ket{\phi_{i}}=-\ket{\phi_{i}}\,.$ (4)

When tracing over the states in the Hilbert space to obtain
$Z^{X_{-}}(\tau,\bar{\tau})$, for example, we will need to consider the signs
from the charged states.

When studying a RCFT, another natural collection of topological defects comes
from Verlinde line operators Verlinde:1988sn ; Petkova:2000 ; Chang_2019 .
Verlinde lines are in one-to-one correspondence with primaries, and for a
diagonal RCFT, a Verlinde line $X_{i}$ commutes with the whole chiral algebra
and acts on a primary by

$\hat{X}_{i}\ket{\phi_{j}}=\frac{S_{ij}}{S_{0j}}\ket{\phi_{j}}\,,$ (5)

where $S_{ij}$ is the modular $S$-matrix of the CFT. It is a straightforward
application of Verlinde’s formula to show that the Verlinde lines then fuse
according to the fusion rules of the associated primaries, i.e.

$X_{i}\otimes X_{j}=\bigoplus_{k}N_{ij}^{k}X_{k}\,.$ (6)

Given any topological defect line $X$ encircling a local operator $\phi$ in
the plane, we can take an equivalent view where the line $X$ squashes down
leaving us with only the local operator $X\phi$ in the plane. If, instead of a
general $\phi$, we had chosen the identity operator, it’s clear we are just
computing the expectation value of a loop of $X$ in the plane, aka the quantum
dimension of $X$. Throughout, we will denote this as $\dim X$ rather than
$\expectationvalue{X}$.

More generally, when a defect line $X$ sweeps past a local operator $\phi$,
the result is not necessarily a local operator, but can be written as a local
operator plus a defect operator $\phi^{\prime}$ that lies on the end of a
topological tail adjoined to $X$, as depicted in Figure 2.

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Figure 2: On the left, a topological defect line $X$ encircles a $\phi$
insertion in the plane. Since the line is topological, this is equivalent
instead to the local operator $X\phi$. On the right, a topological defect line
$X$ sweeps past the $\phi$ insertion leaving behind a local operator and a
defect operator $\phi^{\prime}$. The defect operator is connected to $X$ by a
topological tail (dotted).

There are some interesting and important subtleties about all of this
machinery, which are described in Section 2 of Chang_2019 . We have overlooked
them because they are not particularly pressing for our discussion, but we
will mention two immediate ones.

The first is that the diagram on the right of Figure 2 simply does not reduce
to the diagram on the left, we are left with the naive answer plus a
topological tadpole. The vanishing of this tadpole can be proven if one
assumes that the topological defect lines act faithfully on the bulk local
operators. This condition can be violated in the case the CFT has multiple
ground states, but such CFTs are just a direct sum of theories and won’t be
relevant for us.222There exists some recent literature on this topic and its
interplay with higher-form symmetries komargodski2020symmetries ;
Sharpe:2019ddn ; Yu:2020twi .

The second subtlety is that in 2d there are local curvature counterterms which
may be added to the action. Since we are only really concerned with producing
torus partition functions, this will not play any role because the torus is
flat. In any case, following such terms would just change overall phases.

###  2.2 Orbifolding Abelian Symmetries

To warm-up, consider a 2d CFT $T$ on $M$ with a global finite Abelian symmetry
$A$. If the symmetry is non-anomalous, then we can couple the theory to a
background $A$-connection in a well-defined way. Since $A$ is finite, this
connection is necessarily flat and is specified by an $\alpha\in H^{1}(M,A)$
which labels the holonomies around the different cycles of $M$.

Framed in terms of topological defects, we can represent a background
$A$-connection by a network of topological $A$-defect lines meeting at
trivalent junctions (see Tachikawa:finite  and diagrams within for an
introduction). When a local operator passes the topological defect line
labelled by $g\in A$, it applies the appropriate (linear) $g$-action.

For each background connection $\alpha\in H^{1}(M,A)$ we obtain a “twisted
partition function” of the theory $Z_{T}[\alpha]$, which is like the “regular”
partition function with (invertible) topological symmetry defect lines
inserted around non-contractible loops. For example, on the torus, we have the
$\absolutevalue{A}^{2}$ twisted partition functions

$Z_{T}[g,h]:=\Tr_{\mathcal{H}_{g}}\hat{X}_{h}q^{L_{0}-\frac{c}{24}}\bar{q}^{\bar{L}_{0}-\frac{c}{24}}\,.$
(7)

Here $g,h\in A$, $\mathcal{H}_{g}=\mathcal{H}_{X_{g}}$ is the $g$-twisted
Hilbert space, and we have suppressed dependence on $\tau$ and $\bar{\tau}$.

The orbifold theory $[T/A]$ is the theory obtained by gauging the discrete $A$
symmetry of $T$. As is familiar from the path integral, gauging a symmetry is
making the background connection dynamical, so the orbifold is obtained by
summing over the twisted partition functions. Moreover, when we orbifold there
is an emergent dual symmetry, given by the Pontryagin dual $\hat{A}$ of $A$,
which comes from the action of the Wilson lines for the $A$ gauge fields
vafa:quantumSymmetry . This new dual symmetry can be coupled to a background
flat connection $\beta\in H^{1}(M,\hat{A})$, and we obtain a formula relating
the twisted partition functions in the different theories

$Z_{[T/A]}[\beta]=\frac{1}{\sqrt{\absolutevalue{H^{1}(M,A)}}}\sum_{\alpha\in
H^{1}(M,A)}e^{i(\beta,\alpha)}Z_{T}[\alpha]\,.$ (8)

The exponential denotes the intersection pairing between $H^{1}(M,A)$ and
$H^{1}(M,\hat{A})$. Non-Abelian symmetries can be treated with only a bit more
work.333Even more generally, we can gauge by “symmetric Frobenius algebras” in
a theory with fusion category symmetry by inserting the corresponding
triangulating trivalent mesh of topological defect lines as described in
FR_HLICH_2010 ; Carqueville_2016 ; Bhardwaj_2018 . We discuss fusion
categories more in Section 2.4.

Equation (8) is not necessarily the most general possible way to orbifold by a
symmetry. In particular, one can include extra $U(1)$ weights
$\epsilon(\alpha)$ in front of the twisted partition functions, and still
obtain a gauge invariant partition function. In the CFT language, such a
choice is known as a choice of “discrete torsion” vafa:Torsion1 ;
vafa:Torsion2 ; sharpeTorsion . Such a choice of $U(1)$ phase is classified by
a group cohomology class $\nu_{2}\in H^{2}(A,U(1))$, so that most generally we
have

$Z_{[T/_{\nu_{2}}A]}[\beta]=\frac{1}{\sqrt{\absolutevalue{H^{1}(M,A)}}}\sum_{\alpha\in
H^{1}(M,A)}e^{i(\beta,\alpha)}\epsilon_{\nu_{2}}(\alpha)Z_{T}[\alpha]\,.$ (9)

$\epsilon_{\nu_{2}}$ is obtained from $\nu_{2}$ by evaluating it on a
triangulation of $M$, e.g. on the torus with fluxes in the cycles given by
$(\alpha_{1},\alpha_{2})$, so
$\epsilon_{\nu_{2}}(\alpha_{1},\alpha_{2})=\nu_{2}(\alpha_{1},\alpha_{2})/\nu_{2}(\alpha_{2},\alpha_{1})$.

The two (commensurate) modern viewpoints on this discrete torsion phase are as
follows: Anomalies of 2d theories manifest as phase ambiguities in coupling
$T$ to a background connection, which in turn are encoded in a cohomology
class $\mu_{3}\in H^{3}(A,U(1))$. When a theory is non-anomalous, we don’t
just need to know that the cohomology class is trivial, but must pick a
trivialization $\nu_{2}\in H^{2}(A,U(1))$ that consistently resolves all phase
ambiguities. Alternatively, we may view the phase as the partition function
for a 2d SPT phase $\epsilon_{\nu_{2}}(\alpha)\sim e^{iS_{\nu_{2}}[\alpha]}$.
Then orbifolding with discrete torsion is equivalent to taking a non-anomalous
theory $T$, with phase ambiguities already resolved, and then stacking with
such a 2d SPT phase before orbifolding Chen:2011pg ; wenAnomalies  (see also
Kapustin:2014tfa ).

####  2.2.1 Example: $\operatorname{\mathbb{Z}}_{2}$ Orbifold

To get a better appreciation of what happens at the level of the Hilbert
spaces, and see where the dual symmetry comes from, consider a theory $T$ with
a non-anomalous (bosonic) $\operatorname{\mathbb{Z}}_{2}$ symmetry. The
untwisted orbifold partition function on the torus is

$\displaystyle Z_{[T/\operatorname{\mathbb{Z}}_{2}]}[0,0]$
$\displaystyle=\frac{1}{2}\sum_{g,h\in\operatorname{\mathbb{Z}}_{2}}Z_{T}[g,h]$
(10)
$\displaystyle=\sum_{g\in\operatorname{\mathbb{Z}}_{2}}\Tr_{\mathcal{H}_{g}}\left[\left(\frac{\hat{X}_{+}+\hat{X}_{-}}{2}\right)q^{L_{0}-\frac{c}{24}}\bar{q}^{\bar{L}_{0}-\frac{c}{24}}\right]\,.$
(11)

The term in round brackets just projects us onto the states which are
invariant under the $\operatorname{\mathbb{Z}}_{2}$ symmetry when we trace
over $\mathcal{H}_{g}$. This shows that the operators that contribute to the
untwisted orbifold partition function are those in both the untwisted and
twisted Hilbert spaces, $\mathcal{H}_{+}$ and $\mathcal{H}_{-}$ respectively,
but only those which are invariant under the $\operatorname{\mathbb{Z}}_{2}$
action.

Said differently, we can decompose these Hilbert spaces into sums of even and
odd subsectors

$\displaystyle\mathcal{H}_{+}$
$\displaystyle=\mathcal{H}_{+}^{\text{Even}}\oplus\mathcal{H}_{+}^{\text{Odd}}\,,$
(12) $\displaystyle\mathcal{H}_{-}$
$\displaystyle=\mathcal{H}_{-}^{\text{Even}}\oplus\mathcal{H}_{-}^{\text{Odd}}\,.$
(13)

Then the untwisted orbifold Hilbert space contains just even (i.e. gauge-
invariant) operators, while the twisted orbifold Hilbert space contains just
odd operators

$\displaystyle\mathcal{H}_{\textrm{Orbi.}}^{\textrm{Even}}$
$\displaystyle=\mathcal{H}_{+}^{\text{Even}}\oplus\mathcal{H}_{-}^{\text{Even}}\,,$
(14) $\displaystyle\mathcal{H}_{\textrm{Orbi.}}^{\textrm{Odd}}$
$\displaystyle=\mathcal{H}_{+}^{\text{Odd}}\oplus\mathcal{H}_{-}^{\text{Odd}}\,.$
(15)

Hence we see the emergent $\hat{\operatorname{\mathbb{Z}}}_{2}$ symmetry
appear in the definition of this new twisted orbifold Hilbert space. For
$A=\operatorname{\mathbb{Z}}_{m}$ orbifolds we have to replace “Even” and
“Odd” by the appropriate $\operatorname{\mathbb{Z}}_{m}$ projectors, which are
labelled by representations of $A$, i.e. elements of $\hat{A}$.

###  2.3 Example: The Ising CFT

Consider the (full) $c=\frac{1}{2}$ Ising CFT, which describes the Ising model
at criticality. The theory has 3 Virasoro primaries: the identity operator
$\mathds{1}$, the energy operator $\epsilon$, and the spin operator $\sigma$,
with conformal weights
$(h,\bar{h})=(0,0),(\frac{1}{2},\frac{1}{2}),(\frac{1}{16},\frac{1}{16})$
respectively BPZ . The fusion rules are

$[\sigma][\sigma]=[\mathds{1}]\oplus[\epsilon]\,,\quad[\sigma][\epsilon]=[\sigma]\,,\quad[\epsilon][\epsilon]=[\mathds{1}]\,.$
(16)

The Ising CFT also has exactly 3 simple topological defect lines, these are
the Verlinde lines naturally associated with the 3 primaries:
$X_{\mathds{1}}$, $X_{\epsilon}$, and $X_{\sigma}$. They fuse according to the
fusion rules of the primaries in Equation (16) above, and so it is clear that
the $\sigma$ line is not-invertible. The action of $X_{\mathds{1}}$ on
primaries is straightforward. $X_{\epsilon}$ is the
$\operatorname{\mathbb{Z}}_{2}$ symmetry line, so acts only non-trivially on
the $\operatorname{\mathbb{Z}}_{2}$ charged operator(s) ($\sigma$, $\psi$,
$\bar{\psi}$, etc.) as in Equation (4). The action of $X_{\sigma}$ on the
Ising CFT primaries are depicted in Figure 3.

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Figure 3: Left, the $X_{\sigma}$ defect sweeps past a $\sigma$ primary,
turning it into the disorder operator $\mu$, plus a topological
$\operatorname{\mathbb{Z}}_{2}$ tail. Such a $\mu$ is traditionally called a
“twist-field.” Right, the $X_{\sigma}$ defect sweeps past the relevant
operator $\epsilon$ and turns it into $-\epsilon$.

Famously, in the Ising model (not necessarily at criticality), the correlator
of spins at some inverse temperature $\beta$ is equal to the correlator of
disorder/twist operators at the Kramers-Wannier dual inverse temperature
$\tilde{\beta}$, i.e.
$\langle\sigma(z_{1})\cdots\sigma(z_{n})\rangle_{\beta}=\langle\mu(z_{1})\cdots\mu(z_{n})\rangle_{\tilde{\beta}}$.
This exhibits some duality between weak and strong couplings and can be used
to understand the critical point of the Ising model KramWan ; Frohlich_2004
(see also Aasen:2016dop ; aasen2020topological ).

However, the disorder fields $\mu$ are not local operators in the theory
alone, but have to be accompanied by a topological “line of frustration,”
which has to end similarly on another $\operatorname{\mathbb{Z}}_{2}$ “twist-
field.” In the CFT context above, this line of frustration is the
$\operatorname{\mathbb{Z}}_{2}$ symmetry line $X_{\epsilon}$, and the
$\operatorname{\mathbb{Z}}_{2}$ twisted Hilbert space is
$\mathcal{H}_{\epsilon}=\mathrm{span}\\{\mu,\psi,\bar{\psi}\\}$.

A statistical interpretation of the $X_{\sigma}$ line is not immediately
clear, but can be realized by first studying how it acts on the primaries in
the CFT picture. In particular, when we sweep the $X_{\sigma}$ line past a
$\sigma$ insertion, we are left with a $\mu$ insertion and a topological
$X_{\epsilon}$ tail. But in the orbifold CFT
$[T/\operatorname{\mathbb{Z}}_{2}]$, $\mu$ is a local operator (the
$\operatorname{\mathbb{Z}}_{2}$ line becomes invisible when we gauge), and
$\sigma$ must be accompanied by a topological tail.

Moreover, when $X_{\sigma}$ sweeps past an $\epsilon$ insertion, it leaves
behind a $-\epsilon$ insertion. The energy operator $\epsilon$ is a relevant
operator in the Ising CFT and deformations by it flow the Ising CFT to the
high-temperature (unbroken) or low-temperature (broken) phases. The action of
$X_{\sigma}$ reflects the fact that the Ising model and its orbifold see this
deformation oppositely. This also explains why Kramers-Wannier duality holds
away from criticality.

For these reasons, $X_{\sigma}$ is the archetypal duality defect. It extends
the algebra of topological defects in the Ising model from a collection of
$\operatorname{\mathbb{Z}}_{2}$ group-like defects to a Tambara-Yamagami
category (see Section 2.4).

It is important to stress two basic facts about the duality defect line. The
first is that it is not the non-simple topological defect line
$X_{\textrm{Proj.}}:=X_{\mathds{1}}\oplus X_{\epsilon}$. Inserting a
triangulating mesh of $X_{\textrm{Proj.}}$ into the original theory $T$ just
produces $[T/\operatorname{\mathbb{Z}}_{2}]$. Indeed, $X_{\sigma}$ separates
the theory $T$ from $[T/\operatorname{\mathbb{Z}}_{2}]$. The second basic fact
is that it is not the orbifold duality defect line $X_{\textrm{Proj.}}$… but
it does square to it! This is important because when the $X_{\sigma}$ line
collides with itself (e.g. when deformed around a cycle of the torus) it does
produce a $X_{\textrm{Proj.}}$ defect.

It is also instructive to compare the twisted and “defected” partition
functions for this theory. The twisted partition functions can be written in
terms of the usual Virasoro characters as

$\displaystyle Z_{T}[0,0]$
$\displaystyle=\lvert\chi_{0}\rvert^{2}+\lvert\chi_{\frac{1}{2}}\rvert^{2}+\lvert\chi_{\frac{1}{16}}\rvert^{2}\,,$
(17) $\displaystyle Z_{T}[0,1]$
$\displaystyle=\lvert\chi_{0}\rvert^{2}+\lvert\chi_{\frac{1}{2}}\rvert^{2}-\lvert\chi_{\frac{1}{16}}\rvert^{2}\,,$
(18) $\displaystyle Z_{T}[1,0]$
$\displaystyle=\chi_{0}\bar{\chi}_{\frac{1}{2}}+\chi_{\frac{1}{2}}\bar{\chi}_{0}+\lvert\chi_{\frac{1}{16}}\rvert^{2}\,,$
(19) $\displaystyle Z_{T}[1,1]$
$\displaystyle=-\chi_{0}\bar{\chi}_{\frac{1}{2}}-\chi_{\frac{1}{2}}\bar{\chi}_{0}+\lvert\chi_{\frac{1}{16}}\rvert^{2}\,.$
(20)

While the defected partition function, with an $X_{\sigma}$ inserted around
the spatial $S^{1}$, is easily computable with the help of Equation (5) for
Verlinde lines on primaries

$Z_{T}[0,X_{\sigma}]=\sqrt{2}\lvert\chi_{0}\rvert^{2}-\sqrt{2}\lvert\chi_{\frac{1}{2}}\rvert^{2}\,.$
(21)

Such a structure for the defect partition function is typical, and can be
understood from the second important fact about $X_{\sigma}$. Since
$X_{\sigma}\otimes X_{\sigma}=X_{\mathds{1}}\oplus
X_{\epsilon}=X_{\mathrm{Proj.}}$ we can compute (without identifying
$X_{\sigma}$ with the Verlinde line) that
$\hat{X}_{\mathrm{Proj}.}\ket{\phi}=2\ket{\phi}$ if $\phi$ corresponds to a
$\operatorname{\mathbb{Z}}_{2}$ uncharged operator, and is $0$ otherwise. This
means that

$\hat{X}_{\sigma}\ket{\phi}=\pm\sqrt{2}\ket{\phi}$ (22)

if $\phi$ is uncharged under the $\operatorname{\mathbb{Z}}_{2}$, and is $0$
otherwise. Hence the defected partition function is just a sum over all of the
uncharged primaries, and all we have to do is resolve a sign.

###  2.4 Fusion Categories for Physics

Here we will recap the pertinent properties of the 2 most important fusion
categories for our purposes: the fusion category of $G$ graded vector spaces
$\operatorname{\mathrm{Vec}}_{G}$ and Tambara-Yamagami categories
$\operatorname{\mathrm{TY}}(A,\chi,\tau)$.

####  2.4.1 Fusion Category: Graded Vector Spaces
$\operatorname{\mathrm{Vec}}_{G}$

Given a finite symmetry group $G$, the topological defects associated to the
symmetry group form a unitary fusion category in a natural way. Simple objects
of the category are oriented lines labelled by group elements $g,h\in G$. If
two such lines are brought sufficiently close together, in parallel, with the
same orientation, then they fuse according to the group multiplication law

$X_{g}\otimes X_{h}=X_{gh}\,\qquad\textrm{for $g,h\in G$.}$ (23)

The category is unitary because the orientation reversal of a line is (by
definition) the inverse $X_{g}^{*}=X_{g}^{-1}=X_{g^{-1}}$.444Since we are
technically dealing with objects in a category, by $=$ we mean that there
exists a natural isomorphism between objects in the category.

Given three simple lines labelled by $g,h,k\in G$, the data of an associator
isomorphism $\alpha$ is needed to relate the object $(X_{g}\otimes
X_{h})\otimes X_{k}$ to $X_{g}\otimes(X_{h}\otimes X_{k})$. In practice, this
means that there can be an overall phase ambiguity between the background
connection depicted with $X_{g}$ and $X_{h}$ fusing first or $X_{h}$ and
$X_{k}$ fusing first as in Figure 4.

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Figure 4: Mathematically, the associator $\alpha$ is an element of
$\operatorname{Hom}((X_{g}\otimes X_{h})\otimes
X_{k},X_{g}\otimes(X_{h}\otimes X_{k}))$ that defines in what way the two
tensor products are equal. Physically, given a theory with non-trivializable
anomaly $\alpha\in H^{3}(G,U(1))$, two networks of symmetry defect lines can
be different up to a $U(1)$ phase.

Such an $\alpha$ is required to satisfy the familiar “pentagon identity,”
which asserts that whatever $\alpha$ is, it must agree when we consider the
multiple ways to fuse four $G$ defects as in Figure 5.555Readers may be
familiar with the data of an associator presented in the form of an $F$-symbol
or $6j$-symbol. In this case, such $\alpha$ are classified by an element of
the group cohomology $H^{3}(G,U(1))$ (see e.g. Faddeev:1985iz ; DW90 ). Such
phase ambiguities in coupling a theory to a background $G$ connection, which
cannot be removed by adding local counterterms, are the ’t Hooft anomalies
from the physics literature.

${(X_{g}\otimes X_{h})\otimes(X_{k}\otimes X_{\ell})}$${((X_{g}\otimes
X_{h})\otimes X_{k})\otimes X_{\ell}}$${X_{g}\otimes(X_{h}\otimes(X_{k}\otimes
X_{\ell}))}$${(X_{g}\otimes(X_{h}\otimes X_{k}))\otimes
X_{\ell}}$${X_{g}\otimes((X_{h}\otimes X_{k})\otimes
X_{\ell})}$$\scriptstyle{\alpha}$$\scriptstyle{\alpha}$$\scriptstyle{\alpha\otimes
1}$$\scriptstyle{\alpha}$$\scriptstyle{1\otimes\alpha}$ Figure 5: Given 4
defect lines $X_{g}$, $X_{h}$, $X_{k}$, and $X_{\ell}$, we only demand that
the pentagon diagram commutes. This is familiar to those in RCFT as the
pentagon identity for F-symbols.

All considered, such a collection of defects labelled by elements of $G$ with
a choice of group cohomology class $\alpha\in H^{3}(G,U(1))$ form the fusion
category $\operatorname{\mathrm{Vec}}_{G}^{\alpha}$ of $G$-graded vector
spaces. In practice, we are going to be interested in the case where the
symmetry in question does not have an anomaly, so that we can orbifold by that
symmetry, the fusion category $\operatorname{\mathrm{Vec}}_{G}$.

It may not be a surprising fact, when paralleled with our discussion of
coupling a CFT in the regular group symmetry framework, that the group of
“autoequivalences” of the fusion category $\operatorname{\mathrm{Vec}}_{G}$
(basically, automorphisms of $\operatorname{\mathrm{Vec}}_{G}$ as a fusion
category) is $\operatorname{Aut}(G)\ltimes H^{2}(G,U(1))$. The
$\operatorname{Aut}(G)$ roughly corresponding to renaming of $G$ lines, and
$H^{2}(G,U(1))$ corresponding to stacking with 2d SPT phases Bhardwaj_2018 .

####  2.4.2 Fusion Category: Tambara-Yamagami
$\operatorname{\mathrm{TY}}(A,\chi,\tau)$

In the previous subsubsection we started with a set of desired fusion rules
(and unitary structure) and then were left with classifying the associators
compatible with those fusion rules/unitary structure and pentagon identity.
This is the same thing that Tambara and Yamagami do in a ground-up approach in
their seminal paper tambara1998tensor  (see also tambara2000representations ),
except their fusion rules are minimally enriched by a duality defect.

Start with a finite Abelian group $A$, the simple objects of
$\operatorname{\mathrm{TY}}(A,\chi,\tau)$ are again associated with elements
$a\in A$, but the fusion algebra is now extended by an additional object $m$
of quantum dimension $\sqrt{\absolutevalue{A}}$.666In fact, such a fusion ring
makes sense if the group is not Abelian, but it is only categorifiable if $A$
is Abelian (see tensorCatBook  Example 4.10.5). In addition to the familiar
$\operatorname{\mathrm{Vec}}_{A}$ fusion rules, we also demand

$\displaystyle X_{a}\otimes X_{m}$ $\displaystyle=X_{m}\,,$ (24)
$\displaystyle X_{m}\otimes X_{m}$ $\displaystyle=\bigoplus_{a\in A}X_{a}\,.$
(25)

What Tambara and Yamagami show, by direct computation, is that the data of
potential associators is classified by a symmetric non-degenerate bicharacter
$\chi:A\times A\to U(1)$, and a sign or “Frobenius-Schur indicator” $\tau=\pm
1/\sqrt{\absolutevalue{A}}$ tambara1998tensor ; tambara2000representations
(see also homotopyFusion ; Bhardwaj_2018 ; thorngren2019fusion ;
aasen2020topological ). In particular, the non-trivial associators are

$\displaystyle\alpha_{a,m,b}$
$\displaystyle=\chi(a,b)\operatorname{\mathds{1}}_{m}\,,$ (26)
$\displaystyle\alpha_{m,a,m}$
$\displaystyle=\bigoplus_{b}\chi(a,b)\operatorname{\mathds{1}}_{b}\,,$ (27)
$\displaystyle\alpha_{m,m,m}$
$\displaystyle=(\tau\chi(a,b)^{-1}\operatorname{\mathds{1}}_{m})_{a,b}\,.$
(28)

A first point of note, which will be relevant later, is that TY categories are
a $\operatorname{\mathbb{Z}}_{2}$-extension of
$\operatorname{\mathrm{Vec}}_{A}$. In general, a $G$-extension of a fusion
category $\mathcal{D}$ is a $G$-graded fusion category

$\mathcal{F}=\bigoplus_{g\in G}\mathcal{F}_{g}\,,$ (29)

satisfying $\mathcal{F}_{e}\cong\mathcal{D}$.

Another thing to note is that $\chi:A\times A\to U(1)$, as opposed to the more
familiar pairing of $A$ with $\hat{A}$. This means $\chi$ is an assignment of
(magnetic) $A$ charges to $A$ defect lines, in contradistinction to the
natural Fourier-like pairing in orbifolds $\rho:A\times\hat{A}\to U(1)$.
Pushing further, a bicharacter $\chi$ defines a homomorphism $A\to\hat{A}$,
and non-degeneracy implies that this is an isomorphism. Since $T$ and $[T/A]$
have $A$ and $\hat{A}$ symmetry respectively, this $\chi$ reflects the freedom
of choice in picking an isomorphism from $A$ to $\hat{A}$ (or rather reflects
the non-canonicality of $A\cong\hat{A}$). By the same logic, $\chi$ determines
an isomorphism from $\hat{A}\to A$, and while $A$ and its dual $\hat{A}$ are
not canonically isomorphic, $A$ is canonically isomorphic to its double dual
$\hat{\hat{A}}$. The symmetric property ensures that whatever isomorphism
$\chi$ determines from $A$ to $\hat{A}$, and $\hat{A}$ to $A$, that it
respects the canonicality $A=\hat{\hat{A}}$ i.e. it is honestly the identity
on $A$ and not some random automorphism of $A$.

The role of $\tau$ is less physically obvious, and shows up in the crossing
relations for $m$ thorngren2019fusion . It originates categorically as an
associativity-like constraint similar in spirit to the associator in
$\operatorname{\mathrm{Vec}}_{A}^{\alpha}$ homotopyFusion , but could be
better understood physically.

As a closing remark, the Ising Fusion category that we are familiar with forms
$\operatorname{\mathrm{TY}}(\operatorname{\mathbb{Z}}_{2},1,+1/\sqrt{2})$ (see
Section 4.1 of thorngren2019fusion  for elaboration).

##  3 Duality Defects in $(E_{8})_{1}$

The main goal of this paper is to classify and construct
$\operatorname{\mathbb{Z}}_{m}$ duality defects in the chiral WZW-model
$(E_{8})_{1}$. Said differently, we want to understand the actions of
$\operatorname{\mathbb{Z}}_{m}$ Tambara-Yamagami categories on the holomorphic
lattice VOA $V_{E_{8}}$.

Vertex Operator Algebras are widespread in physics as the axiomatization of
“chiral algebras,” which glue together to form a “full CFT” in two dimensions.
We refer readers to the standard mathematical texts on VOAs frenkel1989vertex
; dong1993generalized ; kac1998vertex ; Frenkel:2004jn ;
lepowsky2004introduction  or any standard textbook on 2d CFT or string theory
for an introduction (a nice review for both communities appeared in
Bae:2020pvv ). Lattice VOAs are also prominent in the physics literature, but
understanding subtleties of their construction, automorphisms, and characters
is important for our purposes, so we review them briefly in the following
sections. Some particularly helpful reviews on lattice VOAs besides those
already listed include Dolan:1989vr ; dolan1996 ; van_Ekeren_2020 ;
moller2017cyclic .

###  3.1 Lattice Vertex Operator Algebras

Given a positive definite even lattice $L$ one can construct a lattice VOA
$V_{L}$. Mathematically,
$V_{L}=M_{\hat{\mathfrak{h}}}(1)\otimes\operatorname{\mathbb{C}}_{\epsilon}[L]$
where the first factor is the “Heisenberg VOA” describing oscillator modes of
chiral bosons (also called $\mathrm{Bos}(\mathfrak{h})$), and the second
factor is the “twisted group algebra” describing “quantized momentum.”777See
e.g. Section 5 of moller2017cyclic  and references listed within for the
formal mathematical construction of these objects and proof $V_{L}$ is
rational, $C_{2}$-cofinite, self-contregredient, and of CFT type.

We start the construction with the quantized momentum modes: for every vector
$\alpha\in L$ we have a state $\ket{\alpha}$ which is created by the vertex
operator $\Gamma_{\alpha}$. By definition, the states are orthonormal
$\innerproduct{\alpha}{\beta}=\delta_{\alpha\beta}$. There are also chiral
bosonic oscillators $a_{n}^{i}$, $n\in\operatorname{\mathbb{Z}}$ and $1\leq
i\leq\operatorname{\mathrm{rk}}(L)$, satisfying the usual Heisenberg
commutation relations

$[a_{m}^{i},a_{n}^{j}]=m\delta^{ij}\delta_{m+n,0}\,,$ (30)

with $a_{m}^{i{\dagger}}=a_{-m}^{i}$.

Combining the two, the positive oscillator modes should annihilate the
$\ket{\alpha}$

$a_{n}^{i}\ket{\alpha}=0\,,\qquad\textrm{if $n>0$,}$ (31)

and $a_{0}^{i}$ should actually behave as the momentum operator, i.e.

$a_{0}^{i}\ket{\alpha}=\alpha^{i}\ket{\alpha}.$ (32)

The physical Hilbert space is thusly generated by the $a_{-n}^{i}$, $n>0$,
acting on the $\ket{\alpha}$, so that a general basis state is of the form

$a^{i_{N}}_{-n_{N}}\cdots a^{i_{2}}_{-n_{2}}a^{i_{1}}_{-n_{1}}\ket{\alpha}\,.$
(33)

Such a state has integral conformal weight

$h=\frac{1}{2}(\alpha,\alpha)+\sum_{i=1}^{N}n_{i}\,,$ (34)

where $(\,.\,,\,.\,)$ denotes the inner product on the lattice $L$.

As mentioned, the vertex operator $\Gamma_{\alpha}$ creates the state
$\ket{\alpha}$, and one expects that
$\Gamma_{\alpha}\Gamma_{\beta}\propto\Gamma_{\alpha+\beta}$. However, asking
for $\Gamma$ to be a representation of the lattice is too strong; locality
requires us to consider projective representations, i.e.

$\Gamma_{\alpha}\Gamma_{\beta}=\epsilon(\alpha,\beta)\Gamma_{\alpha+\beta}\,,$
(35)

where $\epsilon:L\times L\to\\{\pm 1\\}$ is a (normalized) 2-cocycle
satisfying
$\epsilon(\alpha,\beta)=(-1)^{(\alpha,\beta)}\epsilon(\beta,\alpha)$. This
skew condition on the 2-cocycle determines $\epsilon$ up to a coboundary, and
so $\epsilon\in H^{2}(L,\\{\pm 1\\})$, which will play a role for us in
understanding automorphisms.888In other words, the $\Gamma_{\alpha}$ furnish,
at minimum, a representation of a double cover $\hat{L}$ of the root lattice
$L$. The freedom of choice in $\epsilon$ (due to the central extension
defining the double cover not splitting) reflects the fact that there is no
functor from root lattices to vertex algebras, i.e. a choice of $\epsilon$ is
required theobook . Double covers of $L$ coming from differing choices of
$\epsilon$ are isomorphic, which is why one can non-canonically write
$V_{L}=M_{\hat{\mathfrak{h}}}(1)\otimes\operatorname{\mathbb{C}}[L]$ if they
settle some signs once and for all.

When $L$ is the root lattice of a simply laced Lie algebra $\mathfrak{g}$,
this construction gives a vertex-representation of the
$\hat{\mathfrak{g}}_{1}$ WZW model (see e.g. Section 15.6.3 of
francesco2012conformal ). In this case, the sign ambiguities which occur in
trying to build $V_{L}$ from the root lattice $L$ are precisely the same as
those in trying to build the Lie algebra $\mathfrak{g}$ from $L$. For non-
simply laced algebras (or $k>1$), one must add a free fermion (or parafermion)
respectively.

Up to isomorphism, the untwisted irreducible modules of $V_{L}$ are labelled
by elements of the discriminant group $D_{L}:=L^{*}/L$, here $L^{*}$ is the
dual lattice to $L$ dong1993vertex . The MTC of representations
$\operatorname{\mathrm{Rep}}(V_{L})$ has $D_{L}$ group-like fusion in the
obvious way. For some $\lambda\in L^{*}$, the character of the irreducible
$V_{L}$-module $V_{\lambda+L}$

$\chi_{V_{\lambda+L}}(\tau)=\tr_{V_{\lambda+L}}q^{L_{0}-\frac{c}{24}}=\frac{\theta_{\lambda+L}(\tau)}{\eta(\tau)^{\operatorname{\mathrm{rk}}(L)}}\,,$
(36)

where $\theta_{\lambda+L}(\tau)$ is the theta function of the shifted lattice
$\lambda+L$, i.e.

$\theta_{\lambda+L}(\tau)=\sum_{\xi\in\lambda+L}q^{\frac{1}{2}(\xi,\xi)}\,.$
(37)

####  3.1.1 Example: $V_{E_{8}}$ as a $V_{E_{8}}$ Module

The $E_{8}$ root lattice is positive-definite, even, and self-dual, therefore
$V_{E_{8}}$ itself is the only irreducible $V_{E_{8}}$ module (aka the
“adjoint module”). VOAs whose only irreducible module are the adjoint module
are called holomorphic, i.e.
$\operatorname{\mathrm{Rep}}(V_{E_{8}})=\operatorname{\mathrm{Vec}}$.

Using the description of the $E_{8}$ lattice as points in
$\operatorname{\mathbb{Z}}^{8}\cup(\operatorname{\mathbb{Z}}+\frac{1}{2})^{8}$,
whose coordinates sum to an even integer, we obtain the $E_{8}$ partition
function

$\displaystyle\tr_{V_{E_{8}}}q^{L_{0}-\frac{c}{24}}$
$\displaystyle=\frac{1}{\eta(\tau)^{8}}\sum_{\xi\in
E_{8}}q^{\frac{1}{2}(\xi,\xi)}$ (38)
$\displaystyle=\frac{1}{2\eta(\tau)^{8}}\sum_{x\in\operatorname{\mathbb{Z}}^{8}}\left(1+(-1)^{\sum{x_{i}}}\right)\left(q^{\frac{1}{2}\sum{x_{i}^{2}}}+q^{\frac{1}{2}\sum{(x_{i}+\frac{1}{2})^{2}}}\right)$
(39) $\displaystyle Z_{E_{8}}(\tau)$
$\displaystyle=\frac{1}{2\eta(\tau)^{8}}\left(\theta_{1}(\tau)^{8}+\theta_{2}(\tau)^{8}+\theta_{3}(\tau)^{8}+\theta_{4}(\tau)^{8}\right)$
(40)

Note: even though this expression is not modular invariant, since
$c_{L}-c_{R}\in 8\operatorname{\mathbb{Z}}$ its modular non-invariance can be
cured by a (2+1)d gravitational Chern-Simons term Alvarez-Gaume:1983ihn ;
Witten:1988hf  (see Kaidi:2021gbs  for a recent discussion).

###  3.2 Automorphisms and Twisted Partition Functions

#### Describing Automorphisms

Recall that a VOA $V$ has an automorphism/symmetry given by an (invertible)
linear map $\hat{g}:V\to V$, if the action of $\hat{g}$ preserves the vacuum
$\hat{g}\ket{0}=\ket{0}$, preserves the (chiral) stress-tensor
$\hat{g}T(z)=T(z)$, and commutes with the state operator map

$(\hat{g}\phi)(z)=\hat{g}\phi(z)\hat{g}^{-1}\,.$ (41)

Suppose $V_{L}$ is a lattice VOA constructed from the even lattice $L$. The
subgroup of $\operatorname{Aut}(V_{L})$ which maps the subalgebra of bosonic
oscillators to itself is given by the (usually) non-split extension $T.O(L)$.
Here $O(L)$ are the symmetries of the lattice itself and
$T:=\operatorname{\mathbb{R}}^{\operatorname{\mathrm{rk}}(L)}/L^{*}$ is the
torus dual to $L$. $T$ carries the natural action of $O(L)$ viewed as a
subgroup of $O(\operatorname{\mathrm{rk}}(L),\operatorname{\mathbb{R}})$
dong1999automorphism .

The subgroup $T<T.O(L)$ acts trivially on the bosonic oscillators, and simply
modifies the group algebra of quantized momentum modes by phases. That is, for
any $x\in T$ we have the induced map

$\displaystyle a^{i}_{n}$ $\displaystyle\mapsto a^{i}_{n}\,,$ (42)
$\displaystyle\Gamma_{\alpha}$ $\displaystyle\mapsto e^{2\pi
i(x,\alpha)}\Gamma_{\alpha}\,.$ (43)

Note again that this map is actually trivial if $x\in L^{*}$, and so we are
only interested in $x\,\mathrm{mod}\,L^{*}$. Conversely, the $O(L)$ factor
acts on the bosons and also permutes the $\Gamma_{\alpha}$
lepowsky1985calculus ; dong1999automorphism ; moller2017cyclic ;
van_Ekeren_2020 ; Hohn:2020xfe  (see also Section 4.4 of
evans2020tambarayamagami ).

In this vein, suppose $g\in O(L)$ is any automorphism of the lattice $L$, then
$g$ preserves the skew form $(-1)^{(\,\cdot\,,\,\cdot\,)}$. Since the skew
condition determines the 2-cocycle $\epsilon$ up to a 2-coboundary, we see
that $\epsilon(\alpha,\beta)$ and $\epsilon(g\alpha,g\beta)$ are in the same
cohomology class; and so

$\frac{\epsilon(\alpha,\beta)}{\epsilon(g\alpha,g\beta)}=\frac{u(\alpha)u(\beta)}{u(\alpha+\beta)}\,,$
(44)

for all $\alpha,\beta\in L$ and some 1-cocycle $u:L\to\\{\pm 1\\}$.

It is a classic result that such a $g\in O(L)$ and choice of 1-cocycle $u$
define a lift to an automorphism $\hat{g}_{u}\in
T.O(L)<\operatorname{Aut}(V_{L})$ lepowsky1985calculus ; dong1999automorphism
. This automorphism also acts in a nice way

$\displaystyle\hat{g}_{u}(a_{n})$ $\displaystyle=(ga)_{n}\,,$ (45)
$\displaystyle\hat{g}_{u}\Gamma_{\alpha}$
$\displaystyle=u(\alpha)\Gamma_{g\alpha}\,.$ (46)

Furthermore, one can always choose “a standard lift” such that $u(\alpha)=1$
on $L^{g}$ lepowsky1985calculus . We will not work with standard lifts right
away, but with foresight we quote the fact that: any two standard lifts are
conjugate in $\operatorname{Aut}(V_{L})$ van_Ekeren_2020 .999The term standard
lift is a bit of a misnomer and does not have the properties one might expect:
for example, there can be more than one “standard” lift; moreover, since the
extension $T.O(L)$ is not split (i.e. it is not a semi-direct product), the
product of standard lifts is not necessarily standard. All of this richness
can be traced back to the fact that there is no canonical way to construct a
Lie algebra and lattice VOA from the root lattice $L$. For further discussion
see van_Ekeren_2020 ; moller2017cyclic ; theobook  and references within.

Given a positive definite even lattice $L$ and a lifted automorphism
$\hat{g}_{u}\in\operatorname{Aut}(V_{L})$, the irreducible
$\hat{g}_{u}$-twisted modules can be constructed and classified similar to the
untwisted modules dong1996algebraic ; bakalov2004twisted ; moller2017cyclic .
In short, they are labelled by elements of the discriminant $L^{*}/L$ fixed
under the $g$ action. i.e. they are of the form $V_{\lambda+L}$, where
$\lambda+L\in(L^{*}/L)^{g}=\\{\lambda\in
L^{*}/L:(\operatorname{\mathds{1}}-g)\lambda\in L\\}$.

The twisted characters are

$\tr_{V_{L}}\hat{g}_{u}q^{L_{0}-\frac{c}{24}}=\frac{\theta_{L^{g},u}(\tau)}{\eta_{g}(\tau)}\,,$
(47)

where we’ve defined the generalized theta function

$\theta_{L^{g},u}(\tau):=\sum_{\alpha\in
L^{g}}u(\alpha)q^{(\alpha,\alpha)/2}\,.$ (48)

The eta product $\eta_{g}(\tau)$ is given by

$\eta_{g}(\tau):=\prod_{t|m}\eta(t\tau)^{b_{t}},$ (49)

when $g\in O(L)$ has cycle shape $\prod_{t|m}t^{b_{t}}$. The cycle shape can
be determined by calculating the characteristic polynomial of the matrix
representing the automorphism, which is of the form
$\prod_{t|m}(\lambda^{t}-1)^{b_{t}}$.

More physically, factors of $\eta(\tau)$ arise in characters/partition
functions from the contributions of each independent tower of (chiral) bosonic
modes. If $g$ is, for example, a $\operatorname{\mathbb{Z}}_{2}$ symmetry
swapping the $a^{1}_{m}$ and $a^{2}_{n}$ towers, then one expects to get
factors of $\eta(2\tau)$ in the vacuum character of the invariant sub-VOA due
to the “extra length” of the combined
$\operatorname{\mathbb{Z}}_{2}$-invariant tower.

Finally, given an order $m$ automorphism $g\in O(L)$, a standard lift
$\hat{g}_{u}\in\operatorname{Aut}(V_{L})$ has: order $m$ if $m$ is odd; order
$m$ if $m$ is even and $(\alpha,g^{m/2}\alpha)\in 2\operatorname{\mathbb{Z}}$
for all $\alpha$; and order $2m$ otherwise (see e.g. Corollary 5.3.6 of
moller2017cyclic ).

All of these technicalities about canonicality, lifts, and order doubling may
seem overkill, but actually plays an important role in constructing as many
duality defects as possible.

#### Finding Automorphisms

To understand how to find automorphisms of lattice VOAs, it is helpful to
expand on the structure of the automorphism group of a VOA and its
relationship to Lie algebras and physics.

We use the following fact for orientation: For a VOA $V$ of “CFT-type” (like
lattice VOAs or the Monster), the weight one subspace $V_{1}$ is a (possibly
trivial) complex Lie algebra with bracket $[u,v]=u_{0}v$. Moreover, all the
weight spaces of $V$-modules are Lie algebra modules for $V_{1}$ (including
the weight spaces of the adjoint module, i.e. the $V_{n}$), so that we can see
the VOA and its modules as strata of $V_{1}$ Lie algebra modules
frenkel1993axiomatic . From a physics perspective, we are not surprised that
the currents form a Lie algebra under which things transform, this is just
Noether’s theorem and Wigner’s theorem.

There are two extreme examples for the Lie algebra $V_{1}$ in relation to $V$:
the symmetry currents $V_{1}$ generate $V$ and
$\operatorname{Aut}(V_{1})\cong\operatorname{Aut}(V)$, as in affine Kac-Moody
VOAs or their semi-simple quotients; or alternatively, there are no currents
and $V_{1}=0$, as with the Virasoro algebra
$\operatorname{Aut}(\mathrm{Vir})=\\{\mathds{1}\\}$, and the Monster CFT
$\operatorname{Aut}(V^{\natural})=\mathbb{M}$ dong1999automorphism .

The key move is to note that any automorphism/symmetry
$\hat{g}\in\operatorname{Aut}(V)$ restricts to a Lie algebra automorphism of
$V_{1}$.101010A supersymmetry would relate operators of $V_{1}$ to operators
with different spins. Hence we ask: to what extent can we understand
$\operatorname{Aut}(V)$ by lifting automorphisms of the Lie algebra
$\operatorname{Aut}(V_{1})$?

The answer to this question is known for lattice VOAs. For lattice VOAs, the
weight one subspace is

$(V_{L})_{1}=\mathcal{H}\oplus\bigoplus_{\alpha^{2}=2}\operatorname{\mathbb{C}}\\{\Gamma_{\alpha}\\}\,,$
(50)

where $\mathcal{H}$ is a Cartan subalgebra spanned by the weight-1 chiral
bosons and the $\operatorname{\mathbb{C}}\\{\Gamma_{\alpha}\\}$ play the role
of the root spaces. The exact Lie algebra structure constants depend on the
choice of $\epsilon$, but all choices are isomorphic to a Lie algebra
$\mathfrak{g}$ with root lattice $L$.

Using this, in dong1999automorphism  the authors prove that
$\operatorname{Aut}(V_{L})$ is a non-split product

$\operatorname{Aut}(V_{L})=K.O(\hat{L})\,,$ (51)

where $K=\langle\\{e^{u_{0}}:u\in(V_{L})_{1}\\}\rangle$ is the inner
automorphism group generated by (exponentials of) “currents.” In other words,
we have that
$\mathrm{der}(V_{L}):=\mathrm{Lie}(\operatorname{Aut}(V_{L}))=\mathfrak{g}$,
and so we will be interested in finite order automorphisms of
$\mathfrak{g}$.111111If we were studying the derivations of a non-rational
VOA, whose representations need not form an MTC and thus not be so discrete,
its possible that $\mathrm{der}(V)$ isn’t entirely formed by currents. In this
case we could ask about infinitesimal outer automorphisms
$\mathrm{out}(V):=\mathrm{der}(V)/\\{\mathrm{currents}\\}=\mathrm{der}(\operatorname{\mathrm{Rep}}(V))$,
which correspond to smooth deformations of the representation category. This
is not exotic: consider $n$ free chiral bosons, $V=\mathrm{Bos}(n)$, in this
case $\operatorname{Aut}(\operatorname{\mathrm{Rep}}(V))\cong O(n)$. Note that
what we called $T$ before is the maximal toral subgroup, i.e.
$T=\langle\\{e^{u_{0}}:u\in\mathcal{H}\\}\rangle$.

Beautifully, conjugacy classes of finite order automorphisms of (semi-)simple
Lie algebras were classified by Kac kats1969automorphisms ; kac_1990  (also
see Section 8.3.3 of theobook  and moller2017cyclic  for examples). This
includes both inner and outer automorphisms! Even better, the classification
gives an easy way to read off the fixed point Lie subalgebra under that
symmetry! The theorem is as follows:

######  Theorem 1 (Theorem 8.6, kac_1990 ).

Let $\mathfrak{g}$ a finite dimensional simple Lie algebra with Dynkin diagram
$X_{n}$, choose $k=1,2,3$ and write $X^{(k)}_{n}$ for the corresponding affine
Dynkin diagram (or “twisted Dynkin diagram” if $k>1$) with $\ell+1$ nodes.
Then choose non-negative relatively prime integers $s=(s_{0},\cdots,s_{\ell})$
and set

$m=k\sum_{i=0}^{\ell}a_{i}s_{i}\,,$ (52)

where $a_{i}$ are the marks of $X_{n}^{(k)}$. Then the following statements
are true:

  1. 1.

The choices $(k,s)$ define an order $m$ automorphism
$g_{k,s}\in\operatorname{Aut}(\mathfrak{g})$. For $k=1$, if one chooses a set
of simple roots and works in the usual Chevalley basis, the action is given by

$g_{1,s}(E^{j})=e^{2\pi is_{j}/m}E^{j}\,.$ (53)

For $k=2,3$ the definition of $E^{j}$ is slightly more complicated (see
kac_1990 ).

  2. 2.

Up to conjugation, all order $m$ automorphisms of $\mathfrak{g}$ are obtained
this way.

  3. 3.

Two automorphisms $g_{k,s}$ and $g_{k^{\prime},s^{\prime}}$ obtained in this
way are conjugate by an automorphism of $\mathfrak{g}$ if and only if
$k=k^{\prime}$ and the sequence $s$ can be transformed into $s^{\prime}$ by an
automorphism of the diagram $X^{(k)}_{n}$.

  4. 4.

The number $k$ is the least positive integer for which $g^{k}_{k,s}$ is an
inner automorphism. i.e. $k=1$ automorphisms are inner, and $k=2,3$
automorphisms are outer.

  5. 5.

Let $i_{1},\cdots,i_{p}$ be all the indices such that
$s_{i_{1}}=\cdots=s_{i_{p}}=0$. Then the fixed-point Lie subalgebra
$\mathfrak{g}^{g_{k,s}}$ is isomorphic to a direct sum of: the
$(\ell-p)$-dimensional center $\cong\operatorname{\mathbb{C}}^{\ell-p}$, and a
semisimple Lie algebra whose Dynkin diagram is the subdiagram of the affine
diagram $X^{(k)}_{n}$ consisting of the nodes $i_{1},\cdots,i_{p}$.

Kac’s theorem is great for 3 purposes: enumerating finite order automorphisms
of $\mathfrak{g}$, finding the fixed Lie subalgebra $\mathfrak{g}^{g_{k,s}}$,
and understanding the systematic construction of $x\in
T<\operatorname{Aut}(V_{L})$ describing inner automorphisms of lattice VOAs
like Equation (43). In particular, for an order $m$ inner automorphism, $x$ is
constructed from the sequence $s$ using the basis of fundamental weights
$\omega_{1},\dots,\omega_{l}$, dual to the basis of simple coroots:

$x=\frac{1}{m}\sum_{j=1}^{l}s_{j}\,\omega_{j}\,.$ (54)

On the other hand, it is systematically easier for calculations of twisted
characters of outer automorphisms to use a slightly different procedure from
Hohn:2020xfe , which we describe in the remainder of this section.

Let $V_{L}$ be a lattice VOA and fix a choice of Cartan subalgebra
$\operatorname{\mathcal{H}}$ of $(V_{L})_{1}$ as in Equation (50), further fix
a choice of simple roots $\Delta$. Then it can be shown that: up to conjugacy,
every finite-order $\hat{\sigma}\in\operatorname{Aut}(V_{L})$ is of the form

$\hat{\sigma}=e^{2\pi i(x,\,\cdot\,)}.\hat{g}$ (55)

where $\hat{g}$ is a (fixed) choice of standard lift of $g\in H_{\Delta}$ and
$x\in L^{g}\otimes\mathbb{Q}$ is some phase understood modulo $(L^{g})^{*}$.
By $H_{\Delta}:=O(L)_{\\{\Delta\\}}$ we mean the setwise stabilizer of
$\Delta$ in $O(L)$, note that $H_{\Delta}\cong O(L)/W$ which is in turn
isomorphic to the automorphism group of the Dynkin diagram associated to $L$,
its just key that we’ve fixed a base $\Delta$. See dong1999automorphism  and
Theorem 2.13 of Hohn:2020xfe  for details.

This theorem allows for a simple computation of the $\hat{\sigma}$-twisted
traces

$\tr_{V_{L}}\hat{\sigma}\,q^{L_{0}-\frac{c}{24}}=\frac{1}{\eta_{g}(\tau)}\sum_{\alpha\in
L^{g}}e^{2\pi i(x,\alpha)}q^{(\alpha,\alpha)/2}\,.$ (56)

Importantly, if $\hat{\sigma}$ is built from the standard lift $\hat{g}$ of
$g\in O(L)$ as above, then $\hat{\sigma}$ has order $\absolutevalue{g}$ if and
only if

$x\in\begin{cases}0+(1/m)(L^{*})^{g}&\text{and $\absolutevalue{\hat{g}}=m$}\\\
\zeta+(1/m)(L^{*})^{g}&\text{and $\absolutevalue{\hat{g}}=2m$}\\\ \end{cases}$
(57)

where $\zeta\in(1/2m)(L^{*})^{g}$.

The short of all this is that: rather than studying finite order automorphisms
of $(V_{L})_{1}$ and lifting to automorphisms of the VOA, we can take a more
ground-up approach. For example, suppose we want to find all
$\hat{\sigma}$-twisted characters of a VOA $V_{L}$ up to conjugacy, and we
want $\hat{\sigma}$ to have order $2$. Then we fix a set of simple roots for
$L$ and enumerate all $g\in H_{\Delta}$ of order $1$ or $2$. For each $g$, we
make some choice of standard lift $\hat{g}\in\operatorname{Aut}(V_{L})$, so
$\hat{g}$ has order $1$, $2$, or $4$. Then we enumerate all inner
automorphisms of order up to $4$ (by making choices of vectors $x$ as
described previously). Finally, we combine all of our choices as in (55), and
throw away whichever do not have order $2$. The theorem guarantees that we
have produced all $\hat{\sigma}$ of order $2$ up to conjugacy.

###  3.3 Orbifolds of $V_{E_{8}}$

Now that we understand automorphisms of lattice VOAs, we may study their
associated orbifolds. Throughout, let $V:=V_{E_{8}}$ be the holomorphic
VOA121212Also simple, rational, $C_{2}$-cofinite, self-contregredient, CFT-
type. We will ignore such technical details from here out in this section.
constructed from the $E_{8}$ root lattice $L$ as in Example 3.1.1. $V$ is the
unique (up to isomorphism) holomorphic CFT with chiral central charge
$c_{L}=8$, and gauging by any non-anomalous $\operatorname{\mathbb{Z}}_{m}$
symmetry will simply produce another holomorphic CFT with the same central
charge, so $[V/\operatorname{\mathbb{Z}}_{m}]\cong V$.

For this reason, computing orbifold partition functions isn’t particularly
illuminating, they are all just as in Equation (40). However, while the
orbifold theory $[V/\operatorname{\mathbb{Z}}_{m}]$ is isomorphic to $V$, they
are not identical (just as in the Ising model). To appreciate this, we should
look at what it means to orbifold $V$ at the level of lattices.

For this, we will follow the conventions of conway2013sphere . In particular,
$L$ will be spanned by simple roots

$\displaystyle\alpha_{1}$ $\displaystyle=(-1,1,0^{6}),$ (58)
$\displaystyle\alpha_{2}$ $\displaystyle=(0,-1,1,0^{5}),$
$\displaystyle\vdots$ $\displaystyle\alpha_{7}$ $\displaystyle=(0^{6},-1,1),$
$\displaystyle\alpha_{8}$
$\displaystyle=\quantity(\frac{1}{2}^{5},-\frac{1}{2}^{3}),$

which fit into the Dynkin diagram

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{}{}{{}}{}\pgfsys@moveto{8.5359pt}{0.0pt}\pgfsys@lineto{22.76228pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
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{}{}{{}}{}\pgfsys@moveto{135.1506pt}{8.5359pt}\pgfsys@lineto{135.1506pt}{22.76228pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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This diagram can be extended to its affine version with the help of the
highest root

$\alpha_{\text{high}}=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+5\alpha_{4}+6\alpha_{5}+4\alpha_{6}+2\alpha_{7}+3\alpha_{8}\,,$
(60)

which yields

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}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\alpha_{5}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{{}}{}\pgfsys@moveto{179.25246pt}{0.0pt}\pgfsys@lineto{193.47884pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\alpha_{6}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{{}}{}\pgfsys@moveto{213.39569pt}{0.0pt}\pgfsys@lineto{227.62207pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\alpha_{7}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}{{}}{}\pgfsys@moveto{169.29382pt}{8.5359pt}\pgfsys@lineto{169.29382pt}{22.76228pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{164.69531pt}{28.09973pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{
{}{}{}{}{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,.$ (61)

Now, let $G=\langle g_{s}\rangle\cong\operatorname{\mathbb{Z}}_{m}$ be a
subgroup of $T<\operatorname{Aut}(V)$ generated by the sequence $s$ as in the
previous section (and with $k=1$) i.e. $G$ acts by a phase on the group
algebra. Explicitly, write $x:=\sum x_{j}\omega_{j}$ with $x_{j}:=s_{j}/m$,
then

$\displaystyle a^{i}_{n}$ $\displaystyle\mapsto a^{i}_{n}\,,$ (62)
$\displaystyle\Gamma_{\alpha}$ $\displaystyle\mapsto e^{2\pi
i(x,\alpha)}\Gamma_{\alpha}\,,$

with $(x,\alpha_{i})\geq 0$, $(x,\alpha_{\mathrm{high}})\leq 1$, and
$x_{i}\in\frac{1}{m}\operatorname{\mathbb{Z}}$.