Such a symmetry is non-anomalous and can be orbifolded if we can extend the
invariant sub-VOA $V^{G}$ by twisted sectors to a distinct VOA.131313Note: If
$V$ is a lattice VOA, then under an inner automorphism the fixed sub-VOA
$V^{G}$ is always another lattice VOA. In physics terms, the orbifold Hilbert
space is obtained as an extension of the subspace of unscreened/invariant
operators, as in Equation (14) for a $\operatorname{\mathbb{Z}}_{2}$ symmetry.
The anomaly classifies the obstruction to this extension procedure.

The group action above leaves the chiral Cartan bosons unscreened as well as
the vertex operators with $(x,\alpha)\in\operatorname{\mathbb{Z}}$. This last
condition distinguishes a sublattice $L_{0}\subset L$ of index $m$, so that
the fixed sub-VOA $V^{G}$ is the lattice VOA constructed from $L_{0}$. Said in
reverse, $\operatorname{\mathrm{rk}}(L_{0})=\operatorname{\mathrm{rk}}(L)$
both define the same Heisenberg module of chiral bosons, but in $V^{G}$ only
the momentum states corresponding to $L_{0}$ are admissible.141414Note: in the
mathematics literature, $V^{G}$ is often called the “orbifold VOA,” but we
will always mean the full extension.

Our question then becomes: can we extend $L_{0}$ to an even self-dual lattice
$L^{\prime}$, distinct from $L$, inside $L_{0}^{*}$?151515Dropping the even
requirement is how one produces fermionic orbifolds. This can be depicted

$\leavevmode\hbox to101.63pt{\vbox
to60.45pt{\pgfpicture\makeatletter\hbox{\hskip
36.58855pt\lower-30.28972pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-33.25554pt}{-2.51443pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$L_{0}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-19.70547pt}{9.70874pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\rotatebox{45.0}{$\subseteq$}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-19.70547pt}{-13.24428pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\rotatebox{-45.0}{$\subseteq$}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.6458pt}{18.69424pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$L=L^{*}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-0.79694pt}{-26.95671pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$L^{\prime}=L^{\prime*}$}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{37.20004pt}{-18.74402pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\rotatebox{45.0}{$\subseteq$}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{37.20004pt}{15.20848pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{\rotatebox{-45.0}{$\subseteq$}}} }}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{52.10274pt}{-3.16582pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$L_{0}^{*}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,.$ (63)

Such an $L^{\prime}$ exists (and is unique) if and only if

$q(x):=\frac{(x,x)}{2}\in\frac{1}{m}\operatorname{\mathbb{Z}}\,.$ (64)

Generally, $q(x)\in\frac{1}{m^{2}}\operatorname{\mathbb{Z}}$ because $mx\in
L$, so the anomaly class is the class

$[q(x)]\in\left(\frac{1}{m}\operatorname{\mathbb{Z}}\right)/\left(\frac{1}{m^{2}}\operatorname{\mathbb{Z}}\right)\cong\operatorname{\mathbb{Z}}_{m}\,.$
(65)

Note:
$H^{3}(\operatorname{\mathbb{Z}}_{m},U(1))=\operatorname{\mathbb{Z}}_{m}$, so
the matchup with anomalies is evident.

| $[V/\operatorname{\mathbb{Z}}_{m}]$ | $[V/\operatorname{\mathbb{Z}}_{m}]^{1}$ | $\cdots$ | $[V/\operatorname{\mathbb{Z}}_{m}]^{m-1}$
---|---|---|---|---
$V$ | $\operatorname{\mathcal{H}}^{0}_{0}$ | $\operatorname{\mathcal{H}}^{1}_{0}$ | $\cdots$ | $\operatorname{\mathcal{H}}^{m-1}_{0}$
$V_{1}$ | $\operatorname{\mathcal{H}}^{0}_{1}$ | $\operatorname{\mathcal{H}}^{1}_{1}$ | $\cdots$ | $\operatorname{\mathcal{H}}^{m-1}_{1}$
$\vdots$ | $\vdots$ | $\vdots$ | $\ddots$ | $\vdots$
$V_{m-1}$ | $\operatorname{\mathcal{H}}^{0}_{m-1}$ | $\operatorname{\mathcal{H}}^{1}_{m-1}$ | $\cdots$ | $\operatorname{\mathcal{H}}^{m-1}_{m-1}$
Table 1: The holomorphic VOA $V$ decomposes as a direct sum of the untwisted
Hilbert spaces, where the upper grading denotes the “electric”
$\operatorname{\mathbb{Z}}_{m}$ charge. The orbifold VOA decomposes as a
direct sum of the gauge invariant Hilbert spaces, where the lower index
grading denotes the dual “magnetic” $\hat{\operatorname{\mathbb{Z}}}_{m}$
charge.

In the language of VOAs we are just saying that the irreducible
representations $\operatorname{\mathrm{Irr}}(V^{G})$ have group like fusion,
with a fusion group given by a central extension of
$\operatorname{\mathbb{Z}}_{m}$ by $\operatorname{\mathbb{Z}}_{m}$, reflecting
the structure of $L_{0}^{*}/L_{0}$ when we try to factor through $L$. The
structure of the fusion group is controlled by a class living in
$H^{2}(\operatorname{\mathbb{Z}}_{m},\hat{\operatorname{\mathbb{Z}}}_{m})=\operatorname{\mathbb{Z}}_{m}$,
but should not be confused with the group of anomalies. The anomaly does
influence this group, but the map between anomalies and fusion rules is not
1-to-1 (see van_Ekeren_2020 ), as can be seen in following examples with
$\operatorname{\mathbb{Z}}_{2}$. However, in the case that $G$ acts non-
anomalously, then
$\operatorname{\mathrm{Irr}}(V^{G})=\operatorname{\mathbb{Z}}_{m}\times\operatorname{\mathbb{Z}}_{m}$
for sure.

If we suggestively label the elements of $\operatorname{\mathrm{Irr}}(V^{G})$
as $\mathcal{H}_{j}^{i}$, mimicking the decomposition following Equation (12),
then $\operatorname{\mathrm{Irr}}(V^{G})$ can be organized like Table 1. The
original VOA decomposes into the “electrically charged” sectors

$V=\bigoplus_{i=1}^{m}\mathcal{H}_{0}^{i}\,,$ (66)

and the orbifold VOA, if it exists, decomposes into the “magnetically charged”
sectors as

$[V/\operatorname{\mathbb{Z}}_{m}]=\bigoplus_{j=1}^{m}\mathcal{H}_{j}^{0}\,.$
(67)

Each of the
$\operatorname{\mathcal{H}}^{i}_{j}\in\operatorname{\mathrm{Irr}}(V^{G})$ is
associated with a coset of $L_{0}^{*}/L_{0}$, and the discriminant $q(x)$ is
simply telling us the conformal weight in that sector. In the non-anomalous
case, the conformal weight of a vector in $\operatorname{\mathcal{H}}_{j}^{i}$
is $\frac{ij}{m}+\operatorname{\mathbb{Z}}$ which gives
$\operatorname{\mathrm{Irr}}(V^{G})$ the structure of a “metric Abelian group”
(see Section 4 of moller2017cyclic  for more details).

We record the automorphisms of $V$, anomaly, and corresponding Lie algebra for
small $m$ in Table 2. In Table 3 we record just the non-anomalous symmetries
with the marked Dynkin diagram.

_Order_ |  _Automorphism_ |  $\operatorname{\mathbb{Z}}_{m}$ _Anomaly_ |  _Fixed Subalgebra_
---|---|---|---
2 |  (0,0,0,0,0,0,0,1,0) |  0 |  $D_{8}$
|  (0,1,0,0,0,0,0,0,0) |  1 |  $A_{1}\times E_{7}$
3 |  (0,0,0,0,0,0,0,0,1) |  1 |  $A_{8}$
|  (1,0,0,0,0,0,0,1,0) |  2 |  $D_{7}\times\mathfrak{u}(1)$
|  (0,0,1,0,0,0,0,0,0) |  0 |  $A_{2}\times E_{6}$
|  (1,1,0,0,0,0,0,0,0) |  1 |  $E_{7}\times\mathfrak{u}(1)$
4 |  (1,0,0,0,0,0,0,0,1) |  0 |  $A_{7}\times\mathfrak{u}(1)$
|  (2,0,0,0,0,0,0,1,0) |  2 |  $D_{7}\times\mathfrak{u}(1)$
|  (0,0,0,0,0,0,1,0,0) |  3 |  $A_{1}\times A_{7}$
|  (0,0,0,1,0,0,0,0,0) |  2 |  $A_{3}\times D_{5}$
|  (1,0,1,0,0,0,0,0,0) |  3 |  $A_{1}\times E_{6}\times\mathfrak{u}(1)$
|  (2,1,0,0,0,0,0,0,0) |  1 |  $E_{7}\times\mathfrak{u}(1)$
|  (0,1,0,0,0,0,0,1,0) |  1 |  $A_{1}\times D_{6}\times\mathfrak{u}(1)$
5 |  (2,0,0,0,0,0,0,0,1) |  4 |  $A_{7}\times\mathfrak{u}(1)$
|  (3,0,0,0,0,0,0,1,0) |  2 |  $D_{7}\times\mathfrak{u}(1)$
|  (0,0,0,0,0,0,0,1,1) |  1 |  $A_{7}\times\mathfrak{u}(1)$
|  (1,0,0,0,0,0,0,2,0) |  3 |  $D_{7}\times\mathfrak{u}(1)$
|  (1,0,0,0,0,0,1,0,0) |  2 |  $A_{1}\times A_{6}\times\mathfrak{u}(1)$
|  (0,0,0,0,1,0,0,0,0) |  0 |  $A_{4}\times A_{4}$
|  (1,0,0,1,0,0,0,0,0) |  1 |  $A_{2}\times D_{5}\times\mathfrak{u}(1)$
|  (2,0,1,0,0,0,0,0,0) |  3 |  $A_{1}\times E_{6}\times\mathfrak{u}(1)$
|  (0,0,1,0,0,0,0,1,0) |  4 |  $A_{2}\times D_{5}\times\mathfrak{u}(1)$
|  (3,1,0,0,0,0,0,0,0) |  1 |  $E_{7}\times\mathfrak{u}(1)$
|  (0,1,0,0,0,0,0,0,1) |  3 |  $A_{1}\times A_{6}\times\mathfrak{u}(1)$
|  (1,1,0,0,0,0,0,1,0) |  0 |  $D_{6}\times\mathfrak{u}(1)^{2}$
|  (0,1,1,0,0,0,0,0,0) |  2 |  $A_{1}\times E_{6}\times\mathfrak{u}(1)$
|  (1,2,0,0,0,0,0,0,0) |  4 |  $E_{7}\times\mathfrak{u}(1)$
Table 2: The finite order automorphisms of the $E_{8}$ Lie algebra can be enumerated, up to conjugacy, by Theorem 1. The automorphism is given by a sequence $s$ which describes the automorphism. In physics terms, $s$ describes how the operators of $V_{E_{8}}$ are screened and the anomaly of that symmetry. _Order_ |  _Automorphism_ |  _Fixed Subalgebra_
---|---|---
2 |  000000010 |  $D_{8}$
3 |  001000000 |  $A_{2}\times E_{6}$
4 |  100000001 |  $A_{7}\times\mathfrak{u}(1)$
5 |  000010000 |  $A_{4}\times A_{4}$
|  110000010 |  $D_{6}\times\mathfrak{u}(1)^{2}$
Table 3: List of non-anomalous symmetries of the $E_{8}$ Lie algebra and fixed
point Lie subalgebras up to order $m=5$. Instead of listing the corresponding
sequence $s$ for the automorphism, we mark $s$ onto the (affine) $E_{8}$
Dynkin diagram, illustrating the origin of the fixed Lie subalgebras.

####  3.3.1 Example: $\operatorname{\mathbb{Z}}_{2}$ Orbifold of $V_{E_{8}}$

Let $V:=V_{E_{8}}$ throughout. By Kac’s theorem and the discussion of Section
3.2, $V$ has two automorphisms of order $2$ up to conjugacy (see Table 2).
Call the non-anomalous one 2A and the anomalous one 2B, they are generated by

$\displaystyle x^{A}$ $\displaystyle:=\frac{1}{2}\omega_{7}$ (68)
$\displaystyle x^{B}$ $\displaystyle:=\frac{1}{2}\omega_{1}\,.$ (69)

Let us start with the 2A symmetry. In this case, our fixed sub-VOA is
$V^{\operatorname{\mathbb{Z}}_{2}^{A}}=V_{D_{8}}$. To see this at the level of
lattices, first note $(x^{A},\alpha_{i})=0$ for all $i$ except $i=7$, so the
only nontrivial action is $\Gamma_{\alpha_{7}}\mapsto-\Gamma_{\alpha_{7}}$.
Vertex operators like $\Gamma_{2\alpha_{7}}$ do remain invariant though, so
the invariant VOA is the lattice VOA constructed from the span of
$\\{\alpha_{1},\dots,\alpha_{6},2\alpha_{7},\alpha_{8}\\}$. The reason why
Kac’s theorem works is because of the form of the highest root
$\alpha_{\mathrm{high}}$ in Equation (60), which guarantees that the fixed
sublattice is also spanned by
$\\{\alpha_{1},\dots,\alpha_{6},-\alpha_{\mathrm{high}},\alpha_{8}\\}$. It is
then clear from the way these vectors fit inside of the affine Dynkin diagram
in Equation (61) that they span a $D_{8}$ lattice.

Since 2A is non-anomalous, the preceding discussion tells us that the fusion
group is a metric Abelian group
$\operatorname{\mathrm{Irr}}(V^{\operatorname{\mathbb{Z}}_{2}^{A}})=\operatorname{\mathbb{Z}}_{2}\times\operatorname{\mathbb{Z}}_{2}$
with two isotropic subgroups. To see this at the level of lattices, we note
that the $E_{8}$ lattice can be presented as the span of the rows of

$\begin{pmatrix}2&0&0&0&0&0&0&0\\\ -1&1&0&0&0&0&0&0\\\ 0&-1&1&0&0&0&0&0\\\
0&0&-1&1&0&0&0&0\\\ 0&0&0&-1&1&0&0&0\\\ 0&0&0&0&-1&1&0&0\\\
0&0&0&0&0&-1&\,\,1\,\,&0\\\
\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&\,\,\tfrac{1}{2}\,\,\end{pmatrix}\,,$
(70)

as in Chapter 4 of conway2013sphere  (these are not roots). Similarly, the
$D_{8}$ lattice and $D_{8}^{*}$ lattice can be written as the span of the rows
of

$\begin{pmatrix}-1&-1&0&0&0&0&0&0\\\ 1&-1&0&0&0&0&0&0\\\ 0&1&-1&0&0&0&0&0\\\
0&0&1&-1&0&0&0&0\\\ 0&0&0&1&-1&0&0&0\\\ 0&0&0&0&1&-1&0&0\\\
0&0&0&0&0&1&-1&0\\\
0&0&0&0&0&0&1&-1\end{pmatrix}\quad\textrm{and}\quad\begin{pmatrix}\,\,1\,\,&0&0&0&0&0&0&0\\\
0&\,\,1\,\,&0&0&0&0&0&0\\\ 0&0&\,\,1\,\,&0&0&0&0&0\\\
0&0&0&\,\,1\,\,&0&0&0&0\\\ 0&0&0&0&\,\,1\,\,&0&0&0\\\
0&0&0&0&0&\,\,1\,\,&0&0\\\ 0&0&0&0&0&0&\,\,1\,\,&0\\\
\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&\tfrac{1}{2}&\,\,\tfrac{1}{2}\,\,&\tfrac{1}{2}&\,\,\tfrac{1}{2}\,\,\end{pmatrix}\
$ (71)

respectively. Conveniently for us, conway2013sphere  also records the
following elements of $D_{8}^{*}$ which are representatives of separate
classes in $D_{8}^{*}/D_{8}$:

$\displaystyle[0]$ $\displaystyle:=(0,0,\dots,0)\,,$ (72) $\displaystyle[1]$
$\displaystyle:=(\tfrac{1}{2},\tfrac{1}{2},\dots,\tfrac{1}{2})\,,$ (73)
$\displaystyle[2]$ $\displaystyle:=(0,0,\dots,1)\,,$ (74) $\displaystyle[3]$
$\displaystyle:=(\tfrac{1}{2},\tfrac{1}{2},\dots,-\tfrac{1}{2})\,.$ (75)

From this we see that
$D_{8}^{*}/D_{8}\cong\operatorname{\mathbb{Z}}_{2}\times\operatorname{\mathbb{Z}}_{2}$
with decomposition and discriminant

$D_{8}^{*}=\begin{tabular}[]{|c|c|}\hline\cr$D_{8}+[0]$&$D_{8}+[1]$\\\
\hline\cr$D_{8}+[3]$&$D_{8}+[2]$\\\
\hline\cr\end{tabular}\quad\stackrel{{\scriptstyle\mathrm{Weight}}}{{\rightsquigarrow}}\quad
q^{A}(x)=\begin{tabular}[]{|c|c|}\hline\cr$0$&$0$\\\
\hline\cr$0$&$\tfrac{1}{2}$\\\ \hline\cr\end{tabular}\,.$ (76)

Note that $(D_{8}+[0])\cup(D_{8}+[1])$ and $(D_{8}+[0])\cup(D_{8}+[3])$ both
give $E_{8}$ lattices in $D_{8}^{*}$, and that they’re not the same (their
intersection is only a $D_{8}$ lattice).

The characters can be obtained as in Section 3.1.1 and are

$\displaystyle\tr_{V_{D_{8}+[0]}}q^{L_{0}-\frac{c}{24}}$
$\displaystyle=\frac{1}{2\eta(\tau)^{8}}\quantity(\theta_{3}^{8}(q)+\theta_{4}^{8}(q))\,,$
(77) $\displaystyle\tr_{V_{D_{8}+[1]}}q^{L_{0}-\frac{c}{24}}$
$\displaystyle=\frac{1}{2\eta(\tau)^{8}}\quantity(\theta_{2}^{8}(q)+\theta_{1}^{8}(q))\,,$
(78) $\displaystyle\tr_{V_{D_{8}+[2]}}q^{L_{0}-\frac{c}{24}}$
$\displaystyle=\frac{1}{2\eta(\tau)^{8}}\quantity(\theta_{3}^{8}(q)-\theta_{4}^{8}(q))\,,$
(79) $\displaystyle\tr_{V_{D_{8}+[3]}}q^{L_{0}-\frac{c}{24}}$
$\displaystyle=\frac{1}{2\eta(\tau)^{8}}\quantity(\theta_{2}^{8}(q)-\theta_{1}^{8}(q))\,.$
(80)

Since $\theta_{1}(\tau)=0$, we can clearly see how to decompose $V$ into
$V_{D_{8}}$ modules in two distinct ways just by comparing twisted partition
functions.

Looking at the 2B symmetry,
$V^{\operatorname{\mathbb{Z}}_{2}^{B}}=V_{A_{1}\times E_{7}}$ and has fusion
group $\operatorname{\mathbb{Z}}_{2}\times\operatorname{\mathbb{Z}}_{2}$ as
well. We may write the discriminant in this case and it’s

$q^{B}(x)=\begin{tabular}[]{|c|c|}\hline\cr$0$&$0$\\\
\hline\cr$\tfrac{3}{4}$&$\tfrac{1}{4}$\\\ \hline\cr\end{tabular}\,.$ (81)

As expected, the only isotropic subgroup corresponds to the decomposition we
started with initially.

It would be remiss not to point out that $q^{A}(x)$ and $q^{B}(x)$ are the
spins of simple anyons in the $\operatorname{\mathbb{Z}}_{2}$ gauge theory and
twisted $\operatorname{\mathbb{Z}}_{2}$ gauge theory (also called semion-
antisemion) respectively.

Generally, since $V$ is well-behaved,
$\operatorname{\mathrm{Rep}}(V^{\operatorname{\mathbb{Z}}_{m}})$ is a modular
tensor category (this follows rigorously from technical results of
miyamoto2004uniform ; miyamoto2015c_2 ; carnahan2016regularity ) whose simple
objects, the irreducible modules $\operatorname{\mathcal{H}}_{n_{m}}^{n_{e}}$,
correspond to simple anyons in a (2+1)d TFT which is interpretable as a
(possibly twisted) $\operatorname{\mathbb{Z}}_{m}$ gauge theory. If we couple
$V^{\operatorname{\mathbb{Z}}_{m}}$ to the bulk TFT as a 2d boundary theory,
then the anyon associated to $\operatorname{\mathcal{H}}_{n_{m}}^{n_{e}}$ may
terminate on operators from the corresponding module. We will elaborate on
these (2+1)d points further in Section 5 (see also gaiotto2020orbifold ).

###  3.4 Computing Defected Partition Functions in $(E_{8})_{1}$

In this section we use our general results about automorphisms of lattice VOAs
to explain how to compute $\operatorname{\mathbb{Z}}_{m}$ “duality defected”
partition functions in the chiral $(E_{8})_{1}$ WZW model, i.e. we compute
twisted characters for
$\operatorname{\mathrm{TY}}(\operatorname{\mathbb{Z}}_{m})$ actions on the
lattice VOA $V:=V_{E_{8}}$.

Suppose we want to find partition functions with a duality defect line $X$
twisting the Euclidean time direction, $Z_{V}[0,X]$, separating $V$ from some
$\operatorname{\mathbb{Z}}_{m}$ orbifold $[V/\operatorname{\mathbb{Z}}_{m}]$.
In practice, we proceed as follows for order $m$ symmetries:

  1. 1.

Following Kac’s theorem, we find all inner automorphisms of the $E_{8}$ Lie
algebra of order $m$. Using Equation (64), we compute which symmetries of $V$
are non-anomalous and thus can actually be gauged. Note: there may be multiple
conjugacy classes of non-anomalous symmetries e.g. for $m=5$ there are two
(see Table 3).

  2. 2.

Taking one of the non-anomalous symmetry groups
$G\cong\operatorname{\mathbb{Z}}_{m}$, the fixed-point sub-VOA $V^{G}$ is a
lattice VOA and is obtained as described in Section 3.3. The irreps
$A:=\operatorname{\mathrm{Irr}}(V^{G})$ form a metric Abelian group $(A,h)$,
with
$A\cong\operatorname{\mathbb{Z}}_{m}\times\hat{\operatorname{\mathbb{Z}}}_{m}$
and metric function $h(i,j)=ij/m$. $V$ and $[V/G]$ decompose as direct sums
over the isotropic subgroups with irrep labels $(i,0)$ and $(0,j)$
respectively, i.e.

$\displaystyle V$
$\displaystyle=\bigoplus_{i}\operatorname{\mathcal{H}}_{0}^{i}\,,$ (82)
$\displaystyle[V/G]$
$\displaystyle=\bigoplus_{j}\operatorname{\mathcal{H}}_{j}^{0}\,.$ (83)

  3. 3.

Finding automorphisms of $V^{G}$ which “swap the axes” corresponding to $V$
and $[V/G]$ in $A$, we obtain the defected partition functions as the twisted
characters for this second automorphism.

We’ve already seen the first two points in detail, so we elaborate on the
final point.

Write $O(A,h)$ for the group of automorphisms of $A$ which preserve $h$. For
simplicity, consider the case that $m$ is prime, then we have that

$O(A,h)=\operatorname{\mathbb{Z}}_{m}^{\times}\rtimes\operatorname{\mathbb{Z}}_{2}\,.$
(84)

The $\operatorname{\mathbb{Z}}_{m}^{\times}$ factor corresponds to the freedom
in redefining the action of $G$ on $V$ and is the normal subgroup $SO(A,h)$
which preserves the axes of $A$. This leaves
$\operatorname{\mathbb{Z}}_{2}=O(A,h)/SO(A,h)$ to act by swapping the axes of
$A$. The $\operatorname{\mathbb{Z}}_{2}$ factor acts by electric-magnetic
duality in the (2+1)d TFT. We can work this group out for more general $m$,
but generally $O(A,h)$ is just the group of symmetries of
$\operatorname{\mathbb{Z}}_{m}$ Dijkgraaf-Witten theory DW90  aka the group of
braided auto-equivalences of
$\operatorname{\mathcal{Z}}(\operatorname{\mathrm{Vec}}_{G})$ (see
gaiotto2020orbifold ; jaumeDiego  for physics discussions, and homotopyFusion
; FPSV:brauerGroupAbelDW ; nikshychRiepel:catLagGrass  for mathematics
discussions).

Now, any automorphism of $V^{G}$ induces an automorphism of $A$ which
preserves the conformal weight $h$. Note: many elements of
$\operatorname{Aut}(V^{G})$ induce the same automorphism in $O(A,h)$.

Our principal claim for this paper is that:
$(V^{G})^{\operatorname{\mathbb{Z}}_{2}}=V^{\operatorname{\mathrm{TY}}(G)}$ if
and only if the $\operatorname{\mathbb{Z}}_{2}$ action on $V^{G}$ switches the
axes of $A$. We prove this in Theorem 2. At the level of partition functions,
this means if $\hat{\sigma}\in\operatorname{Aut}(V^{G})$ induces a
$\operatorname{\mathbb{Z}}_{2}$ action on $A$ which “swaps the axes” of
$\operatorname{\mathrm{Irr}}(V^{G})$, then there is a
$\operatorname{\mathbb{Z}}_{m}$ Tambara-Yamagami line $X_{\hat{\sigma}}$ in
$V$, whose partition function is the $\hat{\sigma}$-twisted character of
$V^{G}$

$Z_{V}[0,X_{\hat{\sigma}}]=\tr_{V^{G}}\hat{\sigma}q^{L_{0}-\frac{c}{24}}\,.$
(85)

In practice, this is where the technical details about lifts becomes relevant.
We will identify $\operatorname{\mathbb{Z}}_{2}$ symmetries of the underlying
lattices associated to $V^{G}$ which should switch the axes, and then consider
their lifts to the VOA. This means even after we have picked a non-anomalous
$\operatorname{\mathbb{Z}}_{m}$ symmetry and a $\operatorname{\mathbb{Z}}_{2}$
which switches the axes, there is still degeneracy in the partition function
depending on how the $\operatorname{\mathbb{Z}}_{2}$ lifts to a VOA
automorphism, leading to different TY-lines.

Comparing to the end of Section 2.3, we see a 2d shadow of this equation. We
already saw that the duality defected partition function involves only
uncharged operators, and that we just had to resolve a sign in the partition
function which tracks “if $V$ and $[V/G]$ see the primary $\phi$ differently.”
This role of the $\hat{\sigma}$-twisted character is to track this sign.

###  3.5 $\operatorname{\mathbb{Z}}_{2}$ Duality Defects

Given $V:=V_{E_{8}}$, there is one non-anomalous
$\operatorname{\mathbb{Z}}_{2}$ symmetry up to conjugation, and there are 4
duality defected partition functions, i.e. $\operatorname{\mathbb{Z}}_{2}$
Tambara-Yamagami lines or duality defects. In this section we will compute
them in detail using our VOA-theoretic techniques and then compare to the
result using fermionization as in Thorngren:2018bhj ; Ji:2019ugf ; MonsterCFT
.

####  3.5.1 $\operatorname{\mathbb{Z}}_{2}$ Duality Defects from Lie Theory

We start by following our general procedure outlined in the previous section.
There is one non-anomalous $\operatorname{\mathbb{Z}}_{2}$ action on $V$,
generated by $x^{A}=\frac{1}{2}\omega^{7}$ in the sense of Equation (62),
fixing the subalgebra $V^{\operatorname{\mathbb{Z}}_{2}^{A}}\cong V_{D_{8}}$.
We found this by taking the affine $E_{8}$ Dynkin diagram and “chopping” the
7th root to produce the $D_{8}$ Dynkin diagram.

We know from our discriminant calculations in Section 3.3.1 that there are two
ways to extend the $D_{8}$ root lattice into the $E_{8}$ lattice. In fact, we
can see these two extensions directly from the Dynkin diagram

(86)

To obtain a TY-category, we want a $\operatorname{\mathbb{Z}}_{2}$ action on
$V^{\operatorname{\mathbb{Z}}_{2}}$ which swaps the axes of
$\mathrm{Irr}(V^{\operatorname{\mathbb{Z}}_{2}})$. At the level of root
lattices, this means swapping the two $E_{8}$’s in $D_{8}^{*}$, and the
$\operatorname{\mathbb{Z}}_{2}$ which does this is obvious: the $D_{8}$ Dynkin
diagram automorphism.

We start by looking at the conjugacy classes of
$\operatorname{\mathbb{Z}}_{2}$ outer automorphisms of $\mathfrak{so}(16)$,
which all come from the $D_{8}$ Dynkin diagram automorphism. Using Kac’s
theorem with $m=2$ and $k=2$, we need to study the Dynkin diagram
$D_{8}^{(2)}$. This is given (with its marks) by

$1$$1$$1$$1$$1$$1$$1$$1$$\alpha_{0}$$\alpha_{1}$$\alpha_{2}$$\alpha_{3}$$\alpha_{4}$$\alpha_{5}$$\alpha_{6}$$\alpha_{7}$
(87)

Kac’s theorem tells us that there are four different outer
$\operatorname{\mathbb{Z}}_{2}$ automorphisms, which are given by sequences
$(s_{0},\dots,s_{7})$, where all components vanish except $s_{i}=1$ with
$i\in\\{0,1,2,3\\}$.161616To be very explicit, if we think of
$\mathfrak{so}(n)$ as being $n\times n$ antisymmetric matrices, then these
four classes of outer automorphisms can be thought of as coming from the
adjoint actions of the four matrices $\mathrm{diag}(+1^{2i+1},-1^{15-2i})\in
O(16)$. The theorem also produces the fixed Lie subalgebras

$\mathfrak{so}(1)\oplus\mathfrak{so}(15)\,,\quad\mathfrak{so}(3)\oplus\mathfrak{so}(13)\,,\quad\mathfrak{so}(5)\oplus\mathfrak{so}(11)\,,\quad\mathfrak{so}(7)\oplus\mathfrak{so}(9)\,.$
(88)

The fixed Lie subalgebras tell us about the weight-one subspace of the four
different $(V_{D_{8}})^{\operatorname{\mathbb{Z}}_{2}}$s. Of course,
$\mathfrak{so}(1)$ doesn’t really exist, but the demand for its appearance in
the pattern becomes clear if we recall that $\mathfrak{so}(n)_{1}$ is the
theory of $n$ free fermions. The patterns above reflect that the four
different $(V_{D_{8}})^{\operatorname{\mathbb{Z}}_{2}}$s are just breaking
into theories of $(1,15)$, $(3,13)$, $(5,11)$, and $(7,9)$ left-moving free
fermions. We address this further in Section 3.5.3.

Moving on, let’s compute the appropriate
$\operatorname{\mathbb{Z}}_{2}$-twisted characters of $V_{D_{8}}$. By the
result of Hohn:2020xfe  they all take the form $\hat{\sigma}_{i}=e^{2\pi
i(x^{i},\,\cdot\,)}.\hat{g}$ and have twisted character formula given by
Equation (56). The first piece for the computation is the twisted eta product
$\eta_{g}(\tau)$. Using (49), the $D_{8}$ Dynkin diagram automorphism gives

$\eta_{g}(\tau)=\eta(\tau)^{6}\eta(2\tau)=\frac{\eta(\tau)^{8}}{\sqrt{\theta_{3}(\tau)\theta_{4}(\tau)}}.$
(89)

Moreover, the invariant root lattice $L^{g}$ can be read off from the $D_{8}$
Dynkin diagram

$\dots$ Invariant $\dots$ (90)

Note that the fixed root lattice is actually a $C_{7}$ lattice, not the
$B_{7}$ lattice that we might naively expect from looking at the fixed point
Lie subalgebras.

The Dynkin diagram automorphism does not experience order doubling, so we can
obtain the weights $e^{2\pi i(x^{i},\alpha)}$ by finding vectors in
$(1/2)({D_{8}}^{*})^{g}$ as described in Theorem 2.13 of Hohn:2020xfe . We
find four choices:

$\displaystyle\mathfrak{so}(1)\oplus\mathfrak{so}(15):\quad$ $\displaystyle
x^{0}=0\,,$ (91) $\displaystyle\mathfrak{so}(3)\oplus\mathfrak{so}(13):\quad$
$\displaystyle x^{1}=\frac{1}{2}\omega_{1}\,,$ (92)
$\displaystyle\mathfrak{so}(5)\oplus\mathfrak{so}(11):\quad$ $\displaystyle
x^{2}=\frac{1}{2}\omega_{2}\,,$ (93)
$\displaystyle\mathfrak{so}(7)\oplus\mathfrak{so}(9):\quad$ $\displaystyle
x^{3}=\frac{1}{2}\omega_{3}\,.$ (94)

We can then finally sum over the $C_{7}$ root lattice, with these weights
inserted, and combine with the twisted eta functions to obtain the following
four duality defects:

$\displaystyle\tr_{V_{D_{8}}}\hat{\sigma_{i}}\,q^{L_{0}-\frac{c}{24}}=\frac{\sqrt{\theta_{3}(\tau)\theta_{4}(\tau)}}{2\eta(\tau)^{8}}\quantity(\theta_{3}^{i}(\tau)\theta_{4}^{7-i}(\tau)+\theta_{3}^{7-i}(\tau)\theta_{4}^{i}(\tau))\,.$
(95)

In terms of $\mathfrak{so}(2r+1)_{1}$ characters

$\displaystyle\chi^{(r)}_{\hat{\omega}_{0}}(\tau)$
$\displaystyle=\frac{1}{2}\left(\frac{\theta_{3}(\tau)^{r+\frac{1}{2}}+\theta_{4}(\tau)^{r+\frac{1}{2}}}{\eta(\tau)^{r+\frac{1}{2}}}\right)\,,$
(96) $\displaystyle\chi^{(r)}_{\hat{\omega}_{1}}(\tau)$
$\displaystyle=\frac{1}{2}\left(\frac{\theta_{3}(\tau)^{r+\frac{1}{2}}-\theta_{4}(\tau)^{r+\frac{1}{2}}}{\eta(\tau)^{r+\frac{1}{2}}}\right)\,,$
(97) $\displaystyle\chi^{(r)}_{\hat{\omega}_{r}}(\tau)$
$\displaystyle=\frac{1}{\sqrt{2}}\frac{\theta_{2}(\tau)^{r+\frac{1}{2}}}{\eta(\tau)^{r+\frac{1}{2}}}\,,$
(98)

we have (up to normalization)

$Z_{V}[0,X_{i}]=\chi^{(i)}_{\hat{\omega}_{0}}(\tau)\chi^{(7-i)}_{\hat{\omega}_{0}}(\tau)-\chi^{(i)}_{\hat{\omega}_{1}}(\tau)\chi^{(7-i)}_{\hat{\omega}_{1}}(\tau)\,.$
(99)

####  3.5.2 Fermionization

Given a bosonic theory $T_{b}$ with a non-anomalous global
$\operatorname{\mathbb{Z}}_{2}$ symmetry, we may produce a fermionic theory
$T_{f}$ by turning the $\operatorname{\mathbb{Z}}_{2}$ symmetry into a
$\operatorname{\mathbb{Z}}_{2}^{f}$ “Grassmann parity” i.e. $(-1)^{F}$
symmetry, whose background connection is the spin structure on the manifold
(an affine $\operatorname{\mathbb{Z}}_{2}$ connection).171717We assume
unitarity so that spin and statistics are related. This is a generalization of
the classic “Jordan-Wigner transformation.” In practice, on a spacetime $M$,
this Jordan-Wigner transformation is given by

$Z_{T_{f}}[\rho]=\frac{1}{\sqrt{\absolutevalue{H^{1}(M,\operatorname{\mathbb{Z}}_{2})}}}\sum_{\alpha\in
H^{1}(M,\operatorname{\mathbb{Z}}_{2})}(-1)^{\operatorname{\textrm{Arf}}[\alpha+\rho]+\operatorname{\textrm{Arf}}[\rho]}Z_{T_{b}}[\alpha]\,,$
(100)

where $\operatorname{\textrm{Arf}}[\rho]=0$ for even spin structures and
$\operatorname{\textrm{Arf}}[\rho]=1$ for odd spin structures. In other words,
it is the number of zero modes of the Dirac operator mod 2
johnson:quadraticSpin ; atiyah:spinStruct ; seibergWitten:spinStructInString ;
karchTongTurner:webOf2d .

In modern language, the partition function
$(-1)^{\operatorname{\textrm{Arf}}[\rho]}$ is the generator of the group
$\operatorname{Hom}(\Omega_{2}^{\textrm{Spin}}(pt),U(1))=\operatorname{\mathbb{Z}}_{2}$
of invertible topological phases that can be stacked with a theory with
$(-1)^{F}$ symmetry fermionicSPTCobordism . It can also be thought of as the
effective action for the continuum version of the Majorana-Kitaev chain.
Practically, the effect of stacking with the $\operatorname{\textrm{Arf}}$
phase is to change the relative sign of the even and odd partition functions.

Conversely, given any fermionic theory $T_{f}$ (with $c_{L}-c_{R}\in
8\operatorname{\mathbb{Z}}$), it is always possible to gauge the $(-1)^{F}$
symmetry, i.e. sum over spin-structures, and produce a bosonic theory $T_{b}$
with a non-anomalous global $\operatorname{\mathbb{Z}}_{2}$ symmetry by the
inverse of the Jordan-Wigner transformation, or “GSO projection” (see
seibergWitten:spinStructInString ; worldsheetGSO ). This can be done in two
ways, by

$Z_{T_{b}^{A}}[\alpha]=\frac{1}{\sqrt{\absolutevalue{H^{1}(M,\operatorname{\mathbb{Z}}_{2})}}}\sum_{\rho}(-1)^{\operatorname{\textrm{Arf}}[\alpha+\rho]+\operatorname{\textrm{Arf}}[\rho]}Z_{T_{f}}[\rho]\,,$
(101)

or by first stacking with $\operatorname{\textrm{Arf}}$

$Z_{T_{b}^{B}}[\alpha]=\frac{1}{\sqrt{\absolutevalue{H^{1}(M,\operatorname{\mathbb{Z}}_{2})}}}\sum_{\rho}(-1)^{\operatorname{\textrm{Arf}}[\alpha+\rho]}Z_{T_{f}}[\rho]\,.$
(102)

The two bosonizations are related by gauging the emergent
$\operatorname{\mathbb{Z}}_{2}$ symmetry KramWan ; karchTongTurner:webOf2d ;
Thorngren:2018bhj , i.e.

$[\mathrm{GSO}[T_{f}]/\operatorname{\mathbb{Z}}_{2}]=\mathrm{GSO}[T_{f}\times\operatorname{\textrm{Arf}}]\,.$
(103)

For more examples of this procedure in action, see MonsterCFT ; Ji:2019ugf ;
fermionicMinimalModels ; moreMinimalModels ; lessonsFromRamond ; okuda2020u1 ;
Smith:2021luc .

When we gauge $(-1)^{F}$ we can ask what happens to the chiral fermion
parities $(-1)^{F_{L}}$ and $(-1)^{F_{R}}$. As explained in Thorngren:2018bhj
; MonsterCFT ; Ji:2019ugf  they become
$\operatorname{\mathrm{TY}}(\operatorname{\mathbb{Z}}_{2})$ duality defect
lines separating $T_{b}^{A}$ from $T_{b}^{B}$.

To make all this concrete, recall the partition function(s) for a chiral
(left-moving say) Majorana-Weyl fermion in it’s different sectors

$\displaystyle\tr_{\textrm{NS}}q^{L_{0}-\frac{c}{24}}$
$\displaystyle=\sqrt{\frac{\theta_{3}(\tau)}{\eta(\tau)}}\,,$ (104)
$\displaystyle\tr_{\textrm{R}}q^{L_{0}-\frac{c}{24}}$
$\displaystyle=\sqrt{\frac{\theta_{2}(\tau)}{\eta(\tau)}}\,,$ (105)
$\displaystyle\tr_{\textrm{NS}}(-1)^{F}q^{L_{0}-\frac{c}{24}}$
$\displaystyle=\sqrt{\frac{\theta_{4}(\tau)}{\eta(\tau)}}\,,$ (106)
$\displaystyle\tr_{\textrm{R}}(-1)^{F}q^{L_{0}-\frac{c}{24}}$
$\displaystyle=\sqrt{\frac{\theta_{1}(\tau)}{\eta(\tau)}}\,.$ (107)

A “full” Majorana fermion (with both left and right-moving contribution) has
the spin-structure dependent partition function

$Z_{\textrm{Full}}[\rho]=Z_{\textrm{Maj.}}[\rho]\bar{Z}_{\textrm{Maj.}}[\rho]\,.$
(108)

For example, we can GSO project the Majorana-Weyl fermion using the formulas
above, and in either case we get the Ising partition function

$Z_{T_{b}^{A,B}}[0,0]=\frac{1}{2}\left(\absolutevalue{\frac{\theta_{3}}{\eta}}+\absolutevalue{\frac{\theta_{2}}{\eta}}+\absolutevalue{\frac{\theta_{4}}{\eta}}\pm\absolutevalue{\frac{\theta_{1}}{\eta}}\right)\,.$
(109)

To recover the Ising duality defected partition function, we need to gauge the
diagonal spin-structure but first insert a line twisting by the chiral fermion
parity i.e. we have

$\sum_{\rho}Z_{\textrm{Maj.}}[\rho_{1},\rho_{2}+1]\bar{Z}_{\textrm{Maj.}}[\rho_{1},\rho_{2}]\propto\sqrt{2}\lvert\chi_{0}\rvert^{2}-\sqrt{2}\lvert\chi_{\frac{1}{2}}\rvert^{2}\,.$
(110)

More generally, all of this comes from the fact that $(-1)^{F_{L}}$ has one
unit of the “mod 8 anomaly” coming from
$\operatorname{Hom}(\Omega_{3}^{\textrm{Spin}}(B\operatorname{\mathbb{Z}}_{2}),U(1))=\operatorname{\mathbb{Z}}_{8}$.
For $n$ Majorana fermions the bosonization is the $\mathrm{Spin}(n)_{1}$ WZW
model, and the $(-1)^{F_{L}}$ symmetry line bosonizes to
$\operatorname{\mathrm{TY}}(\operatorname{\mathbb{Z}}_{2},1,+1/\sqrt{2})$ if
$n=1,7\,\,\mathrm{mod}\,8$ and
$\operatorname{\mathrm{TY}}(\operatorname{\mathbb{Z}}_{2},1,-1/\sqrt{2})$ if
$n=3,5\,\,\mathrm{mod}\,8$. See Thorngren:2018bhj ; Ji:2019ugf ; Qi_2013 ;
Ryu_2012  and references within for more technical details.

####  3.5.3 Duality Defects from Fermionization

From Section 3.1.1 we see that chiral theory $V:=V_{E_{8}}$ is just 16 chiral
Majorana-Weyl fermions. From the discussion in the previous section, we see
that we have $4$ choices of duality defect based on if we’re going to view the
$E_{8}$ theory as coming from the GSO projection of

$Z_{\textrm{Maj.}}^{1}[\rho]Z_{\textrm{Maj.}}^{15}[\rho]\,,\quad
Z_{\textrm{Maj.}}^{3}[\rho]Z_{\textrm{Maj.}}^{13}[\rho]\,,\quad
Z_{\textrm{Maj.}}^{5}[\rho]Z_{\textrm{Maj.}}^{11}[\rho]\,,\quad
Z_{\textrm{Maj.}}^{7}[\rho]Z_{\textrm{Maj.}}^{9}[\rho]\,.$ (111)

Let $X_{p}$ ($p=1,3,5,7$) be the duality defect obtained from bosonizing
$(-1)^{F_{L}}$ in the
$Z_{\textrm{Maj.}}^{p}[\rho]Z_{\textrm{Maj.}}^{16-p}[\rho]$ setup. Then the
partition function with $X_{p}$ inserted along the spatial $S^{1}$ is

$\displaystyle Z_{V}[0,X_{p}]$
$\displaystyle\propto\frac{1}{2}\sum_{\rho}Z_{\textrm{Maj.}}^{p}[\rho_{1},\rho_{2}+1]Z_{\textrm{Maj.}}^{16-p}[\rho_{1},\rho_{2}]\,,$
(112)
$\displaystyle=\frac{(\theta_{3}(\tau)\theta_{4}(\tau))^{\frac{p}{2}}}{2\eta(\tau)^{8}}\left(\theta_{3}(\tau)^{8-p}+\theta_{4}(\tau)^{8-p}\right)\,.$
(113)

This precisely matches the results of Equation (95), where no intimate
knowledge of superfusion categories, bosonization, or fermionic anomalies were
required.

##  4 Higher order duality defects

In this section, we investigate duality defects for higher order cyclic
symmetries. Along the way we highlight a number of potential phenomena that
one might encounter in enacting our defect hunting procedure described in
Section 3.4, such as order doubling and $\mathfrak{u}(1)$ factors. We compute
the $q$-expansions of the defect partition functions using Magma Magma .

###  4.1 $\operatorname{\mathbb{Z}}_{3}$ Duality Defects

As explained in Section 3.2, there is only one non-anomalous
$\operatorname{\mathbb{Z}}_{3}$ symmetry up to conjugation. We will see that
this symmetry gives rise to seven $\operatorname{\mathbb{Z}}_{3}$ Tambara-
Yamagami lines, which we will investigate using both numerical and analytical
methods.

In Section 3.3 we found that the non-anomalous $\operatorname{\mathbb{Z}}_{3}$
symmetry is given by $x=\frac{1}{3}\omega_{2}$. Following our procedure, we
obtain the fixed sub-VOA by “chopping” the second root of the affine $E_{8}$
Dynkin diagram

(114)

Therefore, we find that $(V_{E_{8}})^{\operatorname{\mathbb{Z}}_{3}}\cong
V_{A_{2}\times E_{6}}$.

Before starting the analysis of the Tambara-Yamagami lines, let’s study
$V_{A_{2}\times E_{6}}$ more carefully and verify that we can extend it into
two different versions of $V_{E_{8}}$. For this, recall the following theta
functions for $A_{2}$ and $E_{6}$ (for details see conway2013sphere ):

$\displaystyle\theta_{A_{2}}(\tau)$
$\displaystyle=\theta_{2}(2\tau)\theta_{2}(6\tau)+\theta_{3}(2\tau)\theta_{3}(6\tau)\,,$
(115) $\displaystyle\theta_{E_{6}}(\tau)$
$\displaystyle=\frac{1}{4}f(\tau)^{3}+\theta_{A_{2}}(\tau)^{3}\,,$ (116)

where

$f(\tau)=\theta_{A_{2}}\left(\frac{\tau}{3}\right)-\theta_{A_{2}}(\tau)\,.$
(117)

The $A_{2}$ and $E_{6}$ lattices have three cosets inside their respective
duals. These cosets are parametrized by shift vectors which we will call $[1]$
and $[2]$ in both cases. This gives two shifted theta functions for each
lattice given by

$\displaystyle\theta_{A_{2}+[1]}(\tau)=\theta_{A_{2}+[2]}(\tau)$
$\displaystyle=\frac{1}{2}f(\tau),$ (118)
$\displaystyle\theta_{E_{6}+[1]}(\tau)=\theta_{E_{6}+[2]}(\tau)$
$\displaystyle=\frac{3}{4}f(\tau)^{2}\,\theta_{A_{2}}(\tau).$ (119)

We organize the cosets of $A_{2}\times E_{6}$ inside its dual as a $3\times 3$
square, where the first column and the top row give two different extensions
to an $E_{8}$ lattice inside the dual of the $A_{2}\times E_{6}$ lattice:

$(A_{2}\times E_{6})^{*}=\begin{tabular}[]{|c|c|c|}\hline\cr$A_{2}\oplus
E_{6}$&$A_{2}^{[1]}\oplus E_{6}^{[2]}$&$A_{2}^{[2]}\oplus E_{6}^{[1]}$\\\
\hline\cr$A_{2}^{[1]}\oplus E_{6}^{[1]}$&$A_{2}^{[2]}\oplus
E_{6}$&$A_{2}\oplus E_{6}^{[2]}$\\\ \hline\cr$A_{2}^{[2]}\oplus
E_{6}^{[2]}$&$A_{2}\oplus E_{6}^{[1]}$&$A_{2}^{[1]}\oplus E_{6}$\\\
\hline\cr\end{tabular}\quad\stackrel{{\scriptstyle\mathrm{Weight}}}{{\rightsquigarrow}}\quad
q^{A}(x)=\begin{tabular}[]{|c|c|c|}\hline\cr$0$&$0$&$0$\\\
\hline\cr$0$&$\tfrac{1}{3}$&$\tfrac{2}{3}$\\\
\hline\cr$0$&$\tfrac{2}{3}$&$\tfrac{1}{3}$\\\ \hline\cr\end{tabular}\,.$ (120)

This shows explicitly that this order three symmetry is non-anomalous. Summing
the relevant theta functions we obtain

$\theta_{E_{8}}(\tau)=\theta_{A_{2}}(\tau)\left(f(\tau)^{3}+\theta_{A_{2}}(\tau)^{3}\right).$
(121)

Now we turn towards the duality defects of this
$\operatorname{\mathbb{Z}}_{3}$-symmetry. For this purpose, we need to study
order two symmetries of the $A_{2}\times E_{6}$ lattice and their possible
lifts to the corresponding VOA. After that, we investigate which of the
$\operatorname{\mathbb{Z}}_{2}$ symmetries switch the two $E_{8}$ extensions
and swap the axes of
$\operatorname{\mathrm{Irr}}(V^{\operatorname{\mathbb{Z}}_{3}})$.

We actually start by looking at the automorphisms of the relevant Lie
algebras. The $A_{2}$ Lie algebra has two inner automorphisms of order $\leq
2$ (up to conjugation) and one outer automorphism of order 2. To see this,
note the affine and twisted Dynkin diagrams are

$A_{2}^{(1)}\,=\leavevmode\hbox to57.88pt{\vbox
to42.68pt{\pgfpicture\makeatletter\hbox{\hskip 11.86601pt\lower-6.55522pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} ; ;
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{2.82861pt}{0.0pt}\pgfsys@curveto{2.82861pt}{1.56221pt}{1.56221pt}{2.82861pt}{0.0pt}{2.82861pt}\pgfsys@curveto{-1.56221pt}{2.82861pt}{-2.82861pt}{1.56221pt}{-2.82861pt}{0.0pt}\pgfsys@curveto{-2.82861pt}{-1.56221pt}{-1.56221pt}{-2.82861pt}{0.0pt}{-2.82861pt}\pgfsys@curveto{1.56221pt}{-2.82861pt}{2.82861pt}{-1.56221pt}{2.82861pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{0.0pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{19.90044pt}{29.5681pt}\pgfsys@curveto{19.90044pt}{31.13031pt}{18.63403pt}{32.39671pt}{17.07182pt}{32.39671pt}\pgfsys@curveto{15.50961pt}{32.39671pt}{14.24321pt}{31.13031pt}{14.24321pt}{29.5681pt}\pgfsys@curveto{14.24321pt}{28.00589pt}{15.50961pt}{26.73949pt}{17.07182pt}{26.73949pt}\pgfsys@curveto{18.63403pt}{26.73949pt}{19.90044pt}{28.00589pt}{19.90044pt}{29.5681pt}\pgfsys@closepath\pgfsys@moveto{17.07182pt}{29.5681pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{17.07182pt}{29.5681pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{36.97183pt}{0.0pt}\pgfsys@curveto{36.97183pt}{1.56221pt}{35.70543pt}{2.82861pt}{34.14322pt}{2.82861pt}\pgfsys@curveto{32.58101pt}{2.82861pt}{31.3146pt}{1.56221pt}{31.3146pt}{0.0pt}\pgfsys@curveto{31.3146pt}{-1.56221pt}{32.58101pt}{-2.82861pt}{34.14322pt}{-2.82861pt}\pgfsys@curveto{35.70543pt}{-2.82861pt}{36.97183pt}{-1.56221pt}{36.97183pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{34.14322pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{34.14322pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{1.51434pt}{2.6228pt}\pgfsys@lineto{15.55748pt}{26.9453pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.02861pt}{0.0pt}\pgfsys@lineto{31.11461pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{32.62892pt}{2.6228pt}\pgfsys@lineto{18.58617pt}{26.9453pt}\pgfsys@stroke\pgfsys@invoke{
} \par{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-8.533pt}{-3.22221pt}\pgfsys@invoke{
}\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{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{20.60483pt}{26.34589pt}\pgfsys@invoke{
}\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{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{37.67622pt}{-3.22221pt}\pgfsys@invoke{
}\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{{$1$}}
}}\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}}\qquad
A_{2}^{(2)}\,=\leavevmode\hbox to45.81pt{\vbox
to20.11pt{\pgfpicture\makeatletter\hbox{\hskip-199.02676pt\lower-13.31044pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} ; ;
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{207.68839pt}{0.0pt}\pgfsys@curveto{207.68839pt}{1.56221pt}{206.42198pt}{2.82861pt}{204.85977pt}{2.82861pt}\pgfsys@curveto{203.29756pt}{2.82861pt}{202.03116pt}{1.56221pt}{202.03116pt}{0.0pt}\pgfsys@curveto{202.03116pt}{-1.56221pt}{203.29756pt}{-2.82861pt}{204.85977pt}{-2.82861pt}\pgfsys@curveto{206.42198pt}{-2.82861pt}{207.68839pt}{-1.56221pt}{207.68839pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{204.85977pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{204.85977pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{241.8316pt}{0.0pt}\pgfsys@curveto{241.8316pt}{1.56221pt}{240.5652pt}{2.82861pt}{239.00299pt}{2.82861pt}\pgfsys@curveto{237.44078pt}{2.82861pt}{236.17438pt}{1.56221pt}{236.17438pt}{0.0pt}\pgfsys@curveto{236.17438pt}{-1.56221pt}{237.44078pt}{-2.82861pt}{239.00299pt}{-2.82861pt}\pgfsys@curveto{240.5652pt}{-2.82861pt}{241.8316pt}{-1.56221pt}{241.8316pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{239.00299pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{239.00299pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{5.0679pt}\pgfsys@invoke{
}{}\pgfsys@moveto{207.88838pt}{0.0pt}\pgfsys@lineto{235.97438pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{4.26791pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{207.88838pt}{0.0pt}\pgfsys@lineto{235.97438pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.22263pt}\pgfsys@invoke{
}{}\pgfsys@moveto{207.88838pt}{0.0pt}\pgfsys@lineto{235.97438pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{207.88838pt}{0.0pt}\pgfsys@lineto{235.97438pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}} {{}}
{{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{220.23407pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}
{{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope} \par{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{202.35977pt}{-9.97743pt}\pgfsys@invoke{
}\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{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{236.50299pt}{-9.97743pt}\pgfsys@invoke{
}\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{{$1$}}
}}\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}}$ (122)

The inner automorphisms can be read off from the affine Dynkin diagram. Kac’s
Theorem tells us we can either chop one node with weight $s_{i}=2$, or we can
chop two nodes with weights $s_{i}=s_{j}=1$; by the symmetry of the affine
Dynkin diagram, it doesn’t matter which we chop. In the first case we get the
trivial automorphism, and in the second case we get an inner automorphism of
order $2$ fixing an $A_{1}\times\mathfrak{u}(1)$ Lie algebra. There is only
one outer automorphism which comes from chopping the node marked $1$ with
weight $1$, fixing an $A_{1}$ Lie algebra. Standard lifts of this automorphism
experience order doubling (as we will explain in Section 4.1.1).

The $E_{6}$ case is similar, we have

$E_{6}^{(1)}\,=\leavevmode\hbox to148.24pt{\vbox
to88.15pt{\pgfpicture\makeatletter\hbox{\hskip-130.74031pt\lower-13.31044pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} ; ;
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{139.40193pt}{0.0pt}\pgfsys@curveto{139.40193pt}{1.56221pt}{138.13553pt}{2.82861pt}{136.57332pt}{2.82861pt}\pgfsys@curveto{135.01111pt}{2.82861pt}{133.7447pt}{1.56221pt}{133.7447pt}{0.0pt}\pgfsys@curveto{133.7447pt}{-1.56221pt}{135.01111pt}{-2.82861pt}{136.57332pt}{-2.82861pt}\pgfsys@curveto{138.13553pt}{-2.82861pt}{139.40193pt}{-1.56221pt}{139.40193pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{136.57332pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{136.57332pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{173.54517pt}{0.0pt}\pgfsys@curveto{173.54517pt}{1.56221pt}{172.27876pt}{2.82861pt}{170.71655pt}{2.82861pt}\pgfsys@curveto{169.15434pt}{2.82861pt}{167.88794pt}{1.56221pt}{167.88794pt}{0.0pt}\pgfsys@curveto{167.88794pt}{-1.56221pt}{169.15434pt}{-2.82861pt}{170.71655pt}{-2.82861pt}\pgfsys@curveto{172.27876pt}{-2.82861pt}{173.54517pt}{-1.56221pt}{173.54517pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{170.71655pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{170.71655pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{207.68839pt}{0.0pt}\pgfsys@curveto{207.68839pt}{1.56221pt}{206.42198pt}{2.82861pt}{204.85977pt}{2.82861pt}\pgfsys@curveto{203.29756pt}{2.82861pt}{202.03116pt}{1.56221pt}{202.03116pt}{0.0pt}\pgfsys@curveto{202.03116pt}{-1.56221pt}{203.29756pt}{-2.82861pt}{204.85977pt}{-2.82861pt}\pgfsys@curveto{206.42198pt}{-2.82861pt}{207.68839pt}{-1.56221pt}{207.68839pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{204.85977pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{204.85977pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{241.8316pt}{0.0pt}\pgfsys@curveto{241.8316pt}{1.56221pt}{240.5652pt}{2.82861pt}{239.00299pt}{2.82861pt}\pgfsys@curveto{237.44078pt}{2.82861pt}{236.17438pt}{1.56221pt}{236.17438pt}{0.0pt}\pgfsys@curveto{236.17438pt}{-1.56221pt}{237.44078pt}{-2.82861pt}{239.00299pt}{-2.82861pt}\pgfsys@curveto{240.5652pt}{-2.82861pt}{241.8316pt}{-1.56221pt}{241.8316pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{239.00299pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{239.00299pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{275.97527pt}{0.0pt}\pgfsys@curveto{275.97527pt}{1.56221pt}{274.70886pt}{2.82861pt}{273.14665pt}{2.82861pt}\pgfsys@curveto{271.58444pt}{2.82861pt}{270.31804pt}{1.56221pt}{270.31804pt}{0.0pt}\pgfsys@curveto{270.31804pt}{-1.56221pt}{271.58444pt}{-2.82861pt}{273.14665pt}{-2.82861pt}\pgfsys@curveto{274.70886pt}{-2.82861pt}{275.97527pt}{-1.56221pt}{275.97527pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{273.14665pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{273.14665pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{207.68839pt}{34.14322pt}\pgfsys@curveto{207.68839pt}{35.70543pt}{206.42198pt}{36.97183pt}{204.85977pt}{36.97183pt}\pgfsys@curveto{203.29756pt}{36.97183pt}{202.03116pt}{35.70543pt}{202.03116pt}{34.14322pt}\pgfsys@curveto{202.03116pt}{32.58101pt}{203.29756pt}{31.3146pt}{204.85977pt}{31.3146pt}\pgfsys@curveto{206.42198pt}{31.3146pt}{207.68839pt}{32.58101pt}{207.68839pt}{34.14322pt}\pgfsys@closepath\pgfsys@moveto{204.85977pt}{34.14322pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{204.85977pt}{34.14322pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{207.68839pt}{68.28644pt}\pgfsys@curveto{207.68839pt}{69.84865pt}{206.42198pt}{71.11505pt}{204.85977pt}{71.11505pt}\pgfsys@curveto{203.29756pt}{71.11505pt}{202.03116pt}{69.84865pt}{202.03116pt}{68.28644pt}\pgfsys@curveto{202.03116pt}{66.72423pt}{203.29756pt}{65.45782pt}{204.85977pt}{65.45782pt}\pgfsys@curveto{206.42198pt}{65.45782pt}{207.68839pt}{66.72423pt}{207.68839pt}{68.28644pt}\pgfsys@closepath\pgfsys@moveto{204.85977pt}{68.28644pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{204.85977pt}{68.28644pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{139.60193pt}{0.0pt}\pgfsys@lineto{167.68794pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{173.74516pt}{0.0pt}\pgfsys@lineto{201.83116pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{207.88838pt}{0.0pt}\pgfsys@lineto{235.97438pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{242.0316pt}{0.0pt}\pgfsys@lineto{270.11804pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{204.85977pt}{31.11461pt}\pgfsys@lineto{204.85977pt}{3.02861pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{204.85977pt}{37.17183pt}\pgfsys@lineto{204.85977pt}{65.25783pt}\pgfsys@stroke\pgfsys@invoke{
} \par{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{134.07332pt}{-9.97743pt}\pgfsys@invoke{
}\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{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{168.21655pt}{-9.97743pt}\pgfsys@invoke{
}\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{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{202.35977pt}{-9.97743pt}\pgfsys@invoke{
}\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{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{236.50299pt}{-9.97743pt}\pgfsys@invoke{
}\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{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{270.64665pt}{-9.97743pt}\pgfsys@invoke{
}\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{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{208.39278pt}{30.921pt}\pgfsys@invoke{
}\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{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{208.39278pt}{65.06422pt}\pgfsys@invoke{
}\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{{$1$}}
}}\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}}\quad
E_{6}^{(2)}\,=\leavevmode\hbox to148.24pt{\vbox
to20.11pt{\pgfpicture\makeatletter\hbox{\hskip-130.74031pt\lower-13.31044pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} ; ;
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{139.40193pt}{0.0pt}\pgfsys@curveto{139.40193pt}{1.56221pt}{138.13553pt}{2.82861pt}{136.57332pt}{2.82861pt}\pgfsys@curveto{135.01111pt}{2.82861pt}{133.7447pt}{1.56221pt}{133.7447pt}{0.0pt}\pgfsys@curveto{133.7447pt}{-1.56221pt}{135.01111pt}{-2.82861pt}{136.57332pt}{-2.82861pt}\pgfsys@curveto{138.13553pt}{-2.82861pt}{139.40193pt}{-1.56221pt}{139.40193pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{136.57332pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{136.57332pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{173.54517pt}{0.0pt}\pgfsys@curveto{173.54517pt}{1.56221pt}{172.27876pt}{2.82861pt}{170.71655pt}{2.82861pt}\pgfsys@curveto{169.15434pt}{2.82861pt}{167.88794pt}{1.56221pt}{167.88794pt}{0.0pt}\pgfsys@curveto{167.88794pt}{-1.56221pt}{169.15434pt}{-2.82861pt}{170.71655pt}{-2.82861pt}\pgfsys@curveto{172.27876pt}{-2.82861pt}{173.54517pt}{-1.56221pt}{173.54517pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{170.71655pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{170.71655pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{207.68839pt}{0.0pt}\pgfsys@curveto{207.68839pt}{1.56221pt}{206.42198pt}{2.82861pt}{204.85977pt}{2.82861pt}\pgfsys@curveto{203.29756pt}{2.82861pt}{202.03116pt}{1.56221pt}{202.03116pt}{0.0pt}\pgfsys@curveto{202.03116pt}{-1.56221pt}{203.29756pt}{-2.82861pt}{204.85977pt}{-2.82861pt}\pgfsys@curveto{206.42198pt}{-2.82861pt}{207.68839pt}{-1.56221pt}{207.68839pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{204.85977pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{204.85977pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{241.8316pt}{0.0pt}\pgfsys@curveto{241.8316pt}{1.56221pt}{240.5652pt}{2.82861pt}{239.00299pt}{2.82861pt}\pgfsys@curveto{237.44078pt}{2.82861pt}{236.17438pt}{1.56221pt}{236.17438pt}{0.0pt}\pgfsys@curveto{236.17438pt}{-1.56221pt}{237.44078pt}{-2.82861pt}{239.00299pt}{-2.82861pt}\pgfsys@curveto{240.5652pt}{-2.82861pt}{241.8316pt}{-1.56221pt}{241.8316pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{239.00299pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{239.00299pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\pgfsys@moveto{275.97527pt}{0.0pt}\pgfsys@curveto{275.97527pt}{1.56221pt}{274.70886pt}{2.82861pt}{273.14665pt}{2.82861pt}\pgfsys@curveto{271.58444pt}{2.82861pt}{270.31804pt}{1.56221pt}{270.31804pt}{0.0pt}\pgfsys@curveto{270.31804pt}{-1.56221pt}{271.58444pt}{-2.82861pt}{273.14665pt}{-2.82861pt}\pgfsys@curveto{274.70886pt}{-2.82861pt}{275.97527pt}{-1.56221pt}{275.97527pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{273.14665pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{273.14665pt}{0.0pt}\pgfsys@invoke{
}\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{}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{139.60193pt}{0.0pt}\pgfsys@lineto{167.68794pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{173.74516pt}{0.0pt}\pgfsys@lineto{201.83116pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{2.22263pt}\pgfsys@invoke{
}{}\pgfsys@moveto{207.88838pt}{0.0pt}\pgfsys@lineto{235.97438pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}\pgfsys@beginscope\pgfsys@invoke{
}{\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@invoke{
}\pgfsys@moveto{207.88838pt}{0.0pt}\pgfsys@lineto{235.97438pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{\pgfsys@beginscope\pgfsys@invoke{ }
{}{{}{}}{}{}{}{{}}{{}}{{}{}}{{}{}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}}
}}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}\pgfsys@beginscope\pgfsys@invoke{
} {\pgfsys@beginscope\pgfsys@invoke{ }{{}}{{}} {{}}
{{}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{220.23407pt}{0.0pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}}{{{{}{}{{}} }}{{}}
{{}} }{{}{}}{{}{}}{{{{}{}{{}} }}{{}} {{}} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{242.0316pt}{0.0pt}\pgfsys@lineto{270.11804pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \par\par{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{134.07332pt}{-9.97743pt}\pgfsys@invoke{
}\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{{$1$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{168.21655pt}{-9.97743pt}\pgfsys@invoke{
}\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{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{202.35977pt}{-9.97743pt}\pgfsys@invoke{
}\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{{$3$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{236.50299pt}{-9.97743pt}\pgfsys@invoke{
}\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{{$2$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}{}{{}}{}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{270.64665pt}{-9.97743pt}\pgfsys@invoke{
}\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{{$1$}}
}}\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}}$ (123)

We find 3 inner automorphisms of the $E_{6}$ Lie algebra up to conjugation,
given by: chopping a node marked $1$ with weight $2$ (the trivial
automorphism); chopping a node marked $2$ with weight $1$, fixing $A_{1}\times
A_{5}$; or chopping two nodes marked $1$ with weight $1$, fixing
$D_{5}\times\mathfrak{u}(1)$. We find $2$ outer automorphisms of $E_{6}$,
fixing an $F_{4}$ or a $C_{4}$ Lie algebra.

Putting everything together, we have 15 automorphisms of the $A_{2}\times
E_{6}$ Lie algebra of order less than or equal to 2. These automorphisms can
be organised as in Table 4.

|  $E_{6}$ inner (3) |  $E_{6}$ outer (2)
---|---|---
$A_{2}$ inner (2) | 6 | 4
$A_{2}$ outer (1) | 3 | 2
Table 4: There are 15 automorphisms of the $A_{2}\times E_{6}$ Lie algebra of
order less than or equal to 2. The row and column labels denote the origin of
the $A_{2}\times E_{6}$ automorphisms. For example, there are $4$
$\operatorname{\mathbb{Z}}_{2}$ actions on $A_{2}\times E_{6}$ which come from
one of the (2) inner actions on $A_{2}$ times one of the (2) outer actions on
$E_{6}$.

Similar to the $D_{8}$ case, we can read off automorphisms which switch the
axes of $\operatorname{\mathrm{Irr}}(V^{\operatorname{\mathbb{Z}}_{3}})$ by
looking at the chopped $E_{8}$ Dynkin diagram (114). We see that there are two
ways to recombine the $A_{2}$ and $E_{6}$ into an (affine) $E_{8}$: by gluing
the chopped node back in, or by acting with one of (the lifts of) the Dynkin
diagram automorphisms of $A_{2}$ or $E_{6}$, but not both, and then gluing.
From the off-diagonals of Table 4, this gives a total of seven automorphisms
switching the axes and hence seven Tambara-Yamagami lines. However, before
computing the defect partition functions, we need to look at order-doubling in
the $A_{2}$ lattice.

####  4.1.1 Example: Order Doubling in $A_{2}$

Before moving forward, let’s consider a standard lift of the one non-trivial
outer automorphism of $A_{2}$, as it is a prototypical example of order-
doubling.

So let $g$ be the automorphism of the $A_{2}$ lattice that comes from the
Dynkin diagram automorphism; switching the two simple roots $\alpha_{1}$ and
$\alpha_{2}$. As described in Section 3.2, $g$ can be lifted to a symmetry
$\hat{g}$ of the lattice VOA in a “standard way”: swapping the chiral bosons
$a^{1}_{n}\leftrightarrow a^{2}_{n}$, and sending $\Gamma_{\alpha}\mapsto
u(\alpha)\Gamma_{\alpha}$, with $u(\alpha)=1$ on the fixed sublattice. In this
case, the fixed sublattice is a “long $A_{1}$” spanned by
$\alpha_{1}+\alpha_{2}$

Invariant (124)

Clearly $\hat{g}^{2}$ is the identity on the chiral bosons, but

$\hat{g}^{2}\Gamma_{\alpha}=u(\alpha)u(g\alpha)\Gamma_{\alpha}\,.$ (125)

Using Equation (44) we have

$u(\alpha)u(g\alpha)=\frac{\epsilon(\alpha,g\alpha)}{\epsilon(g\alpha,g^{2}\alpha)}u(\alpha+g\alpha)=(-1)^{(g\alpha,\alpha)}u(\alpha+g\alpha)\,.$
(126)

Thus the order of $\hat{g}$ is doubled if and only if $(g\alpha,\alpha)\notin
2\operatorname{\mathbb{Z}}$.

Better yet, we explicitly see the origin of order doubling here: we get order
doubling because the Dynkin diagram automorphism folds together two simple
roots which were not originally orthogonal. Hence, just by looking at Dynkin
diagrams, we can tell which automorphisms will have order doubling.

At the end of Section 3.2 we saw all outer order $2$ automorphisms of
$V_{A_{2}}$ can be viewed as a product of a (fixed) standard lift of this
Dynkin diagram automorphism, and a twist by an “inner” automorphism (see
Equation (55)). Now we see the importance of allowing twists by inner
automorphisms of order up to $4$. If we had not, we might falsely conclude
that “there are no outer automorphisms of $V_{A_{2}}$ of order 2,” which would
directly contradict the results of dong1999automorphism  and the fact known to
physicists that the lattice automorphism $\alpha\mapsto-\alpha$ always lifts
to a “charge conjugation automorphism” of order $2$ in the VOA (note:
$\alpha\mapsto-\alpha$ is not in the Weyl group of the $A_{2}$ lattice).

If instead we choose $u$ so that $u(\alpha_{1}+\alpha_{2})=-1$ the lift has
order two, and we can compute the appropriate twisted theta function. The
fixed sublattice is a long $A_{1}$ lattice with theta function
$\theta_{3}(2\tau)$, but, due to the non-trivial lift, we adjust some signs
and obtain $\theta_{4}(2\tau)$.

####  4.1.2 $\operatorname{\mathbb{Z}}_{3}$ Duality Defects from Lie Theory

The partition functions corresponding to the seven Tambara-Yamagami lines can
be computed using similar techniques to the order two case. A careful study of
the lattices and partition functions gives the following list of theta
functions

$\displaystyle\theta_{A_{2},\text{inner}}(\tau)$
$\displaystyle=\theta_{3}(2\tau)\theta_{3}(6\tau)-\theta_{2}(2\tau)\theta_{2}(6\tau)$
(127) $\displaystyle\theta_{A_{2},\text{outer}}(\tau)$
$\displaystyle=\theta_{4}(2\tau)$ (128)
$\displaystyle\theta_{E_{6},\text{inner 1}}(\tau)$
$\displaystyle=2\theta_{A_{5}}(\tau)\theta_{3}(2\tau)-\theta_{E_{6}}(\tau)$
(129) $\displaystyle\ \theta_{E_{6},\text{inner 2}}(\tau)$
$\displaystyle=1+8q-50q^{2}+80q^{3}-88q^{4}+\dots$ (130)
$\displaystyle\theta_{E_{6},\text{outer 1}}(\tau)$
$\displaystyle=\tfrac{1}{2}(\theta_{3}^{4}(\tau)+\theta_{4}^{4}(\tau))$ (131)
$\displaystyle\theta_{E_{6},\text{outer 2}}(\tau)$
$\displaystyle=\theta_{4}^{4}(2\tau)\,.$ (132)

The missing theta function corresponding to an inner $E_{6}$ symmetry is
particularly messy, so we display the first few terms of its $q$-expansion.
The twisted eta products are easily computed being $\eta(\tau)^{6}\eta(2\tau)$
for the outer $A_{2}$ symmetries and $\eta(\tau)^{4}\eta(2\tau)^{2}$ for the
outer $E_{6}$ symmetries. In summary, we have the following seven defected
partition functions:

$\displaystyle Z_{V}[0,X_{A_{2}}^{(1)}]$
$\displaystyle=\frac{\theta_{A_{2},\text{outer}}(\tau)\theta_{E_{6}}(\tau)}{\eta(\tau)^{6}\eta(2\tau)}\,,$
(133) $\displaystyle Z_{V}[0,X_{A_{2}}^{(2)}]$
$\displaystyle=\frac{\theta_{A_{2},\text{outer}}(\tau)\theta_{E_{6},\text{inner
1}}(\tau)}{\eta(\tau)^{6}\eta(2\tau)}\,,$ (134) $\displaystyle
Z_{V}[0,X_{A_{2}}^{(3)}]$
$\displaystyle=\frac{\theta_{A_{2},\text{outer}}(\tau)\theta_{E_{6},\text{inner
2}}(\tau)}{\eta(\tau)^{6}\eta(2\tau)}\,,$ (135) $\displaystyle
Z_{V}[0,X_{E_{6}}^{(1)}]$
$\displaystyle=\frac{\theta_{A_{2}}(\tau)\theta_{E_{6},\text{outer
1}}(\tau)}{\eta(\tau)^{4}\eta(2\tau)^{2}}\,,$ (136) $\displaystyle
Z_{V}[0,X_{E_{6}}^{(2)}]$
$\displaystyle=\frac{\theta_{A_{2},\text{inner}}(\tau)\theta_{E_{6},\text{outer
1}}(\tau)}{\eta(\tau)^{4}\eta(2\tau)^{2}}\,,$ (137) $\displaystyle
Z_{V}[0,X_{E_{6}}^{(3)}]$
$\displaystyle=\frac{\theta_{A_{2}}(\tau)\theta_{E_{6},\text{outer
2}}(\tau)}{\eta(\tau)^{4}\eta(2\tau)^{2}}\,,$ (138) $\displaystyle
Z_{V}[0,X_{E_{6}}^{(4)}]$
$\displaystyle=\frac{\theta_{A_{2},\text{inner}}(\tau)\theta_{E_{6},\text{outer
2}}(\tau)}{\eta(\tau)^{4}\eta(2\tau)^{2}}\,.$ (139)

We have recorded the $q$-expansions of these defect partition functions (in
the same order) with their fixed Lie subalgebras in Table 5. We also obtain
the defect in Equation (136) from the Potts CFT in Appendix A.

_$((V^{\operatorname{\mathbb{Z}}_{3}})^{\operatorname{\mathbb{Z}}_{2}})_{1}$_ |  _$q$ -Expansion of $\operatorname{\mathbb{Z}}_{3}$ Duality Defects_
---|---
$A_{1}\times E_{6}$ |  $q^{-\frac{1}{3}}(1+76q+574q^{2}+3000q^{3}+O(q^{4}))$
$A_{1}\times A_{1}\times A_{5}$ |  $q^{-\frac{1}{3}}(1-4q+14q^{2}-40q^{3}+O(q^{4}))$
$A_{1}\times D_{5}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1+12q-2q^{2}+56q^{3}+O(q^{4}))$
$A_{2}\times F_{4}$ |  $q^{-\frac{1}{3}}(1+34q+304q^{2}+1446q^{3}+O(q^{4}))$
$A_{1}\times F_{4}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1+26q+80q^{2}+350q^{3}+O(q^{4}))$
$A_{2}\times C_{4}$ |  $q^{-\frac{1}{3}}(1+2q-16q^{2}+38q^{3}+O(q^{4}))$
$A_{1}\times C_{4}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1-6q+16q^{2}-34q^{3}+O(q^{4}))$
Table 5: List of the defect partition functions of order 3, with defect
twisting time. On the left are the weight one subspaces of
$(V^{\operatorname{\mathbb{Z}}_{3}})^{\operatorname{\mathbb{Z}}_{2}}$; these
are Lie algebras, not invariant lattices. On the right are $q$-expansions of
the defect partition functions from Equations (133 – 139). The first 3 defects
are lifted from outer $A_{2}$ lattice actions, the next 4 are lifted from
outer $E_{6}$ lattice actions.

As a final check, we’ve obtained the same $q$-expansions using computer
algebra software Magma (in a way that we will explain in the following
section) that matches with (at least) the first 11 terms of the $q$-expansions
of the exact theta series above. This convinces us that our numerical methods
work in the order three case, and justifies the computation of
$\operatorname{\mathbb{Z}}_{4}$ and $\operatorname{\mathbb{Z}}_{5}$ defected
partition functions using numerics only. In principle, exact formulas are
obtainable in those cases following the techniques above.

We comment on data relevant to the symmetric non-degenerate bicharacter in
Appendix B.

###  4.2 $\operatorname{\mathbb{Z}}_{4}$ Duality Defects and Computer
Implementation

Moving to the order 4 defects, our direct Lie algebra methods become less
useful. As seen in Table 3, the only non-anomalous order 4 symmetry of $E_{8}$
has fixed Lie algebra $A_{7}\times\mathfrak{u}(1)$. This is not a simple Lie
algebra and thus does not have an associated root system.

####  4.2.1 An Algorithm for Computing Defects

In this case, we can still construct the fixed sublattice as in the beginning
of Section 3.3.1. The symmetry is generated by the vector

$x=\frac{1}{4}\omega_{8}.$ (140)

Accordingly, the fixed sublattice is

$L_{0}=\mathbb{Z}\alpha_{1}\oplus\cdots\oplus\mathbb{Z}\alpha_{7}\oplus\mathbb{Z}4\alpha_{8}.$
(141)

We can now compute $q$-expansions of our defect partition functions with the
aid of Magma Magma .181818We thank Sven Möller for providing us with our
original Magma code and helpful Magma lessons. The code is effectively an
implementation of Theorem 2.13 of Hohn:2020xfe . The algorithm is as follows:

  1. Step 0.

Input an even lattice $L_{0}$. This can be done in Magma by providing a basis
for $L_{0}$ as above with built-in function BasisWithLattice( ).

  2. Step 1.

Compute the automorphism group $O(L_{0})$ with built-in function
AutomorphismGroup( ) and the setwise stabilizer of a fixed set of roots
$H_{\Delta}$. To obtain $H_{\Delta}$ is more involved:

    1. (a)

Enumerate a full set of simple roots (of length 2) $\Delta$ and split them
into positive and negative roots. Note: this collection of simple roots may
not be full rank, e.g. if there are $\mathfrak{u}(1)$’s involved and the
lattice is not a root-lattice.

    2. (b)

Find the subgroup of $O(L_{0})$ stabilizing some choice of simple roots by
looking at group orbits with built-in function OrbitAction( , ). It is
possible that $O(L_{0})$ won’t act faithfully on the collection $\Delta$, in
which case continually append $\Delta$ with vectors of higher norm until
$O(L_{0})$ acts faithfully, then obtain $H_{\Delta}$.

In view of Hohn:2020xfe , every finite-order automorphism of the VOA $V_{L}$
can be constructed by considering a fixed set of representatives $\nu$ of the
conjugacy classes of $H_{\Delta}$. Obtained with built-in function
ConjugacyClasses( ).

  3. Step 2.

Compute the centralizer $C_{O(L)}(\nu)$ of $\nu$ in $O(L)$ or, more precisely,
a fixed set of orbit representatives of its action on
$L^{\nu}\otimes_{\mathbb{Z}}\mathbb{Q}/{\pi_{\nu}(L^{\prime})}$. Here
$\pi_{\nu}$ is the projection of $L\otimes_{\mathbb{Z}}\mathbb{C}$ onto
$L^{\nu}\otimes_{\mathbb{Z}}\mathbb{C}$, with $L^{\nu}$ the sublattice fixed
by $\nu$. These are our “outer $\operatorname{\mathbb{Z}}_{2}$” actions.

Proposition 2.15 of Hohn:2020xfe  guarantees that in order to find
automorphisms of order 2 swapping the axes, we only need to look at order 1
and 2 lattice automorphisms, $\nu$, and restrict the choice of $h$’s according
to whether or not the standard lift of $\nu$ has order doubling.

  4. Step 3.

Loop through all $\nu$ and compute all possible independent $h$ for each
$\nu$. For each $\nu$ loop over all $h$ to create the automorphism
$\hat{\nu}e^{2\pi ih(0)}$.

  5. Step 4.

Implementing the formula in Equation (56), compute the $q$-expansions of our
$\operatorname{\mathbb{Z}}_{2}$-twisted characters for
$V^{\operatorname{\mathbb{Z}}_{m}}$ by summing over weighted lattice vectors.

At this point, we have enumerated all order 2 automorphisms of our VOA
$V_{L_{0}}$ and the (q-expansions of the) $\operatorname{\mathbb{Z}}_{2}$
twisted characters. However, not all order 2 automorphisms swap the axes of
the metric Abelian group
$A\cong\operatorname{\mathbb{Z}}_{m}\times\hat{\operatorname{\mathbb{Z}}}_{m}$.
Fortunately, this is simple to check within the code described above. To do
this, note that our automorphisms extend to all of
$\operatorname{\mathbb{R}}^{8}$ and hence to our original $E_{8}$ lattice $L$.
Thus, by looking at the image of the basis vectors under $\nu$, we can
determine if the automorphism leaves the original $E_{8}$ invariant. In the
case that it is not left invariant, then $L$ must be mapped to a different
lattice, which must be the second $E_{8}$ extension $L^{\prime}$ by
orthogonality of the automorphism. A similar procedure also tells us about the
symmetric non-degenerate bicharacter, see Appendix B.

####  4.2.2 The $\operatorname{\mathbb{Z}}_{4}$ Defects

Let $b_{i}=\alpha_{i}$ for $i=1,\dots,7$, and let $b_{8}=4\alpha_{8}$.
Following our algorithm above, we find three non-trivial lattice automorphisms
$\nu$ of $L_{0}=\operatorname{\mathbb{Z}}\\{b_{i}\\}_{i=1}^{8}$. As matrices
$b_{i}\mapsto(\nu_{k})_{ij}b_{j}$, we have $\nu_{1}^{(4)}$, $\nu_{2}^{(4)}$
and $\nu_{3}^{(4)}$, which are respectively

$\displaystyle\begin{pmatrix}0\,&0\,&0\,&0\,&0\,&0\,&1\,&-4\\\
0&0&0&0&0&1&0&-8\\\ 0&0&0&0&1&0&0&-12\\\ 0&0&0&1&0&0&0&-12\\\
0&0&1&0&0&0&0&-12\\\ 0&1&0&0&0&0&0&-8\\\ 1&0&0&0&0&0&0&-4\\\
0&0&0&0&0&0&0&-1\end{pmatrix},\quad\begin{pmatrix}1\,&0\,&0\,&0\,&0\,&0\,&0\,&-3\\\
0&1&0&0&0&0&0&-6\\\ 0&0&1&0&0&0&0&-9\\\ 0&0&0&1&0&0&0&-12\\\
0&0&0&0&1&0&0&-15\\\ 0&0&0&0&0&1&0&-10\\\ 0&0&0&0&0&0&1&-5\\\
0&0&0&0&0&0&0&-1\end{pmatrix},\quad\begin{pmatrix}0\,&0\,&0\,&0\,&0\,&0\,&1\,&-1\\\
0&0&0&0&0&1&0&-2\\\ 0&0&0&0&1&0&0&-3\\\ 0&0&0&1&0&0&0&0\\\ 0&0&1&0&0&0&0&3\\\
0&1&0&0&0&0&0&2\\\ 1&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&0&1\end{pmatrix}.$ (142)

To reiterate, these are representatives for the 3 conjugacy classes of
$\operatorname{\mathbb{Z}}_{2}$ automorphisms of the lattice $L_{0}$. We can
check the actions of these matrices on the $E_{8}$ basis vectors spanning $L$,
and find that $\nu_{2}^{(4)}$ and $\nu_{3}^{(4)}$ do not leave $L$ invariant,
and hence swap the two distinct $E_{8}$’s in $L_{0}^{*}/L_{0}$. Neither
$\nu_{2}^{(4)}$ nor $\nu_{3}^{(4)}$ experience order doubling.

Combining everything, we find six defected partition functions, with the
weight one subspaces and $q$-expansions recorded in Table 6 for the
$\operatorname{\mathbb{Z}}_{4}$ case.

_$((V^{\operatorname{\mathbb{Z}}_{4}})^{\operatorname{\mathbb{Z}}_{2}})_{1}$_ |  _$q$ -Expansion of $\operatorname{\mathbb{Z}}_{4}$ Duality Defects_
---|---
$A_{7}$ |  $q^{-\frac{1}{3}}(1+62q+784q^{2}+5088q^{3}+O(q^{4}))$
$A_{1}\times A_{5}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1+14q+16q^{2}+96q^{3}+O(q^{4}))$
$A_{3}\times A_{3}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1-2q+16q^{2}-32q^{3}+O(q^{4}))$
$C_{4}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1+10q+80q^{2}+224q^{3}+O(q^{4}))$
$C_{4}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1+10q+16q^{2}+96q^{3}+O(q^{4}))$
$D_{4}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1-6q+16q^{2}-32q^{3}+O(q^{4}))$
Table 6: List of the defect partition functions of order 4, with defect
twisting time. The first three come from the lattice automorphism
$\nu_{2}^{(4)}$ while the rest come from $\nu_{3}^{(4)}$.

These results are plausible from Lie theory. Ignoring the $\mathfrak{u}(1)$
factor there are several symmetries of the $A_{7}$ Lie algebra. First one can
leave the roots of the $A_{7}$ Dynkin diagram unchanged. The corresponding
phase factors can then be chosen to obtain the fixed Lie subalgebras
$A_{6-i}\times A_{i}\times\mathfrak{u}(1)$. The other option is to swap the
roots in the $A_{7}$ Dynkin diagram. In this situation the fixed subalgebra is
given by $C_{4}$. This compares well to the structure presented in Table 6,
where the first three lines correspond to the first case and the next two
lines to the second case.

More generally, in the case of $\operatorname{\mathbb{Z}}_{m}$ symmetries with
$m$ not prime, its also important to keep in mind that we need to “totally
swap” the axes, since there can be $\operatorname{\mathbb{Z}}_{2}$
automorphisms of
$\operatorname{\mathrm{Irr}}(V^{\operatorname{\mathbb{Z}}_{m}})$ which only
swap a subgroup of $\operatorname{\mathbb{Z}}_{m}$ with a subgroup of
$\hat{\operatorname{\mathbb{Z}}}_{m}$.

We record results on the symmetric non-degenerate bicharacter at the end of
Appendix B.

###  4.3 $\operatorname{\mathbb{Z}}_{5}$ Duality Defects

There are two non-anomalous $\operatorname{\mathbb{Z}}_{5}$ symmetries up to
conjugacy, and so two families of duality defects of order $5$. We will
arbitrarily call the two $\operatorname{\mathbb{Z}}_{5}$ conjugacy classes
$\operatorname{\mathbb{Z}}_{5}^{A}$ and $\operatorname{\mathbb{Z}}_{5}^{B}$
with generators

$\displaystyle x^{A}$ $\displaystyle:=\frac{1}{5}\omega_{4}\,,$ (143)
$\displaystyle x^{B}$
$\displaystyle:=\frac{1}{5}\omega_{1}+\frac{1}{5}\omega_{7}\,.$ (144)

The sub-VOAs $V^{\operatorname{\mathbb{Z}}_{5}^{A}}$ and
$V^{\operatorname{\mathbb{Z}}_{5}^{B}}$ have weight one subspaces isomorphic
to $A_{4}\times A_{4}$ and $D_{6}\times\mathfrak{u}(1)^{2}$ respectively.

####  4.3.1 $\operatorname{\mathbb{Z}}_{5}^{A}$ Duality Defects and
$A_{4}\times A_{4}$

The invariant sublattice $L_{0}^{A}$ under the
$\operatorname{\mathbb{Z}}_{5}^{A}$ symmetry is a genuine root lattice spanned
by the $A_{4}\times A_{4}$ inside $E_{8}$, so we can proceed in a
straightforward way. Consider the chopped Dynkin diagram

(145)

Focusing on just one $A_{4}$, the $A_{4}^{(1)}$ and $A_{4}^{(2)}$ Dynkin
diagrams tell us that the $A_{4}$ Lie algebra has 3 inner automorphisms of
order less than or equal to $2$, with fixed Lie subalgebras: $A_{4}$
(trivial); $A_{3}\times\mathfrak{u}(1)$; and $A_{2}\times
A_{1}\times\mathfrak{u}(1)$. Similarly, there is one order 2 outer
automorphism with a fixed $B_{2}$ subalgebra.

Since we have two $A_{4}$ Dynkin diagrams life is slightly more complicated
than the $\operatorname{\mathbb{Z}}_{3}$ case. First, there are the usual
commuting $\operatorname{\mathbb{Z}}_{2}$ actions on each $A_{4}$ Dynkin
diagram $\sigma_{1}$ and $\sigma_{2}$ say; but we can also exchange the
diagrams themselves in different ways, this gives another
$\operatorname{\mathbb{Z}}_{2}$ action $\tau$. The three together generate a
dihedral group of order $8$
$\langle\,\sigma_{1},\sigma_{2},\tau\,\rangle\cong\mathrm{Di}_{8}$.

However, we are not interested in all eight elements of this group, but in the
conjugacy classes. There are three classes which do not switch the axes of
$\operatorname{\mathrm{Irr}}(V^{\operatorname{\mathbb{Z}}_{5}})$ given by
$\\{e\\}$, $\\{\sigma_{1}\sigma_{2}\\}$ and
$\\{\sigma_{1}\tau,\sigma_{2}\tau\\}$ and two classes switching the axes given
by $\\{\sigma_{1},\sigma_{2}\\}$ and $\\{\tau,\sigma_{1}\sigma_{2}\tau\\}$.

As before, we can obtain explicit matrix representatives of these classes. If
we represent the $A_{4}\times A_{4}$ lattice as the span of $b_{i}=\alpha_{i}$
for $i\neq 4$ and $b_{4}=5\alpha_{4}$, then representatives for the non-
trivial lattice automorphisms $\nu^{(5)}_{[\sigma_{1}\sigma_{2}]}$,
$\nu^{(5)}_{[\sigma_{1}\tau]}$, $\nu^{(5)}_{[\sigma_{1}]}$, and
$\nu^{(5)}_{[\tau]}$ are given respectively by

$\displaystyle\begin{pmatrix}1&0&0&0&0&0&1&0\\\ 2&0&1&-5&0&0&0&0\\\
3&1&0&-5&0&0&0&0\\\ 1&0&0&-1&0&0&0&0\\\ 6&0&0&-10&0&1&0&0\\\
4&0&0&-10&1&0&0&0\\\ 2&0&0&-5&0&0&0&1\\\ 3&0&0&-5&0&0&1&0\end{pmatrix},$
$\displaystyle\begin{pmatrix}0&0&0&-5&0&0&1&1\\\ 0&0&0&-10&1&0&2&0\\\
0&0&0&-10&0&1&3&0\\\ 0&0&0&-2&0&0&1&0\\\ 0&0&1&-15&0&0&6&0\\\
0&1&0&-10&0&0&4&0\\\ 1&0&0&-5&0&0&2&0\\\ 0&0&0&-5&0&0&3&0\end{pmatrix},\quad$
(146) $\displaystyle\begin{pmatrix}1&0&0&0&0&0&1&0\\\ 2&0&1&-5&0&0&0&0\\\
3&1&0&-5&0&0&0&0\\\ 1&0&0&-1&0&0&0&0\\\ 6&0&0&-12&1&0&0&0\\\
4&0&0&-8&0&1&0&0\\\ 2&0&0&-4&0&0&1&0\\\ 3&0&0&-6&0&0&0&1\end{pmatrix},$
$\displaystyle\begin{pmatrix}0&0&0&-5&0&0&1&1\\\ 0&0&0&-10&1&0&2&0\\\
0&0&0&-10&0&1&3&0\\\ 0&0&0&-2&0&0&1&0\\\ 0&1&0&-14&0&0&6&0\\\
0&0&1&-11&0&0&4&0\\\ 0&0&0&-3&0&0&2&0\\\ 1&0&0&-7&0&0&3&0\end{pmatrix}.$

The elements of this $A_{4}$ “flipping” conjugacy class
$\\{\sigma_{1},\sigma_{2}\\}$ flip roots of one $A_{4}$ while leaving one copy
of $A_{4}$ invariant. In this case, the fixed Lie subalgebra is one of the
inners multiplied by a $B_{2}$ coming from the $A_{4}$ outer automorphism,
giving 3 duality defects for this conjugacy class. Note that standard lifts
from this class experience order doubling because the outer automorphism folds
roots together which are not orthogonal. The “exchange” conjugacy class $\tau$
fixes a “diagonal” $A_{4}$ Lie subalgebra giving 1 final duality defect. It
does not experience order doubling. We enumerate these 4 cases with their
$q$-expansions in Table 7.

_$((V^{\operatorname{\mathbb{Z}}_{5}^{A}})^{\operatorname{\mathbb{Z}}_{2}})_{1}$_ |  _$q$ -Expansion of $\operatorname{\mathbb{Z}}_{5}^{A}$ Duality Defects_
---|---
$A_{4}\times B_{2}$ |  $q^{-\frac{1}{3}}(1+20q+34q^{2}+140q^{3}+O(q^{4}))$
$A_{3}\times B_{2}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1+4q-14q^{2}+28q^{3}+O(q^{4}))$
$A_{1}\times A_{2}\times B_{2}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1-4q+10q^{2}-28q^{3}+O(q^{4}))$
$A_{4}$ |  $q^{-\frac{1}{3}}(1+24q+124q^{2}+500q^{3}+O(q^{4}))$
Table 7: List of the defect partition functions of order 5A, with defect
twisting time. The first 3 defects come from the $\\{\sigma_{1},\sigma_{2}\\}$
“flip” conjugacy class lattice action. The final defect lifts from the
$\\{\tau,\sigma_{1}\sigma_{2}\tau\\}$ “exchange” conjugacy class lattice
action.

####  4.3.2 $\operatorname{\mathbb{Z}}_{5}^{B}$ Duality Defects and
$D_{6}\times\mathfrak{u}(1)^{2}$

For the case of $D_{6}\times\mathfrak{u}(1)^{2}$ the solution is slightly less
elegant and cannot be read off just by looking at Dynkin diagrams. Proceeding
as in Section 4.2, we can realize the fixed sublattice as the span of
$b_{i}=\alpha_{i}$ for $i\neq 1$, $7$, and $b_{1}=5\alpha_{1}$ and
$b_{7}=4\alpha_{1}+\alpha_{7}$. We use our previous methods to obtain
representatives of the conjugacy classes of non-trivial automorphisms
$\nu^{(5)}_{1}$, $\nu^{(5)}_{2}$, $\nu^{(5)}_{3}$, $\nu^{(5)}_{4}$, given
respectively by

$\displaystyle\begin{pmatrix}-1&0&0&0&0&0&0&0\\\ -10&1&0&0&0&0&-9&0\\\
-10&0&1&0&0&0&-10&0\\\ -10&0&0&1&0&0&-11&0\\\ -10&0&0&0&1&0&-12&0\\\
-5&0&0&0&0&1&-7&0\\\ 0&0&0&0&0&0&-1&0\\\ -5&0&0&0&0&0&-6&1\end{pmatrix},$
$\displaystyle\begin{pmatrix}-5&0&0&0&0&0&-4&0\\\ -3&1&0&0&0&0&-2&0\\\
0&0&1&0&0&0&0&0\\\ 3&0&0&1&0&0&2&0\\\ 6&0&0&0&1&0&4&0\\\ 6&0&0&0&0&1&4&0\\\
6&0&0&0&0&0&5&0\\\ 3&0&0&0&0&0&2&1\end{pmatrix},\quad$ (147)
$\displaystyle\begin{pmatrix}5&0&0&0&0&0&3&0\\\ -16&1&0&0&0&0&-12&0\\\
-20&0&1&0&0&0&-15&0\\\ -24&0&0&1&0&0&-18&0\\\ -28&0&0&0&1&0&-21&0\\\
-18&0&0&0&0&0&-13&1\\\ -8&0&0&0&0&0&-5&0\\\ -14&0&0&0&0&1&-11&0\end{pmatrix},$
$\displaystyle\begin{pmatrix}-7&0&0&0&0&0&-5&0\\\ 5&1&0&0&0&0&2&0\\\
10&0&1&0&0&0&5&0\\\ 15&0&0&1&0&0&8&0\\\ 20&0&0&0&1&0&11&0\\\
15&0&0&0&0&0&9&1\\\ 10&0&0&0&0&0&7&0\\\ 10&0&0&0&0&1&5&0\end{pmatrix}.$

The automorphisms switching the axes of
$\operatorname{\mathrm{Irr}}(V^{\operatorname{\mathbb{Z}}_{5}^{B}})$ are
$\nu^{(5)}_{2}$ and $\nu^{(5)}_{3}$. The standard lifts of both of these
experience order doubling. In total, there are 7 defected partitions
functions, 4 which lift from $\nu_{2}^{(5)}$ and 3 which lift from
$\nu_{3}^{(5)}$. They are recorded in Table 8.

_$((V^{\operatorname{\mathbb{Z}}_{5}^{B}})^{\operatorname{\mathbb{Z}}_{2}})_{1}$_ |  _$q$ -Expansion of $\operatorname{\mathbb{Z}}_{5}^{B}$ Duality Defects_
---|---
$D_{5}\times\mathfrak{u}(1)^{2}$ |  $q^{-\frac{1}{3}}(1+26q+80q^{2}+352q^{3}+O(q^{4}))$
$A_{5}\times\mathfrak{u}(1)^{2}$ |  $q^{-\frac{1}{3}}(1+6q+32q^{2}+32q^{3}+O(q^{4}))$
$A_{5}\times\mathfrak{u}(1)^{2}$ |  $q^{-\frac{1}{3}}(1+6q+32q^{2}-44q^{3}+O(q^{4}))$
$A_{3}\times A_{3}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1-6q+16q^{2}-32q^{3}+O(q^{4}))$
$B_{5}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1+44q+266q^{2}+1268q^{3}+O(q^{4}))$
$A_{1}\times B_{4}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1+12q+10q^{2}+84q^{3}+O(q^{4}))$
$B_{2}\times B_{3}\times\mathfrak{u}(1)$ |  $q^{-\frac{1}{3}}(1-4q+10q^{2}-28q^{3}+O(q^{4}))$
Table 8: List of the defect partition functions of order 5B, with defect
twisting time. The first four come from the automorphism $\nu^{(5)}_{2}$,
while the rest come from $\nu^{(5)}_{3}$.

##  5 A (2+1)d TFT Perspective

There is a natural (2+1)d perspective which sheds light on our previous
discussions and the procedure in Section 3.4 and Equation (85). This all
follows from the theory of modular tensor categories, fusion categories, and
their relationship to gapped boundary conditions for (2+1)d TFTs.

The relevant pieces of this formalism are well-described in mathematical
physics works such as Fuchs_2013 ; Kong:2013aya  and are well-known in the
condensed-matter literature (see e.g. Barkeshli:2014cna ; Cong_2017 ). For
this reason, we will provide only a small recap of the beautiful connection
between (2+1)d TFTs, their gapped boundary conditions, and the relevant
mathematics, before turning to duality defects. Helpful mathematical
references include tensorCatBook2 ; tensorCatBook .

###  5.1 Topological Boundaries of (2+1)d TFTs

Given a fusion category $\operatorname{\mathcal{A}}$ it’s Drinfeld center
$\operatorname{\mathcal{Z}}(\operatorname{\mathcal{A}})$ is naturally a
braided fusion category. This leads to the natural mathematical question: are
all braided fusion categories the Drinfeld center of some fusion category? The
answer is no, as already demonstrated by various Chern-Simons theories
Freed:2020qfy .

The next simplest question in this line of thinking is: if one has a braided
(non-degenerate191919Non-degenerate means that the S-matrix is non-degenerate.
By definition this is true for a MTC, so applies to e.g. reps of a rational
VOA. More broadly, non-degeneracy is satisfied iff there are no anyons which
are invisible to braiding with the entire rest of the category (Theorem 3.4 of
DGNO ), known as “remote detectability” Levin_2013 ; kong2014braided ;
johnsonfreyd2020classification . So-called “slightly degenerate” braided
fusion categories are also of interest in the study of fermionic topological
orders DNO ; johnsonfreyd2020classification .) fusion category
$\operatorname{\mathcal{C}}$, when is
$\operatorname{\mathcal{C}}=\operatorname{\mathcal{Z}}(\operatorname{\mathcal{A}})$
for some fusion category $\operatorname{\mathcal{A}}$? The answer to this
question is also known: the data of a braided equivalence
$\operatorname{\mathcal{C}}\cong\operatorname{\mathcal{Z}}(\operatorname{\mathcal{A}})$
determines a Lagrangian algebra object $A\in\operatorname{\mathcal{C}}$; and
inversely, each Lagrangian algebra object $A\in\operatorname{\mathcal{C}}$
determines a braided equivalence
$\operatorname{\mathcal{C}}\cong\operatorname{\mathcal{Z}}(\operatorname{\mathcal{C}}_{A})$
where $\operatorname{\mathcal{C}}_{A}$ is the category of right-$A$ modules in
$\operatorname{\mathcal{C}}$. Moreover, equivalences
$\operatorname{\mathcal{Z}}(\operatorname{\mathcal{A}})\cong\operatorname{\mathcal{Z}}(\operatorname{\mathcal{B}})$
are known to be in bijection with indecomposable
$\operatorname{\mathcal{A}}$-module categories ETINGOF2011176 ; MUGER200381 .
Altogether, one has the following result:

######  Proposition 1 (DMNO , Proposition 4.8).

Let $\operatorname{\mathcal{A}}$ be a fusion category and
$\operatorname{\mathcal{C}}=\operatorname{\mathcal{Z}}(\operatorname{\mathcal{A}})$.
There is a bijection between the sets of (isomorphism classes of) Lagrangian
algebras in $\operatorname{\mathcal{C}}$ and (equivalence classes of)
indecomposable $\operatorname{\mathcal{A}}$-module categories.

These questions arise naturally in physics as well, as the data of a (2+1)d
TFT are encoded as a Modular Tensor Category of topological line defects or
anyons freedQGfromPI ; tensorCatBook2 ; bartlett2015modular . A question of
interest when studying a (2+1)d TFT is: does the (2+1)d theory admit any
topological boundary conditions?

In the physical case, the MTC of anyons $\operatorname{\mathcal{C}}$ is the
braided fusion category of interest, and it can be shown that if
$\operatorname{\mathcal{C}}\cong\operatorname{\mathcal{Z}}(\operatorname{\mathcal{A}})$
for some $\operatorname{\mathcal{A}}$, then the bulk admits a topological
boundary condition $B$ with boundary line defects modelled by
$\operatorname{\mathcal{A}}$. So we see that the natural question for
physicists is essentially the same as those studying braided fusion categories
Fuchs_2013 ; Kong:2013aya .

More generally, the theory of “anyon condensation” allows one to describe
topological interfaces between two TFTs, described by MTCs
$\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$, in the case
that the anyons of $\operatorname{\mathcal{C}}$ “condense” to a new vacuum for
$\operatorname{\mathcal{D}}$ at the separating interface Bais:2008ni ;
Kong:2013aya . The formalism allows one to easily compute condensation
interfaces for $\operatorname{\mathcal{C}}$, the new phase
$\operatorname{\mathcal{D}}$, and the excitations on the interface (see
Theorem 4.7 of Kong:2013aya  and mattAnyon  for computational references). In
particular, the search for topological boundary conditions is a condensation
to vacuum, controlled by Lagrangian algebra objects
$A\in\operatorname{\mathcal{C}}$.202020We only care about bosonic anyon
condensations and bosonic topological boundary conditions. For results on
fermionic anyon condensation and fermionic topological orders see Lou:2020gfq
and references within.

It is common in the physics literature to talk about a particular realization
of a (2+1)d TFT modeled by the MTC $\operatorname{\mathcal{C}}$, rather than
just discuss abstract MTC data. For example, in Turaev-Viro/Levin-Wen models
Turaev:1992hq ; Levin:2004mi  one chooses a particular “input fusion category”
$\operatorname{\mathcal{A}}$ and constructs the (2+1)d TFT whose anyons are
modeled by the MTC
$\operatorname{\mathcal{C}}=\operatorname{\mathcal{Z}}(\operatorname{\mathcal{A}})$.
From this frame, the boundary condition with excitations modeled by
$\operatorname{\mathcal{A}}$ is particularly distinguished as a “reference” or
“Dirichlet” boundary condition, and one identifies gapped boundaries with
indecomposable $\operatorname{\mathcal{A}}$-module categories Cong_2017
.212121We will use the term Dirichlet boundary condition to mean the canonical
boundary condition for the Turaev-Viro theory based on
$\operatorname{\mathcal{A}}$. By Proposition 1 these are in bijection with the
Lagrangian algebras.

Similarly, if the theory admits a gauge theory description, making such
distinctions are equivalent to saying “who the Wilson line is” in the TFT. The
choice of a particular boundary condition $B$ defines the Wilson line by
declaring that $B$ is the Dirichlet boundary condition for the bulk gauge
fields in that frame. Mathematically, the Wilson lines are the kernel of our
particular bulk-boundary map
$\operatorname{\mathcal{Z}}(\operatorname{\mathcal{A}})\to\operatorname{\mathcal{A}}$,
i.e. their image is a sum of copies of the vacuum operator on the boundary.

The simplest and most well known example of this relationship in physics is in
the toric code Kitaev_2003 . Algebraically, the simple anyons of the (2+1)d
TFT are described by the irreducible representations of the untwisted quantum
double $D(\operatorname{\mathbb{Z}}_{2})$, or more geometrically as (2+1)d
Dijkgraaf-Witten theory with gauge group $\operatorname{\mathbb{Z}}_{2}$ and
trivial topological action DW90 . In short, the simple anyons in this model
are $\\{1,e,m,f\\}$ with spins $\\{0,0,0,1/2\\}$ respectively and non-trivial
fusion relations

$e\otimes m=f\,,\quad e\otimes f=m\,,\quad m\otimes f=e\,.$ (148)

Famously, such a model admits two bosonic indecomposable topological boundary
conditions as described in bravyi1998quantum , where they manifest in the
microscopic description as “smooth” and “rough” boundaries of the square
lattice. Such a description makes it pictorially clear how certain anyons of
the toric code “condense” as they reach the boundary. For example, the $e$
anyon living along a string of vertices is condensed at a rough boundary, but
the $m$ anyon gets stuck and becomes a boundary defect line (see also
Beigi_2011  and Kitaev_2012 ). In other words, the “rough” topological
boundary condition $B_{e}$ condenses the $e$ anyon to the vacuum on the
topological boundary, and corresponds to the Lagrangian algebra object
$1\oplus e$. The topological boundary condition is populated with boundary
excitations $\\{1\oplus e,m\oplus f\\}$, which behave like
$\mathcal{A}_{e}=\operatorname{\mathrm{Vec}}_{\operatorname{\mathbb{Z}}_{2}}$.

###  5.2 Duality Defects

Gauging 2d theories has a simple (2+1)d interpretation in terms of boundary
conditions and interfaces in topological gauge theories (see e.g.
Freed:2018cec ; gaiotto2020orbifold ; theoMoonshine  for an extended
explanation). In particular, given any 2d theory $T$ with non-anomalous
$\operatorname{\mathbb{Z}}_{m}$ symmetry, the theory provides an “enriched
Neumann” boundary condition $B[T]$ for a (2+1)d
$\operatorname{\mathbb{Z}}_{m}$ Dijkgraaf-Witten theory by coupling the
boundary value of the bulk connection to $T$. The local operators of $B[T]$
are effectively the local operators of $T^{\operatorname{\mathbb{Z}}_{m}}$.

a) $B[T$]
---
b) $\mathrm{Dir}^{\mathrm{e}}$
$I_{em}$ Electric $\operatorname{\mathbb{Z}}_{m}$
---
Gauge Theory
Magnetic $\hat{\operatorname{\mathbb{Z}}}_{m}$
---
Gauge Theory
a) $B[[T/\operatorname{\mathbb{Z}}_{m}]]$
---
b) $\mathrm{Neu}^{\mathrm{m}}$
Magnetic $\hat{\operatorname{\mathbb{Z}}}_{m}$
---
Gauge Theory
Collide $I_{em}$ Figure 6: On the left, a slab of
$\operatorname{\mathbb{Z}}_{m}$ gauge theory and
$\hat{\operatorname{\mathbb{Z}}}_{m}$ gauge theory separated by an (invertible
topological) electric-magnetic duality interface $I_{em}$ (yellow). If one
moves the interface $I_{em}$ all the way to the left, then all that remains is
$\hat{\operatorname{\mathbb{Z}}}_{m}$ gauge theory with a new boundary
condition (green). a) If the left boundary condition (blue) is an enriched
Neumann boundary condition $B[T]$, then when it collides with the interface it
appears as the enriched Neumann boundary condition
$B[[T/\operatorname{\mathbb{Z}}_{m}]]$. b) If the left boundary condition
(blue) is a Dirichlet boundary condition for the
$\operatorname{\mathbb{Z}}_{m}$ gauge theory, then it appears as a Neumann
boundary condition for the $\hat{\operatorname{\mathbb{Z}}}_{m}$ gauge theory.

In this case, the 2d theory $T$ (on $M$ say) can be understood as a
compactification of a slab of this $\operatorname{\mathbb{Z}}_{m}$ gauge
theory on $M\times[0,1]$, with the boundary condition $B[T]$ for one boundary,
and the reference (topological) Dirichlet boundary condition for the other.
Similarly, $[T/\operatorname{\mathbb{Z}}_{m}]$ can be understood as the same
setup but with the (topological) fully-Neumann boundary condition instead.

However, by the same logic, $[T/\operatorname{\mathbb{Z}}_{m}]$ can also be
viewed as a sandwich of $\hat{\operatorname{\mathbb{Z}}}_{m}$ gauge theory
with boundary condition $B[[T/\operatorname{\mathbb{Z}}_{m}]]$ and Dirichlet
boundary condition for $\hat{\operatorname{\mathbb{Z}}}_{m}$ on the other.
These two descriptions are dual, what has changed is the identification of the
bulk TFT as an “electric” $\operatorname{\mathbb{Z}}_{m}$ gauge theory or a
“magnetic” $\hat{\operatorname{\mathbb{Z}}}_{m}$. By our previous section, the
MTC data is the same, but we have presented a different topological boundary
condition as “Dirichlet”. The two pictures are related by an electric-magnetic
duality interface $I_{em}$, which composes with $B[T]$ to produce
$B[[T/\operatorname{\mathbb{Z}}_{m}]]$ and turns the electric Dirichlet
boundary condition into a magnetic Neumann boundary condition. We depict all
of this in Figure 6.

Since a duality defect separates a theory from its orbifold, we recognize a
duality defect as a potential ending line for the interface $I_{em}$,
separating the electric and magnetic Dirichlet boundary conditions in a slab
picture i.e. as a $\operatorname{\mathbb{Z}}_{2}^{em}$ twist line operator,
see Figure 7(a). Gauging the electric-magnetic
$\operatorname{\mathbb{Z}}_{2}^{em}$ 0-form symmetry of the bulk TFT (with
Neumann boundary conditions on the left and Dirichlet boundary conditions on
the right of Figure 7(a)), produces a new bulk TFT where the interface
$I_{em}$ “becomes invisible,” leaving the twist line behind, as in Figure
7(b). In Section XI of Barkeshli:2014cna  (also Section 3.5 of
Bhardwaj:2016clt ) the authors show explicitly how the boundary twist line
left behind after gauging has the fusion rules of
$\operatorname{\mathrm{TY}}(\operatorname{\mathbb{Z}}_{m})$.

This is simply squaring two commensurate pictures. In one case, we are
thinking of $T$ as having a $\operatorname{\mathbb{Z}}_{m}$ symmetry with
topological line defects for the symmetry modeled by the fusion category
$\operatorname{\mathcal{D}}=\operatorname{\mathrm{Vec}}_{\operatorname{\mathbb{Z}}_{m}}$,
and so we can couple it to a
$\operatorname{\mathcal{Z}}(\operatorname{\mathrm{Vec}}_{\operatorname{\mathbb{Z}}_{m}})$
bulk and view the duality defect line as the endpoint of an electric-magnetic
duality wall. Alternatively, we can view $T$ as having “categorical symmetry”
$\operatorname{\mathcal{F}}=\operatorname{\mathrm{TY}}(\operatorname{\mathbb{Z}}_{m})$,
in which case it can be coupled to a bulk of
$\operatorname{\mathcal{Z}}(\operatorname{\mathrm{TY}}(\operatorname{\mathbb{Z}}_{m}))$
theory, and the duality defect line is understood simply as a condensate of
bulk lines on the topological boundary.

In general, suppose $\operatorname{\mathcal{D}}$ admits a $G$-extension

$\operatorname{\mathcal{F}}=\bigoplus_{g\in G}\operatorname{\mathcal{F}}_{g}$
(149)

with identity piece
$\operatorname{\mathcal{F}}_{e}\cong\operatorname{\mathcal{D}}$, then the
(2+1)d TFT $Q$ with MTC
$\operatorname{\mathcal{Z}}(\operatorname{\mathcal{D}})$ and reference
Dirichlet boundary condition $\operatorname{\mathcal{D}}$ necessarily has
topological surfaces with a $G$-composition law. If we gauge this $G$ symmetry
to produce a new (2+1)d TFT $[Q/G]$, then the new MTC data is given by
$\operatorname{\mathcal{Z}}(\operatorname{\mathcal{F}})$ with reference
boundary condition $\operatorname{\mathcal{F}}$. This follows from the work of
gelaki2009centers ; Barkeshli:2014cna .

$\leavevmode\hbox to44.3pt{\vbox
to120.73pt{\pgfpicture\makeatletter\hbox{\hskip
39.81227pt\lower-46.76227pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{{}} {}{}{{}} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor{pgffillcolor}{rgb}{0.8,0.8,1}\pgfsys@color@rgb@fill{0.8}{0.8}{1}\pgfsys@invoke{
}\pgfsys@stroke@opacity{0.8}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{-16.43147pt}{-16.43147pt}\pgfsys@lineto{-16.43147pt}{40.47404pt}\pgfsys@lineto{-32.86295pt}{24.04257pt}\pgfsys@lineto{-32.86295pt}{-32.86295pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}} {}{}{{}}
{}{}{{}} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor{pgffillcolor}{rgb}{0.95,0.8,0.85}\pgfsys@color@rgb@fill{0.95}{0.8}{0.85}\pgfsys@invoke{
}\pgfsys@stroke@opacity{0.8}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{56.90552pt}\pgfsys@lineto{-16.43147pt}{40.47404pt}\pgfsys@lineto{-16.43147pt}{-16.43147pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}} {}{}{{}}
{}{}{{}} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{56.90552pt}\pgfsys@lineto{-32.86295pt}{24.04257pt}\pgfsys@lineto{-32.86295pt}{-32.86295pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{}}{}{{}}{}{{}}{}\pgfsys@moveto{-32.86295pt}{-32.86295pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-36.47926pt}{-43.42926pt}\pgfsys@invoke{
}\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{{$T$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{-14.22638pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-29.60667pt}{63.13852pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$[T/\operatorname{\mathbb{Z}}_{m}]$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.6pt}\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}{}\pgfsys@moveto{-16.43147pt}{-16.43147pt}\pgfsys@lineto{-16.43147pt}{40.47404pt}\pgfsys@stroke\pgfsys@invoke{
} \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}}\quad\longleftrightarrow\leavevmode\hbox
to241.03pt{\vbox to125.72pt{\pgfpicture\makeatletter\hbox{\hskip
47.35378pt\lower-48.86227pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ }
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{{}} {}{}{{}} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\definecolor{pgffillcolor}{rgb}{1,1,0.8}\pgfsys@color@cmyk@fill{0}{0}{0.2}{0}\pgfsys@invoke{
}\pgfsys@stroke@opacity{0.8}\pgfsys@invoke{
}\pgfsys@fill@opacity{0.8}\pgfsys@invoke{
}{}\pgfsys@moveto{-16.43147pt}{-16.43147pt}\pgfsys@lineto{-16.43147pt}{40.47404pt}\pgfsys@lineto{154.28508pt}{40.47404pt}\pgfsys@lineto{154.28508pt}{-16.43147pt}\pgfsys@closepath\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{{}}{}{{}}{}\pgfsys@moveto{68.9268pt}{21.50539pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{62.57434pt}{10.93907pt}\pgfsys@invoke{
}\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{{$I_{em}$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{52.49533pt}{-32.86295pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{13.1827pt}{-43.54039pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\hbox{{$\,\operatorname{\mathbb{Z}}_{m}$ Gauge Theory}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{}{{}}{}{{}}{}\pgfsys@moveto{85.35828pt}{56.90552pt}\pgfsys@stroke\pgfsys@invoke{
}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{45.4901pt}{62.58295pt}\pgfsys@invoke{
}\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{{$\,\,\hat{\operatorname{\mathbb{Z}}}_{m}$ Gauge Theory}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}{}{{}}
{}{}{{}} {}{}{{}} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{-32.86295pt}{-32.86295pt}\pgfsys@lineto{137.8536pt}{-32.86295pt}\pgfsys@lineto{170.71655pt}{0.0pt}\pgfsys@closepath\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{{}}{}{{}} {}{}{{}}
{}{}{{}} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{