††institutetext:  Department of Applied Mathematics and Theoretical Physics,
University of Cambridge, Cambridge, CB3 OWA, UK

# Boundary States and Anomalous Symmetries of Fermionic Minimal Models

Philip Boyle Smith<EMAIL_ADDRESS>
###### Abstract

The fermionic minimal models are a recently-introduced family of two-
dimensional spin conformal field theories. We determine all of their conformal
boundary states and potentially anomalous $\mathbb{Z}_{2}$ global symmetries.
The latter task hinges upon on a conjecture about $\mathfrak{su}(2)$ affine
parities generalising an earlier result known to have an interpretation in
terms of Fermat curves. Our results indicate a close connection between
several properties of the models, including the matching of the sizes of the
SPT classes of boundary states, the existence of anomalous $\mathbb{Z}_{2}$
symmetries, and the vanishing of the Ramond-Ramond sector, for which we
provide an explanation.

##  1 Introduction

The fermionic minimal models are a family of two-dimensional conformal field
theories recently introduced in watts ; tachikawa ; kulp . They are close
relatives of the bosonic minimal models that have been known for a long time
bpz ; zuber ; kato ; ruelle ; minimalstates , but differ in one key respect:
instead of depending solely on a Riemann surface with metric, they also depend
on a choice of spin structure.

The fact that these theories depend upon a spin structure opens up the
possibility of new fermionic phenomena that are not seen in the ordinary
bosonic minimal models. For a start, there can be an issue with their boundary
conditions. While in the bosonic minimal models, there exists a complete set
of boundary conditions such that when one places the theory on a spatial
interval and imposes different boundary conditions at the two ends, one always
obtains a physically sensible theory minimalstates , in fermionic theories
this no longer need be the case. Instead, one may find that for certain
choices of boundary conditions, the spectrum on an interval includes an
unpaired Majorana zero mode, rendering the theory inconsistent bcft . As we
review later in this introduction, the fact this issue arises only for
fermionic theories is closely tied to the classification of SPT phases in two
dimensions, where it is reflected in the mathematical fact that
$\Omega_{2}^{\text{SO}}(\text{pt})=0$ is trivial while
$\Omega_{2}^{\text{Spin}}(\text{pt})=\mathbb{Z}_{2}$ is not kapustin1 ;
kapustin2 .

Second, there is also the possibility of supporting symmetries with nontrivial
’t Hooft anomalies. For the bosonic minimal models, this is not an option: all
global symmetries are necessarily non-anomalous cftrelativeanom . But for
fermionic theories, it is a possibility. We will limit ourselves to
$\mathbb{Z}_{2}$ global symmetries. In fermionic theories, such symmetries can
carry a mod-8 valued anomaly. As was explained in ryuzhang ; qi , this anomaly
is related to SPT phases in three dimensions, and the fact it arises only for
fermionic theories is encoded in the facts
$\Omega_{3}^{\text{SO}}(B\mathbb{Z}_{2})=0$ while
$\Omega_{3}^{\text{Spin}}(B\mathbb{Z}_{2})=\mathbb{Z}_{8}$.

The above two phenomena are easy to exhibit for the simplest fermionic minimal
model, the Majorana fermion. Here, one can simply use free-field techniques to
explicitly show everything one could possibly want to show. But for the other
fermionic minimal models, which are all interacting, things are not so
transparent. Our goal is to extend the analysis of boundary conditions and
anomalous symmetries to these remaining fermionic minimal models. We would
like to see which features of the Majorana fermion generalise to the family as
a whole, and which are artefacts of a free theory. Therefore, to set the
scene, it will be useful to first review the basic facts about the Majorana
fermion we want to generalise. A summary of our results follows, and after
that the organisation of the paper.

###  1.1 A Simple Illustration

We begin with a motivating discussion about SPT phases. A Majorana fermion of
mass $m$ in $d=1+1$ dimensions has action

$S=\frac{i}{2}\int\\!dtdx\,\chi_{+}\partial_{+}\chi_{+}+\chi_{-}\partial_{-}\chi_{-}+m\chi_{+}\chi_{-}$

where $\partial_{\pm}=\partial_{t}\pm\partial_{x}$. When $m\neq 0$ the theory
is gapped, and the two possible gapped phases with $m>0$ and $m<0$ are
examples of two distinct SPT phases. To see the distinction, we promote
$m\rightarrow m(x)$ and consider a domain-wall profile for $m(x)$, say one
with $m(x=\pm\infty)=\pm M$ for some unimportant constant $M>0$. As was
famously shown by Jackiw and Rebbi jr , the spectrum of the theory then
includes a quantum-mechanical zero mode localised on the domain wall, obeying

$\chi=\chi^{\dagger}\qquad\\{\chi,\chi\\}=1$

This system exhibits an anomaly under fermion parity $(-1)^{F}$. (One way to
see this is that $\chi$ acquires a nonzero VEV on a periodic temporal circle.)
The appearance of this anomalous degree of freedom on the interface signals
that the $m>0$ and $m<0$ phases are distinct SPT phases, as promised
wittentalk .

As stressed in ryu ; han ; shen1 ; shen2 , this story has an alternative guise
in the language of boundary conditions. Recall that for a massless Majorana
fermion there are two possible boundary conditions, which we denote as

$\displaystyle+\,$ $\displaystyle{:}\quad\chi_{+}=+\chi_{-}$
$\displaystyle-\,$ $\displaystyle{:}\quad\chi_{+}=-\chi_{-}$

Consider a mass profile $m(x)$ that interpolates from $0$ to $\pm M$. On the
left side lives a massless Majorana fermion, while the right side is gapped at
a scale $M$. At low energies, the massless fermion experiences this set-up as
a boundary condition. Pictorially, the map from gapped phases to boundary
conditions is

Similarly when the gapped phase sits on the left, one obtains a left boundary
condition. However, due to an annoying technicality, the outcome is now
reversed:

($\star$)

The analog of the previous story is now that on a spatial interval with $++$
boundary conditions at both ends, the spectrum again includes an unpaired
Majorana zero mode. A lattice version of this mechanism also exists, and was
the subject of kitaev .

In conformal field theories, boundary conditions are more naturally described
as boundary states. It is these that will form the main focus of the paper. To
each right boundary condition, there is an associated boundary state, which
lives in the NS sector of the theory:

$\chi_{+}=\pm\chi_{-}\text{ on right}\quad\rightarrow\quad|{\pm}\rangle$

Meanwhile, boundary conditions on the left are described by dual states. One
might have thought that the correct dual state that describes boundary
condition $A$ on the left is simply the dual of the state that describes
boundary condition $A$ on the right. But this isn’t quite right. Annoyingly,
there is an extra sign flip, and the correct mapping of boundary conditions to
dual states is actually

$\chi_{+}=\pm\chi_{-}\text{ on left}\quad\rightarrow\quad\langle{\mp}|$

Because of the similar sign flip in ($\star$ ‣ 1.1), however, a gapped phase
$m=\pm M$ corresponds to the same boundary state $\langle{\pm}|$ or
$|{\pm}\rangle$ regardless of whether we sit at a left or a right boundary. It
is for this reason that boundary states more precisely correspond to SPT
phases than boundary conditions. In any case, once the boundary states are
known, there is a simple algebraic procedure to calculate the partition
function for states on an interval. Doing this for the boundary states
$\langle{-}|$ and $|{+}\rangle$ yields the partition function

$\mathcal{Z}_{-+}(\tau)=\sqrt{2}\,\chi_{1/16}(\tau)$

where $\chi_{1/16}(\tau)$ is a Virasoro character. The most important feature
of this partition function is the overall factor of $\sqrt{2}$. This signals
the unpaired Majorana mode, hence the distinctness of the SPT phases
underlying $\langle{-}|$ and $|{+}\rangle$. For more details of this point of
view on unpaired Majorana modes, see also bcft ; flows ; delmastro .

For the second half of our story, we will be interested in the
$\mathbb{Z}_{2}$ global symmetry known as chiral fermion parity. It acts by
flipping the sign of only one of the fermions, which without loss of
generality we can take to be the left-movers:

$\mathllap{\mathbb{Z}_{2}\,{:}\quad}\chi_{+}\rightarrow\chi_{+}\qquad\chi_{-}\rightarrow-\chi_{-}$

Under this symmetry, the mass parameter $m$ is odd. The symmetry therefore
exchanges the two SPT phases. One can check that the symmetry also exchanges
the corresponding boundary states

$\mathllap{\mathbb{Z}_{2}\,{:}\quad}|{\pm}\rangle\rightarrow|{\mp}\rangle$

as expected.

Importantly, the symmetry also carries an anomaly whose strength is 1 mod 8.
To see this, one places the theory on a background with a defect line for the
$\mathbb{Z}_{2}$ symmetry. It will be sufficient for us to consider a torus.
We will denote the partition function on such a background by

$\mathcal{Z}\Big{[}\tau;\,\leavevmode\hbox to24.5pt{\vbox
to20.55pt{\pgfpicture\makeatletter\hbox{\hskip 11.49431pt\lower-7.544pt\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@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.01988pt}{-7.544pt}\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{{\scriptsize{P}}}
}}\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}{-11.49431pt}{4.01016pt}\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{{\scriptsize{AP}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{6.40182pt}{0.0pt}\pgfsys@lineto{6.40182pt}{12.80365pt}\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}}\;\Big{]}$

where $\tau$ is the modular parameter describing a choice of flat metric on
the torus, the diagram labels the spin structure, and the dashed line, if
present, labels the defect. Under the shift $\tau\rightarrow\tau+2$, we have
the transformation law

$\mathcal{Z}\Big{[}\tau+2;\,\leavevmode\hbox to24.5pt{\vbox
to20.55pt{\pgfpicture\makeatletter\hbox{\hskip 11.49431pt\lower-7.544pt\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@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.01988pt}{-7.544pt}\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{{\scriptsize{P}}}
}}\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}{-11.49431pt}{4.01016pt}\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{{\scriptsize{AP}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{6.40182pt}{0.0pt}\pgfsys@lineto{6.40182pt}{12.80365pt}\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}}\;\Big{]}\;=\;e^{2\pi
i/8}\;\mathcal{Z}\Big{[}\tau;\,\leavevmode\hbox to24.5pt{\vbox
to20.55pt{\pgfpicture\makeatletter\hbox{\hskip 11.49431pt\lower-7.544pt\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@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.01988pt}{-7.544pt}\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{{\scriptsize{P}}}
}}\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}{-11.49431pt}{4.01016pt}\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{{\scriptsize{AP}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{6.40182pt}{0.0pt}\pgfsys@lineto{6.40182pt}{12.80365pt}\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}}\;\Big{]}$

The presence of the phase $e^{2\pi i/8}$ indicates the anomaly. In general, it
could be any eighth root of unity $e^{2\pi ik/8}$; the fact that for the
Majorana fermion $k=1$ indicates that the strength of the anomaly is 1 mod 8,
as claimed anomalyinterplay ; wittenphases . The anomaly also manifests itself
in a far more obvious way, which becomes clear if we look at the partition
function. It turns out to be

$\mathcal{Z}\Big{[}\tau;\,\leavevmode\hbox to24.5pt{\vbox
to20.55pt{\pgfpicture\makeatletter\hbox{\hskip 11.49431pt\lower-7.544pt\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@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.01988pt}{-7.544pt}\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{{\scriptsize{P}}}
}}\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}{-11.49431pt}{4.01016pt}\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{{\scriptsize{AP}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{6.40182pt}{0.0pt}\pgfsys@lineto{6.40182pt}{12.80365pt}\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}}\;\Big{]}\;=\;\sqrt{2}\;\overline{(\chi_{0}+\chi_{1/2})(\tau)}\,\chi_{1/16}(\tau)$

Once again we find a notorious factor of $\sqrt{2}$ in the partition function.
This time it is telling us that the R sector, when frustrated by a
$\mathbb{Z}_{2}$ symmetry defect, contains an unpaired Majorana zero mode.
(This statement would also have been true of the NS sector.)

###  1.2 Summary of Results

We now consider generalising these facts to the other models in the family.
Each model should have a complete set of conformal boundary states. We expect
that all of these states will arise from a deformation to a gapped phase. If
so, then each boundary state will fall into one of two distinct classes
depending on which SPT class its gapped phase sits in.

Our first main claim is that this expectation is borne out. We determine, for
each fermionic minimal model, the complete list of conformal boundary states.
We show how these naturally fall into two classes. When boundary conditions
$a$ and $b$ are taken from the same class, we find the partition function

$\mathcal{Z}_{ab}(\tau)=\sum_{i\in\text{KT}}n_{ab}^{i}\,\chi_{i}(\tau)$

where KT denotes the Kac table, as we review in Section 2. The coefficients
$n_{ab}^{i}$ are non-negative integers, so this is a manifestly sensible
partition function. In contrast, when $a$ and $b$ are from different classes,
we find

$\mathcal{Z}_{ab}(\tau)=\sqrt{2}\,\sum_{i\in\text{KT}}n_{ab}^{i}\,\chi_{i}(\tau)$

which contains a factor of $\sqrt{2}$, signalling an unpaired Majorana mode,
but other than that the partition function is perfectly sensible, with the
coefficients $n_{ab}^{i}$ again given by non-negative integers. (For a
previous example of this phenomenon in a different family of models, see also
bcft .) All partition functions are explicitly presented in Section 3.

Figure 1: The classification of fermionic minimal models. Both the infinite
series and exceptionals are labelled by a choice of integer $m$.

To describe these results in a little more detail, we recall that there are
two kinds of fermionic minimal models: an infinite series, and two
exceptionals. Both kinds are labelled by an integer $m$; for the infinite
series, this integer takes the values $m\geq 3$, while for the exceptionals,
it takes the values $m=11,12$. This situation is depicted in Figure 1. We
explain our results for each kind of model in turn.

##### Infinite series

Here the boundary states are labelled by pairs $(r,s)$ in the bottom-left
quadrant of the Kac table, defined by the inequalities

$r\leq m/2\quad\text{and}\quad s\leq(m+1)/2$

These states are shown in Figure 2 for the first four models in the series.
The states fall into two classes, with class 1 consisting of the points in the
interior of the region defined by the above inequalities, and class 2
consisting of the border. It is straightforward to count the number of states
in the two classes. They have sizes

class 1: $\displaystyle\qquad\lfloor(m-1)^{2}/4\rfloor$ class 2:
$\displaystyle\qquad\lfloor m/2\rfloor$

$m=3$

(a) *

$m=4$

(b) *

$m=5$

(c) *

$m=6$

(d) *

Figure 2: Boundary states for the infinite series of models are labelled by
points in the bottom-left quadrant of the Kac table. The two classes are shown
in blue and red. Note that for $m=3,4$ but no other values, the classes have
equal sizes.

##### Exceptionals

In this case the labelling of the boundary states is a little more complicated
and will be left to Section 3. However, the counting is straightforward. Now
both classes have the same size. For the $m=11$ exceptional, both classes have
size 10, while for the $m=12$ exceptional, both have size 12.

Figure 3: The models with an extra $\mathbb{Z}_{2}$ global symmetry, shown in
red. Note that these are the same models which had matching class sizes
earlier, as well as the models with vanishing RR sector.

Our second main result is a classification of which fermionic minimal models
have a global $\mathbb{Z}_{2}$ symmetry, modulo $(-1)^{F}$. We find that the
first two models in the infinite series and both exceptionals have a unique
such symmetry, generalising the chiral fermion parity of the Majorana fermion,
while the remaining models have none. This situation is depicted in Figure 3.

In the four models where the symmetry exists, we also find that the strength
of the anomaly takes the same value 1 mod 8 for all models; that the twisted-
sector partition functions contain a Majorana zero mode whenever there is a
symmetry defect crossing the equal-time contour; and that the symmetry
exchanges the two classes of boundary states. This forces them to have equal
sizes, which is indeed what we found earlier. We also notice that these models
are the same as the ones that have a vanishing RR sector partition function.

In view of the above results, the four special models form a close
generalisation of the Majorana fermion, with all the facts we saw in Section
1.1 continuing to hold. The remaining models, on the other hand, do not form
such a close generalisation, and the only fact that survives is the existence
of two incompatible classes of boundary states.

Finally we mention earlier related research. The ‘chiral fermion parity’ in
the $m=3,4$ models is already well known. The paper watts  also obtains some
of our results on boundary states for a subset of the infinite series, with a
different choice of normalisation. The main novelties of this paper are: a
uniform treatment of all fermionic minimal models, the normalisation, the
interplay between boundary states and symmetries, and the perspective of SPT
phases.

###  1.3 The Plan of the Paper

In Section 2 we review a handful of facts about the minimal models and their
fermionic counterparts that we will need to use, for the purposes of self-
containment. In Section 3, we write down the two classes of boundary states,
and explicitly compute all interval partition functions. In Section 4, we
determine all $\mathbb{Z}_{2}$ global symmetries, and in doing so compute
their anomalies and all twisted-sector partition functions. We also compute
their action on the boundary states found earlier. Finally, in Section 5, we
give some brief arguments showing various implications between our results.

##  2 Review of Fermionic Minimal Models

In this section we review the barest essentials from the theory of minimal
models that we will need to use in this paper. We will not review this
material from scratch; instead, for reviews of minimal models see for example
ruelle , whose conventions we have strived to match, while for those of the
fermionic ones see tachikawa ; kulp .

The fermionic minimal models are defined to be the set of all unitary spin-
CFTs that are rational with respect to the Virasoro algebra. Their
classification has been carried out, and the results are as in Figure 1. The
classification consists of

  * •

An infinite series, with $m\geq 3$.

  * •

Two exceptionals, with $m=11,12$.

The integer $m$ dictates the chiral data of the theory. For example, the
central charge is determined via

$c=1-\frac{6}{m(m+1)}$

Of special importance is the Kac table, which is defined as a set of pairs of
integers modulo an equivalence relation,

$\text{KT}\;=\;\frac{\big{\\{}\,(r,s)\,:\,1\leq r\leq m-1,\,1\leq s\leq
m\,\big{\\}}}{(r,s)\sim(m-r,m+1-s)}$

The relevance of the Kac table is that it labels the available Virasoro
characters at this central charge. We will denote them by $\chi_{r,s}(\tau)$,
and their conformal dimensions by $h_{r,s}$. Sometimes, we will find it
economical to compress an element of KT down to a single composite index,
which we will denote by a letter such as $i,j,k,\dots$.

For the special values $m=11,12$, one also faces a dichotomy between the
infinite series and the exceptionals. This choice affects the way the
characters are combined in the partition function. We will follow the
presentation of tachikawa . We arrange the states of even/odd fermion parity
on an antiperiodic/periodic111Some alternative common synonyms are AP/P, NS/R,
and bounding/non-bounding. circle into a $2\times 2$ table. For the infinite
series, this table is222Here we regard $\mathcal{Z}$ and
$(-1)^{\text{Arf}}\mathcal{Z}$ as equivalent theories, so we only show a
single table for both of them. We revisit their distinction more carefully in
Section 5.

$\begin{array}[]{r|ll}&\text{even}&\text{odd}\\\
\hline\cr\text{AP}&\pagecolor[HTML]{e5f6ff}\sum_{1\leq s\leq
r<m}^{(m+1)r+ms\text{
odd}}|\chi_{r,s}|^{2}&\pagecolor[HTML]{fff9d9}\sum_{1\leq s\leq
r<m}^{(m+1)r+ms+m(m+1)/2\text{ odd}}\overline{\chi_{m-r,s}}\chi_{r,s}\\\
\text{P}&\pagecolor[HTML]{ffecf9}\sum_{1\leq s\leq
r<m}^{(m+1)r+ms+m(m+1)/2\text{
even}}\overline{\chi_{m-r,s}}\chi_{r,s}&\pagecolor[HTML]{e7ffcd}\sum_{1\leq
s\leq r<m}^{(m+1)r+ms\text{ even}}|\chi_{r,s}|^{2}\end{array}$

For the $m=11$ exceptional, it is

$\begin{array}[]{r|ll}&\text{even}&\text{odd}\\\
\hline\cr\text{AP}&\pagecolor[HTML]{e5f6ff}\sum_{r=1,\text{odd}}^{9}|\chi_{r,1}+\chi_{r,7}|^{2}+|\chi_{r,5}+\chi_{r,11}|^{2}&\pagecolor[HTML]{fff9d9}\sum_{r=1,\text{odd}}^{9}(\overline{\chi_{r,1}+\chi_{r,7}})(\chi_{r,5}+\chi_{r,11})+\text{c.c.}\\\
\text{P}&\pagecolor[HTML]{ffecf9}\sum_{r=1,\text{odd}}^{9}|\chi_{r,4}+\chi_{r,8}|^{2}&\pagecolor[HTML]{e7ffcd}\sum_{r=1,\text{odd}}^{9}|\chi_{r,4}+\chi_{r,8}|^{2}\end{array}$

while for the $m=12$ exceptional, it is

$\begin{array}[]{r|ll}&\text{even}&\text{odd}\\\
\hline\cr\text{AP}&\pagecolor[HTML]{e5f6ff}\sum_{s=1,\text{odd}}^{11}|\chi_{1,s}+\chi_{7,s}|^{2}+|\chi_{5,s}+\chi_{11,s}|^{2}&\pagecolor[HTML]{fff9d9}\sum_{s=1,\text{odd}}^{11}(\overline{\chi_{1,s}+\chi_{7,s}})(\chi_{5,s}+\chi_{11,s})+\text{c.c.}\\\
\text{P}&\pagecolor[HTML]{ffecf9}\sum_{s=1,\text{odd}}^{11}|\chi_{4,s}+\chi_{8,s}|^{2}&\pagecolor[HTML]{e7ffcd}\sum_{s=1,\text{odd}}^{11}|\chi_{4,s}+\chi_{8,s}|^{2}\end{array}$

##  3 Boundary States

In this section we write down complete sets of boundary states for the
fermionic minimal models. We compute their interval partition functions to
confirm their consistency, and use them to show which states lie in which
class.

First we review the general formalism that we will use. For our purposes,
boundary states live in the AP sector of the theory. As they preserve the
conformal symmetry, and this includes $(-1)^{F}$, they in fact belong to the
AP-even sector. This Hilbert space decomposes into a direct sum of Verma
modules

$\text{\hbox{\pagecolor[HTML]{e5f6ff}$\mathcal{H}_{\text{AP}}^{\text{even}}$}}=\bigoplus_{i,j\in\text{KT}}M_{ij}\overline{\mathcal{V}_{i}}\otimes\mathcal{V}_{j}$

where the multiplicities $M_{ij}$ can be read off from the partition function
tables listed in Section 2. To preserve the conformal symmetry, boundary
states can only contain states coming from terms with $i=j$, so must lie in
the subspace

$\text{\hbox{\pagecolor[HTML]{e5f6ff}$\mathcal{H}_{\text{AP}}^{\text{even}}\big{|}_{\text{diagonal}}$}}=\bigoplus_{i\in\text{KT}}M_{ii}\overline{\mathcal{V}_{i}}\otimes\mathcal{V}_{i}=\bigoplus_{\begin{subarray}{c}i\in\text{KT}\\\
M_{ii}=1\end{subarray}}\overline{\mathcal{V}_{i}}\otimes\mathcal{V}_{i}$

where in the last step, we’ve taken advantage of the fact that $M_{ii}=0$ or 1
to simplify our answer. In each
$\overline{\mathcal{V}_{i}}\otimes\mathcal{V}_{i}$, there is a unique
conformally-invariant state $\|{i}\rangle\\!\rangle$ whose inner product with
the vacuum state is 1, called an Ishibashi state ishibashi . Any boundary
state must therefore take the form

$|{a}\rangle=\sum_{\begin{subarray}{c}i\in\text{KT}\\\
M_{ii}=1\end{subarray}}a_{i}\|{i}\rangle\\!\rangle$

for some set of coefficients $a_{i}$, which we assume to be real.333The phases
of the $\|{i}\rangle\\!\rangle$ are actually ambiguous. We assume that these
choices of phase have been fixed to make the $a_{i}$ real. As we will see,
this can always be consistently done.

When we impose boundary states $\langle{a}|$ and $|{b}\rangle$ on an interval,
the partition function that counts states on the interval is

$\text{Tr}_{\mathcal{H}_{ab}}(q^{(L/\pi)H})=\sum_{i\in\text{KT}}\chi_{i}(\tau)\sum_{\begin{subarray}{c}j\in\text{KT}\\\
M_{jj}=1\end{subarray}}\mathcal{S}_{ij}a_{j}b_{j}$

where $q=e^{2\pi i\tau}$ and $L$ is the length of the interval. We demand that
the coefficients

$n_{ab}^{i}=\sum_{j}\mathcal{S}_{ij}a_{j}b_{j}$ (1)

occurring in this expression are either positive integers or $\sqrt{2}$ times
positive integers. This is a weakened version of Cardy’s condition cardy  that
allows for the presence of unpaired Majorana modes. Finally, we will look for
a basis of such solutions, known as fundamental boundary states minimalstates
, defined by imposing the additional requirement that

$n_{ab}^{0}=\delta_{ab}$

Here $i=0\in\text{KT}$ denotes the identity module, $(r,s)=(1,1)$. All other
solutions can then be expressed as a linear combination of the fundamental
ones with nonnegative-integer coefficients.

Our goal in what follows will be to simply write down a complete set of
fundamental boundary states consistent with all of the above properties. We do
this for the infinite series and the exceptionals in turn.

###  3.1 Infinite Series

For these models, the quickest way to get the boundary states is to start with
those of the underlying diagonal bosonic minimal model, and project onto their
even part under the model’s unique $\mathbb{Z}_{2}$ global symmetry. After
discarding duplicates, and suitably adjusting the normalisation, we will have
our answer.

We begin by recalling that the boundary states of the $m$th diagonal bosonic
minimal model are given by the Cardy ansatz cardy

$|{i}\rangle_{\text{B}}=\sum_{j\in\text{KT}}\frac{\mathcal{S}_{ij}}{\sqrt{\mathcal{S}_{0j}}}\|{j}\rangle\\!\rangle$
(2)

where $i\in\text{KT}$ labels the states, and the ‘B’ stands for bosonic. As it
stands, these are _not_ valid boundary states of the fermionic theory, because
the sum includes all states $\|{j}\rangle\\!\rangle$, whereas it should only
include those with $M_{jj}=1$. Looking back at the partition function tables
in Section 2, we see the latter condition is equivalent to

$(m+1)r+ms=1\mod 2$

where $j=(r,s)$. But this is the same as the condition for
$\|{j}\rangle\\!\rangle$ to be even under the $\mathbb{Z}_{2}$ global symmetry
of the bosonic model ruelle . Indeed, this symmetry acts as

$U\|{j}\rangle\\!\rangle=(-1)^{(m+1)r+ms+1}\|{j}\rangle\\!\rangle$

It follows that if we simply define our fermionic boundary states to be the
$\mathbb{Z}_{2}$-even projections of the bosonic ones, then the undesirable
states $\|{j}\rangle\\!\rangle$ disappear from the sum, and we are left with

$|{i}\rangle_{F}\;\coloneqq\;\frac{1+U}{2}\,|{i}\rangle_{B}\;=\;\sum_{\begin{subarray}{c}j\in\text{KT}\\\
M_{jj}=1\end{subarray}}\frac{\mathcal{S}_{ij}}{\sqrt{\mathcal{S}_{0j}}}\|{j}\rangle\\!\rangle$
(3)

which, for each $i\in\text{KT}$, defines a valid fermionic boundary state.

Next, we discard duplicates. Currently the states $|{i}\rangle_{F}$ are
overcomplete. We wish to restrict the range of $i$ to eliminate this
redundancy. This can be done by considering the action of the $\mathbb{Z}_{2}$
symmetry on the bosonic boundary states (2), which is

$U|{i}\rangle_{B}=|{i^{\prime}}\rangle_{B}$

Here $i\rightarrow i^{\prime}$ is the involution of the Kac Table defined by

$(r,s)\rightarrow(r,s)^{\prime}=(m-r,s)$ (4)

which corresponds to fusion with the special primary $(m-1,1)$. From (3), we
see that $|{i}\rangle_{F}=|{i^{\prime}}\rangle_{F}$. It follows that we can
eliminate the redundancy by restricting $i$ to lie in a set of representatives
for the equivalence classes of $\text{KT}/(i\sim i^{\prime})$. One possible
choice of a set of representatives, which we will use, is to take the bottom-
left quadrant of the Kac Table, defined by the inequalities

$r\leq m/2\quad\text{and}\quad s\leq(m+1)/2$

This is because the involution (4) acts as a horizontal reflection of the Kac
Table, while the Kac Table is itself defined modulo a combined horizontal +
vertical reflection.

Finally we adjust the normalisation to ensure $n_{ab}^{0}=\delta_{ab}$ between
all pairs of states. To do this it will first prove useful to write the
boundary states in the form

$|{i}\rangle_{F}=\frac{|{i}\rangle_{B}+|{i^{\prime}}\rangle_{B}}{2}$

This makes it easy to calculate the coefficients $n_{ij}^{k}$ between the
fermionic boundary states ${}_{F}\langle{i}|$ and $|{j}\rangle_{F}$ using
known results for the bosonic boundary states cardy . We find

$n_{ij}^{k}=\frac{\mathcal{N}_{ijk}+\mathcal{N}_{i^{\prime}jk}+\mathcal{N}_{ij^{\prime}k}+\mathcal{N}_{i^{\prime}j^{\prime}k}}{4}=\frac{\mathcal{N}_{ijk}+\mathcal{N}_{ij^{\prime}k}}{2}$

From this we can extract the multiplicity of the identity module,

$n_{ij}^{0}=\delta_{ij}\begin{cases}1&i=i^{\prime}\\\ 1/2&i\neq
i^{\prime}\end{cases}$

This is telling us that in order to achieve $n_{ij}^{0}=\delta_{ij}$, we must
rescale the boundary states corresponding to the second case by $\sqrt{2}$.
After performing this rescaling, we arrive at our final result

$|{i}\rangle_{F}=\begin{cases}|{i}\rangle_{B}&i=i^{\prime}\\\
\frac{|{i}\rangle_{B}+|{i^{\prime}}\rangle_{B}}{\sqrt{2}}&i\neq
i^{\prime}\end{cases}$

Not surprisingly, the two cases will turn out to correspond to the two classes
of boundary states. We also take the opportunity to remark that the two cases
have a very simple graphical interpretation: the first corresponds to the
border of the quadrant, the second to the interior.

We are now in a position where we can list the two classes of boundary states,
and demonstrate the consistency of their interval partition functions.

  * •

Class 1 consists of the points $(r,s)$ in the interior of the bottom-left
quadrant of the Kac Table, defined by the inequalities $1\leq r<m/2$ and
$1\leq s<(m+1)/2$. The states take the form

$|{(r,s)}\rangle_{F}=\sqrt{2}\sum_{\begin{subarray}{c}1\leq s^{\prime}\leq
r^{\prime}<m\\\ (m+1)r^{\prime}+ms^{\prime}\text{
odd}\end{subarray}}\frac{\mathcal{S}_{(r,s),(r^{\prime},s^{\prime})}}{\sqrt{\mathcal{S}_{(1,1),(r^{\prime},s^{\prime})}}}\,\|{(r^{\prime},s^{\prime})}\rangle\\!\rangle$

  * •

Class 2 consists of the points $(r,s)$ on the border of the bottom-left
quadrant of the Kac Table, defined by the inequalities $1\leq r\leq m/2$ and
$1\leq s\leq(m+1)/2$ with one of the upper bounds saturated. The states take
an almost identical form, differing only in normalisation:

$|{(r,s)}\rangle_{F}=\sum_{\begin{subarray}{c}1\leq s^{\prime}\leq
r^{\prime}<m\\\ (m+1)r^{\prime}+ms^{\prime}\text{
odd}\end{subarray}}\frac{\mathcal{S}_{(r,s),(r^{\prime},s^{\prime})}}{\sqrt{\mathcal{S}_{(1,1),(r^{\prime},s^{\prime})}}}\,\|{(r^{\prime},s^{\prime})}\rangle\\!\rangle$

On an interval with boundary states $i$ and $j$, the answer for the
coefficients $n_{ij}^{k}$ depends on the classes of $i$ and $j$. There are
three possible combinations to consider. From our earlier results, they are

$\displaystyle\text{1--1:}\qquad n_{ij}^{k}$
$\displaystyle=\mathcal{N}_{ijk}+\mathcal{N}_{ijk^{\prime}}$
$\displaystyle\text{1--2 or 2--1:}\qquad n_{ij}^{k}$
$\displaystyle=\sqrt{2}\,\mathcal{N}_{ijk}$ $\displaystyle\text{2--2:}\qquad
n_{ij}^{k}$ $\displaystyle=\mathcal{N}_{ijk}$

As promised, we see that since the fusion numbers $\mathcal{N}_{ijk}$ are
nonnegative integers (in fact 0 or 1), these answers have the desired
integrality or $\sqrt{2}$-integrality properties.

###  3.2 Exceptionals

For the exceptional models, instead of starting from the underlying bosonic
model, it is perhaps simplest to write down the boundary states directly. We
do this by choosing a pair of seed boundary states and applying fusion to
generate the rest. Most of the details will be the same between the $m=11$ and
the $m=12$ models, so we show the details only for the $m=11$ model.

We start by consulting the table in Section 2 to see when $M_{ii}=1$. We see
that this is true when $i=(r,s)$ with $s=1,5,7,11$ and $r$ odd. It follows
that the admissible boundary states of the fermionic model must lie in the
span

$|{a}\rangle\in\text{span}\Big{\\{}\|{(r,1)}\rangle\\!\rangle,\,\|{(r,5)}\rangle\\!\rangle,\,\|{(r,7)}\rangle\\!\rangle,\,\|{(r,11)}\rangle\\!\rangle:r\text{
odd}\Big{\\}}$ (5)

Our first goal will be to write down the simplest consistent boundary state
obeying this property. To do this, we assume that we already know in advance
what the coefficients $n_{aa}^{i}$ will be. We can then reverse-engineer a
boundary state that does the job:

$a_{i}=\sqrt{\mathcal{S}_{ij}n_{aa}^{j}}$

For most choices of the coefficients $n_{aa}^{i}$, the resulting boundary
state $|{a}\rangle$ will not be allowed, due to containing extra states
$\|{i}\rangle\\!\rangle$ not in the span (5). We would like to make the
simplest choice of $n_{aa}^{i}$ such that it is allowed. Our claim is that
this choice is

$n_{aa}^{i}=\delta_{i,(1,1)}+\delta_{i,(1,5)}+\delta_{i,(1,7)}+\delta_{i,(1,11)}$

##### Proof

To demonstrate the above assertion, we need to show that

$\mathcal{S}_{(r,s),(1,1)}+\mathcal{S}_{(r,s),(1,5)}+\mathcal{S}_{(r,s),(1,7)}+\mathcal{S}_{(r,s),(1,11)}=0\quad\text{for
}s\not\in\\{1,5,7,11\\}$

For this we invoke the explicit formula for the modular $\mathcal{S}$-matrix

$\mathcal{S}_{(r,s),(r^{\prime},s^{\prime})}=\sqrt{\frac{8}{m(m+1)}}\,\sin\\!\left(\frac{\pi
rr^{\prime}}{m}\right)\sin\\!\left(\frac{\pi
ss^{\prime}}{m+1}\right)(-1)^{(r+s)(r^{\prime}+s^{\prime})}$

and well as the trigonometric identity

$\sum_{s^{\prime}=1,5,7,11}\sin\\!\left(\frac{\pi
ss^{\prime}}{12}\right)=\begin{cases}\sqrt{6}&s\in\\{1,5,7,11\\}\\\
0&\text{otherwise}\end{cases}$

Putting these together gives the promised result. $\square$

So far we have obtained a single consistent seed state $|{a}\rangle$ with
coefficients

$a_{i}=\sqrt{\mathcal{S}_{i,(1,1)}+\mathcal{S}_{i,(1,5)}+\mathcal{S}_{i,(1,7)}+\mathcal{S}_{i,(1,11)}}$

Next we cook up a second seed state $|{b}\rangle$ by making a different set of
sign choices in the coefficients. We choose

$b_{(r,s)}=a_{(r,s)}\begin{cases}+1&s=1,7\\\ -1&s=5,11\end{cases}$

where we recall when we discuss the coefficients of a fermionic boundary
state, $r$ is odd and $s=1,5,7,11$. Our next claim is that the state
$|{b}\rangle$ gives rise to the interval partition functions

$n_{bb}^{i}=n_{aa}^{i}\quad\text{and}\quad
n_{ab}^{i}=\sqrt{2}\,(\delta_{i,(1,4)}+\delta_{i,(1,8)})$ (6)

This is telling us that the two seed states $|{a}\rangle$, $|{b}\rangle$ lie
in different SPT classes, but are otherwise consistent.

##### Proof

The first equation in (6) is trivial since $a_{i}$ and $b_{i}$ differ only by
signs. The computation of $n_{ab}^{i}$ is somewhat trickier. This time the key
fact we need is

$(-1)^{[s=5,11]}\sum_{s^{\prime}=1,5,7,11}\sin\\!\left(\frac{\pi
ss^{\prime}}{12}\right)=\sqrt{2}\,\sum_{s^{\prime}=4,8}\sin\\!\left(\frac{\pi
ss^{\prime}}{12}\right)$

Using the above identity, we compute

$a_{(r,s)}b_{(r,s)}=(-1)^{[s=5,11]}\sum_{s^{\prime}=1,5,7,11}\mathcal{S}_{(r,s),(1,s^{\prime})}=\sqrt{2}\,\sum_{s^{\prime}=4,8}\mathcal{S}_{(r,s),(1,s^{\prime})}$

By (1) the claimed result for $n_{ab}^{i}$ immediately follows. $\square$

With the two seed states in hand, all remaining boundary states can now be
generated by fusion affleck ; fuchs . This is a recipe which takes as input a
boundary state, which for us will be either $|{a}\rangle$ or $|{b}\rangle$, as
well as a primary operator $i\in\text{KT}$, and produces a new consistent
boundary state. The recipe for the fused boundary states is

$|{a;i}\rangle=\sum_{\begin{subarray}{c}j\in\text{KT}\\\
M_{jj}=1\end{subarray}}a_{j}\,\frac{\mathcal{S}_{ij}}{\mathcal{S}_{0j}}\,\|{j}\rangle\\!\rangle\qquad|{b;i}\rangle=\sum_{\begin{subarray}{c}j\in\text{KT}\\\
M_{jj}=1\end{subarray}}b_{j}\,\frac{\mathcal{S}_{ij}}{\mathcal{S}_{0j}}\,\|{j}\rangle\\!\rangle$

which reduce back to the seed states when $i=0$. Our final claim is that to
obtain a complete basis of fundamental boundary states, we should let $i$
range over

$i=(r,s)\quad\text{where}\quad r=1,3,5,7,9\text{ and }s=1,2$

This assertion will become evident in a moment when we list the interval
partition functions between all pairs of boundary states. For now, we simply
note that the counting works out: the above family contains $2\cdot 5\cdot
2=20$ states, which coincides with the number of allowed Ishibashi states
$\|{j}\rangle\\!\rangle$, as these correspond to $j=(r,s)$ with $r=1,3,5,7,9$
and $s=1,5,7,11$.

We are now in a position to list the states for the $m=11$ exceptional model,
and their interval partition functions:

  * •

Both classes are labelled by pairs $(r,s)$ with $r=1,3,5,7,9$ and $s=1,2$.
Regarded as elements of the Kac Table, these are all distinct elements. The
two classes of states, after some algebra, are

$\displaystyle|{(r,s)}\rangle_{1}$
$\displaystyle=\sum_{\begin{subarray}{c}r^{\prime}=1,3,5,7,9\\\
s^{\prime}=1,5,7,11\end{subarray}}\left\\{\\!\begin{matrix}[l]\alpha&:\;s^{\prime}=1\\\
\beta&:\;s^{\prime}=5\\\ \beta&:\;s^{\prime}=7\\\
\alpha&:\;s^{\prime}=11\end{matrix}\\!\right\\}\frac{\mathcal{S}_{(r,s),(r^{\prime},s^{\prime})}}{\sqrt{\mathcal{S}_{(1,1),(r^{\prime},s^{\prime})}}}\,\|{(r^{\prime},s^{\prime})}\rangle\\!\rangle$
$\displaystyle|{(r,s)}\rangle_{2}$
$\displaystyle=\sum_{\begin{subarray}{c}r^{\prime}=1,3,5,7,9\\\
s^{\prime}=1,5,7,11\end{subarray}}\left\\{\\!\begin{matrix}[l]\alpha&:\;s^{\prime}=1\\\
-\beta&:\;s^{\prime}=5\\\ \beta&:\;s^{\prime}=7\\\
-\alpha&:\;s^{\prime}=11\end{matrix}\\!\right\\}\frac{\mathcal{S}_{(r,s),(r^{\prime},s^{\prime})}}{\sqrt{\mathcal{S}_{(1,1),(r^{\prime},s^{\prime})}}}\,\|{(r^{\prime},s^{\prime})}\rangle\\!\rangle$

where $\alpha=\sqrt{2(3+\sqrt{3})}$ and $\beta=\sqrt{2(3-\sqrt{3})}$.

  * •

The interval partition functions are

$\displaystyle\text{1--1 or 2--2:}\qquad n_{ij}^{k}$
$\displaystyle=\mathcal{N}_{ij}^{x}\,[\,\mathcal{N}_{x,(1,1)}^{k}+\mathcal{N}_{x,(1,5)}^{k}+\mathcal{N}_{x,(1,7)}^{k}+\mathcal{N}_{x,(1,11)}^{k}\,]$
$\displaystyle\text{1--2 or 2--1:}\qquad n_{ij}^{k}$
$\displaystyle=\sqrt{2}\;\mathcal{N}_{ij}^{x}\,[\,\mathcal{N}_{x,(1,4)}^{k}+\mathcal{N}_{x,(1,8)}^{k}\,]$

where the appearance of the fusion numbers is from Verlinde’s formula verlinde
.

Before we go on, we return to an earlier point and verify the completeness of
the states. Using the above formulas, the multiplicity of the identity module
is

$\displaystyle\text{1--1 or 2--2:}\qquad n_{ij}^{0}$
$\displaystyle=\delta_{ij}+\mathcal{N}_{ij}^{(1,5)}+\mathcal{N}_{ij}^{(1,7)}+\mathcal{N}_{ij}^{(1,11)}$
$\displaystyle\text{1--2 or 2--1:}\qquad n_{ij}^{0}$
$\displaystyle=\sqrt{2}\,[\,\mathcal{N}_{ij}^{(1,4)}+\mathcal{N}_{ij}^{(1,8)}\,]$

One can show that for the range of values of $i$ and $j$ allowed, all the
fusion numbers in the above expression vanish identically, and only the
$\delta_{ij}$ term survives. This shows that $n_{ab}^{0}=\delta_{ab}$ between
all pairs of states $a$ and $b$, establishing completeness.

We now briefly turn to the $m=12$ exceptional. Here, the results are virtually
identical, except with the roles of $r$ and $s$ swapped around:

  * •

Both classes are labelled by pairs $(r,s)$ with $r=1,2$ and $s=1,3,5,7,9,11$.
Regarded as elements of the Kac Table, these are again distinct. The states
are

$\displaystyle|{(r,s)}\rangle_{1}$
$\displaystyle=\sum_{\begin{subarray}{c}r^{\prime}=1,5,7,11\\\
s^{\prime}=1,3,5,7,9,11\end{subarray}}\left\\{\\!\begin{matrix}[l]\alpha&:\;r^{\prime}=1\\\
\beta&:\;r^{\prime}=5\\\ \beta&:\;r^{\prime}=7\\\
\alpha&:\;r^{\prime}=11\end{matrix}\\!\right\\}\frac{\mathcal{S}_{(r,s),(r^{\prime},s^{\prime})}}{\sqrt{\mathcal{S}_{(1,1),(r^{\prime},s^{\prime})}}}\,\|{(r^{\prime},s^{\prime})}\rangle\\!\rangle$
$\displaystyle|{(r,s)}\rangle_{2}$
$\displaystyle=\sum_{\begin{subarray}{c}r^{\prime}=1,5,7,11\\\
s^{\prime}=1,3,5,7,9,11\end{subarray}}\left\\{\\!\begin{matrix}[l]\alpha&:\;r^{\prime}=1\\\
-\beta&:\;r^{\prime}=5\\\ \beta&:\;r^{\prime}=7\\\
-\alpha&:\;r^{\prime}=11\end{matrix}\\!\right\\}\frac{\mathcal{S}_{(r,s),(r^{\prime},s^{\prime})}}{\sqrt{\mathcal{S}_{(1,1),(r^{\prime},s^{\prime})}}}\,\|{(r^{\prime},s^{\prime})}\rangle\\!\rangle$

with the same constants $\alpha$ and $\beta$ as before.

  * •

The interval partition functions are

$\displaystyle\text{1--1 or 2--2:}\qquad n_{ij}^{k}$
$\displaystyle=\mathcal{N}_{ij}^{x}\,[\,\mathcal{N}_{x,(1,1)}^{k}+\mathcal{N}_{x,(5,1)}^{k}+\mathcal{N}_{x,(7,1)}^{k}+\mathcal{N}_{x,(11,1)}^{k}\,]$
$\displaystyle\text{1--2 or 2--1:}\qquad n_{ij}^{k}$
$\displaystyle=\sqrt{2}\;\mathcal{N}_{ij}^{x}\,[\,\mathcal{N}_{x,(4,1)}^{k}+\mathcal{N}_{x,(8,1)}^{k}\,]$

##  4 Anomalous Symmetries

Here we turn to the second main goal of the paper, which is to list all
$\mathbb{Z}_{2}$ global symmetries of the fermionic minimal models – including
potentially those with an anomaly. The motivation for performing this task
comes from looking at the boundary states of the four special models for which
the classes have equal sizes. In Section 3, we found these states to be

  * •

Infinite series, $m=3$:

$\|{0}\rangle\\!\rangle\pm\|{\tfrac{1}{2}}\rangle\\!\rangle$

(Here and in the next example, we label Ishibashi states by conformal
dimensions rather than elements of the Kac Table.)

  * •

Infinite series, $m=4$:

$\displaystyle\tfrac{(5-\sqrt{5})^{1/4}}{10^{1/4}}\Big{(}\|{0}\rangle\\!\rangle\pm\|{\tfrac{3}{2}}\rangle\\!\rangle\Big{)}+\tfrac{(5+\sqrt{5})^{1/4}}{10^{1/4}}\Big{(}\|{\tfrac{3}{5}}\rangle\\!\rangle\pm\|{\tfrac{1}{10}}\rangle\\!\rangle\Big{)}$
$\displaystyle\tfrac{(5+\sqrt{5})^{3/4}}{2^{3/4}\sqrt{5}}\Big{(}\|{0}\rangle\\!\rangle\pm\|{\tfrac{3}{2}}\rangle\\!\rangle\Big{)}-\tfrac{(5-\sqrt{5})^{3/4}}{2^{3/4}\sqrt{5}}\Big{(}\|{\tfrac{3}{5}}\rangle\\!\rangle\pm\|{\tfrac{1}{10}}\rangle\\!\rangle\Big{)}$

  * •

Exceptional, $m=11$:

$\sum_{\begin{subarray}{c}r^{\prime}=1\dots 9,\text{odd}\\\
s^{\prime}=1,5,7,11\end{subarray}}\left\\{\\!\begin{matrix}[l]\alpha&:\;s^{\prime}=1\\\
\pm\beta&:\;s^{\prime}=5\\\ \beta&:\;s^{\prime}=7\\\
\pm\alpha&:\;s^{\prime}=11\end{matrix}\\!\right\\}\frac{\mathcal{S}_{(r,s),(r^{\prime},s^{\prime})}}{\sqrt{\mathcal{S}_{(1,1),(r^{\prime},s^{\prime})}}}\,\|{(r^{\prime},s^{\prime})}\rangle\\!\rangle$

  * •

Exceptional, $m=12$:

$\sum_{\begin{subarray}{c}r^{\prime}=1,5,7,11\\\ s^{\prime}=1\dots
11,\text{odd}\end{subarray}}\left\\{\\!\begin{matrix}[l]\alpha&:\;r^{\prime}=1\\\
\pm\beta&:\;r^{\prime}=5\\\ \beta&:\;r^{\prime}=7\\\
\pm\alpha&:\;r^{\prime}=11\end{matrix}\\!\right\\}\frac{\mathcal{S}_{(r,s),(r^{\prime},s^{\prime})}}{\sqrt{\mathcal{S}_{(1,1),(r^{\prime},s^{\prime})}}}\,\|{(r^{\prime},s^{\prime})}\rangle\\!\rangle$

where, in all cases, the upper choice of sign for the $\pm$ corresponds to the
first class and the lower choice to the second. The above formulas give rise
to an important observation: the two classes differ only by flipping the sign
of a certain subset of the Ishibashi states. This strongly suggests that the
classes are related by the action of a $\mathbb{Z}_{2}$ global symmetry. Our
goal in this section will be to show that these models do indeed have a unique
$\mathbb{Z}_{2}$ symmetry that exchanges the classes, while the remaining
models have no such symmetry.

#### Anomalous $\mathbb{Z}_{2}$ Symmetries

We start by reviewing some basic formalism about $\mathbb{Z}_{2}$ symmetries.
Our perspective is similar to that described in znbootstrap . The Hilbert
space in the AP sector is

$\mathcal{H}_{\text{AP}}=\text{\hbox{\pagecolor[HTML]{e5f6ff}$\mathcal{H}_{\text{AP}}^{\text{even}}$}}\oplus\text{\hbox{\pagecolor[HTML]{fff9d9}$\mathcal{H}_{\text{AP}}^{\text{odd}}$}}=\bigoplus_{i,j\in\text{KT}}N_{ij}\overline{\mathcal{V}_{i}}\otimes\mathcal{V}_{j}=\bigoplus_{\begin{subarray}{c}i,j\in\text{KT}\\\
N_{ij}=1\end{subarray}}\overline{\mathcal{V}_{i}}\otimes\mathcal{V}_{j}$

where the multiplicities $N_{ij}$ can be read off from the partition function
tables in Section 2. Again, these only take the values $N_{ij}=0,1$, allowing
us to make the final step. A $\mathbb{Z}_{2}$ symmetry must act on each
$\overline{\mathcal{V}_{i}}\otimes\mathcal{V}_{j}$ by a sign, which we will
denote as $s_{ij}=\pm 1$. These signs determine the partition function on a
background with an AP-AP spin structure and a symmetry defect along the space
direction:

$\mathcal{Z}\Big{[}\tau;\,\leavevmode\hbox to24.5pt{\vbox
to20.55pt{\pgfpicture\makeatletter\hbox{\hskip 11.49431pt\lower-7.544pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.39488pt}{-7.544pt}\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{{\scriptsize{AP}}}
}}\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}{-11.49431pt}{4.01016pt}\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{{\scriptsize{AP}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{6.40182pt}\pgfsys@lineto{12.80365pt}{6.40182pt}\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}}\;\Big{]}\;=\;\sum_{\begin{subarray}{c}i,j\in\text{KT}\\\
N_{ij}=1\end{subarray}}s_{ij}\,\overline{\chi_{i}(\tau)}\,\chi_{j}(\tau)$

Knowledge of this one partition function then allows further partition
functions to be determined. This is because the partition functions on various
backgrounds are related among each other by acting on $\tau$ with modular
transformations $\mathcal{S}(\tau)=-1/\tau$ and $\mathcal{T}(\tau)=\tau+1$.
The relationships we need are encoded by the diagram

${{\mathcal{Z}\Big{[}\tau;\leavevmode\hbox to24.5pt{\vbox
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} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{6.40182pt}{0.0pt}\pgfsys@lineto{6.40182pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\Big{]}}}$${{\mathcal{Z}\Big{[}\tau;\leavevmode\hbox
to19.25pt{\vbox to20.55pt{\pgfpicture\makeatletter\hbox{\hskip
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} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
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}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$${\scriptsize{AP}}$$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\Big{]}}}$$\scriptstyle{\mathcal{T}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\mathcal{T}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\mathcal{T}}$

which allows all partition functions to be determined from the one on the top-
left. A consistent $\mathbb{Z}_{2}$ symmetry is one for which all these
partition functions admit a sensible expansion into Virasoro characters. For
most of the backgrounds,

APAP APP APP PAP

this means a sum weighted by integers. However for the two special backgrounds

APAP PAP

the partition function is untwisted by any symmetries, and so we impose the
stronger constraint that the weights be _nonnegative_ integers.

Actually, what we have described is not quite correct for anomalous
symmetries. In this case we must weaken the above requirements in several
ways. First, the diagram need only hold projectively, meaning some of the
relationships expressed by the edges are violated by a phase. Second, the
partition functions themselves are ambiguously defined up to a phase. By
adjusting these phases if necessary, which amounts to making a choice of gauge
for the diagram, we can always cast the diagram into the form

${{\mathcal{Z}\Big{[}\tau;\leavevmode\hbox to24.5pt{\vbox
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{}{}\pgfsys@beginscope\pgfsys@invoke{
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ik/8}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\mathcal{S}}$$\scriptstyle{\mathcal{T}\,=\,e^{-\pi
ik/8}}$ (7)

where the phase violations are expressed by the notation
$\mathcal{Z}[\tau;A]\ext@arrow
3359\arrowfill@-\relbar{-\vphantom{\rightarrow}}{}{\mathcal{T}=e^{i\theta}}\mathcal{Z}[\tau;B]$,
which means that $\mathcal{Z}[\tau+1;A]=e^{i\theta}\mathcal{Z}[\tau;B]$ and
vice-versa.444The argument for why this pattern of phases is universal is that
the holonomy of a closed loop in the diagram is the value of
$\exp(-\frac{i\pi}{2}\eta(\mathcal{D}))^{k}$ on a suitable mapping torus,
where $\mathcal{D}$ is a certain 3d Dirac operator. Then because the phases
are universal, they can be read off from the Majorana fermion. For more
details, see wittenphases , and for this particular example, also
anomalyinterplay . Another, more concrete way to see the pattern of phases is
to use the identities $\mathcal{S}^{2}=(\mathcal{ST})^{3}=1$, and the fact
that the top-left partition function is invariant under $\mathcal{T}^{2}$ –
though with this approach the quantisation of $k$ is less obvious. The integer
$k$ is the strength of the anomaly, and is valued mod 8.555Although $k$
appears as the exponent of a 16th root of unity, the shift $k\rightarrow k+8$
is a gauge transformation, so $k$ is valued mod 8 not mod 16. We could have
made $k$ manifestly mod-8 valued by making a different gauge choice. But the
one we have chosen is more convenient in the long run.

To use the diagram, we start with the top-left partition function, and
determine the other partition functions and the integer $k$ by insisting that
the diagram commutes. If no such integer $k$ can be found, we do not have a
consistent symmetry. Otherwise we demand that the partition functions are
sensible, as before. But because we may have needed to adjust their phases to
gauge-fix the diagram, it only makes sense to demand that they have a sensible
expansion into characters up to an overall phase.

There is one final consequence of the anomaly, and that is the possible
appearance of an unpaired Majorana mode in the frustrated sectors. This means
that for the backgrounds

APAP APP PAP PAP

with a symmetry defect that wraps vertically, we should allow a possible
overall factor of $\sqrt{2}$ in the partition function. We previously saw an
example of this phenomenon in Section 1.1.

#### Solving the Constraints

We turn now to the task of solving the above constraints. To do this, we will
recast a subset of the constraints as a set of matrix equations, solve them,
and then check that the resulting solutions satisfy the remaining constraints.

We begin by deriving the set of matrix equations. The partition function on
APAP is encoded by the matrix

$A_{ij}=N_{ij}s_{ij}$

where we recall that $N_{ij}$ is a known matrix of 0s and 1s encoding the
partition function on APAP , and the $s_{ij}$ are a set of unknown signs –
unknown except for that of the identity operator, which we set to be
$s_{11}=+1$. Meanwhile, the corresponding matrix for the APAP partition
function must take the form

$[\sqrt{2}\,]\,e^{i\theta}B_{ij}$

where $B$ is an unknown matrix of nonnegative integers, the phase $\theta$ is
arbitrary, and the $\sqrt{2}$ may or may not be present. Our first equation
arises from the fact that APAP and APAP are related by an
$\mathcal{S}$-transformation. This fact is expressed by

$\mathcal{S}A\,\mathcal{S}=[\sqrt{2}\,]\,e^{i\theta}B$

Actually, since $\mathcal{S}$ is real, the phase $e^{i\theta}$ must equal $\pm
1$, so we obtain

$\mathcal{S}A\,\mathcal{S}=\pm[\sqrt{2}\,]B$ (8)

A second equation arises by considering a $\mathcal{T}^{2}$ transformation of
APAP . Such a transformation preserves the background, but contributes an
anomalous phase of $e^{2\pi ik/8}$. This fact is expressed by

$\mathcal{T}^{-2}B\,\mathcal{T}^{2}=e^{2\pi ik/8}B$ (9)

Equations (8) and (9) will be all we need to determine the symmetries. We
shall analyse them separately depending on whether the $\sqrt{2}$ is present
or absent from (8).

###  4.1 The Case of No Majorana Mode

If (8) contains no $\sqrt{2}$, then we can easily show that the only solutions
are the trivial symmetry and fermion parity. To do this, we will make use of
various Galois-theoretic results that were used extensively to solve the
bosonic version of this problem ruelle . As these results will play an
important role both in this section and the next, we begin with a brief review
of these ideas.

The entries of the modular $\mathcal{S}$-matrix belong to the cyclotomic field
$\mathbb{Q}(\zeta_{n})$ where $n=2m(m+1)$, and $\zeta_{n}=e^{2\pi i/n}$ is an
$n$th root of unity. This field is acted on by Galois transformations
$\sigma_{h}$, labelled by elements $h\in\mathbb{Z}_{n}^{*}$, via

$\sigma_{h}(\zeta_{n})=\zeta_{n}^{h}$

The action of $\sigma_{h}$ on the modular $\mathcal{S}$-matrix is costegannon

$\sigma_{h}(\mathcal{S}_{ij})=\epsilon_{h}(i)\,\mathcal{S}_{\sigma_{h}(i)j}$

where $\epsilon_{h}(i)$ is a sign given by

$\epsilon_{h}(r,s)=\eta_{h}\,\epsilon_{m}(hr)\,\epsilon_{m+1}(hs)$

while $\sigma_{h}$ is a permutation that will not be of interest to us. Here
we have also introduced $\eta_{h}=\sigma_{h}(\sqrt{n})/\sqrt{n}$, a computable
but irrelevant sign, as well as another sign

$\epsilon_{m}(x)=\mathrm{sign}\sin\\!\left(\frac{\pi x}{m}\right)$

known as an $\mathfrak{su}(2)$ affine parity, defined for all $x\neq 0$ mod
$m$.

We can use these facts to derive a useful constraint on $B_{ij}$. Starting
from (8), applying $\sigma_{h}$, and comparing the result back to (8) gives

$B_{ij}=\epsilon_{h}(i)\,\epsilon_{h}(j)\,B_{\sigma_{h}(i)\sigma_{h}(j)}$

Crucially, all the entries of $B_{ij}$ are nonnegative. This means that if
ever the product of signs $\epsilon_{h}(i)\epsilon_{h}(j)$ equals $-1$ for any
$h$, then both sides must be zero. We learn that $B_{ij}$ obeys the _parity
rule_

$B_{ij}\neq 0\quad\text{only if}\quad\epsilon_{h}(i)=\epsilon_{h}(j)\;\forall
h$ (10)

The set of all pairs $(i,j)$ obeying this condition was determined in ruelle ,
and by Result 4 of that paper, they have conformal dimensions satisfying

$h_{i}-h_{j}\in\tfrac{1}{2}\mathbb{Z}\cup\tfrac{1}{3}\mathbb{Z}\cup\tfrac{1}{5}\mathbb{Z}$

Now we recall (9), whose $(i,j)$th component reads

$e^{4\pi i(h_{j}-h_{i})}B_{ij}=e^{2\pi ik/8}B_{ij}$

The phase on the left hand side can never be a nontrivial power of an eight
root of unity. We learn that $k=0$, or in other words, that the symmetry is
non-anomalous.

The existence of a non-anomalous symmetry is a powerful statement. It implies
that if we perform a GSO projection, then the symmetry survives in the
resulting bosonic theory spintqfts ; statesum ; web2d ; orbifoldgroupoids .
Let us call the original symmetry $\alpha$, and denote by $\iota(\alpha)$ its
image in the bosonic theory. Then $\iota(\alpha)$ commutes with
$\iota((-1)^{F})$. In the bosonic minimal models, the only such symmetry is
$\iota((-1)^{F})$ itself, hence $\iota(\alpha)=\iota((-1)^{F})$, which in turn
forces $\alpha=(-1)^{F}$. We therefore learn that the original symmetry must
have been either trivial or fermion parity, as claimed.

We note that it would have also been possible to derive this conclusion
directly from equations (8) and (9), using technical arguments entirely
parallel to those in ruelle . However the above argument, similar to those
used in lessonsramond , is more transparent.

###  4.2 The Case of a Majorana Mode

In the more novel case that (8) contains a $\sqrt{2}$, we show that solutions
exist only for $m=3,4$ in the infinite series and $m=11,12$ in the
exceptionals.

As a first step we show that $m=3,4$ mod 8. Our starting point is (8), from
which we can immediately deduce

$\sqrt{2}\in\mathbb{Q}(\zeta_{n})$

This is true if and only if $n$ is a multiple of 8, which in turn requires

$m=0,3\text{ mod }4$

Assuming from now on that this is the case, we can return to (8) and repeat
our earlier derivation of the parity rule (10) obeyed by $B_{ij}$. We find
that it is modified by the presence of the $\sqrt{2}$, and now takes the form

$B_{ij}\neq 0\quad\text{only
if}\quad\epsilon_{h}(i)=f(h)\,\epsilon_{h}(j)\;\forall h$ (11)

Here $f(h)=\sigma_{h}(\sqrt{2})/\sqrt{2}$ is a sign which comes from the
Galois transformation of the factor of $\sqrt{2}$, and takes the explicit form

$f(h)=\begin{cases}+1&h=1,7\text{ mod }8\\\ -1&h=3,5\text{ mod }8\end{cases}$

To solve condition (11) for $(i,j)$, we write it in terms of affine parities
as

$\epsilon_{m}(hr_{i})\,\epsilon_{m+1}(hs_{i})=f(h)\,\epsilon_{m}(hr_{j})\,\epsilon_{m+1}(hs_{j})\quad\forall
h\in\mathbb{Z}_{2n}^{*}$

Let us assume that $m=0$ mod 4, since the case $m=3$ mod 4 can be treated
almost identically. Then the above condition is equivalent to

$\displaystyle\epsilon_{m}(hr_{i})$
$\displaystyle=f(h)\,\epsilon_{m}(hr_{j})\quad\forall h\in\mathbb{Z}_{2m}^{*}$
$\displaystyle\epsilon_{m+1}(hs_{i})$
$\displaystyle=\epsilon_{m+1}(hs_{j})\quad\forall h\in\mathbb{Z}_{2(m+1)}^{*}$

The solutions to the second equation were determined in ruelle , where Result
3 states that $s_{i}=s_{j}$ or $s_{i}=m+1-s_{j}$. Meanwhile, the solutions to
the first equation are determined by the following conjecture:

##### Conjecture

Suppose $n$ is a multiple of 4, $1\leq x,y\leq n-1$, and
$\epsilon_{n}(hx)=f(h)\,\epsilon_{n}(hy)$ for all $h\in\mathbb{Z}_{2n}^{*}$.
Then the only solutions are

  * •

$n=4$ and $(x,y)=(1,2)$,

  * •

$n=8$ and $(x,y)=(1,3)$,

  * •

$n=12$ and $(x,y)=(1,4),(2,3),(4,5)$,

  * •

$n=24$ and $(x,y)=(1,7),(5,11)$,

and those that can be obtained from them by the solution-preserving
transformations

$(n,x,y)\;\rightarrow\;(n,y,x),\,(n,n-x,y),\,(n,x,n-y),\,(kn,kx,ky)$

It is computationally trivial to verify this conjecture up to $n=1000$, and we
expect, but have not proved, that it holds for all $n$.

We now know all solutions to (11). Without loss of generality, $s_{i}=s_{j}$.
(This uses the equivalence $(r,s)\sim(m-r,m+1-s)$ of the Kac Table.) The
values of $r_{i}$ and $r_{j}$ are given by

  * •

if $m=4a$, $(r_{i},r_{j})=(a,2a)$,

  * •

if $m=8a$, $(r_{i},r_{j})=(a,3a)$,

  * •

if $m=12a$, $(r_{i},r_{j})=(a,4a),(2a,3a),(4a,5a)$,

  * •

if $m=24a$, $(r_{i},r_{j})=(a,7a),(5a,11a)$,

and those related to them under
$(r_{i},r_{j})\rightarrow(r_{j},r_{i}),(m-r_{i},r_{j}),(r_{i},m-r_{j})$.

Next, we use this result to compute the possible values of the phase $e^{4\pi
i(h_{i}-h_{j})}$ among all solutions. Using the identity
$2(h_{i}-h_{j})=(m+1)\frac{r_{i}^{2}-r_{j}^{2}}{2m}$ mod 1, we find the
following contributions to the set of values taken by $e^{4\pi
i(h_{i}-h_{j})}$:

  * •

$m=4a\;\rightarrow\;\zeta_{8}^{\pm a}$

  * •

$m=8a\;\rightarrow\;(-1)^{a}$

  * •

$m=12a\;\rightarrow\;\zeta_{8}^{\pm a},\,\zeta_{24}^{\pm 7a}$

  * •

$m=24a\;\rightarrow\;1$

The payoff of these computations comes from considering the identity (9),
which in components takes the form

$e^{4\pi i(h_{j}-h_{i})}B_{ij}=e^{2\pi ik/8}B_{ij}$

In view of the possible phases that can arise on the left hand side when
$B_{ij}\neq 0$, it is easy to see that the anomaly $k$ must satisfy

$\displaystyle m=4\text{ mod }8\quad$ $\displaystyle\implies\quad
k=\text{odd}$ $\displaystyle m=0\text{ mod }8\quad$
$\displaystyle\implies\quad k=\text{even}$

All of the above analysis also goes through unchanged if instead $m=3$ mod 4.
Combining the results from the two cases, we conclude that

$\displaystyle m=3,4\text{ mod }8\quad$ $\displaystyle\implies\quad
k=\text{odd}$ $\displaystyle m=0,7\text{ mod }8\quad$
$\displaystyle\implies\quad k=\text{even}$

An even value of $k$ is incompatible with our assumption of a $\sqrt{2}$ in
the frustrated partition function delmastro . Indeed, if $k$ were even, then
we could stack with an even number $8-k$ of copies of the Majorana fermion,
obtaining a system with no anomaly yet still with an unpaired Majorana zero in
the frustrated sector. This gives a contradiction, since the presence of an
unpaired zero mode is an exclusive feature of anomalous theories. The only way
to avoid this contradiction is if

$m=3,4\text{ mod }8$

which is what we wanted to show.

While we have just ruled out $m=0,7$ mod 8 by a physical argument, one can
also ask how the mathematical requirements (8) and (9) fail in this case. The
answer is that (8) is generically violated, with $B_{ij}$ failing to be a
nonnegative-integer matrix. The only exceptions are $m=7,8$, where there is a
single solution for $B_{ij}$, but this then goes on to fail (9).

To make further progress with the remaining cases, we deal with the infinite
series and the exceptional models in turn.

####  4.2.1 Infinite Series

For the infinite series when $m=3,4$ mod 8, the matrix $N_{ij}$ takes the form

$N=\bigoplus_{\begin{subarray}{c}i\in\text{KT}/^{\prime}\\\
\xi_{i}=1\end{subarray}}\blockarray{ccc}&ii^{\prime}\\\ \block{r(cc)}i11\\\
i^{\prime}11\\\ $

This notation requires a little explanation. To each weight
$i=(r,s)\in\text{KT}$ there is an associated sign

$\xi_{i}=(-1)^{(m+1)r+ms+1}$

One can show that $\xi_{i}=\xi_{i^{\prime}}$, and that $\xi_{i}=+1\implies
i\neq i^{\prime}$. (Here $(r,s)^{\prime}=(m-r,s)$ is the involution (4)
introduced in Section 3.) These facts are necessary to ensure the above block
decomposition makes sense.

First we use the consistency conditions to constrain the form of $A_{ij}$, the
matrix of unknown signs corresponding to APAP , using arguments analogous to
ruelle . Because $\mathcal{S}N\mathcal{S}=N$, which follows from
$\mathcal{S}$-invariance of APAP , $N_{ij}$ obeys the parity rule (10). Since
$A_{ij}=N_{ij}s_{ij}$, so too does $A_{ij}$. By acting on (8) with a Galois
transformation, comparing it back to (8) and invoking the previous fact, we
find that $A_{ij}$ obeys a _modified permutation rule_

$A_{\sigma_{h}(i)\sigma_{h}(j)}=f(h)A_{ij}$

This states that the many unknown signs in $A_{ij}$ are in fact related.
Rather than using the full power of this equation, though, we shall only use
it for the special case $h=m(m+1)-1$. In this case $\sigma_{h}(i)=i^{\prime}$
and $f(h)=-1$, so we obtain

$A_{i^{\prime}j^{\prime}}=-A_{ij}$

This states that the unknown signs within each $2\times 2$ block of $A_{ij}$
are related, and that $A_{ij}$ takes the form

$A=\bigoplus_{\begin{subarray}{c}i\in\text{KT}/^{\prime}\\\
\xi_{i}=1\end{subarray}}\blockarray{ccc}&ii^{\prime}\\\
\block{r(cc)}i\epsilon_{i}\eta_{i}\\\ i^{\prime}-\eta_{i}-\epsilon_{i}\\\ $
(12)

where $\epsilon_{i}$ and $\eta_{i}$ are a set of unknown signs associated to
the elements of $\text{KT}/^{\prime}$, and the sign corresponding to the
identity operator is $\epsilon_{1}=+1$.

We are now in a position to rule out all but a finite number of models as
having no symmetries. Recall that $B_{ij}$ obeys the modified parity rule
(11). By the conjecture, this implies that unless $m\in\\{3,4,11,12\\}$, we
have

$B_{1i}=B_{1i}=0$

By (8) this tells us that $A$ annihilates the vector $\mathcal{S}_{1i}$ from
both sides. Using (12) and the fact
$\mathcal{S}_{1i}=\mathcal{S}_{1i^{\prime}}>0$, we obtain the contradictory
equations

$\epsilon_{i}=\pm\eta_{i}$

showing immediately there are no solutions.

Finally we return to the special cases $m\in\\{3,4,11,12\\}$ that were exempt
from the above no-go analysis. Listing the solutions for these cases is a
purely finite problem, which can easily be done manually. We find the
following results:

  * •

$m=3$ The $m=3$ model has two symmetries. We specify them by writing out the
partition function diagram (7) explicitly. For the first one, we have

${{\overline{(\chi_{0}+\chi_{\frac{1}{2}})}(\chi_{0}-\chi_{\frac{1}{2}})}}$${{\sqrt{2}\,\overline{(\chi_{0}+\chi_{\frac{1}{2}})}\,\chi_{\frac{1}{16}}}}$${{\sqrt{2}\,\overline{(\chi_{0}-\chi_{\frac{1}{2}})}\,\chi_{\frac{1}{16}}}}$${{\overline{(\chi_{0}-\chi_{\frac{1}{2}})}(\chi_{0}+\chi_{\frac{1}{2}})}}$${{\sqrt{2}\,\overline{\vphantom{(}\chi_{\frac{1}{16}}}(\chi_{0}+\chi_{\frac{1}{2}})}}$${{\sqrt{2}\,\overline{\vphantom{(}\chi_{\frac{1}{16}}}(\chi_{0}-\chi_{\frac{1}{2}})}}$

with anomaly $k=1$. The other symmetry is given by flipping the diagram upside
down, an operation which corresponds to composing with $(-1)^{F}$, and has the
opposite anomaly $k=-1$. These symmetries are of course simply left and right
chiral fermion parity of the Majorana fermion.

  * •

$m=4$ The $m=4$ model has a similar structure, with two symmetries related by
$(-1)^{F}$ and opposite anomalies. We shall therefore only give details of the
first one. The partition function diagram (7) in this case takes the form

${{\begin{aligned}
&\overline{(\chi_{0}+\chi_{\frac{3}{2}})}(\chi_{0}-\chi_{\frac{3}{2}})\\\
{}+{}&\overline{(\chi_{\frac{3}{5}}+\chi_{\frac{1}{10}})}(\chi_{\frac{3}{5}}-\chi_{\frac{1}{10}})\end{aligned}}}$${{\begin{aligned}
\sqrt{2}\,[\,&\overline{(\chi_{0}+\chi_{\frac{3}{2}})}\,\chi_{\frac{7}{16}}\\\
{}+{}&\overline{(\chi_{\frac{3}{5}}+\chi_{\frac{1}{10}})}\,\chi_{\frac{3}{80}}\,]\end{aligned}}}$${{\begin{aligned}
\sqrt{2}\,[\,&\overline{(\chi_{0}-\chi_{\frac{3}{2}})}\,\chi_{\frac{7}{16}}\\\
{}+{}&\overline{(\chi_{\frac{3}{5}}-\chi_{\frac{1}{10}})}\,\chi_{\frac{3}{80}}\,]\end{aligned}}}$${{\begin{aligned}
&\overline{(\chi_{0}-\chi_{\frac{3}{2}})}(\chi_{0}+\chi_{\frac{3}{2}})\\\
{}+{}&\overline{(\chi_{\frac{3}{5}}-\chi_{\frac{1}{10}})}(\chi_{\frac{3}{5}}+\chi_{\frac{1}{10}})\end{aligned}}}$${{\begin{aligned}
\sqrt{2}\,[\,&\overline{\chi_{\frac{7}{16}}}\,(\chi_{0}+\chi_{\frac{3}{2}})\\\
{}+{}&\overline{\chi_{\frac{3}{80}}}\,(\chi_{\frac{3}{5}}+\chi_{\frac{1}{10}})\,]\end{aligned}}}$${{\begin{aligned}
\sqrt{2}\,[\,&\overline{\chi_{\frac{7}{16}}}\,(\chi_{0}-\chi_{\frac{3}{2}})\\\
{}+{}&\overline{\chi_{\frac{3}{80}}}\,(\chi_{\frac{3}{5}}-\chi_{\frac{1}{10}})\,]\end{aligned}}}$

and the anomaly is $k=-1$.

  * •

$m=11,12$ The models with $m=11,12$ turn out to have no symmetries.

####  4.2.2 Exceptionals

The above analysis can also be carried out for the two exceptional models at
$m=11,12$. The results have the same structure as for $m=3,4$ earlier: there
are two symmetries, related to each other by $(-1)^{F}$, with opposite
anomalies. Below we list the symmetries for the two models. This time we will
only list the partition functions on APAP and APAP , as all others are related
to these by the same pattern as in previous cases.

  * •

$m=11$ For the $m=11$ exceptional, we have

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$\displaystyle={}$
$\displaystyle\sum_{\mathclap{r=1,\text{odd}}}^{9}\overline{(\chi_{r,1}+\chi_{r,7}+\chi_{r,5}+\chi_{r,11})}(\chi_{r,1}+\chi_{r,7}-\chi_{r,5}-\chi_{r,11})$
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$\displaystyle={}$ $\displaystyle\sqrt{2}\,$
$\displaystyle\sum_{\mathclap{r=1,\text{odd}}}^{9}\overline{(\chi_{r,1}+\chi_{r,7}+\chi_{r,5}+\chi_{r,11})}(\chi_{r,4}+\chi_{r,8})$

with anomaly $k=-1$.

  * •

$m=12$ The $m=12$ exceptional is similar, but with the roles of $r$ and $s$
reversed

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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\;\Big{]}$
$\displaystyle={}$
$\displaystyle\sum_{\mathclap{s=1,\text{odd}}}^{11}\overline{(\chi_{1,s}+\chi_{7,s}+\chi_{5,s}+\chi_{11,s})}(\chi_{1,s}+\chi_{7,s}-\chi_{5,s}-\chi_{11,s})$
$\displaystyle\mathcal{Z}\Big{[}\tau;\leavevmode\hbox to24.5pt{\vbox
to20.55pt{\pgfpicture\makeatletter\hbox{\hskip 11.49431pt\lower-7.544pt\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@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.39488pt}{-7.544pt}\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{{\scriptsize{AP}}}
}}\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}{-11.49431pt}{4.01016pt}\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{{\scriptsize{AP}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{6.40182pt}{0.0pt}\pgfsys@lineto{6.40182pt}{12.80365pt}\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}}\;\Big{]}$
$\displaystyle={}$ $\displaystyle\sqrt{2}\,$
$\displaystyle\sum_{\mathclap{s=1,\text{odd}}}^{11}\overline{(\chi_{1,s}+\chi_{7,s}+\chi_{5,s}+\chi_{11,s})}(\chi_{4,s}+\chi_{8,s})$

and anomaly $k=1$.

Before going on, we pause to note that the earlier results for the infinite
series can also be rewritten to look more like the results above. For $m=3$,
we have

$\displaystyle\mathcal{Z}\Big{[}\tau;\leavevmode\hbox to24.5pt{\vbox
to20.55pt{\pgfpicture\makeatletter\hbox{\hskip 11.49431pt\lower-7.544pt\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@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.39488pt}{-7.544pt}\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{{\scriptsize{AP}}}
}}\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}{-11.49431pt}{4.01016pt}\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{{\scriptsize{AP}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{6.40182pt}\pgfsys@lineto{12.80365pt}{6.40182pt}\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}}\;\Big{]}$
$\displaystyle={}$
$\displaystyle\sum_{\mathclap{r=1,\text{odd}}}^{1}\overline{(\chi_{r,1}+\chi_{r,3})}(\chi_{r,1}-\chi_{r,3})$
$\displaystyle\mathcal{Z}\Big{[}\tau;\leavevmode\hbox to24.5pt{\vbox
to20.55pt{\pgfpicture\makeatletter\hbox{\hskip 11.49431pt\lower-7.544pt\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@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.39488pt}{-7.544pt}\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{{\scriptsize{AP}}}
}}\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}{-11.49431pt}{4.01016pt}\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{{\scriptsize{AP}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{6.40182pt}{0.0pt}\pgfsys@lineto{6.40182pt}{12.80365pt}\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}}\;\Big{]}$
$\displaystyle={}$ $\displaystyle\sqrt{2}\,$
$\displaystyle\sum_{\mathclap{r=1,\text{odd}}}^{1}\overline{(\chi_{r,1}+\chi_{r,3})}\,\chi_{r,2}$

while for $m=4$ we have

$\displaystyle\mathcal{Z}\Big{[}\tau;\leavevmode\hbox to24.5pt{\vbox
to20.55pt{\pgfpicture\makeatletter\hbox{\hskip 11.49431pt\lower-7.544pt\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@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.39488pt}{-7.544pt}\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{{\scriptsize{AP}}}
}}\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}{-11.49431pt}{4.01016pt}\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{{\scriptsize{AP}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{6.40182pt}\pgfsys@lineto{12.80365pt}{6.40182pt}\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}}\;\Big{]}$
$\displaystyle={}$
$\displaystyle\sum_{\mathclap{s=1,\text{odd}}}^{3}\overline{(\chi_{1,s}+\chi_{3,s})}(\chi_{1,s}-\chi_{3,s})$
$\displaystyle\mathcal{Z}\Big{[}\tau;\leavevmode\hbox to24.5pt{\vbox
to20.55pt{\pgfpicture\makeatletter\hbox{\hskip 11.49431pt\lower-7.544pt\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@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.39488pt}{-7.544pt}\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{{\scriptsize{AP}}}
}}\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}{-11.49431pt}{4.01016pt}\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{{\scriptsize{AP}}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{6.40182pt}{0.0pt}\pgfsys@lineto{6.40182pt}{12.80365pt}\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}}\;\Big{]}$
$\displaystyle={}$ $\displaystyle\sqrt{2}\,$
$\displaystyle\sum_{\mathclap{s=1,\text{odd}}}^{3}\overline{(\chi_{1,s}+\chi_{3,s})}\,\chi_{2,s}$

##  5 Discussion

We conclude with some comments on our results. First we return to the earlier
claim that the anomalous symmetries, where they exist, exchange the two
classes of boundary states. This follows at a glance from the APAP partition
functions of Section 4.2. Indeed the coefficient of $|\chi_{i}|^{2}$ in this
partition function determines the sign with which the symmetry acts on the
Ishibashi state $\|{i}\rangle\\!\rangle$. For example, in the $m=11$
exceptional, the charges of Ishibashi states are

$\|{(r,1)}\rangle\\!\rangle$ | $\|{(r,5)}\rangle\\!\rangle$ | $\|{(r,7)}\rangle\\!\rangle$ | $\|{(r,11)}\rangle\\!\rangle$
---|---|---|---
$+1$ | $-1$ | $+1$ | $-1$

These are precisely the charges needed to exchange the boundary states we
wrote down in Section 3. The same conclusion also holds in the other models.

Independently of the details of any particular model, the fact the symmetry
exchanges the classes can also be understood from the anomaly $k$ being odd.
This follows by a stacking argument. Suppose a theory has a boundary state
$|{a}\rangle$ and a symmetry $U$ with odd anomaly $k$. Then if we stack the
theory with $-k$ mod 8 copies of the Majorana fermion, the resulting theory
has a boundary state $|{a}\rangle\otimes|{+}\rangle^{-k}$ and a non-anomalous
symmetry $U\otimes(-1)^{F_{L}}$. Acting on the state with the symmetry gives a
new state $U|{a}\rangle\otimes|{-}\rangle^{-k}$, which must lie in the same
class since the symmetry is non-anomalous. From Section 1.1, we know that
$|{+}\rangle^{-k}$ and $|{-}\rangle^{-k}$ are in different classes when $k$ is
odd. Therefore so too must $|{a}\rangle$ and $U|{a}\rangle$, as claimed.

We would also like to return to a subtlety we felt was best left unaddressed
in Section 4, but are now in a position to close. A theory with a
$\mathbb{Z}_{2}$ symmetry actually has six more partition functions that we
did not consider. These fit into an orbit diagram, analogous to (7), which
looks like

${{\mathcal{Z}\Big{[}\tau;\leavevmode\hbox to24.5pt{\vbox
to20.55pt{\pgfpicture\makeatletter\hbox{\hskip 11.49431pt\lower-7.544pt\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@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.01988pt}{-7.544pt}\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{$${\scriptsize{P}}$$}
}}\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}{-11.49431pt}{3.90182pt}\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{$${\scriptsize{AP}}$$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setdash{3.0pt,2.0pt}{0.0pt}\pgfsys@invoke{
}{}\pgfsys@moveto{0.0pt}{6.40182pt}\pgfsys@lineto{12.80365pt}{6.40182pt}\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}}\;\Big{]}}}$${{\mathcal{Z}\Big{[}\tau;\leavevmode\hbox
to19.25pt{\vbox to20.55pt{\pgfpicture\makeatletter\hbox{\hskip
6.24431pt\lower-7.544pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{}
{}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{12.80365pt}\pgfsys@lineto{12.80365pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{12.80365pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{1.39488pt}{-7.544pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$${\scriptsize{AP}}$$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{{}{}}}{{}{}}
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-6.24431pt}{3.90182pt}\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{$${\scriptsize{P}}$$}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {}{{}}{}
{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{6.40182pt}{0.0pt}\pgfsys@lineto{6.40182pt}{12.80365pt}\pgfsys@stroke\pgfsys@invoke{
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