Fermionization and boundary states in 1+1 dimensions

Yoshiki Fukusumi1, Yuji Tachikawa2, and Yunqin Zheng2,3

1 | Department of Physics, Faculty of Science,
---|---
| University of Zagreb, HR-10000 Zagreb, Croatia
2 | Kavli Institute for the Physics and Mathematics of the Universe,
| University of Tokyo, Kashiwa, Chiba 277-8583, Japan
3 | Institute for Solid State Physics,
| University of Tokyo, Kashiwa, Chiba 277-8581, Japan

In the last few years it was realized that every fermionic theory in 1+1
dimensions is a generalized Jordan-Wigner transform of a bosonic theory with a
non-anomalous $\mathbb{Z}_{2}$ symmetry. In this note we determine how the
boundary states are mapped under this correspondence. We also interpret this
mapping as the fusion of the original boundary with the fermionization
interface.

###### Contents

  1. 1 Introduction and summary
  2. 2 Mapping of the boundary states
    1. 2.1 Gauging $\mathbb{Z}_{2}$: $\mathsf{A}\leftrightarrow\mathsf{D}$
    2. 2.2 Fermionizing $\mathbb{Z}_{2}$: $\mathsf{A}\to\mathsf{F},\mathsf{F}^{\prime}$
    3. 2.3 Comments
  3. 3 Examples
    1. 3.1 Fermionic $SU(2)_{k}$ WZW model
    2. 3.2 The Ising model and a massless Majorana fermion
    3. 3.3 $Spin(N)_{1}$ WZW Model and $N$ Majorana Fermions
    4. 3.4 Maldacena-Ludwig boundary state of eight Majorana fermions
  4. 4 Interfaces and boundary states
    1. 4.1 $\mathsf{A}\leftrightarrow\mathsf{D}$
    2. 4.2 $\mathsf{A}\leftrightarrow\mathsf{F},\mathsf{F}^{\prime}$
    3. 4.3 The special case $\mathsf{A}\simeq\mathsf{D}$ and the anomalous $\mathbb{Z}_{2}$ symmetry of $\mathsf{F}\simeq\mathsf{F}^{\prime}$
  5. A RCFTs and their $\mathbb{Z}_{2}$ symmetries

##  1 Introduction and summary

It is a classic result that the Jordan-Wigner transformation [1] allows us to
map any 1-dimensional spin chain written in terms of $\sigma_{X,Y,Z}^{(i)}$
preserving a $\mathbb{Z}_{2}$ symmetry generated by $\prod\sigma_{Z}^{(i)}$ in
terms of fermion operators $\psi^{(i)}$, $\bar{\psi}^{(i)}$. One of the most
famous applications of this transformation is the solution of the Ising model
by mapping it into a free fermionic chain [2]. This transformation was so
effective that in the older literature the fermionic model and the bosonic
model related in this manner were not carefully distinguished. It is to be
stressed, however, that the models related in this manner do differ, and have
different energy spectrum once boundary conditions are carefully taken
account.

Therefore, there are much that could be gained by maintaining this
distinction. For example, we now know that an abstract version of the Jordan-
Wigner transformation can be formulated in the continuum, and any continuum
fermionic theory in 1+1 dimensions is obtained from a bosonic
(1+1)-dimensional system with a non-anomalous $\mathbb{Z}_{2}$ symmetry by
this transformation [3, 4, 5, 6, 7, 8], generalizing the relation between the
Ising conformal field theory (CFT) and the Majorana fermion CFT. It was also
learned recently that fermionic versions of unitary minimal models can be
constructed in this manner [9, 10, 11]111There should also be a generalization
from $\mathbb{Z}_{2}$ to $\mathbb{Z}_{N}$, where $\mathbb{Z}_{N}$ symmetric
critical points are related to $\mathbb{Z}_{N}$ parafermionic models.
Historically, such generalization was first suggested in [12] in 1985, where
it was conjectured that the $m$-multicritical points of $\mathbb{Z}_{N}$
Fateev-Zamolodchikov model should be described by the second parafermionic
models with parameter $m$ for $N\geq 3$ in [12]; the corresponding bosonic
coset models were later given in [13, 14]. From this perspective, it was
equally natural to expect that the $m$-multicritical Ising model should also
have a fermionic counterpart, whose explicit lattice construction was given
only very recently in [10]. It would be interesting to revisit and develop
these points further. See e.g. [15, 16, 17] for recent works. 222See also [18]
for a recent study of fermionic RCFTs with small number of irreducible
representations. . The aim of this paper is to study how this map between
fermionic models and bosonic models with $\mathbb{Z}_{2}$ symmetry affects the
boundary states.

Let us first recall the basic facts of the continuum version of the Jordan-
Wigner transformation. We start from a bosonic theory $\mathsf{A}$ with a non-
anomalous $\mathbb{Z}_{2}$ global symmetry. Another bosonic theory
$\mathsf{D}$ can be introduced by

$\mathsf{D}:=\mathsf{A}/\mathbb{Z}_{2},$ (1.1)

where dividing by $\mathbb{Z}_{2}$ stands for orbifolding, or equivalently
gauging of the $\mathbb{Z}_{2}$ symmetry. This theory $\mathsf{D}$ is known to
possess a dual $\mathbb{Z}_{2}$ symmetry [19] and gauging it reproduces the
original theory: $\mathsf{A}=\mathsf{D}/\mathbb{Z}_{2}$. The fermionic theory
$\mathsf{F}$ is obtained with the help of another theory $\mathsf{Kitaev}$,
the low-energy limit of the topologically nontrivial phase of Kitaev’s
Majorana chain [20].333This theory is also known as the Arf theory, since its
partition function on a closed two-dimensional surface with spin structure is
given by its Arf invariant.  The formula is

$\mathsf{F}:={(\mathsf{A}\times\mathsf{Kitaev})}/{\mathbb{Z}_{2}}$ (1.2)

where the orbifold is with respect to the diagonal $\mathbb{Z}_{2}$ symmetry.
We can also define a closely related theory $\mathsf{F}^{\prime}$ starting
from $\mathsf{D}$

$\mathsf{F}^{\prime}:={(\mathsf{D}\times\mathsf{Kitaev})}/{\mathbb{Z}_{2}}$
(1.3)

which satisfies444 In terms of the partition function coupled to
$\mathbb{Z}_{2}$ background field $B$ and spin structure $\rho$, we have
$Z_{\mathsf{F}}(\rho)=2^{-g}\sum_{B}Z_{\mathsf{A}}(B)(-1)^{\mathrm{Arf}[\rho+B]}.$
Similarly, we have
$Z_{\mathsf{F}^{\prime}}(\rho)=2^{-g}\sum_{B}Z_{\mathsf{A}}(B)(-1)^{q_{\rho}[B]}$
where $q_{\rho}[B]\equiv\mathrm{Arf}[\rho+B]-\mathrm{Arf}[\rho]$ is the
quadratic form associated to the spin structure. These two are related as
follows:
$Z_{\mathsf{F}^{\prime}}[\rho]=Z_{\mathsf{F}}[\rho](-1)^{\mathrm{Arf}[\rho]}.$
In the works [6, 10] by one of the authors (YT) and other collaborators,
$\mathsf{F}$ is defined to be the fermionization of $\mathsf{A}$, whereas in
most of the other works of this topic e.g. [3, 5, 21, 8],
$\mathsf{F}^{\prime}$ is defined to be the fermionization of $\mathsf{A}$
instead. We note that when $\mathsf{A}$ is the completely trivial theory, the
theory $\mathsf{F}$ is trivial while $\mathsf{F}^{\prime}$ is equal to
$\mathsf{Kitaev}$.

$\mathsf{F}^{\prime}=\mathsf{F}\times\mathsf{Kitaev}.$ (1.4)

We place these theories on a circle. The bosonic theories can be put on a
circle without or with $\mathbb{Z}_{2}$ twist. Similarly, the fermionic
theories can be put on the Neveu-Schwarz (NS) sector, or equivalently the
antiperiodic sector, and the Ramond (R) sector, or equivalently the periodic
sector. The Hilbert spaces of theories $\mathsf{A}$, $\mathsf{D}$,
$\mathsf{F}$, $\mathsf{F}^{\prime}$ are known to decompose in the following
manner:

$\begin{array}[]{cc}\begin{array}[]{c|cc}\mathsf{A}&\text{untwisted}&\text{twisted}\\\
\hline\cr\text{even}&\pagecolor{lightblue}S&\pagecolor{lightpink}U\\\
\text{odd}&\pagecolor{lightgreen}T&\pagecolor{lightyellow}V\\\
\hline\cr\hline\cr\mathsf{F}&\text{NS}&\text{R}\\\
\hline\cr(-1)^{F}=+1&\pagecolor{lightblue}S&\pagecolor{lightpink}U\\\
(-1)^{F}=-1&\pagecolor{lightyellow}V&\pagecolor{lightgreen}T\end{array}&\begin{array}[]{c|cc}\mathsf{D}&\text{untwisted}&\text{twisted}\\\
\hline\cr\text{even}&\pagecolor{lightblue}S&\pagecolor{lightgreen}T\\\
\text{odd}&\pagecolor{lightpink}U&\pagecolor{lightyellow}V\\\
\hline\cr\hline\cr\mathsf{F}^{\prime}&\text{NS}&\text{R}\\\
\hline\cr(-1)^{F}=+1&\pagecolor{lightblue}S&\pagecolor{lightgreen}T\\\
(-1)^{F}=-1&\pagecolor{lightyellow}V&\pagecolor{lightpink}U\end{array}\end{array}$
(1.5)

We denote the Hilbert spaces in the respective sectors by
$\mathcal{H}_{S,T,U,V}$, so that the untwisted and twisted Hilbert spaces of
the theory $\mathsf{A}$ have the decomposition

$\displaystyle\mathcal{H}^{\mathsf{A}}$
$\displaystyle=\mathcal{H}_{S}\oplus\mathcal{H}_{T},$ (1.6)
$\displaystyle\mathcal{H}^{\mathsf{A}}_{\text{tw}}$
$\displaystyle=\mathcal{H}_{U}\oplus\mathcal{H}_{V}$ (1.7)

for example. It should be noted that the only difference of the theories
$\mathsf{F}$ and $\mathsf{F}^{\prime}$ are in the assignment of the fermionic
parity on the R-sector. This makes it tricky to distinguish them in the
lattice realizations on closed chains.

We now consider boundary conditions of the bosonic theory $\mathsf{A}$, which
can be broadly classified into two types by their behavior under the
$\mathbb{Z}_{2}$ symmetry. Namely, there are $\mathbb{Z}_{2}$ invariant ones,
which we denote by $i$, $j$, …, and the ones which break the $\mathbb{Z}_{2}$
symmetry, which therefore form pairs we can denote by $a\pm$, $b\pm$, …. We
denote their boundary states as $\ket{i}^{\mathsf{A}}$,
$\ket{a\pm}^{\mathsf{A}}$, etc. Note that the $\mathbb{Z}_{2}$-invariant
states can be placed on a circle which is twisted by the $\mathbb{Z}_{2}$
operation, which defines twisted boundary states
$\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$ taking values in the twisted Hilbert
space.

Our main result is the expressions of the boundary states of the theories
$\mathsf{D}$, $\mathsf{F}$ and $\mathsf{F}^{\prime}$, which will be derived in
detail later. Here we will simply summarize them. In the
$\mathbb{Z}_{2}$-orbifold theory $\mathsf{D}$, the $\mathbb{Z}_{2}$-invariant
boundary condition $i$ of the original theory $\mathsf{A}$ splits into two
types $i\pm$, which are exchanged under the emergent
$\widehat{\mathbb{Z}}_{2}$ symmetry. Conversely, the pair of
$\mathbb{Z}_{2}$-breaking boundary conditions $a\pm$ is combined into a single
$\mathbb{Z}_{2}$-invariant condition $a$. Their boundary states are given as
follows:  $\displaystyle\ket{i+}^{\mathsf{D}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{i}^{\mathsf{A}}+\ket{i}^{\mathsf{A}}_{\mathrm{tw}}),$
$\displaystyle\ket{i-}^{\mathsf{D}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{i}^{\mathsf{A}}-\ket{i}^{\mathsf{A}}_{\mathrm{tw}}),$
(1.8) $\displaystyle\ket{a}^{\mathsf{D}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}}),$
$\displaystyle\ket{a}^{\mathsf{D}}_{\mathrm{tw}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}}).$
Here $\widehat{\mathbb{Z}}_{2}$ exchanges $\ket{i+}^{\mathsf{D}}$ and
$\ket{i-}^{\mathsf{D}}$, while $\ket{a}^{\mathsf{D}}$ and
$\ket{a}_{\mathrm{tw}}^{\mathsf{D}}$ are both $\widehat{\mathbb{Z}}_{2}$ even.

In the fermionic theories, boundary conditions can be classified into two
types, those with and without an unpaired Majorana fermion zero mode. In
particular, $\mathsf{Kitaev}$ itself has such an unpaired Majorana fermion
zero mode [22]. Let us start from a $\mathbb{Z}_{2}$-invariant boundary
condition $i$ of the theory $\mathsf{A}$. When stacked with $\mathsf{Kitaev}$,
this boundary has a Majorana fermion $\psi$, but the $\mathbb{Z}_{2}$ quotient
removes it since this fermion is $\mathbb{Z}_{2}$ odd. Conversely, when we
start from a $\mathbb{Z}_{2}$-breaking boundary condition $a\pm$, this
additional $\mathbb{Z}_{2}$-odd Majorana fermion $\psi$ is kept since the
$\mathbb{Z}_{2}$ gauge symmetry is broken here, resulting in a boundary with a
Majorana fermion. We denote the resulting boundary states as
$\ket{i}^{\mathsf{F}}$ and $\ket{a\psi}^{\mathsf{F}}$, etc. The corresponding
boundary states are given as follows:
$\displaystyle\ket{i}_{\mathrm{NS}}^{\mathsf{F}}$
$\displaystyle=\ket{i}^{\mathsf{A}},$
$\displaystyle\ket{i}_{\mathrm{R}}^{\mathsf{F}}$
$\displaystyle=\ket{i}_{\mathrm{tw}}^{\mathsf{A}},$ (1.9)
$\displaystyle\ket{a\psi}_{\mathrm{NS}}^{\mathsf{F}}$
$\displaystyle=\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}},$
$\displaystyle\ket{a\psi}_{\mathrm{R}}^{\mathsf{F}}$
$\displaystyle=\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}}.$ Here
$\ket{a\psi}_{\text{R}}^{\mathsf{F}}$ has $(-1)^{F}=-1$ while the other three
have $(-1)^{F}=+1$. We note that the boundary state in the R-sector of a
boundary condition with an unpaired Majorana fermion vanishes due to the
Majorana fermion zero mode coming from the periodicity. Our
$\ket{a\psi}_{\text{R}}^{\mathsf{F}}$ is defined with an insertion of the
Majorana fermion operator to absorb this zero mode. We do not repeat this
comment unless absolutely necessary.

In the theory $\mathsf{F}^{\prime}$, the assignment is reversed, and we find
$\ket{i\psi}^{\mathsf{F}^{\prime}}$ and $\ket{a}^{\mathsf{F}^{\prime}}$. These
boundary states are given as follows:
$\displaystyle\ket{i\psi}_{\mathrm{NS}}^{\mathsf{F}^{\prime}}$
$\displaystyle=\sqrt{2}\ket{i}^{\mathsf{A}},$
$\displaystyle\ket{i\psi}_{\mathrm{R}}^{\mathsf{F}^{\prime}}$
$\displaystyle=\sqrt{2}\ket{i}_{\mathrm{tw}}^{\mathsf{A}},$ (1.10)
$\displaystyle\ket{a}_{\mathrm{NS}}^{\mathsf{F}^{\prime}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}}),$
$\displaystyle\ket{a}_{\mathrm{R}}^{\mathsf{F}^{\prime}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}}).$
Again $\ket{i\psi}_{\mathrm{R}}^{\mathsf{F}}$ has $(-1)^{F}=-1$ while the
other three have $(-1)^{F}=+1$. Note that the theory $\mathsf{F}$ and
$\mathsf{F}^{\prime}$ have rather different sets of boundary conditions if
$\mathsf{A}\neq\mathsf{D}$, even though the NS sectors of $\mathsf{F}$ and
$\mathsf{F}^{\prime}$ are the same and the R sectors of $\mathsf{F}$ and
$\mathsf{F}^{\prime}$ only differ in their assignments of the fermion parity.

We find it also useful to present our results in a different way, showing
explicitly to which sector $\mathcal{H}_{S,T,U,V}$ the boundary states belong.
We first organize the boundary states of theory $\mathsf{A}$ into four
blocks555Note that $\ket{a}_{S}$ and $\ket{a}_{T}$ are not boundary states of
$\mathsf{A}$. They are $\mathbb{Z}_{2}$ even and odd linear combinations of
boundary states in the untwisted sector. Likewise, $\ket{i}_{S}$ and
$\ket{i}_{U}$ are not boundary states of $\mathsf{D}$, but instead are linear
combinations of them.  (1.5):

$\begin{array}[]{c|lc}\mathsf{A}&\lx@intercol\hfil\text{untwisted}\hfil\lx@intercol&\text{twisted}\\\
\hline\cr\text{even}&\pagecolor{lightblue}|i\rangle_{S}=|i\rangle^{\mathsf{A}}&\pagecolor{lightpink}|i\rangle_{U}=|i\rangle_{\mathrm{tw}}^{\mathsf{A}}\\\
&\pagecolor{lightblue}|a\rangle_{S}=\frac{1}{\sqrt{2}}\left(|a+\rangle^{\mathsf{A}}+|a-\rangle^{\mathsf{A}}\right)&\pagecolor{lightpink}\hfil\\\
\text{odd}&\pagecolor{lightgreen}|a\rangle_{T}=\frac{1}{\sqrt{2}}\left(|a+\rangle^{\mathsf{A}}-|a-\rangle^{\mathsf{A}}\right)&\pagecolor{lightyellow}\hfil\end{array}$
(1.11)

Then the boundary states of theory $\mathsf{D}$ can be presented as:

$\begin{array}[]{c|lc}\mathsf{D}&\lx@intercol\hfil\text{untwisted}\hfil\lx@intercol&\text{twisted}\\\
\hline\cr\text{even}&\pagecolor{lightblue}|a\rangle_{S}=|a\rangle^{\mathsf{D}}&\pagecolor{lightgreen}|a\rangle_{T}=|a\rangle_{\mathrm{tw}}^{\mathsf{D}}\\\
&\pagecolor{lightblue}|i\rangle_{S}=\frac{1}{\sqrt{2}}\left(|i+\rangle^{\mathsf{D}}+|i-\rangle^{\mathsf{D}}\right)&\pagecolor{lightgreen}\hfil\\\
\text{odd}&\pagecolor{lightpink}|i\rangle_{U}=\frac{1}{\sqrt{2}}\left(|i+\rangle^{\mathsf{D}}-|i-\rangle^{\mathsf{D}}\right)&\pagecolor{lightyellow}\hfil\end{array}$
(1.12)

For the theory $\mathsf{F}$ and $\mathsf{F}^{\prime}$ they are given by

$\begin{array}[]{c|rlrl}\mathsf{F}&\lx@intercol\hfil\text{NS}\hfil\lx@intercol&\lx@intercol\hfil\text{R}\hfil\lx@intercol\\\
\hline\cr(-1)^{F}=+1&\pagecolor{lightblue}|i\rangle_{S}&\pagecolor{lightblue}=\ket{i}_{\text{NS}}^{\mathsf{F}}&\pagecolor{lightpink}|i\rangle_{U}&\pagecolor{lightpink}=|i\rangle_{\text{R}}^{\mathsf{F}}\\\
&\pagecolor{lightblue}\sqrt{2}|a\rangle_{S}&\pagecolor{lightblue}=|a\psi\rangle_{\text{NS}}^{\mathsf{F}}&\pagecolor{lightpink}\hfil&\pagecolor{lightpink}\\\
(-1)^{F}=-1&\pagecolor{lightyellow}\hfil&\pagecolor{lightyellow}&\pagecolor{lightgreen}\sqrt{2}\ket{a}_{T}&\pagecolor{lightgreen}=\ket{a\psi}_{\text{R}}^{\mathsf{F}}\\\
\hline\cr\end{array}$ (1.13)

and

$\begin{array}[]{c|rlrl}\mathsf{F}^{\prime}&\lx@intercol\hfil\text{NS}\hfil\lx@intercol&\lx@intercol\hfil\text{R}\hfil\lx@intercol\\\
\hline\cr(-1)^{F}=+1&\pagecolor{lightblue}|a\rangle_{S}&\pagecolor{lightblue}=|a\rangle_{\text{NS}}^{\mathsf{F}^{\prime}}&\pagecolor{lightgreen}\ket{a}_{T}&\pagecolor{lightgreen}=\ket{a}_{\text{R}}^{\mathsf{F}^{\prime}}\\\
&\pagecolor{lightblue}\sqrt{2}|i\rangle_{S}&\pagecolor{lightblue}=\ket{i\psi}_{\text{NS}}^{\mathsf{F}^{\prime}}&\pagecolor{lightgreen}\hfil&\pagecolor{lightgreen}\\\
(-1)^{F}=-1&\pagecolor{lightyellow}\hfil&\pagecolor{lightyellow}&\pagecolor{lightpink}\sqrt{2}|i\rangle_{U}&\pagecolor{lightpink}=|i\psi\rangle_{\text{R}}^{\mathsf{F}^{\prime}}\end{array}$
(1.14)

We see that the states are exchanged as in (1.5).

The rest of the note is organized as follows. In Sec. 2, we provide a detailed
derivation of the expressions of the boundary states of the theories
$\mathsf{D}$, $\mathsf{F}$ and $\mathsf{F}^{\prime}$ we listed above. We will
not use the property that the theories are conformal. Instead, we will only
use the fact that the theory on a torus can be consistently interpreted with
either direction as the Euclidean time direction. We then provide in Sec. 3
various examples. Finally in Sec. 4 we interpret our findings in terms of
interfaces between our theories $\mathsf{A}$, $\mathsf{D}$, $\mathsf{F}$ and
$\mathsf{F}^{\prime}$. We also see that when the original theory $\mathsf{A}$
has a duality interface implementing the $\mathbb{Z}_{2}$ orbifold, so that
$\mathsf{A}\simeq\mathsf{D}$, the fermionized theory
$\mathsf{F}\simeq\mathsf{F}^{\prime}$ has an anomalous $\mathbb{Z}_{2}$
symmetry. We have an appendix A collecting various facts on diagonal
invariants of rational conformal field theories (RCFTs) which can be used to
construct their fermionic versions and fermionic boundary conditions.

Before proceeding, the authors note that a paper with a large overlap [23]
appeared when this work was being completed, in which the fermionic boundary
states of fermionic minimal models and the effect of anomalous chiral
$\mathbb{Z}_{2}$ symmetry were studied in detail; he also carefully discusses
the boundary states of exceptional fermionic minimal models introduced in
[11]. The authors also note that another paper with a significant overlap [24]
will appear on the arXiv on the same day as this one. The authors thank the
authors of [24] for discussions and for coordinating the submission.

##  2 Mapping of the boundary states

In this section we study how the boundary states change under gauging and
fermionization. The objective is to give a detailed derivation of the mappings
we already announced in Sec. 1.

###  2.1 Gauging $\mathbb{Z}_{2}$: $\mathsf{A}\leftrightarrow\mathsf{D}$

Let us start with a bosonic theory $\mathsf{A}$ with $\mathbb{Z}_{2}$ global
symmetry. We demand that $\mathbb{Z}_{2}$ is non-anomalous, so that there is
no obstruction to gauge it. We denote the resulting $\mathbb{Z}_{2}$ gauged
theory by $\mathsf{D}$, whose emergent $\mathbb{Z}_{2}$ symmetry we denote by
$\widehat{\mathbb{Z}}_{2}$. We study how the boundary states of theory
$\mathsf{A}$ transform under gauging. The content of this section is
essentially known in the vast existing literature on the boundary conditions
of $\mathbb{Z}_{2}$ orbifolds, but we find it useful to discuss in detail as a
preparation of our discussion of the fermionization.

#### Setup:

We start with the general definition of gauging, and will first assume that
the spacetime is a torus without boundary. Denote the partition function of
theory $\mathsf{A}$ as $Z_{\mathsf{A}}^{(r,s)}$ where $r,s\in\\{0,1\\}$ count
the number of $\mathbb{Z}_{2}$ defect lines in the spatial and temporal
directions. We use the symbols $Z_{\mathsf{D}}^{(m,n)}$ similarly for the
gauged theory $\mathsf{D}$, where $m,n$ now count the number of defect lines
for the emergent $\widehat{\mathbb{Z}}_{2}$ global symmetry. We then have the
relation666More generally, the partition functions of $\mathsf{A}$ and
$\mathsf{D}$ are related as
$Z_{\mathsf{D}}[\widehat{B}]=\frac{1}{2^{g}}\sum_{B}Z_{\mathsf{A}}[B](-1)^{\int
B\cup\widehat{B}}$, where $B$ and $\widehat{B}$ are the $\mathbb{Z}_{2}$ and
dual $\mathbb{Z}_{2}$ background fields in $\mathsf{A}$ and $\mathsf{D}$
respectively. The pair $(r,s)$ in (2.1) labels the nontrivial holomonies of
the background field $\int_{x}B=s$, $\int_{y}B=r$ mod 2 where the integrals
are along the non-contractible cycles in the direction $x$ and $y$,
respectively. The pair $(m,n)$ is obtained from $\widehat{B}$ in a similar
manner.

$Z_{\mathsf{D}}^{(m,n)}=\frac{1}{2}\sum_{(r,s)}(-1)^{rn-
ms}Z_{\mathsf{A}}^{(r,s)}.$ (2.1)

Next, we demand that the spacetime has nontrivial boundaries in the temporal
direction, i.e. time direction is open. We place two boundary states
$\ket{\alpha}$ and $\ket{\beta}$ in the initial time and the final time
respectively, and denote the partition function as
$Z_{\mathsf{A}}^{(r,s)}[\braket{\beta}{\alpha}]$. As discussed in the
introduction, there are broadly speaking two types of boundaries: we denote
the $\mathbb{Z}_{2}$-invariant ones by $\ket{i}^{\mathsf{A}}$,
$\ket{j}^{\mathsf{A}}$, …and $\mathbb{Z}_{2}$-breaking pairs by
$\ket{a\pm}^{\mathsf{A}}$, $\ket{b\pm}^{\mathsf{A}}$, …. We also denote the
twisted sector boundary states by
$\ket{i}_{\mathrm{tw}}^{\mathsf{A}},\ket{j}_{\mathrm{tw}}^{\mathsf{A}}$ …. We
also employ analogous notations for theory $\mathsf{D}$.

####  $\mathbb{Z}_{2}$ charge of $\ket{i}^{\mathsf{A}}$:

Since the boundary state $\ket{i}^{\mathsf{A}}$ and
$\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$ are invariant under $\mathbb{Z}_{2}$, we
need to find their $\mathbb{Z}_{2}$ charge. The boundary state
$\ket{i}^{\mathsf{A}}$ in the untwisted Hilbert space is $\mathbb{Z}_{2}$ even
rather than $\mathbb{Z}_{2}$ odd. To see this, we use Cardy’s analysis to
transform the partition function to the open string channel. Consider the
theory on a cylinder of size $L_{1}\times L_{2}$ where $L_{1}$ is along the
open direction and $L_{2}$ is along the closed direction. Suppose
$\ket{i}^{\mathsf{A}}$ has $\mathbb{Z}_{2}$ charge $s=\pm 1$, thus
$g\ket{i}^{\mathsf{A}}=s\ket{i}^{\mathsf{A}}$. Then

$Z_{\mathsf{A}}^{(1,0)}[^{\mathsf{A}}\langle
j|i\rangle^{\mathsf{A}}]={}^{\mathsf{A}}\\!\bra{j}e^{-L_{1}H}g\ket{i}^{\mathsf{A}}=s{}^{\mathsf{A}}\\!\bra{j}e^{-L_{1}H}\ket{i}^{\mathsf{A}}=s\mathop{\mathrm{Tr}}\nolimits_{\mathcal{H}_{j|i}}e^{-L_{2}H}.$
(2.2)

We also have

$Z_{\mathsf{A}}^{(1,0)}[^{\mathsf{A}}\langle
j|i\rangle^{\mathsf{A}}]=\mathop{\mathrm{Tr}}\nolimits_{\mathcal{H}_{j|i}^{g}}e^{-L_{2}H}$
(2.3)

where $\mathcal{H}_{j|i}^{g}$ is the defect Hilbert space in the open string
channel with two boundary conditions labeled by $j$ and $i$ with an explicit
insertion of $\mathbb{Z}_{2}$ line in the middle. Now,
$\mathop{\mathrm{Tr}}\nolimits_{\mathcal{H}}e^{-L_{2}H}$ for any $\mathcal{H}$
should be a sum of exponentials with non-negative coefficients, therefore we
need to have $s=+1$.777The authors thank Shu-Heng Shao for suggesting this
argument.  Summarizing, we see $g\ket{i}^{\mathsf{A}}$ is
$+\ket{i}^{\mathsf{A}}$ instead of $-\ket{i}^{\mathsf{A}}$, where $g$ is the
$\mathbb{Z}_{2}$ generator.

$\vbox{\hbox{\includegraphics[width=43.36464pt]{fig1}}}=\vbox{\hbox{\includegraphics[width=43.36464pt]{fig2}}}=e^{2\pi
is}\vbox{\hbox{\includegraphics[width=43.36464pt]{fig3}}}$
Figure 1: $\mathbb{Z}_{2}$ defect, $\mathbb{Z}_{2}$ charge and the spin of the
twisted Hilbert space. The $\mathbb{Z}_{2}$ charge of a state in the
$\mathbb{Z}_{2}$ is given by the leftmost figure, which can be deformed to the
middle figure. Therefore the charge is related to the spin of the state. When
we put a $\mathbb{Z}_{2}$-invariant boundary condition on the top and bottom
slices, we can always unwind the defect, and the spin is always integer
valued.

####  $\mathbb{Z}_{2}$ charge of $\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$:

To find the $\mathbb{Z}_{2}$ charge of $\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$,
we make use of the following trick [25, 26]. By definition, the
$\mathbb{Z}_{2}$ charge of a state in the twisted Hilbert space can be found
by considering the configuration of the $\mathbb{Z}_{2}$ line operators in the
leftmost figure of Fig. 1. Assuming that the $\mathbb{Z}_{2}$ symmetry is non-
anomalous, this configuration can be continuously deformed to a line operator
winding the cylinder once, therefore the $\mathbb{Z}_{2}$-charge equals
$e^{2\pi is}$ where $s$ is the spin of the state
$\ket{\phi}\in\mathcal{H}_{\text{tw}}$:

$g\ket{\phi}=e^{2\pi is}\ket{\phi}$ (2.4)

where $g$ is the $\mathbb{Z}_{2}$ generator. When the state in question is the
boundary state for a $\mathbb{Z}_{2}$-invariant boundary condition, we can
always unwind $g$, say at the top time slice, to make it straight, hence
$e^{2\pi is}=1$. In summary, both the states
$\ket{i}^{\mathsf{A}},\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$ are $\mathbb{Z}_{2}$
even.

#### Rough properties of the boundary states:

We first look for the boundary states in theory $\mathsf{D}$ that is
$\mathbb{Z}_{2}$-invariant under the emergent $\widehat{\mathbb{Z}}_{2}$
symmetry. Let us symbolically denote it as $\ket{\alpha}$ and place it in the
initial time slice. We also put a reference state $\ket{R}$ in the final time
slice. (The choice of reference state does not affect our discussion). We have
the following relations

$\begin{split}Z_{\mathsf{D}}^{(0,0)}[\braket{R}{\alpha}]&=Z_{\mathsf{D}}^{(1,0)}[\braket{R}{\alpha}],\\\
Z_{\mathsf{D}}^{(0,1)}[\braket{R}{\alpha}]&=Z_{\mathsf{D}}^{(1,1)}[\braket{R}{\alpha}]=0.\end{split}$
(2.5)

The first equality follows from $\ket{\alpha}$ being
$\widehat{\mathbb{Z}}_{2}$ symmetric, where we can move the
$\widehat{\mathbb{Z}}_{2}$ line along the spatial direction down to the
initial time slice and let it be absorbed by the initial state $\ket{\alpha}$.
The second and third equations follow from the fact that $\ket{\alpha}$ do not
live in the twisted sector. Combining (2.1) and (2.5), we have

$\begin{split}Z_{\mathsf{A}}^{(0,0)}[\braket{R}{\alpha}]&=Z_{\mathsf{A}}^{(1,0)}[\braket{R}{\alpha}],\\\
Z_{\mathsf{A}}^{(0,1)}[\braket{R}{\alpha}]&=Z_{\mathsf{A}}^{(1,1)}[\braket{R}{\alpha}]=0.\end{split}$
(2.6)

The relations (2.6) imply that $\ket{\alpha}$ can not be
$\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$ because otherwise
$Z_{\mathsf{A}}^{(0,0)}[\braket{R}{\alpha}]=Z_{\mathsf{A}}^{(1,0)}[\braket{R}{\alpha}]=0$,
thus $Z_{\mathsf{A}}^{(r,s)}[\braket{R}{\alpha}]=0$ for all $r,s$, which is
absurd. The first equality in (2.6) also implies that $\ket{\alpha}$ can not
be proportional to $\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}}$ neither,
because the relations (2.6) imply that $\ket{\alpha}$ should have
$\mathbb{Z}_{2}$ eigenvalue $+1$. Therefore $\ket{\alpha}$ can only be
proportional to $\ket{i}^{\mathsf{A}}$ or
$\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}}$.

We further look for the symmetric boundary state $\ket{\beta}$ in the twisted
sector of theory $\mathsf{D}$. Applying an analysis that was carried out in
the last paragraph, we find

$\begin{split}Z_{\mathsf{D}}^{(0,1)}[\braket{R}{\beta}]&=Z_{\mathsf{D}}^{(1,1)}[\braket{R}{\beta}],\\\
Z_{\mathsf{D}}^{(0,0)}[\braket{R}{\beta}]&=Z_{\mathsf{D}}^{(1,0)}[\braket{R}{\beta}]=0.\end{split}$
(2.7)

Combining with (2.1), we have

$\begin{split}Z_{\mathsf{A}}^{(0,0)}[\braket{R}{\beta}]&=-Z_{\mathsf{A}}^{(1,0)}[\braket{R}{\beta}],\\\
Z_{\mathsf{A}}^{(0,1)}[\braket{R}{\beta}]&=Z_{\mathsf{A}}^{(1,1)}[\braket{R}{\beta}]=0.\end{split}$
(2.8)

This implies that $\ket{\beta}$ can only be proportional to
$\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}}$. This in fact would suggest that
$\ket{\alpha}$ should be proportional to
$\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}}$, but not $\ket{i}^{\mathsf{A}}$.
Hence we will denote

$\begin{split}\ket{a}^{\mathsf{D}}&\propto\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}},\\\
\ket{a}_{\mathrm{tw}}^{\mathsf{D}}&\propto\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}}.\\\
\end{split}$ (2.9)

We further look for the $\mathbb{Z}_{2}$-odd boundary state $\ket{\gamma}$ in
the untwisted sector of theory $\mathsf{D}$. This means that

$\begin{split}Z_{\mathsf{D}}^{(0,0)}[\braket{R}{\gamma}]&=-Z_{\mathsf{D}}^{(1,0)}[\braket{R}{\gamma}],\\\
Z_{\mathsf{D}}^{(0,1)}[\braket{R}{\gamma}]&=Z_{\mathsf{D}}^{(1,1)}[\braket{R}{\gamma}]=0.\end{split}$
(2.10)

Combining with (2.1), we have

$\begin{split}Z_{\mathsf{A}}^{(0,1)}[\braket{R}{\gamma}]&=Z_{\mathsf{A}}^{(1,1)}[\braket{R}{\gamma}],\\\
Z_{\mathsf{A}}^{(0,0)}[\braket{R}{\gamma}]&=Z_{\mathsf{A}}^{(1,0)}[\braket{R}{\gamma}]=0.\end{split}$
(2.11)

Hence $\ket{\gamma}$ should be the twisted sector state of theory
$\mathsf{A}$, $\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$, which indeed is
$\mathbb{Z}_{2}$ even.

Lastly we look for the $\mathbb{Z}_{2}$-odd boundary state $\ket{\eta}$ in the
twisted sector of $\mathsf{D}$. This means that

$\begin{split}Z_{\mathsf{D}}^{(0,1)}[\braket{R}{\eta}]&=-Z_{\mathsf{D}}^{(1,1)}[\braket{R}{\eta}],\\\
Z_{\mathsf{D}}^{(0,0)}[\braket{R}{\eta}]&=Z_{\mathsf{D}}^{(1,0)}[\braket{R}{\eta}]=0.\end{split}$
(2.12)

Combining with (2.1), we have

$\begin{split}Z_{\mathsf{A}}^{(0,1)}[\braket{R}{\eta}]&=-Z_{\mathsf{A}}^{(1,1)}[\braket{R}{\eta}],\\\
Z_{\mathsf{A}}^{(0,0)}[\braket{R}{\eta}]&=Z_{\mathsf{A}}^{(1,0)}[\braket{R}{\eta}]=0.\end{split}$
(2.13)

Because there is no $\mathbb{Z}_{2}$ odd state in the twisted sector of theory
$\mathsf{A}$, there is no solution to (2.13). Hence all the boundary states in
theory $\mathsf{D}$ that are related to $\ket{i}^{\mathsf{A}}$ and
$\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$ do not live in the twisted sector of
$\widehat{\mathbb{Z}}_{2}$. This suggests that there are
$\ket{i+}^{\mathsf{D}}$ and $\ket{i-}^{\mathsf{D}}$ which are mapped to each
other under $\widehat{\mathbb{Z}}_{2}$, which satisfy

$\begin{split}\ket{i+}^{\mathsf{D}}+\ket{i-}^{\mathsf{D}}&\propto\ket{i}^{\mathsf{A}},\\\
\ket{i+}^{\mathsf{D}}-\ket{i-}^{\mathsf{D}}&\propto\ket{i}_{\mathrm{tw}}^{\mathsf{A}}.\\\
\end{split}$ (2.14)

#### Fixing the normalization constants:

To determine the coefficients in (2.9) and (2.14), we need to make use of
Cardy’s conditions. We consider a theory on a cylinder of size $L_{1}\times
L_{2}$, with two boundary conditions $\alpha$ and $\beta$. We then have
Cardy’s conditions

$\bra{\alpha}e^{-L_{1}H}\ket{\beta}=\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{\alpha|\beta}}e^{-L_{2}H},\qquad\prescript{}{\mathrm{tw}}{\bra{\alpha}}e^{-L_{1}H}\ket{\beta}_{\mathrm{tw}}=\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{\alpha|\beta}}ge^{-L_{2}H}$
(2.15)

where $g$ is the $\mathbb{Z}_{2}$ generator. Below, we will drop $e^{-L_{1}H}$
and $e^{-L_{2}H}$ for brevity.

Let us first consider (2.14). Suppose
$\ket{i+}^{\mathsf{D}}+\ket{i-}^{\mathsf{D}}=P\ket{i}^{\mathsf{A}}$ and
$\ket{i+}^{\mathsf{D}}-\ket{i-}^{\mathsf{D}}=Q\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$.
Then the relations (2.15) imply that

$\begin{split}{}^{\mathsf{A}}\braket{i}{j}^{\mathsf{A}}&=\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|j}^{+}}+\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|j}^{-}},\\\
{}_{\mathrm{tw}}^{\mathsf{A}}\braket{i}{j}_{\mathrm{tw}}^{\mathsf{A}}&=\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|j}^{+}}-\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|j}^{-}}\end{split}$
(2.16)

where $\mathcal{H}_{i|j}^{\pm}$ is the part of the open Hilbert space with
boundary conditions $i$, $j$ placed on two ends, with the $\mathbb{Z}_{2}$
charge $\pm 1$. Hence

$\begin{split}{}^{\mathsf{D}}\braket{i\pm}{j\pm}^{\mathsf{D}}&=\frac{1}{4}(|P|^{2}+|Q|^{2})\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|j}^{+}}+\frac{1}{4}(|P|^{2}-|Q|^{2})\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|j}^{-}},\\\
{}^{\mathsf{D}}\braket{i\pm}{j\mp}^{\mathsf{D}}&=\frac{1}{4}(|P|^{2}-|Q|^{2})\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|j}^{+}}+\frac{1}{4}(|P|^{2}+|Q|^{2})\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|j}^{-}}.\\\
\end{split}$ (2.17)

For $\ket{i\pm}^{\mathsf{D}}$ to be legal boundary states, we should have
$\frac{1}{4}(|P|^{2}\pm|Q|^{2})$ to be non-negative integers. It is then
reasonable to set $P=Q=\sqrt{2}$, which is the minimal solution of $P$ and
$Q$. We thus get

$\begin{split}\ket{i\pm}^{\mathsf{D}}=\frac{1}{\sqrt{2}}(\ket{i}^{\mathsf{A}}\pm\ket{i}_{\mathrm{tw}}^{\mathsf{A}}).\end{split}$
(2.18)

Since gauging twice goes back to the original theory itself, we also finds the
normalization in (2.9)

$\ket{a}^{\mathsf{D}}=\frac{1}{\sqrt{2}}(\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}}),~{}~{}~{}~{}~{}\ket{a}_{\mathrm{tw}}^{\mathsf{D}}=\frac{1}{\sqrt{2}}(\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}}).$
(2.19)

These results were presented in a slightly different manner in Sec. 1.

###  2.2 Fermionizing $\mathbb{Z}_{2}$:
$\mathsf{A}\to\mathsf{F},\mathsf{F}^{\prime}$

#### Setup:

Let us further consider fermionizing the $\mathbb{Z}_{2}$ global symmetry of
$\mathsf{A}$, and denote the resulting theory as $\mathsf{F}$. More precisely,
we first stack the theory $\mathsf{A}$ with a Kitaev chain in the nontrivial
phase, and gauge the diagonal $\mathbb{Z}_{2}$ global symmetry:
$\mathsf{F}=({\mathsf{A}\times{\mathsf{Kitaev}}})/{\mathbb{Z}_{2}}.$ The
partition functions on the torus are related as follows:888See the footnote 4
for the expression on a general surface of genus $g$.

$Z_{\mathsf{F}}^{(m,n)}=\frac{1}{2}\sum_{(r,s)}(-1)^{(m+r)(n+s)}Z_{\mathsf{A}}^{(r,s)},$
(2.20)

where the pair $(m,n)$ now specifies the periodicity of the spin structure on
the torus, in the convention where $m,n=0$ is the NS sector and $m,n=1$ is the
R sector.

We first need to recall the argument we already gave in the introduction:
consider a $\mathbb{Z}_{2}$-invariant boundary condition $i$ of the theory
$\mathsf{A}$. When stacked with $\mathsf{Kitaev}$, this boundary has a
Majorana fermion $\psi$, which is removed by the $\mathbb{Z}_{2}$ quotient
since it is $\mathbb{Z}_{2}$ odd. In contrast, when we start from a
$\mathbb{Z}_{2}$-breaking boundary condition $a\pm$, this $\mathbb{Z}_{2}$-odd
Majorana fermion $\psi$ is retained since the $\mathbb{Z}_{2}$ gauge symmetry
is broken here, resulting in a boundary with a Majorana fermion. Furthermore,
acting by $\psi$ essentially converts $a+$ to $a-$ and vice versa. Therefore
we find two types of boundary conditions: $\ket{i}^{\mathsf{F}}$ without
Majorana fermion which comes from a $\mathbb{Z}_{2}$-invariant boundary
condition $i$ of theory $\mathsf{A}$, and $\ket{a\psi}^{\mathsf{F}}$ with a
Majorana fermion which comes from a pair of $\mathbb{Z}_{2}$-breaking boundary
conditions $a\pm$ of theory $\mathsf{A}$.

#### Rough properties of the boundary conditions:

The way we determine the boundary states is analogous to what we did in the
case $\mathsf{A}\leftrightarrow\mathsf{D}$ studied above. When there are
nontrivial boundaries along the temporal direction, we again put a reference
state $\ket{R}$ in the final time slice, and put a boundary state
$\ket{\alpha}$ in the initial slice which we will study. Now we need to
consider four choices: $\text{NS}/\text{R}$ sectors and $(-1)^{F}=\pm 1$,
which we study in turn.

Let us first look into the state $\ket{\alpha}$ in the NS sector with fermion
parity $(-1)^{F}=1$. We have

$\begin{split}Z_{\mathsf{F}}^{(0,0)}[\braket{R}{\alpha}]&=Z_{\mathsf{F}}^{(1,0)}[\braket{R}{\alpha}],\\\
Z_{\mathsf{F}}^{(0,1)}[\braket{R}{\alpha}]&=Z_{\mathsf{F}}^{(1,1)}[\braket{R}{\alpha}]=0.\\\
\end{split}$ (2.21)

The first equality follows from $(-1)^{F}=1$ and the last two equalities are
because $\ket{\alpha}$ is in the NS sector. Combining with (2.20), we have

$\begin{split}Z_{\mathsf{A}}^{(0,0)}[\braket{R}{\alpha}]&=Z_{\mathsf{A}}^{(1,0)}[\braket{R}{\alpha}],\\\
Z_{\mathsf{A}}^{(0,1)}[\braket{R}{\alpha}]&=Z_{\mathsf{A}}^{(1,1)}[\braket{R}{\alpha}]=0,\\\
\end{split}$ (2.22)

which implies that

$\ket{\alpha}\propto\ket{i}^{\mathsf{A}}~{}~{}~{}\mathrm{or}~{}~{}~{}\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}}.$
(2.23)

Let us further look into the state $\ket{\beta}$ in the R sector with
$(-1)^{F}=1$. We have

$\begin{split}Z_{\mathsf{F}}^{(0,1)}[\braket{R}{\beta}]&=Z_{\mathsf{F}}^{(1,1)}[\braket{R}{\beta}],\\\
Z_{\mathsf{F}}^{(0,0)}[\braket{R}{\beta}]&=Z_{\mathsf{F}}^{(1,0)}[\braket{R}{\beta}]=0.\\\
\end{split}$ (2.24)

Combining with (2.20), we have

$\begin{split}Z_{\mathsf{A}}^{(0,1)}[\braket{R}{\beta}]&=Z_{\mathsf{A}}^{(1,1)}[\braket{R}{\beta}],\\\
Z_{\mathsf{A}}^{(0,0)}[\braket{R}{\beta}]&=Z_{\mathsf{A}}^{(1,0)}[\braket{R}{\beta}]=0.\\\
\end{split}$ (2.25)

The only solution $\ket{\beta}$ to (2.25) is
$\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$. We will denote $\ket{\beta}$ as
$\ket{i}_{\text{R}}^{\mathsf{F}}$, i.e.

$\ket{i}_{\text{R}}^{\mathsf{F}}\propto\ket{i}_{\mathrm{tw}}^{\mathsf{A}}.$
(2.26)

Let us further look into the boundary state $\ket{\gamma}$ in the R sector
with $(-1)^{F}=-1$. This means

$\begin{split}Z_{\mathsf{F}}^{(0,1)}[\braket{R}{\gamma}]&=-Z_{\mathsf{F}}^{(1,1)}[\braket{R}{\gamma}],\\\
Z_{\mathsf{F}}^{(0,0)}[\braket{R}{\gamma}]&=Z_{\mathsf{F}}^{(1,0)}[\braket{R}{\gamma}]=0.\\\
\end{split}$ (2.27)

Combining with (2.20), we have

$\begin{split}Z_{\mathsf{A}}^{(0,0)}[\braket{R}{\gamma}]&=-Z_{\mathsf{A}}^{(1,0)}[\braket{R}{\gamma}],\\\
Z_{\mathsf{A}}^{(0,1)}[\braket{R}{\gamma}]&=Z_{\mathsf{A}}^{(1,1)}[\braket{R}{\gamma}]=0.\\\
\end{split}$ (2.28)

The only solution $\ket{\gamma}$ to (2.31) is
$\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}}$. We will denote $\ket{\gamma}$
as $\ket{a\psi}_{\text{R}}^{\mathsf{F}}$, thus

$\ket{a\psi}_{\text{R}}^{\mathsf{F}}\propto\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}}.$
(2.29)

Finally let us look into the boundary state $\ket{\eta}$ in the NS sector with
$(-1)^{F}=-1$. This means

$\begin{split}Z_{\mathsf{F}}^{(0,0)}[\braket{R}{\eta}]&=-Z_{\mathsf{F}}^{(1,0)}[\braket{R}{\eta}],\\\
Z_{\mathsf{F}}^{(0,1)}[\braket{R}{\eta}]&=Z_{\mathsf{F}}^{(1,1)}[\braket{R}{\eta}]=0.\\\
\end{split}$ (2.30)

Combining with (2.20), we have

$\begin{split}Z_{\mathsf{A}}^{(0,1)}[\braket{R}{\eta}]&=-Z_{\mathsf{A}}^{(1,1)}[\braket{R}{\eta}],\\\
Z_{\mathsf{A}}^{(0,0)}[\braket{R}{\eta}]&=Z_{\mathsf{A}}^{(1,0)}[\braket{R}{\eta}]=0.\\\
\end{split}$ (2.31)

We note, however, that there is no boundary state in the twisted sector of
theory $\mathsf{A}$ which is $\mathbb{Z}_{2}$ odd. Hence there is no boundary
state in NS sector with $(-1)^{F}=-1$ in theory $\mathsf{F}$.

To encode the $(-1)^{F}$ quantum number to the boundary state, we introduce
$\ket{\pm}$, representing $(-1)^{F}=\pm 1$ respectively. This can be thought
of as coming from the fermion parity of the vacuum of the nontrivial phase of
the Kitaev chain. In summary, we find

$\begin{split}\ket{i}_{\text{NS}}^{\mathsf{F}}&\propto\ket{i}^{\mathsf{A}}\otimes\ket{+},\\\
\ket{a\psi}_{\text{NS}}^{\mathsf{F}}&\propto(\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}})\otimes\ket{+},\\\
\ket{i}_{\text{R}}^{\mathsf{F}}&\propto\ket{i}_{\mathrm{tw}}^{\mathsf{A}}\otimes\ket{+},\\\
\ket{a\psi}_{\text{R}}^{\mathsf{F}}&\propto(\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}})\otimes\ket{-}.\end{split}$
(2.32)

where we remind the reader that our $\ket{a\psi}^{\mathsf{F}}_{\text{R}}$
includes an insertion of a Majorana fermion to absorb the zero mode, which is
required to make the state nonzero.

#### Fixing the normalization constants:

We need to further determine the normalization constants in (2.41). Suppose
the normalization constants are $P_{1},P_{2},P_{3},P_{4}$ for the four
relations in (2.41) respectively. Exchanging the time directions, we find

${}_{\text{NS}}^{\mathsf{F}}\braket{i}{j}_{\text{NS}}^{\mathsf{F}}$
$\displaystyle=|P_{1}|^{2}(\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|j}^{+}}+\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|j}^{-}}),$
(2.33)
${}_{\text{NS}}^{\mathsf{F}}\braket{a\psi}{b\psi}_{\text{NS}}^{\mathsf{F}}$
$\displaystyle=|P_{2}|^{2}(\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a+|b+}}+\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a-|b-}}+\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a+|b-}}+\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a-|b+}})$
(2.34)
$\displaystyle=2|P_{2}|^{2}(\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a+|b+}}+\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a+|b-}}),$
(2.35) ${}_{\text{R}}^{\mathsf{F}}\braket{i}{j}_{\text{R}}^{\mathsf{F}}$
$\displaystyle=|P_{3}|^{2}(\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|j}^{+}}-\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|j}^{-}}),$
(2.36)
${}_{\text{R}}^{\mathsf{F}}\braket{a\psi}{b\psi}_{\text{R}}^{\mathsf{F}}$
$\displaystyle=|P_{4}|^{2}(\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a+|b+}}+\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a-|b-}}-\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a+|b-}}-\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a-|b+}})$
(2.37)
$\displaystyle=2|P_{4}|^{2}(\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a+|b+}}-\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a+|b-}}),$
(2.38) ${}_{\text{NS}}^{\mathsf{F}}\braket{i}{a\psi}_{\text{NS}}^{\mathsf{F}}$
$\displaystyle=P_{1}^{*}P_{2}(\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|a+}}+\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{i|a-}}).$
(2.39)

where we used the fact that the $\mathbb{Z}_{2}$ symmetry of the
$\mathsf{A}$-theory implies that
$\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a+|b+}}=\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a-|b-}}$
and
$\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a+|b-}}=\mathop{\mathrm{Tr}}\nolimits_{{\mathcal{H}}_{a-|b+}}$
to convert (2.34), (2.37) to (2.35), (2.38), respectively.

To have consistent Hamiltonian interpretations, we simply take

$P_{1}=P_{2}=P_{3}=P_{4}=1.$ (2.40)

The equations (2.33) and (2.36) then reiterate the fact that for
$\mathbb{Z}_{2}$-invariant boundary condition $i$, $j$ of theory $\mathsf{A}$,
the corresponding boundary conditions of the theory $\mathsf{F}$ is obtained
by attaching a Majorana fermion $\psi$ and simply removing it by the
$\mathbb{Z}_{2}$ projection. Therefore the open Hilbert space in the fermionic
theory with boundary conditions $i$ and $j$ is simply equal to the Hilbert
space of the original theory $\mathcal{H}_{i|j}$, and the fermion parity
$(-1)^{F}$ equals the original $\mathbb{Z}_{2}$-charge $U$.

To interpret (2.35) and (2.38), we note that each of these boundary conditions
hosts a single Majorana zero mode. Therefore, with two boundaries, we need to
quantize a pair of such modes, providing a factor of $2$. We note that the
relative sign in (2.38) does not correspond to $(-1)^{F}$, since in our
definition we inserted a Majorana fermion operator on the boundaries
$\ket{a\psi}^{\mathsf{F}}$ and $\ket{b\psi}^{\mathsf{F}}$.

In summary, we found999As one of the authors emphasized in [27], the linear
combination of Cardy states appears naturally as zero mode of fermionic
model[28, 29]. This type of states has also captured attention in the
condensed matter physics community [30, 31].

$\begin{split}\ket{i}_{\text{NS}}^{\mathsf{F}}&=\ket{i}^{\mathsf{A}}\otimes\ket{+},\\\
\ket{a\psi}_{\text{NS}}^{\mathsf{F}}&=(\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}})\otimes\ket{+},\\\
\ket{i}_{\text{R}}^{\mathsf{F}}&=\ket{i}_{\mathrm{tw}}^{\mathsf{A}}\otimes\ket{+},\\\
\ket{a\psi}_{\text{R}}^{\mathsf{F}}&=(\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}})\otimes\ket{-}.\end{split}$
(2.41)

We can further append a nontrivial Kitaev chain to $\mathsf{F}$ to obtain
$\mathsf{F}^{\prime}$. As the relation between $\mathsf{F}$ and $\mathsf{A}$
is equal to that of the relation of $\mathsf{F}^{\prime}$ and $\mathsf{D}$, we
can easily find from (2.18) and (2.19) that

$\begin{split}\ket{i\psi}_{\text{NS}}^{\mathsf{F}^{\prime}}&=\sqrt{2}\ket{i}^{\mathsf{A}}\otimes\ket{+},\\\
\ket{a}_{\text{NS}}^{\mathsf{F}^{\prime}}&=\frac{1}{\sqrt{2}}(\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}})\otimes\ket{+},\\\
\ket{i\psi}_{\text{R}}^{\mathsf{F}^{\prime}}&=\sqrt{2}\ket{i}_{\mathrm{tw}}^{\mathsf{A}}\otimes\ket{-},\\\
\ket{a}_{\text{R}}^{\mathsf{F}^{\prime}}&=\frac{1}{\sqrt{2}}(\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}})\otimes\ket{+}.\end{split}$
(2.42)

As a consistency check, attaching a Kitaev chain amounts to change the number
of Majorana zero modes in the R sector states, hence the amplitude of the
boundary states are exchanged between $1$ and $\sqrt{2}$, exactly as shown in
(2.42) and (2.41). Our final results (2.41) and (2.42) were already presented
in Sec. 1.

###  2.3 Comments

Before proceeding, we would like to make two remarks.

#### On two common normalizations:

Recall that during the derivation, we demanded that the overlap of any pair of
boundary states has a consistent Hamiltonian interpretation in the open
channel to deduce the proportionality coefficients. This is different from the
standard viewpoint in hep-th in the following sense.

Consider a massless free Majorana fermion in 1+1d, which allows two boundary
conditions $\psi_{L}=\pm\psi_{\text{R}}$ on the boundary. In hep-th, it is
often stated that if we choose a wrong pair of boundary conditions $\alpha$,
$\beta$ on the two ends of a segment, there is a bulk fermionic Majorana zero
mode, which makes the system not quantizable. This problem can be cured by
putting a Majorana fermion by hand. In such a case, it is
$\sqrt{2}\braket{\alpha}{\beta}$ which has a sensible Hamiltonian
interpretation in the open channel. The recent paper [23] uses this
normalization.

Instead, in this paper we chose to use a viewpoint which is more in line with
cond-mat, or which happens when we realize the Majorana fermion on the lattice
for example. In this case, if $\psi_{L}=+\psi_{\text{R}}$ is a boundary
condition without a boundary Majorana fermion, then the boundary condition
$\psi_{L}=-\psi_{\text{R}}$ comes with a boundary Majorana zero mode. In this
case any pair of boundary conditions is quantizable, but then we need to
introduce additional normalization constants of $\sqrt{2}^{\pm 1}$ in front of
the boundary states. Then the boundary states for a single Majorana fermion
with boundary conditions $\psi_{L}=+\psi_{\text{R}}$ and
$\psi_{L}=-\psi_{\text{R}}$ would have different norms. This $\sqrt{2}$ factor
can also be observed as the $\sqrt{2}$ fold degeneracy of Majorana chain [28,
29, 27].

#### Relation with the lattice Jordan-Wigner transformations:

Recall also that during the derivation, we saw the relation between the open
Hilbert spaces of the original bosonic model $\mathsf{A}$ and the transformed
fermionic model $\mathsf{F}$, which we gave in (2.33) to (2.39). We also saw
that $\mathbb{Z}_{2}$-invariant boundary conditions give rise to boundary
conditions without localized Majorana fermions, and that
$\mathbb{Z}_{2}$-breaking boundary conditions to boundary conditions with
localized Majorana fermions.

Here we show that this is also what we have when we perform the Jordan-Wigner
transformation on an open chain. In this sense, our transformation form
$\mathsf{A}$ to $\mathsf{F}$ is indeed the Jordan-Wigner transformation in
disguise.

Let us first consider the case of standard $\mathbb{Z}_{2}$-invariant boundary
conditions on open chains. Namely, we consider an open chain of $L$ sites with
the operators $\sigma^{(s)}_{X,Y,Z}$, ($s=1,\ldots,L$). We take a Hamiltonian
$H$ commuting with the $\mathbb{Z}_{2}$ symmetry generated by
$\prod_{s}\sigma^{(s)}_{Z}$. The boundary conditions at both ends are
$\mathbb{Z}_{2}$-invariant. Such a Hamiltonian can be rewritten in terms of
Majorana fermions $\psi^{(t)}$, ($t=1,\ldots,2L$) using the standard formulas.
Then almost by definition, the Hamiltonian written in the fermionic variables
is equal to the Hamiltonian written in terms of $\sigma^{(s)}_{X,Y,Z}$, and
the fermion parity is equal to $\prod_{s}\sigma^{(s)}_{Z}$. This reproduces
the relation (2.33) and (2.36).

Next, we consider a particular class of $\mathbb{Z}_{2}$-breaking boundary
conditions on an open chain, and see it reproduces (2.34). The existence of
localized Majorana zero modes is also clearly visible.

We again start from the same Hamiltonian $\mathbb{Z}_{2}$-invariant $H$ on a
chain of $L$ sites, and express it in terms of $\sigma^{(s)}_{X,Z}$ without
$\sigma^{(s)}_{Y}$. At the leftmost site $i=1$, we drop the terms from $H$
involving $\sigma^{(1)}_{Z}$, and also perform an analogous operation on the
other end $i=L$. Let us denote the resulting Hamiltonian by $H^{\prime}$. Such
a Hamiltonian $H^{\prime}$ commutes with $\sigma^{(1)}_{X}$ and
$\sigma^{(L)}_{X}$, and therefore $H^{\prime}$ and $\sigma^{(1,L)}_{X}$ can be
simultaneously diagonalized. In other words, $H^{\prime}$ splits into four
sectors $H^{\prime}_{\pm\pm}$ depending on the eigenvalues of
$\sigma^{(1,L)}_{X}$, and the four sectors describe $\mathbb{Z}_{2}$-breaking
boundary conditions imposed on both ends. Now we perform the standard Jordan-
Wigner transformation for $H^{\prime}$, which is still
$\mathbb{Z}_{2}$-symmetric. The construction guarantees that $H^{\prime}$ does
not involve $\psi^{(1)}$ and $\psi^{(2L)}$, meaning that they commute with
$H^{\prime}$. We found that each end has an unpaired Majorana zero mode.

To make this abstract discussion more concrete, take for example, the standard
critical Ising chain

$-H_{\text{Ising}}=\sum_{s=1}^{L}\sigma^{(s)}_{Z}+\sum_{s=1}^{L-1}\sigma^{(s)}_{X}\sigma^{(s+1)}_{X}.$
(2.43)

The modified chain has the Hamiltonian

$-H^{\prime}_{\text{Ising}}=\sum_{s=2}^{L-1}\sigma^{(s)}_{Z}+\sum_{s=1}^{L-1}\sigma^{(s)}_{X}\sigma^{(s+1)}_{X}$
(2.44)

which splits into four sectors described by

$-H^{\prime}_{\text{Ising}}{}_{\pm\pm}=(\sum_{s=2}^{L-1}\sigma^{(s)}_{Z})\pm\sigma^{(2)}_{X}+(\sum_{s=2}^{L-2}\sigma^{(s)}_{X}\sigma^{(s+1)}_{X})\pm\sigma^{(L-1)}_{X}$
(2.45)

acting on sites $s=2,\ldots,L-1$. This $H^{\prime}_{\text{Ising}}{}_{\pm\pm}$
indeed breaks $\mathbb{Z}_{2}$ at both boundaries and has the standard forms
there.

Now $H^{\prime}$ written in the Majorana fermion variables is simply

$-H^{\prime}=\sum_{t=2}^{2L-2}i\psi^{(t)}\psi^{(t+1)}$ (2.46)

and commutes with $\psi^{(1)}$ and $\psi^{(2L)}$. We indeed found localized
Majorana zero modes, and also reproduced the relation (2.34).

##  3 Examples

In section 2, we discussed the transformation of boundary states under gauging
and fermionizing the $\mathbb{Z}_{2}$ global symmetry on general grounds. In
this section, we apply the general results in section 2 to concrete examples.
We often use standard results in the theory of RCFTs, such as Ishibashi and
Cardy states [32, 33]; for general aspects of BCFT, see e.g. [34, 35, 36]. We
refer the readers to Appendix A for a brief review and further references.

###  3.1 Fermionic $SU(2)_{k}$ WZW model

Let us first discuss the fermionic version of the $SU(2)_{k}$ Wess-Zumino-
Witten (WZW) model. Recall that $SU(2)_{k}$ WZW model has $k+1$ primaries
labeled by $j=0,\frac{1}{2},\cdots,\frac{k}{2}$ with
$\Delta_{j}=\frac{j(j+1)}{k+2}$. The fusion rule is

$[j]\otimes[j^{\prime}]=[|j-j^{\prime}|]\oplus[|j-j^{\prime}|+1]\oplus\cdots\oplus[\min(j+j^{\prime},k-j-j^{\prime})].$
(3.1)

The modular $S$ matrix is

$S_{jj^{\prime}}=\sqrt{\frac{2}{k+2}}\sin\frac{\pi(2j+1)(2j^{\prime}+1)}{k+2}.$
(3.2)

There is a $\mathbb{Z}_{2}$ global symmetry, generated by the Verlinde line
${\mathcal{L}}_{\frac{k}{2}}$ associated with the primary operator
$[\frac{k}{2}]$. This is non-anomalous when $k$ is even and anomalous when $k$
is odd. In the following we only consider the case $k$ is even.

For a given $j$, the conformal primary and its descendants $\ket{j,N}$ all
have the same $\mathbb{Z}_{2}$ eigenvalue
$\frac{S_{\frac{k}{2}j}}{S_{0j}}=(-1)^{2j}$. The Ishibashi states are
schematically given by

$|j\rangle\\!\rangle\sim\sum_{N}\ket{j,N}\otimes\ket{\bar{j},\bar{N}}$ (3.3)

which are $\mathbb{Z}_{2}$ eigenstates with eigenvalue $(-1)^{2j}$, i.e.
$g|j\rangle\\!\rangle=(-1)^{2j}|j\rangle\\!\rangle$. Therefore
$\mathbb{Z}_{2}$ acts on the Cardy state as

$g\ket{j}=\ket{\tfrac{k}{2}-j}.$ (3.4)

The only $\mathbb{Z}_{2}$-invariant Cardy state is $\ket{\frac{k}{4}}$.

The twisted Cardy state $\ket{\frac{k}{4}}_{\mathrm{tw}}$ can be determined
using the general formula given in Appendix A. Here we use an ad hoc but more
elementary method. The twisted Hilbert space has the form

$\mathcal{H}=\oplus_{j}V_{j}\otimes\overline{V_{k/2-j}}.$ (3.5)

Therefore, the only possible Ishibashi state which can appear in this twisted
Cardy state is $|\frac{k}{4}\rangle\\!\rangle_{\text{tw}}$:

$\ket{\frac{k}{4}}_{\mathrm{tw}}\propto|\frac{k}{4}\rangle\\!\rangle_{\mathrm{tw}}.$
(3.6)

Let us note that we have

$\braket{\frac{k}{4}}{e^{-2\pi
H/\delta}}{\frac{k}{4}}=\chi_{0}(i\delta)+\chi_{1}(i\delta)+\chi_{2}(i\delta)+\chi_{3}(i\delta)+\cdots+\chi_{\frac{k}{2}}(i\delta),$
(3.7)

while

$\langle\\!\langle\frac{k}{4}|e^{-2\pi
H/\delta}|\frac{k}{4}\rangle\\!\rangle\propto\chi_{0}(i\delta)-\chi_{1}(i\delta)+\chi_{2}(i\delta)-\chi_{3}(i\delta)+\cdots+(-1)^{\frac{k}{2}}\chi_{\frac{k}{2}}(i\delta).$
(3.8)

Therefore we should have

${}_{\text{tw}}\langle\frac{k}{4}|e^{-2\pi
H/\delta}|\frac{k}{4}\rangle_{\mathrm{tw}}=\chi_{0}(i\delta)-\chi_{1}(i\delta)+\chi_{2}(i\delta)-\chi_{3}(i\delta)+\cdots+(-1)^{\frac{k}{2}}\chi_{\frac{k}{2}}(i\delta).$
(3.9)

from which we can determine the proportionality coefficient:

$\ket{\frac{k}{4}}{}_{\text{tw}}=\left(\frac{k+2}{2}\right)^{\frac{1}{4}}|\frac{k}{4}\rangle\\!\rangle_{\mathrm{tw}}.$
(3.10)

So far we discussed the untwisted and twisted Cardy states in the original
diagonal $SU(2)_{k}$ model. For even $k$, we can orbifold the $\mathbb{Z}_{2}$
symmetry and then obtain the D-type modular invariants of the $SU(2)_{k}$ WZW
model. Equivalently, for even $k$, we can consider the WZW sigma model whose
target space is the group manifold $SO(3)$ rather than $SU(2)$, and the D-type
modular invariants are the infrared limit.101010 Depending on whether $k=4n$
and $k=4n+2$, these models are called $D_{\text{even}}$ and $D_{\text{odd}}$
models, respectively, and they show rather distinct behaviors, in that the
chiral algebra enhances in the former while it does not in the latter.
Historically, this made the construction of the boundary states for the latter
more difficult, and was initiated in [37, 38]. This paved the way for a more
general method applicable to arbitrary simple current orbifolds e.g. in [39,
40, 41, 42].  Denoting the $SU(2)_{k}$ WZW model as theory $\mathsf{A}$, we
can now determine the boundary states of theories $\mathsf{D},\mathsf{F}$ and
$\mathsf{F}^{\prime}$ following our general prescription.

  1. 1.

The boundary states of theory $\mathsf{D}$ is obtained by gauging
$\mathbb{Z}_{2}$ (i.e. $\mathbb{Z}_{2}$ orbifolding). Using

$\begin{split}\ket{j}^{\mathsf{D}}&=\frac{1}{\sqrt{2}}(\ket{j}^{\mathsf{A}}+\ket{\frac{k}{2}-j}{}^{\mathsf{A}}),\\\
\ket{j}_{\mathrm{tw}}^{\mathsf{D}}&=\frac{1}{\sqrt{2}}(\ket{j}^{\mathsf{A}}-\ket{\frac{k}{2}-j}{}^{\mathsf{A}}),~{}~{}~{}~{}~{}j=0,\frac{1}{2},...,\frac{k-2}{4}\\\
\ket{\frac{k}{4}+}^{\mathsf{D}}&=\frac{1}{\sqrt{2}}(\ket{\frac{k}{4}}^{\mathsf{A}}+\ket{\frac{k}{4}}_{\mathrm{tw}}^{\mathsf{A}}),\\\
\ket{\frac{k}{4}-}^{\mathsf{D}}&=\frac{1}{\sqrt{2}}(\ket{\frac{k}{4}}^{\mathsf{A}}-\ket{\frac{k}{4}}_{\mathrm{tw}}^{\mathsf{A}}).\\\
\end{split}$ (3.11)

These boundary states have been discussed in the context of orbifolding [43].

  2. 2.

The boundary states of theory $\mathsf{F}$ is obtained by stacking a Kitaev
chain (Arf invariant) before gauging $\mathbb{Z}_{2}$. Using (2.41),

$\begin{split}\ket{j}^{\mathsf{F}}_{\text{NS}}&=(\ket{j}^{\mathsf{A}}+\ket{\frac{k}{2}-j}{}^{\mathsf{A}})\otimes\ket{+},\\\
\ket{j}^{\mathsf{F}}_{\text{R}}&=(\ket{j}^{\mathsf{A}}-\ket{\frac{k}{2}-j}{}^{\mathsf{A}})\otimes\ket{-},~{}~{}~{}~{}~{}j=0,\frac{1}{2},...,\frac{k-2}{4}\\\
\ket{\frac{k}{4}}^{\mathsf{F}}_{\text{NS}}&=\ket{\frac{k}{4}}^{\mathsf{A}}\otimes\ket{+},\\\
\ket{\frac{k}{4}}^{\mathsf{F}}_{\text{R}}&=\ket{\frac{k}{4}}_{\mathrm{tw}}^{\mathsf{A}}\otimes\ket{+}.\\\
\end{split}$ (3.12)

  3. 3.

The boundary states of theory $\mathsf{F}^{\prime}$ is obtained by stacking a
Kitaev chain (Arf invariant) before gauging $\mathbb{Z}_{2}$, and then further
stacking another Kitaev chain. Using (2.42),

$\begin{split}\ket{j}^{\mathsf{F}^{\prime}}_{\text{NS}}&=\frac{1}{\sqrt{2}}(\ket{j}^{\mathsf{A}}+\ket{\frac{k}{2}-j}{}^{\mathsf{A}})\otimes\ket{+},\\\
\ket{j}^{\mathsf{F}^{\prime}}_{\text{R}}&=\frac{1}{\sqrt{2}}(\ket{j}^{\mathsf{A}}-\ket{\frac{k}{2}-j}{}^{\mathsf{A}})\otimes\ket{+},~{}~{}~{}~{}~{}j=0,\frac{1}{2},...,\frac{k-2}{4}\\\
\ket{\frac{k}{4}}^{\mathsf{F}^{\prime}}_{\text{NS}}&=\sqrt{2}\ket{\frac{k}{4}}^{\mathsf{A}}\otimes\ket{+},\\\
\ket{\frac{k}{4}}^{\mathsf{F}^{\prime}}_{\text{R}}&=\sqrt{2}\ket{\frac{k}{4}}_{\mathrm{tw}}^{\mathsf{A}}\otimes\ket{-}.\\\
\end{split}$ (3.13)

Before proceeding, we should pause here to mention that the boundary states of
the fermionic versions of diagonal unitary minimal models are given
essentially by the same formulas, since the primaries $(r,s)$ of the Virasoro
minimal models111111The indices $(r,s)$ here are different from those we have
used in the previous section to denote the number of $\mathbb{Z}_{2}$ defects
on $T^{2}$.  have the fusion rule which is isomorphic to the fusion rule of
the $SU(2)$ affine algebra acting separately on $r$ and $s$, and the
$\mathbb{Z}_{2}$ quotient acts only on a single index [44, 45, 46]. The
resulting boundary states agree with those determined in [23], up to the
factor of $\sqrt{2}$ explained at the end of Sec. 2. The $SU(2)_{k}$ WZW
models also have exceptional invariants [47, 48]. We have not studied the
boundary states of fermionic versions of these, but again it should be
possible to recover them from the discussion of boundary states of fermionic
exceptional minimal models in [23].

###  3.2 The Ising model and a massless Majorana fermion

It is well known that the bosonization of a free massless Majorana fermion is
the Ising CFT. We will briefly review the boundary states to set up the
notation. A careful discussion was given in [49], but our interpretation is
slightly different from the one given there.

The Cardy states of the Ising CFT were discussed in [33]. They are
$\ket{0},\ket{\frac{1}{2}}$ and $\ket{\frac{1}{16}}$ and can be written in
terms of the Ishibashi states
$|0\rangle\\!\rangle,|\frac{1}{2}\rangle\\!\rangle$ and
$|\frac{1}{16}\rangle\\!\rangle$ via (A.11):

$\begin{split}\ket{0}&=\frac{1}{\sqrt{2}}|0\rangle\\!\rangle+\frac{1}{\sqrt{2}}|\frac{1}{2}\rangle\\!\rangle+\frac{1}{2^{\frac{1}{4}}}|\frac{1}{16}\rangle\\!\rangle,\\\
\ket{\frac{1}{2}}&=\frac{1}{\sqrt{2}}|0\rangle\\!\rangle+\frac{1}{\sqrt{2}}|\frac{1}{2}\rangle\\!\rangle-\frac{1}{2^{\frac{1}{4}}}|\frac{1}{16}\rangle\\!\rangle,\\\
\ket{\frac{1}{16}}&=|0\rangle\\!\rangle-|\frac{1}{2}\rangle\\!\rangle.\end{split}$
(3.14)

Under the $\mathbb{Z}_{2}$ generator $g$ we have

$g\ket{0}=\ket{\frac{1}{2}},~{}~{}~{}g\ket{\frac{1}{2}}=\ket{0},~{}~{}~{}g\ket{\frac{1}{16}}=\ket{\frac{1}{16}}.$
(3.15)

Hence $\ket{\frac{1}{16}}$ is invariant under $\mathbb{Z}_{2}$, while the
other two states are exchanged by $\mathbb{Z}_{2}$. In additional to the three
Cardy states, there is also a twisted state
$\ket{\frac{1}{16}}_{\mathrm{tw}}$.

For the free Majorana fermion, there are two boundary conditions,
$\psi_{L}=\pm\psi_{\text{R}}$. Following [49], we will denote the boundary
states by $\ket{B_{\pm}}_{NS/R}$. To write down the boundary states, we
consider the mode expansion of the fermions:
$\psi_{L}(z)=\sum_{\text{R}}\psi_{\text{R}}z^{-r},\psi_{\text{R}}(\bar{z})=\sum_{\text{R}}\tilde{\psi}_{\text{R}}\bar{z}^{-r}$
where $r\in\mathbb{Z}$ for the R sector, and $r\in\mathbb{Z}+\frac{1}{2}$ for
the NS sector. Note that we have zero modes in the R sector. Boundary states
can then be written down by noticing that we have the conditions $\psi=\pm
i\tilde{\psi}$ on the boundary along the spatial direction [49]:

$\begin{split}\ket{B_{\pm}}_{\text{NS}}&\propto\exp\left(\pm
i\sum_{r>0,r\in\mathbb{Z}+\frac{1}{2}}\psi_{-r}\tilde{\psi}_{-r}\right)\ket{0},\\\
\ket{B_{\pm}}_{\text{R}}&\propto\exp\left(\pm
i\sum_{n>0,n\in\mathbb{Z}}\psi_{-n}\tilde{\psi}_{-n}\right)\ket{\pm_{0}},\end{split}$
(3.16)

where the sign $\pm$ in the exponent and in the zero mode state
$\ket{\pm_{0}}$ are correlated with the choice of boundary condition
$\psi_{L}=\pm\psi_{\text{R}}$. The vacuum $\ket{0}$ in the NS sector is
defined to be annihilated by all the fermion modes with positive index
$\psi_{r}\ket{0}=\tilde{\psi}_{r}\ket{0}=0,r>0$. Let us further specify
$\ket{\pm_{0}}$ in the R sector. There are two fermion zero modes, $\psi_{0}$
and $\tilde{\psi}_{0}$. We can now form a complex fermion
$c=\psi_{0}+i\widetilde{\psi}_{0}$, which then defines a two dimension Hilbert
space $\ket{\pm_{0}}$ via

$c\ket{+_{0}}=0,~{}~{}~{}\ket{-_{0}}=c^{\dagger}\ket{+_{0}},~{}~{}~{}c^{\dagger}\ket{-_{0}}=0,~{}~{}~{}\psi_{n}\ket{\pm_{0}}=\tilde{\psi}_{n}\ket{\pm_{0}}=0,~{}~{}~{}\forall
n>0.$ (3.17)

To see the fermion parity, note that the state in the NS sector does not have
fermion zero modes, and non zero modes come in pairs of left handed and right
handed modes. Thus $(-1)^{F}=1$ for $\ket{B_{\pm}}_{\text{NS}}$. For the R
sector states, the nonzero modes in the exponent come in pairs, hence they do
not contribute to nontrivial fermion parity either. However, the zero modes do
contribute to nontrivial fermion parity. We declare that the zero mode state
$\ket{+_{0}}$ is even, $(-1)^{F}\ket{+_{0}}=\ket{+_{0}}$, then

$(-1)^{F}\ket{-_{0}}=(-1)^{F}c^{\dagger}\ket{+_{0}}=-c^{\dagger}(-1)^{F}\ket{+_{0}}=-\ket{-_{0}}$
(3.18)

which shows that $\ket{-_{0}}$ has odd fermion parity. In summary, we have

$(-1)^{F}\ket{\pm_{0}}=\pm\ket{\pm_{0}}.$ (3.19)

The fermion parities of various states can now be summarized as follows:

$\begin{split}(-1)^{F}=+1:&~{}~{}~{}\ket{B_{+}}_{\text{NS}},~{}~{}~{}\ket{B_{-}}_{\text{NS}},~{}~{}~{}\ket{B_{+}}_{\text{R}}\\\
(-1)^{F}=-1:&~{}~{}~{}\ket{B_{-}}_{\text{R}}.\end{split}$ (3.20)

Notice that this is consistent with our general results (2.41) and (2.42).
There are two possible identifications:

$\begin{split}\ket{B_{-}}_{\text{NS}}&=(\ket{0}+\ket{\frac{1}{2}})\otimes\ket{+},\\\
\ket{B_{+}}_{\text{NS}}&=\vphantom{\frac{1}{\sqrt{2}}}\ket{\frac{1}{16}}\otimes\ket{+},\\\
\ket{B_{-}}_{\text{R}}&=(\ket{0}-\ket{\frac{1}{2}})\otimes\ket{-},\\\
\ket{B_{+}}_{\text{R}}&=\vphantom{\frac{1}{\sqrt{2}}}\ket{\frac{1}{16}}_{\mathrm{tw}}\otimes\ket{+}.\\\
\end{split}\quad\text{or}\quad\begin{split}\ket{B_{-}}_{\text{NS}}&=\sqrt{2}\ket{\frac{1}{16}}\otimes\ket{+},\\\
\ket{B_{+}}_{\text{NS}}&=\frac{1}{\sqrt{2}}(\ket{0}+\ket{\frac{1}{2}})\otimes\ket{+},\\\
\ket{B_{-}}_{\text{R}}&=\sqrt{2}\ket{\frac{1}{16}}_{\mathrm{tw}}\otimes\ket{-},\\\
\ket{B_{+}}_{\text{R}}&=\frac{1}{\sqrt{2}}(\ket{0}-\ket{\frac{1}{2}})\otimes\ket{+}.\\\
\end{split}$ (3.21)

These two choices are equally valid. This is because the Ising theory is self-
dual under the Krammers-Wannier duality, and therefore when we take
$\mathsf{A}$ to be the Ising theory we have $\mathsf{A}\simeq\mathsf{D}$,
which then leads to $\mathsf{F}\simeq\mathsf{F}^{\prime}$. We discuss more
details in Sec. 4.3.

###  3.3 $Spin(N)_{1}$ WZW Model and $N$ Majorana Fermions

We move on to consider $N$ Majorana fermions. As is well-known, this is a
fermionization of the $Spin(N)_{1}$ WZW model. We start by checking this
statement.

#### The bulk spectrum:

The $Spin(N)_{1}$ WZW model has three primaries $0$, $v$, $s$ for odd $N$ and
four primaries $0$, $v$, $s$, $c$ for even $N$, where the letters specify how
the primary transforms under $Spin(N)$: adjoint ($0$), vector ($v$), spinor
($s$), and conjugate spinor ($c$). Their spins are $0$, $\frac{1}{2}$,
$\frac{N}{16}$, $\frac{N}{16}$ in this order, and we also denote the primaries
by their spins, distinguishing the conjugate spinor by a tilde if necessary.
We consider the diagonal modular invariant as the theory $\mathsf{A}$, and use
the $\mathbb{Z}_{2}$ symmetry generated by the primary $v$. We then have

$\displaystyle\mathcal{H}_{S}$
$\displaystyle=V_{0}\otimes\overline{V_{0}}\oplus
V_{v}\otimes\overline{V_{v}},$ $\displaystyle\mathcal{H}_{U}$
$\displaystyle=V_{s}\otimes\overline{V_{s}},$ (3.22)
$\displaystyle\mathcal{H}_{T}$ $\displaystyle=V_{s}\otimes\overline{V_{s}},$
$\displaystyle\mathcal{H}_{V}$
$\displaystyle=V_{0}\otimes\overline{V_{v}}\oplus
V_{v}\otimes\overline{V_{0}},$

for odd $N$ and

$\displaystyle\mathcal{H}_{S}$
$\displaystyle=V_{0}\otimes\overline{V_{0}}\oplus
V_{v}\otimes\overline{V_{v}},$ $\displaystyle\mathcal{H}_{U}$
$\displaystyle=V_{s}\otimes\overline{V_{c}}\oplus
V_{c}\otimes\overline{V_{s}},$ (3.23) $\displaystyle\mathcal{H}_{T}$
$\displaystyle=V_{s}\otimes\overline{V_{s}}\oplus
V_{c}\otimes\overline{V_{c}},$ $\displaystyle\mathcal{H}_{V}$
$\displaystyle=V_{0}\otimes\overline{V_{v}}\oplus
V_{v}\otimes\overline{V_{0}}$

for even $N$.

Now the theory of $N$ Majorana fermions have the Hilbert space

$\displaystyle\mathcal{H}_{\text{NS}}$ $\displaystyle=(V_{0}\oplus
V_{v})\otimes(\overline{V_{0}}\oplus\overline{V_{v}}),$ (3.24)
$\displaystyle\mathcal{H}_{\text{R}}$ $\displaystyle=\begin{cases}(V_{s}\oplus
V_{c})\otimes(\overline{V_{s}}\oplus\overline{V_{c}})&(\text{$N$ even}),\\\
V_{s}\otimes\overline{V_{s}}\oplus V_{s}\otimes\overline{V_{s}}&(\text{$N$
odd}).\end{cases}$

We can now use (1.5) to confirm that this can be identified either with the
theory $\mathsf{F}$ or $\mathsf{F}^{\prime}$ for odd $N$. There is a subtlety
for even $N$: the standard assignment of $(-1)^{F}$ in the R sector is to
assign $(-1)^{F}=+1$ for $V_{s}\otimes\overline{V_{s}}\oplus
V_{c}\otimes\overline{V_{c}}$ and $(-1)^{F}=-1$ for
$V_{s}\otimes\overline{V_{c}}\oplus V_{c}\otimes\overline{V_{s}}$. This is the
assignment for the theory $\mathsf{F}^{\prime}$ but not for $\mathsf{F}$.

#### Cardy states of the bosonic model:

Let us now discuss the Cardy states of the bosonic $Spin(N)_{1}$ WZW model.
For odd $N$, there are three primary fields, with conformal weights
$(0,0),(\frac{1}{2},\frac{1}{2}),(\frac{N}{16},\frac{N}{16})$. The $S$ matrix
is the same as the one for the Ising model, hence the three Cardy states have
the same expression (3.14) in terms of the Ishibashi states, where we need to
replace the label $1/16$ by $N/16$. Under $\mathbb{Z}_{2}$ we have

$g\ket{0}=\ket{\frac{1}{2}},~{}~{}~{}g\ket{\frac{1}{2}}=\ket{0},~{}~{}~{}g\ket{\frac{N}{16}}=\ket{\frac{N}{16}},$
(3.25)

in particular $\ket{\frac{N}{16}}$ is $\mathbb{Z}_{2}$ invariant. This means
that we also have a $\mathbb{Z}_{2}$ twisted sector state
$\ket{\frac{N}{16}}_{\mathrm{tw}}$.

For even $N$, the modular $S$ matrix is

$S=\frac{1}{2}\begin{pmatrix}1&1&1&1\\\ 1&e^{i\pi N/4}&-1&-e^{i\pi N/4}\\\
1&-1&1&-1\\\ 1&-e^{i\pi N/4}&-1&e^{i\pi N/4}\end{pmatrix}$ (3.26)

where the columns from the left to the right are for the primaries $0$,
$\frac{N}{16}$, $\frac{1}{2}$, $\widetilde{\frac{N}{16}}$ in this order. Hence
in terms of the Ishibashi states, we have

$\begin{split}\ket{0}&=\frac{1}{\sqrt{2}}\left(|0\rangle\\!\rangle+|\frac{N}{16}\rangle\\!\rangle+|\frac{1}{2}\rangle\\!\rangle+\widetilde{|\frac{N}{16}\rangle\\!\rangle}\right),\\\
\ket{\frac{N}{16}}&=\frac{1}{\sqrt{2}}\left(|0\rangle\\!\rangle+e^{i\pi
N/4}|\frac{N}{16}\rangle\\!\rangle-|\frac{1}{2}\rangle\\!\rangle-e^{i\pi
N/4}\widetilde{|\frac{N}{16}\rangle\\!\rangle}\right),\\\
\ket{\frac{1}{2}}&=\frac{1}{\sqrt{2}}\left(|0\rangle\\!\rangle-|\frac{N}{16}\rangle\\!\rangle+|\frac{1}{2}\rangle\\!\rangle-\widetilde{|\frac{N}{16}\rangle\\!\rangle}\right),\\\
\widetilde{\ket{\frac{N}{16}}}&=\frac{1}{\sqrt{2}}\left(|0\rangle\\!\rangle-e^{i\pi
N/4}|\frac{N}{16}\rangle\\!\rangle-|\frac{1}{2}\rangle\\!\rangle+e^{i\pi
N/4}\widetilde{|\frac{N}{16}\rangle\\!\rangle}\right).\\\ \end{split}$ (3.27)

Under $\mathbb{Z}_{2}$, $|0\rangle\\!\rangle$ and
$|\frac{1}{2}\rangle\\!\rangle$ are even, and $|\frac{N}{16}\rangle\\!\rangle$
and $\widetilde{|\frac{N}{16}\rangle\\!\rangle}$ are odd. In terms of Cardy
states we then have

$g\ket{0}=\ket{\frac{1}{2}},~{}~{}~{}g\ket{\frac{1}{2}}=\ket{0},~{}~{}~{}g\ket{\frac{N}{16}}=\widetilde{\ket{\frac{N}{16}}},~{}~{}~{}g\widetilde{\ket{\frac{N}{16}}}=\ket{\frac{N}{16}}.$
(3.28)

Hence they form two $\mathbb{Z}_{2}$ conjugate pairs. There is no
$\mathbb{Z}_{2}$ invariant boundary state, and therefore there are no boundary
states in the $\mathbb{Z}_{2}$ twisted sector.

#### Boundary states of free fermions:

We further discuss the boundary states in the fermionic theory. We will
preserve the $SO(N)$ global symmetry, hence all flavors of fermions will be in
the same sector, and will have the same sign in $\psi^{i}_{+}=\pm\psi^{i}_{-}$
for all $i=1,...,N$. The boundary states are direct generalization of (3.16),

$\begin{split}\ket{B_{\pm}}^{N}_{\text{NS}}&\propto\exp\left(\pm
i\sum_{r>0,r\in\mathbb{Z}+\frac{1}{2}}\sum_{j=1}^{N}\psi_{-r}^{j}\tilde{\psi}_{-r}^{j}\right)\ket{0},\\\
\ket{B_{\pm}}^{N}_{\text{R}}&\propto\exp\left(\pm
i\sum_{n>0,n\in\mathbb{Z}}\sum_{j=1}^{N}\psi_{-n}^{j}\tilde{\psi}_{-n}^{j}\right)\ket{\pm_{0}},\end{split}$
(3.29)

where $\ket{0}$ is defined to be annihilated by all $\psi_{\text{R}}^{i}$’s
with $r>0$, and $\ket{\pm_{0}}$ is defined as
$\otimes_{j=1}^{N}\ket{\pm_{0}^{j}}$ where $\ket{\pm_{0}^{j}}$ is determined
in (3.17) for the $j$-th flavor. In summary, there are still four boundary
conditions, i.e.
$\ket{B_{+}}^{N}_{\text{NS}},\ket{B_{-}}^{N}_{\text{NS}},\ket{B_{-}}^{N}_{\text{R}},\ket{B_{+}}^{N}_{\text{R}}$.

Let us analyze the fermion parity for the boundary states. For NS states
$\ket{B_{\pm}}_{\text{NS}}$, the non-zero modes do not contribute to fermion
parity. For R states $\ket{B_{\pm}}_{\text{R}}$, only the zero mode
contributes the fermion parity. The fermion number is the sum of fermion
number for each flavor, hence

$\begin{cases}\ket{+_{0}^{1}}\otimes\cdots\otimes\ket{+_{0}^{N}},&~{}~{}~{}(-1)^{F}=1,\\\
\ket{-_{0}^{1}}\otimes\cdots\otimes\ket{-_{0}^{N}},&~{}~{}~{}(-1)^{F}=(-1)^{N}.\\\
\end{cases}$ (3.30)

Thus when $N$ is odd, $\ket{B_{-}}_{\text{R}}$ should have $(-1)^{F}=-1$ and
$\ket{B_{+}}_{\text{NS}},\ket{B_{-}}_{\text{NS}},\ket{B_{-}}_{\text{R}}$ all
have $(-1)^{F}=1$. However, when $N$ is even, all boundary states in (3.30)
have even fermion parity. Thus we conclude as follows:

  1. 1.

When $N$ is odd, the boundary states of fermionic theory are related to those
in the bosonic theory in the same way as in $N=1$ case (3.21).

  2. 2.

When $N$ is even, all the boundary states satisfy $(-1)^{F}=1$. This is
consistent with the fact that $Spin(N)_{1}$ WZW with even $N$ does not have
$\mathbb{Z}_{2}$-invariant Cardy states, therefore there is no boundary states
in the twisted sector. With our rules, each of the boundary states in the
theory $\mathsf{F}$ has unpaired Majorana zero mode, while the boundary states
in the theory $\mathsf{F}^{\prime}$ do not have Majorana zero modes. The
boundary states usually used in the literature are the latter ones, and we
identify the boundary states as follows:

$\displaystyle\ket{B_{+}}^{N}_{\text{NS}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{0}+\ket{\frac{1}{2}})\otimes\ket{+},$
$\displaystyle\ket{B_{+}}^{N}_{\text{R}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{0}-\ket{\frac{1}{2}})\otimes\ket{+},$
(3.31) $\displaystyle\ket{B_{-}}^{N}_{\text{NS}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{\frac{N}{16}}+\widetilde{\ket{\frac{N}{16}}})\otimes\ket{+},$
$\displaystyle\ket{B_{-}}^{N}_{\text{R}}$
$\displaystyle=\frac{1}{\sqrt{2}}(\ket{\frac{N}{16}}-\widetilde{\ket{\frac{N}{16}}})\otimes\ket{+}.$

###  3.4 Maldacena-Ludwig boundary state of eight Majorana fermions

As the last example in this section, let us discuss the boundary state of
eight Majorana fermions studied by Maldacena and Ludwig in [50]. This boundary
state arises when we study the monopole catalysis of baryon decay [51, 52, 52]
and also when we study the Kondo problem [53, 54]. It was also recently
revisited in [55].

Let us start by recalling the bulk system, which is the special case $N=8$ of
$N$ Majorana fermions we discussed above. In this particular case, the
primaries $\frac{1}{2}$, $\frac{N}{16}$ and $\widetilde{\frac{N}{16}}$ all
have the same spin. We distinguish them by labeling them as $v$, $s$ and $c$,
as commonly done. The $spin(8)_{1}$ affine algebra has an $S_{3}$ outer
automorphism permuting them. The Maldacena-Ludwig boundary state is
characterized by the condition that the left-moving $spin(8)_{1}$ and the
right-moving $spin(8)_{1}$ are related by the order-2 automorphism $\omega$
exchanging $v\leftrightarrow s$ and fixing $0$ and $c$. We therefore have
$J=\omega(\tilde{J})$ at the boundary, and any boundary state $\ket{\phi}$
with such a boundary condition needs to satisfy

$(J_{-n}+\omega(\tilde{J}_{n}))\ket{\phi}=0.$ (3.32)

Note that the boundary conditions and the defect operators discussed up to the
previous subsection always used the trivial automorphism to identify the left-
moving and right-moving chiral algebras. Note in particular that the
$\mathbb{Z}_{2}$ lines we used repeatedly, generated by the primary $v$,
commute with the chiral algebra, and are not to be confused with the
automorphisms of $spin(8)_{1}$ affine algebra.

The property of such $\omega$-twisted boundary states was studied in detail in
[56], and we can simply quote the results there. In the $v$-untwisted sector,
there are two $\omega$-twisted Ishibashi states

$|0\rangle\\!\rangle_{\omega}:=R_{0}(\omega)|0\rangle\\!\rangle,\qquad|c\rangle\\!\rangle_{\omega}:=R_{c}(\omega)|c\rangle\\!\rangle$
(3.33)

where $R_{0}(\omega)$, $R_{c}(\omega)$ are the representation matrices of
$\omega$ on $V_{0}$ and $V_{c}$ with the convention that they only act on the
holomorphic side. The two $\omega$-twisted Cardy states are then

$\ket{0}_{\omega}=|0\rangle\\!\rangle_{\omega}+|c\rangle\\!\rangle_{\omega},\qquad\ket{c}_{\omega}=|0\rangle\\!\rangle_{\omega}-|c\rangle\\!\rangle_{\omega},\qquad$
(3.34)

and they are exchanged by the action of the $\mathbb{Z}_{2}$ Verlinde line
operator $v$.

We then convert this pair of $\mathbb{Z}_{2}$-breaking states to the theory
$\mathsf{F}^{\prime}$ to find the Maldacena-Ludwig boundary state121212We were
lucky that the $\omega$-twisted boundary states on the bosonic side turned out
to be $\mathbb{Z}_{2}$-breaking. If it were $\mathbb{Z}_{2}$-preserving, we
would have also needed the $\omega$-twisted boundary state in the $v$-twisted
sector. The general theory of such ‘doubly-twisted’ boundary states do not
seem to be readily available. , which is given by

$\ket{\text{ML}}_{\text{NS}}=\sqrt{2}|0\rangle\\!\rangle_{\omega},\qquad\ket{\text{ML}}_{\text{R}}=\sqrt{2}|c\rangle\\!\rangle_{\omega}.$
(3.35)

We can then find, for example,

$\begin{array}[]{r@{}l@{}l}{}_{\text{NS}}\braket{\text{ML}}{e^{-2\pi
H/\delta}}{\text{ML}}&{}_{\text{NS}}&=\chi_{0}(i\delta)+\chi_{v}(i\delta)+\chi_{s}(i\delta)+\chi_{c}(i\delta),\\\
{}_{\text{R}}\braket{\text{ML}}{e^{-2\pi
H/\delta}}{\text{ML}}&{}_{\text{R}}&=\chi_{0}(i\delta)-\chi_{v}(i\delta)-\chi_{s}(i\delta)+\chi_{c}(i\delta).\end{array}$
(3.36)

##  4 Interfaces and boundary states

In the last section we derived the form of the boundary states of fermionic
theories $\mathsf{F}$ and $\mathsf{F}^{\prime}$. In this section we use this
knowledge to determine the algebra of the interfaces between the theories
$\mathsf{A}$, $\mathsf{D}$, $\mathsf{F}$ and $\mathsf{F}^{\prime}$. This
analysis also allows us to see how the anomalous chiral $\mathbb{Z}_{2}$
symmetry arises in the theory $\mathsf{F}\simeq\mathsf{F}^{\prime}$ when
$\mathsf{A}\simeq\mathsf{D}$.

###  4.1 $\mathsf{A}\leftrightarrow\mathsf{D}$

We start by considering the interface between the theory $\mathsf{A}$ and
$\mathsf{D}$. When placed on a circle, such an interface determines various
operators, some of which are shown in Fig 2. Namely, from $\mathsf{A}$ to
$\mathsf{D}$ we have

$\displaystyle I_{\mathsf{A}\to\mathsf{D}}$
$\displaystyle:\mathcal{H}^{\mathsf{A}}\to\mathcal{H}^{\mathsf{D}},$
$\displaystyle I_{\mathsf{A}\to\mathsf{D}_{\text{tw}}}$
$\displaystyle:\mathcal{H}^{\mathsf{A}}\to\mathcal{H}^{\mathsf{D}}_{\text{tw}},$
(4.1) $\displaystyle I_{\mathsf{A}_{\text{tw}}\to\mathsf{D}}$
$\displaystyle:\mathcal{H}^{\mathsf{A}}_{\text{tw}}\to\mathcal{H}^{\mathsf{D}},$
$\displaystyle I_{\mathsf{A}_{\text{tw}}\to\mathsf{D}_{\text{tw}}}$
$\displaystyle:\mathcal{H}^{\mathsf{A}}_{\text{tw}}\to\mathcal{H}^{\mathsf{D}}_{\text{tw}}.$

The interface operators from $\mathsf{D}$ to $\mathsf{A}$ can be considered in
a similar manner and they are adjoints of the operators given in (4.1).

|  |  |
---|---|---|---
$I_{\mathsf{A}\to\mathsf{D}}$ | $I_{\mathsf{A}_{\text{tw}}\to\mathsf{D}}$ | $I_{\mathsf{A}\to\mathsf{D}_{\text{tw}}}$ | $I_{\mathsf{A}_{\text{tw}}\to\mathsf{D}_{\text{tw}}}$
Figure 2: Interfaces from the theory $\mathsf{A}$ to $\mathsf{D}$.

From the known relations among the Hilbert spaces (1.5) of $\mathsf{A}$ and
$\mathsf{D}$, we know that

$I_{\mathsf{A}\to\mathsf{D}}\propto P_{S},\quad
I_{\mathsf{A}\to\mathsf{D}_{\text{tw}}}\propto P_{T},\quad
I_{\mathsf{A}_{\text{tw}}\to\mathsf{D}}\propto P_{U},\quad
I_{\mathsf{A}_{\text{tw}}\to\mathsf{D}_{\text{tw}}}\propto P_{V},$ (4.2)

where $P_{S,T,U,V}$ are the projections to the respective components in (1.5).

For example, we have
$\mathcal{H}^{\mathsf{A}}=\mathcal{H}_{S}\oplus\mathcal{H}_{T}$ and
$\mathcal{H}^{\mathsf{D}}=\mathcal{H}_{S}\oplus\mathcal{H}_{T}$. Therefore,
$I_{\mathsf{A}\to\mathsf{D}}$ can only map the states in
$\mathcal{H}_{S}\subset\mathcal{H}^{\mathsf{A}}$ to
$\mathcal{H}_{S}\subset\mathcal{H}^{\mathsf{D}}$. We expect that
$I_{\mathsf{D}\to\mathsf{A}}I_{\mathsf{A}\to\mathsf{D}}$ is proportional to
the identity on $\mathcal{H}_{S}$. The two copies of $\mathcal{H}_{S}$ belong
to two distinct Hilbert spaces131313When $\mathsf{A}\simeq\mathsf{D}$,
$I_{\mathsf{A}\to\mathsf{D}}$ can define a non-trivial order-2 operation on
$\mathcal{H}_{S}$. We will discuss this possibility in more detail in Sec.
4.3.  $\mathcal{H}^{\mathsf{A}}$ and $\mathcal{H}^{\mathsf{D}}$, so we use an
appropriate multiple of $I_{\mathsf{D}\leftrightarrow\mathsf{A}}$ to identify
them, resulting in the simple equations (4.2).

| $\ket{i}^{\mathsf{A}}$ | $\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$ | $\ket{a\pm}^{\mathsf{A}}$
---|---|---|---
$I_{\mathsf{A}\to\mathsf{D}}$ | $\ket{i+}^{\mathsf{D}}+\ket{i-}^{\mathsf{D}}$ | - | $\ket{a}^{\mathsf{D}}$
$I_{\mathsf{A}\to\mathsf{D}_{\mathrm{tw}}}$ | 0 | - | $\pm\ket{a}_{\mathrm{tw}}^{\mathsf{D}}$
$I_{\mathsf{A}_{\mathrm{tw}}\to\mathsf{D}}$ | - | $\ket{i+}^{\mathsf{D}}-\ket{i-}^{\mathsf{D}}$ | -
$I_{\mathsf{A}_{\mathrm{tw}}\to\mathsf{D}_{\mathrm{tw}}}$ | - | 0 | -
| $\ket{a}^{\mathsf{D}}$ | $\ket{a}_{\mathrm{tw}}^{\mathsf{D}}$ | $\ket{i\pm}^{\mathsf{D}}$
$I_{\mathsf{D}\to\mathsf{A}}$ | $\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}}$ | - | $\ket{i}^{\mathsf{A}}$
$I_{\mathsf{D}_{\mathrm{tw}}\to\mathsf{A}}$ | - | $\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}}$ | -
$I_{\mathsf{D}\to\mathsf{A}_{\mathrm{tw}}}$ | 0 | - | $\pm\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$
$I_{\mathsf{D}_{\mathrm{tw}}\to\mathsf{A}_{\mathrm{tw}}}$ | - | 0 | -
Table 1: Action of interface operators (between theories $\mathsf{A}$ and
$\mathsf{D}$) on the boundary states.

The proportionality coefficients can be fixed in various ways. Here, we
require that the application of the interface operators to the boundary states
give rise to an integral linear combination of the boundary states. Comparing
(1.11) and (1.12), we find141414Strictly speaking this method does not
determine $I_{\mathsf{A}_{\text{tw}}\to\mathsf{D}_{\text{tw}}}$, since the
projection of the boundary states is zero.

$I_{\mathsf{A}\to\mathsf{D}}=\sqrt{2}P_{S},\quad
I_{\mathsf{A}\to\mathsf{D}_{\text{tw}}}=\sqrt{2}P_{T},\quad
I_{\mathsf{A}_{\text{tw}}\to\mathsf{D}}=\sqrt{2}P_{U},\quad
I_{\mathsf{A}_{\text{tw}}\to\mathsf{D}_{\text{tw}}}=\sqrt{2}P_{V}$ (4.3)

and similarly for the interfaces from $\mathsf{D}$ to $\mathsf{A}$. We then
find, for example,

$I_{\mathsf{A}\to\mathsf{D}}\ket{i}^{\mathsf{A}}=\ket{i+}^{\mathsf{D}}+\ket{i-}^{\mathsf{D}},\qquad
I_{\mathsf{A}\to\mathsf{D}}\ket{a\pm}^{\mathsf{A}}=\ket{a}^{\mathsf{D}}$ (4.4)

and

$I_{\mathsf{D}\to\mathsf{A}}I_{\mathsf{A}\to\mathsf{D}}=1+g$ (4.5)

where $g$ is the $\mathbb{Z}_{2}$ generator. Here the right hand side is
restricted to act on the untwisted sector. This last relation is known to
generalize to

$I_{\mathsf{D}\to\mathsf{A}}I_{\mathsf{A}\to\mathsf{D}}=\sum_{g\in G}g$ (4.6)

when $\mathsf{A}$ is $G$-symmetric and $\mathsf{D}$ is the $G$-gauged theory
[57]. Systematically, we tabulate the action of interface operators between
$\mathsf{A}$ and $\mathsf{D}$ on the boundary states in table 1.

###  4.2 $\mathsf{A}\leftrightarrow\mathsf{F},\mathsf{F}^{\prime}$

Next we consider the interfaces between the original theory and the
fermionized versions. We first consider the ones between $\mathsf{A}$ and
$\mathsf{F}^{\prime}$. We denote the interfaces from $\mathsf{A}$ to
$\mathsf{F}^{\prime}$ as

$\displaystyle I_{\mathsf{A}\to\mathsf{F}^{\prime}_{\text{NS}}}$
$\displaystyle:\mathcal{H}^{\mathsf{A}}\to\mathcal{H}^{\mathsf{F}^{\prime}}_{\text{NS}},$
$\displaystyle I_{\mathsf{A}\to\mathsf{F}^{\prime}_{\text{R}}}$
$\displaystyle:\mathcal{H}^{\mathsf{A}}\to\mathcal{H}^{\mathsf{F}^{\prime}}_{\text{R}},$
(4.7) $\displaystyle
I_{\mathsf{A}_{\text{tw}}\to\mathsf{F}^{\prime}_{\text{NS}}}$
$\displaystyle:\mathcal{H}^{\mathsf{A}}_{\text{tw}}\to\mathcal{H}^{\mathsf{F}^{\prime}}_{\text{NS}},$
$\displaystyle I_{\mathsf{A}_{\text{tw}}\to\mathsf{F}^{\prime}_{\text{R}}}$
$\displaystyle:\mathcal{H}^{\mathsf{A}}_{\text{tw}}\to\mathcal{H}^{\mathsf{F}^{\prime}}_{\text{R}}.$

Again they are proportional to the projectors $P_{S,T,U,V}$:

$I_{\mathsf{A}\to\mathsf{F}^{\prime}_{\text{NS}}}\propto P_{S},\quad
I_{\mathsf{A}\to\mathsf{F}^{\prime}_{\text{R}}}\propto P_{T},\quad
I_{\mathsf{A}_{\text{tw}}\to\mathsf{F}^{\prime}_{\text{NS}}}\propto
P_{V},\quad I_{\mathsf{A}_{\text{tw}}\to\mathsf{F}^{\prime}_{\text{R}}}\propto
P_{U}.$ (4.8)

The proportionality coefficients can be found by studying their actions on
boundary states. We find that

$I_{\mathsf{A}\to\mathsf{F}^{\prime}_{\text{NS}}}=\sqrt{2}P_{S},\quad
I_{\mathsf{A}\to\mathsf{F}^{\prime}_{\text{R}}}=\sqrt{2}P_{T},\quad
I_{\mathsf{A}_{\text{tw}}\to\mathsf{F}^{\prime}_{\text{NS}}}=\sqrt{2}P_{V},\quad
I_{\mathsf{A}_{\text{tw}}\to\mathsf{F}^{\prime}_{\text{R}}}=\sqrt{2}P_{U}.$
(4.9)

These interfaces act on the boundary states for example as

$I_{\mathsf{A}\to\mathsf{F}^{\prime}_{\text{NS}}}\ket{i}^{\mathsf{A}}=\ket{i}^{\mathsf{F}^{\prime}}_{\text{NS}},\quad
I_{\mathsf{A}\to\mathsf{F}^{\prime}_{\text{NS}}}\ket{a\pm}^{\mathsf{A}}=\ket{a}^{\mathsf{F}^{\prime}}_{\text{NS}}$
(4.10)

and

$I_{\mathsf{F}^{\prime}_{\text{NS}}\to\mathsf{A}}I_{\mathsf{A}\to\mathsf{F}^{\prime}_{\text{NS}}}=1+g.$
(4.11)

Again the right hand side is restricted to the NS sector only. We tabulate the
action of interface operators between $\mathsf{A}$ and $\mathsf{F}^{\prime}$
on the boundary states in table 2. These actions were first determined in [58]
for two specific cases when the theory $\mathsf{A}$ is the critical Ising
model or the tricritical Ising model. They were also discussed in [9].

| $\ket{i}^{\mathsf{A}}$ | $\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$ | $\ket{a+}^{\mathsf{A}}$ | $\ket{a-}^{\mathsf{A}}$
---|---|---|---|---
$I_{\mathsf{A}\to\mathsf{F}^{\prime}_{\text{NS}}}$ | $\ket{i\psi}_{\text{NS}}^{\mathsf{F}^{\prime}}$ | - | $\ket{a}_{\text{NS}}^{\mathsf{F}^{\prime}}$ | $\ket{a}_{\text{NS}}^{\mathsf{F}^{\prime}}$
$I_{\mathsf{A}\to\mathsf{F}^{\prime}_{\text{R}}}$ | 0 | - | $+\ket{a}_{\text{R}}^{\mathsf{F}^{\prime}}$ | $-\ket{a}_{\text{R}}^{\mathsf{F}^{\prime}}$
$I_{\mathsf{A}_{\mathrm{tw}}\to\mathsf{F}^{\prime}_{\text{NS}}}$ | - | 0 | - | -
$I_{\mathsf{A}_{\mathrm{tw}}\to\mathsf{F}^{\prime}_{\text{R}}}$ | - | $\ket{i\psi}_{\text{R}}^{\mathsf{F}^{\prime}}$ | - | -
| $\ket{i\psi}_{\text{NS}}^{\mathsf{F}^{\prime}}$ | $\ket{i\psi}^{\mathsf{F}^{\prime}}_{\text{R}}$ | $\ket{a}_{\text{NS}}^{\mathsf{F}^{\prime}}$ | $\ket{a}_{\text{R}}^{\mathsf{F}^{\prime}}$
$I_{\mathsf{F}^{\prime}_{\text{NS}}\to\mathsf{A}}$ | $2\ket{i}^{\mathsf{A}}$ | - | $\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}}$ | -
$I_{\mathsf{F}^{\prime}_{\text{R}}\to\mathsf{A}}$ | - | $0$ | - | $\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}}$
$I_{\mathsf{F}^{\prime}_{\text{NS}}\to\mathsf{A}_{\mathrm{tw}}}$ | 0 | - | 0 | -
$I_{\mathsf{F}^{\prime}_{\text{R}}\to\mathsf{A}_{\mathrm{tw}}}$ | - | $2\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$ | - | 0
Table 2: Action of interface operators (between theories $\mathsf{A}$ and
$\mathsf{F}^{\prime}$) on the boundary states.

The interfaces between $\mathsf{A}$ and $\mathsf{F}$ can be determined in a
similar manner. We find

$I_{\mathsf{A}\to\mathsf{F}_{\text{NS}}}=2P_{S},\quad
I_{\mathsf{A}\to\mathsf{F}_{\text{R}}}=2P_{T},\quad
I_{\mathsf{A}_{\text{tw}}\to\mathsf{F}_{\text{NS}}}=2P_{V},\quad
I_{\mathsf{A}_{\text{tw}}\to\mathsf{F}_{\text{R}}}=2P_{U}.$ (4.12)

Note the difference by the factor $\sqrt{2}$ in (4.9) and (4.12). Because of
this we find that

$I_{\mathsf{F}_{\text{NS}}\to\mathsf{A}}I_{\mathsf{A}\to\mathsf{F}_{\text{NS}}}=2(1+g).$
(4.13)

We tabulate the action of interface operators between $\mathsf{A}$ and
$\mathsf{F}$ on the boundary states in table 3.

| $\ket{i}^{\mathsf{A}}$ | $\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$ | $\ket{a+}^{\mathsf{A}}$ | $\ket{a-}^{\mathsf{A}}$
---|---|---|---|---
$I_{\mathsf{A}\to\mathsf{F}_{\text{NS}}}$ | $2\ket{i}_{\text{NS}}^{\mathsf{F}}$ | - | $\ket{a\psi}_{\text{NS}}^{\mathsf{F}}$ | $\ket{a\psi}_{\text{NS}}^{\mathsf{F}}$
$I_{\mathsf{A}\to\mathsf{F}_{\text{R}}}$ | 0 | - | $+\ket{a\psi}{}_{\text{R}}^{\mathsf{F}}$ | $-\ket{a\psi}{}_{\text{R}}^{\mathsf{F}}$
$I_{\mathsf{A}_{\mathrm{tw}}\to\mathsf{F}_{\text{NS}}}$ | - | 0 | - | -
$I_{\mathsf{A}_{\mathrm{tw}}\to\mathsf{F}_{\text{R}}}$ | - | $2\ket{i}_{\text{R}}^{\mathsf{F}}$ | - | -
| $\ket{i}_{\text{NS}}^{\mathsf{F}}$ | $\ket{i}^{\mathsf{F}}_{\text{R}}$ | $\ket{a\psi}_{\text{NS}}^{\mathsf{F}}$ | $\ket{a\psi}_{\text{R}}^{\mathsf{F}}$
$I_{\mathsf{F}_{\text{NS}}\to\mathsf{A}}$ | $2\ket{i}^{\mathsf{A}}$ | - | $2(\ket{a+}^{\mathsf{A}}+\ket{a-}^{\mathsf{A}})$ | -
$I_{\mathsf{F}_{\text{R}}\to\mathsf{A}}$ | - | $0$ | - | $2(\ket{a+}^{\mathsf{A}}-\ket{a-}^{\mathsf{A}})$
$I_{\mathsf{F}_{\text{NS}}\to\mathsf{A}_{\mathrm{tw}}}$ | 0 | - | 0 | -
$I_{\mathsf{F}_{\text{R}}\to\mathsf{A}_{\mathrm{tw}}}$ | - | $2\ket{i}_{\mathrm{tw}}^{\mathsf{A}}$ | - | 0
Table 3: Action of interface operators (between theories $\mathsf{A}$ and
$\mathsf{F}$) on the boundary states.

The difference between (4.9) and (4.12) can also be understood by considering
the interface between the theories $\mathsf{F}$ and $\mathsf{F}^{\prime}$.
Since the only difference between these two theories is the stacking by the
nontrivial Kitaev chain, the interface simply hosts an unpaired Majorana mode,
and satisfies

$I_{\mathsf{F}_{\text{NS}}\to\mathsf{F}_{\text{NS}}^{\prime}}I_{\mathsf{F}^{\prime}_{\text{NS}}\to\mathsf{F}_{\text{NS}}}=2.$
(4.14)

This is the dimension of the Hilbert space generated by two paired Majorana
modes.

When acting on to the boundary states, we have

$I_{\mathsf{F}_{\text{NS}}\to\mathsf{F}^{\prime}_{\text{NS}}}\ket{i}^{\mathsf{F}}_{\text{NS}}=\ket{i}^{\mathsf{F}^{\prime}}_{\text{NS}},\quad
I_{\mathsf{F}_{\text{NS}}\to\mathsf{F}^{\prime}_{\text{NS}}}\ket{a}^{\mathsf{F}}=2\ket{a}^{\mathsf{F}^{\prime}}.$
(4.15)

The factor $2$ in the second equation comes from the fact that
$\ket{a}^{\mathsf{F}}$ already has an unpaired Majorana zero mode, which
combines with another Majorana zero mode hosted on the interface between
$\mathsf{F}$ and $\mathsf{F}^{\prime}$. We then find

$I_{\mathsf{F}^{\prime}_{\text{NS}}\to\mathsf{F}_{\text{NS}}}I_{\mathsf{A}\to\mathsf{F}^{\prime}_{\text{NS}}}=I_{\mathsf{A}\to\mathsf{F}_{\text{NS}}},\quad
I_{\mathsf{F}_{\text{NS}}\to\mathsf{F}^{\prime}_{\text{NS}}}I_{\mathsf{A}\to\mathsf{F}_{\text{NS}}}=2I_{\mathsf{A}\to\mathsf{F}^{\prime}_{\text{NS}}}.$
(4.16)

$\mathsf{D}$$\mathsf{F}$$\mathsf{F}^{\prime}$$\mathsf{A}$=

$\sqrt{2}$$2$$\sqrt{2}$$\mathsf{A}$

Figure 3: A closed loop of an interface without any operators in it can be
evaluated to a number. This number can be called the quantum dimension of the
interface, generalizing the same quantity for topological line operators of a
single theory.

Finally, we note that the proportionality coefficients

$I_{\mathsf{A}\to\mathsf{D}}=\sqrt{2}P_{S},\quad
I_{\mathsf{A}\to\mathsf{F}}=2P_{S},\quad
I_{\mathsf{A}\to\mathsf{F}^{\prime}}=\sqrt{2}P_{S}$ (4.17)

translate to the property of the interfaces shown in Fig. 3. Namely, when we
have a closed loop of an interface without any operator in it as shown there,
they can be simply evaluated to be an insertion of a number, $\sqrt{2}$, $2$
or $\sqrt{2}$ depending on whether the interface is from $\mathsf{A}$ to
$\mathsf{D}$, $\mathsf{F}$ or $\mathsf{F}^{\prime}$. For topological line
operators in a single theory, the number obtained by evaluating a closed loop
without any operator in it is called the _quantum dimension_ of the loop, and
here we are dealing with its natural generalization to the interfaces. That
the interface between $\mathsf{A}$ and $\mathsf{F}^{\prime}$ has a non-integer
quantum dimension $\sqrt{2}$ is consistent with the fact that when
$\mathsf{A}$ is trivial, $\mathsf{F}^{\prime}$ is the nontrivial Kitaev chain.

###  4.3 The special case $\mathsf{A}\simeq\mathsf{D}$ and the anomalous
$\mathbb{Z}_{2}$ symmetry of $\mathsf{F}\simeq\mathsf{F}^{\prime}$

In some important cases, e.g. the Ising model or the tricritical Ising model,
the $\mathbb{Z}_{2}$ orbifold theory $\mathsf{D}$ of the original bosonic
theory $\mathsf{A}$ is equal to itself, $\mathsf{A}\simeq\mathsf{D}$. In this
case the interface $X:=I_{\mathsf{A}\to\mathsf{D}}$ can be considered as a
defect of the single theory $\mathsf{A}\simeq\mathsf{D}$ which now satisfies
the fusion rule

$X^{2}=1+g,\qquad gX=X,$ (4.18)

see [59] and also more recent works [25, Sec. 4.3.1] and [60, 27]. We note
that, when we wrote in (4.3) above that
$I_{\mathsf{A}\to\mathsf{D}}=\sqrt{2}P_{S}$, this is meant to be a map
obtained by first projecting to the summand
$\mathcal{H}_{S}\subset\mathcal{H}^{\mathsf{A}}$ and then sent isometrically
to the corresponding summand $\mathcal{H}_{S}\subset\mathcal{H}^{\mathsf{D}}$.
We now identify $\mathcal{H}^{\mathsf{A}}\simeq\mathcal{H}^{\mathsf{D}}$, but
this can introduce a nontrivial unitary operator of order two on
$\mathcal{H}_{S}$. Therefore, the duality interface $X$ as acting on
$\mathcal{H}^{\mathsf{A}}$ is a composition

$X=\sqrt{2}h_{S}P_{S}$ (4.19)

of the projector $P_{S}$ to $\mathcal{H}_{S}$ together with a unitary operator

$h_{S}:\mathcal{H}_{S}\to\mathcal{H}_{S}$ (4.20)

which squares to $1$.

Similarly, the interface $X_{\text{tw}}:=I_{\mathsf{A}_{\text{tw}}\to
D_{\text{tw}}}$ has the form

$X_{\text{tw}}=\sqrt{2}h_{V}P_{V}$ (4.21)

where

$h_{V}:\mathcal{H}_{V}\to\mathcal{H}_{V}$ (4.22)

is again a unitary which squares to one.

The operators $h_{S}$ and $h_{V}$ then combine to give a unitary operator

$h:\mathcal{H}^{\mathsf{F}}_{\text{NS}}\to\mathcal{H}^{\mathsf{F}}_{\text{NS}}\qquad\text{where}\quad\mathcal{H}^{\mathsf{F}}_{\text{NS}}=\mathcal{H}_{S}\oplus\mathcal{H}_{V}$
(4.23)

which squares to one. This is the $\mathbb{Z}_{2}$ operator acting on the NS
sector of the theory $\mathsf{F}$.

For example, in the case of the Ising model treated above, we have
$X=\mathcal{L}_{\frac{1}{16}}$, which is the Kramers-Wannier duality defect.
Under $X$,

$\begin{split}X\ket{0}=X\ket{\frac{1}{2}}=\ket{\frac{1}{16}},~{}~{}~{}~{}~{}X\ket{\frac{1}{16}}=\ket{0}+\ket{\frac{1}{2}}.\end{split}$
(4.24)

Using (3.21), one finds that the boundary conditions $B_{-}$ and $B_{+}$ for
the Majorana fermions are exchanged, i.e., $\psi_{L}=\pm\psi_{\text{R}}$
becomes $\psi_{L}=\mp\psi_{\text{R}}$, which is precisely the chiral
$\mathbb{Z}_{2}$ symmetry.

On the R-sector, we see that the operator coming from the duality wall $X$
exchanges $\mathcal{H}_{U}$ and $\mathcal{H}_{T}$, and in particular flips the
sign of $(-1)^{F}$. This means that the $\mathbb{Z}_{2}$ symmetry is
anomalous. Recall that the anomaly of $\mathbb{Z}_{2}$ symmetry of fermionic
(1+1)-dimensional theory is specified by an integer modulo 8; it is known that
the anomalous $\mathbb{Z}_{2}$ symmetry obtained from a duality defect $X$
satisfying the fusion rule (4.18) automatically has the anomaly characterized
by an odd number modulo 8, see e.g. [21, Sec. IV and Sec. VI].

## Acknowledgements

The authors would like to thank the authors of [24] for discussions. The
authors would also like to thank Shu-Heng Shao and Philip Boyle Smith for very
helpful comments on the draft. Y.F. thanks discussions and comments of Yuan
Yao. He also thanks Shumpei Iino for the collaboration on the work [27] which
is closely related to this project. Y.T. is partially supported by JSPS
KAKENHI Grant-in-Aid (Wakate-A), No.17H04837 and JSPS KAKENHI Grant-in-Aid
(Kiban-S), No.16H06335. Y.T. and Y.Z. are partially supported by WPI
Initiative, MEXT, Japan at IPMU, the University of Tokyo. Y.F. acknowledges
the support by the QuantiXLie Center of Excellence, a project co-financed by
the Croatian Government and European Union through the European Regional
Development Fund – the Competitiveness and Cohesion Operational Programme
(Grant KK.01.1.1.01.0004).

##  Appendix A RCFTs and their $\mathbb{Z}_{2}$ symmetries

In this section we collect various facts on RCFTs. More specifically, we
discuss the diagonal modular invariants and their $\mathbb{Z}_{2}$ symmetries
generated by simple currents of order 2. This will allow us to construct a
large class of fermionic models and fermionic boundary states.

#### Diagonal modular invariant:

We start from a chiral algebra $\mathcal{A}$ with irreducible representations
$V_{\alpha}$, where $\alpha=0,\ldots,$ such that the label $0$ corresponds to
the vacuum representation. We denote the fusion rule as

$V_{\alpha}V_{\beta}\sim N^{\gamma}_{\alpha\beta}V_{\gamma}.$ (A.1)

The untwisted Hilbert space $\mathcal{H}_{0}$ has the decomposition

$\mathcal{H}_{0}=\bigoplus_{\alpha}V_{\alpha}\otimes\overline{V_{\alpha}}$
(A.2)

and therefore has the character

$\mathop{\mathrm{Tr}}\nolimits_{\mathcal{H}_{0}}e^{-2\pi
tH}=\sum_{\alpha}\chi_{\alpha}(t)\overline{\chi_{\alpha}(t)}.$ (A.3)

In some references these modular invariants are called charge-conjugation
invariants, since the left-mover and the right-movers are complex conjugates
of each other.

#### Verlinde line operators and generalized twisted sectors:

In this diagonal theory, we have Verlinde line operators labeled by $\alpha$
[61, 62, 59]. When wrapped around the spatial circle, it determines an
operator

$\mathcal{L}_{\alpha}:\mathcal{H}_{0}\to\mathcal{H}_{0}$ (A.4)

satisfying the fusion rule equation

$\mathcal{L}_{\alpha}\mathcal{L}_{\beta}=N_{\alpha\beta}^{\gamma}\mathcal{L}_{\gamma}.$
(A.5)

They are known to act in the following manner: by a multiplication by

$\mathcal{L}_{\alpha}\ket{\phi_{\beta}}=\frac{S_{\alpha\beta}}{S_{0\beta}}\ket{\phi_{\beta}}$
(A.6)

where $\ket{\phi_{\beta}}$ is in the summand
$V_{\beta}\otimes\overline{V_{\beta}}$ of $\mathcal{H}_{0}$. These equations
are compatible because of the celebrated formula of Verlinde [61],

$N_{\alpha\beta}^{\gamma}=\sum_{\delta}\frac{S_{\alpha\delta}S_{\beta\delta}\overline{S_{\gamma\delta}}}{S_{0\delta}}.$
(A.7)

This also means that the Hilbert space $\mathcal{H}_{\gamma}$ on a circle with
an insertion of the Verlinde operator labeled by $\alpha$ is given by

$\mathcal{H}_{\gamma}=\bigoplus_{\alpha,\beta}N_{\alpha\gamma}^{\beta}V_{\alpha}\otimes\overline{V_{\beta}}$
(A.8)

so that the character is

$\mathop{\mathrm{Tr}}\nolimits_{\mathcal{H}_{\gamma}}e^{-2\pi
tH}=\sum_{\alpha,\beta}N_{\alpha\gamma}^{\beta}\chi_{\alpha}(t)\overline{\chi_{\beta}(t)}.$
(A.9)

#### Ishibashi states and Cardy states:

To describe the boundary states, we first consider Ishibashi states
$|\alpha\rangle\\!\rangle\in V_{\alpha}\otimes\overline{V_{\alpha}}$ [32] with
the property

$\langle\\!\langle\alpha|e^{-2\pi
H/\delta}|\beta\rangle\\!\rangle=\delta_{\alpha\beta}\chi_{\alpha}(\frac{i}{\delta}).$
(A.10)

Then the Cardy states are [33]

$\ket{\alpha}=\sum_{\beta}\frac{S_{\alpha\beta}}{\sqrt{S_{0\beta}}}|\beta\rangle\\!\rangle$
(A.11)

which satisfies

$\bra{\alpha}e^{-2\pi
H/\delta}\ket{\beta}=\mathop{\mathrm{Tr}}\nolimits_{\mathcal{H}_{\alpha|\beta}}e^{-2\pi\delta
H}=\sum_{\alpha,\beta}N^{\gamma}_{\alpha\beta}\chi_{\gamma}(i\delta),$ (A.12)

meaning that the open Hilbert space has the decomposition

$\mathcal{H}_{\alpha|\beta}=\bigoplus_{\gamma}N^{\gamma}_{\alpha\beta}V_{\gamma}.$
(A.13)

We also find the action of the Verlinde operators on the Cardy states as
follows:

$\mathcal{L}_{\alpha}\ket{\beta}=N^{\gamma}_{\alpha\beta}\ket{\gamma}.$ (A.14)

#### Order-2 simple currents:

In general, a primary $p$ which has the fusion rule $(V_{p})^{n}\sim V_{0}$ is
called a simple current. The corresponding Verlinde line operator generates a
$\mathbb{Z}_{n}$ symmetry. Here we only consider the case when $n=2$. Denoting
the simple current by $\upsilon$, we have the fusion rule
$V_{\upsilon}V_{\upsilon}\sim V_{0}$. The spin $h_{\upsilon}$ is $0$ or $1/4$
mod $1/2$, and the $\mathbb{Z}_{2}$ is non-anomalous in the former and
anomalous in the latter case.

Under the fusion with $\upsilon$, primaries can be grouped into two types.
Namely, there are those invariant under the multiplication:

$V_{\upsilon}V_{i}\sim V_{i}$ (A.15)

and pairs exchanged by the multiplication:

$V_{\upsilon}V_{a+}\sim V_{a-},\qquad V_{\upsilon}V_{a-}\sim V_{a+}.$ (A.16)

#### Cardy states in the twisted sector:

We now examine the effect of the $\mathbb{Z}_{2}$ symmetry on the boundary
conditions. The equations above translate to the statement that the boundary
condition $i$ is invariant under $\mathbb{Z}_{2}$ while the boundary
conditions $a\pm$ break the $\mathbb{Z}_{2}$ symmetry and are exchanged by it.
This means that one can put $\mathbb{Z}_{2}$-invariant boundary conditions
$i$, $j$, …on a circle twisted by the $\mathbb{Z}_{2}$ symmetry. This
construction should then determine twisted Cardy states

$\ket{i}_{\mathrm{tw}}\in\mathcal{H}_{\upsilon}=\bigoplus_{\alpha}V_{\alpha}\otimes\overline{V_{\upsilon\alpha}}.$
(A.17)

They can be expanded in terms of the twisted Ishibashi states

$|i\rangle\\!\rangle_{\mathrm{tw}}\in
V_{i}\otimes\overline{V_{i}}\subset\mathcal{H}_{\upsilon}$ (A.18)

having the same normalization as in (A.10). We note that the untwisted and
twisted Ishibashi states $|i\rangle\\!\rangle$ and
$|i\rangle\\!\rangle_{\mathrm{tw}}$ labeled by the same symbol $i$ behave in
the same way under the action of the chiral algebra $\mathcal{A}$ and its
antiholomorphic counterpart $\overline{\mathcal{A}}$, but that they live in
different sectors and should better be distinguished.

The expansion of the twisted Cardy states in terms of the twisted Ishibashi
states was studied in the case of $\mathbb{Z}_{2}$ simple currents e.g. in
[40]. A general formula applicable for any simple current orbifold was
conjectured in [41] which was later proved in [42]. In our case the formula
boils down to

$\ket{i}_{\mathrm{tw}}=\sum_{j}\frac{\hat{S}_{ij}}{\sqrt{S_{0j}}}|j\rangle\\!\rangle_{\mathrm{tw}}$
(A.19)

up to a subtle phase which does not concern us in this paper.151515This phase
is an eighth root of unity when the diagonal RCFT is placed on an unoriented
surface [41]. This should be related to the mod-8 classification of the time-
reversal anomaly of the boundary of a pin- surface.  Here, $\hat{S}_{ij}$
describes the action of $S\in SL(2,\mathbb{Z})$ on the space of torus
conformal blocks with an insertion of the simple current $\upsilon$.

The general form of $\hat{S}_{ij}$ was determined e.g. in [63, 64]. They are
also known to satisfy a generalized version of the Verlinde formula [64, 65,
66, 67]

$\hat{N}_{i\alpha}^{j}=\sum_{k}\frac{\hat{S}_{ik}S_{\alpha
k}\overline{\hat{S}_{jk}}}{S_{0k}}$ (A.20)

where the summation is over $\mathbb{Z}_{2}$ invariant primaries and
$\hat{N}_{i\alpha}^{j}$ is the trace of the $\mathbb{Z}_{2}$ action $g$ on the
fusion space among $i$, $j$ and $\alpha$, see also [43]. This relation
guarantees that the overlap of twisted Cardy states satisfies

$\prescript{}{\mathrm{tw}}{\bra{i}}e^{-2\pi
H/\delta}\ket{j}_{\mathrm{tw}}=\mathop{\mathrm{Tr}}\nolimits_{\mathcal{H}_{i|j}}ge^{-2\pi\delta
H}=\sum_{\alpha}\hat{N}^{j}_{i\alpha}\chi_{\alpha}(i\delta)$ (A.21)

and has a consistent Hamiltonian interpretation in the open channel.

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