SciPost Phys. 11, 082 (2021)

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

Abstract
In the last few years it was realized that every fermionic theory in 1+1 dimensions isa generalized Jordan-Wigner transform of a bosonic theory with a non-anomalous Z2
symmetry. In this note we determine how the boundary states are mapped under thiscorrespondence. We also interpret this mapping as the fusion of the original boundarywith the fermionization interface.
Copyright Y. Fukusumi et al.

Received 19-03-2021

This work is licensed under the Creative Commons

Accepted 28-09-2021

Attribution 4.0 International License.

updatesPublished 25-10-2021
doi:10.21468/SciPostPhys.11.4.082

Published by the SciPost Foundation.

Check forContents
1 Introduction and Summary2

2 Mapping of the boundary states
2.1 Gauging Z2 : A ↔ D
2.2 Fermionizing Z2 : A → F, F0
2.3 Comments

6
6
10
13

3 Examples
3.1 Fermionic SU(2)k WZW model
3.2 The Ising model and a massless Majorana fermion
3.3 S pin(N )1 WZW Model and N Majorana Fermions
3.4 Maldacena-Ludwig boundary state of eight Majorana fermions

15
15
17
19
22

4 Interfaces and boundary states
4.1 A ↔ D
4.2 A ↔ F, F0
4.3 The special case A ' D and the anomalous Z2 symmetry of F ' F0

23
23
24
26

A RCFTs and their Z2 symmetries

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SciPost Phys. 11, 082 (2021)

References

130

Introduction and Summary

It is a classic result that the Jordan-Wigner transformation [1] allows us to map any 1-dimenQ (i)(i)σZsional spin chain written in terms of σX ,Y,Z preserving a Z2 symmetry generated byin terms of fermion operators ψ(i) , ψ̄(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 thebosonic 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 beformulated 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 Z2 symmetry by thistransformation [3–8], generalizing the relation between the Ising conformal field theory (CFT)and the Majorana fermion CFT. It was also learned recently1 that fermionic versions of unitary
minimal models can be constructed in this manner [11–13].23 The aim of this paper is to
study how this map between fermionic models and bosonic models with Z2 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 A with a non-anomalous Z2 global symmetry. Anotherbosonic theory D can be introduced by

D := A/Z2 ,(1.1)

where dividing by Z2 stands for orbifolding, or equivalently gauging of the Z2 symmetry. This
theory D is known to possess a dual Z2 symmetry [21] and gauging it reproduces the original
theory: A = D/Z2 . The fermionic theory F is obtained with the help of another theory Kitaev,
the low-energy limit of the topologically nontrivial phase of Kitaev’s Majorana chain [22].4
The formula is
F := (A × Kitaev)/Z2 ,(1.2)
1Note added in proof: The authors learned very recently that fermionic minimal models had already been foundin 1988 [9, 10]. They thank Prof. V. Petkova for information.
2
There should also be a generalization from Z2 to ZN , where ZN symmetric critical points are related to ZN
parafermionic models. Historically, such generalization was first suggested in [14] in 1985, where it was conjectured that the m-multicritical points of ZN Fateev-Zamolodchikov model should be described by the second
parafermionic models with parameter m for N ≥ 3 in [14]; the corresponding bosonic coset models were later
given in [15, 16]. 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 [12].
It would be interesting to revisit and develop these points further. See e.g. [17–19] for recent works.
3
See also [20] for a recent study of fermionic RCFTs with small number of irreducible representations.
4This 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.

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SciPost Phys. 11, 082 (2021)

where the orbifold is with respect to the diagonal Z2 symmetry. We can also define a closely
related theory F0 starting from D

F0 := (D × Kitaev)/Z2 ,(1.3)

F0 = F × Kitaev .

(1.4)

which satisfies5
We place these theories on a circle. The bosonic theories can be put on a circle without or with
Z2 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 A, D, F, F0 are known to decompose in the following
manner:

A
even
odd

F
(−1) F = +1(−1) F = −1D

untwisted twisted
S
U
T
V
NS
S
V

R
U
T

even
odd

F0
(−1) F = +1(−1) F = −1

untwisted twisted
S
T
U
V
NS
S
V

R
T
U

(1.5)

We denote the Hilbert spaces in the respective sectors by HS,T,U,V , so that the untwisted andtwisted Hilbert spaces of the theory A have the decomposition

H A = HS ⊕ H T ,

(1.6)

A
= HU ⊕ HV ,
Htw

(1.7)

for example. Since F and F0 differ only by a decoupled Z2F SPT, this makes it tricky to distinguishthem in the lattice realizations on closed chains.
We now consider boundary conditions of the bosonic theory A, which can be broadly classified into two types by their behavior under the Z2 symmetry. Namely, there are Z2 invariant
ones, which we denote by i, j, . . . , and the ones which break the Z2 symmetry, which therefore
form pairs we can denote by a±, b±, . . . . We assume the theory is defined on a cylinder ofsize L1 × L2 , where L1 is the length of the open spatial direction and L2 is the circumference ofthe closed temporal direction. We denote the Hilbert space on a finite interval with boundary
conditions α on the left boundary and β on the right boundary as Hα|β . The boundary states
are obtained by 90 degree rotation of the spacetime, hence the temporal direction becomes
open, and the boundary states are defined on the initial and final slices. We denote the corresponding boundary states as |i〉A , |a±〉A , etc. Note that a Z2 invariant boundary condition α
can be placed on a circle which is twisted by the Z2 operation, which defines twisted boundary
states |i〉A
tw taking values in the twisted Hilbert space. The (twisted) boundary conditions andthe (twisted) boundary states are related by the Cardy condition,〈α| e−L1 Hclosed |β〉 = TrHα|β e−L2 H open ,tw 〈α| e−L1 Hclosed|β〉tw = TrHα|β g e−L2 Hopen ,(1.8)

In terms of the partition function coupled to Z2 background field B and spin structure ρ, wePPhave ZF (ρ) = 2−g B ZA (B)(−1)Arf[ρ+B] . Similarly, we have ZF0 (ρ) = 2−g B ZA (B)(−1)qρ [B] whereqρ [B] ≡ Arf[ρ + B] − Arf[ρ] is the quadratic form associated to the spin structure. These two are related asfollows: ZF0 [ρ] = ZF [ρ](−1)Arf[ρ] . In the works [6, 12] by one of the authors (YT) and other collaborators, F isdefined to be the fermionization of A, whereas in most of the other works of this topic e.g. [3,5,8,23], F0 is defined
to be the fermionization of A instead. We note that when A is the completely trivial theory, the theory F is trivial
while F0 is equal to Kitaev.
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SciPost Phys. 11, 082 (2021)

where g is the Z2 action on Hα|β . The Cardy condition (1.8) constrains the normalization of
the boundary states.
Let us further consider the boundary conditions and boundary states of a fermionic theory
F. The discussion is similar to the bosonic theory as the previous paragraph. The global
symmetry is the fermion parity symmetry Z2F . The boundary conditions can be classified to be
Z2F even and odd ones. The fermion along the closed time cycle can have either anti-periodic(NS) spin structure or periodic (R) spin structure. For the R spin structure, there is a Z2F line
inserted along the spatial direction, which is analogous to the twisted boundary condition in
the bosonic case. In summary, the boundary conditions are denoted by α. After 90 degree
rotation, the boundary conditions are mapped to boundary states, which we denote as |α〉NSor |α〉R . The boundary states and boundary conditions satisfy the spin Cardy condition:
NS 〈α| e−L1 Hclosed|β〉NS = TrHα|β e−L2 Hopen ,R 〈α| e−L1 Hclosed|β〉R = TrHα|β (−1) F e−L2 Hopen . (1.9)

In the following sections, we do not specify whether Hamiltonian is defined on the "open" or"closed" string, because it should be evident from the discussion.
Our main result is the expressions of the boundary states of the theories D, F and F0 , which
will be derived in detail later. Here we will simply summarize them. In the Z2 -orbifold theory
D, the Z2 -invariant boundary condition i of the original theory A splits into two types i±,b 2 symmetry. Conversely, the pair of Z2 -breaking
which are exchanged under the emergent Z
boundary conditions a± is combined into a single Z2 -invariant condition a. Their boundary
states are given as follows:
1
1|i+〉D = p (|i〉A + |i〉A
|i−〉D = p (|i〉A − |i〉Atw ) ,tw ) ,
2
2
1
1
A
A|a〉D = p (|a+〉A + |a−〉A ) , |a〉D
tw = p (|a+〉 − |a−〉 ) .
2
2(1.10)

b 2 exchanges |i+〉D and |i−〉D , while |a〉D and |a〉D are both Zb 2 even. The boundary
Here ZtwD
D
D
states |i±〉 , |a〉 and |a〉tw satisfy Cardy’s condition (1.8).
In the fermionic theories, boundary conditions can be classified into two types, those withand without an unpaired Majorana fermion zero mode. In particular, Kitaev itself has suchan unpaired Majorana fermion zero mode [24]. Let us start from a Z2 -invariant boundary
condition i of the theory A. When stacked with Kitaev, this boundary has a Majorana fermion
ψ, but the Z2 quotient removes it since this fermion is Z2 odd. Conversely, when we startfrom a Z2 -breaking boundary condition a±, this additional Z2 -odd Majorana fermion ψ is
kept since the Z2 gauge symmetry is broken here, resulting in a boundary with a Majorana
fermion. We denote the resulting boundary states as |i〉F and |aψ〉F , etc. The corresponding
boundary states are given as follows:|i〉FNS = |i〉A ,|i〉FR = |i〉A
tw ,|aψ〉FNS = |a+〉A + |a−〉A , |aψ〉FR = |a+〉A − |a−〉A .

(1.11)

Here |aψ〉FR 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 dueto the Majorana fermion zero mode coming from the periodicity. Our |aψ〉FR is defined withan insertion of the Majorana fermion operator to absorb this zero mode. We do not repeat

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SciPost Phys. 11, 082 (2021)

this comment unless absolutely necessary. The boundary states |i〉FNS/R and |aψ〉FNS/R satisfy
the spin Cardy condition (1.9).
00
In the theory F0 , the assignment is reversed, and we find |iψ〉F and |a〉F . These boundary
states are given as follows:pp02 |i〉A ,|iψ〉FR = 2 |i〉A
tw ,
0
0
1
1
|a〉FNS = p (|a+〉A + |a−〉A ) ,|a〉FR = p (|a+〉A − |a−〉A ) .
220|iψ〉FNS =(1.12)

Again |iψ〉FR has (−1) F = −1 while the other three have (−1) F = +1. Note that the theory F
and F0 have rather different sets of boundary conditions if A 6= D, even though the NS sectorsof F and F0 are the same and the R sectors of F and F0 only differ in their assignments ofthe fermion parity. The boundary states |iψ〉FNS/R and |a〉FNS/R satisfy the spin Cardy condition(1.9).
We find it also useful to present our results in a different way, showing explicitly to which
sector HS,T,U,V the boundary states belong. We first organize the boundary states of theory A
into four blocks6 (1.5):

A
even
odd

untwisted
twistedA|i〉S = |i〉|i〉U = |i〉Atw
1|a〉S = p2 |a+〉A + |a−〉A|a〉 T = p12 |a+〉A − |a−〉A(1.13)

Then the boundary states of theory D can be presented as:

D
even
odd

untwisted
twistedD|a〉S = |a〉|a〉 T = |a〉Dtw
1|i〉S = p2 |i+〉D + |i−〉D|i〉U = p12 |i+〉D − |i−〉D(1.14)For the theory F and F0 they are given byF(−1) = +1
F(−1) F = −1
andNS|i〉S = |i〉FNS
p2|a〉S = |aψ〉FNSF0(−1) = +1
F(−1) F = −1Rp2 |a〉 T

NS
|a〉S
p
2|i〉S

= |i〉FR|i〉U(1.15)= |aψ〉FRR

F0= |a〉NS0= |iψ〉FNS

0

|a〉 T

= |a〉FRp2|i〉U= |iψ〉FR(1.16)
0

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 D, F and F0 we listed above. We will6

Note that |a〉S and |a〉 T are not boundary states of A. They are Z2 even and odd linear combinations ofboundary states in the untwisted sector. Likewise, |i〉S and |i〉U are not boundary states of D, but instead are linear
combinations of them.

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SciPost Phys. 11, 082 (2021)

not use the property that the theories are conformal. Instead, we will only use the fact thatthe 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 ourfindings in terms of interfaces between our theories A, D, F and F0 . We also see that when
the original theory A has a duality interface implementing the Z2 orbifold, so that A ' D, thefermionized theory F ' F0 has an anomalous Z2 symmetry. We have an appendix A collecting
various facts on diagonal invariants of rational conformal field theories (RCFTs) which can beused to construct their fermionic versions and fermionic boundary conditions.
Before proceeding, the authors note that a paper with a large overlap [25] appeared when
this work was being completed, in which the fermionic boundary states of fermionic minimalmodels and the effect of anomalous chiral Z2 symmetry were studied in detail; he also carefully
discusses the boundary states of exceptional fermionic minimal models introduced in [13]. The
authors also note that another paper with a significant overlap [26] appeared on the arXivon the same day as this one. The authors thank the authors of [26] 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 Z2 : A ↔ D

Let us start with a bosonic theory A with Z2 global symmetry. We demand that Z2 is nonanomalous, so that there is no obstruction to gauge it. We denote the resulting Z2 gaugedb 2 . We study how the boundary
theory by D, whose emergent Z2 symmetry we denote by Z
states of theory A transform under gauging. The content of this section is essentially known
in the vast existing literature on the boundary conditions of Z2 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 space(r,s)
time is a torus without boundary. Denote the partition function of theory A as ZA wherer, s ∈ {0, 1} count the number of Z2 defect lines in the spatial and temporal directions. We use(m,n)
the symbols ZD
similarly for the gauged theory D, where m, n now count the number ofb 2 global symmetry. We then have the relation7
defect lines for the emergent Z
(m,n)

ZD

=

1X
(r,s)
(−1) r n−ms ZA .
2

(2.1)

(r,s)

Next, we demand that the spacetime has nontrivial boundaries in the temporal direction,i.e. time direction is open. We place two boundary states |α〉 and |β〉 in the initial time and the(r,s)
final time respectively, and denote the partition function as ZA [〈β|α〉]. As discussed in the
introduction, there are broadly speaking two types of boundaries: we denote the Z2 -invariant
R
P
b
B∪B
b] = 1g
More generally, the partition functions of A and D are related as ZD [B, where B andB ZA [B](−1)
2b are the Z2 and dual Z2 background fields in A and D respectively. The pair (r, s) in (2.1) labels the nontrivial
B
RR
holomonies of the background field x B = s, y B = r mod 2 where the integrals are along the non-contractible
b in a similar manner.
cycles in the direction x and y, respectively. The pair (m, n) is obtained from B
7

6

SciPost Phys. 11, 082 (2021)= e2πis=Figure 1: Z2 defect, Z2 charge and the spin of the twisted Hilbert space. The Z2
charge of a state in the Z2 is given by the leftmost figure, which can be deformed tothe middle figure. Therefore the charge is related to the spin of the state. When weput a Z2 -invariant boundary condition on the top and bottom slices, we can always
unwind the defect, and the spin is always integer valued.

ones by |i〉A , | j〉A , . . . and Z2 -breaking pairs by |a±〉A , |b±〉A , . . . . We also denote the twisted
A
sector boundary states by |i〉A
tw , | j〉tw . . . . We also employ analogous notations for theory D.
Z2 charge of |i〉A : Since the boundary state |i〉A and |i〉A
tw are invariant under Z2 , we need toAfind their Z2 charge. The boundary state |i〉 in the untwisted Hilbert space is Z2 even rather
than Z2 odd. To see this, we use Cardy’s analysis to transform the partition function to theopen string channel. Consider the theory on a cylinder of size L1 × L2 where L1 is along theopen direction and L2 is along the closed direction. Suppose |i〉A has Z2 charge s = ±1, thusg |i〉A = s |i〉A . Then(1,0) AZA
We also have

[ 〈 j|i〉A ] = A 〈 j| e−L1 H g |i〉A = sA 〈 j| e−L1 H |i〉A = s TrH j|i e−L2 H .
(1,0) AZA

[ 〈 j|i〉A ] = TrH g e−L2 H ,
j|i

(2.2)

(2.3)

g

where H j|i is the defect Hilbert space in the open string channel with two boundary conditions
labeled by j and i with an explicit insertion of Z2 line in the middle. Now, TrH e−L2 H for any
H should be a sum of exponentials with non-negative coefficients, therefore we need to haves = +1.8 Summarizing, we see g |i〉A is + |i〉A instead of − |i〉A , where g is the Z2 generator.
A
Z2 charge of |i〉A
tw : To find the Z2 charge of |i〉tw , we make use of the following trick [27,28].
By definition, the Z2 charge of a state in the twisted Hilbert space can be found by considering
the configuration of the Z2 line operators in the leftmost figure of Fig. 1. Assuming that theZ2 symmetry is non-anomalous, this configuration can be continuously deformed to a line
operator winding the cylinder once, therefore the Z2 -charge equals e2πis where s is the spin of
the state |φ〉 ∈ Htw :g |φ〉 = e2πis |φ〉 ,(2.4)

where g is the Z2 generator. When the state in question is the boundary state for a Z2 -invariant
boundary condition, we can always unwind g, say at the top time slice, to make it straight,
hence e2πis = 1. In summary, both the states |i〉A , |i〉A
tw are Z2 even.
Rough properties of the boundary states: We first look for the boundary states in theory Db 2 symmetry. Let us symbolically denote it as |α〉 andthat is Z2 -invariant under the emergent Z
8

The authors thank Shu-Heng Shao for suggesting this argument.

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SciPost Phys. 11, 082 (2021)

place it in the initial time slice. We also put a reference state |R〉 in the final time slice. (Thechoice of reference state does not affect our discussion). We have the following relations
(0,0)

ZD

(0,1)

ZD

(1,0)

[〈R|α〉] ,(1,1)[〈R|α〉] = 0 .

[〈R|α〉] = ZD[〈R|α〉] = ZD

(2.5)

b 2 symmetric, where we can move the Zb 2 line along
The first equality follows from |α〉 being Z
the spatial direction down to the initial time slice and let it be absorbed by the initial state |α〉.
The second and third equations follow from the fact that |α〉 do not live in the twisted sector.
Combining (2.1) and (2.5), we have(0,0)[〈R|α〉] = ZA

(0,1)

ZA
ZA
The
(0,0)

ZA

relations

(2.6)
(1,0)

[〈R|α〉] = ZA

imply

(1,0)

[〈R|α〉] ,[〈R|α〉] = ZA

(1,1)[〈R|α〉] = 0 .

|α〉

not

that

can

be

(r,s)

(2.6)
|i〉A
tw

becauseotherwise[〈R|α〉] = 0, thus ZA [〈R|α〉] = 0 for all r, s, which is absurd. The

first equality in (2.6) also implies that |α〉 can not be proportional to |a+〉A − |a−〉A neither,because the relations (2.6) imply that |α〉 should have Z2 eigenvalue +1. Therefore |α〉 canonly be proportional to |i〉A or |a+〉A + |a−〉A .
We further look for the symmetric boundary state |β〉 in the twisted sector of theory D.
Applying an analysis that was carried out in the last paragraph, we find(0,1)[〈R|β〉] = ZD

(0,0)[〈R|β〉] = ZD

(0,0)[〈R|β〉] = −ZA(0,1)[〈R|β〉] = ZA

ZD
ZD

(1,1)

[〈R|β〉] ,(1,0)[〈R|β〉] = 0 .

(2.7)

Combining with (2.1), we have
ZA
ZA

(1,0)

(1,1)

[〈R|β〉] ,(2.8)[〈R|β〉] = 0 .

This implies that |β〉 can only be proportional to |a+〉A − |a−〉A . This in fact would suggestthat |α〉 should be proportional to |a+〉A + |a−〉A , but not |i〉A . Hence we will denote|a〉D ∝ |a+〉A + |a−〉A ,

(2.9)

A
A
|a〉D
tw ∝ |a+〉 − |a−〉 .

We further look for the Z2 -odd boundary state |γ〉 in the untwisted sector of theory D. Thismeans that(0,0)[〈R|γ〉] = −ZD(0,1)[〈R|γ〉] = ZD

(0,1)[〈R|γ〉] = ZA

ZD
ZD

(1,0)

[〈R|γ〉] ,(1,1)[〈R|γ〉] = 0 .

(1,1)[〈R|γ〉] ,(1,0)[〈R|γ〉] = 0 .

(2.10)

Combining with (2.1), we have
ZA

(0,0)

ZA

[〈R|γ〉] = ZA

(2.11)

Hence |γ〉 should be the twisted sector state of theory A, |i〉A
tw , which indeed is Z2 even.
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SciPost Phys. 11, 082 (2021)

Lastly we look for the Z2 -odd boundary state |η〉 in the twisted sector of D. This means
that
(0,1)

[〈R|η〉] = −ZD

(0,0)

(1,0)

[〈R|η〉] = ZD

(0,1)[〈R|η〉] = −ZA(0,0)[〈R|η〉] = ZA

ZD

ZD

(1,1)

[〈R|η〉] ,[〈R|η〉] = 0 .

(2.12)

Combining with (2.1), we have
ZA

ZA

(1,1)

(1,0)

[〈R|η〉] ,[〈R|η〉] = 0 .

(2.13)

Because there is no Z2 odd state in the twisted sector of theory A, there is no solution to (2.13).
Hence all the boundary states in theory D that are related to |i〉A and |i〉A
tw do not live in theDDbtwisted sector of Z2 . This suggests that there are |i+〉 and |i−〉 which are mapped to eachb 2 , which satisfy
other under Z|i+〉D + |i−〉D ∝ |i〉A ,|i+〉D − |i−〉D ∝ |i〉A
tw .

(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 L1 × L2 ,with two boundary conditions α and β. We then have Cardy’s conditions〈α| e−L1 H |β〉 = TrHα|β e−L2 H ,tw 〈α|e−L1 H|β〉tw = TrHα|β g e−L2 H ,(2.15)

where g is the Z2 generator. Below, we will drop e−L1 H and e−L2 H for brevity.
Let us first consider (2.14). Suppose |i+〉D + |i−〉D = P |i〉A and |i+〉D − |i−〉D = Q |i〉A
tw .
Then the relations (2.15) imply thatA

〈i| j〉A = TrH+ + TrH− ,
i| j

A

A

i| j

tw 〈i| j〉tw = TrHi| j − TrHi| j ,+(2.16)−

where H±is the part of the open Hilbert space with boundary conditions i, j placed on two
i| jends, with the Z2 charge ±1. Hence
1
1(|P|2 + |Q|2 ) TrH+ + (|P|2 − |Q|2 ) TrH− ,
i| j
i|
j
4
4
1
1
D
〈i ± | j∓〉D = (|P|2 − |Q|2 ) TrH+ + (|P|2 + |Q|2 ) TrH− .
i| j
i|
j
4
4

D

〈i ± | j±〉D =(2.17)

For |i±〉D to be legal boundary states, we should have 14 (|P|2 ±|Q|2 ) to be non-negative integers.
p
It is then reasonable to set P = Q = 2, which is the minimal solution of P and Q. We thusget
1|i±〉D = p (|i〉A ± |i〉Atw ) .
2(2.18)

Since gauging twice goes back to the original theory itself, we also finds the normalization in
(2.9)
1
1
A
A
|a〉D = p (|a+〉A + |a−〉A ),|a〉D(2.19)
tw = p (|a+〉 − |a−〉 ) .
2
2
These results were presented in a slightly different manner in Sec. 1.
9SciPost Phys. 11, 082 (2021)

2.2

Fermionizing Z2 : A → F, F0

Setup: Let us further consider fermionizing the Z2 global symmetry of A, and denote theresulting theory as F. More precisely, we first stack the theory A with a Kitaev chain in the
nontrivial phase, and gauge the diagonal Z2 global symmetry: F = (A × Kitaev)/Z2 . The
partition functions on the torus are related as follows:9
(m,n)

ZF

=

1X
(r,s)
(−1)(m+r)(n+s) ZA ,2(2.20)(r,s)

where the pair (m, n) now specifies the periodicity of the spin structure on the torus, in theconvention 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 Z2 invariant boundary condition i of the theory A. When stacked with Kitaev, this boundary has
a Majorana fermion ψ, which is removed by the Z2 quotient since it is Z2 odd. In contrast,
when we start from a Z2 -breaking boundary condition a±, this Z2 -odd Majorana fermion ψ isretained since the Z2 gauge symmetry is broken here, resulting in a boundary with a Majorana
fermion. Furthermore, acting by ψ essentially converts a+ to a− and vice versa. Therefore
we find two types of boundary conditions: |i〉F without Majorana fermion which comes froma Z2 -invariant boundary condition i of theory A, and |aψ〉F with a Majorana fermion whichcomes from a pair of Z2 -breaking boundary conditions a± of theory A.
Rough properties of the boundary conditions: The way we determine the boundary states
is analogous to what we did in the case A ↔ D studied above. When there are nontrivial
boundaries along the temporal direction, we again put a reference state |R〉 in the final timeslice, and put a boundary state |α〉 in the initial slice which we will study. Now we need toconsider four choices: NS/R sectors and (−1) F = ±1, which we study in turn.
Let us first look into the state |α〉 in the NS sector with fermion parity (−1) F = 1. We have
(0,0)

ZF

(0,1)

ZF

(1,0)

[〈R|α〉] ,(1,1)[〈R|α〉] = 0 .

[〈R|α〉] = ZF[〈R|α〉] = ZF

(2.21)

The first equality follows from (−1) F = 1 and the last two equalities are because |α〉 is in the
NS sector. Combining with (2.20), we have(0,0)[〈R|α〉] = ZA

(0,1)[〈R|α〉] = ZA

ZA
ZA

(1,0)

[〈R|α〉] ,(1,1)[〈R|α〉] = 0 ,(2.22)

which implies that|α〉 ∝ |i〉Aor |a+〉A + |a−〉A .

(2.23)Let us further look into the state |β〉 in the R sector with (−1) F = 1. We have
(0,1)[〈R|β〉] = ZF

(0,0)[〈R|β〉] = ZF

ZF
ZF
9

(1,1)

[〈R|β〉] ,(1,0)[〈R|β〉] = 0 .

See the footnote 5 for the expression on a general surface of genus g.

10(2.24)

SciPost Phys. 11, 082 (2021)

Combining with (2.20), we have
(0,1)

ZA

(0,0)

ZA

(1,1)

[〈R|β〉] ,(1,0)[〈R|β〉] = 0 .

[〈R|β〉] = ZA[〈R|β〉] = ZA

(2.25)F
The only solution |β〉 to (2.25) is |i〉A
tw . We will denote |β〉 as |i〉R , i.e.

|i〉FR ∝ |i〉A
tw .

(2.26)Let us further look into the boundary state |γ〉 in the R sector with (−1) F = −1. This means(0,1)[〈R|γ〉] = −ZF

(0,0)

(1,0)

ZF
ZF

(1,1)

[〈R|γ〉] = ZF[〈R|γ〉] ,[〈R|γ〉] = 0 .

(2.27)

Combining with (2.20), we have(0,0)[〈R|γ〉] = −ZA(0,1)[〈R|γ〉] = ZA

ZA
ZA

(1,0)

(1,1)

[〈R|γ〉] ,[〈R|γ〉] = 0 .

(2.28)The only solution |γ〉 to (2.28) is |a+〉A − |a−〉A . We will denote |γ〉 as |aψ〉FR , thus|aψ〉FR ∝ |a+〉A − |a−〉A .

(2.29)

Finally let us look into the boundary state |η〉 in the NS sector with (−1) F = −1. This
means
(0,0)

[〈R|η〉] = −ZF(0,1)[〈R|η〉] = ZF

ZF
ZF

(1,0)

(1,1)

[〈R|η〉] ,[〈R|η〉] = 0 .

(2.30)

Combining with (2.20), we have
(0,1)[〈R|η〉] = −ZA(0,0)[〈R|η〉] = ZA

ZA
ZA

(1,1)

(1,0)

[〈R|η〉] ,[〈R|η〉] = 0 .

(2.31)

We note, however, that there is no boundary state in the twisted sector of theory A which isZ2 odd. Hence there is no boundary state in NS sector with (−1) F = −1 in theory F.
To encode the (−1) F quantum number to the boundary state, we introduce |±〉, representing (−1) F = ±1 respectively. In summary, we find|i〉FNS ∝ |i〉A ⊗ |+〉 ,|aψ〉FNS ∝ (|a+〉A + |a−〉A ) ⊗ |+〉 ,|i〉FR ∝ |i〉A
tw ⊗ |+〉 ,(2.32)|aψ〉FR ∝ (|a+〉A − |a−〉A ) ⊗ |−〉 .
where we remind the reader that our |aψ〉FR includes an insertion of a Majorana fermion toabsorb the zero mode, which is required to make the state nonzero.

11

SciPost Phys. 11, 082 (2021)

Fixing the normalization constants: We need to further determine the normalization constants in (2.32). Suppose the normalization constants are P1 , P2 , P3 , P4 for the four relationsin (2.32) respectively. Exchanging the time directions, we findF2−+
NS 〈i| j〉NS = |P1 | (TrHi| j + TrHi| j ),FF
2
NS 〈aψ|bψ〉NS = |P2 | (TrHa+|b+ + TrHa−|b− + TrHa+|b− + TrHa−|b+ )= 2|P2 |2 (TrHa+|b+ + TrHa+|b− ),
F

F
2
−
+
R 〈i| j〉R = |P3 | (TrHi| j − TrHi| j ),F

(2.33)(2.34)(2.35)(2.36)

F2
R 〈aψ|bψ〉R = |P4 | (TrHa+|b+ + TrHa−|b− − TrHa+|b− − TrHa−|b+ )= 2|P4 |2 (TrHa+|b+ − TrHa+|b− ),(2.38)F∗
NS 〈i|aψ〉NS = P1 P2 (TrHi|a+ + TrHi|a− ) ,(2.39)

F∗
R 〈i|aψ〉R = P3 P4 (TrHi|a+ − TrHi|a− ) ,(2.40)

F

F

F(2.37)

where we used the fact that the Z2 symmetry of the A-theory implies that TrHa+|b+ = TrHa−|b−and TrHa+|b− = TrHa−|b+ to convert (2.34), (2.37) to (2.35), (2.38), respectively.
The boundary states of F should satisfy the spin Cardy conditions (1.9). This requires thatthe coefficients on the right hand side should be non-negative integers for NS states, while
integers for R states.10 There are two minimal11 solutions of Pi ’s,
P1 = P2 = P3 = P4 = 1(2.41)

andP1 = P3 =p
2,

1
P2 = P4 = p ,2(2.42)

which yield minimal integer coefficients on the right hand sides of (2.33)-(2.39), hence satisfy
the spin Cardy conditions. The correct choice for the theory F is (2.41). To see this, it suffices
to consider the case when A is the completely trivial theory. There is a single Z2 -symmetricboundary condition i, for which Hi|i = 1 is one dimensional and Z2 -even. In this theory
F = (A × Kitaev)/Z2 is also a trivial theory, which has a trivial boundary, for whichFF
NS 〈i|i〉NS = 1. This fixes P1 = 1.
For more general theories, the equations (2.33) and (2.36) simply state the fact that forZ2 -invariant boundary condition i, j of theory A, the corresponding boundary conditions of
the theory F is obtained by attaching a Majorana fermion ψ and simply removing it by the Z2
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 Hi| j , and the fermion parity(−1) F equals the original Z2 -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 pairof such modes, providing a factor of 2. We note that the relative sign in (2.38) does notcorrespond to (−1) F , since in our definition we inserted a Majorana fermion operator on theboundaries |aψ〉F and |bψ〉F .
Here we choose the convention that the coefficient P3∗ P4 in (2.40) is positive. One can alternatively let P3∗ P4
be negative. The two conventions are related by flipping the sign of the boundary Majorana zero mode inserted tomake |aψ〉FR nonzero, and neither sign of the Majorana zero mode is privileged.
11
Minimal means to keep the coefficients on the right hand side of (2.33)-(2.39) to be minimal non-negative
integers.
10

12

SciPost Phys. 11, 082 (2021)

In summary, we found12|i〉FNS = |i〉A ⊗ |+〉 ,|aψ〉FNS = (|a+〉A + |a−〉A ) ⊗ |+〉 ,|i〉FR = |i〉A
tw ⊗ |+〉 ,(2.43)|aψ〉FR = (|a+〉A − |a−〉A ) ⊗ |−〉 .
We can further append a nontrivial Kitaev chain to F to obtain F0 . As the relation between
F and A is equal to that of the relation of F0 and D, we can easily find from (2.18) and (2.19)
thatp0|iψ〉FNS = 2 |i〉A ⊗ |+〉 ,01|a〉FNS = p (|a+〉A + |a−〉A ) ⊗ |+〉 ,2(2.44)pF0|iψ〉R = 2 |i〉A
⊗
|−〉
,
tw
1
F0
|a〉R = p (|a+〉A − |a−〉A ) ⊗ |+〉 .
2
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 exchangedp
between 1 and 2, exactly as shown in (2.44) and (2.43). Our final results (2.43) and (2.44)
were already presented in Sec. 1 as (1.11) and (1.12). There, we suppressed |±〉, while justremarking their fermion parity explicitly.

2.3

Comments

Before proceeding, we would like to make two remarks.
On two common normalizations: Recall that during the derivation, we demanded thatthe 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
ψ L = ±ψR on the boundary. In hep-th, it is often stated that if we choose a wrong pair ofboundary conditions α, β 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 puttingpa Majorana fermion by hand. In such a case, it is 2 〈α|β〉 which has a sensible Hamiltonian
interpretation in the open channel. The recent paper [25] 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 thiscase, if ψ L = +ψR is a boundary condition without a boundary Majorana fermion, then theboundary condition ψ L = −ψR comes with a boundary Majorana zero mode. In this caseany pair of boundary conditions is quantizable, but then we need to introduce additional
p ±1
normalization constants of 2 in front of the boundary states. Then the boundary states fora single Majorana fermion with boundary conditions ψ L = +ψR and ψ L = −ψR would have
As one of the authors emphasized in [29], the linear combination of Cardy states appears naturally as zeromode of fermionic model [30, 31]. This type of states has also captured attention in the condensed matter physics
community [32–34].
12

13

SciPost Phys. 11, 082 (2021)

different norms. This
chain [29–31].

pp
2 factor can also be observed as the 2 fold degeneracy of Majorana

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 A
and the transformed fermionic model F, which we gave in (2.33) to (2.39). We also saw that
Z2 -invariant boundary conditions give rise to boundary conditions without localized Majorana
fermions, and that Z2 -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 A to F is indeed the
Jordan-Wigner transformation in disguise.
Let us first consider the case of standard Z2 -invariant boundary conditions on open chains.
(s)Namely, we consider an open chain of L sites with the operators σX ,Y,Z , (s = 1, . . . , L). We
Q (s)take a Hamiltonian H commuting with the Z2 symmetry generated by s σ Z . The boundary
conditions at both ends are Z2 -invariant. Such a Hamiltonian can be rewritten in terms of Majorana fermions ψ(t) , (t = 1, . . . , 2L) using the standard formulas. Then almost by definition,
the Hamiltonian written in the fermionic variables is equal to the Hamiltonian written in terms
Q (s)
(s)
of σX ,Y,Z , and the fermion parity is equal to s σ Z . This reproduces the relation (2.33) and(2.36).
Next, we consider a particular class of Z2 -breaking boundary conditions on an open chain,and see it reproduces (2.34). The existence of localized Majorana zero modes is also clearlyvisible.
We again start from the same Hamiltonian Z2 -invariant H on a chain of L sites, and ex(s)(s)
press it in terms of σX ,Z without σY . At the leftmost site i = 1, we drop the terms from H(1)

involving σ Z , and also perform an analogous operation on the other end i = L. Let us denote
(1)

(L)

the resulting Hamiltonian by H 0 . Such a Hamiltonian H 0 commutes with σX and σX , and(1,L)

therefore H 0 and σX

can be simultaneously diagonalized. In other words, H 0 splits into four
(1,L)

0
sectors H±±
depending on the eigenvalues of σX , and the four sectors describe Z2 -breaking
boundary conditions imposed on both ends. Now we perform the standard Jordan-Wignertransformation for H 0 , which is still Z2 -symmetric. The construction guarantees that H 0 does
not involve ψ(1) and ψ(2L) , meaning that they commute with H 0 . 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
L
L−1
X
X
(s)
(s) (s+1)−HIsing =
σZ +
σX σX
.
(2.45)s=1

s=1

The modified chain has the Hamiltonian0−HIsing
=

L−1
X

(s)

σZ +

s=2

L−1
X

(s)

(s+1)

σX σX

,

(2.46)

s=1

which splits into four sectors described by
L−1
L−2
X
X
(s)
(2)
(s) (s+1)(L−1)
0−HIsing
=
(
σ
)
±
σ
+
(
σX σX ) ± σX
,
±±
Z
X
s=2

s=2

14

(2.47)

SciPost Phys. 11, 082 (2021)

0
acting on sites s = 2, . . . , L − 1. This HIsing±± indeed breaks Z2 at both boundaries and has thestandard forms there.
Now H 0 written in the Majorana fermion variables is simply−H 0 =

2L−2
X

iψ(t) ψ(t+1) ,(2.48)

t=2

and commutes with ψ(1) and ψ(2L) . We indeed found localized Majorana zero modes, andalso reproduced the relation (2.34).

3

ExamplesIn section 2, we discussed the transformation of boundary states under gauging and fermionizing the Z2 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, suchas Ishibashi and Cardy states [35, 36]; for general aspects of BCFT, see e.g. [37–39]. 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.
j( j+1)
Recall that SU(2)k WZW model has k +1 primaries labeled by j = 0, 21 , · · · , 2k with ∆ j = k+2 .
The fusion rule is[ j] ⊗ [ j 0 ] = [| j − j 0 |] ⊕ [| j − j 0 | + 1] ⊕ · · · ⊕ [min( j + j 0 , k − j − j 0 )] .

(3.1)

The modular S matrix isv
t 2π(2 j + 1)(2 j 0 + 1)sin.
S j j0 =
k+2
k+2

(3.2)

There is a Z2 global symmetry, generated by the Verlinde line L k associated with the primary2

operator [ 2k ]. This is non-anomalous when k is even and anomalous when k is odd. In thefollowing we only consider the case k is even.
For a given j, the conformal primary and its descendants | j, N 〉 all have the same Z2 eigenSk

jvalue S20 j = (−1)2 j . The Ishibashi states are schematically given by| j〉〉 ∼X

| j, N 〉 ⊗ | j̄, N̄ 〉 ,

(3.3)

N

which are Z2 eigenstates with eigenvalue (−1)2 j , i.e. g| j〉〉 = (−1)2 j | j〉〉. Therefore Z2 acts on
the Cardy state as
g | j〉 = | 2k − j〉 .
(3.4)
The only Z2 -invariant Cardy state is | 4k 〉.
The twisted Cardy state | 4k 〉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
H = ⊕ j Vj ⊗ Vk/2− j .
(3.5)
15

SciPost Phys. 11, 082 (2021)

Therefore, the only possible Ishibashi state which can appear in this twisted Cardy state is| 4k 〉〉tw :
k
k
(3.6)
| 〉 ∝ | 〉〉tw .
4 tw
4Let us note that we havekk
〈 |e−2πH/δ | 〉 = χ0 (iδ) + χ1 (iδ) + χ2 (iδ) + χ3 (iδ) + · · · + χ k (iδ) ,
2
4
4

(3.7)

while
k
k
k
〈〈 |e−2πH/δ | 〉〉 ∝ χ0 (iδ) − χ1 (iδ) + χ2 (iδ) − χ3 (iδ) + · · · + (−1) 2 χ k (iδ) .
2
4
4

(3.8)

Therefore we should havetw 〈kk −2πH/δ k|e| 〉tw = χ0 (iδ) − χ1 (iδ) + χ2 (iδ) − χ3 (iδ) + · · · + (−1) 2 χ k (iδ) .
2
4
4

(3.9)

from which we can determine the proportionality coefficient:1kk+2 4 k| 〉 tw =| 〉〉tw .
4
2
4(3.10)

So far we discussed the untwisted and twisted Cardy states in the original diagonal SU(2)kmodel. For even k, we can orbifold the Z2 symmetry and then obtain the D-type modular
invariants of the SU(2)k WZW model. Equivalently, for even k, we can consider the WZWsigma model whose target space is the group manifold SO(3) rather than SU(2), and the Dtype modular invariants are the infrared limit.13 Denoting the SU(2)k WZW model as theory
A, we can now determine the boundary states of theories D, F and F0 following our generalprescription.
1. The boundary states of theory D is obtained by gauging Z2 (i.e. Z2 orbifolding). Using
1k| j〉D = p (| j〉A + | − j〉 A ) ,
2
2
1
k
A
A
| j〉D
tw = p (| j〉 − | − j〉 ) ,
2
2

1
k−2
j = 0, , ... ,
2
4

1 k A
k A
k D
| +〉 = p (| 〉 + | 〉 ) ,
4
4 tw
2 4

(3.11)

k D
1 k A
k A
| −〉 = p (| 〉 − | 〉 ) .
4
4 tw
2 4
These boundary states have been discussed in the context of orbifolding [46].
Depending on whether k = 4n and k = 4n + 2, these models are called Deven and Dodd 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 [40,41]. This paved the way for a more general method applicable to arbitrary simple current orbifolds
e.g. in [42–45].
13

16

SciPost Phys. 11, 082 (2021)

2. The boundary states of theory F is obtained by stacking a Kitaev chain (Arf invariant)before gauging Z2 . Using (2.43),k| j〉FNS = (| j〉A + | − j〉 A ) ⊗ |+〉 ,2k| j〉FR = (| j〉A − | − j〉 A ) ⊗ |−〉 ,
2
F
A
k
k
| 〉 = | 〉 ⊗ |+〉 ,
4 NS
4
k F
k A
| 〉 = | 〉 ⊗ |+〉 .
4 R
4 tw

1
k−2
j = 0, , ...,,
2
4

(3.12)

3. The boundary states of theory F0 is obtained by stacking a Kitaev chain (Arf invariant)before gauging Z2 , and then further stacking another Kitaev chain. Using (2.44),
0k
1| j〉FNS = p (| j〉A + | − j〉 A ) ⊗ |+〉 ,
2
2
0
1
k
| j〉FR = p (| j〉A − | − j〉 A ) ⊗ |+〉 ,
2
2
F0
A
p k
k
| 〉 = 2 | 〉 ⊗ |+〉 ,
4 NS
4
F0
p
k
k A
| 〉 = 2 | 〉 ⊗ |−〉 .
4 R
4 tw

1
k−2
j = 0, , ...,,
2
4

(3.13)

Before proceeding, we should pause here to mention that the boundary states of thefermionic versions of diagonal unitary minimal models are given essentially by the same formulas, since the primaries (r, s) of the Virasoro minimal models14 have the fusion rule which
is isomorphic to the fusion rule of the SU(2) affine algebra acting separately on r and s, andthe Z2 quotient acts only on a single index [47–49]. The resulting boundary states agree withp
those determined in [25], up to the factor of 2 explained at the end of Sec. 2. The SU(2)k
WZW models also have exceptional invariants [50, 51]. We have not studied the boundary
states of fermionic versions of these, but again it should be possible to recover them from thediscussion of boundary states of fermionic exceptional minimal models in [25].

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 [52, 53], but our interpretation is slightly different from the one given there.
Cardy States for the Ising Model: The Cardy states of the Ising CFT were discussed in [36].
11〉 and can be written in terms of the Ishibashi states |0〉〉, | 12 〉〉 and | 16〉〉
They are |0〉 , | 12 〉 and | 16
The indices (r, s) here are different from those we have used in the previous section to denote the number ofZ2 defects on T 2 .
14

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SciPost Phys. 11, 082 (2021)

via (A.11):
1
1 1
1 1
|0〉 = p |0〉〉 + p | 〉〉 + 1 | 〉〉 ,22 2
2 4 1611 1
11 1| 〉 = p |0〉〉 + p | 〉〉 − 1 | 〉〉 ,222 2
2 4 161
1| 〉 = |0〉〉 − | 〉〉 .
16
2

(3.14)

Under the Z2 generator g we have1g |0〉 = | 〉 ,2

1g | 〉 = |0〉 ,
2

g|

1
1
〉=| 〉.
16
16(3.15)

1
Hence | 16
〉 is invariant under Z2 , while the other two states are exchanged by Z2 . In additional1〉tw .
to the three Cardy states, there is also a twisted state | 16

Ishibashi States for the Massless Majorana Fermion: For the free Majorana fermion, there
are two boundary conditions, ψ L = ±ψR . Following [52], we will denote the boundary statesby |B± 〉N S/R . To write down the boundary states, we consider the mode expansion of thePP11
fermions: ψ L (z) = r ψ r z −r− 2 , ψR (z̄) = r ψ̃ r z̄ −r− 2 where r ∈ Z for the R sector, andr ∈ Z + 12 for the NS sector. Note that we have zero modes in the R sector. Boundary Ishibashi
states for the free fermion can then be written down by demanding (ψ r ∓ i ψ̃−r )|B± 〉〉NS/R = 0
[52]:


X
|B± 〉〉NS = exp ±iψ−r ψ̃−r  |0〉 ,r>0,r∈Z+ 21(3.16)!

|B± 〉〉R = exp ±iX

ψ−n ψ̃−n |±0 〉 ,n>0,n∈Z

where the sign ± in the exponent and in the zero mode state |±0 〉 are correlated with the
choice of boundary condition ψ L = ±ψR . The vacuum |0〉 in the NS sector is defined to beannihilated by all the fermion modes with positive index ψ r |0〉 = ψ̃ r |0〉 = 0, r > 0. Let us
further specify |±0 〉 in the R sector. There are two fermion zero modes, ψ0 and ψ̃0 . We can
e0 , which then defines a two dimension Hilbert space
now form a complex fermion c = ψ0 + i ψ|±0 〉 via
c |+0 〉 = 0 ,|−0 〉 = c † |+0 〉 ,c † |−0 〉 = 0 , ψn |±0 〉 = ψ̃n |±0 〉 = 0, ∀n > 0 .

(3.17)To see the fermion parity, note that the state in the NS sector does not have fermion zeromodes, and non zero modes come in pairs of left handed and right handed modes. Thus(−1) F = 1 for |B± 〉NS . For the R sector states, the nonzero modes in the exponent come inpairs, hence they do not contribute to nontrivial fermion parity either. However, the zeromodes do contribute to nontrivial fermion parity. We declare that the zero mode state |+0 〉 iseven, (−1) F |+0 〉 = |+0 〉, then(−1) F |−0 〉 = (−1) F c † |+0 〉 = −c † (−1) F |+0 〉 = − |−0 〉 ,(3.18)

which shows that |−0 〉 has odd fermion parity. In summary, we have(−1) F |±0 〉 = ± |±0 〉 .
18(3.19)

SciPost Phys. 11, 082 (2021)

The fermion parities of various Ishibashi states can now be summarized as follows:(−1) F = +1 : |B+ 〉〉NS , |B− 〉〉NS , |B+ 〉〉R ,(−1) F = −1 : |B− 〉〉R .

(3.20)

Cardy States for the Massless Majorana Fermion: We further construct the Cardy statesfrom the Ishibashi states |B± 〉〉NS/R . As discussed in the introduction, we label the boundary
states of the fermionic theory with definite quantum numbers. The boundary state is one
to one correspondence with the boundary condition labeled by ±, and each boundary state iseither in the NS sector or R sector. 15 This means that the boundary states of the free Majorana
fermion is proportional to the Ishibashi states, i.e.
|B+ 〉NS ∝ |B+ 〉〉NS ,|B− 〉NS ∝ |B− 〉〉NS ,|B+ 〉R ∝ |B+ 〉〉R ,|B− 〉R ∝ |B− 〉〉R .

(3.21)

To determine the proportionality constants in (3.21), one needs to compute the overlap of the
Ishibashi states, and demand the boundary states to satisfy the spin Cardy condition (1.9).
Since it does not affect the discussion below, we will not work out the normalization constanthere. What is important is that the Z2F quantum numbers of the boundary states |B± 〉NS/R are
the same as those in the Ishibashi states:(−1) F = +1 : |B+ 〉NS ,|B− 〉NS ,|B+ 〉R ,F(−1) = −1 : |B− 〉R .

(3.22)

By matching the quantum numbers, we find that it is consistent to relate the fermion boundary
1
1states |B± 〉NS/R and Ising boundary states |0, 21 , 16
〉 and | 16
〉tw via either (2.43) or (2.44):
p 1
1|B− 〉NS = 2 | 〉 ⊗ |+〉 ,|B− 〉NS = (|0〉 + | 〉) ⊗ |+〉 ,
16
2
1
1
1
|B+ 〉NS = p (|0〉 + | 〉) ⊗ |+〉 ,|B+ 〉NS = | 〉 ⊗ |+〉 ,
2
16
2
or
p
1
1
|B− 〉R = 2 | 〉 ⊗ |−〉 ,|B− 〉R = (|0〉 − | 〉) ⊗ |−〉 ,
16 tw
2
1
1
1
|B+ 〉R = | 〉 ⊗ |+〉 ,|B+ 〉R = p (|0〉 − | 〉) ⊗ |+〉 .
16 tw
2
2

(3.23)

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 A to be the Ising theory we have
A ' D, which then leads to F ' F0 . We discuss more details in Sec. 4.3.

3.3

S pin(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 S pin(N )1 WZW model. We start by checking this statement.
The bulk spectrum: The S pin(N )1 WZW model has three primaries 0, v, s for odd N andfour primaries 0, v, s, c for even N , where the letters specify how the primary transforms underN
S pin(N ): adjoint (0), vector (v), spinor (s), and conjugate spinor (c). Their spins are 0, 21 , 16,N
16 in this order, and we also denote the primaries by their spins, distinguishing the conjugate
This is in contrast with the discussion in [52,53], where they allow the boundary states in the fermionic theory
to be a superposition of NS sector and R sector Ishibashi states.
15

19

SciPost Phys. 11, 082 (2021)

spinor by a tilde if necessary. We consider the diagonal modular invariant as the theory A, anduse the Z2 symmetry generated by the primary v. We then have

HS = V0 ⊗ V0 ⊕ Vv ⊗ Vv , HU = Vs ⊗ Vs ,HV = V0 ⊗ Vv ⊕ Vv ⊗ V0

H T = Vs ⊗ Vs ,for odd N and

HS = V0 ⊗ V0 ⊕ Vv ⊗ Vv , HU = Vs ⊗ Vc ⊕ Vc ⊗ Vs ,H T = Vs ⊗ Vs ⊕ Vc ⊗ Vc ,HV = V0 ⊗ Vv ⊕ Vv ⊗ V0

(3.24)(3.25)

for even N .
Now the theory of N Majorana fermions have the Hilbert space

HNS = (V0 ⊕ Vv ) ⊗ (V0 ⊕ Vv ) ,((Vs ⊕ Vc ) ⊗ (Vs ⊕ Vc ) (N even) ,HR =Vs ⊗ Vs ⊕ Vs ⊗ Vs
(N odd) .

(3.26)

We can now use (1.5) to confirm that this can be identified either with the theory F or F0 for
odd N . There is a subtlety for even N : the standard assignment of (−1) F in the R sector isto assign (−1) F = +1 for Vs ⊗ Vs ⊕ Vc ⊗ Vc and (−1) F = −1 for Vs ⊗ Vc ⊕ Vc ⊗ Vs . This is theassignment for the theory F0 but not for F.
Cardy states of the bosonic model: Let us now discuss the Cardy states of the bosonic
S pin(N )1 WZW model. For odd N , there are three primary fields, with conformal weights
N N(0, 0), ( 12 , 12 ), ( 16
, 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 needto replace the label 1/16 by N /16. Under Z2 we have:1g |0〉 = | 〉 ,2

1g | 〉 = |0〉 ,
2

g|

N
N
〉=| 〉,
16
16(3.27)

NN
in particular | 16
〉 is Z2 invariant. This means that we also have a Z2 twisted sector state | 16
〉tw .
For even N , the modular S matrix is


1
1
1
1
1 1 e iπN /4 −1 −e iπN /4 ,
(3.28)
S= 
−1
1
−1 
2 1
1 −e iπN /4 −1 e iπN /4
N 1 f
Nwhere the columns from the left to the right are for the primaries 0, 16, 2 , 16
in this order.
Hence in terms of the Ishibashi states, we have


à
1
N
1
N
|0〉 = p |0〉〉 + | 〉〉 + | 〉〉 + | 〉〉 ,
16
2
16
2


à
N
1
1
iπN /4 N
iπN /4 N| 〉 = p |0〉〉 + e| 〉〉 − | 〉〉 − e| 〉〉 ,
16
16
2
16
2(3.29)


à
1
1
N
1
N
| 〉 = p |0〉〉 − | 〉〉 + | 〉〉 − | 〉〉 ,
2
16
2
16
2


ß
à
N
1
1
iπN /4 N
iπN /4 N| 〉 = p |0〉〉 − e| 〉〉 − | 〉〉 + e| 〉〉 .
16
16
2
16
2

20

SciPost Phys. 11, 082 (2021)

NN〉〉 and |ßUnder Z2 , |0〉〉 and | 21 〉〉 are even, and | 1616 〉〉 are odd. In terms of Cardy states we then
have
ß
ß
1
1
N
N
N
N
g |0〉 = | 〉 , g | 〉 = |0〉 , g | 〉 = | 〉 , g | 〉 = | 〉 .
(3.30)
2
2
16
16
16
16
Hence they form two Z2 conjugate pairs. There is no Z2 invariant boundary state, and therefore
there are no boundary states in the Z2 twisted sector.

Boundary States of free fermions: We further discuss the Ishibashi states in the fermionictheory. We will preserve the SO(N ) global symmetry, hence all flavors of fermions will be iniithe same sector, and will have the same sign in ψ+= ±ψ−for all i = 1, ..., N . The Ishibashi
states are direct generalization of (3.16),


N
X X
j
j
N
ψ−r ψ̃−r  |0〉 ,|B± 〉〉NS= exp ±i
r>0,r∈Z+ 21 j=1|B± 〉〉RN = exp ±i

N
X X(3.31)!
jj

ψ−n ψ̃−n |±0 〉 ,n>0,n∈Z j=1
jwhere |0〉 is defined to be annihilated by all ψRi ’s with r > 0, and |±0 〉 is defined as ⊗Nj=1 |±0 〉jwhere |±0 〉 is determined in (3.17) for the j-th flavor. In summary, there are still four Ishibashi
NN
states, i.e. |B+ 〉〉NS, |B− 〉〉NS, |B− 〉〉RN , |B+ 〉〉RN .
N
Let us analyze the fermion parity for the Ishibashi states. For NS states |B± 〉〉NS
, the non-zeroNmodes do not contribute to fermion parity. For R states |B± 〉〉R , only the zero mode contributes
the fermion parity. The fermion number is the sum of fermion number for each flavor, hence(|+10 〉 ⊗ · · · ⊗ |+0N 〉 ,(−1) F = 1 ,(3.32)|−10 〉 ⊗ · · · ⊗ |−0N 〉 ,(−1) F = (−1)N .
N
N
Thus when N is odd, |B− 〉〉RN should have (−1) F = −1 and |B+ 〉〉NS, |B− 〉〉NS, |B− 〉〉RN all haveF(−1) = 1. However, when N is even, all Ishibashi states in (3.31) have even fermion parity.
Similar to the discussion in Sec.3.2, the boundary states for N Majorana fermions, denoted
NNN
as |B± 〉NS/R, are proportional to the corresponding Ishibashi states, |B± 〉NS/R∝ |B± 〉〉NS/R.
This means that the quantum numbers of the Ishibashi states discussed in the previous paragraph also apply to the boundary states: when N is odd, |B− 〉RN should have (−1) F = −1 andN
N|B+ 〉NS, |B− 〉NS, |B− 〉RN all have (−1) F = 1. However, when N is even, all boundary states
N|B± 〉NS/R have even fermion parity.
By matching the quantum numbers of the boundary states, we conclude as follows:

1. When N is odd, the boundary states of the fermionic theory are related to those in thebosonic theory in the same way as in N = 1 case (3.23).
2. When N is even, all the boundary states satisfy (−1) F = 1. The fermion boundary states
N|B± 〉NS/R
and the Ising boundary states can only be identified via (2.44):
1
1
1
1
N
|B+ 〉NS= p (|0〉 + | 〉) ⊗ |+〉 ,|B+ 〉RN = p (|0〉 − | 〉) ⊗ |+〉 ,
2
2
2
2
ß
ß
1 N
N
1 N
N
N
|B− 〉NS= p (| 〉 + | 〉) ⊗ |+〉 , |B− 〉RN = p (| 〉 − | 〉) ⊗ |+〉 .
16
16
2 16
2 16
21

(3.33)

SciPost Phys. 11, 082 (2021)

To see why (2.43) does not work, we note that all the boundary states in the S pin(N )1
theory are Z2 breaking states, while (2.43) pairs certain superposition of Z2 breaking
states with (−1) F = −1 fermion boundary states, which is in contradiction with the factthat there are no (−1) F = −1 boundary states in the Majorana fermion theory. Thismeans that if S pin(N )1 WZW is theory A, then N massless Majorana fermions is theory
F0 :N Majorana Fermions =3.4

S pin(N )1 WZW × Kitaev
× Kitaev .
Z2(3.34)

Maldacena-Ludwig boundary state of eight Majorana fermions

As the last example in this section, let us discuss the boundary state of eight Majorana fermionsstudied by Maldacena and Ludwig in [54]. This boundary state arises when we study themonopole catalysis of baryon decay [55, 56] and also when we study the Kondo problem[57, 58]. It was also recently revisited in [59].
Let us start by recalling the bulk system, which is the special case N = 8 of N MajoN
allrana fermions we discussed above. In this particular case, the primaries 1 , N and f
2

16

16

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 S3 outer automorphism permuting them. The MaldacenaLudwig boundary state is characterized by the condition that the left-moving spin(8)1 andthe right-moving spin(8)1 are related by the order-2 automorphism ω exchanging v ↔ s andfixing 0 and c. We therefore have J = ω(J̃) at the boundary, and any boundary state |φ〉 withsuch a boundary condition needs to satisfy(J−n + ω(J̃n )) |φ〉 = 0 .

(3.35)

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 Z2 lines we used repeatedly, generated by the primary v, commute with the chiral algebra, and are not to be confused with the automorphismsof spin(8)1 affine algebra.
The property of such ω-twisted boundary states was studied in detail in [60], and we cansimply quote the results there. In the v-untwisted sector, there are two ω-twisted Ishibashi
states|0〉〉ω := R0 (ω)|0〉〉 ,|c〉〉ω := R c (ω)|c〉〉 ,(3.36)
where R0 (ω), R c (ω) are the representation matrices of ω on V0 and Vc with the convention
that they only act on the holomorphic side. The two ω-twisted Cardy states are then|0〉ω = |0〉〉ω + |c〉〉ω ,|c〉ω = |0〉〉ω − |c〉〉ω ,(3.37)and they are exchanged by the action of the Z2 Verlinde line operator v.
We then convert this pair of Z2 -breaking states to the theory F0 to find the MaldacenaLudwig boundary state,16 which is given by|ML〉NS =p2|0〉〉ω ,|ML〉R =

p2|c〉〉ω .

(3.38)

We were lucky that the ω-twisted boundary states on the bosonic side turned out to be Z2 -breaking. If it
were Z2 -preserving, we would have also needed the ω-twisted boundary state in the v-twisted sector. The general
theory of such ‘doubly-twisted’ boundary states do not seem to be readily available.
16

22

SciPost Phys. 11, 082 (2021)

DA

I A→ D

IAtw →DIA→Dtw

IAtw →Dtw

Figure 2: Interfaces from the theory A to D.

We can then find, for example,−2πH/δ|ML〉NS = χ0 (iδ) + χ v (iδ) + χs (iδ) + χc (iδ) ,−2πH/δ
〈ML|e|ML〉
R
R = χ0 (iδ) − χ v (iδ) − χs (iδ) + χc (iδ) .

NS 〈ML|e4(3.39)

Interfaces and boundary states

In the last section we derived the form of the boundary states of fermionic theories F and
F0 . In this section we use this knowledge to determine the algebra of the interfaces between
the theories A, D, F and F0 . This analysis also allows us to see how the anomalous chiral Z2
symmetry arises in the theory F ' F0 when A ' D.

4.1

A↔D

We start by considering the interface between the theory A and D. When placed on a circle,
such an interface determines various operators, some of which are shown in Fig 2. Namely,
from A to D we have
I A→ D : H A → H D ,D,
IA→Dtw : HA → Htw

AIAtw →D : Htw→ HD ,DA.
→ Htw
IAtw →Dtw : Htw(4.1)

The interface operators from D to A can be considered in a similar manner and they are adjointsof the operators given in (4.1).
From the known relations among the Hilbert spaces (1.5) of A and D, we know that
IA→D ∝ PS ,IA→Dtw ∝ PT ,IAtw →D ∝ PU ,IAtw →Dtw ∝ PV ,(4.2)

where PS,T,U,V are the projections to the respective components in (1.5).
For example, we have HA = HS ⊕ H T and HD = HS ⊕ H T . Therefore, IA→D can only map
the states in HS ⊂ HA to HS ⊂ HD . We expect that ID→A IA→D is proportional to the identity
on HS . The two copies of HS belong to two distinct Hilbert spaces17 HA and HD , so we use
an appropriate multiple of ID↔A to identify them, resulting in the simple equations (4.2).
The proportionality coefficients can be fixed in various ways. Here, we require that theapplication of the interface operators to the boundary states give rise to an integral linear
17
When A ' D, IA→D can define a non-trivial order-2 operation on HS . We will discuss this possibility in moredetail in Sec. 4.3.

23

SciPost Phys. 11, 082 (2021)

Table 1: Action of interface operators (between theories A and D) on the boundary
states.

I A→ DIA→Dtw
IAtw →D
IAtw →Dtw
I D→ A
IDtw →A
ID→Atw
IDtw →Atw

|i〉A

|i〉A
tw

|a±〉A

|i+〉D + |i−〉D0-D|i+〉 − |i−〉D0|a〉D± |a〉D
tw
-

|a〉D

|a〉D
tw

|i±〉D

|a+〉A + |a−〉A0-|a+〉A − |a−〉A0|i〉A± |i〉Atw-combination of the boundary states. Comparing (1.13) and (1.14), we find18pp
p
p
IA→D = 2PS , IA→Dtw = 2PT , IAtw →D = 2PU , IAtw →Dtw = 2PV .

(4.3)

and similarly for the interfaces from D to A. We then find, for example,
IA→D |i〉A = |i+〉D + |i−〉D ,IA→D |a±〉A = |a〉D

(4.4)

and
ID→A IA→D = 1 + g ,(4.5)

where g is the Z2 generator. Here the right hand side is restricted to act on the untwisted
sector. This last relation is known to generalize to
X
ID→A IA→D =
g,
(4.6)
g∈G

when A is G-symmetric and D is the G-gauged theory [61]. Systematically, we tabulate theaction of interface operators between A and D on the boundary states in table 1.

4.2

A ↔ F, F0

Next we consider the interfaces between the original theory and the fermionized versions. We
first consider the ones between A and F0 . We denote the interfaces from A to F0 as
00F
IA→F0NS : HA → HNS,

IA→F0R : HA → HRF ,

0

0

A
F
IAtw →F0NS : Htw
→ HNS
,

A
IAtw →F0R : Htw→ HRF .

(4.7)

Again they are proportional to the projectors PS,T,U,V :IA→F0NS ∝ PS ,IA→F0R ∝ PT ,IAtw →F0NS ∝ PV ,IAtw →F0R ∝ PU .

(4.8)

The proportionality coefficients can be found by studying their actions on boundary states. We
find that
p
p
p
p
(4.9)
IA→F0NS = 2PS , IA→F0R = 2PT , IAtw →F0NS = 2PV , IAtw →F0R = 2PU .
18

Strictly speaking this method does not determine IAtw →Dtw , since the projection of the boundary states is zero.

24

SciPost Phys. 11, 082 (2021)

Table 2: Action of interface operators (between theories A and F0 ) on the boundary
states.
|i〉A
0

|i〉A
tw

|a+〉A
0

IA→F0NS|iψ〉FNS-|a〉FNS

IA→F0R0-+ |a〉FR

IAtw →F0NS-

0

IAtw →F0R

-

F0

|iψ〉R
F0

F0

0

0

− |a〉FR
-

-

-

F0

|a〉FR

|iψ〉R

|a〉NS

IF0NS →A

2 |i〉A-|a+〉A + |a−〉A

IF0R →A-

0

IF0NS →Atw

0-

IF0R →Atw-

2 |i〉tw|a〉FNS-|iψ〉NS

A

|a−〉A
0

0

A

-

|a+〉 − |a−〉A

0

-

-

0

These interfaces act on the boundary states for example as0

IA→F0NS |i〉A = |i〉FNS ,0

IA→F0NS |a±〉A = |a〉FNS(4.10)

andIF0NS →A IA→F0NS = 1 + g .

(4.11)

Again the right hand side is restricted to the NS sector only. We tabulate the action of interface operators between A and F0 on the boundary states in table 2. These actions were first
determined in [53] for two specific cases when the theory A is the critical Ising model or thetricritical Ising model. They were also discussed in [11].
The interfaces between A and F can be determined in a similar manner. We find
IA→FNS = 2PS ,IA→FR = 2PT , IAtw →FNS = 2PV , IAtw →FR = 2PU .
(4.12)
p
Note the difference by the factor 2 in (4.9) and (4.12). Because of this we find that
IFNS →A IA→FNS = 2(1 + g) .

(4.13)

We tabulate the action of interface operators between A and F on the boundary states in table
3.
The difference between (4.9) and (4.12) can also be understood by considering the interface between the theories F and F0 . 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
IFNS →F0NS IF0NS →FNS = 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 have0

IFNS →F0NS |i〉FNS = |i〉FNS ,0

IFNS →F0NS |a〉F = 2 |a〉F .

(4.15)

The factor 2 in the second equation comes from the fact that |a〉F already has an unpaired Majorana zero mode, which combines with another Majorana zero mode hosted on the interface
between F and F0 . We then findIF0NS →FNS IA→F0NS = IA→FNS ,IFNS →F0NS IA→FNS = 2IA→F0NS .
25(4.16)

SciPost Phys. 11, 082 (2021)

Table 3: Action of interface operators (between theories A and F) on the boundary
states.

IA→FNSI A→ FR
IAtw →FNS
IAtw →FRIFNS →AI FR → A
IFNS →Atw
IFR →Atw

|i〉A

|i〉A
tw

|a+〉A

|a−〉A

2 |i〉FNS0-02 |i〉FR|aψ〉FNS+ |aψ〉 FR-|aψ〉FNS− |aψ〉 FR-|i〉FNS|i〉FR|aψ〉FNS|aψ〉FR2 |i〉A0-02 |i〉Atw

2(|a+〉A + |a−〉A )0-2(|a+〉A − |a−〉A )
0

A

A

D

F

F

0

=

p
2

2

p
2

Figure 3: A closed loop of an interface without any operators in it can be evaluatedto 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 A→ D =p
2PS ,IA→F = 2PS ,I A→ F0 =p
2PS ,(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 topp
be an insertion of a number, 2, 2 or 2 depending on whether the interface is from A to D,
F or F0 . For topological line operators in a single theory, the number obtained by evaluating aclosed 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 betweenp
A and F0 has a non-integer quantum dimension 2 is consistent with the fact that when A istrivial, F0 is the nontrivial Kitaev chain.

4.3

The special case A ' D and the anomalous Z2 symmetry of F ' F0

In some important cases, e.g. the Ising model or the tricritical Ising model, the Z2 orbifold
theory D of the original bosonic theory A is equal to itself, A ' D. In this case the interface
X := IA→D can be considered as a defect of the single theory A ' D which now satisfies the26

SciPost Phys. 11, 082 (2021)

fusion rule
X 2 = 1 + g,

gX = X ,(4.18)

see [62] and also more recent works [27, Sec. 4.3.1] and [29, 63]. We note that, when wepwrote in (4.3) above that IA→D = 2PS , this is meant to be a map obtained by first projecting to
the summand HS ⊂ HA and then sent isometrically to the corresponding summand HS ⊂ HD .
We now identify HA ' HD , but this can introduce a nontrivial unitary operator of order two
on HS . Therefore, the duality interface X as acting on HA is a compositionp(4.19)
X = 2hS PS ,of the projector PS to HS together with a unitary operator
hS : HS → HS ,(4.20)

which squares to 1.
Similarly, the interface X tw := IAtw →Dtw has the formp
X tw = 2hV PV ,(4.21)

wherehV : HV → HV ,(4.22)

is again a unitary which squares to one.
The operators hS and hV then combine to give a unitary operator
F
F
→ HNS
h : HNS

where

F= HS ⊕ HV ,HNS(4.23)

which squares to one. This is the Z2 operator acting on the NS sector of the theory F.
For example, in the case of the Ising model treated above, we have X = L 1 , which is the
16
Kramers-Wannier duality defect. Under X ,1
1
X |0〉 = X | 〉 = | 〉 ,
2
16

X|

1
1
〉 = |0〉 + | 〉 .
16
2(4.24)

Using (3.23), one finds that the boundary conditions B− and B+ for the Majorana fermions areexchanged, i.e., ψ L = ±ψR becomes ψ L = ∓ψR , which is precisely the chiral Z2 symmetry.
On the R-sector, we see that the operator coming from the duality wall X exchanges HU and
H T , and in particular flips the sign of (−1) F . This means that the Z2 symmetry is anomalous.
Recall that the anomaly of Z2 symmetry of fermionic (1+1)-dimensional theory is specified
by an integer modulo 8; it is known that the anomalous Z2 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. [23, Sec. IV and Sec. VI].

Acknowledgements
The authors would like to thank the authors of [26] for discussions. The authors would also
like to thank Shu-Heng Shao, Philip Boyle Smith, Gerard Watts and an anonymous referee forvery 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 [29] which is closely related to thisproject. Y.T. is partially supported by JSPS KAKENHI Grant-in-Aid (Wakate-A), No.17H04837
27

SciPost Phys. 11, 082 (2021)

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 – theCompetitiveness and Cohesion Operational Programme (Grant KK.01.1.1.01.0004).

A

RCFTs and their Z2 symmetries

In this section we collect various facts on RCFTs. More specifically, we discuss the diagonal
modular invariants and their Z2 symmetries generated by simple currents of order 2. This willallow us to construct a large class of fermionic models and fermionic boundary states.
Diagonal modular invariant: We start from a chiral algebra A with irreducible representations Vα , where α = 0, . . . , such that the label 0 corresponds to the vacuum representation.
We denote the fusion rule asγ
Vα Vβ ∼ Nαβ Vγ .
(A.1)
The untwisted Hilbert space H0 has the decompositionM
H0 =Vα ⊗ Vα ,

(A.2)

α

and therefore has the character
TrH0 e−2πt H =Xαχα (t)χα (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 α [62, 64, 65]. When wrapped around the spatial circle, it
determines an operator
Lα : H0 → H0 ,(A.4)
satisfying the fusion rule equationγ

Lα Lβ = Nαβ Lγ .

(A.5)

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

Lα |φβ 〉 =

Sαβ
S0β

|φβ 〉 ,

(A.6)

where |φβ 〉 is in the summand Vβ ⊗ Vβ of H0 . These equations are compatible because of thecelebrated formula of Verlinde [64],
γ

Nαβ =

X Sαδ Sβδ Sγδ
S0δ

δ

28

.

(A.7)

SciPost Phys. 11, 082 (2021)

This also means that the Hilbert space Hγ on a circle with an insertion of the Verlinde operator
labeled by α is given by
M
β
Hγ =
Nαγ
Vα ⊗ Vβ ,
(A.8)
α,β

so that the character is
TrHγ e−2πt H =

X
α,β

β
Nαγ
χα (t)χβ (t) .

(A.9)

Ishibashi states and Cardy states: To describe the boundary states, we first consider Ishibashi states |α〉〉 ∈ Vα ⊗ Vα [35] with the propertyi〈〈α|e−2πH/δ |β〉〉 = δαβ χα ( ) .
δThen the Cardy states are [36]|α〉 =(A.10)X Sαβ|β〉〉 ,
ÆS0β
β(A.11)

which satisfies〈α| e−2πH/δ |β〉 = TrHα|β e−2πδH =

X
α,β

γ

Nαβ χγ (iδ) ,meaning that the open Hilbert space has the decomposition
M γ
Hα|β =
Nαβ Vγ .

(A.12)(A.13)γWe also find the action of the Verlinde operators on the Cardy states as follows:γLα |β〉 = Nαβ |γ〉 .

(A.14)

Order-2 simple currents: In general, a primary p which has the fusion rule (Vp )n ∼ V0 iscalled a simple current. The corresponding Verlinde line operator generates a Zn symmetry.
Here we only consider the case when n = 2. Denoting the simple current by υ, we have thefusion rule Vυ Vυ ∼ V0 . The spin hυ is 0 or 1/4 mod 1/2, and the Z2 is non-anomalous in theformer and anomalous in the latter case.
Under the fusion with υ, primaries can be grouped into two types. Namely, there are thoseinvariant under the multiplication:Vυ Vi ∼ Vi ,(A.15)and pairs exchanged by the multiplication:
Vυ Va+ ∼ Va− ,Vυ Va− ∼ Va+ .

(A.16)

Cardy states in the twisted sector: We now examine the effect of the Z2 symmetry on
the boundary conditions. The equations above translate to the statement that the boundary
condition i is invariant under Z2 while the boundary conditions a± break the Z2 symmetry andare exchanged by it. This means that one can put Z2 -invariant boundary conditions i, j, . . . ona circle twisted by the Z2 symmetry. This construction should then determine twisted Cardy
statesM|i〉tw ∈ Hυ =Vα ⊗ Vυα .
(A.17)
α29

SciPost Phys. 11, 082 (2021)

They can be expanded in terms of the twisted Ishibashi states|i〉〉tw ∈ Vi ⊗ Vi ⊂ Hυ ,(A.18)

having the same normalization as in (A.10). We note that the untwisted and twisted Ishibashi
states |i〉〉 and |i〉〉tw labeled by the same symbol i behave in the same way under the actionof the chiral algebra A and its antiholomorphic counterpart 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 wasstudied in the case of Z2 simple currents e.g. in [43]. A general formula applicable for any
simple current orbifold was conjectured in [44] which was later proved in [45]. In our casethe formula boils down to
X Ŝi j(A.19)|i〉tw =
Æ | j〉〉tw ,S0 jj
up to a subtle phase which does not concern us in this paper.19 Here, Ŝi j describes the actionof S ∈ S L(2, Z) on the space of torus conformal blocks with an insertion of the simple currentυ.
The general form of Ŝi j was determined e.g. in [66, 67]. They are also known to satisfy ageneralized version of the Verlinde formula [67–70]jN̂iα =

X Ŝik Sαk Ŝ jk
S0k

k

,

(A.20)jwhere the summation is over Z2 invariant primaries and N̂iα is the trace of the Z2 action g on
the fusion space among i, j and α, see also [46]. This relation guarantees that the overlap oftwisted Cardy states satisfiesXj−2πH/δ
−2πδH
〈i|e
|
j〉
=
Tr
g
e
=
N̂iα χα (iδ) ,(A.21)
H
tw
tw
i| j
α

and has a consistent Hamiltonian interpretation in the open channel.

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