11institutetext: Yukawa Institute for Theoretical Physics, Kyoto University,
Kyoto 606-8502, Japan22institutetext: Department of Physics, North Carolina
State University, Raleigh, NC 27607, USA

# Semiclassics with ’t Hooft flux background for QCD with $2$-index quarks

Yuya Tanizaki<EMAIL_ADDRESS>2 Mithat Ünsal
<EMAIL_ADDRESS>
###### Abstract

We study quantum chromodynamics including the two-index symmetric or anti-
symmetric quark (QCD(Sym/ASym)) on small $\mathbb{R}^{2}\times T^{2}$ with a
suitable magnetic flux. We first discuss the ’t Hooft anomaly of these
theories and claim that discrete chiral symmetry should be spontaneously
broken completely to satisfy the anomaly matching condition. The $T^{2}$
compactification with the magnetic flux preserves the ’t Hooft anomaly, and
the $2$d effective theory is constrained by the same anomaly of $4$d
QCD(Sym/ASym). We demonstrate the spontaneous breakdown of chiral symmetry
using the dilute gas of center vortices, which confirms the prediction of the
’t Hooft anomaly. We also find that each vacuum maintains the charge
conjugation symmetry, and this gives affirmative support for the
nonperturbative large-$N$ orientifold equivalence between QCD(Sym/ASym) and
$\mathcal{N}=1$ supersymmetric $SU(N)$ Yang-Mills theory.

††preprint: YITP-22-45

##  1 Introduction

In our recent work Tanizaki:2022ngt , we have proposed a new semiclassical
formalism to study four-dimensional non-Abelian gauge theories by
compactifying the theory on small $\mathbb{R}^{2}\times T^{2}$ including ’t
Hooft flux. Our framework provides the weak-coupling description of
confinement and reproduces many qualitative features of strongly coupled
confining vacua, and thus it is reasonable to expect that its dynamics show
adiabatic continuity to the strong dynamics on $\mathbb{R}^{4}$.

Technically, the ’t Hooft flux lifts the classical moduli space of gauge
holonomies on $T^{2}$ completely tHooft:1979rtg , and the $0$-form center
symmetry is unbroken. As the $T^{2}$ size is much smaller than the strong
scale, the center-symmetric vacuum is stable under perturbative corrections.
On the other hand, the $1$-form center symmetry is spontaneously broken at the
classical level, and we need to consider the nonperturbative corrections to
restore it.

The center vortex tHooft:1977nqb ; Cornwall:1979hz ; Nielsen:1979xu ;
Ambjorn:1980ms  is the key player that dominates the leading nonperturbative
correction. In this $T^{2}$ compactified setup with ’t Hooft flux, center
vortices are realized as the BPS configuration having the fractional
topological charge $Q=\pm\frac{1}{N}$, and its classical action is given by
$S=\frac{S_{I}}{N}$, where $S_{I}=\frac{8\pi^{2}}{g^{2}}$ is the 4d instanton
action Gonzalez-Arroyo:1998hjb ; Montero:1999by ; Montero:2000pb . The
distinguishing feature of the center vortex is its nontrivial mutual
statistics with Wilson loops. We can show that the dilute-gas approximation of
center vortices describes the confinement for $1$-form center symmetry, and
this provides the first semiclassical computations of $4$d confining gauge
theories based on center vortices.

In this paper, we generalize our earlier work to quantum chromodynamics (QCD)
with two-index representation fermions, e.g. symmetric or anti-symmetric
representation, and we denote them as QCD(Sym/ASym). QCD(ASym) provides an
alternative large-$N$ limit of QCD because of the simple kinematic fact that
fundamental and anti-symmetric representations are the same for $N=3$. Unlike
the ’t Hooft large-$N$ limit tHooft:1973alw , quark loops are not suppressed
compared with gluon loops in QCD(ASym), and this feature is similar to the
large-$N$ limit of QCD with $N_{f}$ fundamental quarks with $N_{f}/N$ fixed.
QCD(Sym/ASym) received much attention in the context of orientifold
equivalence with ${\cal N}=1$ supersymmetric Yang-Mills theory Armoni:2003gp ;
Armoni:2004ub ; Unsal:2006pj .

We examine the nonperturbative dynamics of $N_{f}=1$ QCD(Sym/ASym) by using
compactification with magnetic flux. We shall demonstrate that the discrete
chiral symmetry is spontaneously broken completely, by performing the dilute
gas approximation with center vortices dressed with gapless fermions. By
adding a soft mass deformation, we obtain the multi-branch structure of the
ground states as in the case of confining vacua of pure Yang-Mills theory. A
remarkable feature here is that even though the theta angle is fractionalized
as $\theta/N$, the dilute gas of center vortices produces the correct $N+2$
(and $N-2$) vacua and multi-branched structure once a soft mass term is added.
As these theories are non-supersymmetric, there have not been powerful
techniques to study QCD(Sym/ASym) directly (except with double-trace
deformations on $\mathbb{R}^{3}\times S^{1}$ Shifman:2008ja ), and thus our
result provides a novel insight into their dynamics and the validity of
nonperturbative orientifold equivalence.

##  2 QCD with $2$-index (anti-)symmetric representation

In this section, we discuss the dynamics of $1$-flavor QCD(Sym) and QCD(ASym).
Since the analyses on these theories become almost parallel, we pay attention
to QCD(Sym) first, and then we briefly discuss the case of QCD(ASym) later.

###  2.1 Global symmetry and ’t Hooft anomaly of QCD(Sym)

We consider the $4$d $SU(N)$ gauge theory coupled to the Dirac fermion with
$2$-index symmetric representation. This model is denoted as QCD(Sym), and we
set $N\geq 3$ in the following discussion.111For $N=2$, the $2$-index
symmetric representation is nothing but the adjoint representation, which is a
real representation. For $N\geq 3$, the representation is a complex
representation. This causes the significant difference on the global symmetry,
so we only consider the case with $N\geq 3$.  The Lagrangian of QCD(Sym) is
given by

$S=\frac{1}{g^{2}}\int\mathrm{tr}(f\wedge\star
f)+\frac{\mathrm{i}\theta}{8\pi^{2}}\int\mathrm{tr}(f\wedge
f)+\int\overline{\psi}\gamma_{\mu}D_{\mu}\psi,$ (1)

where $f=\mathrm{d}a+\mathrm{i}a\wedge a$ is the $SU(N)$ gauge field strength,
and $D=\mathrm{d}+\mathrm{i}a$ is the covariant derivative. Since $\psi$
belongs to the $2$-index symmetric representation, the fermion kinetic term
takes the following form:

$\overline{\psi}\gamma_{\mu}D_{\mu}\psi=\overline{\psi}\gamma_{\mu}\left(\partial_{\mu}+\mathsf{sym}(\mathrm{i}a_{\mu})\right)\psi.$
(2)

The classical theory has the chiral symmetry, $U(1)_{\mathrm{L}}\times
U(1)_{\mathrm{R}}\simeq[U(1)_{\mathrm{V}}\times
U(1)_{\mathrm{A}}]/\mathbb{Z}_{2}$, but its axial part is explicitly broken by
the Adler-Bell-Jackiw (ABJ) anomaly. Under the $U(1)_{\mathrm{A}}$
transformation,

$\psi\mapsto\mathrm{e}^{\mathrm{i}\alpha\gamma_{5}}\psi,\,\overline{\psi}\mapsto\overline{\psi}\,\mathrm{e}^{\mathrm{i}\alpha\gamma_{5}},$
(3)

the fermionic path-integral measure is changed as

${\mathcal{D}}\overline{\psi}{\mathcal{D}}\psi\mapsto{\mathcal{D}}\overline{\psi}{\mathcal{D}}\psi\exp\left(\mathrm{i}\frac{2\alpha}{8\pi^{2}}\int\mathrm{tr}_{\mathsf{sym}}(f\wedge
f)\right),$ (4)

where $\mathrm{tr}_{\mathsf{sym}}$ denotes the trace operation over the
$2$-index symmetric representation. Using the identity,
$\mathrm{tr}_{\mathsf{sym}}(T^{a}T^{b})=(N+2)\mathrm{tr}(T^{a}T^{b})$, we find
that

${\mathcal{D}}\overline{\psi}{\mathcal{D}}\psi\mapsto{\mathcal{D}}\overline{\psi}{\mathcal{D}}\psi\exp\left(\mathrm{i}\frac{2(N+2)\alpha}{8\pi^{2}}\int\mathrm{tr}(f\wedge
f)\right),$ (5)

and then the measure is invariant if $\alpha\in\frac{2\pi}{2(N+2)}\mathbb{Z}$.
Therefore, the axial symmetry $U(1)_{\mathrm{A}}$ is explicitly broken to its
discrete subgroup, $\bigl{(}\mathbb{Z}_{2(N+2)}\bigr{)}_{\mathrm{A}}$.

We also need to care about the periodicity of the vector symmetry and its
relation to the $1$-form symmetry. When the matter fields are absent, the
model becomes the $SU(N)$ pure Yang-Mills theory, which enjoys the
$\mathbb{Z}_{N}^{[1]}$ symmetry, or the center symmetry. When the $2$-index
fermion is introduced, the vector-like symmetry including the $1$-form
symmetry is given by

$G_{\mathrm{V}}=\left\\{\begin{array}[]{cc}U(1)/\mathbb{Z}_{\frac{N}{2}}\times\mathbb{Z}_{2}^{[1]}&(\mathrm{even}\,N),\\\
U(1)/\mathbb{Z}_{N}&(\mathrm{odd}\,N).\end{array}\right.$ (6)

For even $N$, $\mathbb{Z}_{2}^{[1]}\subset\mathbb{Z}_{N}^{[1]}$ survives as
the global symmetry since the $2$-index representation is neutral, while for
odd $N$, the $1$-form symmetry is completely gone. This also indicates that
the part of the $U(1)_{\mathrm{V}}$ transformation has an overlap with a part
of the center subgroup of the $SU(N)$ gauge transformations, and the faithful
$U(1)$ symmetry group has the fractional periodicity in terms of the quark
language.

In order to discuss the ’t Hooft anomaly, let us introduce the background
gauge field for the vector-like symmetry $G_{\mathrm{V}}$ and we shall check
if the discrete chiral symmetry becomes anomalous (For recent developments of
’t Hooft anomalies, see Refs. Gaiotto:2017yup ; Tanizaki:2017bam ;
Komargodski:2017dmc ; Komargodski:2017smk ; Shimizu:2017asf ; Gaiotto:2017tne
). Taking the above expression for the symmetry structure naively, the
structure of the background gauge fields seems to depend on $N\bmod 2$.
However, thanks to the subtle $2$-group extension for even $N$, the background
gauge fields become almost identical to the case with those for odd $N$. In
order to see this, it is convenient to introduce the $U(1)$ background gauge
field $A$ first, and then the fermion kinetic term becomes

$\overline{\psi}\gamma^{\mu}(\partial_{\mu}+\mathrm{i}(\mathsf{sym}(a_{\mu})+A_{\mu}))\psi.$
(7)

This has the $\mathbb{Z}_{N}^{[1]}$ symmetry as it is invariant under the
following transformation,

$a\mapsto a+\frac{1}{N}\varepsilon,\quad A\mapsto A-2\frac{1}{N}\varepsilon,$
(8)

where $\frac{1}{N}\varepsilon$ is the flat $\mathbb{Z}_{N}$ gauge field.
Therefore, we can introduce the $\mathbb{Z}_{N}$ $2$-form gauge field, which
is realized as the $U(1)$ $2$-form gauge field with the constraint,

$NB=\mathrm{d}C,$ (9)

where $C$ is the auxiliary $U(1)$ gauge field. To have the coupling between
$(B,C)$ and QCD(Sym), we use the trick that $a$ is promoted to the $U(N)$
gauge field $\widetilde{a}$ given by Kapustin:2014gua

$\widetilde{a}=a+\frac{1}{N}C\bm{1}_{N}.$ (10)

This should be understood as the local expression, and what it actually means
is that the path integral $\int{\mathcal{D}}\widetilde{a}$ is performed over
the $U(N)$ gauge fields with the constraint

$\mathrm{tr}(\widetilde{a})=C.$ (11)

We postulate the $U(1)$ $1$-form gauge transformation,

$B\mapsto B+\mathrm{d}\lambda,\,C\mapsto C+N\lambda,$ (12)

and thus

$\widetilde{a}\mapsto\widetilde{a}+\lambda\bm{1}_{N},\,A\mapsto A-2\lambda.$
(13)

All of these operations are valid for both even and odd $N$, and the
background gauge fields for $G_{\mathrm{V}}$ are successfully coupled to
QCD(Sym).

How can we see the difference of the symmetry group in the language of these
background gauge fields? To see this, we have to note that the gauge-invariant
$U(1)$ field strength is given by

$\mathrm{d}A+2B,$ (14)

since $A$ is transformed under the $U(1)$ $1$-form gauge transformation. Since
$B$ is quantized in $\frac{1}{N}$, this does not satisfy the proper Dirac
quantization for $U(1)$ gauge fields. If we denote the corresponding $U(1)$
gauge field with the proper Dirac quantization as $A_{\mathrm{B}}$, then its
field strength should be given by Tanizaki:2018wtg

$\mathrm{d}A_{\mathrm{B}}=\left\\{\begin{array}[]{cc}\frac{N}{2}(\mathrm{d}A+2B)&\quad(\mbox{even}\,N),\\\
N(\mathrm{d}A+2B)&\quad(\mbox{odd}\,N),\end{array}\right.$ (15)

and we here encounter the distinction between even and odd $N$. The
coefficient in front on the right-hand-side discriminates that the minimal
gauge-invariant baryonic operator is given by $\psi^{\otimes(N/2)}$ for even
$N$ while $\psi^{\otimes N}$ for odd $N$. When $N$ is a multiple of $4$, the
baryon operator is bosonic, which corresponds to the fact that the fermion
parity $(-1)^{F}$ is a part of $SU(N)$ gauge redundancy for $N\in
4\mathbb{Z}$.

Next, we discuss the ’t Hooft anomaly for QCD(Sym) at the massless point.
Under the presence of the $G_{\mathrm{V}}$ gauge field, the index of the Dirac
operator
$\not{D}=\gamma^{\mu}(\partial_{\mu}+\mathrm{i}(\mathsf{sym}(\tilde{a}_{\mu})+A_{\mu}))$
is given by

$\displaystyle\mathrm{Ind}(\not{D})$
$\displaystyle=\frac{1}{8\pi^{2}}\int\left(\mathrm{tr}_{\mathsf{sym}}((\widetilde{f}-B)^{2})+\dim(\mathsf{sym})(\mathrm{d}A+2B)^{2}\right)$
$\displaystyle=\frac{1}{8\pi^{2}}\int\left((N+2)\mathrm{tr}((\widetilde{f}-B)^{2})+\frac{N(N+1)}{2}(\mathrm{d}A+2B)^{2}\right)$
$\displaystyle=\frac{N+2}{8\pi^{2}}\int\mathrm{tr}(\widetilde{f}\wedge\widetilde{f})+\frac{1}{8\pi^{2}}\int\left(\frac{N(N+1)}{2}(\mathrm{d}A)^{2}+2(N+1)\mathrm{d}A\wedge
NB+(NB)^{2}\right).$ (16)

Under the discrete axial transformation,

$\bigl{(}\mathbb{Z}_{2(N+2)}\bigr{)}_{\mathrm{A}}:\psi\mapsto\mathrm{e}^{\frac{2\pi}{2(N+2)}\mathrm{i}\gamma_{5}}\psi,\,\overline{\psi}\mapsto\overline{\psi}\,\mathrm{e}^{\frac{2\pi}{2(N+2)}\mathrm{i}\gamma_{5}},$
(17)

the partition function with the background gauge fields, $\mathcal{Z}[A,B]$,
acquires the overall phase factor,

$\displaystyle\bigl{(}\mathbb{Z}_{2(N+2)}\bigr{)}_{\mathrm{A}}:$
$\displaystyle\mathcal{Z}[A,B]\mapsto\exp\left(\frac{2\pi\mathrm{i}}{(N+2)}\mathrm{Ind}(\not{D})\right)\mathcal{Z}[A,B]$
$\displaystyle\quad=\mathrm{e}^{\frac{2\pi\mathrm{i}}{8\pi^{2}(N+2)}\int\left(\frac{N(N+1)}{2}(\mathrm{d}A)^{2}+2(N+1)\mathrm{d}A\wedge
NB+(NB)^{2}\right)}\mathcal{Z}[A,B].$ (18)

This is a special case of the baryon-color(-flavor) anomaly discussed in Refs.
Anber:2019nze ; Anber:2021lzb . If we assume that the low-lying states are
described by color-singlet hadrons and that the anomaly is matched by
spontaneous breaking of the axial symmetry, we can conclude the complete
breakdown,

$\bigl{(}\mathbb{Z}_{2(N+2)}\bigr{)}_{\mathrm{A}}\xrightarrow{\mathrm{SSB}}\mathbb{Z}_{2},$
(19)

and there have to be $(N+2)$ distinct vacua.

###  2.2 Semiclassical study of QCD(Sym) on small $\mathbb{R}^{2}\times T^{2}$
with magnetic flux

Having the knowledge on the ’t Hooft anomaly of QCD(Sym), we construct its
anomaly-preserving $T^{2}$ compactification and study its $2$d effective field
theory. This can be achieved by introducing the magnetic flux along small
$T^{2}$ following Ref. Tanizaki:2022ngt , and it allows us to study the
dynamics of QCD(Sym) in a controllable semiclassical calculation. Therefore,
we take the $4$d spacetime to be $M\times T^{2}$, where $M$ is a sufficiently
large closed manifold and the size of $T^{2}$ is set to be much smaller than
the strong scale. We shall see that the semiclassical computation predicts the
complete spontaneous breakdown of the discrete chiral symmetry.

####  2.2.1 Tree-level analysis

We would like to keep the ’t Hooft anomaly as intact as possible under the
$T^{2}$ compactification Tanizaki:2017qhf ; Tanizaki:2017mtm ;
Yamazaki:2017dra . One of the simplest possibilities for this purpose is to
set

$A=0,\,\,B=\frac{2\pi}{N}\frac{\mathrm{d}x_{3}\wedge\mathrm{d}x_{4}}{L^{2}},$
(20)

where $x_{3}\sim x_{3}+L$ and $x_{4}\sim x_{4}+L$.

To make the connection with the conventional approach, let us write down the
Dirac equation and the boundary condition on $T^{2}$ explicitly. The boundary
condition is denoted as

$\displaystyle\psi(x_{3}+L,x_{4})$
$\displaystyle=\mathsf{sym}(g_{3}(x_{4})^{\dagger})\mathrm{e}^{-\mathrm{i}\phi_{3}(x_{4})}\psi(x_{3},x_{4}),$
(21) $\displaystyle\psi(x_{3},x_{4}+L)$
$\displaystyle=\mathsf{sym}(g_{4}(x_{3})^{\dagger})\mathrm{e}^{-\mathrm{i}\phi_{4}(x_{3})}\psi(x_{3},x_{4}).$
(22)

Since we have introduced the minimal ’t Hooft flux,
$\int_{T^{2}}B=\frac{2\pi}{N}$, the color transition functions must satisfy,

$g_{3}^{\dagger}(L)g_{4}^{\dagger}(0)=g_{4}^{\dagger}(L)g_{3}^{\dagger}(0)\mathrm{e}^{\frac{2\pi}{N}\mathrm{i}}.$
(23)

Then, the uniqueness of the matter wavefunction requires that the $U(1)$
transition functions satisfy

$\exp\Bigl{(}\mathrm{i}\bigl{\\{}(\phi_{4}(L)-\phi_{4}(0))-(\phi_{3}(L)-\phi_{3}(0))\bigr{\\}}\Bigr{)}=\exp\left(-\frac{4\pi}{N}\mathrm{i}\right).$
(24)

The net $U(1)$ gauge field is given by $\widetilde{A}=A+\frac{2}{N}C$, and we
choose the gauge so that

$\widetilde{A}=\frac{4\pi}{N}\frac{x_{3}\mathrm{d}x_{4}}{L^{2}}.$ (25)

The $U(1)$ transition functions are given by
$A(x_{3}+L,x_{4})=A(x_{3},x_{4})+\mathrm{d}\phi_{3}(x_{4})$ and
$A(x_{3},x_{4}+L)=A(x_{3},x_{4})+\mathrm{d}\phi_{4}(x_{3})$, and we obtain

$\phi_{3}(x_{4})=\frac{4\pi}{N}\frac{x_{4}}{L},\,\phi_{4}(x_{3})=0,$ (26)

and the transition functions satisfy the above constraint. For the $SU(N)$
color transition functions, we can perform the gauge transformation so that

$g_{3}(x_{4})=S,\,g_{4}(x_{3})=C,$ (27)

where $C_{nm}\propto\mathrm{e}^{2\pi\mathrm{i}n/N}\delta_{nm}$ and
$S_{nm}\propto\delta_{n+1,m}$ are the clock and shift matrices. This gives a
concrete realization of the background magnetic flux (20). With this choice of
the transition functions, the $SU(N)$ gauge fields on $T^{2}$ obey the
boundary conditions, $a(x_{3}+L,x_{4})=S^{\dagger}a(x_{3},x_{4})S$ and
$a(x_{3},x_{4}+L)=C^{\dagger}a(x_{3},x_{4})C$, and we can specifically have
$a=0$ as the classical minimum. Therefore, for the classical vacuum, the gauge
holonomies and the transition functions are identified, and we have

$P_{3}=S,\,P_{4}=C.$ (28)

Since the $SU(N)$ gauge group is Higgsed to $\mathbb{Z}_{N}$, there are no
perturbatively gapless spectrum in the gauge sector.

Next, we have to discuss the gapless modes of the fermionic fields. To solve
it, we have to solve the zero-mode Dirac equation,

$\left[\gamma_{3}\partial_{3}+\gamma_{4}\left(\partial_{4}+\mathrm{i}\frac{4\pi}{N}\frac{x_{3}}{L^{2}}\right)\right]\psi(x_{3},x_{4})=0,$
(29)

with the above boundary condition. We introduce the $2$d chirality operator in
$T^{2}$ as $\mathrm{i}\gamma_{3}\gamma_{4}$, which satisfies
$(\mathrm{i}\gamma_{3}\gamma_{4})^{2}=1$. To find the number of zero modes, we
can use the $2$d index theorem, which gives

$\mathrm{Ind}(D_{T^{2}})=\dim(\mathrm{Rep})\int_{T^{2}}\frac{F_{U(1)}}{2\pi}.$
(30)

In our case, $\dim(\mathrm{Rep})=\frac{N(N+1)}{2}$ and
$\int\frac{F}{2\pi}=\frac{2}{N}$, so we have $N+1$ flavors of $2$d Dirac
fermions.222For the fundamental representation, we have $\dim(\mathrm{Rep})=N$
and $\int\frac{F}{2\pi}=\frac{1}{N}$, and we get a single $2$d Dirac fermion.
This reproduces the result of our previous analysis on QCD with fundamental
quarks Tanizaki:2022ngt .  We confirm this result in Appendix B by solving the
Dirac equation (29).

####  2.2.2 Perturbative analysis

We have $(N+1)$ flavors of $2$d massless fermions, so we naively have
$U(N+1)_{\mathrm{L}}\times U(N+1)_{\mathrm{R}}$ chiral symmetry. However, this
is absent in $4$d, and thus most part of it should be broken by higher-
dimensional operators in the $2$d effective theory. In $2$d, four-fermion
operators are perturbatively marginal, and they can be marginally relevant.
Therefore, except $U(1)_{\mathrm{A}}$, the axial symmetries would be
explicitly broken by integrating out massive Kaluza-Klein (KK) modes.

Let us perform Abelian bosonizations, then we get $N+1$ compact bosons,
$\phi_{1},\ldots,\phi_{N+1}$. The correspondence is given by

$\overline{\psi_{\mathrm{L}}^{i}}{\psi}_{\mathrm{R}}^{i}\sim\mathrm{e}^{\mathrm{i}\phi_{i}},$
(31)

and the tree-level effective action becomes

$S_{\mathrm{tree}}=\sum_{i=1}^{N+1}\int_{M}\frac{1}{8\pi}|\mathrm{d}\phi_{i}|^{2}.$
(32)

Since the $U(1)_{\mathrm{A}}$ transformation corresponds to
$\phi_{i}\mapsto\phi_{i}+2\alpha$, we can introduce the following perturbation
without breaking $U(1)_{\mathrm{A}}$:

$\Delta S=\int_{M}\mathrm{d}^{2}x\sum_{ij}J_{ij}\cos(\phi_{i}-\phi_{j}).$ (33)

This corresponds to the four-fermion vertices in the fermionic theory.

We note that the one-gluon exchange diagram cannot produce $\Delta S$, and
they are generated by the loop diagrams. The tree-level $4$d gluon exchange
can be decomposed into the $2$d vector-boson exchange and the scalar exchange.
The exchange of $2$d vector boson gives the vector-vector four-fermion
coupling, which modifies the kinetic term in the bosonic description instead
of generating the potential term. Therefore, we need to consider the exchange
of scalar $P_{3},P_{4}$ to obtain $\Delta S$, but it turns out that333To see
this, let us take the specific basis
$\gamma_{i}=\begin{pmatrix}0&\mathrm{i}\sigma_{i}\\\
-\mathrm{i}\sigma_{i}&0\end{pmatrix}$ and $\gamma_{4}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}$. Then, the $2$d chirality operator of $T^{2}$ is given by
$\mathrm{i}\gamma_{3}\gamma_{4}=\begin{pmatrix}-\sigma_{3}&0\\\
0&\sigma_{3}\end{pmatrix}$, and the $2$d massless fermion should take the form
$\psi=(0,*,*,0)^{t}$ to satisfy $\mathrm{i}\gamma_{3}\gamma_{4}=1$. This shows
explicitly that $\gamma_{3,4}\psi=(*,0,0,*)^{t}$, which cannot be the zero
mode by the spinor structure.

$\overline{\psi}\gamma_{3,4}T^{a}\psi=0$ (35)

when $\psi$ is restricted to $2$d massless fermions satisfying
$\mathrm{i}\gamma_{3}\gamma_{4}=1$. In other words, by emitting gluons in the
compactified directions, $P_{3}$ or $P_{4}$, the $2$d fermion has to be
excited to non-zero KK mode. To have the scalar-scalar four-fermion vertex for
$2$d massless fermions, we need to consider the loop diagrams, such as

$\displaystyle\includegraphics[scale={0.4}]{loop.pdf},$ (36)

in which the internal fermion loop runs only over the non-zero KK tower. As a
result, the coupling $J_{ij}$ in $\Delta S$ is of the order of $g(\mu)^{4}$,
where the renormalization scale should be set as $\mu\sim(NL)^{-1}$.

As the four-fermion terms (33) are consistent with the global symmetry, loop
diagrams can produce them. Instead of performing their explicit calculations,
let us simply assume that $\Delta S$ is generated and that its classical
moduli is given by

$\phi_{1}=\phi_{2}=\cdots=\phi_{N+1}.$ (37)

Under this assumption, $\Delta S$ gives the marginally relevant perturbation
to $S_{\mathrm{tree}}$ according to the renormalization-group analysis of the
Berezinskii-Kosterlitz-Thouless (BKT) transition. Let us introduce the
coordinate of the classical moduli,

$\phi\equiv\phi_{i},$ (38)

and then the effective action of the perturbative gapless mode is given by

$S_{\mathrm{pert.}}=\int_{M}\frac{N+1}{8\pi}|\mathrm{d}\phi|^{2}.$ (39)

The zero mode $\phi$ cannot be lifted at any orders of the perturbation
theory.

####  2.2.3 Reproducing ’t Hooft anomaly with the compact boson

This is the good moment to take a detour for checking how the anomaly (18) is
realized in the $2$d effective theory. Let $A_{2\mathrm{d}}$ and
$B_{2\mathrm{d}}$ be the $2$d counterpart of background gauge fields defined
on $M$, then the background gauge fields on $M\times T^{2}$ is given by

$A=A_{2\mathrm{d}},\,B=B_{2\mathrm{d}}+\frac{2\pi}{N}\frac{\mathrm{d}x_{3}\wedge\mathrm{d}x_{4}}{L^{2}}.$
(40)

Substituting this expression, the $4$d mixed ’t Hooft anomaly (18) becomes

$\bigl{(}\mathbb{Z}_{2(N+2)}\bigr{)}_{\mathrm{A}}:\mathcal{Z}[A_{2\mathrm{d}},B_{2\mathrm{d}}]\mapsto\mathrm{e}^{\frac{\mathrm{i}}{(N+2)}\int\left\\{(N+1)\mathrm{d}A_{2\mathrm{d}}+NB_{2\mathrm{d}}\right\\}}\mathcal{Z}[A_{2\mathrm{d}},B_{2\mathrm{d}}],$
(41)

and this gives the ’t Hooft anomaly for $2$d effective theory. The first term
comes from the perturbative axial anomaly, and the coefficient $(N+1)$ counts
the number of $2$d massless Dirac fermions. To reproduce it with compact
bosons, we introduce the coupling

$\sum_{i=1}^{N+1}\frac{\mathrm{i}}{2\pi}\int_{M}\phi_{i}\wedge(\mathrm{d}A_{2\mathrm{d}}+2B_{2\mathrm{d}})=\frac{\mathrm{i}(N+1)}{2\pi}\int_{M}\phi\wedge(\mathrm{d}A_{2\mathrm{d}}+2B_{2\mathrm{d}}).$
(42)

This is the standard topological coupling for the vector-like gauge fields in
the bosonized description, where the vector-like gauge field strength is made
manifestly $1$-form gauge invariant. However, this does not reproduce the
second term of the anomaly correctly and we need an extra contribution. To see
where it comes from, we should note that there is the dynamical
$\mathbb{Z}_{N}$ gauge $a^{\prime}$ with the gas of center-vortex defects:
$\mathrm{d}a^{\prime}$ can be nonzero and
$\int_{V_{c}}\mathrm{d}a^{\prime}=\frac{2\pi}{N}$ for the integration over the
vortex core $V_{c}$. This should couple to the $\theta$ parameter in the
following standard form,

$\frac{\mathrm{i}\theta}{2\pi}\int_{M}\mathrm{d}a^{\prime},$ (43)

but this does not respect the $U(1)_{\mathrm{A}}$ spurious symmetry,
$\theta\to\theta+2(N+2)\alpha$, $\phi\to\phi+2\alpha$, and
$\mathrm{d}a^{\prime}$ is not invariant under $1$-form gauge transformation.
These requirements fix the form of couplings in a unique manner:

$\frac{\mathrm{i}}{2\pi}\int_{M}(\theta-(N+2)\phi)\wedge(\mathrm{d}a^{\prime}+B_{2\mathrm{d}}).$
(44)

Combining (42) and (44), the discrete chiral transformation,
$\phi\to\phi+\frac{2\pi}{N+2}$, shifts the local counter term as

$\frac{\mathrm{i}}{N+2}\int\left((N+1)(\mathrm{d}A_{2\mathrm{d}}+2B_{2\mathrm{d}})-(N+2)B_{2\mathrm{d}}\right)=\frac{\mathrm{i}}{N+2}\int\left((N+1)\mathrm{d}A_{2\mathrm{d}}+NB_{2\mathrm{d}}\right),$
(45)

and this reproduces the ’t Hooft anomaly.

####  2.2.4 Center vortices and chiral symmetry breaking for QCD(Sym)

To go beyond the perturbation theory, we should perform the dilute gas
approximation of the semiclassical configuration. In the $T^{2}$ compactified
setup with the ’t Hooft flux, the BPS bound is given by the center vortex
Gonzalez-Arroyo:1998hjb ; Montero:1999by ; Montero:2000pb , which has the
fractional topological charge $Q_{\mathrm{top}}=\frac{1}{N}$ and the
nontrivial mutual statistics with Wilson loops.444 This nontrivial mutual
statistics is important to describe confinement using center vortices. For
numerical studies of center-vortex induced confinement of $4$d gauge theories,
see Refs. DelDebbio:1996lih ; Faber:1997rp ; DelDebbio:1998luz ;
Langfeld:1998cz ; Kovacs:1998xm ; Engelhardt:1999fd ; deForcrand:1999our , and
for a visual guide, see Biddle:2019gke . From the BPS bound, the YM action is
given by $S_{I}/N=\frac{8\pi^{2}}{g^{2}N}$. The center-vortex vertex is then
given by

$K\mathrm{e}^{-\frac{S_{I}}{N}}\mathrm{e}^{\frac{\mathrm{i}}{N}(\theta-(N+2)\phi)}\sim\frac{(\Lambda
L)^{3-\frac{4}{3N}}}{L^{2}}\mathrm{e}^{\frac{\mathrm{i}}{N}(\theta-(N+2)\phi)},$
(46)

where $K$ denotes the fluctuation determinant. From the dimensional analysis,
$K\sim 1/L^{2}$. This corresponds to the fractionalization of Kobayashi-
Maskawa-’t Hooft vertex Kobayashi:1970ji ; Kobayashi:1971qz ; Maskawa:1974vs ;
tHooft:1976snw .555The six-quark determinant vertex,
$\det_{i,j}(\overline{\psi}^{i}(1+\gamma_{5})\psi_{j})$, was first introduced
in Ref. Kobayashi:1970ji  by Kobayashi and Maskawa to phenomenologically
explain heavy mass of $\eta^{\prime}$, and its connection to the ABJ anomaly
was speculated in Maskawa:1974vs . The concrete mechanism to generate it by
instantons was shown by ’t Hooft in Ref. tHooft:1976snw . Accordingly,
although the scalar field $\phi$ is introduced as the $2\pi$-periodic field,
this expression of the center-vortex vertex does not respect its periodicity.
We shall see that this problem is resolved by considering the dilute gas of
center vortices.

As the gluon sector is perturbatively gapped due to the ’t Hooft twisted
boundary condition, the center vortex has the fixed size, and the dilute gas
approximation is reliable without suffering from infrared divergence. Since
the ’t Hooft twist is introduced only in the $x_{3}$-$x_{4}$ direction, the
total topological charge has to be an integer. The numbers of center vortices
and anti-vortices, $n$ and $\overline{n}$, must satisfy $n-\overline{n}\in
N\mathbb{Z}$ for consistency. The summation over the dilute vortex gas gives

$\displaystyle\sum_{n,\overline{n}\geq 0}\frac{\delta_{n-\overline{n}\in
N\mathbb{Z}}}{n!\overline{n}!}\left(\int\mathrm{d}^{2}xK\mathrm{e}^{-\frac{S_{I}}{N}}\mathrm{e}^{\frac{\mathrm{i}}{N}(\theta-(N+2)\phi)}\right)^{n}\left(\int\mathrm{d}^{2}xK\mathrm{e}^{-\frac{S_{I}}{N}}\mathrm{e}^{-\frac{\mathrm{i}}{N}(\theta-(N+2)\phi)}\right)^{\overline{n}}$
$\displaystyle=\sum_{\ell=0}^{N-1}\exp\left(\int\mathrm{d}^{2}x\,2K\mathrm{e}^{-\frac{S_{I}}{N}}\cos\left(\frac{(N+2)\phi+2\pi\ell-\theta}{N}\right)\right),$
(47)

and this full expression satisfy the $2\pi$-periodicity of the field $\phi$.
However, the local effective potential induced by the center vortex,

$V_{\mathrm{vortex},\ell}(\phi)=-2K\mathrm{e}^{-\frac{S_{I}}{N}}\cos\left(\frac{(N+2)\phi+2\pi\ell-\theta}{N}\right),$
(48)

has an extra label $\ell\in\mathbb{Z}_{N}$ and violates the $2\pi$ periodicity
for fixed $\ell$. A better way to understand its periodicity is given by the
following identification,

$\displaystyle(\phi,\ell+N)\sim(\phi,\ell),$ (49)
$\displaystyle(\phi+2\pi,\ell)\sim(\phi,\ell+(N+2))\sim(\phi,\ell+2).$ (50)

When $N$ is odd, we can eliminate the label $\ell$ completely by extending the
periodicity of $\phi$. That is, the branch label can be absorbed by $\phi$ but
the price to pay is to change the periodicity as $\phi\sim\phi+2\pi N$. As a
result, we can set $\ell=0$, and the ground-states are given by

$\phi=\frac{\theta+2\pi Nk}{(N+2)},$ (51)

with $k\sim k+(N+2)$. Thus, we have $(N+2)$ vacua with $k=0,1,\ldots,(N+2)-1$.
Accordingly, the chiral condensate of the $k$-th vacuum is given by

$\langle\overline{\psi}_{\mathrm{L}}\psi_{\mathrm{R}}\rangle_{k}\sim\Lambda^{3}(\Lambda
L)^{-\frac{4}{3N}}\mathrm{e}^{\mathrm{i}\frac{\theta+2\pi Nk}{(N+2)}},$ (52)

and the discrete chiral symmetry is spontaneously broken completely. This
result matches exactly to the one in QCD(Sym) on $\mathbb{R}^{3}\times S^{1}$
Shifman:2008ja , but interestingly the underlying microscopic mechanisms
(center vortices vs. monopole-instantons) are genuinely distinct. This suggest
a nontrivial adiabatic continuity between these two reliable semi-classical
regimes.

When $N$ is even, the label $\ell$ cannot be eliminated completely. The
periodicity of $\phi$ is extended as $\phi\sim\phi+2\pi\frac{N}{2}$, and we
still have the discrete label $\ell\in\\{0,1\\}$. The presence of the
$\mathbb{Z}_{2}$ discrete label $\ell$ is the consequence of the $1$-form
symmetry, $\mathbb{Z}_{2}^{[1]}(\subset\mathbb{Z}_{N}^{[1]})$, and the branch
label $\ell$ discriminates the universes666The $1$-form symmetry in $2$d QFT
requires the presence of topological local operators, and we can consider the
superselection rule associated with those topological local operators. As the
superselection rule in this case is much stronger than the ordinary one for
spontaneous symmetry breaking, the corresponding superselction sectors are
sometimes called as universes to emphasize it.  in the $2$d effective theory
Pantev:2005zs ; Hellerman:2006zs ; Hellerman:2010fv ; Tanizaki:2019rbk ;
Cherman:2019hbq ; Komargodski:2020mxz . That is, the ground states are given
by

$(\phi,\ell)=\left(\frac{\theta+2\pi
Nk}{N+2},0\right),\left(\frac{\theta+2\pi+2\pi Nk}{N+2},1\right),$ (53)

with $k\sim k+\frac{N+2}{2}$. There are two universes labelled by $\ell=0,1$,
and each universe has $\frac{N+2}{2}$ vacua labelled by
$k=0,1,\ldots,\frac{N}{2}$. As a result, we get $(N+2)$ ground states, and the
chiral symmetry is spontaneously broken completely.

We note that the fundamental Wilson line is a genuine line operator that
transforms under $\mathbb{Z}_{2}^{[1]}$ when $N$ is even, and the universe
label $\ell$ jumps to $\ell+1$ by going across the Wilson line. Since two
universes have degenerate ground-state energy densities, it obeys the
perimeter law, and $\mathbb{Z}_{2}^{[1]}$ is spontaneously broken. This
however does not lead extra ground-state degeneracy because of the mixed
anomaly between $\mathbb{Z}_{2}^{[1]}$ and the discrete axial symmetry
$(\mathbb{Z}_{2(N+2)})_{\mathrm{A}}$. The number of ground states is given by
$N+2$ as discussed above.

###  2.3 Mass deformation of QCD(Sym) on $M\times T^{2}$

So far, we have considered the massless fermion and obtained the spontaneous
chiral symmetry breaking,
$\left(\mathbb{Z}_{2(N+2)}\right)_{\mathrm{A}}\xrightarrow{\mathrm{SSB}}\mathbb{Z}_{2}$,
using the semiclassical method. Let us add the mass deformation to the
Lagrangian,

$\Delta\mathcal{L}_{m}=m\left((\overline{\psi}_{\rm L}\psi_{\rm
R})+(\overline{\psi}_{\rm R}\psi_{\rm L})\right),$ (54)

and assume that the mass $m$ is sufficiently small. The deformation breaks the
$\mathbb{Z}_{2N+4}^{[0]}$ discrete chiral symmetry softly. After the
bosonization, the mass deformation becomes

$\Delta\mathcal{L}_{m}\sim-m\frac{(\Lambda
L)^{3-\frac{4}{3N}}}{L^{3}}\cos(\phi).$ (55)

The ground-state energy densities can be obtained by substituting the vacuum
configurations for $\phi$ at $m=0$.

For odd $N$, we then obtain

$E_{k}(\theta)\sim-m\Lambda^{3}(\Lambda
L)^{-\frac{4}{3N}}\cos\left(\frac{\theta+2\pi Nk}{N+2}\right),$ (56)

with $k=0,1,\ldots,N+1$. At generic values of $\theta$, the system has the
unique and gapped ground state. That is, the degeneracy between chirally
broken vacua is lifted by the mass deformation. Concretely, the ground state
is given by $k=0$ when $-\pi<\theta<\pi$. At $\theta=\pi$, there are two
degenerate vacua, $k=0$ and $Nk=-1$. Here, we note that $Nk=-1$ has the unique
solution in $k\in\mathbb{Z}_{N+2}$ as $\gcd(N,N+2)=1$ for odd $N$.

For even $N$, there are two universes due to $\mathbb{Z}_{2}^{[1]}$ labelled
by $\ell\in\\{0,1\\}$. For the $\ell=0$ universe, the multi-branch ground-
state energies are given by

$E^{(\ell=0)}_{k}(\theta)\sim-m\Lambda^{3}(\Lambda
L)^{-\frac{4}{3N}}\cos\left(\frac{\theta+2\pi Nk}{N+2}\right),$ (57)

with $k=0,1,\ldots,\frac{N}{2}$. For the $\ell=1$ universe, they are given by

$E^{(\ell=1)}_{k^{\prime}}(\theta)\sim-m\Lambda^{3}(\Lambda
L)^{-\frac{4}{3N}}\cos\left(\frac{\theta+2\pi(Nk^{\prime}+1)}{N+2}\right),$
(58)

with $k^{\prime}=0,1,\ldots,\frac{N}{2}$. At generic values of $\theta$, the
system again has the unique and gapped ground state. As a result, the
fundamental Wilson line shows the area law. The string tension $T(\theta)$ of
the fundamental Wilson loop is given by

$T(\theta)=L^{2}\left(\min_{k^{\prime}}E^{(\ell=1)}_{k^{\prime}}(\theta)-\min_{k}E^{(\ell=0)}_{k}(\theta)\right),$
(59)

and it is indeed nonzero for generic values of $\theta$.

###  2.4 Semiclassical study of QCD(ASym) on small $\mathbb{R}^{2}\times
T^{2}$ with magnetic flux

In this section, we study the dynamics of QCD(ASym) using the center-vortex
effective theory. As the discussion is almost parallel to that of QCD(Sym), we
explain it very briefly.

The structure of the vector-like symmetry $G_{\mathrm{V}}$ is completely
identical to (6) including the form of its background gauge field. The ABJ
anomaly explicitly breaks $U(1)_{\mathrm{A}}$ to
$\left(\mathbb{Z}_{2(N-2)}\right)_{\mathrm{A}}$ since
$\mathrm{tr}_{\mathsf{asym}}(T^{a}T^{b})=(N-2)\mathrm{tr}(T^{a}T^{b})$, and
thus the factor “$(N+2)$” in the analysis of QCD(Sym) should be replaced by
“$(N-2)$” in the case of QCD(ASym). As the dimension of the anti-symmetric
representation is $\dim(\mathsf{asym})=\frac{N(N-1)}{2}$, the factor “$(N+1)$”
in QCD(Sym) is also replaced by “$(N-1)$”.

To compute the mixed ’t Hooft anomaly between $G_{\mathrm{V}}$ and the
discrete chiral symmetry, let us calculate the index of the Dirac operator
under the presence of the $G_{\mathrm{V}}$ background gauge field $(A,B)$:

$\displaystyle\mathrm{Ind}(\not{D})$
$\displaystyle=\frac{1}{8\pi^{2}}\int\left(\mathrm{tr}_{\mathsf{asym}}((\widetilde{f}-B)^{2})+\dim(\mathsf{asym})(\mathrm{d}A+2B)^{2}\right)$
$\displaystyle=\frac{1}{8\pi^{2}}\int\left((N-2)\mathrm{tr}((\widetilde{f}-B)^{2})+\frac{N(N-1)}{2}(\mathrm{d}A+2B)^{2}\right)$
$\displaystyle=\frac{N-2}{8\pi^{2}}\int\mathrm{tr}(\widetilde{f}\wedge\widetilde{f})+\frac{1}{8\pi^{2}}\int\left(\frac{N(N-1)}{2}(\mathrm{d}A)^{2}+2(N-1)\mathrm{d}A\wedge
NB+(NB)^{2}\right).$ (60)

As a result, under the discrete axial transformation,

$\bigl{(}\mathbb{Z}_{2(N-2)}\bigr{)}_{\mathrm{A}}:\psi\mapsto\mathrm{e}^{\frac{2\pi}{2(N-2)}\mathrm{i}\gamma_{5}}\psi,\,\overline{\psi}\mapsto\overline{\psi}\,\mathrm{e}^{\frac{2\pi}{2(N-2)}\mathrm{i}\gamma_{5}},$
(61)

the partition function of QCD(ASym) is shifted by the local counterterm of the
$G_{\mathrm{V}}$ background field:

$\displaystyle\bigl{(}\mathbb{Z}_{2(N-2)}\bigr{)}_{\mathrm{A}}:$
$\displaystyle\mathcal{Z}[A,B]\mapsto\mathrm{e}^{\frac{2\pi\mathrm{i}}{8\pi^{2}(N-2)}\int\left(\frac{N(N-1)}{2}(\mathrm{d}A)^{2}+2(N-1)\mathrm{d}A\wedge
NB+(NB)^{2}\right)}\mathcal{Z}[A,B].$ (62)

This is the mixed ’t Hooft anomaly, and the anomaly matching condition can be
satisfied by the complete spontaneous breakdown of the discrete axial
symmetry,

$\left(\mathbb{Z}_{2(N-2)}\right)_{\mathrm{A}}\xrightarrow{\mathrm{SSB}}\mathbb{Z}_{2}.$
(63)

In order to apply the semiclassical description with center vortices, we would
like to perform the $T^{2}$ compactification to maintain the above ’t Hooft
anomaly as much as possible. To this end, we again consider the magnetic flux
background (20). It effectively introduces the minimal ’t Hooft flux in the
gauge sector, and thus the gauge field has the perturbative mass gap because
of the Higgsing $SU(N)\to\mathbb{Z}_{N}$. For Dirac fermions, there are
perturbative gapless modes following the index theorem on $T^{2}$ given by
(30), and it predicts the presence of $(N-1)$ flavors of $2$d Dirac fermions
at the classical level.

By performing the Abelian bosonization, $2$d effective theory at the classical
level is given by the theory of $2\pi$-periodic compact scalars, $\phi_{i}$,
with $i=1,\ldots,N-1$. The $U(1)_{\mathrm{A}}$ transformation acts as
$\phi_{i}\mapsto\phi_{i}+2\alpha$, and other axial symmetry in the classical
analysis should be eliminated by the four-fermion vertex (33), which can be
obtained by integrating out massive KK towers. We again assume that its
classical moduli is given by $\phi\equiv\phi_{1}=\cdots=\phi_{N-1}$, and then
we only have a single compact scalar as a result of the perturbative analysis.

The center-vortex vertex takes the following form,

$K\mathrm{e}^{-\frac{S_{I}}{N}}\mathrm{e}^{\frac{\mathrm{i}}{N}(\theta-(N-2)\phi)}\sim\frac{(\Lambda
L)^{3+\frac{4}{3N}}}{L^{2}}\mathrm{e}^{\frac{\mathrm{i}}{N}(\theta-(N-2)\phi)},$
(64)

and this satisfies the spurious symmetry, $\phi\to\phi+2\alpha$ associated
with $\theta\to\theta+2(N-2)\alpha$. By performing the dilute gas
approximation of center vortices, the effective potential,

$V_{\mathrm{vortex},\ell}(\phi)=-2K\mathrm{e}^{-\frac{S_{I}}{N}}\cos\left(\frac{(N-2)\phi+2\pi\ell-\theta}{N}\right),$
(65)

is generated with the discrete label, $\ell\in\mathbb{Z}_{N}$. The periodicity
of the field $\phi$ is then understood as

$\displaystyle(\phi,\ell+N)\sim(\phi,\ell),$ (66)
$\displaystyle(\phi+2\pi,\ell)\sim(\phi,\ell+(N-2))\sim(\phi,\ell-2).$ (67)

With this identification, the field $\phi$ is extended to $2\pi N$-periodic
field for odd $N$, and the discrete label $\ell$ is completely eliminated. For
even $N$, the periodicity of $\phi$ is $2\pi\frac{N}{2}$, and $\ell$ takes
values in $\mathbb{Z}_{2}$. In this case, the label $\ell$ discriminates two
universes of the $1$-form symmetry, $\mathbb{Z}_{2}^{[1]}$, in the $2$d
effective theory. In both cases, we get $(N-2)$ degenerate vacua, which shows
the spontaneous breakdown of discrete chiral symmetry.

##  3 Implications on the large-$N$ orbifold/orientifold equivalence

There is a conjecture called as the large-$N$ orientifold equivalence
Armoni:2003gp ; Armoni:2004ub , which states that $C$-invariant bosonic
sectors of QCD(Sym/ASym) becomes equivalent to those of $\mathcal{N}=1$
$SU(N)$ super Yang-Mills (SYM) theory in the large-$N$ limit. This is a
striking conjecture, which is interesting and important not only theoretically
but also phenomenologically. For $N=3$, two-index anti-symmetric
representation becomes identical to the fundamental representation, and thus
QCD(ASym) for $N=3$ is the ordinary QCD that describes the strong interaction
of our world. If the conjecture is true and $N=3$ is regarded as a large
parameter, the orientifold equivalence may provide a direct path to apply the
knowledge of supersymmetric gauge theories to the actual QCD Armoni:2003fb ;
Armoni:2003yv .

It is therefore an important question if the large-$N$ orientifold equivalence
is true at the nonperturbative level. In Ref. Unsal:2006pj , it is uncovered
that the unbroken charge-conjugation symmetry is the necessary and sufficient
condition for its validity. In order to see this, it is useful to think of
both $\mathcal{N}=1$ $SU(N)$ SYM and QCD(ASym) as $\mathbb{Z}_{2}$ projections
of $\mathcal{N}=1$ $SO(2N)$ SYM theory. To obtain $\mathcal{N}=1$ $SU(N)$ SYM
theory from $SO(2N)$ theory, we introduce the symplectic matrix
$J=\mathrm{i}\sigma_{2}\otimes\bm{1}_{N}\in SO(2N)$ and impose the constraint
$A_{\mu}=JA_{\mu}J^{t}$ and $\lambda=J\lambda J^{t}$ for the gluons and
gluinos of $SO(2N)$ theory. Then, the gauge group becomes $U(N)$ and $\lambda$
becomes the Weyl fermion in the adjoint representation, so we obtain
$\mathcal{N}=1$ $U(N)$ SYM theory. To obtain QCD(ASym), we replace the
constraint for the gluino as $\lambda=-J\lambda J^{t}$, then the fermion
becomes the Dirac fermion in the anti-symmetric representation of $U(N)$. In
the large-$N$ limit, the difference between $U(N)$ and $SU(N)$ becomes
negligible, and we obtain the correspondence. In this way, the large-$N$
orientifold equivalence proposed in Refs. Armoni:2003gp ; Armoni:2004ub  has
been decomposed into two large-$N$ orbifold-type equivalences
Bershadsky:1998cb ; Schmaltz:1998bg ; Kachru:1998ys ; Strassler:2001fs
between the parent and daughter theories with the same parent theory: It is an
example of the daughter-daughter equivalence.

The necessary and sufficient condition for the large-$N$ orbifold equivalence
has been derived in Refs. Kovtun:2003hr ; Kovtun:2004bz ; Kovtun:2005kh , and
it shows that the equivalence holds for correlation functions of neutral
sectors if and only if the ground state also belongs to the neutral sector.
Applying it to the present case, the ground states must have unbroken charge-
conjugation symmetry, and then the orientifold equivalence is true for
observables in the $C$-even bosonic sector. Since it is widely accepted that
$\mathcal{N}=1$ SYM theory does not break $C$ while it breaks the discrete
axial symmetry, the question becomes whether QCD(ASym) maintains $C$-symmetry
or not. This is a nontrivial question as it involves the strong dynamics of
$4$d gauge theories.

So far, we pay attention to QCD(ASym), but we can perform the same analysis
for QCD(Sym) starting from the $\mathcal{N}=1$ $USp(2N)$ SYM theory, and we do
not repeat it here. Again, the criterion for the large-$N$ orientifold
equivalence is given by the unbroken $C$-symmetry for QCD(Sym).

Our analysis on small $\mathbb{R}^{2}\times T^{2}$ predicts that $C$ symmetry
is unbroken. We have also observed the complete spontaneous breakdown of
discrete chiral symmetry, which suggests the adiabatic continuity between our
semiclassical description and the strong dynamics on $\mathbb{R}^{4}$. This
gives positive support for the nonperturbative orientifold equivalence.

We would like to emphasize the magnetic flux on small $T^{2}$ plays the
pivotal role to achieve the adiabatic continuity Tanizaki:2022ngt  (see also
Refs. vanBaal:1984ra ; Yamazaki:2017ulc ; Cox:2021vsa ). In Ref. Unsal:2006pj
, QCD(Sym/ASym) on small $\mathbb{R}^{3}\times S^{1}$ has been studied, and it
was found that these theories have chiral symmetric vacua with broken $C$
symmetry. Since the anomaly matching on $\mathbb{R}^{4}$ requires the
spontaneous breaking of chiral symmetry, this suggests that the naive
compactification must have phase transitions and it is not suitable to study
the dynamics of strongly coupled regimes.

In order to achieve the adiabatic continuity on $\mathbb{R}^{3}\times S^{1}$,
the double-trace deformed gauge theories are more suitable choice Unsal:2007vu
; Unsal:2007jx ; Unsal:2008ch . By adding the double-trace deformations to
QCD(ASym) on $\mathbb{R}^{3}\times S^{1}$, we can find that $C$ becomes
unbroken and instead the chiral symmetry is spontaneously broken
Shifman:2008ja . Therefore, in both $S^{1}$ and $T^{2}$ compactifications with
adiabatic continuity, we obtained the evidence for unbroken $C$ symmetry,
suggesting the validity of nonperturbative large-$N$ orientifold equivalence.

##  4 Summary

We performed the semiclassical analysis of QCD(Sym/ASym) on
$\mathbb{R}^{2}\times T^{2}$ by introducing the magnetic flux in $T^{2}$.
QCD(Sym/ASym) have the mixed ’t Hooft anomaly between its vector symmetry and
discrete chiral symmetry, and the magnetic flux in $T^{2}$ allows us to
preserve the information of ’t Hooft anomaly. As a result, the dynamics of the
$2$d effective theory is constrained by the same anomaly of $4$d
QCD(Sym/ASym).

We find that discrete chiral symmetry is spontaneously broken completely
thanks to the dilute gas of center vortices, and this satisfies the anomaly
matching constraint. This leads to the multi-branch structure of the ground
states after the tiny mass deformation, and this leads to the same vacuum
structure with the pure YM theory.

Each ground state has unbroken $C$ symmetry, and this gives an important
implication for the large-$N$ orientifold equivalence between QCD(Sym/ASym)
and $\mathcal{N}=1$ $SU(N)$ SYM theory. Since the unbroken $C$ symmetry gives
the necessary and sufficient condition for its nonperturbative justification,
our result gives a strong positive support for the nonperturbative orientifold
equivalence, assuming the adiabatic continuity.

As we showed in this work, the chiral condensate obtained in the small
symmetric $T^{2}$ regime (52) matches exactly the result on
$\mathbb{R}^{3}\times S^{1}_{\rm small}$ in Shifman:2008ja . At the same time,
the microscopic configurations which lead to this result are genuinely
distinct, even though they have the same topological charge $Q=1/N$ and
action. Monopole-instantons and center vortices have different mutual
statistics with Wilson loops. It would be very interesting to establish how
these two reliable semiclassical regimes are adiabatically connected.

Lastly, let us comment on several future directions. One of them is about the
domain wall that connects different chiral broken vacua. As the phase of
chiral condensate can be regarded as an effective $\theta$ angle via the ABJ
anomaly, the domain-wall theory can be thought of $3$d QCD(Sym/ASym) with a
Chern-Simons term. This problem has been studied in Ref. Choi:2018tuh , and
they suggest the domain wall connecting $n$-th and $0$-th chiral broken vacua
is described by $U(n)_{(N\pm 2)-n,(N\pm 2)-2n}$ Chern-Simons theory, where
$n=1,2,\ldots,(n\pm 2)-1$. It would be an interesting future work to study if
the $T^{2}$ compactification with magnetic flux can give a nontrivial check of
this proposal on the wall.

Another curious problem is to extend our work of vector-like QCD to some
chiral gauge theories Bolognesi:2017pek ; Bolognesi:2019wfq ;
Bolognesi:2019fej ; Sulejmanpasic:2020zfs . Recently, Ref. Anber:2021iip
considers the $4$d chiral gauge theories that contain both two-index symmetric
and anti-symmetric Weyl fermions, and the anomaly matching constraint is
studied in detail both on spin and non-spin manifolds. It would be fascinating
if we can check those scenarios (even partially) by the reliable semiclassics
with $T^{2}$ compactification, although the extension might not be
straightforward due to the lack of the vector-like $U(1)_{\mathrm{B}}$
symmetry.

###### Acknowledgements.

This work was completed during the workshop “Continuous Advances in QCD 2022”
(CAQCD2022), and the authors thank Aleksey Cherman and Misha Shifman for the
kind invitation and appreciate their hospitality. The work of Y. T. was
supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant
numbers, 22H01218 and 20K22350, and by Center for Gravitational Physics and
Quantum Information (CGPQI) at Yukawa Institute for Theoretical Physics. The
work of M. Ü. was supported by U.S. Department of Energy, Office of Science,
Office of Nuclear Physics under Award Number DE-FG02-03ER41260.

##  Appendix A Group theoretic factors of two-index representations

Let $T^{a}_{\mathsf{rep}}$ be the generators of $\mathfrak{su}(N)$ Lie algebra
with the representation $\mathsf{rep}$. We take the orthonormal basis,

$\mathrm{tr}(T^{a}_{\mathsf{rep}}T^{b}_{\mathsf{rep}})=T(\mathsf{rep})\delta^{ab},$
(68)

where $T(\mathsf{rep})$ is the trace normalization factor. For the defining
(or fundamental) representation, $\mathsf{rep}=\mathsf{fd}$, we set

$T(\mathsf{fd})=\frac{1}{2}.$ (69)

The quadratic Casimir invariant $C_{2}(\mathsf{rep})$ is defined by

$\sum_{a=1}^{N^{2}-1}T^{a}_{\mathsf{rep}}T^{a}_{\mathsf{rep}}=C_{2}(\mathsf{rep})\,\mathrm{id}_{\mathsf{rep}}.$
(70)

By taking the trace of both sides, $C_{2}(\mathsf{rep})$ is related to
$T(\mathsf{rep})$ by the formula,

$C_{2}(\mathsf{rep})\dim(\mathsf{rep})=(N^{2}-1)T(\mathsf{rep}).$ (71)

Let us write down the table of these factors for the fundamental, adjoint,
two-index symmetric and antisymmetric representations:

$\displaystyle\begin{array}[]{c|ccc}\mathsf{rep}&\dim(\mathsf{rep})&T(\mathsf{rep})&C_{2}(\mathsf{rep})\\\
\hline\cr\mathsf{fd}&N&\frac{1}{2}&\frac{N^{2}-1}{2N}\\\
\mathsf{adj}&N^{2}-1&N&N\\\
\mathsf{sym}&\frac{N(N+1)}{2}&\frac{N+2}{2}&\frac{(N-1)(N+2)}{N}\\\
\mathsf{asym}&\frac{N(N-1)}{2}&\frac{N-2}{2}&\frac{(N+1)(N-2)}{N}\end{array}$
(77)

The perturbative beta function,
$\beta(g)=\frac{\mathrm{d}g}{\mathrm{d}\ln\mu}$, is given by

$\beta(g)=-\frac{\beta_{0}}{(4\pi)^{2}}g^{3}-\frac{\beta_{1}}{(4\pi)^{4}}g^{5}+O(g^{7}).$
(78)

The leading coefficient is

$\displaystyle\beta_{0}$
$\displaystyle=\frac{11}{3}N-\frac{4}{3}T(\mathsf{rep})N_{f},$
$\displaystyle\beta_{1}$
$\displaystyle=\frac{34}{3}N^{2}-\frac{20}{3}NT(\mathsf{rep})N_{f}-4C_{2}(\mathsf{rep})T(\mathsf{rep})N_{f},$
(79)

for $SU(N)$ gauge theory with $N_{f}$ Dirac fermions in the representation
$\mathsf{rep}$. The normalization of the coefficient is the standard one for
non-supersymmetric gauge theories, and then the instanton factor within the
$1$-loop renormalization group is given by

$\exp\left(-\frac{8\pi^{2}}{g^{2}(\mu)}\right)=\left(\frac{\Lambda}{\mu}\right)^{\beta_{0}},$
(80)

where $\Lambda$ is the strong scale. For $1$-flavor QCD(Sym/Asym), $\beta_{0}$
and $\beta_{1}$ are given by

$\displaystyle\beta_{0}$ $\displaystyle=3N\mp\frac{4}{3},$ (81)
$\displaystyle\beta_{1}$
$\displaystyle=6N^{2}\mp\left(\frac{38}{3}N-\frac{8}{N}\right),$ (82)

and they are identical to those of $\mathcal{N}=1$ super Yang-Mills theory in
the large-$N$ limit, $\beta_{0}=3N$ and $\beta_{1}=6N^{2}$.

##  Appendix B Dirac zero modes on $T^{2}$ with the magnetic flux

In this Appendix, we confirm the result of the index theorem (30) by solving
the Dirac zero-mode equation (29) explicitly. We first define the chirality of
small $T^{2}$ by

$(\mathrm{i}\gamma_{3}\gamma_{4})\psi_{\pm}=\pm\psi_{\pm},$ (83)

and then the Dirac equation for each chirality becomes

$\left[\partial_{3}\mp\mathrm{i}\left(\partial_{4}+\mathrm{i}\frac{4\pi}{N}\frac{x_{3}}{L^{2}}\right)\right]\psi_{\pm}=0,$
(84)

or

$(\partial_{3}\mp\mathrm{i}\partial_{4})\left(\mathrm{e}^{\pm\frac{2\pi}{N}\frac{x_{3}^{2}}{L^{2}}}\psi_{\pm}\right)=0.$
(85)

This shows that

$\psi_{\pm}(x_{3},x_{4})=\mathrm{e}^{\mp\frac{2\pi}{N}\frac{x_{3}^{2}}{L^{2}}}u_{\pm}((x_{3}\mp\mathrm{i}x_{4})/L).$
(86)

We can check that there is no zero modes for
$\mathrm{i}\gamma_{3}\gamma_{4}=-1$ by the following equality,

$\displaystyle 0$
$\displaystyle=\int_{T^{2}}\left|\left(\partial_{3}\mp\mathrm{i}\left(\partial_{4}+\mathrm{i}Bx_{3}\right)\right)\psi_{\pm}\right|^{2}$
$\displaystyle=\int_{T^{2}}\left[|\partial_{3}\psi_{\pm}|^{2}+|(\partial_{4}+\mathrm{i}Bx_{3})\psi_{\pm}|^{2}\mp
B|\psi_{\pm}|^{2}\right].$ (87)

Here, we set $B=\frac{4\pi}{NL^{2}}$ for simplicity of notation. In the first
equality, we use the equation for Dirac zero modes. The second equality can be
obtained using integration by parts. For $\mathrm{i}\gamma_{3}\gamma_{4}=-1$,
all the three terms are positive semi-definite, and it requires that
$\psi_{-}=0$. We therefore only consider the case
$\mathrm{i}\gamma_{3}\gamma_{4}=1$ in the following discussion.

Introducing the dimensionless coordinate $z=(x_{3}-\mathrm{i}x_{4})/L$, we can
rewrite the boundary condition as

$\displaystyle
u_{+}(z+1)=\mathsf{sym}(S^{\dagger})\mathrm{e}^{\frac{2\pi}{N}+\frac{4\pi}{N}z}u_{+}(z),$
(88) $\displaystyle u_{+}(z-\mathrm{i})=\mathsf{sym}(C^{\dagger})u_{+}(z).$
(89)

Writing color components explicitly, we find that

$\displaystyle
u_{+,ab}(z+1)=\mathrm{e}^{\frac{2\pi}{N}+\frac{4\pi}{N}z}u_{+,a-1\,b-1}(z),$
(90) $\displaystyle
u_{+,ab}(z-\mathrm{i})=\mathrm{e}^{-\frac{2\pi\mathrm{i}}{N}((a-1)+(b-1))}u_{+,ab}(z).$
(91)

We can perform the Fourier expansion to satisfy (91),

$u_{+,ab}(z)=\mathrm{e}^{\frac{2\pi}{N}((a-1)+(b-1))z}\sum_{n}c_{n,ab}\mathrm{e}^{2\pi
nz}.$ (92)

We then use the boundary condition (90) for the $x_{3}$ direction. When
$1<a,b\leq N$, the condition becomes

$\mathrm{e}^{\frac{2\pi}{N}(a-1+b-1)(z+1)}\sum_{n}c_{n,ab}\mathrm{e}^{2\pi
n(z+1)}=\mathrm{e}^{\frac{2\pi}{N}+\frac{4\pi}{N}z}\mathrm{e}^{\frac{2\pi}{N}(a-2+b-2)z}\sum_{n}c_{n,a-1\,b-1}\mathrm{e}^{2\pi
nz},$ (93)

and we have

$c_{n,ab}=\mathrm{e}^{\frac{2\pi}{N}(1-(a-1)-(b-1))-2\pi n}c_{n,a-1\,b-1}.$
(94)

When $a=1$ but $b>1$, the condition becomes

$\mathrm{e}^{\frac{2\pi}{N}(0+b-1)(z+1)}\sum_{n}c_{n,1\,b}\mathrm{e}^{2\pi
n(z+1)}=\mathrm{e}^{\frac{2\pi}{N}+\frac{4\pi}{N}z}\mathrm{e}^{\frac{2\pi}{N}(N-1+b-2)z}\sum_{n}c_{n,N\,b-1}\mathrm{e}^{2\pi
nz},$ (95)

and we have

$c_{n,1\,b}=\mathrm{e}^{\frac{2\pi}{N}(1-(b-1))-2\pi n}c_{n-1,N\,b-1}.$ (96)

Similarly, when $a>1$ and $b=1$,

$c_{n,a\,1}=\mathrm{e}^{\frac{2\pi}{N}(1-(a-1))-2\pi n}c_{n-1,a-1\,N}.$ (97)

When $a=b=1$, we get

$c_{n,11}=\mathrm{e}^{\frac{2\pi}{N}-2\pi n}c_{n-2,NN}.$ (98)

This gives $c_{n,a\,b}\propto\exp\left(-\frac{2\pi
N}{4}n^{2}-\pi((a-1)+(b-1))n\right)$, and we can find that the solutions can
be expressed with theta functions.777The Fourier series expansion of Jacobi’s
theta function is given by
$\theta_{3}(z)\equiv\theta_{3}(z|\tau)=\sum_{n\in\mathbb{Z}}\mathrm{e}^{\mathrm{i}\pi\tau
n^{2}+2\mathrm{i}nz}=1+2\sum_{n\geq 1}\mathrm{e}^{\mathrm{i}\pi\tau
n^{2}}\cos(2nz)$ with $\mathrm{Im}(\tau)>0$. Importantly, it satisfies the
quasi-periodicity, $\theta_{3}(z+(m+n\tau)\pi)=\mathrm{e}^{-\mathrm{i}\pi\tau
n^{2}-2\mathrm{i}nz}\theta_{3}(z)$, and it is the key property to solve the
quasi-periodicity of $u_{+}$ coming out of (90), $u_{+}(z+N)=\mathrm{e}^{2\pi
N+4\pi z}u_{+}(z)$, combined with (91). We note that the theta functions are
holomorphic, i.e. do not have poles or singularities, and thus they give
normalizable zero modes.

Let us count the number of zero modes. By the shift matrix, $a-b\bmod N$ is
preserved, so the diagonal directions are related in mod $N$. That means, when
we consider $k$-diagonal from the top-left to the bottom-right (i.e.
$c_{n,i\,i+k}$ wit $i:1\to N-k$), we jump to the $-(N-k)$-diagonal elements
(i.e. $c_{n+1,N-k+j\,j}$ with $j:1\to k$), and we come back to the
$k$-diagonal elements, $c_{n+2,1\,1+k}$. As a result, $c_{n,ab}$ is related to
$c_{n\pm 2,ab}$, which means that we can have $2$ zero modes, corresponding to
even and odd $n$: For example, the $1$-$1$ component is given by

$u_{+,11}(z)=a\sum_{n\in 2\mathbb{Z}}\mathrm{e}^{-\frac{2\pi
N}{4}n^{2}}\mathrm{e}^{2\pi nz}+b\sum_{n\in
2\mathbb{Z}+1}\mathrm{e}^{-\frac{2\pi N}{4}n^{2}}\mathrm{e}^{2\pi nz},$ (99)

where $a,b\in\mathbb{C}$. For the main diagonal, we thus have $2$ zero modes,
but for off-diagonals, the symmetry condition imposes the restriction, and
thus we only have $1$ zero mode for each. In total, we get $2+(N-1)=N+1$ zero
modes as indicated by the index theorem (30).

The discussion is completely parallel also for the case of the two-index anti-
symmetric representation. The crucial difference appears only in the last step
and it comes from the absence of the diagonal elements. As a result, we get
$0+(N-1)=N-1$ zero modes.

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