Discrete Anomaly Matching

for the Pouliot Type Dualities

Teruhiko Kawano

Department of Physics, Kyoto Prefectural University of Medicine, Kyoto
606-0823, Japan

We compute the ’t Hooft anomalies of discrete symmetries in the Pouliot type
dual theories and check their anomaly matching conditions. The Pouliot type
dual theories we will consider in this paper are two dual pairs; the dual pair
of ${{\cal N}}=1$ supersymmetric theories of a $Spin(7)$ gauge theory with
spinors and a $SU(N_{f}-4)$ gauge theory with a symmetric tensor, fundamentals
and singlets, and the other dual pair of ${{\cal N}}=1$ supersymmetric
theories of a $Spin(10)$ gauge theory with a spinor and vectors and a
$SU(N_{f}-5)$ gauge theory with a symmetric tensor, fundamentals and singlets.
We will show that the both pairs satisfy the discrete anomaly matching
conditions.

###### Contents

  1. 1 Introduction
  2. 2 The dual pair of the $Spin(7)$ theory with spinors
  3. 3 The dual pair of the $Spin(10)$ with a spinor and vectors
  4. 4 Discussions
  5. A The continuous ’t Hooft anomaly matching of the extended theories of the $Spin(10)$ dual pair

##  1 Introduction

In this paper, we will study the ’t Hooft anomalies of independent discrete
symmetries of two dual pairs, an ${{\cal N}}=1$ supersymmetric $Spin(7)$ gauge
theory with $N_{f}$ spinors and its dual [1], and an ${{\cal N}}=1$
supersymmetric $Spin(10)$ gauge theory with a spinor and $N_{f}$ vectors and
its dual [2, 3]. We will show that the two dual pairs satisfy the ’t Hooft
anomaly matching conditions of the independent discrete symmetries.

The concern that quantum gravitational effects might break global
symmetries111See [4] for recent discussions. and the facts that discrete
symmetries play important roles in phenomenological models have led to the
idea of discrete gauge symmetries [5], and further to the anomaly cancellation
conditions of them [6, 7, 8]222See [9] for recent discussions on discrete
(non-)Abelian symmetries.. They have been used to derive the anomaly matching
conditions of discrete symmetries [10, 11] by extending the discssions given
by ’t Hooft [12].

The continuous ’t Hooft anomaly matching conditions are necessary but severe
conditions for possible low-energy theories of a strong coupling high-energy
theory to satisfy. In particular, they yield very strong evidences for the
Seiberg dualities [13]333See [14] for a recent excellent review.. The anomaly
matching conditions for discrete symmetries may give more evidences for the
conjectures, if a dual pair has independent discrete symmetries. In fact, in
the papers [10, 11], for many dual pairs, the discrete anomaly matching have
been checked, and some of them do not pass the tests.

Therefore, it is significant to check the discrete anomaly matching for
unchecked dual pairs. Among those, we will study the above two pairs, the
$Spin(7)$ theory and its dual, and the $Spin(10)$ theory and its dual.

In section 2, we will discuss an independent discrete symmetry of the
$Spin(7)$ theory and embed the discrete symmetry into an anomaly free $U(1)$
symmetry by introducing additional fields so as to define the discrete
anomalies. We will repeat the same discussions for the dual $SU(N_{f}-4)$
theory. In the magnetic theory, the transformation laws of the fields for the
discrete symmetry is not uniquely determined. However, we will show that they
all saturate the discrete ’t Hooft anomaly matching conditions given by the
electric $Spin(7)$ theory. The dual pair of the $Spin(7)$ theory is obtained
by the parent dual pair of an ${{\cal N}}=1$ supersymmetric $Spin(8)$ theory
with a spinor and $N_{f}$ vectors and its dual [15] by higgsing the $Spin(8)$
gauge group by the non-zero vacuum expectation value of the spinor. In the
$Spin(8)$ theory and the dual theory, there is an anomaly free $U(1)$ global
symmetry, and after the higgsing, it is broken into the discrete symmetry of
the $Spin(7)$ theory and its dual. The continuous ’t Hooft anomalies which the
$U(1)$ symmetry takes part in become the discrete anomalies, and therefore,
the fact that the parent dual pair satisfies the continuous ’t Hooft anomaly
matching conditions immediately implies that the $Spin(7)$ and its dual
satisfy the discrete anomaly matching. However, there is a subtlety upon
defining the discrete anomalies from the continuous anomalies because we need
to perform a gauge transformation as well to gain the discrete symmetry. We
will elaborate on this issue and show that the dual theories surely satisfy
the discrete anomaly matching conditions.

In section 3, we will proceed to the dual pair of the $Spin(10)$ theory, and
repeat the same procedure as done for the $Spin(7)$ dual pair. We will embed
the discrete symmetry into an anomaly free $U(1)$ symmetry by adding fields to
the theories and compute the anomalies. Although we can not uniquely determine
the transformation laws of the magnetic fields, we will show that they all
satisfy the discrete anomaly matching conditions, as for the $Spin(7)$ dual
theories. Contrary to the $Spin(7)$ dual theories, there are no known parent
dual pair, from which the $Spin(10)$ dual pair can be derived. Therefore, we
haven’t found the extended theories with the embedding $U(1)$ symmetry of the
dual pair, where all the continuous ’t Hooft anomaly matching conditions
including those with the embedding $U(1)$ symmetry are satisfied. However, in
Appendix A, we will construct extended theories of the $Spin(10)$ dual pair,
where the continuous ’t Hooft anomaly matching conditions which the embedding
$U(1)$ symmetry take part in are satisfied exactly. It means that the discrete
anomaly matching is automatic after the higgsing of the embedding $U(1)$
symmetries on the both sides. The rest of the continuous ’t Hooft anomaly
matching are recovered after the decoupling of the additional fields.

Upon computations of anomalies, we will need to use the Dynkin index
$T_{G}(R)$ of a representation $R$ of a group $G$ defined by

$\displaystyle{{\rm tr}}\left[T_{R}^{a}T_{R}^{b}\right]=T_{G}(R)\delta^{ab},$

where $T_{R}^{a}$ ($a=1,\cdots,\dim{G}$) are the generators of the group $G$
in the representation $R$. We often omit the subscript $G$ of the Dynkin
index, when it is obvious. As for the normalization of $T(R)$, we take
$T(\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}})=1$
for the fundamental representation  of $SU(N)$, and $T(\bm{N})=2$ for the
vector representation of $Spin(N)$. They count the number of zero modes of a
single fermion in the representation of the group, when the one-instanton
background is turned on. For the adjoint representation of $Spin(N)$, we have
$T({{\bf adj}})=2(N-2)$. The properties of the spin representations frequently
depend on the parity of $N$ of $Spin(N)$. For the spin reprensentataion ${\bf
2^{n-1}}$ of both chiralities of $Spin(2n)$, we have $T({\bf
2^{n-1}})=2^{n-3}$. On the other hand, for the spin representation ${\bf
2^{n}}$ of $Spin(2n+1)$, $T({\bf 2^{n}})=2^{n-2}$.

##  2 The dual pair of the $Spin(7)$ theory with spinors

We will consider an ${{\cal N}}=1$ supersymmetric $Spin(7)$ gauge theory with
$N_{f}$ spinors $Q^{i}$ ($i=1,\cdots,N_{f}$) with no superpotentials [1].
Besides the continuous global symmetries $SU(N_{f})\times U(1)_{R}$, there is
an discrete ${{\bf Z}}_{2N_{f}}$ symmetry in the theory. Under the discrete
symmetry transformation, the spinors $Q^{i}$ transform as

$\displaystyle Q^{i}\quad\to\quad\exp\left({2\pi{i}\over 2N_{f}}\right)Q^{i}.$
(1)

The charge assignments of the spinors $Q^{i}$ are listed in Table 1.

| $Spin(7)$ | $SU(N_{f})$ | $U(1)_{R}$ | ${{\bf Z}}_{2N_{f}}$
---|---|---|---|---
$Q^{i}$ | $8$ | $N_{f}$ | $1-5/N_{f}$ | 1
Table 1: The charge assignments of the spinors $Q^{i}$ in the electric
$Spin(7)$ theory

Performing the discrete transformation (1) twice, it gives a transformation
given by an element of the center ${{\bf Z}}_{N_{f}}$ of the flavor
$SU(N_{f})$ symmetry. It implies that the subgroup ${{\bf
Z}}_{N_{f}}\subset{{\bf Z}}_{2N_{f}}$ can be identified with the center of the
flavor group $SU(N_{f})$. Therefore, we need to take the quotient of ${{\bf
Z}}_{2N_{f}}$ by the subgroup ${{\bf Z}}_{N_{f}}$ to find an independent
discrete symmetry.

When $N_{f}$ is an odd integer, we have ${{\bf Z}}_{2N_{f}}\simeq{{\bf
Z}}_{2}\times{{\bf Z}}_{N_{f}}$, where we may identify the cyclic group ${{\bf
Z}}_{N_{f}}$ with the center of the flavor group $SU(N_{f})$. The matter field
$Q^{i}$ transforms into $-Q^{i}$ under the remaining subgroup ${{\bf Z}}_{2}$.
We may identify it with a gauge transformation. In fact, we may take the gamma
matrices for the gauge group $Spin(7)$,

$\begin{split}&\gamma_{1}=\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1},\qquad\gamma_{2}=\sigma_{2}\otimes\sigma_{1}\otimes\sigma_{1},\cr&\gamma_{3}=\sigma_{3}\otimes\sigma_{1}\otimes\sigma_{1},\qquad\gamma_{4}={{\bf
1}}_{2}\otimes\sigma_{2}\otimes\sigma_{1},\cr&\gamma_{5}={{\bf
1}}_{2}\otimes\sigma_{3}\otimes\sigma_{1},\qquad\gamma_{6}={{\bf
1}}_{2}\otimes{{\bf
1}}_{2}\otimes\sigma_{2},\cr&\gamma_{7}=i\gamma_{1}\cdots\gamma_{6}={{\bf
1}}_{2}\otimes{{\bf 1}}_{2}\otimes\sigma_{3},\end{split}$ (2)

so that $\gamma_{12}={i}\sigma_{3}\otimes{{\bf 1}}_{2}\otimes{{\bf 1}}_{2}$,
which is one of the generators of the $Spin(7)$. Under the gauge
transformation generated by $\gamma_{12}$, the matter fields $Q^{i}$ transform
as

$\displaystyle Q^{i}\quad\mapsto\quad\exp\left({1\over
2}\phi\gamma_{12}\right)Q^{i}=\begin{pmatrix}e^{{{i}\over 2}\phi}{{\bf
1}}_{4}&\\\ &e^{-{{i}\over 2}\phi}{{\bf 1}}_{4}\end{pmatrix}Q^{i}.$

When $\phi=2\pi$, we see that the gauge transformation yields
$Q^{i}\mapsto-Q^{i}$. Therefore, we have found that the discrete symmetry
${{\bf Z}}_{2N_{f}}$ is a subgroup of $Spin(7)\times{SU}(N_{f})$, when $N_{f}$
is an odd integer. It implies that there are no independent discrete
symmetries for odd $N_{f}$.

We will compute the discrete ’t Hooft anomalies below, whichever the ${{\bf
Z}}_{2N_{f}}$ symmetry is independent or not. As is explained in [10], in
order to compute Type II discrete anomalies ${{\bf Z}}_{2N_{f}}^{3}$, ${{\bf
Z}}_{2N_{f}}U(1)_{R}^{2}$, ${{\bf Z}}_{2N_{f}}^{2}U(1)_{R}$, we need to
promote the discrete ${{\bf Z}}_{2N_{f}}$ symmetry to a anomaly-free
continuous $U(1)$ symmetry. To this end, we will extend the theory by
introducing a singlet $\Phi$ and a spinor $P$ of the gauge group $Spin(7)$.
Under the promoted $U(1)$ symmetry transformation, the fields transform as

$\displaystyle
Q^{i}~{}\to~{}\exp\left({i}\omega\right)Q^{i},\quad\Phi~{}\to~{}\exp\left({i}2N_{f}\omega\right)\Phi,\quad
P~{}\to~{}\exp\left(-{i}{N_{f}}\omega\right)P,$

and $\Phi$ carries the $U(1)_{R}$ charge 0, and $P$ the $U(1)_{R}$ charge one.
Then, the $U(1)_{R}$ charge assignment for the spinors $Q^{i}$ may be kept
intact and the $U(1)_{R}$ symmetry are still anomaly free. By inspection, we
can verify that the promoted $U(1)$ symmetry is also anomaly free. We will
turn on the superpotential $\Phi{}PP$. When we will promote the $U(1)$
symmetry to a gauge symmetry by introducing a $U(1)$ gauge superfield and turn
on the Fayet-Iliopoulos term in the $D$-term potential of the gauged $U(1)$
symmetry, we will find the vacuum $\left\langle{\Phi}\right\rangle\not=0$,
where the $U(1)$ symmetry is broken into the discrete $Z_{2N_{f}}$ symmetry.
The spinor $P$ will gain a mass through the superpotential
$\left\langle{\Phi}\right\rangle PP$ and decouple from the rest of the theory
in the infrared. We can utilize the $U(1)$ gauge superfield to compute the
Type II discrete anomalies ${{\bf Z}}_{2N_{f}}^{3}$, ${{\bf
Z}}_{2N_{f}}U(1)_{R}^{2}$, ${{\bf Z}}_{2N_{f}}^{2}U(1)_{R}$ by caluculating
the ’t Hooft anomalies $U(1)^{3}$, $U(1)U(1)_{R}^{2}$, $U(1)^{2}U(1)_{R}$. For
the computations, we will also multiply the $U(1)_{R}$ charges by $N_{f}$ to
make them integers.

| $Spin(7)$ | $SU(N_{f})$ | $U(1)_{R}$ | ${{\bf Z}}_{2N_{f}}$
---|---|---|---|---
$Q^{i}$ | ${\bf 8}$ |  | $1-5/N_{f}$ | 1
$\Phi$ | ${{\bf 1}}$ | ${{\bf 1}}$ | $0$ | $2N_{f}$
$P$ | ${\bf 8}$ | ${{\bf 1}}$ | $1$ | $-N_{f}$
Table 2: The charge assignments of the fields in the extended $Spin(7)$ theory

Before proceeding to the computations, we notice that the field with the
discrete charge $q$ in the representation $R$ of the flavor $SU(N_{f})$ group
and in the representation $R_{g}$ of the gauge $Spin(7)$ group contributes to
the discrete anomalies except for the ${{\bf Z}}_{2N_{f}}SU(N_{f})^{2}$
anomaly by a mutiple of $q\dim{R}\dim{R}_{g}$. We see from Table 2 that the
combination $q\dim{R}\dim{R}_{g}$ is a multiple of $2N_{f}$. Since we count
the discrete anomalies modulo $2N_{f}$, all the discrete anomalies except for
the ${{\bf Z}}_{2N_{f}}SU(N_{f})^{2}$ should be zero modulo $2N_{f}$. In fact,
our computations give the discrete ’t Hooft anomalies,

$\displaystyle\bullet~{}{{\bf Z}}_{2N_{f}}SU(N_{f})^{2}:~{}8,$
$\displaystyle\bullet~{}{{\bf Z}}_{2N_{f}}(\mbox{gravity})^{2}:~{}1\times
8N_{f}+2N_{f}\times 1+(-N_{f})\times 8=2N_{f}$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}^{3}:~{}\left(1\right)^{3}\times
8N_{f}+\left(2N_{f}\right)^{3}\times 1+(-N_{f})^{3}\times 8=4\times 2N_{f},$
$\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}U(1)_{R}^{2}:~{}1\times\left({-5}\right)^{2}\times
8N_{f}+{2N_{f}}\times\left({-N_{f}}\right)^{2}=(N_{f}^{2}+100)\times 2N_{f},$
$\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}^{2}U(1)_{R}:~{}\left(1\right)^{2}\times\left({-5}\right)\times
8N_{f}+\left(2N_{f}\right)^{2}\times\left({-N_{f}}\right)=-(20+2N_{f}^{2})\times
2N_{f}.$

Note that we can introduce a vector $P^{\prime}$ of the gauge group $Spin(7)$
instead of the spinor $P$ to cancel the $U(1)$ gauge anomaly and add the term
$\Phi{}P^{\prime}P^{\prime}$ to the superpotential, which becomes the Majorana
mass term after the $U(1)$ symmetry breaking
$\left\langle{\Phi}\right\rangle\not=0$. Since the vector $P^{\prime}$ carries
the $U(1)_{R}$ charge 1, it contributes only to the discrete ’t Hooft
anomalies ${{\bf Z}}_{2N_{f}}(\mbox{gravity})^{2}$ and ${{\bf
Z}}_{2N_{f}}^{3}$. Its contributions increase the anomalies ${{\bf
Z}}_{2N_{f}}(\mbox{gravity})^{2}$ and ${{\bf Z}}_{2N_{f}}^{3}$ computed for
the spinor $P$ by $N_{f}$ and $N_{f}^{3}$, respectively.

When $N_{f}=6$, the $Spin(7)$ theory is in the confining phase without chiral
symmetry breaking [1], and the low-energy physics is described by the mesons
$M^{ij}\sim Q^{i}Q^{j}$ and the baryons $B\sim Q^{4}$ with the superpotential

$\displaystyle\det{M}-M^{ik}M^{jl}B_{ij}B_{kl}-{\rm Pf}B.$ (3)

| $SU(N_{f}=6)$ | $U(1)_{R}$ | ${{\bf Z}}_{2N_{f}}$
---|---|---|---
$M^{ij}$ |  | $2-5/3$ | 2
$B$ | $\overline{\raisebox{-3.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}\hskip-6.9pt\raisebox{3.0pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $4-10/3$ | 4
$X$ | 1 | 0 | $-12$
Table 3: The low-energy theory extended for $N_{f}=6$

In order to embed the discrete symmetry into an anomaly free $U(1)$ symmetry,
we will introduce a singlet $X$, as listed in Table 3, and replace the
superpotential (3) by

$\displaystyle X\left(\det{M}-M^{ik}M^{jl}B_{ij}B_{kl}-{\rm Pf}B\right).$

Upon the $U(1)$ symmetry breaking to the discrete symmetry by
$\left\langle{X}\right\rangle\not=0$, rescaling the meson $M^{ij}$ and the
baryons $B_{ij}$, it is reduced into the original low energy theory of
$M^{ij}$ and $B_{ij}$. Using the fields $M^{ij}$, $B_{ij}$ annd $X$, we find
that the discrete ’t Hooft anomalies are given by

$\displaystyle\bullet~{}{{\bf Z}}_{2N_{f}}SU(N_{f})^{2}:~{}2\times 8+{4}\times
4=8+2\times 12,$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}(\mbox{gravity})^{2}:~{}{2}\times{6\times 7\over
2}+{4}\times{6\times 5\over 2}-12=6+7\times 12,$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}^{3}:~{}2^{3}\times{6\times 7\over 2}+4^{3}\times{6\times 5\over
2}+(-12)^{3}=-50\times 12,\qquad(6^{3}=18\times 12),$
$\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}U(1)_{R}^{2}:~{}{2}\times\left({-4}\right)^{2}\times{6\times
7\over 2}+{4}\times\left(-{2}\right)^{2}\times{6\times 5\over
2}-12\times(-1)^{2}=75\times 12,$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}^{2}U(1)_{R}:~{}\left({2}\right)^{2}\times\left({-4}\right)\times{6\times
7\over 2}+\left({4}\right)^{2}\times\left({-{2}}\right)\times{6\times 5\over
2}+(-12)^{2}\times(-1)=-80\times 12.$

They saturate the discrete ’t Hooft anomalies of the $Spin(7)$ theory extended
by the vector $P^{\prime}$ instead of the spinor $P$. Thus, the low-energy
theory of the meson $M^{ij}$ and the baryons $B$ passes the discrete ’t Hooft
anomaly matching tests.

The dual of the $Spin(7)$ theory for $7\leq N_{f}\leq 14$ [1] is an ${{\cal
N}}=1$ supersymmetric $SU(N_{f}-4)$ gauge theory with a $s$ in  , $N_{f}$
$\tilde{q}_{i}$ in
$\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$,
and singlets $M^{ij}$ with the superpotential

$\displaystyle M^{ij}\tilde{q}_{i}\cdot{s}\cdot\tilde{q}_{j}+\det{s}.$

The charge assignments of the fields are listed in Table 4.

| $SU(N_{f}-4)$ | $SU(N_{f})$ | $U(1)_{R}$
---|---|---|---
$s$ |  | $1$ | $2/(N_{f}-4)$
$\tilde{q}_{i}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $5/N_{f}-1/(N_{f}-4)$
$M^{ij}$ | $1$ |  | $2-10/N_{f}$
Table 4: The field content of the dual of the $Spin(7)$ theory

The correspondence of the gauge invariant operators $M^{ij}\sim
Q^{i}\cdot{Q}^{j}$, $Q^{4}\sim\tilde{q}^{N_{f}-4}$ and the invariance of the
superpotential determine the transformation laws of the discrete symmetry

$\displaystyle M^{ij}~{}\to~{}\exp\left(2\pi{i}{1\over
N_{f}}\right)M^{ij},\quad\tilde{q}_{i}~{}\to~{}\exp\left(2\pi{i}{2+N_{f}p\over
N_{f}(N_{f}-4)}\right)\tilde{q}_{i},\quad
s~{}\to~{}\exp\left(-2\pi{i}{1+2p\over N_{f}-4}\right)s,$

up to an integer $p\in{{\bf Z}}$. The anomaly free condition of the discrete
symmetry by the gauge interaction is

$\displaystyle{2+N_{f}p\over N_{f}(N_{f}-4)}\times N_{f}-{1+2p\over
N_{f}-4}\times(N_{f}-2)=-(p+1)\in{{\bf Z}}.$

The anomaly free condition for the discrete symmetry is obviously satisfied
for any integer $p$, and it is sufficient for computations of Type I discrete
’t Hooft anomalies, but, in order to compute Type II discrete anomalies, we
need to embed the discrete group into an anomaly free $U(1)$ group, as is done
in the electric theory. To this end, we set $p=-1$ so that the gauge anomaly
for the discrete symmetry is strictly vanishing444 This is the simplest
solution for embedding of the discrete symmetry into a $U(1)$ symmetry. In
order to cancel the gauge anomaly, we may introduce an appropriate set of
massive fields in non-trivial representations of the gauge group, as we will
see below.. We thus find the transformation laws of the discrete symmetry

$\displaystyle\tilde{q}_{i}\to\exp\left(-2\pi{i}{N_{f}-2\over
N_{f}(N_{f}-4)}\right)\tilde{q}_{i},\qquad s\to\exp\left({2\pi{i}\over
N_{f}-4}\right)s,\qquad M^{ij}\to\exp\left({2\pi{i}\over N_{f}}\right)M^{ij}.$

We see that the discrete symmetry group is a subgroup of the cyclic group
${{\bf Z}}_{N_{f}(N_{f}-4)}$.

Looking at the exponent of the transformed $\tilde{q}_{i}$,

$\displaystyle-{N_{f}-2\over N_{f}(N_{f}-4)}=-{1\over 2N_{f}}-{1\over
2(N_{f}-4)},$

when we perform the same transformation twice, we see that the resulting
transformation can be given by an element of the center of the flavor group
$SU(N_{f})$ and an element of the center of the gauge group $SU(N_{f}-4)$.
Therefore, it is not an independent discrete symmetry. We have seen that this
is also the case for the discrete ${{\bf Z}}_{2N_{f}}$ symmetry in the
electric theory. In this sence, the independent discrete symmetries on the
both side of the duality are given by the quotient group ${{\bf Z}}_{2}$. When
$N_{f}$ is an odd integer, we have seen that the quotient group ${{\bf
Z}}_{2}$ is given by the center of the gauge group $Spin(7)$ in the electric
theory, and therefore, no independent discrete symmetry in the electric theory
are found. In the magnetic theory, when $N_{f}$ is an odd integer, i.e.,
$N_{f}=2k+1$ ($k\in{{\bf Z}}$), the exponent of the transformed
$\tilde{q}_{i}$ may be rewritten into

$\displaystyle-{N_{f}-2\over N_{f}(N_{f}-4)}={k\over N_{f}}-{k-1\over
N_{f}-4},$

and also we may rewrite the exponents of the other transformed fields into

$\displaystyle\exp\left(2\pi{i}{1\over
N_{f}-4}\right)s=\exp\left(2\pi{i}{N_{f}-3\over
N_{f}-4}\right)s=\exp\left(2\pi{i}{2(k-1)\over N_{f}-4}\right)s,$
$\displaystyle\exp\left(2\pi{i}{1\over
N_{f}}\right)M^{ij}=\exp\left(2\pi{i}{-N_{f}+1\over
N_{f}}\right)M^{ij}=\exp\left(2\pi{i}{-2k\over N_{f}}\right)M^{ij}.$

We find that the transformation laws of the discrete symmetry can be given by
an element of the center of the flavor group $SU(N_{f})$ and an element of the
center of the gauge group $SU(N_{f}-4)$. Therefore, there is no independent
discrete symmetry in the magnetric theory as well for odd $N_{f}$.

This can be understood in a different way. For odd $N_{f}$, $N_{f}$ and
$N_{f}-4$ are relatively prime to each other555 When $N_{f}$ is odd, it is
obivous that $N_{f}$ is prime to $2^{m}$, for a positive integer $m$. It means
that there are two integers $p$, $q$ satisfying $N_{f}p+2^{m}q=1$. Rewriting
it into $N_{f}(p+q)+(N_{f}-2^{m})(-q)=1$, we see that $N_{f}$ and
$N_{f}-2^{m}$ are relatively prime to each other. . In the same way, $N_{f}$
and $N_{f}-2$ are also relatively prime to each other, and shifting $N_{f}$ by
$-2$, we see that $N_{f}-2$ and $N_{f}-4$ are relatively prime to each other
as well. Therefore, since $N_{f}-2$ is prime to $N_{f}(N_{f}-4)$, the discrete
symmetry is a cyclic group ${{\bf Z}}_{N_{f}(N_{f}-4)}$. Furthermore, as
$N_{f}$ and $N_{f}-4$ are relatively prime to each other, the group ${{\bf
Z}}_{N_{f}(N_{f}-4)}$ is isomorphic to ${{\bf Z}}_{N_{f}}\times{{\bf
Z}}_{N_{f}-4}$, the product group of the center ${{\bf Z}}_{N_{f}}$ of the
flavor group $SU(N_{f})$ and the center ${{\bf Z}}_{N_{f}-4}$ of the gauge
group $SU(N_{f}-4)$.

Although there are no independent discrete symmetries for odd $N_{f}$, we will
treat the cases for both odd and even $N_{f}$ on the same footing. We will now
promote the discrete ${{\bf Z}}_{N_{f}(N_{f}-4)}$ symmetry to an anomaly free
$U(1)$ symmetry by introducing a singlet chiral superfield $X$ and replacing
the term $\det{s}$ in the superpotential by $X\det{s}$. We suppose that the
singlet $X$ carries no $U(1)_{R}$ charge to leave the $U(1)_{R}$ charge of the
symmetric tensor $s$ unchanged.

The anomaly free condition of the $U(1)$ symmetry by the $SU(N_{f}-4)$ gauge
interaction and the invariance of the superpotential suggest that the fields
under the $U(1)$ transformation should transform as

$\displaystyle M^{ij}\quad\to\quad
e^{(N_{f}-4)i\omega}M^{ij},\qquad\tilde{q}_{i}\quad\to\quad
e^{-(N_{f}-2)i\omega}\tilde{q}_{i},$ $\displaystyle s\quad\to\quad
e^{{N_{f}}i\omega}s,\qquad X\quad\to\quad e^{-{N_{f}(N_{f}-4)}i\omega}X,$

with a transformation parameter $\omega$. Note that the $U(1)$ transformation
is distinct from the $U(1)$ transformation in the electric theory, as we can
verify from the gauge invariant operators $M^{ij}\sim Q^{i}Q^{j}$,
$Q^{4}\sim\tilde{q}^{N_{f}-4}$. However, the reason that we embed the discrete
symmetry into an anomaly free $U(1)$ symmetry is just to define Type II
discrete ’t Hooft anomalies, but not to find the dual theory of the extended
electric theory with the extra $U(1)$ symmetry.

Let us introduce a $U(1)$ gauge superfield to promote the $U(1)$ symmetry to a
gauge symmetry and introduce the Fayet-Iliopoulos term in the $D$-term
potential of the $U(1)$ gauge symmetry so that there exists a vacuum
$\left\langle{X}\right\rangle\not=0$, where the $U(1)$ symmetry is broken into
the original discrete ${{\bf Z}}_{N_{f}(N_{f}-4)}$ symmetry. As was done for
the electric theory, we can make use of the background $U(1)$ gauge field to
compute Type II discrete anomalies ${{\bf Z}}_{2N_{f}}^{3}$, ${{\bf
Z}}_{2N_{f}}U(1)_{R}^{2}$, ${{\bf Z}}_{2N_{f}}^{2}U(1)_{R}$ by computing the
’t Hooft anomalies $U(1)^{3}$, $U(1)U(1)_{R}^{2}$, $U(1)^{2}U(1)_{R}$. We will
multiply the $U(1)_{R}$ charges by $N_{f}$. However, the $U(1)_{R}$ charges
are not all integers even after multiplying them by $N_{f}$, contrary to the
electric theory.

| $SU(N_{f}-4)$ | $SU(N_{f})$ | $U(1)_{R}$ | ${{\bf Z}}_{N_{f}(N_{f}-4)}$
---|---|---|---|---
$s$ |  | $1$ | $2/(N_{f}-4)$ | $N_{f}$
$\tilde{q}_{i}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $5/N_{f}-1/(N_{f}-4)$ | $-(N_{f}-2)$
$M^{ij}$ | $1$ |  | $2-10/N_{f}$ | $N_{f}-4$
$X$ | 1 | 1 | 0 | $-N_{f}(N_{f}-4)$
Table 5: The field content in the extended magnetic theory

We compute the ’t Hooft anomalies for the magnetic theory,

$\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}SU(N_{f})^{2}:~{}-\left({N_{f}-2}\right)\times(N_{f}-4)+(N_{f}-4)\times(N_{f}+2)=4(N_{f}-4),$
$\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}(\mbox{gravity})^{2}:~{}N_{f}\times{(N_{f}-4)(N_{f}-3)\over
2}+\left(-(N_{f}-2)\right)\times N_{f}\times(N_{f}-4)$ $\displaystyle\hskip
99.58464pt+(N_{f}-4)\times{N_{f}(N_{f}+1)\over 2}-N_{f}(N_{f}-4)=0,$
$\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}^{3}:~{}N_{f}^{3}\times{(N_{f}-4)(N_{f}-3)\over
2}+\left(-(N_{f}-2)\right)^{3}\times N_{f}\times(N_{f}-4)$
$\displaystyle\hskip 51.21504pt+(N_{f}-4)^{3}\times{N_{f}(N_{f}+1)\over
2}+\left(-N_{f}(N_{f}-4)\right)^{3}$ $\displaystyle\hskip
51.21504pt=-(N_{f}^{2}-1)(N_{f}-4)^{2}\times N_{f}(N_{f}-4),$
$\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}U(1)_{R}^{2}:~{}N_{f}\times{(N_{f}-4)(N_{f}-3)\over
2}\times\left(-N_{f}+{2N_{f}\over N_{f}-4}\right)^{2}$ $\displaystyle\hskip
79.6678pt+\left(-(N_{f}-2)\right)\times
N_{f}\times(N_{f}-4)\times\left(-N_{f}+5-{N_{f}\over N_{f}-4}\right)^{2}$
$\displaystyle\hskip 79.6678pt+(N_{f}-4)\times{N_{f}(N_{f}+1)\over
2}\times\left(N_{f}-10\right)^{2}-N_{f}(N_{f}-4)\times(-N_{f})^{2}$
$\displaystyle\hskip 79.6678pt=-4\left(N_{f}^{2}-25\right)\times
N_{f}(N_{f}-4),$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}^{2}U(1)_{R}:~{}N_{f}^{2}\times{(N_{f}-4)(N_{f}-3)\over
2}\times\left(-N_{f}+{2N_{f}\over N_{f}-4}\right)$ $\displaystyle\hskip
79.6678pt+\left(-(N_{f}-2)\right)^{2}\times
N_{f}\times(N_{f}-4)\times\left(-N_{f}+5-{N_{f}\over N_{f}-4}\right)$
$\displaystyle\hskip 79.6678pt+(N_{f}-4)^{2}\times{N_{f}(N_{f}+1)\over
2}\times\left(N_{f}-10\right)+\left(-N_{f}(N_{f}-4)\right)^{2}\times(-N_{f})$
$\displaystyle\hskip 79.6678pt=-2\left(N_{f}^{2}+5\right)(N_{f}-4)\times
N_{f}(N_{f}-4).$

In order to examine whether the dual theories satisfy the discrete ’t Hooft
anomaly matching conditions, we will embed both of the discrete ${{\bf
Z}}_{2N_{f}}$ group in the electric theory and the discrete ${{\bf
Z}}_{N_{f}(N_{f}-4)}$ group into a ${{\bf Z}}_{2N_{f}(N_{f}-4)}$ group by
multiplying the discrete charges in the electric theory by $N_{f}-4$ and those
in the magnetic theory by $2$, respectively. Then, we find that the ${{\bf
Z}}_{2N_{f}(N_{f}-4)}SU(N_{f})^{2}$ anomaly in the electric theory exactly
matches the one in the magnetic theory. For the other discrete anomalies, we
see that the magnetic theory saturates the discrete ’t Hooft anomaly matching
conditions from the electric theory, modulo $2N_{f}(N_{f}-4)$.

| $SU(N_{f}-4)$ | $SU(N_{f})$ | $U(1)_{R}$ | ${{\bf Z}}_{2N_{f}(N_{f}-4)}$
---|---|---|---|---
$s$ |  | $1$ | $2/(N_{f}-4)$ | $-2(2p+1)N_{f}$
$\tilde{q}_{i}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $5/N_{f}-1/(N_{f}-4)$ | $2(N_{f}p+2)$
$M^{ij}$ | $1$ |  | $2-10/N_{f}$ | $2(N_{f}-4)$
$X$ | 1 | 1 | 0 | $2(2p+1)N_{f}(N_{f}-4)$
$\tilde{X}$ | 1 | 1 | 0 | $-2(p+1)N_{f}(N_{f}-4)$
$F$ |  | 1 | $1+1/(N_{f}-4)$ | $(p+1)N_{f}(N_{f}-4)-(1+2p)N_{f}$
$\tilde{F}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | 1 | $1-1/(N_{f}-4)$ | $(p+1)N_{f}(N_{f}-4)+(1+2p)N_{f}$
Table 6: The field content in the extended magnetic theory for $p\not=-1$

We have seen the ambiguity of the magnetic discrete symmetry by an integer
$p$, and we have taken $p=-1$ as the simplest choice for an anomaly free
$U(1)$ symmetry. We will here consider the case $p\not=-1$ and add additional
matter fields for an anomaly free $U(1)$ symmetry. We read a $U(1)$ symmetry
transformation for $p\not=-1$,

$\displaystyle
M^{ij}~{}\to~{}e^{i(N_{f}-4)\omega}M^{ij},\qquad\tilde{q}_{i}~{}\to~{}e^{i(2+N_{f}p)\omega}\tilde{q}_{i},$
$\displaystyle s~{}\to~{}e^{-i(2p+1)N_{f}\omega}s,\qquad
X~{}\to~{}e^{i(2p+1)N_{f}(N_{f}-1)\omega}X,$

with a transformation parameter $\omega$. Since the $U(1)$ symmetry is
anomalous, we will add additional fields $\tilde{X}$, $F$ and $\tilde{F}$,
listed in Table 6, to make it anomaly free. We will add the term
$\tilde{X}F\tilde{F}$ to the superpotential so as to make them massive after
the $U(1)$ symmetry breaking. Then, the discrete symmetry group is a subgroup
of a cyclic group ${{\bf Z}}_{2N_{f}(N_{f}-4)}$, due to the presence of the
extra fundamental pair $F$, $\tilde{F}$. The discrete charges of the magnetic
fields are listed in Table 6.

The introduction of a $U(1)$ gauge superfield to promote the $U(1)$ symmetry
to a $U(1)$ gauge group is the same as for $p=-1$, and we will find the vacuum
where $\langle{X}\rangle\not=0$, $\langle\tilde{X}\rangle\not=0$, breaking the
$U(1)$ symmetry to the discrete ${{\bf Z}}_{2N_{f}(N_{f}-4)}$ symmetry.
Multiplying the $U(1)_{R}$ charges by $N_{f}$, we compute the discrete ’t
Hooft anomalies. We take the $U(1)_{R}$ charges of $F$ and $\tilde{F}$ to be
the values in Table 6 in order for the ${{\bf
Z}}_{2N_{f}(N_{f}-4)}U(1)_{R}^{2}$ anomaly to be a multiple of
$2N_{f}(N_{f}-4)$, therefore, saturating the matching condition of the anomaly
in the electric theory, modulo $2N_{f}(N_{f}-4)$, as we will see soon.

To compare these anomalies with those in the electric theory, we will embed
the electric ${{\bf Z}}_{2N_{f}}$ group into the ${{\bf Z}}_{2N_{f}(N_{f}-4)}$
group by multiplying the discrete charges by $N_{f}-4$, and we find that all
the discrete ’t Hooft anomalies of both electric and magnetic theories match
modulo $2N_{f}(N_{f}-4)$.

In the previous sections, we have promoted the discrete symmetries into
anomaly free $U(1)$ symmetries by extending both of the dual theories to
larger theories. However, those extended theories are not dual to each other
666For $p\not=0$, we have different $U(1)$ symmetries between the dual
theories. For $p=0$, although we have the same $U(1)$ symmetry, the continuous
’t Hooft anomaly matching conditions are not satisfied.. But, as we will see
below, there is an ideal dual pair of extended theories, an ${{\cal N}}=1$
supersymmetric $Spin(8)$ gauge theory with a spinor $S$ and $N_{f}$ vectors
$Q^{i}$ ($i=1,\cdots,N_{f}$) with no superpotential, and its dual theory [15].
See Table 7 for the field content of the $Spin(8)$ theory.

| $Spin(8)$ | $SU(N_{f})$ | $U(1)$ | $U(1)_{R}$
---|---|---|---|---
$S$ | $8_{s}$ | $1$ | $-N_{f}$ | $0$
$Q^{i}$ | $8_{v}$ |  | $1$ | $1-5/N_{f}$
Table 7: The $Spin(8)$ theory with a spinor and $N_{f}$ vectors

Although there are no independent discrete symmetries on the both sides of the
duality, they have an anomaly free $U(1)$ symmetry. After higgsing the
$Spin(8)$ to $Spin(7)$ by $\left\langle{S}\right\rangle\not=0$, $N_{f}$
vectors $Q^{i}$ of $Spin(8)$ become $N_{f}$ spinors of $Spin(7)$. The dual of
the $Spin(8)$ theory is reduced to the dual of the $Spin(7)$ theory. Then, the
non-zero vacuum expectation value $\left\langle{S}\right\rangle\not=0$ also
breaks the anomaly free $U(1)$ symmetry to the discrete symmetry of the
$Spin(7)$ theory. Before the higgsing of the $Spin(8)$ gauge group, all the
continuous ’t Hooft anomaly matching conditions are satisfied by the $Spin(8)$
theory and its dual theory. Therefore, all the discrete ’t Hooft anomaly
matching conditions should be satisfied by the $Spin(7)$ theory and its dual
$SU(N_{f}-4)$ theory. However, it is somewhat less trivial to confirm this, as
we will see below.

The magnetic theory of the $Spin(8)$ theory is an ${{\cal N}}=1$ supersymmetry
$SU(N_{f}-4)$ theory with a symmetric tensor $s$, $N_{f}$ antifundamentals
$\tilde{q}_{i}$ ($i=1,\cdots,N_{f}$), and singlets $M^{ij}$, $X$ with the
superpotential [15]

$\displaystyle M^{ij}\tilde{q}_{i}\cdot{s}\cdot\tilde{q}_{j}+X\det{s}.$

The singlets $M^{ij}$, $X$ correspond to the gauge invariant operators of the
electric theory, $M^{ij}\sim Q^{i}\cdot{Q}^{j}$, $X\sim S\cdot{S}$. There is
also another gauge invariant operator $B\sim
S^{2}Q^{4}\sim\tilde{q}^{N_{f}-4}$. See Table 8 for the field content of the
magnetic theory.

| $SU(N_{f}-4)$ | $SU(N_{f})$ | $U(1)$ | $U(1)_{R}$
---|---|---|---|---
$s$ |  | ${{\bf 1}}$ | $2N_{f}/(N_{f}-4)$ | $2/(N_{f}-4)$
$\tilde{q}_{i}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $-1-N_{f}/(N_{f}-4)$ | $5/N_{f}-1/(N_{f}-4)$
$M^{ij}$ | ${{\bf 1}}$ |  | $2$ | $2-10/N_{f}$
$X$ | ${{\bf 1}}$ | ${{\bf 1}}$ | $-2N_{f}$ | $0$
Table 8: The field content of the dual of the $Spin(8)$ theory

The magnetic theory reminds us of the extended dual theory of the $Spin(7)$
theory with $p=-1$. In order to go down to the dual theory of the $Spin(7)$
theory, we take the vacuum with $\left\langle{X}\right\rangle\not=0$, which
breaks the $U(1)$ symmetry to the discrete symmetry, under which the fields
transform as

$\begin{split}&s~{}\to~{}\exp\left({2\pi{i}\over
N_{f}-4}\right)s,\qquad\tilde{q}_{i}~{}\to~{}\exp\left[-2\pi{i}\left({1\over
2N_{f}}+{1\over
2(N_{f}-4)}\right)\right]\tilde{q}_{i},\cr&M^{ij}~{}\to~{}\exp\left({2\pi{i}\over
N_{f}}\right)M^{ij},\end{split}$ (4)

which is identical to the discrete symmetry for $p=-1$. Rescaling $s$,
$\tilde{q}_{i}$ and $M^{ij}$, the superpotential is reduced into
$M^{ij}\tilde{q}_{i}\cdot{s}\cdot\tilde{q}_{j}+\det{s}$. Thus, we obtain the
dual of the $Spin(7)$ theory.

On the other hand, in the electric $Spin(8)$ theory, the vacuum
$\left\langle{X}\right\rangle\not=0$ corresponds to the vacuum
$\left\langle{S}\right\rangle\not=0$ via the correspondence $X\sim S^{2}$.
Then, on the vacuum $\left\langle{S}\right\rangle\not=0$, the gauge group
$Spin(8)$ is broken to $Spin(7)$. Since $X$ and $S$ transform under the global
$U(1)$ symmetry transformation as

$\displaystyle X~{}\to~{}\exp\left(-4\pi{i}{N}_{f}\omega\right)X,\qquad
S~{}\to~{}\exp\left(-2\pi{i}{N}_{f}\omega\right)S,$

with a transformation parameter $\omega$, the discrete symmetry (4)
corresponds to

$\displaystyle S~{}\to~{}\exp\left(-\pi{i}\right)S,\qquad
Q^{i}~{}\to~{}\exp\left({2\pi{i}\over 2N_{f}}\right)Q^{i}.$ (5)

In order to leave the vacuum expectation value
$\left\langle{S}\right\rangle\not=0$ invariant, we need to perform the gauge
transformation

$\displaystyle S~{}\to~{}-S,\qquad Q^{i}~{}\to~{}Q^{i},$ (6)

at the same time, which is in a ${{\bf Z}}_{2}$ subgroup of the center of the
$Spin(8)$ group. Let us take a closer look at the gauge transformation (6). To
this end, we take the gamma matrices for the gauge group $Spin(8)$

$\displaystyle\Gamma_{m}=\gamma_{m}\otimes\sigma_{1},\qquad(m=1,\cdots,7),\qquad\Gamma_{8}={{\bf
1}}_{8}\otimes\sigma_{2}$

and the chirality matrix
$\Gamma_{9}=\Gamma_{1}\Gamma_{2}\cdots\Gamma_{8}={{\bf
1}}_{8}\otimes\sigma_{3}$, where $\gamma_{m}$ ($m=1,\cdots,7$) are the gamma
matrices of the $Spin(7)$ group in (2). Let us take the spinor $S$ to be of
positive chirality $\Gamma_{9}S=S$. Then, the gauge transformation generated
by $\Gamma_{12}$ transforms the spinor $S$ and the vectors $Q^{i}$ into

$\displaystyle S~{}\to~{}\begin{pmatrix}e^{{i\over 2}\phi}{{\bf 1}}_{4}&\\\
&e^{-{i\over 2}\phi}{{\bf 1}}_{4}\end{pmatrix}S,\qquad
Q^{i}~{}\to~{}\begin{pmatrix}\cos\phi&-\sin\phi&\\\ \sin\phi&\cos\phi&\\\
&&{{\bf 1}}_{6}\end{pmatrix}Q^{i},$ (7)

and the $Spin(8)$ gaugino $\lambda$ into

$\displaystyle\lambda~{}\to~{}\begin{pmatrix}\cos\phi&-\sin\phi&\\\
\sin\phi&\cos\phi&\\\ &&{{\bf
1}}_{6}\end{pmatrix}\lambda\begin{pmatrix}\cos\phi&\sin\phi&\\\
-\sin\phi&\cos\phi&\\\ &&{{\bf 1}}_{6}\end{pmatrix}.$

We may define the gauge transformation (6) as the limit $\phi\to 2\pi$, and
then it is convenient to decompose the representations of the $Spin(8)$ group
under $Spin(2)\times Spin(6)$, where the $Spin(2)$ group is generated by the
$\Gamma_{12}$. The spinor $S$ is decomposed into ${\bf 4}_{1/2}\oplus\bar{\bf
4}_{-1/2}$, and the vectors $Q^{i}$ into ${{\bf 1}}_{1}\oplus{{\bf
1}}_{-1}\oplus{\bf 6}_{0}$. The gaugino is decomposed into ${{\bf
1}}_{0}\oplus{\bf 6}_{1}\oplus{\bf 6}_{-1}\oplus{\bf 15}_{0}$. We may regard
the charges of the $Spin(2)$ as additional discrete charges to accompany with
the discrete transformation (5) in order to leave
$\left\langle{S}\right\rangle\not=0$ invariant, a more refined definition of
(6). The discrete transformation (5) is anomaly free,

$\displaystyle-{1\over 2}\times T({\bf 8}_{s})+{1\over 2N_{f}}\times
N_{f}\times T({\bf 8}_{v})=0.$

Although the naive definition (6) of the accompanying gauge transformation may
be anomalous, we can verify that the refined one keeps it anomaly free. For
example, the spinor $S$ contributes to the gauge anomaly as

$\displaystyle{1\over 2}\times\left[T_{Spin(6)}({\bf 4})+\left({1\over
2}\right)^{2}\right]-{1\over 2}\times\left[T_{Spin(6)}(\bar{{\bf
4}})+\left(-{1\over 2}\right)^{2}\right]=0,$

with the Dynkin indices $T_{Spin(6)}({\bf 4})$ and $T_{Spin(6)}(\bar{\bf 4})$
of the representations ${\bf 4}$ and $\bar{\bf 4}$, respectively, of the
$Spin(6)$ group, where we have used the fact that the generators of the
$Spin(6)$ are traceless. In the same way, we can show that the contributions
from the vectors $Q^{i}$ and the gaugino $\lambda$ to the gauge anomaly also
vanish. The global symmetries $SU(N_{f})\times{U}(1)_{R}\times{U}(1)$ of the
$Spin(8)$ theory are anomaly free, and the generators of the $SU(N_{f})$ group
are traceless. The $Spin(2)$ charges are ‘traceless’ in each of the
irreducible representations of the $Spin(8)$ group. They imply that the
contributions from the $Spin(2)$ charges, i.e., the accompanying gauge
transformation, to the discrete ’t Hooft anomalies are canceled to give
nothing. For example, the ${{\bf Z}}_{2N_{f}}^{3}$ anomaly is computed by
$\left(U(1)+Spin(2)\right)^{3}$, where $U(1)$ denotes the global $U(1)$
charges, which are the same as the discrete charges appearing in the discrete
transformation (5) and $Spin(2)$ means the $Spin(2)$ charges of the
accompanying gauge transformation. Then, $U(1)Spin(2)^{2}$, $U(1)^{2}Spin(2)$,
and $Spin(2)^{3}$ are zero, and it becomes $U(1)^{3}$. Thus, we may ignore the
$Spin(2)$ charges to compute the discrete anomalies, and therefore, the
discrete ’t Hooft anomalies of the $Spin(7)$ theory should be given by the
continuous ’t Hooft anomalies of the $Spin(8)$ theory. For the magnetic
theories, there is no such subtleties to compute the ’t Hooft anomalies, and
the discrete ’t Hooft anomalies of the dual of the $Spin(7)$ theory are also
given by the continuous ’t Hooft anomalies of the dual of the $Spin(8)$
theory. Since the continuous ’t Hooft anomaly matching conditions are
satisfied [15] by the $Spin(8)$ theory and the dual $SU(N_{f}-4)$ theory, the
discrete ’t Hooft anomaly matching conditions are obviously satisfied by the
$Spin(7)$ theory and its dual theory. In fact, we can verify that the discrete
’t Hooft matching conditions are satisfied by the direct computations

$\displaystyle\bullet~{}{{\bf Z}}_{2N_{f}}SU(N_{f})^{2}=U(1)SU(N_{f})^{2}~{}$
$\displaystyle\mbox{the electric theory}:~{}{1\over 2N_{f}}\times 8={4\over
N_{f}},$ $\displaystyle\mbox{the magnetic theory}:~{}-\left({1\over
2N_{f}}+{1\over 2(N_{f}-4)}\right)\times(N_{f}-4)+{1\over
N_{f}}\times(N_{f}+2)={4\over N_{f}},$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}(\mbox{gravity})^{2}=U(1)(\mbox{gravity})^{2}~{}$
$\displaystyle\mbox{the electric theory}:~{}-{1\over 2}\times 8+{1\over
2N_{f}}\times N_{f}\times 8=0,$ $\displaystyle\mbox{the magnetic
theory}:~{}{1\over N_{f}-4}\times{(N_{f}-4)(N_{f}-3)\over 2}-\left({1\over
2N_{f}}+{1\over 2(N_{f}-4)}\right)\times(N_{f}-4)\times N_{f}$
$\displaystyle\hskip 113.81102pt+{1\over N_{f}}\times{N_{f}(N_{f}+1)\over
2}-1=0,$ $\displaystyle\bullet~{}{{\bf Z}}_{2N_{f}}^{3}=U(1)^{3}~{}$
$\displaystyle\mbox{the electric theory}:~{}\left(-{1\over 2}\right)^{3}\times
8+\left({1\over 2N_{f}}\right)^{3}\times N_{f}\times 8={1\over N_{f}^{2}}-1,$
$\displaystyle\mbox{the magnetic theory}:~{}\left({1\over
N_{f}-4}\right)^{3}\times{(N_{f}-4)(N_{f}-3)\over 2}-\left({1\over
2N_{f}}+{1\over 2(N_{f}-4)}\right)^{3}\times(N_{f}-4)\times N_{f}$
$\displaystyle\hskip 113.81102pt+\left({1\over
N_{f}}\right)^{3}\times{N_{f}(N_{f}+1)\over 2}-1={1\over N_{f}^{2}}-1.$
$\displaystyle\bullet~{}{{\bf Z}}_{2N_{f}}U(1)_{R}^{2}=U(1)U(1)_{R}^{2}~{}$
$\displaystyle\mbox{the electric theory}:~{}-{1\over 2}\times(-1)^{2}\times
8+{1\over 2N_{f}}\times\left(-{5\over N_{f}}\right)^{2}\times N_{f}\times
8={100\over N_{f}^{2}}-4,$ $\displaystyle\mbox{the magnetic theory}:~{}{1\over
N_{f}-4}\times\left(-1+{2\over
N_{f}-4}\right)^{2}\times{(N_{f}-4)(N_{f}-3)\over 2}$ $\displaystyle\hskip
113.81102pt-\left({1\over 2N_{f}}+{1\over
2(N_{f}-4)}\right)\times\left(-1+{5\over N_{f}}-{1\over
N_{f}-4}\right)^{2}\times(N_{f}-4)\times N_{f}$ $\displaystyle\hskip
113.81102pt+{1\over N_{f}}\times\left(1-{10\over
N_{f}}\right)^{2}\times{N_{f}(N_{f}+1)\over 2}-1\times(-1)^{2}={100\over
N_{f}^{2}}-4,$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}^{2}U(1)_{R}=U(1)^{2}U(1)_{R}~{}$ $\displaystyle\mbox{the electric
theory}:~{}\left(-{1\over 2}\right)^{2}\times(-1)\times 8+\left({1\over
2N_{f}}\right)^{2}\times\left(-{5\over N_{f}}\right)\times N_{f}\times
8=-{10\over N_{f}^{2}}-2,$ $\displaystyle\mbox{the magnetic
theory}:~{}\left({1\over N_{f}-4}\right)^{2}\times\left(-1+{2\over
N_{f}-4}\right)\times{(N_{f}-4)(N_{f}-3)\over 2}$ $\displaystyle\hskip
113.81102pt+\left(-{1\over 2N_{f}}-{1\over
2(N_{f}-4)}\right)^{2}\times\left(-1+{5\over N_{f}}-{1\over
N_{f}-4}\right)\times(N_{f}-4)\times N_{f}$ $\displaystyle\hskip
113.81102pt+\left({1\over N_{f}}\right)^{2}\times\left(1-{10\over
N_{f}}\right)\times{N_{f}(N_{f}+1)\over 2}+(-1)^{2}\times(-1)=-{10\over
N_{f}^{2}}-2.$

Under the gauge transformation (7), the gauge invariant operator $S^{2}Q^{4}$
is left invariant by definition. Therefore, under the discrete transformation
(5), it transforms as

$\displaystyle S^{2}Q^{4}~{}\to~{}\exp\left[2\pi{i}\left({4\over
2N_{f}}-1\right)\right]S^{2}Q^{4}.$

The contribution from the spinor $S$ to the discrete charge of $S^{2}Q^{4}$
accounts for the discrepancy of the $U(1)$ symmetries between the $Spin(7)$
theory and its dual, upon the promotion of the discrete symmetries into
anomaly free $U(1)$ symmetries, when choosing $p=-1$, as we have seen in the
previous sections.

In summary, we have seen that the discrete ’t Hooft anomaly matching between
the $Spin(7)$ theory and its dual can beautifully be shown by using the parent
dual pair, the Spin(8) theory and its dual.

##  3 The dual pair of the $Spin(10)$ with a spinor and vectors

We will consider an ${{\cal N}}=1$ supersymmetric $Spin(10)$ gauge theory with
a single spinor $S$ and $N_{f}$ vectors $Q^{i}$ ($i=1,\cdots,N_{f}$) with no
superpotentials and its dual theory[2, 3]. See Table 9 for the charge
assignments of the matter fields in the electric $Spin(10)$ theory. There is
an independent ${{\bf Z}}_{2N_{f}}$ symmetry in the theory. The discrete
${{\bf Z}}_{2N_{f}}$ symmetry transformation act on the vectors $Q^{i}$ and
the spinor $S$ as

$\displaystyle Q^{i}\quad\to\quad\exp\left({2\pi{i}\over
2N_{f}}\right)Q^{i},\qquad S\quad\to\quad S,$

which is anomaly free discrete symmetry. We will introduce a singlet $\Phi$
and a vector $P$ with the superpotential $\Phi{PP}$ in order to promote the
discrete symmetry ${{\bf Z}}_{2N_{f}}$ to an anomaly free $U(1)$ symmetry for
computations of Type II discrete ’t Hooft anomalies so that the fields $\Phi$,
$P$ cancel the gauge anomaly of the promoted $U(1)$ symmetry by the $Spin(10)$
gauge interactions.

| $Spin(10)$ | $SU(N_{f})$ | $U(1)$ | $U(1)_{R}$ | ${{\bf Z}}_{2N_{f}}$
---|---|---|---|---|---
$S$ | 16 | $1$ | $-N_{f}$ | $1$ | $0$
$Q^{i}$ | 10 |  | $2$ | $1-8/N_{f}$ | $1$
$\Phi$ | 1 | $1$ | $0$ | $0$ | $2N_{f}$
$P$ | 10 | $1$ | $0$ | $1$ | $-N_{f}$
Table 9: The electric $Spin(10)$ theory with a spinor and $N_{f}$ vectors

In order to make the $U(1)_{R}$ charges integers, we will multiply them by
$N_{f}$. Since the spinor $S$ does not give any contributions to the discrete
anomalies, we may divide the $U(1)$ charges by two. Each of the fields gives a
multiple of $q\dim{R}\dim{R_{g}}$ as its contribution to all the discrete
anomalies except for ${{\bf Z}}_{2N_{f}}SU(N_{f})^{2}$, where $q$ is the
discrete charge of the field, $\dim{R}$ is the dimension of its representation
of the flavor symmetry group $SU(N_{f})$ and $\dim{R_{g}}$ is the dimension of
its representation of the gauge group $Spin(10)$. As we can see from Table 9,
the quantity $q\dim{R}\dim{R_{g}}$ for each of all the fields is always a
multiple of $2N_{f}$, and therefore, all the discrete anomalies except for
${{\bf Z}}_{2N_{f}}SU(N_{f})^{2}$ are zero modulo $2N_{f}$. In fact, we
compute the discrete ’t Hooft anomalies;

$\displaystyle\bullet~{}{{\bf Z}}_{2N_{f}}SU(N_{f})^{2}:~{}10,$
$\displaystyle\bullet~{}{{\bf Z}}_{2N_{f}}(\mbox{gravity})^{2}:~{}1\times
10N_{f}+2N_{f}\times 1+(-N_{f})\times 10=2N_{f},$
$\displaystyle\bullet~{}{{\bf Z}}_{2N_{f}}^{3}:~{}1^{3}\times
10N_{f}+\left(2N_{f}\right)^{3}\times 1+(-N_{f})^{3}\times
10=(-2N_{f}^{2}+5)\times 2N_{f},$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}U(1)_{R}^{2}:~{}1\times\left({-8}\right)^{2}\times
10N_{f}+{2N_{f}}\times\left({-N_{f}}\right)^{2}=(N_{f}^{2}+320)\times 2N_{f},$
$\displaystyle\bullet~{}{{\bf Z}}_{2N_{f}}U(1)^{2}:~{}1\times 1^{2}\times
10N_{f}=5\times 2N_{f},$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}U(1)_{R}U(1):~{}1\times\left({-8}\right)\times 1\times
10N_{f}=-40\times 2N_{f},$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}}^{2}U(1)_{R}:~{}1^{2}\times\left({-8}\right)\times
10N_{f}+\left(2N_{f}\right)^{2}\times\left({-N_{f}}\right)=-(40+2N_{f}^{2})\times
2N_{f}.$ $\displaystyle\bullet~{}{{\bf Z}}_{2N_{f}}^{2}U(1):~{}1^{2}\times
1\times 10N_{f}=5\times 2N_{f}.$

The magnetic theory exists for $7\leq N_{f}\leq 21$, and it is an ${{\cal
N}}=1$ supersymmetric $SU(N_{f}-5)$ gauge theory with $N_{f}$ antifundamentals
$\tilde{q}_{i}$, a single fundamental $q$, a symmetric tensor $s$ and singlets
$M^{ij}$, $Y^{i}$, with the superpotential [2, 3]

$\displaystyle
M^{ij}\tilde{q}_{i}\cdot{s}\cdot\tilde{q}_{j}+Y^{i}\tilde{q}_{i}\cdot{q}+\det{s}.$

The charge assignments for the magnetic fields are listed in Table 10.

| $SU(N_{f}-5)$ | $SU(N_{f})$ | $U(1)$ | $U(1)_{R}$ | ${{\bf Z}}_{2N_{f}(N_{f}-5)}$
---|---|---|---|---|---
$\tilde{q}_{i}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $-2$ | ${8/N_{f}}-{1/(N_{f}-5)}$ | $-(2N_{f}-5)$
${q}$ |  | ${{\bf 1}}$ | $2N_{f}$ | $-1+{1/(N_{f}-5)}$ | $N_{f}$
$s$ |  | ${{\bf 1}}$ | $0$ | ${2/(N_{f}-5)}$ | $2N_{f}$
$M^{ij}$ | ${\bf 1}$ |  | $4$ | $2-{16/N_{f}}$ | $2(N_{f}-5)$
$Y^{i}$ | ${\bf 1}$ |  | $-2(N_{f}-1)$ | $3-{8/N_{f}}$ | $(N_{f}-5)$
$X$ | ${{\bf 1}}$ | ${{\bf 1}}$ | 0 | 0 | $-2N_{f}(N_{f}-5)$
Table 10: The magnetic theory of the $Spin(10)$ with a spinor and vectors

The gauge invariant operators $M^{ij}\sim Q^{i}Q^{j}$, $Y^{i}\sim S^{2}Q^{i}$,
$B\sim S^{2}Q^{5}\sim\tilde{q}^{N_{f}-5}$ are transformed under the discrete
symmetry transformation of the electric theory as

$\displaystyle M^{ij}\to\exp\left({2\pi{i}\over N_{f}}\right)M^{ij},\qquad
Y^{i}\to\exp\left({2\pi{i}\over 2N_{f}}\right)Y^{i},\qquad
B\to\exp\left(2\pi{i}{5\over 2N_{f}}\right)B,$

which, together with the invariance of the superpotential, determines the
discrete symmetry transformation of the other magnetic fields

$\displaystyle\tilde{q}_{i}\to\exp\left[2\pi{i}\left({1+2p\over
2(N_{f}-5)}-{1\over 2N_{f}}\right)\right]\tilde{q}_{i},$
$\displaystyle{s}\to\exp\left(-{2\pi{i}}{1+2p\over
N_{f}-5}\right){s},\qquad{q}\to\exp\left(-{2\pi{i}}{1+2p\over
2(N_{f}-5)}\right){q},$

up to an integer $p$. The discrete symmetry transformations with different
values of $p$ are related to one another by the gauge transformations given by
elements of the center of the gauge group $SU(N_{f}-5)$, and therefore, there
exists only one independent discrete symmetry out of them.

Since our intension for the promotion of the discrete symmetry to an anomaly
free $U(1)$ symmetry is just to define Type II discrete ’t Hooft anomalies in
the magnetic theory, we do not have to take the $U(1)$ symmetry to be the
promoted $U(1)$ symmetry of the electric theory through the correspondence of
the gauge invariant operators. When we choose $p=-1$ and promote it to the
$U(1)$ symmetry transformation of the magnetic theory

$\displaystyle M^{ij}\to\exp\left({2\pi{i}\over
N_{f}}\omega\right)M^{ij},\quad Y^{i}\to\exp\left({2\pi{i}\over
2N_{f}}\omega\right)Y^{i},\quad\tilde{q}_{i}\to\exp\left(-2\pi{i}{2N_{f}-5\over
2N_{f}(N_{f}-5)}\omega\right)\tilde{q}_{i},$
$\displaystyle{s}\to\exp\left({2\pi{i}\over
N_{f}-5}\omega\right){s},\qquad{q}\to\exp\left({2\pi{i}\over
2(N_{f}-5)}\omega\right){q},$

with a transformation parameter $\omega$, we find that the promoted $U(1)$
symmetry is anomaly free, and we need to introduce no more fields to cancel
the gauge anomaly. However, the gauge invariant operator
$B_{mag}\sim\tilde{q}^{N_{f}-5}$ transforms under the magnetic $U(1)$ symmetry
into the transform of $B_{ele}\sim S^{2}Q^{5}$ under the electric $U(1)$
symmetry multiplied by $e^{-2\pi{i}\omega}$.

For the promotion to the anomaly free $U(1)$ symmetry, we will also replace
the term $\det{s}$ in the superpotential by $X\det{s}$ with a singlet $X$,
transforming under the promoted magnetic $U(1)$ symmetry as

$\displaystyle X\to\exp\left(-2\pi{i}\omega\right)X.$

In order to compute the Type II discrete ’t Hooft anomalies, we will introduce
a $U(1)$ gauge superfield for the promoted magnetic $U(1)$ symmetry and will
turn on the Fayet-Iliopoulos term in the $D$-term potential of the $U(1)$
gauge symmetry. Then, we will find a vacuum with
$\left\langle{X}\right\rangle\not=0$, which breaks the $U(1)$ symmetry down
back to the original discrete symmetry in the magnetic theory. In the
infrared, the theory is reduced into the original magnetic theory.

The discrete symmetry group is a subgroup of a cyclic group ${{\bf
Z}}_{2N_{f}(N_{f}-5)}$. However, when we perform the above discrete symmetry
transformation twice, the resulting transformation can be given by an element
of the center of the flavor group and an element of the center of the gauge
group, and therefore, it is not an independent discrete symmetry, anymore.

As is done for the electric theory, we will multiply the $U(1)_{R}$ charges by
$N_{f}$ and divide the $U(1)$ charges by two. Let $\Psi$ be one of the
magnetic fields with the discrete charge $q$ in the representation $R$ of the
flavor symmetry group $SU(N_{f})$ and in the representation $R_{g}$ of the
gauge symmetry group $SU(N_{f}-5)$. The field $\Psi$ gives its contributions
to all the discrete ’t Hooft anomaly except for ${{\bf
Z}}_{2N_{f}(N_{f}-5)}SU(N_{f})^{2}$ by a multiple of $q\dim{R}\dim{R_{g}}$. As
we can see from Table 10, the combination $q\dim{R}\dim{R_{g}}$ is a multiple
of $N_{f}(N_{f}-5)$, and therefore, checking the discrete anomaly matching
except for ${{\bf Z}}_{2N_{f}(N_{f}-5)}SU(N_{f})^{2}$ is to examine whether it
is an even or odd multiple of $N_{f}(N_{f}-5)$ for the anomalies including no
$U(1)_{R}$. Although the $U(1)_{R}$ charges of the magnetic fields are not all
integers, even after multiplying them by $N_{f}$, we find that the anomalies
including the $U(1)_{R}$ symmetry are also integers by the computations,

$\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}(N_{f}-5)}SU(N_{f})^{2}:~{}(-2N_{f}+5)\times(N_{f}-5)+2(N_{f}-5)\times(N_{f}+2)+(N_{f}-5)$
$\displaystyle\hskip 128.0374pt=10(N_{f}-5),$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}(N_{f}-5)}(\mbox{gravity})^{2}:~{}(-2N_{f}+5)\times
N_{f}(N_{f}-5)+N_{f}\times(N_{f}-5)$ $\displaystyle\hskip
128.0374pt+(2N_{f})\times{(N_{f}-5)(N_{f}-4)\over
2}+2(N_{f}-5)\times{N_{f}(N_{f}+1)\over 2}$ $\displaystyle\hskip
128.0374pt+(N_{f}-5)\times
N_{f}+\left(-2N_{f}(N_{f}-5)\right)=2N_{f}(N_{f}-5),$
$\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}(N_{f}-5)}^{3}:~{}(-2N_{f}+5)^{3}\times
N_{f}(N_{f}-5)+N_{f}^{3}\times(N_{f}-5)+(2N_{f})^{3}\times{(N_{f}-5)(N_{f}-4)\over
2}$ $\displaystyle\hskip
76.82234pt+\left(2(N_{f}-5)\right)^{3}\times{N_{f}(N_{f}+1)\over
2}+(N_{f}-5)^{3}\times N_{f}+\left(-2N_{f}(N_{f}-5)\right)^{3}$
$\displaystyle\hskip 76.82234pt=-(N_{f}-5)^{2}(4N_{f}^{2}-5)\times
2N_{f}(N_{f}-5),$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}(N_{f}-5)}U(1)_{R}^{2}:~{}(-2N_{f}+5)\times\left(-N_{f}+8-{N_{f}\over
N_{f}-5}\right)^{2}\times N_{f}(N_{f}-5)$ $\displaystyle\hskip
105.2751pt+N_{f}\times\left(-2N_{f}+{N_{f}\over
N_{f}-5}\right)^{2}\times(N_{f}-5)$ $\displaystyle\hskip
105.2751pt+(2N_{f})\times\left(-N_{f}+{2N_{f}\over
N_{f}-5}\right)^{2}\times{(N_{f}-5)(N_{f}-4)\over 2}$ $\displaystyle\hskip
105.2751pt+2(N_{f}-5)\times\left(N_{f}-16\right)^{2}\times{N_{f}(N_{f}+1)\over
2}$ $\displaystyle\hskip
105.2751pt+(N_{f}-5)\times\left(2N_{f}-8\right)^{2}\times
N_{f}+\left(-2N_{f}(N_{f}-5)\right)\times(-1)^{2}$ $\displaystyle\hskip
105.2751pt=320\times 2N_{f}(N_{f}-5),$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}(N_{f}-5)}U(1)^{2}:~{}(-2N_{f}+5)\times\left(-1\right)^{2}\times
N_{f}(N_{f}-5)+N_{f}\times\left(N_{f}\right)^{2}\times(N_{f}-5)$
$\displaystyle\hskip 105.2751pt+2(N_{f}-5)\times
2^{2}\times{N_{f}(N_{f}+1)\over
2}+(N_{f}-5)\times\left(-(N_{f}-1)\right)^{2}\times N_{f}$
$\displaystyle\hskip 105.2751pt=\left(N_{f}^{2}+5\right)\times
2N_{f}(N_{f}-5),$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}(N_{f}-5)}U(1)_{R}U(1):~{}(-2N_{f}+5)\times\left(-N_{f}+8-{N_{f}\over
N_{f}-5}\right)\times(-1)\times N_{f}(N_{f}-5)$ $\displaystyle\hskip
128.0374pt+N_{f}\times\left(-2N_{f}+{N_{f}\over N_{f}-5}\right)\times
N_{f}\times(N_{f}-5)$ $\displaystyle\hskip
128.0374pt+2(N_{f}-5)\times\left(N_{f}-16\right)\times
2\times{N_{f}(N_{f}+1)\over 2}$ $\displaystyle\hskip
128.0374pt+(N_{f}-5)\times\left(2N_{f}-8\right)^{2}\times\left(-(N_{f}-1)\right)\times
N_{f}$ $\displaystyle\hskip 128.0374pt=-2(N_{f}^{2}+20)\times
2N_{f}(N_{f}-5),$ $\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}(N_{f}-5)}^{2}U(1)_{R}:~{}(-2N_{f}+5)^{2}\times\left(-N_{f}+8-{N_{f}\over
N_{f}-5}\right)\times N_{f}(N_{f}-5)$ $\displaystyle\hskip
105.2751pt+N_{f}^{2}\times\left(-2N_{f}+{N_{f}\over
N_{f}-5}\right)\times(N_{f}-5)$ $\displaystyle\hskip
105.2751pt+(2N_{f})^{2}\times\left(-N_{f}+{2N_{f}\over
N_{f}-5}\right)\times{(N_{f}-5)(N_{f}-4)\over 2}$ $\displaystyle\hskip
105.2751pt+\left(2(N_{f}-5)\right)^{2}\times\left(N_{f}-16\right)\times{N_{f}(N_{f}+1)\over
2}$ $\displaystyle\hskip
105.2751pt+(N_{f}-5)^{2}\times\left(2N_{f}-8\right)\times
N_{f}+\left(-2N_{f}(N_{f}-5)\right)^{2}\times(-1)$ $\displaystyle\hskip
105.2751pt=-4(N_{f}-5)(N_{f}^{2}+10)\times 2N_{f}(N_{f}-5),$
$\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}(N_{f}-5)}^{2}U(1):~{}(-2N_{f}+5)^{2}\times\left(-1\right)\times
N_{f}(N_{f}-5)+N_{f}^{2}\times\left(N_{f}\right)\times(N_{f}-5)$
$\displaystyle\hskip 105.2751pt+\left(2(N_{f}-5)\right)^{2}\times
2\times{N_{f}(N_{f}+1)\over
2}+(N_{f}-5)^{2}\times\left(-(N_{f}-1)\right)\times N_{f}$
$\displaystyle\hskip 105.2751pt=5\left(N_{f}-5\right)\times 2N_{f}(N_{f}-5).$

In order to check the discrete ’t Hooft matching conditions between the dual
theories, we will embed the ${{\bf Z}}_{2N_{f}}$ symmetry group of the
electric theory into the ${{\bf Z}}_{2N_{f}(N_{f}-5)}$ group by multiplying
the discrete charges of the electric fields by $N_{f}-5$. Then, we can see
that the ${{\bf Z}}_{2N_{f}(N_{f}-5)}SU(N_{f})^{2}$ anomaly in the electric
theory is in agreement with the one in the magnetic theory. Recalling that the
other discrete anomalies in the electric theory are zero modulo
$2N_{f}(N_{f}-5)$, we find that the ’t Hooft anomaly matching conditions of
them are also satisfied by the magnetic theory.

Incidentally, let us consider the discrete ’t Hooft anomalies for the other
cases with $p\not=-1$. When $n\equiv p+1\not=0$, the invariance of the term
$X\det{s}$ in the superpotential requires the discrete charge of $X$ to be
$-2N_{f}(N_{f}-5)(1-2n)$, as in Table 11. Then, the promoted $U(1)$ symmetry
is anomalous by the $SU(N_{f}-5)$ gauge interactions, and therefore, we need
to introduce more matter fields listed in Table 11 to cancel the gauge anomaly
in order to promote the discrete ${{\bf Z}}_{2N_{f}(N_{f}-5)}$ symmetry to an
anomaly $U(1)$ symmetry. We will also add the term $\tilde{X}F\tilde{F}$ to
the superpotential so that the extra matter fields $\tilde{X}$, $F$,
$\tilde{F}$ decouple in the vacuum $\langle{\tilde{X}}\rangle\not=0$ at the
low energies. The discrete charges of $F$ and $\tilde{F}$ are determined by
cancellation of the gauge anomaly of the promoted $U(1)$ symmetry and by the
requirement that performing the discrete symmetry transformation twice gives
the gauge transformation of the element of the center of the gauge group
$SU(N_{f}-5)$. We have chosen the $U(1)_{R}$ charges of $F$ and $\tilde{F}$ so
as to saturate the ’t Hooft anomaly matching condition for ${{\bf
Z}}_{2N_{f}(N_{f}-5)}U(1)_{R}^{2}$.

| $SU(N_{f}-5)$ | $SU(N_{f})$ | $U(1)$ | $U(1)_{R}$ | ${{\bf Z}}_{2N_{f}(N_{f}-5)}$
---|---|---|---|---|---
$\tilde{X}$ | ${{\bf 1}}$ | ${{\bf 1}}$ | $0$ | $0$ | $-2N_{f}(N_{f}-5)n$
$F$ |  | ${{\bf 1}}$ | $0$ | $1+{1/(N_{f}-5)}$ | $nN_{f}(N_{f}-5)+N_{f}$
$\tilde{F}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | ${{\bf 1}}$ | $0$ | $1-{1/(N_{f}-5)}$ | $nN_{f}(N_{f}-5)-N_{f}$
${X}$ | ${{\bf 1}}$ | ${{\bf 1}}$ | $0$ | $0$ | $-2N_{f}(N_{f}-5)(1-2n)$
Table 11: The discrete charge of $X$ and the additional fields $\tilde{X}$,
$F$, $\tilde{F}$ for $p\not=-1$ in the magnetic dual of the $Spin(10)$ theory,
where $n=p+1$.

The remaining procedure we have to carry out for the computation of the
discrete anomalies is almost the same as what was done for $p=-1$, except that
we take the vacuum such that $\left\langle{X}\right\rangle\not=0$,
$\langle\tilde{X}\rangle\not=0$.

The ${{\bf Z}}_{2N_{f}(N_{f}-5)}SU(N_{f})^{2}$ anomaly for $p\not=-1$ is
computed to give

$\displaystyle\bullet~{}{{\bf
Z}}_{2N_{f}(N_{f}-5)}SU(N_{f})^{2}:~{}10(N_{f}-5)+2N_{f}(N_{f}-5)n,$

which is equal modulo $2N_{f}(N_{f}-5)$ to the one for the case $p=-1$
($n=0$). For the rest of the discrete anomalies, multiplying the $U(1)_{R}$
charges by $N_{f}$ and dividing the $U(1)$ charges by two, we find that they
are all zero modulo $2N_{f}(N_{f}-5)$, satisfying all the ’t Hooft anomaly
matching conditions.

##  4 Discussions

We have studied the discrete anomaly matching of the two dual pairs. One of
them is the $Spin(7)$ gauge theory with spinors and the $SU(N_{f}-4)$ gauge
theory with a symmetric tensor, fundamentals and singlets [1]. The other is
the $Spin(10)$ gauge theory with a spinor and vectors and the $SU(N_{f}-5)$
gauge theory with a symmetric tensor, fundamentals and singlets [2, 3]. We
have shown that both of the dual pairs satisfy the discrete anomaly matching
conditions.

For the dual pair of the $Spin(7)$ theory, we have done this in two ways. In
one way, we have embedded the discrete symmetries into an anomaly free $U(1)$
symmetries by additional fields, which decouple after the $U(1)$ symmetry
breaking into the discrete symmetries, on the both sides of the duality. The
extended theories are not dual to each other, and we have to compute the
continuous ’t Hooft anomalies in order to ensure the anomaly matching
conditions. In the other way, we take another dual pair [15] of the $Spin(8)$
gauge theory with a spinor and vectors and the $SU(N_{f}-4)$ gauge theory with
a symmetric tensor, anti-fundamentals and singlets, which is reduced to the
$Spin(7)$ dual pair by higgsing the $Spin(8)$ gauge group to $Spin(7)$. The
$Spin(8)$ dual pair has an anomaly free $U(1)$ symmetry, which is broken to
the discrete symmetries of the $Spin(7)$ dual pair upon the higgsing of
$Spin(8)$. Since the continuous ’t Hooft anomaly matching is satisfied by the
$Spin(8)$ dual pair and since the discrete anomalies of the $Spin(7)$ dual
pair are given by the continuous anomalies of the $Spin(8)$ dual pair, we have
seen that the discrete anomaly matching conditions are obviously satisfied by
the $Spin(7)$ dual pair, modulo subtleties with the gauge transformation
involved in the actual computations.

For the $Spin(10)$ dual pair, we have no known parent dual pair and we have
computed the discrete anomalies by embedding the discrete symmetries on the
both sides of the duality into anomaly free $U(1)$ symmetries. In Appendix A,
we have constructed the extended theories with the embedding $U(1)$
symmetries, which satisfy the continuous ’t Hooft anomaly matching conditions
for the anomalies with the $U(1)$ symmetries. Although the $U(1)$ symmetries
of the both sides of the duality are different from each other, the continuous
anomalies with the $U(1)$ symmetries become the discrete anomalies after the
higgsing of the $U(1)$ symmetries, and we easily see that the discrete anomaly
matching is achieved.

Another dual pair of an ${{\cal N}}=1$ supersymmetric $G_{2}$ gauge theory
with fundamentals in the representation ${\bf 7}$ and an ${{\cal N}}=1$
supersymmetric $SU(N_{f}-3)$ gauge theory with a symmetric tensor,
fundamentals and singlets is reduced from the $Spin(7)$ dual pair [1]777See
also [16], for the confining phases of the $G_{2}$ gauge theory. by higgsing
the $Spin(7)$ gauge group. Therefore, it should be straightforward to check
the discrete anomaly matching in a similar way to what we have done for the
$Spin(8)$ dual pair. This is also the case for other Pouliot type dualities
[17] reduced from the $Spin(10)$ dual pair.

There are no known dual pair, from which the $Spin(10)$ dual pair is derived.
Although we wish that the extended theories in Appendix A would give an
insight into the discovery of such a parent dual pair, we guess that the gauge
group of the electric $Spin(10)$ theory should be larger than the $Spin(10)$
group 888See [18] for the confining phases of ${{\cal N}}=1$ supersymmetric
$Spin(11)$ and $Spin(12)$ gauge theories with a spinor and vectors., so that
the higgsing in the electric theory should correspond to the decoupling of
massive states in the magnetic theory.

Finally, we may extend the studies in this paper to a dual pair of an ${{\cal
N}}=1$ supersymmetric $Spin(10)$ gauge theory with more than one spinor and
vectors, and the dual theory [19]. However, we will leave this subject for
future investigation.

Acknowledgement

The author thanks Yukawa Institute for Theoretical Physics, Kyoto University
for hospitality during the course of this project.

Appendix

##  Appendix A The continuous ’t Hooft anomaly matching of the extended
theories of the $Spin(10)$ dual pair

Contrary to the $Spin(7)$ dual pair, there is no known parent dual pair, which
is reduced into the $Spin(10)$ theory and the dual $SU(N_{f}-5)$ theory by
higgsing or decoupling of massive states. Therefore, there are no reasons to
expect that the extended theories of the $Spin(10)$ theory and its dual
satisfy all the continuous ’t Hooft anomaly matching conditions. The extended
theories have the $U(1)$ symmetries, into which the discrete symmetries are
embedded. We will denote the $U(1)$ symmetries as $U(1)_{X}$, although the
$U(1)_{X}$ symmetry of the electric theory is not the same as the one of the
magnetic theory, for $p\not=0$.

For $p=-1$, we will compute the continuous ’t Hooft anomalies of the both
extended theories, and will find that there are discrepancies in the anomalies
with the $U(1)_{X}$ charges between the electric theory and the magnetic
theory. However, we will show that we can eliminate the discrepancies by
adding a set of fields to the electric theory, as shown in Table 12 and to the
magnetic theory as listed in Table 13. After higgsing the $U(1)_{X}$ symmetry
down to the discrete symmetry, the additional fields gain masses and decouple
in the infrared. Then, the anomalies with the $U(1)_{X}$ charges become the
discrete ’t Hooft anomalies of the original dual theories, and we can ensure
that they satisfy the discrete ’t Hooft anomaly matching conditions.

The anomalies without the $U(1)_{X}$ charges are the same as those of the
original dual theories, except for the $U(1)_{R}^{3}$ and
$U(1)_{R}(\mbox{gravity})^{2}$ anomalies. Therefore, they satisfy the ’t Hooft
anomaly matching conditions. For the $U(1)_{R}^{3}$ and
$U(1)_{R}(\mbox{gravity})^{2}$ anomalies, there are additional contributions
from the Higgs fields $\Phi$ in the electric theory and $X$ in the magnetic
theory, respectively. However, their contributions are the same, and the
$U(1)_{R}^{3}$ and $U(1)_{R}(\mbox{gravity})^{2}$ anomalies also satisfy the
anomaly matching conditions between the electric theory and the magnetic
theory. The remaining anomalies are those which the $U(1)_{X}$ charges take
part in, and our results of the computations of them are the following:

$\bullet$ $U(1)_{X}SU(N_{f})^{2}$:

$\displaystyle\mbox{the electric side}:~{}{5\over N_{f}},$
$\displaystyle\mbox{the magnetic side}:~{}\left(-{1\over 2N_{f}}-{1\over
2(N_{f}-5)}\right)(N_{f}-5)+{2\over 2N_{f}}(N_{f}+2)+{1\over 2N_{f}}={5\over
N_{f}}.$

$\bullet$ $U(1)_{X}(\mbox{gravity})^{2}$:

$\displaystyle\mbox{the electric side}:~{}{1\over 2N_{f}}\times
10N_{f}+1-{1\over 2}\times 10=1,$ $\displaystyle\mbox{the magnetic
side}:~{}\left(-{1\over 2N_{f}}-{1\over
2(N_{f}-5)}\right)N_{f}(N_{f}-5)+\left({1\over 2(N_{f}-5)}\right)(N_{f}-5)$
$\displaystyle\hskip 108.12054pt+\left({2\over
2(N_{f}-5)}\right){(N_{f}-5)(N_{f}-4)\over 2}+{2\over
2N_{f}}{N_{f}(N_{f}+1)\over 2}+{1\over 2N_{f}}N_{f}-1$ $\displaystyle\hskip
108.12054pt=1.$

$\bullet$ $U(1)_{X}^{3}$:

$\displaystyle\mbox{the electric side}:~{}\left({1\over
2N_{f}}\right)^{3}\times 10N_{f}+1^{3}+\left(-{1\over 2}\right)^{3}\times
10={5\over 4N_{f}^{2}}-{1\over 4},$ $\displaystyle\mbox{the magnetic
side}:~{}\left(-{1\over 2N_{f}}-{1\over
2(N_{f}-5)}\right)^{3}N_{f}(N_{f}-5)+\left({1\over
2(N_{f}-5)}\right)^{3}(N_{f}-5)$ $\displaystyle\hskip
108.12054pt+\left({2\over 2(N_{f}-5)}\right)^{3}{(N_{f}-5)(N_{f}-4)\over
2}+\left({2\over 2N_{f}}\right)^{3}{N_{f}(N_{f}+1)\over 2}$
$\displaystyle\hskip 108.12054pt+\left({1\over
2N_{f}}\right)^{3}N_{f}-1={5\over 4N_{f}^{2}}-1.$

$\bullet$ $U(1)_{X}U(1)_{R}^{2}$:

$\displaystyle\mbox{the electric side}:~{}{1\over 2N_{f}}\times\left(-{8\over
N_{f}}\right)^{2}\times 10N_{f}+1\times(-1)^{2}=320{1\over N_{f}^{2}}+1,$
$\displaystyle\mbox{the magnetic side}:~{}\left(-{1\over 2N_{f}}-{1\over
2(N_{f}-5)}\right)\left(-1+{8\over N_{f}}-{1\over
N_{f}-5}\right)^{2}N_{f}(N_{f}-5)$ $\displaystyle\hskip
108.12054pt+\left({1\over 2(N_{f}-5)}\right)\left(-2+{1\over
N_{f}-5}\right)^{2}(N_{f}-5)$ $\displaystyle\hskip 108.12054pt+\left({2\over
2(N_{f}-5)}\right)\left(-1+{2\over N_{f}-5}\right)^{2}{(N_{f}-5)(N_{f}-4)\over
2}$ $\displaystyle\hskip 108.12054pt+{2\over 2N_{f}}\left(1-{16\over
N_{f}}\right)^{2}{N_{f}(N_{f}+1)\over 2}+{1\over 2N_{f}}(2-{8\over
N_{f}})^{2}N_{f}-1\times(-1)^{2}$ $\displaystyle\hskip 108.12054pt=320{1\over
N_{f}^{2}}.$

$\bullet$ $U(1)_{X}U(1)^{2}$:

$\displaystyle\mbox{the electric side}:~{}{1\over 2N_{f}}\times 2^{2}\times
10N_{f}=20,$ $\displaystyle\mbox{the magnetic side}:~{}\left(-{1\over
2N_{f}}-{1\over 2(N_{f}-5)}\right)\left(-2\right)^{2}N_{f}(N_{f}-5)$
$\displaystyle\hskip 108.12054pt+\left({1\over
2(N_{f}-5)}\right)\left(2N_{f}\right)^{2}(N_{f}-5)+{2\over 2N_{f}}\times
4^{2}\times{N_{f}(N_{f}+1)\over 2}$ $\displaystyle\hskip 108.12054pt+{1\over
2N_{f}}\Big{(}-2(N_{f}-1)\Big{)}^{2}N_{f}=20+4N_{f}^{2}.$

$\bullet$ $U(1)_{X}U(1)_{R}U(1)$:

$\displaystyle\mbox{the electric side}:~{}{1\over 2N_{f}}\times
2\times\left(-{8\over N_{f}}\right)\times 10N_{f}=-{80\over N_{f}},$
$\displaystyle\mbox{the magnetic side}:~{}\left(-{1\over 2N_{f}}-{1\over
2(N_{f}-5)}\right)(-2)\left(-1+{8\over N_{f}}-{1\over
N_{f}-5}\right)N_{f}(N_{f}-5)$ $\displaystyle\hskip 108.12054pt+\left({1\over
2(N_{f}-5)}\right)(2N_{f})\left(-2+{1\over N_{f}-5}\right)(N_{f}-5)$
$\displaystyle\hskip 108.12054pt+{2\over 2N_{f}}4\left(1-{16\over
N_{f}}\right){N_{f}(N_{f}+1)\over 2}+{1\over
2N_{f}}\big{(}-2(N_{f}-1)\big{)}\left(2-{8\over N_{f}}\right)N_{f}$
$\displaystyle\hskip 108.12054pt=-{80\over N_{f}}-4N_{f}.$

$\bullet$ $U(1)_{X}^{2}U(1)_{R}$:

$\displaystyle\mbox{the electric side}:~{}\left({1\over
2N_{f}}\right)^{2}\times\left(-{8\over N_{f}}\right)\times 10N_{f}=-{20\over
N_{f}^{2}},$ $\displaystyle\mbox{the magnetic side}:~{}\left(-{1\over
2N_{f}}-{1\over 2(N_{f}-5)}\right)^{2}\left(-1+{8\over N_{f}}-{1\over
N_{f}-5}\right)N_{f}(N_{f}-5)$ $\displaystyle\hskip 108.12054pt+\left({1\over
2(N_{f}-5)}\right)^{2}\left(-2+{1\over N_{f}-5}\right)(N_{f}-5)$
$\displaystyle\hskip 108.12054pt+\left({2\over
2(N_{f}-5)}\right)^{2}\left(-1+{2\over N_{f}-5}\right){(N_{f}-5)(N_{f}-4)\over
2}$ $\displaystyle\hskip 108.12054pt+\left({2\over
2N_{f}}\right)^{2}\left(1-{16\over N_{f}}\right){N_{f}(N_{f}+1)\over
2}+\left({1\over 2N_{f}}\right)^{2}\left(2-{8\over N_{f}}\right)N_{f}$
$\displaystyle\hskip 108.12054pt=-{20\over N_{f}^{2}}-1.$

$\bullet$ $U(1)_{X}^{2}U(1)$:

$\displaystyle\mbox{the electric side}:~{}\left({1\over
2N_{f}}\right)^{2}\times 2\times 10N_{f}+1^{2}\times(-1)={5\over N_{f}}-1,$
$\displaystyle\mbox{the magnetic side}:~{}\left(-{1\over 2N_{f}}-{1\over
2(N_{f}-5)}\right)^{2}\left(-2\right)N_{f}(N_{f}-5)$ $\displaystyle\hskip
108.12054pt+\left({1\over
2(N_{f}-5)}\right)^{2}\left(2N_{f}\right)(N_{f}-5)+\left({2\over
2N_{f}}\right)^{2}4{N_{f}(N_{f}+1)\over 2}$ $\displaystyle\hskip
108.12054pt+\left({1\over
2N_{f}}\right)^{2}\Big{(}-2(N_{f}-1)\Big{)}N_{f}+(-1)^{2}\times(-1)={5\over
N_{f}}-1.$

| $Spin(10)$ | $SU(N_{f})$ | $U(1)$ | $U(1)_{R}$ | $U(1)_{X}$
---|---|---|---|---|---
$S$ | 16 | ${{\bf 1}}$ | $-N_{f}$ | $1$ | $0$
$Q^{i}$ | 10 |  | $2$ | $1-8/N_{f}$ | $1/(2N_{f})$
$\Phi$ | 1 | ${{\bf 1}}$ | $0$ | $0$ | $1$
$P$ | 10 | ${{\bf 1}}$ | $0$ | $1$ | $-1/2$
$\tilde{\Phi}$ | 1 | 1 | 0 | 0 | $-1$
${\Psi}$ | 1 | 1 | 0 | 1 | 1/2
$\tilde{\Psi}$ | 1 | 1 | 0 | 1 | 1/2
${\Xi}$ | 1 | 1 | 0 | 0 | $-2$
${G}$ | 1 | 1 | $-N_{f}$ | 2 | 2
${H}$ | 1 | 1 | $N_{f}$ | 0 | 0
$\tilde{\Xi}$ | 1 | 1 | 0 | 0 | $-2$
$\tilde{G}$ | 1 | 1 | $N_{f}$ | 0 | 2
$\tilde{H}$ | 1 | 1 | $-N_{f}$ | 2 | 0
Table 12: The original fields and the set of the fields added to the extended
$Spin(10)$ theory to fill the gaps in the continuous ’t Hooft anomalies

We see that there are discrepancies in the anomalies except for
$U(1)_{X}SU(N_{f})$, $U(1)_{X}(\mbox{gravity})^{2}$ and $U(1)_{X}^{2}U(1)$,
between both the sides. We will add the singlet fields $\tilde{\Phi}$, $\Psi$
and $\tilde{\Psi}$, listed in Table 12, and the term
$\tilde{\Phi}\Psi\tilde{\Psi}$ to the superpotential in the electric theory.
The fields contribute to the ’t Hooft anomalies by

$\displaystyle\bullet~{}U(1)_{X}SU(N_{f})^{2}:~{}0,\qquad\bullet~{}U(1)_{X}:~{}0,\qquad\bullet~{}U(1)_{X}^{3}:~{}-1+2\times\left({1\over
2}\right)^{3}=-1+{1\over 4},$
$\displaystyle\bullet~{}U(1)_{X}U(1)_{R}^{2}:~{}-1,\qquad\bullet~{}U(1)_{X}U(1)^{2}:~{}0,\qquad\bullet~{}U(1)_{X}U(1)U(1)_{R}:~{}0,$
$\displaystyle\bullet~{}U(1)_{X}^{2}U(1)_{R}:~{}-1,\qquad\bullet~{}U(1)_{X}^{2}U(1):~{}0,$

which fill the gap in the anomalies $U(1)_{X}^{3}$, $U(1)_{X}U(1)_{R}^{2}$,
$U(1)_{X}^{2}U(1)_{R}$.

There still remain discrepancies in the anomalies $U(1)_{X}U(1)^{2}$ and
$U(1)_{X}U(1)_{R}U(1)$. To fill the gap, we introduce two sets of fields
$(\Xi,G,H)$, $(\tilde{\Xi},\tilde{G},\tilde{H})$, and add the terms
$\Xi{GH}+\tilde{\Xi}\tilde{G}\tilde{H}$ to the superpotential in the electric
theory. Their contributions to the ’t Hooft anomalies are the following:

$\displaystyle\bullet~{}U(1)_{X}SU(N_{f})^{2}:~{}0,\qquad\bullet~{}U(1)_{X}:~{}0,\qquad\bullet~{}U(1)_{X}^{3}:~{}0,\qquad\bullet~{}U(1)_{X}U(1)_{R}^{2}:~{}0,$
$\displaystyle\bullet~{}U(1)_{X}U(1)^{2}:~{}2\times(-N_{f})^{2}+2\times
N_{f}^{2}=4N_{f}^{2},$ $\displaystyle\bullet~{}U(1)_{X}U(1)_{R}U(1):~{}2\times
1\times(-N_{f})+2\times(-1)\times N_{f}=-4N_{f},$
$\displaystyle\bullet~{}U(1)_{X}^{2}U(1)_{R}:~{}-8,\qquad\bullet~{}U(1)_{X}^{2}U(1):~{}0,$

and we find that they fill the gap in the anomalies $U(1)_{X}U(1)^{2}$,
$U(1)_{X}U(1)_{R}U(1)$. However, unfortunately, they make a gap again in the
anomaly $U(1)_{X}^{2}U(1)_{R}$, and all the other anomalies satisfy the
matching conditions.

| $SU(N_{f}-5)$ | $SU(N_{f})$ | $U(1)$ | $U(1)_{R}$ | $U(1)_{X}$
---|---|---|---|---|---
$\tilde{q}_{i}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $\overline{\raisebox{-0.5pt}{\hbox{\rule{0.4pt}{6.5pt}\hskip-0.4pt\rule{6.5pt}{0.4pt}\hskip-6.5pt\rule[6.5pt]{6.5pt}{0.4pt}}\rule[6.5pt]{0.4pt}{0.4pt}\hskip-0.4pt\rule{0.4pt}{6.5pt}}}$ | $-2$ | ${8\over N_{f}}-{1/(N_{f}-5)}$ | $-1/(2N_{f})-1/(2(N_{f}-5))$
${q}$ |  | ${{\bf 1}}$ | $2N_{f}$ | $-1+{1/(N_{f}-5)}$ | $1/(2(N_{f}-5))$
$s$ |  | ${{\bf 1}}$ | $0$ | ${2/(N_{f}-5)}$ | $1/(N_{f}-5)$
$M^{ij}$ | ${\bf 1}$ |  | $4$ | $2-{16/N_{f}}$ | $1/N_{f}$
$Y^{i}$ | ${\bf 1}$ |  | $-2(N_{f}-1)$ | $3-{8/N_{f}}$ | $1/(2N_{f})$
$X$ | ${{\bf 1}}$ | ${{\bf 1}}$ | 0 | 0 | $-1$
$\tilde{X}^{\prime}$ | ${{\bf 1}}$ | ${{\bf 1}}$ | $0$ | $0$ | $-2$
$F^{\prime}$ | ${{\bf 1}}$ | ${{\bf 1}}$ | $0$ | $0$ | $2$
$\tilde{F}^{\prime}$ | ${{\bf 1}}$ | ${{\bf 1}}$ | $0$ | $2$ | $0$
Table 13: The original magnetic fields and the set of the fields added to the
extended $SU(N_{f}-5)$ theory to fill the gaps in the continuous ’t Hooft
anomalies

In order to fill the gap in the anomaly $U(1)_{X}^{2}U(1)_{R}$, we will add
three fields $\tilde{X}^{\prime}$, $F^{\prime}$, $\tilde{F}^{\prime}$, listed
in Table 13, and add the term $\tilde{X}^{\prime}F^{\prime}\tilde{F}^{\prime}$
to the superpotential in the magnetic extended theory. They contribute to the
anomaly $U(1)_{X}^{2}U(1)_{R}$ by $-8$, and finally, all the anomalies with
the $U(1)_{X}$ charges satisfy the continuous ’t Hooft anomaly matching
conditions.

However, the additional fields to the both sides cause the discrepancies in
the anomalies $U(1)_{R}^{3}$ and $U(1)_{R}(\mbox{gravity})^{2}$. Upon the
higgsing of the $U(1)_{X}$ symmetries by the vacuum expectation values
$\left\langle{\Phi}\right\rangle$, $\langle{\tilde{\Phi}}\rangle$,
$\left\langle{\Xi}\right\rangle$, $\langle{\tilde{\Xi}}\rangle$ on the
electric side, $\left\langle{X}\right\rangle$,
$\left\langle{X^{\prime}}\right\rangle$ on the magnetic side, those extended
theories are reduced into the original dual pair in the infrared, after the
decoupling of the massive fields, and we find that the continuous ’t Hooft
anomalies without the $U(1)_{X}$ charges match with each other. Since the
continuous ’t Hooft anomalies with the $U(1)_{X}$ charges yields the discrete
’t Hooft anomalies of the original dual theories in the infrared, it is
obvious from the above computations that they satisfy the discrete ’t Hooft
anomaly matching conditions.

## References

  * [1] P. Pouliot, “Chiral Duals of Nonchiral SUSY Gauge Theories,” Phys. Lett. B 359, 108-113 (1995), hep-th/9507018.
  * [2] P. Pouliot and M. J. Strassler, “Duality and Dynamical Supersymmetry Breaking in $Spin(10)$ with a Spinor,” Phys. Lett. B 375, 175-180 (1996), hep-th/9602031.
  * [3] T. Kawano, “Duality of ${{\cal N}}=1$ Supersymmetric $SO(10)$ Gauge Theory with Matter in the Spinorial Representation,” Prog. Theor. Phys. 95, 963-968 (1996), hep-th/9602035.
  * [4] D. Harlow and H. Ooguri, “Symmetries in Quantum Field Theory and Quantum Gravity,” Commun. Math. Phys. 383, no.3, 1669-1804 (2021), arXiv:1810.05338 [hep-th].
  * [5] L. M. Krauss and F. Wilczek, “Discrete Gauge Symmetry in Continuum Theories,” Phys. Rev. Lett. 62, 1221 (1989).
  * [6] J. Preskill, S. P. Trivedi, F. Wilczek and M. B. Wise, “Cosmology and Broken Discrete Symmetry,” Nucl. Phys. B 363, 207-220 (1991).
  * [7] L. E. Ibanez and G. G. Ross, “Discrete Gauge Symmetry Anomalies,” Phys. Lett. B 260, 291-295 (1991),
L. E. Ibanez and G. G. Ross, “Discrete Gauge Symmetries and the Origin of
Baryon and Lepton Number Conservation in Supersymmetric Versions of the
Standard Model,” Nucl. Phys. B 368, 3-37 (1992),
L. E. Ibanez, “More about Discrete Gauge Anomalies,” Nucl. Phys. B 398,
301-318 (1993), hep-ph/9210211.

  * [8] T. Banks and M. Dine, “Note on Discrete Gauge Anomalies,” Phys. Rev. D 45, 1424-1427 (1992), hep-th/9109045.
  * [9] T. Araki, “Anomaly of Discrete Symmetries and Gauge Coupling Unification,” Prog. Theor. Phys. 117, 1119-1138 (2007), arXiv:hep-ph/0612306 [hep-ph],
T. Araki, T. Kobayashi, J. Kubo, S. Ramos-Sanchez, M. Ratz and P. K. S.
Vaudrevange, “(Non-)Abelian Discrete Anomalies,” Nucl. Phys. B 805, 124-147
(2008), arXiv:0805.0207 [hep-th],
T. Kobayashi and H. Uchida, “Anomaly of Non-Abelian Discrete Symmetries,”
arXiv:2111.10811 [hep-th],
B. Gripaios, “Gauge Anomalies of Finite Groups,” arXiv:2201.11801 [hep-th].

  * [10] C. Csaki and H. Murayama, “Discrete Anomaly Matching,” Nucl. Phys. B 515, 114-162 (1998), hep-th/9710105.
  * [11] B. Henning, X. Lu, T. Melia and H. Murayama, “Outer Automorphism Anomalies,” arXiv:2111.04728 [hep-th].
  * [12] G. ’t Hooft, “Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking,” NATO Sci. Ser. B 59, 135-157 (1980).
  * [13] N. Seiberg, “Electric-Magnetic Duality in Supersymmetric Non-Abelian Gauge Theories,” Nucl. Phys. B 435, 129-146 (1995), hep-th/9411149,
K. A. Intriligator and N. Seiberg, “Lectures on Supersymmetric Gauge Theories
and Electric-Magnetic Duality,” Nucl. Phys. B Proc. Suppl. 45BC, 1-28 (1996),
hep-th/9509066.

  * [14] Y. Tachikawa, “Lectures on 4$d$ ${{\cal N}}=1$ Dynamics and Related Topics,” arXiv:1812.08946 [hep-th].
  * [15] P. Pouliot and M. J. Strassler, “A Chiral $SU(n)$ Gauge Theory and its Nonchiral $Spin(8)$ Dual,” Phys. Lett. B 370, 76-82 (1996), hep-th/9510228.
  * [16] I. Pesando, “Exact Results for the Supersymmetric $G_{2}$ Gauge Theories,” Mod. Phys. Lett. A 10 (1995), 1871-1886, hep-th/9506139,
S. B. Giddings and J. M. Pierre, “Some Exact Results in Supersymmetric
Theories based on Exceptional Groups,” Phys. Rev. D 52 (1995), 6065-6073, hep-
th/9506196.

  * [17] P. L. Cho, “More on Chiral-Nonchiral Dual Pairs,” Phys. Rev. D 56, 5260-5271 (1997), hep-th/9702059.
  * [18] P. L. Cho, “Exact Results in $SO(11)$ SUSY Gauge Theories with Spinor and Vector Matter,” Phys. Lett. B 400 (1997), 101-108, hep-th/9701020,
N. Maru, “Confining Phase in SUSY $SO(12)$ Gauge Theory with One Spinor and
Several Vectors,” Mod. Phys. Lett. A 13 (1998), 1361-1370, hep-th/9801187.

  * [19] M. Berkooz, P. L. Cho, P. Kraus and M. J. Strassler, “Dual Descriptions of $SO(10)$ SUSY Gauge Theories with Arbitrary Numbers of Spinors and Vectors,” Phys. Rev. D 56, 7166-7182 (1997), hep-th/9705003.