boxsize=.6em,centertableaux ††institutetext: Department of Applied Mathematics
and Theoretical Physics
University Cambridge, CB3 0WA, UK

# Chiral Gauge Dynamics: Candidates for Non-Supersymmetric Dualities

Avner Karasik, Kaan Önder and David Tong<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the dynamics of chiral $SU(N)$ gauge theories. These contain Weyl
fermions in the symmetric or anti-symmetric representation of the gauge group,
together with further fermions in the fundamental and anti-fundamental. We
revisit an old proposal of Bars and Yankielowicz who match the ’t Hooft
anomalies of this theory to free fermions. We show that there are novel and,
in some cases, quite powerful constraints on the dynamics in the large $N$
limit.

In addition, we study these $SU(N)$ theories with an extra Weyl fermion
transforming in the adjoint representation. Here we show that all 21 ’t Hooft
anomalies for global symmetries are matched with those of a Spin(8) gauge
theory. This suggests a non-supersymmetric extension of the duality of Pouliot
and Strassler. Finally, we also discuss some non-supersymmetric dualities with
vector-like matter content for $SO(N)$ and $Sp(N)$ gauge theories and the
constraints imposed by Weingarten inequalities.

##  1 Introduction

Supersymmetry affords exquisite control over the dynamics of 4d gauge
theories. However, for the most part this control has not extended to non-
supersymmetric theories.

Indeed, without supersymmetry our understanding of even the most qualitative
questions remains lacking. For example, it has long been known that
supersymmetric theories exhibit a broad array of interesting dynamics,
including confinement without chiral symmetry breaking, dual theories flowing
to the same critical point, and the emergence of gapless, “magnetic” gauge
bosons in the infra-red seib1 ; seib2 . But it is not yet established whether
any of these phenomena can arise in the absence of supersymmetry. (Proposals
do exist in the literature, some of which we will describe in more detail
below.)

In this paper, we explore some scenarios for the dynamics of certain chiral
gauge theories. These chiral theories have gauge group $SU(N)$ and contain a
Weyl fermion in either the symmetric or anti-symmetric representation,
together with a collection of further fermions in the fundamental, anti-
fundamental and, in some cases, adjoint representations. These theories do not
enjoy supersymmetry and we do not have enough control over their dynamics to
be confident of their low-energy physics. We do, however, have a number of
tools that constrain the possible dynamics, including ’t Hooft anomaly
matching, large $N$ techniques and the a-theorem. Using these, we give
consistent and, we believe, plausible proposals for the dynamics of these
theories that exhibit each of the phenomena described above.

Some of these proposals are not new. In fact, the idea that chiral gauge
theories may confine without breaking their global symmetry group predates our
understanding of supersymmetric theories raby . In particular, Bars and
Yankielowicz made a striking proposal that a certain chiral gauge theory
confines, with the ’t Hooft anomalies saturated by a natural collection of
massless composite fermions by . Subsequently, some doubt was cast on this
suggestion by a study of the theory at large $N$, where it was argued that one
of these composite fermions could not remain massless in the limit
$N\rightarrow\infty$ zepp . We devote much of Section 2 to this proposal of
Bars and Yankielowicz and revisit what happens in the large $N$ limit. By
adapting large $N$ arguments of Coleman and Witten cw , we show that in the
simplest cases the theory must confine without breaking its global symmetry.
Moreover, we will show that the interpretation of the results of zepp , which
rule out this behaviour in general, is incorrect and the idea that these
theories flow to a collection of free, massless composite fermions remains
compelling for certain values of the number of colours $N$ and the number of
flavours. We use the a-theorem to give estimates for when this may occur.

In Section 3, we turn to a slightly different chiral theory, one that includes
an additional Weyl fermion in the adjoint representation. This $SU(N)$ gauge
theory has global symmetry group

$\displaystyle G=SU(q)\times SU(p)\times U(1)^{3}$

There are 21 independent ’t Hooft anomalies associated to $G$. This time there
is no collection of free fermions that match all ’t Hooft anomalies. There is,
however, a Spin(8) gauge theory that does the job, with the full collection of
’t Hooft anomalies matched to matter sitting in the ${\bf 8}_{v}$, ${\bf
8}_{s}$ and ${\bf 8}_{c}$ representations, together with some gauge singlets.
We describe a plausible scenario in which these two theories are dynamically
related, with the Spin(8) gauge bosons emerging in the infra-red from the
strong coupling of the $SU(N)$ gauge theory. (The opposite RG flow is also
possible for certain values of $N$.) If this scenario is realised dynamically,
it would provide an extension of the supersymmetric duality discovered by
Pouliot and Strassler ps1 but, in the present case, the proposed duality does
not admit a supersymmetric completion.

In Section 4, we move from chiral theories to vector-like theories. Here we
have further tools at our disposal. As first stressed in ofer , Vafa-Witten
theorems and the related Weingarten theorem can be used to rule out certain
putative dualities in any theory that has a positive semi-definite fermion
measure. We review these results and extend them. In particular, we look at
some detail at the non-supersymmetric dualities proposed in adi1 ; adi2 .

##  2 The Bars-Yankielowicz Proposal

In this section, we study a pair of chiral gauge theories. Each is built
around an $SU(N)$ gauge group, with a Weyl fermion in either the symmetric or
anti-symmetric representation, together with a collection of further Weyl
fermions in the fundamental and anti-fundamentals.

Despite a great deal of work exploring the properties of these theories, their
dynamics is not fully understood and a number of proposals have been put
forward. Prominent among them is a particularly simple suggestion of Bars and
Yankielowicz by , following earlier work raby , that the theory flows to free
fermions without breaking its global symmetry group. The ’t Hooft anomalies of
these free fermions match in a rather striking fashion. Further work
discussing various aspects of this theory can be found in appel ; konishi ;
mury1 ; mury2 ; us ; avner .

Attractive as the Bars-Yankielowicz proposal is, there has long been a
roadblock to its realisation. In zepp , large $N$ techniques were employed in
the study of chiral gauge theories. It was argued that it is not possible to
realise the Bars-Yankielowicz free fermion states at large $N$. Here we give a
critical review of these arguments and show that this conclusion is not
justified: the Bars-Yankielowicz free fermion proposal is perfectly consistent
at large $N$. Moreover, we adapt the large $N$ arguments to show that certain
composite fermion states must be massless in this limit.

In Section 2.1, we review the $SU(N)$ chiral gauge theories and the Bars-
Yankielowicz proposal and make some obvious remarks about its possible role in
the dynamics of these theories. In Section 2.2, we analyse these theories
through the lens of large $N$ and exhibit a number of constrains in this
limit. Finally, in Section 2.3, we describe a scenario in which these chiral
theories may exhibit a duality, with the symmetric and anti-symmetric both
flowing to the same CFT (together with an additional decoupled sector).

###  2.1 $SU(N)$ Chiral Gauge Theories

Throughout this section, we will consider a pair of chiral $SU(N)$ gauge
theories: one with a Weyl fermion in the symmetric $\ydiagram{2}$
representation, and the other with a Weyl fermion in the anti-symmetric
$\ydiagram{1,1}$ representation. (All Weyl fermions here, and below, can be
taken to be left-handed.) For now, we treat each in turn. Later, we will see
that their low-energy dynamics may be related.

#### Chiral Gauge Theories with a Symmetric

We start with the $SU(N)$ gauge theory with a single Weyl fermion in the
$\ydiagram{2}$. In addition, we have $p$ Weyl fermions in the fundamental
$\Box$ representation and a further $q$ Weyl fermions in the anti-fundamental
$\overline{\Box}$. To ensure cancellation of the gauge anomaly, we must take

$\displaystyle q=N+p+4$ (1)

This theory has non-anomalous, global symmetry group

$\displaystyle G=SU(q)\times SU(p)\times U(1)_{1}\times U(1)_{2}$ (2)

where we have omitted discrete quotients. (These discrete quotients won’t be
important for much of our story, although they play a small role in Section
2.3 so we relegate their discussion to Appendix A. For the theory with $p=0$,
the discrete quotients were previously discussed in konishi ; us .)

The full collection of fermions and their transformation under the global
symmetry group $G$ is

$\displaystyle\begin{array}[]{|c||c|c|c|c|c|c|}\hline\cr&SU(N)&SU(q)&SU(p)&U(1)_{1}&U(1)_{2}\\\
\hline\cr\chi&\ydiagram{2}&\mathbf{1}&\mathbf{1}&-q&-p\\\
\psi&\overline{\Box}&\Box&\mathbf{1}&N+2&0\\\
\tilde{\psi}&\Box&\mathbf{1}&\Box&0&N+2\\\ \hline\cr\end{array}$ (7)

We’ll refer to this as the symmetric theory. It is a simple matter to compute
the ’t Hooft anomalies of $G$. The two non-Abelian ’t Hooft anomalies are:

$\displaystyle{\cal A}[SU(q)^{3}]={\cal A}[SU(p)^{3}]=N$ (8)

In addition, there are ten further anomalies of the form $X^{2}\cdot U(1)_{a}$
with $a=1,2$ and $X$ either gravity or one of the factors of $G$. These
anomalies are summarised in the following table:

$\displaystyle\begin{array}[]{|c||c|c|}\hline\cr{\cal A}&U(1)_{1}&U(1)_{2}\\\
\hline\cr\hline\cr SU(q)^{2}&N(N+2)&0\\\ \hline\cr SU(p)^{2}&0&N(N+2)\\\
\hline\cr
U(1)^{2}_{1}&Nq[(N+2)^{3}-\frac{1}{2}(N+1)q^{2}]&-\frac{1}{2}N(N+1)pq^{2}\\\
\hline\cr
U(1)^{2}_{2}&-\frac{1}{2}N(N+1)qp^{2}&Np[(N+2)^{3}-\frac{1}{2}(N+1)p^{2}]\\\
\hline\cr{\rm grav}^{2}&\frac{1}{2}qN(N+3)&\frac{1}{2}pN(N+3)\\\
\hline\cr\end{array}$ (15)

where $q$, $p$ and $N$ are not all independent: they are related by $q=N+p+4$.

Long ago, Bars and Yankielowicz proposed that, at least for certain values of
$N$ and $p$, this theory may confine, leaving behind a collection of three
different massless, composite fermions, each of which is an $SU(N)$ singlet,

$\displaystyle\rho_{A}=\psi\chi\psi\ \ \ ,\ \ \
\rho_{S}=\tilde{\psi}^{\dagger}\chi\tilde{\psi}^{\dagger}\ \ \ ,\ \ \
\rho_{B}=\psi^{\dagger}\chi^{\dagger}\tilde{\psi}$ (16)

The indices are contracted such that the fermions have the following
transformation properties under $G$,

$\displaystyle\begin{array}[]{|c||c|c|c|c|}\hline\cr&SU(q)&SU(p)&U(1)_{1}&U(1)_{2}\\\
\hline\cr\rule{0.0pt}{10.76385pt}\rho_{A}&\ydiagram{1,1}&\mathbf{1}&N-p&-p\\\
\rho_{S}&\mathbf{1}&\overline{\ydiagram{2}}&-q&-(N+q)\\\
\rho_{B}&\overline{\Box}&\Box&p+2&q-2\\\ \hline\cr\end{array}$ (21)

Bars and Yankielowicz pointed out that the ’t Hooft anomalies of these free
fermions reproduce (8) and (15).

#### Chiral Gauge Theories with an Anti-symmetric

There is a similar story for a companion theory, consisting of an
$SU(\tilde{N})$ gauge group with a Weyl fermion in the anti-symmetric
representation. In fact, it is useful to place this fermion in the
$\overline{\ydiagram{1,1}}$ representation. In addition, we have $q$ Weyl
fermions in the $\overline{\Box}$ representation and $p$ Weyl fermions in the
${\Box}$. This time, the gauge anomaly vanishes only if we take

$\displaystyle p=\tilde{N}+q-4$ (22)

The global symmetry group $G$ is again given by (2) and the matter content is
summarised by

$\begin{array}[]{|c||c|c|c|c|c|c|}\hline\cr\rule{0.0pt}{10.76385pt}&SU(\tilde{N})&SU(q)&SU(p)&U(1)_{1}&U(1)_{2}\\\
\hline\cr\rule{0.0pt}{13.3472pt}\chi&\overline{\raisebox{-4.73611pt}{\epsfbox{anti.eps}}}&\mathbf{1}&\mathbf{1}&-q&-p\\\
\psi&{\tiny{{{\yng(1)}}}}&{\bf 1}&{\tiny{\overline{\yng(1)}}}&0&\tilde{N}-2\\\
\tilde{\psi}&{\tiny{\overline{\yng(1)}}}&{\tiny{\overline{\yng(1)}}}&\mathbf{1}&\tilde{N}-2&0\\\
\hline\cr\end{array}$ (23)

We refer to this as the anti-symmetric theory. Formally, the condition (22)
agrees with (1) if we take

$\displaystyle\tilde{N}=-N$

Moreover, this relation ensures that the ’t Hooft anomalies of the anti-
symmetric theory are again given by (8) and (15). In this sense, the anti-
symmetric chiral theory (23) can be thought of as the continuation of the
symmetric theory (7) to negative values of $N$.

Since the ’t Hooft anomalies for the anti-symmetric theory are given by (8)
and (15), it is natural to think that they too have a free fermion realisation
(21). The only novelty is the identification of the composite fermions
$\rho_{S}$, $\rho_{A}$ and $\rho_{B}$ in terms of the UV degrees of freedom.
This time the dictionary is

$\displaystyle\rho_{S}=\psi\chi\psi\ \ \ ,\ \ \
\rho_{A}=\tilde{\psi}^{\dagger}\chi\tilde{\psi}^{\dagger}\ \ \ ,\ \ \
\rho_{B}=\psi^{\dagger}\chi^{\dagger}\tilde{\psi}$

where the indices are contracted to give the transformation properties shown
in (21).

A better way to think of this collection of different theories is through the
lens of the global symmetry $G$. Any global symmetry group, with its attendant
’t Hooft anomalies, can be realised in different field theories. The free
fermion avatar (21) holds for any $q$ and $p$. In contrast, the symmetric and
anti-symmetric chiral gauge theories hold only for certain ranges of $q$ and
$p$. The requirement that $N\geq 2$ means that the symmetric theory provides a
realisation of the symmetry group only when

$\displaystyle q\geq p+6$

(For $N=2$, the gauge theory has a larger symmetry group, but this can be
broken to $G$ through suitable interactions with scalar fields or through
irrelevant six-fermion operators.) In contrast, the anti-symmetric theory
provides a realisation of $G$ only when

$\displaystyle q\leq p+2$

(This time the cases $\tilde{N}=2,3$ and 4 are all special in the sense that
they are non-chiral and may have enhanced symmetries.). Note that the pair of
chiral theories do not cover all possible values of $q$ and $p$.

The anomaly matching of Bars and Yankielowicz is certainly striking. The
obvious question is: are these massless free fermions realised dynamically?

The honest answer is: we don’t know. At stake is the question of whether the
global symmetry group $G$ is dynamically broken or survives in the far infra-
red. A firm answer to this is beyond our current understanding of strongly
interacting chiral gauge theories. If the global symmetry survives, then
anomaly matching means that it must flow to some phase with gapless fermions.
This could be an interacting CFT, or it could be the free massless fermions
described above. As we now review, the latter is a very plausible scenario for
low $p$ (for the symmetric theory) and low $q$ (for the anti-symmetric
theory).

#### Confinement without Chiral Symmetric Breaking

The question of whether the theories confine without breaking chiral symmetry
is a little different for the cases of $q=0$ (in the anti-symmetric theory)
and $p=0$ (in the symmetric theory) and the cases with both $p,q\neq 0$. Here
we briefly summarise what’s known, first in the former case and then in the
latter, before turning to more details.

The anti-symmetric theory with $q=0$ was first discussed in raby where it was
proposed that the theory does indeed confine, without breaking the global
symmetry group $G$, leaving behind the single massless, composite fermion
$\rho_{S}$. This is compelling in part because of the sheer simplicity of the
proposal. Further evidence comes from the fact that the same physics is
observed in the Higgs phase, as well as the confining phase, an observation
known, rather grandly, as “complementarity”. The authors of raby also discuss
the symmetric theory with $p=0$, again proposing that the theory confines,
preserving $G$, now with $\rho_{A}$ massless.

For both of the symmetric and anti-symmetric theories, confinement without
chiral symmetry breaking remains the simplest dynamical scenario, but it not
the only one. It is also possible that the theory develops a condensate and
spontaneously breaks $G$. Various proposals are discussed in mury1 ; mury2 ;
us . No doubt other phases are possible. These different candidates cannot be
ruled out using anomalies us , so the question of which is favoured is a
dynamical one. And we do not have good control over this dynamics.

Some help comes from large $N$. A previous large $N$ study zepp argued that
the fermions $\rho_{S}$ (in the anti-symmetric theory) and $\rho_{A}$ (in the
symmetric theory) are obliged to massless. We review this argument in Section
2.2 and provide another (and, we believe, better) argument that reaches the
same conclusion in a more direct fashion. Hence, it appears that at large $N$
the issue is settled: the theory does indeed confines, leaving behind massless
composite fermions.

What about the symmetric theory with $p>0$ and the anti-symmetric theory with
$q>0$, where the full trinity of Bars-Yankielowicz fermions is needed to match
anomalies? At first glance, there seems to be no obstacle to realising these
massless fermions dynamically, at least for small $p$ (for the symmetric
theory) and small $q$ (for the anti-symmetric theory). We still do not have
control over the breaking of $G$, but assuming that it is unbroken in the
infra-red, the free massless fermion phase is surely an appropriate candidate.

There is, however, a fly in the ointment. Or, more precisely, in the
literature. This time, the authors of zepp argued that, in contrast to the
$p=0$ and $q=0$ cases, the Bars-Yankielowicz fermions cannot be realised in
the planar diagram expansion, casting doubt on their dynamical viability. We
will review this calculation in Section 2.2 where we claim that the conclusion
is flawed: there is no large $N$ obstacle to realising the free fermion Bars-
Yankielowicz phase and this remains a plausible candidate for the dynamics for
suitably small $p$ and $q$.

#### The Conformal Window

Of course, when the number of additional fermionic flavours becomes large,
there is no mystery about the dynamics. The symmetric theory ceases to be
asymptotically free when

$\displaystyle p\geq p_{AF}=\frac{9}{2}N-3$

For $p$ slightly below this value, the theory will flow to a weakly coupled
Banks-Zaks fixed point, with $G$ unbroken. As $p$ is decreased still further,
this CFT is expected to become more strongly coupled and persist in some
conformal window

$\displaystyle p_{\star}<p<p_{AF}$

We define this window to have $G$ unbroken. The lower end of the conformal
window is, like for QCD, unknown. Moreover, in contrast to QCD, we do not have
the luxury of lattice simulations to guide us. For $p\leq p_{\star}$, we
presumably return to the story above, with either the $G$ broken phase or the
free massless fermions as candidates for the low-energy physics.

Although we do not have a good handle on $p_{\star}$, some insight comes from
the a-theorem cardy ; koma . This captures the fact that the number of degrees
of freedom must decrease under RG through the statement that the “$a$ anomaly”
obeys $a_{UV}>a_{IR}$. (The authors of appel did a similar analysis, but using
the free energy as a proxy for the number of degrees in these theories.
However, the free energy is not necessarily decreasing under RG.)

For free theories, the contribution from various fields is (after absorbing a
factor of $90(8\pi)^{2}$),

$\displaystyle a=\mbox{\\# scalars}+\frac{11}{2}\mbox{\\# Weyl
fermions}+62\,\mbox{\\# Gauge Fields}$

Suppose that the chiral symmetric theory (7) flows to the free massless
fermions (21). Then we have

$\displaystyle\mbox{Symmetric Theory:}\ \ \
a_{UV}-a_{IR}=\frac{135}{2}N^{2}-11p^{2}+\frac{11}{2}N-44p-95$

This flow is only possible if $a_{UV}>a_{IR}$, which puts an upper bound on
$p$. For even reasonably small values of $N$ and $p$, the $N^{2}$ and $p^{2}$
terms dominate giving an approximately linear bound

$\displaystyle p\lesssim 2.48N$

Under the assumption that the theory exits the conformal window to the free
fermion phase, this puts an upper bound on the lower end of the conformal
window111For what it’s worth, when applied to Seiberg duality of $SU(N_{c})$
SQCD with $N_{f}$ flavours, the a-theorem gives a bound on the lower end of
the conformal window of $N_{f}\lesssim 1.85N_{c}$. The true value is, of
course, $N_{f}=1.5N_{c}$.

$\displaystyle p_{\star}\lesssim 2.48N$

Again, there is a similar story for the $SU(\tilde{N})$ anti-symmetric theory.
This ceases to be asymptotically free when

$\displaystyle q\geq q_{AF}=\frac{9}{2}N+3$

Now the a-theorem requires

$\displaystyle\mbox{Anti-Symmetric Theory:}\ \ \
a_{UV}-a_{IR}=\frac{135}{2}\tilde{N}^{2}-11q^{2}-\frac{11}{2}\tilde{N}+44q-95$

Again, for large(ish) $N$, the a-theorem tells us that we can only flow to the
free-fermion phase when $q\lesssim 2.48\tilde{N}$. Again, this could be viewed
as an upper bound for the end of the conformal window.

###  2.2 Large N

The large $N$ expansion is one of the most important theoretical tools in the
analysis of strongly interacting, 4d non-Abelian gauge theories thooft ;
witten . The purpose of this section is to apply large $N$ techniques to the
pair of chiral gauge theories (7) and (23). A previous study of these theories
at large $N$ was performed by Eichten et al zepp . We will explain their
results, disagree with some of them, and provide some new ones. We start this
section with a review of large $N$ in general before we turn to the chiral
theories of interest.

#### A Review of Large $N$ QCD

To start, consider pure $SU(N)$ gauge theory with coupling $g^{2}$ and ’t
Hooft coupling $\lambda=g^{2}N$. The Yang-Mills action can be written as

$\displaystyle S_{YM}=\frac{N}{2\lambda}\int d^{4}x\ {\rm
tr}\,F_{\mu\nu}F^{\mu\nu}$

with
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-i[A_{\mu},A_{\nu}]$.
The large $N$ limit consists of taking $N\rightarrow\infty$ with $\lambda$
held fixed.

Figure 1: Three diagrams contributing to the vacuum energy of Yang-Mills.
Diagrams $a$ and $b$ are both planar and scale like $N^{2}$. Diagram $a$ has 2
vertices, 3 propagators, and 3 colour loops and so scales as
$N^{2}N^{-3}N^{3}=N^{2}$. Diagram $b$ has 6 vertices, 10 propagators, and 6
colour loops resulting again in $N^{6}N^{-10}N^{6}=N^{2}$ scaling. In
contrast, diagram $c$ is non-planar: it has 4 vertices, 6 propagators, and 2
colour loops, resulting in $N^{4}N^{-6}N^{2}=N^{0}$ and so is suppressed in
the large $N$ limit.

The Feynman rules can be read from this action. Every gluon propagator is
proportional to $1/N$ while every vertex is proportional to $N$ (ignoring the
$\lambda$ dependence). Feynman diagrams are drawn using the double line
representation for the propagators, where each line carries a single colour
index. We can then count the additional powers of $N$ that come from summing
colour indices: they are equal to the number of loops. The famous observation
of ’t Hooft is that the leading contribution to any correlation function comes
from planar diagrams. Examples of the contribution to the vacuum energy are
shown in Figure 1.

Local operators in this theory are glueballs, which we write schematically as

$\displaystyle{G}\sim{\rm tr}\,F^{k}$

with $k\geq 2$. The normalisation of these operators, with no explicit factors
of $N$, ensures that the 2-point function scales as

$\displaystyle\langle{G(x)G(y)}\rangle\sim{\cal O}(1)$

Higher order $n$-point functions then scale as $N^{2-n}$. An example is
depicted in the left hand diagram in Figure 2.

Figure 2: Diagrams with insertions of glueballs (in blue) and mesons (in
green). The solid and dashed lines represent colour and flavour indices
respectively. Every insertion reduces the number of vertices by 1 or increases
the number of propagators by 1. In any case, this gives a factor of $1/{N}$.
In addition, a meson insertion comes with a factor of $\sqrt{N}$ due to the
normalization. This gives the rules described in the text: an $n$-point
function of glueballs scales as $N^{2-n}$, as shown in diagram $a$; an
$n$-point functions of mesons scales as $N^{1-n/2}$, as shown in in diagram
$b$; and a mixed correlator of $n_{G}$ glueballs and $n_{M}$ mesons scales as
$N^{1-n_{G}-n_{M}/2}$, as shown in diagram $c$.

Next we add matter to our theory. We start by considering vector-like matter,
resulting in a large $N$ version of QCD. This consists of $N_{f}$ Weyl
fermions, each transforming in the $\Box$ representation of $SU(N)$, and a
further $N_{f}$ Weyl fermions transforming in the $\overline{\Box}$. We denote
these fermions as $\psi$ and $\tilde{\psi}$ respectively and pair them
together into $N_{f}$ Dirac fermions $\Psi$. The action is

$\displaystyle S_{\rm Dirac}=N\int d^{4}x\
\sum_{i=1}^{N_{f}}i\bar{\Psi}_{i}\\!\,\,{\raise
0.6458pt\hbox{/}\mkern-12.0muD}\Psi_{i}$

The overall factor of $N$ means that fermion propagators, like their gluonic
counterparts, scale as $1/N$ while gluon-fermion vertices carry a factor of
$N$. The key difference between these fermions and the gluons is that the
former carry a single colour index $a=1,\ldots,N$ and a single flavour index
$i=1,\ldots N_{f}$, while the latter carry two colour indices. This has two
consequences. The first is that a fermion loop is weighted by a factor of
${N_{f}}/{N}$ relative to a gluon loop. If $N_{f}\ll N$ fermion loops will be
suppressed. This is referred to as the ’t Hooft limit. If, however, $N_{f}\sim
N$ then there is no such suppression: this is referred to as the Veneziano
limit ven and will be relevant for the chiral theories that we discuss
shortly.

The second consequence is that external states carry fixed flavour index and
so these are not summed over. To compute the scaling, we define the meson
operators

$\displaystyle M\sim\sqrt{N}\tilde{\psi}F^{k}\psi$ (24)

The $\sqrt{N}$ normalization is again chosen so that the two-point function
scales as

$\displaystyle\langle M(x)M(y)\rangle\sim{\cal O}(1)$

Higher order $n$-point correlators of mesons then scale as $N^{1-n/2}$. An
example is depicted in the middle diagram in Figure 2.

Using these results, we can determine the stability of massive states.
Glueballs decay as $G\rightarrow GG$ or $G\rightarrow MM$, both of which scale
as $1/N$. This gives a rate of ${\rm diagram}^{2}\sim 1/N^{2}$ for the decay
to gluons or, after summing over all possible decay products, $\sim
N_{f}^{2}/N^{2}$ for the decay to mesons. If we are working in the limit of
fixed $N_{f}$, this results the well-known statement that the strict
$N\rightarrow\infty$ limit enjoys an infinite tower of stable, massive
glueball states. If, however, $N_{f}\sim N$, then glueballs decay with a rate
of ${\cal O}(1)$.

Figure 3: Leading 4-baryon diagrams. On the left diagram the baryons exchange
a quark without any interactions. The choice of colour gives a factor of $N$.
On the right diagram the baryons exchange a gluon. The choice of colour now
gives a factor of $N^{2}$, the gluon propagator and two vertices give $1/{N}$,
resulting again in a total factor of $N$.

For mesons, the decay $M\rightarrow MM$ is determined by the diagram
$\langle{M_{ij}(x_{1})M_{ik}(x_{2})M_{kj}(x_{3})}\rangle\sim 1/{\sqrt{N}}$.
Squaring and summing over all the external states results in a decay rate
$\sim{N_{f}}/{N}$ (because one flavour index is summed over). Again, if
$N_{f}$ is held fixed in the large $N$ limit then there is an infinite tower
of stable mesons, while if $N_{f}\sim N$ then then excited meson states, like
the glueballs, decay with an ${\cal O}(1)$ rate.

Large $N$ QCD also has a spectrum of baryons that require special attention
witten . A baryon is an operator whose colours indices are contracted using
the epsilon tensor,

$\displaystyle
B^{i_{1}...i_{N}}\sim\epsilon^{a_{1}a_{2}...a_{N}}\psi_{a_{1}}^{i_{1}}\psi_{a2}^{i_{2}}\ldots\psi_{a_{N}}^{i_{N}}$

A result that we will need later concerns the strength of 4-baryon
interaction. As shown by Witten witten , the leading contribution to this
interaction comes from diagrams in Figure 3 and scales as

$\displaystyle\langle
B(x_{1})B^{\dagger}(x_{2})B(x_{3})B^{\dagger}(x_{4})\rangle\sim{\cal O}(N)$
(25)

Finally, we review the possibilities for symmetry breaking in large $N$ QCD,
following the work of Coleman and Witten cw . The global symmetry of this
theory includes the factor

$\displaystyle SU(N_{f})_{L}\times SU(N_{f})_{R}$

under which the meson operator $M_{i}^{j}=\sqrt{N}\tilde{\psi}_{i}\psi^{j}$
transforms in the bi-fundamental representation. This operator provides an
order parameter for symmetry breaking. The Coleman-Witten theorem restricts
the kind of symmetry breaking that is possible in the large $N$ limit.

For this analysis, it is important to assume that $N_{f}\ll N$. As described
above, the $2n$-point meson interaction is dominated by single trace diagrams
and scales as $\langle(M^{\dagger}M)^{n}\rangle\sim N^{1-n}$. The leading
terms in the effective potential for the meson then takes the form

$\displaystyle V(M)=N\sum_{n}N^{-n}a_{n}{\rm tr}(M^{\dagger}M)^{n}$ (26)

with $a_{n}\sim{\cal O}(1)$. We can use the $SU(N_{f})_{L}\times
SU(N_{f})_{R}$ transformation to diagonalize the meson operator, so
$M_{i}^{j}=\lambda_{i}\delta_{i}^{j}$. The potential now takes the form,

$\displaystyle V(\lambda_{i})=N\sum_{n}N^{-n}a_{n}\sum_{i}|\lambda_{i}|^{2n}$
(27)

The key observation is that the $\lambda_{i}$ are decoupled: each has the same
equation of motion, solved by $\lambda_{i}=\lambda$. Therefore,
$\langle{M}_{i}^{j}\rangle=\lambda\delta_{i}^{j}$ and the only possible
symmetry breaking in the large $N$ limit is $SU(N_{f})_{L}\times
SU(N_{f})_{R}\to SU(N_{f})_{diag}$. This is the Coleman-Witten theorem cw .

The story above holds when $N_{f}/N\ll 1$. What happens when $N_{f}\sim N$? In
this case multi-trace operators can become as important as the single trace
ones. As an example, we will compare the single-trace and double-trace
$(MM^{\dagger})^{2}$ operators,

$\displaystyle V_{2}(M)$ $\displaystyle=$ $\displaystyle\frac{1}{N}a_{2}{\rm
tr}(MM^{\dagger}MM^{\dagger})+\frac{1}{N^{2}}b_{2}{\rm tr}(MM^{\dagger})\,{\rm
tr}(MM^{\dagger})$ (28) $\displaystyle=$
$\displaystyle\frac{1}{N}a_{2}\sum_{i=1}^{N_{f}}|\lambda_{i}|^{4}+\frac{1}{N^{2}}b_{2}\sum_{i,j=1}^{N_{f}}|\lambda_{i}\lambda_{j}|^{2}$

The second term is suppressed by $1/N$ relative to the first, but has a double
sum over flavour. So if $N_{f}\sim N$ the two terms are equally important. In
this case, the eigenvalues are not decoupled and other symmetry breaking
pattern are, in principle, allowed.

#### Large $N$ for Chiral Theories

After this review, we turn to the large $N$ limit of the chiral theories (7)
and (23). Our goal is to shed some light on the dynamics of these theories. A
large $N$ analysis of these theories was previously given in zepp and we
review some of their results below. We focus on the symmetric chiral theory
(7). All results carry over immediately to the anti-symmetric theory.

The fermion $\chi$ transforms in the two-index representation $\ydiagram{2}$
and so, in the large $N$ limit, behaves much like the gluons in the adjoint
representation. In particular, loops of $\chi$ in planar diagrams are
unsuppressed. The only minor difference is that $\chi$ carries two fundamental
indices, in contrast to the gluon which carries one fundamental and one anti-
fundamental index. This can be accounted for by adding arrows to propagators
and ensuring that they are matched correctly.

In addition, our theory has $q$ fermions $\psi$ in the $\overline{\Box}$
representation and $p$ fermions $\tilde{\psi}$ in the $\Box$. Anomaly
cancellation (1) means that we necessarily have $q\sim{\cal O}(N)$. Our
interest in this section will focus on the regime in which $p\sim{\cal O}(1)$.
In this sense, the large $N$ limit of the chiral theory shares some features
with both the ’t Hooft and Veneziano limits of QCD.

The Bars-Yankielowicz proposal is that the chiral gauge theory confines,
resulting in the collection of massless fermions $\rho_{S}$, $\rho_{A}$ and
$\rho_{B}$ given in (16). Each of these composite fermions behaves like a
meson in QCD. Including the appropriate meson normalisation (24), we should
replace (16) with

$\displaystyle\rho_{A}=\sqrt{N}\psi\chi\psi\ \ \ ,\ \ \
\rho_{S}=\sqrt{N}\tilde{\psi}^{\dagger}\chi\tilde{\psi}^{\dagger}\ \ \ ,\ \ \
\rho_{B}=\sqrt{N}\psi^{\dagger}\chi^{\dagger}\tilde{\psi}$

The additional $\sqrt{N}$ factors ensure that the two-point functions are
correctly normalised: $\langle\rho(x)\rho(y)\rangle\sim{\cal O}(1)$, with
higher point functions suppressed at large $N$. All other meson-like
operators, in which a string of $F$ and $\chi$ fields are sandwiched between
some a pair of $\psi$ or $\tilde{\psi}$ fundamentals, are similarly normalised
with a $\sqrt{N}$.

Although higher point functions are suppressed, excited mesons can still decay
with rate of order ${\cal O}(1)$ because the number of $\psi$ flavour is
$q\sim{\cal O}(N)$ and these should be summed over in final states. For
example, $\psi_{i}\tilde{\psi}^{I}$ can decay to
$(\rho_{A})_{ij}(\rho_{B})^{jI}$ where $i,j$ are $SU(q)$ indices and $I$ is an
$SU(p)$ index. The probability to decay to a fixed $j$ is
$\sim\left(1/{\sqrt{N}}\right)^{2}=1/{N}$. Summing over $j$ we get an ${\cal
O}(1)$ decay rate.

#### The $p=0$ Theory

We start by looking at the $p=0$ theory, where there are $q=N+4$ $\psi$ fields
and no $\tilde{\psi}$ fields. In this case, the global symmetry group is just

$\displaystyle G=SU(q)\times U(1)$

and the conjectured dynamics is that the theory confines, preserving $G$,
resulting in a single gapless fermion $\rho_{A}$.

The authors of zepp studied this theory in the large $N$ limit and provided
an, admittedly slightly circuitous, argument that $\rho_{A}$ must indeed be
massless. Their argument proceeds by showing that the global $U(1)\subset G$
cannot be spontaneously broken in the large $N$ limit because the associated
Goldstone bosons cannot be seen in cuts of planar diagrams222This argument was
recently criticised in yetagain by working with non-gauge invariant operators
in the planar limit. We do not agree with this criticism.. The $U(1)$ ’t Hooft
anomalies mean that there must be $\frac{1}{2}q(q-1)$ massless fermions, and
it would be perverse if these did not sit in the anti-symmetric representation
of $SU(q)$.

Here we give an alternative, more direct argument that shows the $\rho_{A}$
fermion must be massless in the large $N$ limit. To see this, first note that,
by anomaly matching, if the fermion $\rho_{A}$ is not massless then the
symmetry $G$ must be spontaneously broken. We introduce a gauge invariant
order parameter $\Phi$ for this symmetry breaking. The fermion gets its mass
through interactions with the condensate $\Phi$, which can then be determined
by computing an appropriate correlation function, such as
$\langle\Phi\rho_{A}\rho_{A}\rangle$. To determine how the mass scales with
$N$, we need to compute both the scaling of these correlation functions, and
the scaling of the expectation value of $\langle\Phi\rangle$. Both are
straightforward.

#####

It’s useful to think of the correlation function
$\langle\Phi\rho_{A}\rho_{A}\rangle$ as giving rise to an effective Yukawa
interaction. The most general form is

$\displaystyle{\cal L}_{\rm Yuk}\sim
y\Phi_{ijkl}\,\rho_{A}^{ij}\rho_{A}^{kl}+{\rm h.c.}$ (29)

with $i,j,k,l=1,\ldots q$. Here the Yukawa coupling $y$ will have some scaling
on $N$ determined by $\langle\Phi\rho_{A}\rho_{A}\rangle$. The exact form of
this scaling depends on the choice of operator $\Phi$ so the next question is:
what kind of gauge invariant operators $\Phi$ can we construct?

First note that there is no single trace meson-like operator with 4 $SU(q)$
indices. But we can construct “multi-trace” (really, “multi-meson” in this
context) operators and we can construct baryonic operators. We start with the
former.

An example of a multi-trace operator is simply
$\Phi=\rho_{A}^{\dagger}\rho_{A}^{\dagger}$. There is nothing to prohibit such
an operator getting an expectation value of, say, the form

$\displaystyle\langle\Phi_{ijkl}\rangle=v[\delta_{ik}\delta_{jl}-\delta_{il}\delta_{jk}]$

which has the ability to gap the fermion $\rho_{A}$. From the general rules
described previously, we know that the Yukawa coupling in this case must scale
as $y\sim 1/N$. This, in turn, means that if the mass in (29) is to survive in
the large $N$ limit, then the expectation value $v$ would have to scale as
$v\sim N$. But this is forbidden. We can see this by looking at the effective
potential (28) for $\Phi$. This includes terms

$\displaystyle
V_{2}(\Phi,\Phi^{\dagger})\sim\frac{1}{N}\Phi^{\dagger\,ijkl}\Phi_{ijkl}\sim\frac{q^{2}v^{2}}{N}$

If $v\sim N$, then this contribution to the vacuum energy would scale as
$V_{2}\sim q^{2}N\sim N^{3}$. But that’s illegal! The vacuum energy must scale
as $N^{2}$. We learn that $v\sim{N}$ is not allowed and, from (29), the
fermionic mass is suppressed by $1/N$. The double-trace operator
$\Phi=\rho_{A}^{\dagger}\rho_{A}^{\dagger}$ cannot gap the fermions in the
large $N$ limit.

The same conclusion is reached if we consider other multi-meson operators. But
what about baryons? For example, we can construct a scalar operator of the
form $\Phi_{B}\sim\chi^{N-4}\psi^{N-8}$, with the hanging colour indices
contracted with an $SU(N)$ epsilon symbol. Is it possible to gap the fermions
through some expectation $\langle\Phi_{B}\rangle$? The answer, again, is no.
From (25), the effective potential for $\Phi_{B}$ scales as

$\displaystyle
V(\Phi_{B}^{\dagger},\Phi_{B})\sim|\Phi_{B}|^{2}+N|\Phi_{B}|^{4}+\ldots$

and so any expectation value must scale as $\langle\Phi_{B}\rangle\sim
1/\sqrt{N}$. The Yukawa term (29) then gives a mass for the fermions that
scales as ${\rm mass}\sim y/\sqrt{N}$. This mass vanishes in the large $N$
limit provided that $y$ does not grow with $N$. This is surely the case: the
vertex $\Phi_{B}\rho_{A}\rho_{A}$, with $\Phi_{B}$ a baryon, vanishes in
perturbation theory because the total charge under the anomalous $U(1)$ is
non-zero. Such a vertex should emerge non-perturbatively but is certainly
expected to be suppressed in the large $N$ limit.

The upshot is that the fermion $\rho_{A}$ must be massless in the large $N$
limit.

The same arguments go through for the anti-symmetric theory with $q=0$. In
this case, the fermion $\rho_{S}$ must be massless in the large $N$ limit.

#### The $p>0$ Theories

We can apply the same kind of arguments to the Bars-Yankielowicz proposal that
all three composite fermions, $\rho_{A}$, $\rho_{S}$ and $\rho_{B}$ remain
massless. Importantly, we work in the limit $p/N\rightarrow 0$ and
$N\rightarrow\infty$. This time, as we shall see, the results are not so
strong. There is no obstacle to a collection of these fermions becoming
gapped, with the commensurate symmetry breaking. However, in contrast to zepp
, we will argue that nor is there any reason to believe that the fermions are
necessarily gapped.

First, we repeat the arguments above and show that they now allow for the
possibility that the fermions are gapped in the large $N$ limit. This is
achieved through the scalar operator

$\displaystyle\Phi_{iI}=\sqrt{N}\psi_{i}\tilde{\psi}_{I}$

with $i=1,\ldots,q$ and $I=1,\ldots,p$. The interactions with the mesons are
captured by the Yukawa interactions

$\displaystyle{\cal L}_{\rm
Yuk}\sim\frac{1}{\sqrt{N}}y_{A}\Phi^{\dagger}_{iI}(\rho_{A})^{ij}(\rho_{B})_{j}^{I}+\frac{1}{\sqrt{N}}y_{S}\Phi^{iI}(\rho_{S})_{IJ}(\rho_{B})_{i}^{J}+{\rm
h.c}$ (30)

with $y_{S,A}\sim 1$. The overall $1/{\sqrt{N}}$ comes from the scaling of
three-meson diagrams.

The expectation value of $\Phi$ can be determined by looking at the effective
potential. In the large $N$ limit, this is given by a sum of single trace
terms as in (26). (The multi-trace terms that arose in the Veneziano limit in
(28) are once again suppressed here because ${\rm rank}(\Phi)\leq p\ll N$.)
This tells us that any expectation value scales as
$\langle\Phi\rangle\sim\sqrt{N}$. Substituting this back into the potential
gives an entirely expected, and entirely legal, contribution to the vacuum
energy $V\sim Np$. Meanwhile, substituting this expectation value into (30)
gives a collection of the fermions a mass that is unsuppressed in the large
$N$ limit.

The fact that single trace operators dominate in the potential for $\Phi$
means that a version of the Coleman-Witten theorem applies. After
diagonalisation by $SU(q)\times SU(p)$, we can write the scalar in the form

$\displaystyle\Phi=\left(\begin{array}[]{ccc}\lambda_{1}&\ldots&0\\\
&\ddots&\\\ 0&\ldots&\lambda_{p}\\\ \hdashline 0&\ldots&0\end{array}\right)$
(35)

The potential then splits into independent terms for each $\lambda_{i}$, as in
(27). This tells us that each $\lambda_{i}$ takes the same value in the ground
state, and the symmetry breaking pattern is

$\displaystyle{SU(q)\times SU(p)\times U(1)_{1}\times U(1)_{2}\to
SU(q-p)\times SU(p)_{diag}\times U(1)^{2}\ ,}$ (36)

where $U(1)^{2}\subset SU(q)\times U(1)_{1}\times U(1)_{2}$. The remaining
$SU(q-p)$ means that the fermions $(\rho_{A})^{ij}$, $i,j=1,...,q-p$
necessarily remain massless, transforming in the anti-symmetric representation
of $SU(q-p)$ and saturating the surviving ’t Hooft anomaly.

This argument shows that the symmetry broken phase is consistent in the large
$N$ limit, but it doesn’t whether it is dynamically preferred. We don’t know
the answer to that. The authors of zepp argue something stronger: that it is
not possible to realise the full spectrum of massless fermions $\rho_{A}$,
$\rho_{S}$ and $\rho_{A}$ in the large $N$ limit. The offending fermion is
$\rho_{S}$, transforming in the $\ydiagram{2}$ of $SU(p)$. This is special
because it has a finite number of components in the $N\rightarrow\infty$
limit, in contrast to both $\rho_{A}$ and $\rho_{B}$, each of which transforms
under $SU(q)$ and so have an infinite number of components as
$N\rightarrow\infty$. The argument of zepp is that it is not possible to find
$\rho_{S}$ in the cut of a planar diagram. Translated into correlation
functions, this is the same statement that the decay of an excited meson
$M^{I}_{J}$ into $\rho_{S}\rho_{S}^{\dagger}$ is suppressed in the large $N$
limit

$\displaystyle\langle
M^{I}_{J}\,\rho_{S}^{JK}\,(\rho^{\dagger}_{S})_{IK}\rangle\sim{\cal
O}\left(\frac{1}{\sqrt{N}}\right)$ (37)

This means that the decay rate scales as $\sim p/N$. The lack of such a decay
was interpreted as the statement that $\rho_{S}$ cannot be massless, since
then such a decay would be expected to proceed at a rate of order ${\cal
O}(1)$. We see no reason to adopt this interpretation. The situation is
entirely analogous to pions in QCD. A generic massive state in the ’t Hooft
limit does not decay to pions, but this is not because pions are heavy; it is
because the interaction is suppressed.

The same is true of the result (37). It does not restrict the mass of
$\rho_{S}$. Indeed, the same $1/\sqrt{N}$ scaling is seen for appropriate
decays to $\rho_{A}$ and $\rho_{B}$ as well, but these are unsuppressed after
summing over the possible final states. The upshot is that the large $N$ limit
does not rule out a phase of gapless fermions with unbroken chiral symmetry
$G$. It does, however, predict that if this occurs then the $\rho_{S}$ sector
is decoupled from heavy states.

The argument in this section was restricted to $p\sim{\cal O}(1)$. For
$p\sim{\cal O}(N)$, symmetry breaking patterns other than (36) may occur.
Alternatively, if $G$ is unbroken then $\rho_{S}$ no longer decouples from the
dynamics in the $N\rightarrow\infty$ limit.

###  2.3 A Speculative Duality

Above, we described various possible phases of the symmetric and anti-
symmetric chiral theories. These theories share the same global symmetry group
$G=SU(q)\times SU(p)\times U(1)^{2}$, but exist for complementary values of
$q$ and $p$. For this reason, there’s no way that the two theories can flow to
the same interacting CFT. The purpose of this section is to present a
deformation of these theories that may induce them to flow to the same fixed
point. We stress that the results that we present here are plausible, but rely
on several assumptions about the dynamics that may not be realised in
practice.

#### The Symmetric Theory

We start by returning to the $SU(N)$ symmetric theory, but with the addition
of a singlet fermion in the UV. The matter content is

$\displaystyle\begin{array}[]{|c||c|c|c|c|c|c|}\hline\cr&SU(N)&SU(q)&SU(p)&U(1)_{1}&U(1)_{2}\\\
\hline\cr\chi&\ydiagram{2}&\mathbf{1}&\mathbf{1}&-q&-p\\\
\psi&\overline{\Box}&\Box&\mathbf{1}&N+2&0\\\
\tilde{\psi}&\Box&\mathbf{1}&\Box&0&N+2\\\ \nu&{\bf
1}&{\Box}&\overline{\Box}&-(p+2)&2-q\\\ \hline\cr\end{array}$ (43)

As before, we must have $q=N+p+4$ to ensure consistency of the theory. We now
add the 4-fermion term

$\displaystyle{\cal L}_{\rm
4-fermi}\sim\psi^{\dagger}\chi^{\dagger}\tilde{\psi}\nu+{\rm h.c.}$ (44)

This is, of course, an irrelevant operator in the UV. However, it is plausible
that, in certain situations, it is a dangerously irrelevant operator, with the
strong coupling dynamics giving it a large anomalous dimension in the infra-
red. Since the theory is weakly coupled for large $p$, this can only take
place for suitably small values of $p$

$\displaystyle p<p_{DI}$

where $p_{DI}<p_{AF}$. The exact value of $p_{DI}$ is, of course, not known.

For $p\geq p_{DI}$, the infra-red physics coincides with that in the previous
section, now with the additional fermion $\nu$ as a bystander. But for
$p<p_{DI}$, the 4-fermion term drives us to a new fixed point.

If we are in a phase where the theory flows to the free fermions then it’s
clear what happens: after confinement, the 4-fermion term (44) descends to a
mass term ${\cal L}_{\rm 4-fermi}\sim\rho_{B}\nu+{\rm h.c.}$ in the infra-red
where $\rho_{B}$ is the bi-fundamental composite defined in (21). Both of
these fermions then get a mass and we’re left just with the pair of massless
fermions $\rho_{A}$ and $\rho_{S}$, with charges

$\displaystyle\begin{array}[]{|c||c|c|c|c|}\hline\cr&SU(q)&SU(p)&U(1)_{1}&U(1)_{2}\\\
\hline\cr\rule{0.0pt}{10.76385pt}\rho_{A}&\ydiagram{1,1}&\mathbf{1}&N-p&-p\\\
\rho_{S}&\mathbf{1}&\overline{\ydiagram{2}}&-q&-(N+q)\\\ \hline\cr\end{array}$
(48)

At this point, it is useful to redefine the $U(1)$ charges. We introduce new
charges $Q_{1}^{\prime}$ and $Q_{2}^{\prime}$, related to the previous charges
$Q_{1}$ and $Q_{2}$ as

$\displaystyle Q_{1}^{\prime}=\frac{N+q}{N+2}Q_{1}-\frac{q}{N+2}Q_{2}\ \ \
{\rm and}\ \ \ Q_{2}^{\prime}=\frac{p}{N+2}Q_{1}+\frac{N-p}{N+2}Q_{2}$

It’s not immediately obvious that this transformation is legal because of the
denominator $1/(N+2)$. But one can check that the charges of all UV fermions
remain integer. The advantage of this new basis that it diagonalises the
charges of the IR composites, which read

$\displaystyle\begin{array}[]{|c||c|c|c|c|}\hline\cr&SU(q)&SU(p)&U(1)^{\prime}_{1}&U(1)^{\prime}_{2}\\\
\hline\cr\rule{0.0pt}{10.76385pt}\rho_{A}&\ydiagram{1,1}&\mathbf{1}&2N&0\\\
\rho_{S}&\mathbf{1}&\overline{\ydiagram{2}}&0&-2N\\\ \hline\cr\end{array}$
(52)

This reflects the fact that, without the bi-fundamental fermion $\rho_{B}$,
the group structure factorises in the infra-red, with

$\displaystyle G=G_{1}\times G_{2}=[SU(q)\times
U(1)_{1}^{\prime}]\times[SU(p)\times U(1)^{\prime}_{2}]$ (53)

Here, factorisation is the statement that there are are no mixed ’t Hooft
anomalies between $G_{1}$ and $G_{2}$.

In fact, a more careful analysis shows that the factorisation is not quite
complete: there are discrete quotients that forbid certain representations. To
avoid distraction from the main narrative, we describe this in Appendix A.

The factorisation (53) is trivial to see from the free fermion perspective,
simply because there are no degrees of freedom with charges under both $G_{1}$
and $G_{2}$. But it is much more surprising result when viewed from the
perspective of the original $U(1)$ theory which does have degrees of freedom
charged under both subgroups. (For example, one can check that the
$U(1)_{1}^{\prime}$ and $U(1)_{2}^{\prime}$ charges of $\chi$, $\psi$ and
$\tilde{\psi}$ are all non-vanishing.) The existence of the factorisation is
an interesting feature of the chiral gauge theory with the four-fermion term.
The question that we would like to ask is: does it hold a further clue for the
dynamics?

As we’ve seen, the effect of the dangerously irrelevant operator (44) is
straightforward in the free fermion phase. What might its effect be in the
conformal window, assuming that $p_{DI}>p_{\star}$, so that the 4-fermion term
does indeed provide a relevant nudge in the infra-red. It will make the theory
flow to a new fixed point and, once again, there are various options: we may
flow to the free fermion theory, or break the global group $G$. But we could
flow to a different interacting fixed point. And because the global symmetry
group and its anomalies factorise, it is natural to postulate that so too does
the infra-red CFT, which takes the form of two decoupled sectors

$\displaystyle{\rm CFT}\Big{[}q;\ydiagram{1,1}\Big{]}\otimes{\rm
CFT}\Big{[}p;\ydiagram{2}\Big{]}$

Here CFT$[q;\ydiagram{1,1}]$ saturates the ’t Hooft anomalies of
$G_{1}=SU(q)\times U(1)_{1}^{\prime}$, which equal those of the anti-symmetric
fermion $\rho_{A}$. Meanwhile, CFT$[p;\ydiagram{2}]$ saturates the ’t Hooft
anomalies of $G_{2}=SU(p)\times U(1)_{2}^{\prime}$, which equal those of the
symmetric fermion $\rho_{S}$.

#### The Anti-Symmetric Theory

We can repeat the same story for the anti-symmetric theory. For our purposes,
it will be useful to think of the $SU(\tilde{N})$ theory, with global symmetry
group

$\displaystyle\tilde{G}=SU(\tilde{q})\times SU(\tilde{p})\times U(1)_{1}\times
U(1)_{2}$

We will leave $\tilde{N}$, $\tilde{q}$ and $\tilde{p}$ arbitrary for now; only
later will we see how they relate to $N$, $p$ and $q$ of the symmetric theory.
We again add a gauge singlet, so the full collection of fermions is

$\begin{array}[]{|c||c|c|c|c|c|c|}\hline\cr\rule{0.0pt}{10.76385pt}&SU(\tilde{N})&SU(\tilde{q})&SU(\tilde{p})&U(1)_{1}&U(1)_{2}\\\
\hline\cr\rule{0.0pt}{13.3472pt}\chi&\overline{\raisebox{-4.73611pt}{\epsfbox{anti.eps}}}&\mathbf{1}&\mathbf{1}&-\tilde{q}&-\tilde{p}\\\
\psi&{\tiny{{{\yng(1)}}}}&{\bf 1}&{\tiny{\overline{\yng(1)}}}&0&\tilde{N}-2\\\
\tilde{\psi}&{\tiny{\overline{\yng(1)}}}&{\tiny{\overline{\yng(1)}}}&\mathbf{1}&\tilde{N}-2&0\\\
\nu&{\bf 1}&{\Box}&\overline{\Box}&-(\tilde{p}+2)&2-\tilde{q}\\\
\hline\cr\end{array}$

The gauge anomaly cancels only if $\tilde{p}=\tilde{N}+\tilde{q}-4$. The
singlet $\nu$ couples to the other fermions through the same (possibly
dangerously) irrelevant four fermion term (44).

The story is entirely analogous to the symmetric theory. For large
$\tilde{q}$, the theory is weakly coupled and the four-fermion term is
irrelevant. For small $\tilde{q}$, there is the possibility that it becomes
dangerously irrelevant. This happens for some

$\displaystyle\tilde{q}<\tilde{q}_{DI}$

If the free fermion phase is realised, it is again clear that the role of the
4-fermion interaction (44) is to gap the bi-fundamental composite $\rho_{B}$.
Here, our interest is in the region of the conformal window in which the
4-fermion operator is relevant. Under the assumption that it flows again to
two decoupled CFTs, we have

$\displaystyle\widetilde{\rm
CFT}\Big{[}\tilde{q};\ydiagram{1,1}\Big{]}\otimes\widetilde{\rm
CFT}\Big{[}\tilde{p};\ydiagram{2}\Big{]}$

At this point, an obvious duality suggests itself. Whenever one of the CFTs
arising from the symmetric theory and the anti-symmetric theory share the same
symmetry group and anomalies, it is natural to conjecture that they are the
same CFT. For the $U(1)$ charges to match, we require

$\displaystyle N=\tilde{N}$

We then conjecture that when $q=\tilde{q}$ or when $p=\tilde{p}$, one of the
factors coincide

$\displaystyle{\rm CFT}\Big{[}q;\ydiagram{1,1}\Big{]}=\widetilde{\rm
CFT}\Big{[}q;\ydiagram{1,1}\Big{]}\ \ \ {\rm and}\ \ \ {\rm
CFT}\Big{[}p;\ydiagram{2}\Big{]}=\widetilde{\rm
CFT}\Big{[}p;\ydiagram{2}\Big{]}$

Note that it’s not possible for both infra-red factors to coincide in the two
theories because, for example, $q=\tilde{q}\Rightarrow p\neq\tilde{p}$. This
is trivial to see because $q>p$ while $\tilde{p}>\tilde{q}$. Instead, when
either $q=\tilde{q}$ or $p=\tilde{p}$, and both theories flow to a pair of
CFTs, one of this pair share the same symmetries and anomalies and so are
plausibly the same.

##  3 Chiral Gauge Theories and Spin(8) Theories

In this section, we study a slight variant of the chiral $SU(N)$ theories
described in the previous section. They differ only by the addition of an
extra adjoint fermion. We will show that they share the same global symmetry
and ’t Hooft anomalies as a non-chiral theory with Spin(8) gauge group. (As we
will see, strictly the gauge group must be ${\rm
Spin}(8)\times{\mathds{Z}}_{2}$.) This anomaly matching is an extension of the
supersymmetric duality of Pouliot and Strassler ps1 but, as we will see, with
matter content that does not admit a supersymmetric completion. Nonetheless,
we will explore the possibility that the $SU(N)$ theory flows in the infra-red
to the Spin(8) gauge theory.

###  3.1 Anomaly Matching

The theories in this section have global symmetry group

$\displaystyle G=SU(q)\times SU(p)\times U(1)_{1}\times U(1)_{2}\times
U(1)_{R}$

for some $q$ and $p$. We start by ignoring discrete quotients of $G$, but they
will be important later and will be described in some detail in Section 3.2.
The name of the $U(1)_{R}$ group is taken from the supersymmetric context,
where it coincides with the R-symmetry of the theory.

#### Chiral Gauge Theories

The first theory that realises this global symmetry has $SU(N)$ gauge group
and a collection of left-handed Weyl fermions, notably with one transforming
in the symmetric $\ydiagram{2}$ representation of $SU(N)$. The full collection
transforms as

$\displaystyle\begin{array}[]{|c||c|c|c|c|c|c|}\hline\cr&SU(N)&SU(q)&SU(p)&U(1)_{1}&U(1)_{2}&U(1)_{R}\\\
\hline\cr\lambda&\operatorname{adj}&\mathbf{1}&\mathbf{1}&0&0&N+2\\\
\chi&{\tiny{{\yng(2)}}}&\mathbf{1}&\mathbf{1}&-q&-p&-2N\\\
\psi&{\tiny{\overline{{\yng(1)}}}}&{\tiny{{\yng(1)}}}&\mathbf{1}&N+2&0&0\\\
\tilde{\psi}&{\tiny{{\yng(1)}}}&\mathbf{1}&{\tiny{{\yng(1)}}}&0&N+2&0\\\
\hline\cr\end{array}$ (59)

Clearly this is an extension of the symmetric theory (7) that we met
previously. It differs only by the addition of the adjoint fermion $\lambda$.
In the supersymmetric context, $\lambda$ plays the role of the gaugino. There
is nothing to prohibit extending this theory to have ${\cal N}=1$
supersymmetry and we briefly describe some of the known results about
supersymmetric dynamics below. Nonetheless, we will consider the non-
supersymmetric version, motivated by the fact that the Spin(8) theory that
shares the anomalies does not have a supersymmetric completion.

As in Section 2, the theory is consistent only if

$\displaystyle q=N+p+4$ (60)

The symmetric chiral theory only provides a realisation of the global symmetry
group $G$ for $q\geq p+6$. For $q$ and $p$ outside this range, there is a
second, different chiral theory that has the same $G$ and the same anomalies.
This theory has gauge group $SU(\tilde{N})$, now with a Weyl fermion in the
anti-symmetric $\ydiagram{1,1}$ rather than symmetric representation. The full
collection of left-handed fermions transforms as

$\displaystyle\begin{array}[]{|c||c|c|c|c|c|c|}\hline\cr\rule{0.0pt}{10.76385pt}&SU(\tilde{N})&SU(q)&SU(p)&U(1)_{1}&U(1)_{2}&U(1)_{R}\\\
\hline\cr\rule{0.0pt}{10.76385pt}\lambda&\operatorname{adj}&\mathbf{1}&\mathbf{1}&0&0&2-\tilde{N}\\\
\chi&\overline{\raisebox{-4.73611pt}{\epsfbox{anti.eps}}}&\mathbf{1}&\mathbf{1}&-q&-p&2\tilde{N}\\\
\psi&{\tiny{{{\yng(1)}}}}&\mathbf{1}&{\tiny{\overline{\yng(1)}}}&0&\tilde{N}-2&0\\\
\tilde{\psi}&{\tiny{\overline{\yng(1)}}}&{\tiny{\overline{\yng(1)}}}&{\bf
1}&\tilde{N}-2&0&0\\\ \hline\cr\end{array}$ (66)

Again, this is an obvious extension of the anti-symmetric theory of Section 2,
with the addition of the adjoint fermion $\lambda$. The theory is consistent
if we take

$\displaystyle p=\tilde{N}+q-4$

This coincides with (60) if we formally identify $\tilde{N}=-N$. This anti-
symmetric theory carries a representation of $G$ in the range $q\leq p+2$.

As in Section 2, the identificatation $N=-\tilde{N}$ allows us to write the ’t
Hooft anomalies of the symmetric and anti-symmetric theories in a unified
form. The two non-Abelian ’t Hooft anomalies are again given by (8). In
addition, there are 18 further anomalies of the form ${\cal A}[X^{2}\cdot
U(1)_{a}]$ with $a=1,2,R$ and $X$ either gravity or one of the global
symmetries. These anomalies are summarised in the following table:

$\displaystyle\begin{array}[]{|c||c|c|c|}\hline\cr{\cal
A}&U(1)_{1}&U(1)_{2}&U(1)_{R}\\\ \hline\cr\hline\cr SU(q)^{2}&N(N+2)&0&0\\\
\hline\cr SU(p)^{2}&0&N(N+2)&0\\\ \hline\cr
U(1)^{2}_{1}&Nq[(N+2)^{3}-\frac{1}{2}(N+1)q^{2}]&-\frac{1}{2}N(N+1)pq^{2}&-N^{2}(N+1)q^{2}\\\
\hline\cr
U(1)^{2}_{2}&-\frac{1}{2}N(N+1)qp^{2}&Np[(N+2)^{3}-\frac{1}{2}(N+1)p^{2}]&-N^{2}(N+1)p^{2}\\\
\hline\cr U(1)_{R}^{2}&-2N^{3}(N+1)q&-2N^{3}(N+1)p&\mbox{See Below}\\\
\hline\cr{\rm grav}^{2}&\frac{1}{2}qN(N+3)&\frac{1}{2}pN(N+3)&(N+1)(N-2)\\\
\hline\cr\end{array}\ \ \ \ $ (74)

The one, missing entry is given by

$\displaystyle{\cal A}[U(1)_{R}^{3}]=-(N+1)(N-2)(3N^{3}+N^{2}-4N-4)$ (75)

Lastly there is the mixed anomaly involving all three $U(1)$’s,

$\displaystyle{\cal A}[U(1)_{1}\cdot U(1)_{2}\cdot U(1)_{R}]=-N^{2}(N+1)pq$
(76)

#### A Spin(8) Gauge Theory

There is a very different gauge theory which realises the same symmetry group
$G$ and ’t Hooft anomalies. This is, roughly speaking, a ${\rm Spin}(8)$ gauge
theory with the following matter content. (We will explain what the subtlety
hidden in “roughly speaking” in Section 3.2.)

$\begin{array}[]{|c||c|c|c|c|c|c|}\hline\cr&{\rm
Spin}(8)&SU(q)&SU(p)&U(1)_{1}&U(1)_{2}&U(1)_{R}\\\
\hline\cr\rule{0.0pt}{10.76385pt}\tilde{\lambda}&\operatorname{adj}=\mathbf{28}&\mathbf{1}&\mathbf{1}&0&0&N+2\\\
v&\mathbf{8}_{v}&\overline{{\tiny{{\yng(1)}}}}&\mathbf{1}&-\frac{1}{2}(N-p)&\frac{1}{2}p&-\frac{1}{2}(N+6)\\\
s&\mathbf{8}_{s}&\mathbf{1}&{\overline{\tiny{{\yng(1)}}}}&-\frac{1}{2}q&-\frac{1}{2}(N+q)&\frac{1}{2}(N+6)\\\
c&\mathbf{8}_{c}&\mathbf{1}&\mathbf{1}&\frac{1}{2}Nq&\frac{1}{2}Np&\frac{1}{2}N(N-2)\\\
\sigma_{S}&\mathbf{1}&{\tiny{{\yng(2)}}}&\mathbf{1}&N-p&-p&4\\\
\sigma_{A}&\mathbf{1}&\mathbf{1}&\overline{\raisebox{-4.73611pt}{\epsfbox{anti.eps}}}&-q&-(N+q)&4\\\
\rho_{B}&\mathbf{1}&\overline{{\tiny{{\yng(1)}}}}&{\tiny{{\yng(1)}}}&p+2&q-2&-4\\\
u&\mathbf{1}&\mathbf{1}&\mathbf{1}&-qN&-pN&-(N^{2}-N+2)\\\
\hline\cr\end{array}$

One can check that all 21 ’t Hooft anomalies of this theory are again given by
(8), (74), (75) and (76).

The ${\rm Spin}(8)$ theory includes a number of gauge singlets. They must be
endowed with their global quantum numbers under $G$ through appropriate
interaction terms. These could come about by introducing scalars to the
theory, with appropriate Yukawa couplings, or by considering the irrelevant
4-fermion terms

$\displaystyle\mathcal{L}_{4}\sim(c\tilde{\lambda})(cu)+s^{\dagger}s^{\dagger}\tilde{\lambda}\sigma_{A}+v\tilde{\lambda}v\sigma_{S}+\rho_{B}\sigma_{S}\rho_{B}\sigma_{A}$

with the indices contracted appropriately to impose the desired
(anti)-symmetric structure on $\sigma_{A}$ and $\sigma_{S}$.

Note that the gauge singlet $\rho_{B}$ carries the same quantum numbers as its
namesake in the Bars-Yankielowicz free fermion theory. But the quantum numbers
of $\sigma_{A}$ and $\sigma_{S}$ above are not the same as the Bars-
Yankielowicz $\rho_{A}$ and $\rho_{S}$: the anti-symmetric under $SU(q)$ is
exchanged for a symmetric, and vice-versa for $SU(p)$. The $SU(q)^{3}$ anomaly
for the anti-symmetric representation is $N-4$; for the symmetric
representation it is $N+4$. Roughly speaking, the Spin(8) theory is an
extension of the Bars-Yankielowicz free fermions whose anomaly matching works
because $N-4+8=N+4$. However, the matching of all 21 anomalies appears quite
non-trivial333As a note of caution: there are examples in supersymmetric
theories where anomaly matching is known to be misleading mislead . There are
also non-supersymmetric theories whose anomalies match whose dynamical meaning
remains unclear terning .. The main purpose of this section is to explore
whether this matching can be realised dynamically, with one theory flowing to
the other.

In the case of $p=0$, there is a straightforward answer. On the chiral side,
the $p=0$ symmetry can be realised only by the symmetric theory. Meanwhile,
the Spin(8) theory has neither $s$, $\sigma_{A}$ nor $\rho_{B}$ fermions. In
this case, there is no obstacle to supersymmetrising either of these theories,
and the anomaly matching finds a natural home in the supersymmetric Pouliot-
Strassler duality ps1 .

For other values of $p$ the chiral theories – with both symmetric and anti-
symmetric fermions – can be supersymmetrised and there is a long literature
discussing the resulting dynamics. There is a known dual for the $p=1$
supersymmetric chiral theory, which is based around a Spin(10) gauge group
ps10 . There are also known duals for the supersymmetrised chiral theories
involving an anti-symmetric tensor in which the anti-symmetric chiral
multiplet arises as a composite from the confining dynamics of an auxiliary
gauge group micha ; poul . (See also hooked ; hooke .) Much of the interest in
these theories and their duals revolves around the fact that they tend to
dynamically break supersymmetry ads ; erich . However, to our knowledge, the
anomaly matching between the chiral theories and the Spin(8) theory described
above – which suggests a natural extension of the Pouliot-Strassler duality
ps1 – has not been previously noted. Indeed, as we now explain, if there is a
duality between these two theories then it is not one that enjoys
supersymmetry.

While there is no difficulty in adding scalars to make the chiral $SU(N)$
theories supersymmetric, this is not possible for the Spin(8) theory when
$p\neq 0$. At first glance, this seems surprising. After all, the fermion
$\tilde{\lambda}$ is nicely placed to play the role of the gaugino, while all
other fermions would seem to sit nicely in chiral multiplets. Nonetheless,
naive attempts at making the theory supersymmetric fail. Here we explain why.

Suppose that $\tilde{\lambda}$ is a gaugino of a supersymmetric Spin(8)
theory, with all other fermions sitting in chiral multiplets. We will denote
these superfields with hats, so $\hat{v}$ is the chiral superfield containing
the fermion $v$, and so on. We would like to write down interaction terms so
that the global symmetry of the theory is $G$. This is straightforward for
$v$, $s$ and $c$ but, as we’ve seen above, we must write down further
interactions that imprint the correct symmetry transformations on the gauge
singlets. One can check that there is a superpotential term that correctly
implements the transformation for $\hat{\sigma}_{S}$ and $\hat{u}$: these are

$\displaystyle W=\hat{u}\hat{c}\hat{c}+\hat{\sigma}_{S}\hat{v}\hat{v}$

This is neutral under $SU(p)\times SU(q)\times U(1)_{1}\times U(1)_{2}$ and
has charge +2 under $R_{\rm susy}=R/(N+2)$. Here the normalisation ensures
that $\tilde{\lambda}$ has $R_{\rm susy}$ charge $+1$. (And we must remember
that $R_{\rm susy}({\rm boson})=R_{\rm susy}{(\rm fermion})+1$ for chiral
superfields.) However, no further superpotential terms are possible preserving
all symmetries, and with only positive powers of fields. For example, one can
construct the term
$(\hat{s}\hat{v}\hat{c}\hat{\rho}_{B}\hat{\sigma}_{S})^{2}\hat{u}$, which
imposes the right $SU(q)\times SU(p)\times U(1)_{1}\times U(1)_{2}$ symmetry
structure on $\hat{\rho}_{B}$, but not the right R-charge. There is no
holomorphic term that imposes the right symmetry structure on
$\hat{\sigma}_{A}$. (The choice
$\hat{\sigma}_{A}\hat{\rho}_{B}\hat{\rho}_{B}\hat{\sigma}_{S}$ vanishes on
symmetry grounds, and also carries the wrong R-charge.)

One may wonder if we can make progress by introducing yet further fields.
These additional fields cannot contribute to the anomalies of $G$, and so
should be vector-like. But then we can always give them a mass and integrate
them out and any constraints that they impose should be reproducible using
just irrelevant operators for the massless fields. And, as we have seen, no
such superpotential is possible. It appears that any duality that exists
between these theories must be non-supersymmetric444One may wonder if the
symmetry transformations can be imprinted by irrelevant terms arising from a
non-trivial Kähler potential. Even this appears challenging, simply because
the obvious combination
$\hat{s}^{i}\hat{s}^{j}(\hat{\sigma}_{A})_{ij}^{\dagger}$ vanishes on anti-
symmetry grounds..

###  3.2 Discrete Symmetries

As we’ve seen, the ’t Hooft anomalies for the continuous symmetry group $G$
are shared by the $SU(N)$ and ${\rm Spin}(8)$ theories. But that alone does
not guarantee that the symmetry structure matches. One must worry about
discrete aspects of the symmetry. In this section we look more carefully at
these discrete quotients.

Before we embark on a full analysis, we can flag up the issue of concern. In
the $SU(N)$ chiral theory, all fermions have integer $U(1)$ charges, while in
the ${\rm Spin}(8)$ theory, for certain values of $N$, $p$ and $q$, some
fermions have half-integer charges. This, in itself, is not necessarily a
worry since only gauge invariant operators should match across the duality.
However, one can check that the gauge invariant composite fermion $vsc$ has a
half-integer charge under some $U(1)\subset G$ whenever $N$ is odd. This means
that there is no possibility of a duality between the $SU(N)$ theory and ${\rm
Spin}(8)$ theory as currently stated, at least for $N$ odd.

This issue, it turns out, lies at the heart of a mismatch between the discrete
symmetry structure of the two theories. There is, however, a simple
resolution. The ${\rm Spin}(8)$ theory has an additional $\mathds{Z}_{2}$
symmetry which can be taken to act as

$\displaystyle\mathds{Z}_{2}:c\rightarrow-c$ (77)

If we promote this to a gauge symmetry, then it projects out the offending
composite $vsc$ from the spectrum. As we now show, after making this
projection the discrete symmetry structure between the two theories matches.

In the rest of this section, we explain this in more detail. Specifically we
will show that, after gauging the $\mathds{Z}_{2}$ symmetry (77), the
faithfully acting global symmetry group of both theories is (roughly)

$\displaystyle G=\frac{SU(q)\times SU(p)\times U(1)_{1}\times U(1)_{2}\times
U(1)_{R}}{\mathds{Z}_{p}\times\mathds{Z}_{q}}$ (78)

(There are a couple of further flourishes that should be included when $N$,
$p$ and $q$ are even that turn the “roughly” into “exactly”. We describe these
below.)

#### Discrete Symmetries for the Chiral Theories

We start with the chiral theories. First we show that there are no additional
discrete symmetries. These can arise if there is a $\mathds{Z}_{k}$ subgroup
of the anomalous $U(1)$ that changes the $\theta$ angle by $2\pi$. Under a
general $U(1)$ transformation,

${\begin{split}\chi\to e^{i\omega_{\chi}}\chi\ ,\ \psi\to
e^{i\omega_{\psi}}\psi\ ,\ \tilde{\psi}\to
e^{i\omega_{\tilde{\psi}}}\tilde{\psi}\ ,\ \lambda\to
e^{i\omega_{\lambda}}\lambda\ ,\end{split}}$

the $\theta$ angle is shifted by

${\begin{split}\delta\theta=(N+2)\omega_{\chi}+q\omega_{\psi}+p\omega_{\tilde{\psi}}+2N\omega_{\lambda}\
.\end{split}}$

Is there a choice of $\omega_{i}$ which is not part of the continuous global
symmetry such that $\delta\theta=2\pi$? Using the 3 $U(1)$s, we can always set
$\omega_{\chi}=\omega_{\tilde{\psi}}=\omega_{\lambda}=0$, so we have

${\begin{split}q\omega_{\psi}=2\pi\ \Rightarrow\ \omega_{\psi}=\frac{2\pi}{q}\
.\end{split}}$

But this is equivalent to an $SU(q)$ centre transformation. Hence, there are
no additional discrete symmetries.

Next, we will find the faithful symmetry group acting on gauge invariant
operators. In order to do that we need to write the most general
transformation acting on the fermions and look for non-trivial solutions that
leave all the fermions invariant. We analyse the different factors of $G$ in
turn. First, we check that the $U(1)^{3}$ symmetries are freely acting. Then
we show that some combination of $U(1)$ transformations can be undone by a
suitable $SU(q)\times SU(p)$ transformation, resulting in the quotient in
(78).

First $U(1)_{R}$. It is simple to check that this is freely acting if $N$ is
odd, but can be quotiented by $\mathds{Z}_{2}$ if $N$ is even, and by
$\mathds{Z}_{4}$ if $N=2\ mod\ 4$. This same behaviour occurs for the Spin(8)
theory too. This is a rather trivial quotient and we ignore it in what
follows.

Next, we turn to $U(1)_{1}\times U(1)_{2}$. For the symmetric theory, we
define the linear combination of the $U(1)_{1}\times U(1)_{2}$ charges $Q_{1}$
and $Q_{2}$

$\displaystyle Q_{A}=\frac{1}{N}(Q_{2}-Q_{1})\ \ \ {\rm and}\ \ \
Q_{B}=\frac{q}{N(N+2)}(Q_{1}-Q_{2})+\frac{1}{N+2}Q_{1}$ (79)

For the anti-symmetric theory, we take the same linear combination but with
$N=-\tilde{N}$. The charges of the fermions $\chi$, $\psi$ and $\tilde{\psi}$
under these redefined $U(1)$’s are given by

$\displaystyle\begin{array}[]{|c||c|c|}\hline\cr&U(1)_{A}&U(1)_{B}\\\
\hline\cr\rule{0.0pt}{10.76385pt}\chi&(N+4)/{N}&-{2q}/{N}\\\
\psi&-(N+2)/{N}&(N+q)/{N}\\\ \tilde{\psi}&(N+2)/{N}&-{q}/{N}\\\
\hline\cr\end{array}$ (84)

These are fractional. However, all gauge invariant operators have integer
charges. To see this, it suffices to check that a $2\pi$ rotation by either
$U(1)_{A}$ or $U(1)_{B}$ is proportional to the action of the centre of the
$SU(N)$ gauge group. Furthermore, the two gauge invariant operators
$\tilde{\psi}\chi^{\dagger}\tilde{\psi}$ and $\psi\tilde{\psi}$ have
$U(1)_{A}\times U(1)_{B}$ charges of $(1,0)$ and $(0,1)$ respectively. Since
there exist gauge invariant operators with the minimal charges, no quotient
acting solely on these two $U(1)$s is possible.

It is, however, possible that some combination of $U(1)$ transformations
coincides with the centres of the non-Abelian groups. The transformation of
$\lambda$ means that there is no possibility that a $U(1)_{R}$ can participate
in this game, but we must look more carefully at $U(1)_{A}\times U(1)_{B}$. We
parameterise the centres by $k,l,m\in\mathds{Z}$ for $SU(N)$, $SU(q)$ and
$SU(p)$ respectively, and denote the $U(1)$ transformations as $e^{2\pi
iQ_{A}\alpha}$ and $e^{2\pi iQ_{B}\beta}$. The action of these combined
symmetries leave $\chi$, $\psi$ and $\tilde{\psi}$ invariant only if the
following conditions hold:

$\displaystyle\chi:\ \ \ $
$\displaystyle\frac{2k}{N}+\frac{N+4}{N}\alpha-\frac{2q}{N}\beta\ \in\
\mathds{Z}$ $\displaystyle\psi:\ \ \ $
$\displaystyle-\frac{k}{N}+\frac{l}{q}-\frac{N+2}{N}\alpha+\frac{N+q}{N}\beta\
\in\ \mathds{Z}$ $\displaystyle\tilde{\psi}:\ \ \ $
$\displaystyle\frac{k}{N}+\frac{m}{p}+\frac{N+2}{N}\alpha-\frac{q}{N}\beta\
\in\ \mathds{Z}$

There are two solutions. The first is

$\displaystyle k=-1\ \ ,\ \ l=1\ \ ,\ \ m=0\ \ ,\ \ \alpha=0\ \ ,\ \
\beta=-\frac{1}{q}$ (85)

This is the quotient by $\mathds{Z}_{q}$ in (78). The second is

$\displaystyle k=-1\ \ ,\ \ l=0\ \ ,\ \ m=1\ \ ,\ \ \alpha=-\frac{2}{p}\ \ ,\
\ \beta=-\frac{1}{p}$ (86)

This is the quotient by $\mathds{Z}_{p}$ in (78).

If both $p$ and $q$ are even then there is a
$\mathds{Z}_{2}\subset\mathds{Z}_{q}\times\mathds{Z}_{p}$ quotient that acts
only on the centres of the non-Abelian groups, without involving the
$U(1)^{3}$. There is one final subtlety. It turns out that some combination of
the global symmetry transformation coincides with the fermion number
$(-1)^{F}$. This $\mathds{Z}_{2}$ quotient arises from $k=-1$ $SU(N)$ centre
transformation, coupled with a $U(1)_{A}\times U(1)_{R}$ transformation which,
combined, satisfy

$\displaystyle\frac{k}{N}+\left(\frac{1}{2}-\frac{N}{N+2}\right)Q_{A}+\frac{1}{2(N+2)}{Q_{R}}=\left\\{\begin{array}[]{cc}\frac{1}{2}\
{\rm mod}\ \mathds{Z}&\ \ \mbox{for fermions}\\\ 0\ {\rm mod}\ Z&\ \ \mbox{for
bosons}\end{array}\right.$ (89)

where we should take $k=-1$. This means that the true symmetry group takes the
form

$\displaystyle\frac{Spin(1,3)\times G}{\mathds{Z}_{2}}$

#### Discrete Symmetries for the Spin(8) Theory

Now we turn to the Spin(8) theory. For the proposed duality to have a chance
of success, the global structure of the symmetry group should agree. We will
see that this is indeed the case after we gauge a $\mathds{Z}_{2}$ symmetry of
the Spin(8) symmetry.

As with the chiral theories, we will start by checking whether there are
additional discrete transformations that change $\theta$ by $2\pi$. Under a
general $U(1)$ rotation of the charged fields,

${\begin{split}v\to e^{i\omega_{v}}v\ ,\ s\to e^{i\omega_{s}}s\ ,\ c\to
e^{i\omega_{c}}c\ ,\ \tilde{\lambda}\to
e^{i\omega_{\tilde{\lambda}}}\tilde{\lambda}\ ,\end{split}}$

the Spin(8) $\theta$-angle is shifted by

${\begin{split}\delta\theta=12\omega_{\tilde{\lambda}}+2\omega_{c}+2p\omega_{s}+2q\omega_{v}\
.\end{split}}$

Using the 3 $U(1)$s we can set
$\omega_{v}=\omega_{s}=\omega_{\tilde{\lambda}}=0$. We are left with

${\begin{split}2\omega_{c}=2\pi\ \Rightarrow\ \omega_{c}=\pi\ .\end{split}}$

We see that the $\mathds{Z}_{2}$ transformation (77), which acts as $c\to-c$,
is a global symmetry of the theory. In what follows we will find various
discrepancies between the global symmetry structure of the $SU(N)$ theory and
the Spin(8) theory. All of them are resolved if this $\mathds{Z}_{2}$ symmetry
is gauged.

First, we ask: is this $\mathds{Z}_{2}$ contained inside the other global
symmetries? If $N$ is odd, a $U(1)_{A}$ transformation by $2\pi$ acts as
$(v,s,c)\to(-v,-s,-c)$. If we accompany this by a centre transformation of
Spin(8), it is equivalent to $c\to-c$. Hence, in this case the
$\mathds{Z}_{2}$ is contained in $U(1)_{A}$. This is related to the fact,
mentioned above, that the gauge invariant operator $vsc$ has half-integer
charge when $N$ is odd. In contrast, when $N$ is even the $\mathds{Z}_{2}$ is
not contained within the other symmetry. To see this, we exhibit gauge
invariant operators that are charged only under $c\to-c$. Such an operator is
given by $v^{q}s^{p}c\,tr(\tilde{\lambda}^{6})$ if $q,p$ are odd, and
$v^{q-1}s^{p+1}c\,\rho_{B}tr(\tilde{\lambda}^{5})$ if $q,p$ are even. Although
the underlying reasons are different, for both $N$ odd and $N$ even we should
gauge the $\mathds{Z}_{2}$ symmetry (77).

Now we can move on to the quotients. If $N=0,2\ mod\ 4$ then we should
quotient the $U(1)_{R}$ symmetry by a $\mathds{Z}_{2},\ \mathds{Z}_{4}$
respectively to ensure that it acts faithfully. This is the same structure
that we saw in the $SU(N)$ theory.

Next, under the linear combinations of charges (79) that form $U(1)_{A}$ and
$U(1)_{B}$, the various fields of the Spin(8) theory transform as

$\displaystyle\begin{array}[]{|c||c|c|}\hline\cr&U(1)_{A}&U(1)_{B}\\\
\hline\cr\rule{0.0pt}{10.76385pt}v&1/2&-1\\\ s&-1/2&0\\\ c&-(N+4)/2&q\\\
\sigma_{S}&-1&2\\\ \sigma_{A}&-1&0\\\ \rho_{B}&1&-1\\\ u&N+4&-2q\\\
\hline\cr\end{array}$ (98)

The adjoint fermion has $A[\tilde{\lambda}]=B[\tilde{\lambda}]=0$. A number of
these fields have $1/2$ integer $U(1)_{A}$ charges. One can check that all
gauge invariant fields have integer charges with one exception: the composite
$vsc$ has $1/2$ integer $U(1)_{A}$ charge when $N$ is odd. Hence, as it
stands, this field can have no counterpart in the chiral description. As we
mentioned previously, this discrepancy is resolved by gauging the
$\mathds{Z}_{2}$ symmetry (77). This removes the offending states from the
spectrum.

After this gauging, the remaining symmetry structure coincides with that of
the chiral theory. First, one can check that there are gauge invariant
operators with minimal $U(1)_{A}$ and $U(1)_{B}$ charges (e.g. $\sigma_{A}$
and $\sigma_{A}\rho_{B}$). Furthermore, one can show that the transformations
(85) and (86) act trivially on all fields and so the faithful action of the
global symmetry again enjoys the quotient $\mathds{Z}_{p}\times\mathds{Z}_{q}$
seen in (78). (Note: one should neglect the $k=-1$ centre of $SU(N)$ when
implementing (85) and (86) on the Spin(8) fields.)

The last remaining subtlety involves the fermion number. The constraint (89)
again holds for almost all gauge invariant operators (this time with $k=0$ as
there is no $SU(N)$ gauge symmetry). The “almost” is because there is one
exception: the fermion $vsc$ obeys

$\displaystyle\left(\frac{1}{2}-\frac{N}{N+2}\right)Q_{A}[vsc]+\frac{1}{2(N+2)}Q_{R}[vsc]=\frac{N}{2}$

This is inconsistent with the spin relation when $N$ is even. Again, this
discrepancy is resolved if we act with the $\mathds{Z}_{2}$ quotient (77).

The upshot of this analysis is that we should gauge the additional
$\mathds{Z}_{2}$ symmetry both for $N$ even and $N$ odd. The global symmetry
structure of the two conjectured dual theories then agrees.

#### There Are No Further Discrete Anomalies

One may worry that there may be further discrete ’t Hooft anomalies that must
also be matched between the two theories. In fact, as we now explain, the
matching of perturbative ’t Hooft anomalies for $SU(q)\times SU(p)\times
U(1)^{3}$ is sufficient to ensure the matching of the faithful symmetry group
$G$ given by (78).

This follows from the following, general result. Suppose that
$\mathcal{T}_{1}$ and ${\cal T}_{2}$ are two theories that share the same
symmetry group $G={H}/{\Gamma}$, where $\Gamma$ is a discrete subgroup of $H$.
If $\Pi_{0}(H)=0$, then ’t Hooft anomaly matching for $G$ follows from anomaly
matching for $H$. To understand why, we can define the combined theory
$\mathcal{T}_{+}=\mathcal{T}_{1}+\mathcal{T}_{2}^{\dagger}$. $\mathcal{T}_{+}$
has no anomalies for $H$. This means that if we turn on background gauge
fields for $H$, symmetry transformations don’t generate any $\theta$-term for
these background fields. We write these as $\bar{\theta}$ to distinguish them
from the dynamical theta terms that appeared in our earlier analysis. The lack
of anomalies for $H$ means that $\delta\bar{\theta}=0$. The quotient by
$\Gamma$ can change the periodicity of $\bar{\theta}$, but this doesn’t change
the fact that $\delta\bar{\theta}=0$ exactly. Hence, $\mathcal{T}_{+}$ has no
anomalies for $G$.555The requirement that $\Pi_{0}(H)=0$ is somewhat hidden in
the discussion. This requirement is needed because, even though
$\mathcal{T}_{+}$ has no anomalies for $H$, a non-trivial element of
$\Pi_{0}(H)$ can act as $\delta\bar{\theta}=2\pi$. The quotient by $\Gamma$
can change the periodicity of $\bar{\theta}$ such that
$\delta\bar{\theta}=2\pi$ is non-trivial and $\mathcal{T}_{+}$ has an anomaly
for $G$ even when there is no anomaly for $H$.

###  3.3 Putative Dynamics

The $SU(N)$ and ${\rm Spin}(8)$ gauge theories share the same symmetry
structure and the same anomalies. The obvious question is: so what? Is there
some way in which these theories are dynamically related, presumably for some
particular range of $N$, $p$ and $q$?

This is a difficult question to answer. It may be that the two theories are
dynamically related, but only after suitable scalars are added, with a well-
designed potential coaxing one theory to flow to the other. Supersymmetric
theories are, of course, in this category but understanding similarly detailed
phenomena in the absence of supersymmetry lies well beyond our current grasp
of strongly coupled quantum field theory. Nonetheless, given the anomaly
matching it seems reasonable to conjecture that, for a certain range of
parameters, the chiral $SU(N)$ gauge theories may flow to the ${\rm Spin}(8)$
theory at low energies. In this section and the next, we make a few obvious
comments about possible relations between the two theories.

We start with the chiral $SU(N)$ theories. We will consider the theories
without additional scalars; these change some of the specific numbers below,
but not the overall story. The $SU(N)$ theory with a symmetric fermion is
asymptotically free provided:

$\displaystyle q>q_{AF}=\frac{1}{7}(9p+34)$

(Recall that $q=N+p+4$ so this inequality is telling us that the number of
colours $N$ should be suitably larger than the number of flavours $p$, as
expected.) For $q\leq q_{AF}$, the $SU(N)$ symmetric theory is infra-red free.
For $q$ sitting slightly above the value $q_{AF}$, the theory flows to an
interacting, chiral Banks-Zaks fixed point, with the global symmetry group $G$
unbroken. This conformal behaviour, with unbroken $G$ will persist for some
range $q\in(q_{AF},q_{\rm crit})$ that defines the conformal window. As in
QCD, the value of $q_{\rm crit}$ that determines the strongly coupled end of
the conformal window is unknown. Our interest here lies in those values
$q>q_{\rm crit}$ that are beyond the conformal window. The theory may, of
course, break the global symmetry $G$. Alternatively, it may flow to the ${\rm
Spin}(8)$ theory described above.

We have one data point where we do understand quantitatively what happens.
This is the case $p=0$ where we add scalars so the theory becomes
supersymmetric. Here the chiral $SU(N)$ theory flows to the supersymmetric
${\rm Spin}(8)$ theory only for $N\geq 13$ or, equivalently, $q\geq 17$ ps1 .

If there is indeed a flow from the non-supersymmetric chiral theories to the
Spin(8) theory, then it should be possible to match gauge invariant operators
across the duality. Indeed, there is a fairly natural conjecture for the gauge
singlet fermions in the ${\rm Spin}(8)$ theory. Their quantum numbers coincide
with the following gauge-invariant, composite operators of the $SU(N)$ theory,

$\displaystyle\sigma_{S}\sim\chi\lambda^{2}\psi^{2}\ \ \ ,\ \ \
\sigma_{A}\sim\chi\lambda^{2}\tilde{\psi}^{2}\ \ \ ,\ \ \
\rho\sim\chi^{\dagger}(\lambda^{\dagger})^{2}\psi^{\dagger}\tilde{\psi}\ \ \
,\ \ \ u\sim\chi^{N}\lambda^{N-1}$ (99)

where
$u\sim\epsilon_{i_{1}...i_{N}}\epsilon_{j_{1}...j_{N}}\chi^{i_{1}j_{1}}...\chi^{i_{N}j_{N}}\text{tr}(\lambda...\lambda)$.

There is a similar story for the $SU(N)$ chiral theory with an anti-symmetric
fermion. Now the asymptotic freedom bound is

$\displaystyle q<\tilde{q}_{AF}=\frac{1}{9}(7p+34)$

Note that the sign of the inequality is reversed. For $q\geq\tilde{q}_{AF}$,
the chiral theory is infra-red free. For $q\in(\tilde{q}_{\rm
crit},\tilde{q}_{AF})$, the theory flows to an interacting fixed point with
$\tilde{q}_{\rm crit}$ unknown. Again, our interest is in the strong coupling
dynamics for $q<\tilde{q}_{\rm crit}$ where the ${\rm Spin}(8)$ theory
provides a putative end point for the flow.

This story is largely consistent with the dynamics of the ${\rm Spin}(8)$
theory, which is asymptotically free only if

$\displaystyle q+p<34$

This suggests that there are large swathe of parameter space in which the
chiral $SU(N)$ theory may flow to an infra-red free ${\rm Spin}(8)$ theory.

Figure 4: Candidate low energy phases of the chiral $SU(N)$ theories, plotted
as a function of $q$ and $p$. The upper region is the chiral theory with a
symmetric fermion and the lower with an anti-symmetric fermion. In the white
strip, there are no chiral theories carrying a representation of the global
symmetry group $G$. In the blue strip, the $SU(N)$ chiral theory is IR free.
The green region is where flows to the Spin(8) theory are inconsistent and
form a potential conformal window for the chiral gauge theory. The orange and
light red regions are where the flow from the chiral theories to the ${\rm
Spin}(8)$ theory is consistent with the $a$-theorem; in the orange region the
Spin(8) theory is IR free and in the light red region it is asymptotically
free. Finally, the dark red region shows where a flow from Spin(8) to the
infra-red free chiral theory is feasible.

Further constraints on the allowed RG flows come from the a-theorem. For the
symmetric chiral theory, we have (again, absorbing a factor of
$90(8\pi)^{2}$),

$\displaystyle
a_{s}=\dfrac{1}{4}\left[303q^{2}-2325q-562pq+7p(37p+307)+4182\right]$

Meanwhile, for the Spin(8) theory,

$\displaystyle
a_{8}=\dfrac{1}{4}\left[11(p^{2}+q^{2})+187q+11p(2q+15)+8430\right]$

In the large $N$ limit, this tells us that the symmetric chiral theory can
flow to the Spin(8) theory provided

$\displaystyle N\gtrsim 0.39p$

Similarly the anti-symmetric chiral theory has

$\displaystyle
a_{a}=\dfrac{1}{4}\left[303p^{2}+7q(37q-307)+p(-562q+2325)+4182\right]$

At large $N$, this theory can flow to the Spin(8) theory provided that

$\displaystyle\tilde{N}\gtrsim 0.39q$

These results are plotted in the $(q,p)$-plane in Figure 4.

Figure 5: Possible RG flows of the chiral $SU(N)$ theories as they are
deformed by a mass term for the adjoint fermion.

It has been known since the initial work of Pouliot ps0 that chiral gauge
theories may have non-chiral duals. In the known supersymmetric examples, a
chiral theory in the UV always flows to a non-chiral theory in the IR. It is
interesting to ask whether the inverse flow is possible, with chirality – in
the sense that no gauge invariant mass terms are allowed – emerging in the IR.
There is nothing to prohibit such a flow in principle, and it would be
interesting as it may offer an alternative avenue to putting chiral gauge
theories on the lattice. Taken at face value, the results above suggest that
there is a small window, shown by the red region in Figure 4, in which the
${\rm Spin}(8)$ theory is asymptotically free and may flow to a (admittedly
weakly coupled) chiral $SU(N)$ theory. The window is not parametrically large
and it quickly closes if one starts to introduce further scalars. Nonetheless,
this behaviour remains as an interesting possibility.

#### Mass Deformations

If the non-supersymmetric flow from the chiral theory to the Spin(8) theory
does take place for some range of parameters, then it has an interesting
consequence. Suppose we deform the chiral theory by giving a mass to the
adjoint fermion $\lambda$,

$\displaystyle\delta\mathcal{L}=m\,{\rm tr}\,\lambda\lambda+{\rm h.c.}$

From (59) or (66), we see that this breaks the $U(1)_{R}$ symmetry, leaving
the other global symmetry groups intact. What is the effect of this mass
deformation in the infra-red?

For $m\ll\Lambda$, the strong coupling scale of the chiral gauge theory, it
should be possible to see the effect directly in the low-energy effective
theory. If this is the Spin(8) dual, then the deformation naturally maps to a
mass term for the adjoint fermion $\tilde{\lambda}$, again breaking the
$U(1)_{R}$ symmetry. No other relevant deformations are possible without
further breaking the global symmetry.

In particular, the gauge singlets $\sigma_{S}$, $\sigma_{A}$, $\rho_{B}$ and
$u$ must all remain massless after this deformation. Yet, as shown in (99),
each is a composite operator that contains the now-massive fermion $\lambda$.
If the duality holds there must be massless composite fermions that contain a
massive constituent. This behaviour is surprising and is in violation of the
so-called “persistence of mass condition” preskill . (The fact that non-
supersymmetric dualities violate the persistence of mass condition was
previously noted in ofer .) Indeed, such behaviour is ruled out in QCD-like
theories by the Vafa-Witten theorem vw . But the Vafa-Witten theorem relies on
the positive definiteness of the fermionic measure, which is true in QCD but
not in the chiral gauge theories that are of interest here.

It is unclear what to make of this. Depending on taste, this behaviour could
be viewed as a hint that the duality does not, in fact, hold, or as a hint
that strongly interacting chiral gauge theories can do interesting things.

In the limit $m\gg\Lambda$, we can integrate out the adjoint fermion and
recover the Bars-Yankielowicz chiral theories of Section 2. There does not
appear to be any relevant operator that flows from the Spin(8) theory to the
free fermions (21). This means that there should be a phase transition as the
mass is increased. This is shown schematically in Figure 5. Presumably, at the
separatrix between the different end points, there should be a novel fixed
point.

As an aside: one might naively think that one could integrate out
$\tilde{\lambda}$ to be left with a Spin(8) theory without an adjoint fermion
that shares the same ’t Hooft anomalies as the $SU(N)$ chiral gauge theories
that we met in Section 2. While it is true that the perturbative ’t Hooft
anomalies agree, the global structure of the symmetry group does not. For this
reasons, we do not know of a Spin(8) dual for the Bars-Yankielowicz theories.

##  4 Deformations of Supersymmetric Dualities

In this final section, we review and slightly extend a number of interesting
comments and proposals for non-supersymmetric dualities that can be found in
the literature.

###  4.1 Vector-Like Dualities Without Scalars

One way to search for non-supersymmetric dualities is to start from well known
supersymmetric dualities and softly break supersymmetry on both sides. There
have been many attempts at doing so, including ofer ; yael ; ah ; mury . But
retaining control in the strong coupling regime after supersymmetry breaking
is challenging and, to date, there are no well-established dualities beyond
those that arise trivially through soft supersymmetry breaking.

One may try to be bolder. The ’t Hooft anomaly matching conditions that hold
for Seiberg dualities also trivially hold for the theories in which all scalar
superpartners are simply discarded. For example, from the original Seiberg
duality seib2 , the QCD-like theory with the following fermions

$\displaystyle{\rm Electric\ Theory:}$
$\displaystyle\begin{array}[]{|c||c|c|c|c|c|}\hline\cr&SU(N_{c})&SU(N_{f})_{L}&SU(N_{f})_{R}&U(1)_{B}&U(1)_{R}\\\
\hline\cr\lambda&\operatorname{adj}&\mathbf{1}&\mathbf{1}&0&N_{f}\\\
\psi&\Box&\Box&\mathbf{1}&+1&-N_{c}\\\
\tilde{\psi}&\overline{\Box}&\mathbf{1}&\overline{\Box}&-1&-N_{c}\\\
\hline\cr\end{array}$ (104)

has the same ’t Hooft anomalies as the following fermions of the magnetic
theory

$\displaystyle{\rm Magnetic\ Theory:}$
$\displaystyle\begin{array}[]{|c||c|c|c|c|c|}\hline\cr&SU(N_{f}-N_{c})&SU(N_{f})_{L}&SU(N_{f})_{R}&U(1)_{B}&U(1)_{R}\\\
\hline\cr\tilde{\lambda}&\operatorname{adj}&\mathbf{1}&\mathbf{1}&0&N_{f}\\\
q&\Box&\overline{\Box}&\mathbf{1}&+1&N_{c}-N_{f}\\\
\tilde{q}&\overline{\Box}&\mathbf{1}&\Box&-1&N_{c}-N_{f}\\\
\mu&\mathbf{1}&\Box&\overline{\Box}&0&N_{f}-2N_{c}\\\ \hline\cr\end{array}$
(110)

Might it be that, for some values of $N_{c}$ and $N_{f}\geq N_{c}$, the non-
supersymmetric electric theory flows to the magnetic theory, preserving the
$SU(N_{f})_{L}\times SU(N_{f})_{R}\times U(1)_{B}\times U(1)_{R}$ global
symmetry?

The answer is an unequivocal no ofer . This follows from various QCD theorems.
The key observation is that the determinant that arises from integrating out
the fermions is positive definite. (This is not true of the chiral theories
described in earlier section, and nor is it true of supersymmetric Seiberg
dualities where the presence of scalars with Yukawa couplings is sufficient to
ensure that the fermion determinant can have both positive and negative sign.)
This fact allows us to invoke various Vafa-Witten-Weingarten theorems.

There are two such arguments ofer . The first uses the Weingarten inequality
wein . Roughly speaking, in the context of QCD, the Weingarten inequality
states that the proton is always heavier than the pion. Crucially, the proof
of the inequality assumes neither confinement nor chiral symmetry breaking. In
the present context, the Weingarten inequality can be applied to the gauge
invariant singlet in the infa-red. This is a composite fermion, with UV
constituents given by

$\displaystyle\mu=\tilde{\psi}\lambda\psi$ (111)

The Weingarten inequality can be used to show that $\mu$ is always heavier
than the scalar formed from $\tilde{\psi}\psi$. But if all global symmetries
are unbroken, and in the absence of fine-tuning, then the scalar
$\tilde{\psi}\psi$ is surely massive. The Weingarten inequality then states
that $\mu$ must also be massive and so the ’t Hooft anomalies cannot be
saturated by the matching above.

The Weingarten inequalities also apply, with some small amendments, to the
$SO(N)$ and $Sp(N)$ dualities of is and ip in the absence of scalars. We
describe this in some detail in Appendix B.

The second argument of ofer arises if we deform the electric theory by
providing a small mass for the adjoint fermion $\lambda$. This breaks the
$U(1)_{R}$ symmetry, leaving the remaining $SU(N_{f})_{L}\times
SU(N_{f})_{R}\times U(1)_{B}$ untouched. In the infra-red, magnetic theory the
only such deformation consistent with these symmetries is a mass for the
adjoint fermion $\tilde{\lambda}$. In particular, the gauge singlet $\mu$ must
remain massless. But $\mu$ is a composite fermion, given in terms of UV
constituents as (111) and includes the now massive $\lambda$. The proposal
that $\mu$ remains massless after the deformation is therefore in
contradiction with the Vafa-Witten theorem of vw . This states that – under
the assumption that the fermion determinant is positive definite – a massless,
composite fermion cannot contain a massive constituent.

We note that both of these objections disappear if a gauge singlet
$\tilde{\mu}$, with quantum numbers conjugate to $\mu$, is added to the
electric theory which is then deformed by a four fermion term ${\cal
L}\sim\tilde{\mu}\tilde{\psi}\lambda\psi$. For values of $N_{c}$ and $N_{f}$
for which this is dangerously irrelevant, it is plausible that the theory
flows to the magnetic theory, now without the gauge singlet $\mu$.

###  4.2 Non-Supersymmetric $Sp(N)$ and $SO(N)$ Dualities

A number of proposals for non-supersymmetric dualities exist in the
literature, inspired by various orientifold constructions in string theory
moreadi ; hook . In particular, motivated by D-branes, Armoni suggested a non-
supersymmetric modification of Seiberg duality for $Sp(N)$ and $SO(N)$ gauge
groups adi1 ; adi2 . Here we make some comments on this proposal. We first
describe the $Sp(N)$ duality, and then the $SO(N)$ duality.

A well-established duality for supersymmetric $Sp(N)$ gauge theories was
discovered by Intriligator and Pouliot ip . For $Sp(N)$ gauge group, the
adjoint representation coincides with the $\ydiagram{2}$. The proposal of
Armoni is to exchange the gaugino in the $\ydiagram{2}$ representation for a
Weyl fermion in the $\ydiagram{1,1}$ representation. The resulting theories
have global symmetry

$\displaystyle G=SU(2N_{f})\times U(1)$

The electric theory has fermionic matter content

$\displaystyle\mbox{Electric Theory:}\ \ \ \
\begin{array}[]{|c||c|c|c|}\hline\cr&Sp(N_{c})&SU(2N_{f})&U(1)\\\
\hline\cr\rule{0.0pt}{14.63881pt}\lambda&{\tiny{{\yng(1,1)}}}&\mathbf{1}&-N_{f}\\\
\psi&{\tiny{{\yng(1)}}}&{\tiny{{\yng(1)}}}&N_{c}-1\\\ \hline\cr\end{array}$
(115)

For $N_{f}>N_{c}-2$, the ’t Hooft anomalies for $G$ are matched by a putative
dual, magnetic theory

$\displaystyle\mbox{Magnetic Theory:}\ \ \ \
\begin{array}[]{|c||c|c|c|}\hline\cr&Sp({N_{f}-N_{c}+2})&SU(2N_{f})&U(1)\\\
\hline\cr\rule{0.0pt}{14.63881pt}\tilde{\lambda}&{\tiny{{\yng(1,1)}}}&\mathbf{1}&-N_{f}\\\
\tilde{\psi}&{\tiny{{\yng(1)}}}&\overline{{\tiny{{\yng(1)}}}}&N_{f}-N_{c}+1\\\
\mu&\mathbf{1}&{\tiny{{\yng(2)}}}&2N_{c}-2-N_{f}\\\ \hline\cr\end{array}$
(120)

When $N_{f}=N_{c}-2$, the ’t Hooft anomalies are matched by the free fermions
$\mu$ and $(\psi\lambda)^{\dagger}\psi$ in the $\mathbf{1}_{(N_{c}-1)}$ of
$G$.

There is a similar story for $SO(N)$ gauge theories. The supersymmetric
duality of Intriligator and Seiberg is can be deformed by replacing the
adjoint gaugino, which sits in the $\ydiagram{1,1}$ representation, with a
fermion in the $\ydiagram{2}$. Now the global symmetry group is

$\displaystyle G=SU(N_{f})\times U(1)$

This time, the electric theory has fermionic matter content

$\displaystyle\mbox{Electric Theory:}\ \ \ \
\begin{array}[]{|c||c|c|c|}\hline\cr&SO(N_{c})&SU(N_{f})&U(1)\\\
\hline\cr\lambda&{\tiny{{\yng(2)}}}&\mathbf{1}&-N_{f}\\\
\psi&{\tiny{{\yng(1)}}}&{\tiny{{\yng(1)}}}&N_{c}+2\\\ \hline\cr\end{array}$
(124)

For $N_{f}>N_{c}+4$, there is a second, magnetic theory whose ’t Hooft
anomalies match those of $G$,

$\displaystyle\mbox{Magnetic Theory:}\ \ \ \
\begin{array}[]{|c||c|c|c|}\hline\cr&SO(N_{f}-N_{c}-4)&SU(N_{f})&U(1)\\\
\hline\cr\rule{0.0pt}{10.76385pt}\tilde{\lambda}&{\tiny{{\yng(2)}}}&\mathbf{1}&-N_{f}\\\
\tilde{\psi}&{\tiny{{\yng(1)}}}&\overline{{\tiny{{\yng(1)}}}}&N_{f}-N_{c}-2\\\
\mu&\mathbf{1}&{\tiny{{\yng(1,1)}}}&2N_{c}-N_{f}+4\\\ \hline\cr\end{array}$
(129)

Under what circumstances might the electric theories above flow to the
magnetic theories (or, indeed, vice versa)? First, the Vafa-Witten-Weingarten
theorems described in the previous section and in more detail in Appendix B
ensure that the purely fermionic electric theory cannot flow to the magnetic
theory. We should, therefore, add scalars to the electric theory.

It is natural to take these, as in the supersymmetric theories, to sit in the
representation $(\Box,\Box)_{q}$, with $q$ the same $U(1)$ charge as $\psi$.
We then add additional terms to the action

$\displaystyle{\cal L}\sim\phi^{\dagger}\lambda\psi+{\rm h.c.}-V(\phi)$

where we are free to tune the potential $V(\phi)$ in the hope of coaxing the
electric theory to flow to the magnetic theory.

As an aside: motivated by the D-brane set-up, Armoni set $V(\phi)=0$, relying
on the induced Coleman-Weinberg potential. This immediately rules out the
$SO(N)$ duals because the resulting potential pushes $\phi\rightarrow\infty$
on the pseudo-moduli space. This is because bosons give rise to an attractive
potential while fermions give rise to a repulsive potential, but for the
$SO(N)$ theory there are more fermionic degrees of freedom in the
$\ydiagram{2}$ than there are gauge boson degrees of freedom in the adjoint
$\ydiagram{1,1}$. For the $Sp(N)$ theory, it’s the other way around and the
Coleman-Weinberg potential pushes the theory to the strongly coupled origin of
the moduli space where interesting things may happen. However, in the absence
of supersymmetry there’s nothing to stop us exploring all possible potentials
$V(\phi)$.

As always, the question that we would like to ask is: what is the low-energy
dynamics of the theories? As usual, this is a difficult question. If the
global symmetry group $G$ is unbroken in the infra-red and the magnetic theory
is infra-red free, then the magnetic theory provides a good candidate for the
low-energy physics. There is certainly no contradiction with the a-theorem
which, assuming that there are no light scalars, allows a flow from the
electric theory to magnetic theory whenever $N_{f}\lesssim 1.9N_{c}$ for both
$Sp(N_{c})$ and $SO(N_{c})$ theories in the largeish $N_{c}$ limit.

##  Appendix A Appendix: Discrete Quotients

In Section 2.3, we described a slight variant on the Bars-Yankielowicz (BY)
theories that resulted in a factorised symmetry (53)

$\displaystyle G=G_{1}\times G_{2}=[SU(q)\times
U(1)_{1}^{\prime}]\times[SU(p)\times U(1)_{2}^{\prime}]$ (130)

Here the factorisation is the statement that the theory has no mixed ’t Hooft
anomaly between $G_{1}$ and $G_{2}$. This observation allowed for a
speculative, but plausible, duality proposal.

In this appendix, we point out that the factorisation (130) is not quite
complete. There are discrete quotients that relate $G_{1}$ and $G_{2}$. We
describe these discrete quotients and their implications for the speculative
duality of Section 2.3.

The analysis is almost identical to that of Section 3.2 where we studied the
discrete structure of the theories with an additional adjoint $\lambda$.
Without $\lambda$ we don’t have $U(1)_{R}$, but otherwise the story is the
same. As in Section 3.2, it will be convenient to define $U(1)_{A}$ and
$U(1)_{B}$ as in (79), under which the fermions have charges

$\displaystyle\begin{array}[]{|c||c|c|}\hline\cr&U(1)_{A}&U(1)_{B}\\\
\hline\cr\rule{0.0pt}{10.76385pt}\chi&(N+4)/{N}&-{2q}/{N}\\\
\psi&-(N+2)/{N}&(N+q)/{N}\\\ \tilde{\psi}&(N+2)/{N}&-{q}/{N}\\\
\hline\cr\end{array}$ (135)

These two $U(1)$s act faithfully on gauge invariant operators. The
factorisation of the global symmetry group (130) is realised only by defining
linear combinations of $U(1)_{1}^{\prime}$ and $U(1)_{2}^{\prime}$; these are
related to $U(1)_{A}$ and $U(1)_{B}$ as

$\displaystyle Q_{1}^{\prime}=NQ_{B}\ \ \ {\rm and}\ \ \
Q_{2}^{\prime}=2NQ_{A}+NQ_{B}$ (136)

The centres of the two non-abelian global symmetries $\mathds{Z}_{q,p}\subset
SU(q),\ SU(p)$ are equivalent to a certain $U(1)_{A}\times U(1)_{B}$
transformations as described in equations (85) and (86). In terms of
$U(1)^{\prime}_{1}$ and $U(1)^{\prime}_{2}$ this works as follows: A
$\mathds{Z}_{q}\subset SU(q)$ is gauge equivalent to $e^{2\pi
iQ_{1}^{\prime}\omega_{1}^{\prime}+2\pi iQ_{2}^{\prime}\omega_{2}^{\prime}}$
with $\omega_{1}^{\prime}=\frac{1}{qN}\ ,\ \omega_{2}^{\prime}=0$, and a
$\mathds{Z}_{p}\subset SU(p)$ is gauge equivalent to $e^{2\pi
iQ_{1}^{\prime}\omega_{1}^{\prime}+2\pi iQ_{2}^{\prime}\omega_{2}^{\prime}}$
with $\omega_{1}^{\prime}=0\ ,\ \omega_{2}^{\prime}=\frac{1}{Np}$. If $q,p$
are both even, the $\mathds{Z}_{2}\subset\mathds{Z}_{q}\times\mathds{Z}_{p}$
is gauge equivalent to the identity without the need of using $U(1)$
transformations.

The last point we want to emphasize is the mixing with fermion number. It is
straight forward to show that $e^{2\pi iQ_{1}^{\prime}\omega_{1}^{\prime}+2\pi
iQ_{2}^{\prime}\omega_{2}^{\prime}}$ with
$\omega_{1}^{\prime}=-\omega_{2}^{\prime}=\frac{1}{4N}$ is equivalent to
fermion number. What does it tell us about the factorized spectrum? Assuming
all the operators are charged only under one of the $G_{1}\times G_{2}$
factors, the discrete structure puts some constraints on the allowed spectrum.
For example, if $p,q$ are both even, the
$\mathds{Z}_{2}\subset\mathds{Z}_{q}\times\mathds{Z}_{p}$ implies there are no
fundamentals of $SU(q)$ or $SU(p)$ (more generally, no representations with
odd number of indices are allowed). The relation (136) implies that all the
$U(1)^{\prime}_{1}$ and $U(1)^{\prime}_{2}$ charges are multiples of $2N$
which is not true for general operators charged under both $U(1)$s (recall
that $U(1)_{AB}$ and $U(1)_{B}$ act faithfully). Happily, the mixing with
fermion numbers implies that every operator charged only under one of
$U(1)^{\prime}_{1}$ and $U(1)^{\prime}_{2}$ must have charge
$2N(1+2\mathds{Z})$ for fermions, and $(4N\mathds{Z})$ for bosons. So the two
requirements are consistent with each other and allow for factorized operators
of any spin to exist.

##  Appendix B Appendix: Weingarten Inequalities

In this Appendix, we describe in some detail the application of the Weingarten
inequality wein to rule out certain non-supersymmetric dualities in vector-
like theories, as first proposed in ofer . We also provide generalisations to
$Sp(N)$ and $SO(N)$ dualities, which require small tweaks to the arguments.

#### When Is the Fermion Determinant Positive Definite?

The key assumption underlying both Vafa-Witten and Weingarten theorems is that
the fermion determinant is positive definite. We first review the arguments
for when, and why, this occurs.

The most common setting for the Weingarten inequalities is in QCD, where we
are dealing with Dirac fermions $\Psi$. Consider the Euclidean partition
function for a massless Dirac fermion in $d=4$ coupled to gauge fields in an
arbitrary representation,

$\displaystyle\mathcal{Z}=\int\mathcal{D}A{\cal D}\psi{\cal D}\bar{\psi}\
e^{-S_{\text{YM}}-S_{\text{Fermion}}}=\int\mathcal{D}A\,\operatorname{det}(\not{D})\,e^{-S_{\text{YM}}}$

The Dirac operator $i\not{D}$ is Hermitian and so its eigenvalues are real

$\displaystyle
i\not{D}\psi_{n}=\lambda_{n}\psi_{n}\;\;\;\;\lambda_{n}\in\mathbb{R}$

Because $\\{\gamma^{5},\not{D}\\}=0$, we also have
$i\not{D}(\gamma^{5}\psi_{n})=-\lambda_{n}(\gamma^{5}\psi_{n})$.This means
that all non-zero eigenvalues come in plus-minus pairs. There may also be some
number, $m$, of zero modes $\tilde{\lambda}$. The general form of the Dirac
determinant is then

$\displaystyle\operatorname{det}(\not{D})=\tilde{\lambda}^{m}\prod_{n}(i\lambda_{n})=\tilde{\lambda}^{m}\prod_{\lambda_{n}>0}(i\lambda_{n})(-i\lambda_{n})=\tilde{\lambda}^{m}\prod_{\lambda_{n}>0}(\lambda_{n})^{2}\geq
0$ (137)

which is manifestly positive semi-definite.

If we couple scalars through a Yukawa coupling, then the Dirac determinant
becomes complex. This means that the Weingarten theorem (or, indeed, the Vafa-
Witten theorem) does not apply in the supersymmetric context. This is
fortunate because, as we will see, these theorems can be used to rule out
certain dualities ofer .

What about Weyl fermions that are not in Dirac pairs? Under what circumstances
is the path integral measure non-negative?

To answer this, we first return to a Dirac fermion in $\Psi$ in Lorentzian
signature, coupled to a background gauge fields $A$ in some complex
representation $R$ of the gauge group $G$. We can write $\Psi$ as two left
handed Weyl fermions $\psi$ and $\tilde{\psi}$ in conjugate representations of
$G$. Then the partition functions are related by
$\mathcal{Z}_{\Psi}=\mathcal{Z}_{\psi}\mathcal{Z}_{\tilde{\psi}}$.
Furthermore, $\mathcal{Z}_{\tilde{\psi}}=\mathcal{Z}_{\psi}^{*}$ so we can
write $\mathcal{Z}_{\Psi}=|\mathcal{Z}_{\psi}|^{2}$. (See, for example,
wittenfermion , for a careful study of fermion determinants.) The path
integral for a single Weyl fermion then takes the form

$\displaystyle\mathcal{Z}_{\psi}[A]=e^{iW[A]}|\mathcal{Z}_{\Psi}[A]|=e^{iW[A]}\operatorname{det}(\not{D})^{1/2}$

where we take the positive square-root of the Dirac determinant. It was shown
in ag that the phase $W[A]$ is given by

$\displaystyle W[A]-W[A_{0}]=\pi\eta[H_{5}]+2\pi Q_{5}[A_{t}]$

where $H_{5}$ is a five dimensional Dirac operator and $A_{t}=(1-t)A_{0}+A$,
$t\in[0,1]$ linearly interpolates between a reference configuration $A_{0}$ to
$A$. Here $Q_{5}[A_{t}]$ is the Chern-Simons form and encodes chiral
anomalies.

If the representation $R$ of the fermion is complex, then the partition
function will in general carry a complex phase. This is a feature of chiral
gauge theories and means that Vafa-Witten-Weingarten theorems do not apply to
chiral gauge theories.

In contrast, if we have Weyl fermions in a real or pseudo-real representation
of $G$, then $\mathcal{Z}_{\tilde{\psi}}=\mathcal{Z}_{\psi}$ and
$\mathcal{Z}_{\Psi}=\mathcal{Z}_{\psi}^{2}$ such that
$\mathcal{Z}_{\psi}=\pm\operatorname{det}(\not{D})^{1/2}$. We see that the
path integral is real, but it is not necessarily positive semi-definite.

To proceed, we pick a reference gauge configuration $A_{0}$ and define the
sign to be positive $\mathcal{Z}_{\psi}[A_{0}]\geq 0$. This corresponds to
taking the product of the positive eigenvalues of $i\not{D}$. We then ask how
the sign varies as we deform $A_{0}$ to physically distinct gauge
configurations $A$. If the sign turns negative for any $A$, the partition
function is not positive semi-definite. As we now explain, the partition
function obeys $\mathcal{Z}_{\psi}[A]\geq 0$ for all gauge configurations $A$
for a Weyl fermion in a real representation of $G$666We thanks Nakarin
Lohitsiri and Tin Sulejmanpasic for explaining to us the importance of
Kramers’ doubling in the argument below.. This is not necessarily the case for
pseudo-real representations.

To this end, firstly recall that in $d=4$, we can write a Weyl fermion in a
real representation of $G$ as a Dirac-Majorana fermion $\Psi$ with the action

$\displaystyle S=i\int d^{4}x\Psi^{T}C\not{D}\Psi$

where $C$ is the unitary, anti-symmetric charge conjugation matrix satisfying
$C\gamma^{\mu}C^{-1}=-(\gamma^{\mu})^{T}$ and $D_{\mu}^{*}=D_{\mu}$ as $\Psi$
is in a real representation. The partition function formally evaluates to the
Pfaffian of the anti-symmetric bilinear form $C\not{D}$

$\displaystyle\mathcal{Z}[A]=\operatorname{Pf}[C\not{D}]$ (138)

(B) still holds for the Dirac operator $i\not{D}$ but in addition we have an
extra degeneracy. We can choose the Euclidean gamma matrices
$(\gamma^{\mu})^{\dagger}=\gamma^{\mu}$ such that for real representations
$C\not{D}C^{-1}=-\not{D}^{*}$. Then conjugating (B) gives

$\displaystyle
iC\not{D}C^{-1}\psi_{n}^{*}=\lambda_{n}\psi_{n}^{*}\;\;\;\Rightarrow\;\;\;i\not{D}(C^{-1}\psi^{*}_{n})=\lambda_{n}(C^{-1}\psi^{*}_{n})$
(139)

We see that for every eigenvector $\psi_{n}$, $C^{-1}\psi^{*}_{n}$ is also an
eigenvector with the same eigenvalue. As discussed in wittenfermion , in
Euclidean signature, there is no field that corresponds the the complex
conjugate of $\psi_{n}$. However, once we formally write the partition
function as a Pfaffian (138), it makes sense to consider the behaviour of the
eigenmodes of the Dirac operator under complex conjugation. Then $\psi_{n}$
and $C^{-1}\psi^{*}_{n}$ are orthogonal to each other due to the anti-symmetry
of $C$.

Next, we define the anti-linear operator $\mathcal{T}=C^{-1}K$ with $K$ the
complex conjugation operator, we have $[\mathcal{T},i\not{D}]=0$ and
$\mathcal{T}^{2}=C^{-1}KC^{-1}K=C^{-1}C^{T}=-C^{-1}C=-1$. Therefore we can see
this as a Kramers degeneracy wittenfermion .

Now if we chose the basis
$(\psi_{1},C^{-1}\psi^{*}_{1},...,\psi_{n},C^{-1}\psi^{*}_{n},...)$, the
operator $C\not{D}$ takes the form

$\displaystyle
C\not{D}=\bigoplus_{n}\left(\begin{array}[]{cc}0&i\lambda_{n}\\\
-i\lambda_{n}&0\end{array}\right)$ (142)

and the partition function (138) evaluates to

$\displaystyle\mathcal{Z}[A]={\prod_{n}}^{\prime}(i\lambda_{n})$

where by ${\prod_{n}}^{\prime}$ we mean in the product we take only one
eigenvalue from each Kramers doublet pair. The doubling argument we used in
(137) by anti-commuting $\gamma^{5}$ also still holds so we can write the
partition function as

$\displaystyle\mathcal{Z}={\prod_{\lambda_{n}\geq
0}}^{\prime}(\lambda_{n})^{2}$

which is manifestly positive semi-definite for all $A$ and is unaffected by a
change of basis as discussed in stone .

The Kramers doubling argument in (139) fails for pseudo-real representations
and the partition function of these theories are not positive semi-definite.
This of course had to be the case in order to be consistent with the Witten
anomaly wittensu2 where the partition function changes sign under a gauge
transformation.

#### Weingarten Inequalities for Putative $Sp(N_{c})$ Duals

We move onto consider a Weingarten analysis of the purely fermionic electric
theories introduced in Section 4.2. We focus on the $Sp(N_{c})$ theory as this
case is slightly more subtle. The same results trivially apply to the
$SO(N_{c})$ theories.

Our aim is to show that the gauge singlet composite fermion
$\mu\sim\psi\chi\psi$ must be heavier than the meson $\pi\sim\psi\psi$. For
this, we firstly need to make sure that the fermionic path integral is
positive definite: this reads

$\displaystyle\mathcal{Z}_{\text{fermion}}[A]=\operatorname{det}(\not{D}_{\lambda})^{1/2}\operatorname{det}(\not{D}_{\psi})^{N_{f}}$

The anti-symmetric representation of $Sp(N_{c})$ is real so the discussion
above shows $\mathcal{Z}_{\text{fermion}}[A]\geq 0$ for all gauge
configurations $A$.

The inequalities involving the $Sp(N_{c})$ theory are slightly more subtle
than the $SU(N_{c})$ and $SO(N_{c})$ cases. This is simply because the colour
indices in $\rho$ and $\pi$ are contracted via $J^{ab}$, the invariant anti-
symmetric tensor of $Sp(N_{c})$ as opposed to the identity matrix in the
$SU(N_{c})$ and $SO(N_{c})$ cases. We will however see that this does not
invalidate the inequalities.

It is convenient to work with Majorana fermions rather than Weyl fermions, so
we take the Majorana fermions $\Lambda$ and $N_{f}$ flavours of $\Psi^{i}$,
which in the Weyl basis is

$\displaystyle\Lambda_{ab}=\begin{pmatrix}\lambda_{\alpha,ab}\\\
J_{ac}J_{bd}\bar{\lambda}^{\dot{\alpha},cd}\end{pmatrix},\;\;\;\;\Psi_{a}=\begin{pmatrix}\psi_{\alpha,a}\\\
J_{ab}\bar{\psi}^{\dot{\alpha}b}\end{pmatrix}$ (143)

with explicit colour and spinor indices. We are interested in the correlation
functions of the composite fermion operator

$\displaystyle\mu^{ij}_{\alpha}=J^{ab}J^{cd}\Gamma^{\beta\gamma\delta}_{\alpha}\Psi^{i}_{\beta,a}\Lambda_{\gamma,bc}\Psi^{j}_{\delta,d}$

where $\Gamma$ is a tensor acting on the spinor indices which we may chose to
symmetrise over the flavour indices and project appropriately onto the left-
handed spinors. For simplicity, take the flavour indices $i\neq j$ (when they
are equal we simply get and extra factor from Wicks theorem) and consider the
Euclidean propagator (with no $i$, $j$ sum)

$\displaystyle\begin{split}\langle\mu_{\alpha}^{ij}(x)\bar{\mu}^{\alpha^{\prime}}_{ij}(y)\rangle=&-\Gamma^{\beta\gamma\delta}_{\alpha}\overline{\Gamma}_{\beta^{\prime}\gamma^{\prime}\delta^{\prime}}^{\alpha^{\prime}}J^{ab}J^{cd}J^{a^{\prime}b^{\prime}}J^{c^{\prime}d^{\prime}}\\\
&\int
d\mu\left(S_{\Psi}(x,y)^{\beta^{\prime}}_{\beta}\right)_{aa^{\prime}}\left(S_{\Lambda}(x,y)^{\gamma^{\prime}}_{\gamma}\right)_{bc,b^{\prime}c^{\prime}}\left(S_{\Psi}(x,y)^{\delta^{\prime}}_{\delta}\right)_{dd^{\prime}}\end{split}$

Here we have integrated out the fermions such that $S$ are the fermion
propagators in the presence of a fixed background gauge field and $d\mu$ is
the path integral measure including the fermion determinants. We take the
Frobenius norm over the spinor indices and bound the correlator from above

$\displaystyle\begin{split}|\langle\mu^{ij}(x)\bar{\mu}_{ij}(y)\rangle|_{F}:=&\left[\sum_{\alpha\alpha^{\prime}}|\langle\mu_{\alpha}^{ij}(x)\bar{\mu}^{\alpha^{\prime}}_{ij}(y)\rangle|^{2}\right]^{1/2}\\\
\leq&C\left[\sum_{\text{All Indices}}\left|\int
d\mu\left(S_{\Psi\beta}^{\beta^{\prime}}\right)_{aa^{\prime}}\left(S_{\Lambda\gamma}^{\gamma^{\prime}}\right)_{bc,b^{\prime}c^{\prime}}\left(S^{\delta^{\prime}}_{\Psi\delta}\right)_{dd^{\prime}}\right|^{2}\right]^{1/2}\end{split}$

where $C$ is a positive constant. Importantly here we have buried the details
of the colour and spinor contractions in this constant. Next we use the Hölder
inequality which says for any positive definite measure

$\displaystyle\left|\int d\mu(fg)\right|\leq\left(\int
d\mu|f|^{2}\right)^{\frac{1}{2}}\left(\int d\mu|g|^{2}\right)^{\frac{1}{2}}$

such that

$\displaystyle\begin{split}|\langle\mu^{ij}(x)\bar{\mu}_{ij}(y)\rangle|_{F}\leq&C\Bigg{[}\int
d\mu\sum_{\begin{subarray}{c}\beta\beta^{\prime}\\\
aa^{\prime}\end{subarray}}\left|\left(S_{\Psi\beta}^{\beta^{\prime}}\right)_{aa^{\prime}}\right|^{2}\Bigg{]}^{1/2}\\\
&\times\Bigg{[}\int
d\mu\sum_{\begin{subarray}{c}\gamma\gamma^{\prime},\delta\delta^{\prime}\\\
bb^{\prime},cc^{\prime},dd^{\prime}\end{subarray}}\left|\left(S_{\Lambda\gamma}^{\gamma^{\prime}}\right)_{bc,b^{\prime}c^{\prime}}\left(S^{\delta^{\prime}}_{\Psi\delta}\right)_{dd^{\prime}}\right|^{2}\Bigg{]}^{1/2}\end{split}$

We now want to interpret each integral, both of which are gauge invariant, as
a correlation function. The first integral is equal to the propagator of the
meson $\pi=\bar{\Psi}^{a}\gamma_{5}\Psi_{a}$. Crucially the colour indices of
$\pi$ are contracted via the the identity which allows this identification.
Explicitly, in Lorentzian signature, the meson can be written in terms of Weyl
fermions as $\pi=J^{ab}\psi_{b}\psi_{a}+\text{h.c.}$ as the $J$ tensors are
burried in (143). The second integral can, at worst, approach a constant
asymptotically. In the $|x-y|\rightarrow\infty$ limit, the mass of the
lightest particle carrying the same quantum numbers as the operators dominates
and we get the inequality

$\displaystyle e^{-m_{\pi}|x-y|}\geq
e^{-m_{\mu}|x-y|}\;\;\;\;\Rightarrow\;\;\;\;m_{\pi}\leq m_{\mu}$ (144)

There is no reason for the pion to be massless without symmetry breaking, so
(144) prohibits $\rho$ being massless in the absence of symmetry breaking. A
very similar analysis goes through for the fermionic $SO(N_{c})$ theory.

This result excludes an s-confining phase where we need the massless fermion
$\mu$ for anomaly matching. It also prohibits the electric to magnetic flow of
the putative duals in Section 4.2 if the electric theory has no scalars. This
follows as the singlet $\mu$ must be massless for anomalies to match across
the duality but the map $\mu=\psi\lambda\psi$ implies $\mu$ must be massive
which leads to a contradiction. As explained in Section 4.1, this is closely
tied with the “persistent mass condition”.

###### Acknowledgements.

We would like to thank Micha Berkooz, Nakarin Lohitsiri, and Tin Sulejmanpasic
for discussions and correspondence on various topics. We are supported by the
STFC consolidated grant ST/P000681/1 and the EPSRC grant EP/V047655/1 “Chiral
Gauge Theories: From Strong Coupling to the Standard Model”. KO is further
supported by a Cambridge Trust International Scholarship and DT by a Simons
Investigator Award. For the purpose of open access, the authors have applied a
Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript
version arising from this submission.

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