![](https://csdnimg.cn/release/download_crawler_static/12304816/bg8.jpg)
Note also we can adjust the strength of the interaction terms
3
which has the effect
of moving the location of the diagonal critical line of figure 3. In [29], we worked in the
limit where SU quiver has no interaction terms. This necessitated very large interactions
in some of the other theories down the duality chain. For the purposes of the first part of
this paper, we will always assume we are working with finite interaction strengths.
2 Extending the master duality
In this section we will argue that the master duality can be extended beyond the flavor
bound, similar to the case of QCD
3
discussed in [25]. The natural extension is to attempt
to look in the regime where k < N
f
< N
∗
(N, k) and N
s
< N.
We will begin by reviewing the construction of [25], which corresponds to the N
s
= 0
limit. Like the ordinary master duality, the phase diagrams for the N
s
< N and N
s
= N
cases look fairly different due to particular cancellations which occur in the latter case. We
will begin with the phase diagram for the more general N
s
< N since it is slightly simpler
to analyze. The N = N
s
case will be considered second and it will be applicable to the
quivers we are constructing to model the domain wall behavior of QCD
4
. Finally, we will
discuss extending the master duality to the double saturated case in section 2.3.
2.1 Review of QCD
3
symmetry-breaking
As was mentioned in the introduction, the flavor bounds of (1.1a) are imposed because
when N
f
> k, the Higgs phase of the U side of the theory becomes a non-linear sigma
model. This does not appear to be matched on the SU side, because the corresponding
fermion mass deformation simply shifts the Chern-Simons level. Hence the bound N
f
≤ k
was imposed to avoid the mismatch from the non-linear sigma model phase.
The proposal of [25] is that when k < N
f
< N
∗
(N, k), for small fermion masses,
i.e. |m
ψ
| < m
∗
, the SU Chern-Simons theory causes the fermions to condense, yielding a
non-linear sigma model whose target space is the complex Grassmannian manifold
4
M(N
f
, k) =
U(N
f
)
U(k) × U(N
f
− k)
. (2.1)
Said another way, the SU side exhibits “chiral” symmetry breaking, yielding as its low
energy “pion” Lagrangian the target space of (2.1). Importantly, it is the number of
3
Here, and in our previous work, we have been treating the strengths of the interactions as an adjustable
parameters. Of course this it not really true. It is not believed that the coefficient of the scalar
2
fermion
2
coupling is a marginal parameter. In fact, for the master duality to hold it needs to be a relevant operator
and so the corresponding coupling constants flow to their IR fixed values. They do not correspond to
adjustable parameters. When we say we adjust the strength of the coupling we mean that by rescaling the
scalars relative to the fermions we zoom into a region of the phase diagram in which the couplings appear to
dominate or be unimportant. That is, when we are talking about a zero coupling constant we mean that we
are focusing on the regions in parameter space where the explicit mass terms are chosen to be larger than
the ones induced by the interaction and the other way around when we are talking about infinite coupling.
But, in the end, there is really only one theory.
4
Note, here we are using a slightly different convention than [25] for the levels of the Chern-Simons
terms. The SU levels we use do not have the additional −N
f
/2 shift used in [25] to make the phases look
more symmetric.
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