QCD3对称破缺与味颤抖对偶性研究进展

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"QCD3对称破缺和味颤抖对偶性的推广" 这篇研究论文主要探讨了量子色动力学（Quantum Chromodynamics, QCD）在特定维度下的对称性破缺现象以及相关的对偶理论。QCD是描述强相互作用的理论，通常在四维空间时间中被研究，但在较低维度的研究可以简化问题并揭示一些基本的物理原理。标题中的"QCD3"指的是在三维空间和一维时间（3+1d）的QCD，这种简化版本常用于理论研究。研究者将之前提出的QCD3对称性破坏模型扩展到了（2+1）维的玻色化双重性，即“主”（2+1）d玻化双重性。这种双重性涉及到在理论的两端同时存在玻色子（bosons）和费米子（fermions）的物质。通过对称性破缺，他们发现了具有多个新颖相区的相图，这些相区可能对应着不同的物理状态或行为。描述中提到的"味颤抖对偶性"（flavored quiver dualities）是一种在2+1维空间时间中构造的对偶性理论。"味"（flavor）在粒子物理学中通常指夸克的不同种类，颤抖对偶则涉及将理论分解为节点并逐一进行对偶化的过程。这种对偶性适用于那些存在于四维QCD类理论的畴壁（domain walls）上的理论，这些理论有动态夸克参与。动态夸克意味着夸克不仅作为背景场存在，还能参与相互作用。此外，研究还涉及到了基于正交（orthogonal）和辛（symplectic）规范群的颤抖对偶性。正交和辛群是两种不同的数学结构，在规范理论中扮演重要角色，它们可以描述不同类型的粒子相互作用。这些对偶性的建立有助于我们理解不同规范群下的物理现象。最后，研究者通过全息原理（Holography）来支持他们的二元性猜想。全息原理是一个强大的工具，它将高维的量子场论与低维的引力理论联系起来，即使对于这个全息构造的一些方面，目前的理解仍然停留在定性的阶段。这篇论文深化了我们对QCD在不同维度下对称性破缺和对偶性理论的理解，尤其是在理解和模拟四维QCD中动态夸克效应的边界条件时提供了新的视角。通过这些理论工具，物理学家可以更有效地探索强相互作用的复杂世界，并可能找到新奇的物理现象。

JHEP02(2020)060

Note also we can adjust the strength of the interaction terms

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

discussed in [25]. The natural extension is to attempt

to look in the regime where k < N

< N

∗

(N, k) and N

< N.

We will begin by reviewing the construction of [25], which corresponds to the N

= 0

limit. Like the ordinary master duality, the phase diagrams for the N

< N and N

= 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

< N since it is slightly simpler

to analyze. The N = N

case will be considered second and it will be applicable to the

quivers we are constructing to model the domain wall behavior of QCD

. Finally, we will

discuss extending the master duality to the double saturated case in section 2.3.

2.1 Review of QCD

symmetry-breaking

As was mentioned in the introduction, the ﬂavor bounds of (1.1a) are imposed because

when N

> 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

≤ k

was imposed to avoid the mismatch from the non-linear sigma model phase.

The proposal of [25] is that when k < N

< N

∗

(N, k), for small fermion masses,

i.e. |m

| < 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

M(N

, k) =

U(N

)

U(k) × U(N

− 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

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

fermion

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.

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

/2 shift used in [25] to make the phases look

more symmetric.

– 7 –

JHEP02(2020)060

fermions with |m

| < m

∗

which determines whether or not the associated gauge group

conﬁnes, resulting in the Grassmannian (2.1). Speciﬁcally, the assumption is that if the

number of light fermions exceeds the bare level of the gauge group (the level without the

η-invariant oﬀset), the fermions still condense. When we construct quivers below we will

further split the N

symmetry into smaller subgroups and individually tune their masses.

This can lead to a rich structure with many diﬀerent Grassmannians, but in this work

we will generally be concerned with phases outside of this region. More details on such

subtleties of the phase diagram can be found in refs. [36, 37].

A potential issue with this assumption is that the Grassmannian of the SU side only

exists for small mass deformations, see ﬁgure 1. This is not matched on the U side if we use

the phase diagram of the ﬂavor-bounded duality (1.1a), which has a Grassmannian phase

for unbounded negative scalar mass deformations. To rectify this, the authors of ref. [25]

propose that there are two scalar theories dual to the SU end that are patched together to

describe the full phase diagram. More explicitly, the duality proposed is

SU(N)

−k+N

with N

ψ ↔

(

U(k)

with N

= −m

∗

(blue)

U(N

− k)

−N

with N

= m

∗

(red)

, (2.2)

where the colors in parenthesis correspond to those in ﬁgure 1 and ﬁgures which follow.

Note that in both scalars theories, when the mass becomes negative we obtain the same

aforementioned Grassmannian manifold of (2.1). Accompanying the Grassmannian is a

Wess-Zumino-Witten term with a coeﬃcient that is determined by the level of the Chern-

Simons theories on the U side. This is important for matching of the operators across

the duality, but will not play an essential role in our story. For more details on this term

see [25]. This model is summarized in ﬁgure 1.

One feature of this construction that we will see occurs also in the master duality case

is that the SU side of the duality does not yield a good description of the quantum phase

or the theories which exist at the phase transition. That is, the true IR description on the

SU side is hidden by strong dynamics and so we must pass to the U side to obtain a full

description of the phase diagram.

2.2 Flavor-violated master duality

We now generalize the analysis of [25] to the master duality. In particular, we are interested

in mapping out the phase diagram of

SU(N)

−k+N

with N

ψ and N

φ (2.3)

when N

> k and N

≤ N as a function of the scalar and fermion masses. We will begin

our analysis in the asymptotic regimes where the mass deformations are large compared to

the strong scale and we don’t expect to see quantum phases. When one or both mass defor-

mations are small, we again expect to ﬁnd phases that may be described by Grassmannian

manifolds.

When we give an asymptotically large positive mass to the scalars, we can integrate

them out and they have no eﬀect on the gauge group. The theory we are left with is

SU(N)

−k+N

with N

ψ. (2.4)

– 8 –

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