2+1维QCD对偶性与F定理：对称破坏的新洞察

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"QCD3对偶与F定理" 这篇研究论文主要探讨了2+1维度量子色动力学（QCD3）中的对偶性和F定理的应用。在物理学领域，尤其是在理论物理和高能物理中，对偶性是理解和连接不同理论的重要工具。2+1维度的量子色动力学是一个简化版本的四维QCD，它在凝聚态物理和弦理论中有重要应用。这篇论文关注的是在2+1维QCD中出现的对称性破缺现象，这是一种物理系统中基本对称性的非平凡行为，可能与物质的相变和临界现象相关。 F定理是理论物理学中的一个重要概念，特别是在研究二维和三维量子场论的renormalization group（RG）流时。这个定理指出，在一个二维的共形场论（CFT）向一个具有更低超对称性的CFT的RG流过程中，一个被称为中央 charge 或 F 函数的量总是减少的。这里的F函数衡量了理论的复杂度或熵。在3+1维度以上的理论中，没有直接对应的F定理，但在2+1维度，它可以用来分析相变并确定理论的稳定性和对称性破缺的可能性。在论文中，作者Adar Sharon运用F定理来确定2+1维QCD中可能出现对称性破缺的参数范围。他给出了SU(2)规范理论的精确界限，以及SU(N)规范理论（N>2）的近似界限。SU(N)群是粒子物理中常见的对称性群，特别是与强相互作用相关的颜色对称性。这里的对称性破缺可能涉及到SU(N)群的非平凡表示，导致规范玻色子获得质量，这与标准模型中的电弱对称性破缺有类似之处。关键词包括对偶性、规范场论、 Chern-Simons 理论和超对称规范理论，表明论文还可能涉及了与拓扑性质相关的内容，如Chern-Simons项，这是2+1维理论中常见的一个拓扑项，它可以影响理论的相结构和对称性。超对称性则是理论物理中的另一个关键概念，它关联着粒子的费米子和玻色子性质，并且在稳定理论的真空状态和计算中发挥重要作用。这篇论文通过深入研究2+1维QCD的对偶性和F定理，揭示了理论中的相变可能性和对称性破缺的界限，这对理解高能物理中的基本相互作用以及凝聚态物理中的新型相变现象具有重要意义。此外，开放获取的特性使得该研究对学术界和公众都是可访问的，促进了科学知识的传播和交流。

JHEP08(2018)078

is even). Throughout this paper, k will always denote the QCD

Chern-Simons level as

deﬁned above, while

k will denote the corresponding Chern-Si m ons level for SUSY theori e s.

One could ask what are the low-energy phases of this theory as a function of the fermion

masses m. It turns out that diﬀerent phases appe ar in diﬀerent regimes of the theory:

• For 2k ≥ N

, the two phases of the theory are SU(N)

k±N

, with a single phase

transition between them [

2–8, 31, 34–69].

• For 2k < N

< N

⋆

(N, k), where the theory is strongly coupled, the following phase

diagram was proposed in [1]: when the fermion masses are large, the phases ar e still

SU(N)

k±N

. However, for small masses a new phase appears, which is a NLSM

with a Wess- Zum i no term. For N > 2 its target space is

U(N

)

U(N

/2+k)×U(N

/2−k)

, while

for N = 2 its target space i s

Sp(N

)

Sp(N

/2+k)×Sp(N

/2−k)

. This is consistent with some

conjectured dualities at the transition points. There are two such critical points, One

where the dual t he ory is U



+ k



−N

with N

bosons,, and another where the

dual theory is U



− k



with N

bosons.

• for N

≥ N

⋆

the exact behavior is not known, apart from the fact that at large

enough N

the phases should once again contain only Chern-Simons term s, as in the

regime 2k ≥ N

[

27].

Here, N

⋆

is some upper bound on the symmetry-breaking (SB) phase. This bound

must exist, since in the limit N

→ ∞, the theory does not develop dynamical masses for

the fermions and thus symmetry breaking cannot occur [

27]. Intuitively, one might say

that for very large N

, the theory becomes weak l y coupled (we will see explicitly that this

is true by calculating the dimension of the chiral mul ti plets and the F-coeﬃcients for these

theories for lar ge N

The purpose of this paper is to ﬁnd a bound on N

⋆

. That is, we attempt to constrain

the val ues of the parameters N

, N, k for which symmetry breaking can o cc ur . In order

to ﬁnd this bound , we shall use the method described in [

12], where a similar bound for

QED

was calculated. Let us describe this method in the present context.

Our m et hod wi l l rely on the F-theorem. Consider the theory of a 2+1d N = 2

supersymmetric SU(N)

gauge theory wi t h N

chiral multiplets (which we call SQCD

By adding appropriate deformati ons, we can ﬂow from SQCD

to a non-supersymmetric

SU(N)

gauge theory with N

fermions (QCD

), and from there the proposed ﬂow i n [

1] is

to the IR symmetry-breaking phase. We then calculate the F-coeﬃcient of the SUSY theor y

SUSY

= F

SQCD

, and the F-coeﬃcient of the symmetry-breaking IR theory F

= F

. The

F-theorem then states that the proposal is valid only when F

SQCD

≥ F

(with F

SQCD

, F

functions of N

, N, k). Now, if we deﬁne N

bound

(N, k) as the value of N

for which we

have F

SQCD

= F

, the n for any N

such that N

> N

bound

, we must have F

SQCD

< F

This is in conﬂi c t with the F-theorem, an d so we conclude that we must have N

⋆

≤ N

bound

(this is summari z ed in ﬁgure

1).

This pape r will foc us on ﬁnding values of the bound N

bound

(N, k), that is, we ﬁnd

the value of N

for which F

SUSY

= F

. In order to ﬁnd these values, one must com put e

– 4 –

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