大规模量子色动力学中的时间反转断裂：二维双关性与三维相变

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本文探讨的是四维量子色动力学(QCD4)中的时间反转破缺现象，特别是在2+1维空间维度下的特殊行为。作者们聚焦于3+1维的SU(N) QCD，考虑了Nf个具有相同质量m的费米子味夸克以及θ参数的影响。在一般情况下，理论存在一个单一的间隙真空。然而，当质量m较大且θ参数取值接近π时，理论经历了一个第一阶相变，这体现了QCD4中非平凡的拓扑性质。对于Nf等于1的情况，当第一阶相变线终止时，会出现一个无质量的η'粒子，这个特性是普遍存在的，不依赖于N的值。而对于Nf大于1，情况有所不同，第一阶相变线会在m=0处终结，此时理论的红外(IR)行为取决于Nf的具体数值。如果Nf足够大，IR理论会表现出自由的Nambu-Goldstone玻色子，可能是相互作用的共形场理论或者自由规范理论。值得注意的是，即使在四维理论的整体行为平滑的情况下，领域的墙壁和界面也承载着丰富的相变现象，它们将不同的三维相区分开来。这些领域的结构与最近研究的三维Chern-Simons物质理论相联系，强化了四维与三维之间动力学的关联。特别是，当无质量的四维理论表现出SU(Nf) sigma模型时，低质量领域的理论会支持三维无质量的复投影空间ℂℙNf-1上的非线性σ模型，这种模型在2+1维的猜想动力学中得到了一致性的验证。此外，文章还提到了这篇研究的时间接收日期、修订日期和接受日期，以及出版信息，显示了其学术交流和公开访问的特性。研究者们来自Perimeter Institute for Theoretical Physics、Weizmann Institute of Science和Simons Center for Geometry and Physics等知名机构，他们的合作揭示了QCD4与量子场论之间的复杂而深刻的对称性和相互作用。总结来说，这篇文章深入探讨了QCD4中时间反转破缺现象如何通过二维和三维之间的对称性关联，展示了不同维度之间的理论连接，并对量子色动力学在不同空间维度的物理行为进行了细致的研究。

JHEP01(2018)110

where we used Euclidean signature (as we will throughout this note) and hence the factor

of i in front of the θ-term. We will study this theory on a four-dimensional manifold M

The instanton number is quantized as

8π

Tr(F ∧ F ) ∈ Z . (2.2)

As a result, the theory with θ is equivalent to the theory with θ + 2π. More precisely, the

Hamiltonians with θ and θ+2π are similar and the similarity transformation is implemented

by the unitary operator

U = e

4π

CS(A)

, (2.3)

with CS(A) being the standard Chern-Simons action for an SU(N) gauge ﬁeld and M

a space-like slice.

The symmetries of the theory include the one-form global symmetry Z

associated

with the center of the gauge group [33, 34] (and see references therein). It acts on the

fundamental Wilson loop W

= Tr

P e

→ e

2πi

, (2.4)

and at θ = 0, π there is in addition also a time-reversal symmetry (equivalently, CP

symmetry). The theory at θ = 0 has no anomalies and it is believed to have a trivial,

unique, gapped ground state. At θ = π there is a mixed ’t Hooft anomaly between the

time-reversal symmetry and the Z

one-form global symmetry [6]. As a result, the theory

cannot have a trivial ground state. One way to saturate this anomaly in the infrared is

to break time-reversal symmetry spontaneously. In fact, this occurs in the planar limit

N = ∞ [1]. We will assume that this is how the anomaly is saturated also at ﬁnite N.

(For low values of N, especially for N = 2, there are also other plausible scenarios [6].)

As a result of the spontaneous breaking of time-reversal symmetry at θ = π, the theory

admits a domain wall. The domain wall theory cannot be trivial in the infrared because

of anomaly inﬂow. Indeed, the bulk has a mixed ’t Hooft anomaly involving time-reversal

symmetry and the Z

one-form symmetry. Since time-reversal symmetry is broken in the

bulk, the domain wall theory must have an ’t Hooft anomaly for its one-form Z

symmetry.

In more detail, let B be the two-form Z

gauge ﬁeld. Then, the 3d theory on the wall

can be coupled to B, but the partition function is not gauge invariant. The non-gauge

invariance can be canceled by a bulk term

2πi(1 − N)

B ∪ B . (2.5)

(More precisely, the Pontryagin square has to be used. Furthermore, this expression is

valid only for even N; for odd N see [6] and some comments below. See also [35] for an

earlier discussion of similar ideas.)

– 5 –

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