非阿贝尔西格玛模型：Yang-Mills理论在圆周上的低能行为

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本文探讨的是Yang-Mills理论在特定几何背景下的一个有趣现象，即当将理论限制在R2,1（二维空间-时间）与一个空间圆S1的组合上时，其在低能极限下的行为。SU(N) Yang-Mills理论在这种情况下表现为一个非阿贝尔西格玛模型，这是理论物理中的一个重要概念，它描述了规范场的低能量行为。在S1的小半径红外区域，原始的Yang-Mills作用简化为在R2,1上的西格玛模型，其目标空间不再是SU(N)群本身，而是经历了一个结构上的变化。具体来说，这个目标空间被定义为Weyl群作用下的2(N-1)维环面，即2(N-1)维拓扑torus，但受到了Weyl群的对称变换的影响。Weyl群是特殊单位ary群的一个对称子群，对于不同的N值，其作用会产生不同的几何特征。值得注意的是，作者强调了在选择规范丛（gauge bundle）框架时的自由度，这导致了对目标空间的更广泛理解。通过这种选择，他们能够证明低能量限制导致的目标空间实际上是SU(N)与它的Langlands对偶SU(N)/ZN的直积。Langlands对偶是数学中一种重要的对应关系，它揭示了不同群之间的相互关系。在这个直积中，SU(N)与SU(N)/ZN并存，其中后者是SU(N)群与其商群的结合，而商群ZN是中心化子，包含了所有可能的阶数为N的元素。这个结构允许我们看到目标空间中最大的阿贝尔子群，即那些可以通过中心元素生成的子群，它与原始的环面相对应。这个发现对于非阿贝尔规范理论有着广泛的意义，因为它揭示了这些理论在特定边界条件和几何约束下可能出现的新特征。研究这样的限制和简化不仅有助于深化我们对基础理论的理解，还可能为量子场论、弦理论和高能物理的其他分支提供新的洞察。通过这种方法，物理学家能够探索不同尺度上的对称性破缺和规范场的行为，这对于粒子物理学和宇宙学等领域具有潜在的应用价值。

Physics Letters B 781 (2018) 322–326

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Non-Abelian sigma models from Yang–Mills theory compactiﬁed

on a circle

Tatiana A. Ivanova

, Olaf Lechtenfeld

b,c,∗

, Alexander D. Popov

Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Region, Russia

Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany

Riemann Center for Geometry and Physics, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany

a r t i c l e i n f o a b s t r a c t

Article history:

Received

29 March 2018

Accepted

6 April 2018

Available

online 9 April 2018

Editor:

M. Cveti

We consider SU(N) Yang–Mills theory on R

2,1

× S

, where S

is a spatial circle. In the infrared limit of

a small-circle radius the Yang–Mills action reduces to the action of a sigma model on R

2,1

whose target

space is a 2(N − 1)-dimensional torus modulo the Weyl-group action. We argue that there is freedom

in the choice of the framing of the gauge bundles, which leads to more general options. In particular,

we show that this low-energy limit can give rise to a target space SU(N)×SU(N)/Z

. The latter is the

direct product of SU(N) and its Langlands dual SU(N)/Z

, and it contains the above-mentioned torus as

its maximal Abelian subgroup. An analogous result is obtained for any non-Abelian gauge group.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction and summary

Pure Yang–Mills or QCD-like theories in four spacetime dimen-

sions

are strongly coupled in the infrared limit. It is known that

one can partially overcome this diﬃculty by compactifying Yang–

Mills

theory on a circle S

with small radius R (see e.g. [1,2]and

references therein). In the adiabatic limit, when the metric on S

is scaled down, the d = 4SU(N) Yang–Mills action can be reduced

(already on the classical level) to the action of a d = 3sigma model

whose target space is T ×T

∨

/W . Here, T = U(1)

N−1

is the Cartan

torus in SU(N) corresponding to Wilson loops around S

, and T

∨

is the Cartan torus in the Langlands dual SU(N)/Z

. The torus T

∨

parametrizes the dual (magnetic) photons on R

2.1

and corresponds

to ’t Hooft loops around S

[3,4]. Finally, W is the Weyl group,

which for SU(N) is the ﬁnite permutation group S

The

above-mentioned action on R

2,1

may be augmented by

an effective potential for the sigma-model scalar ﬁelds, which ap-

pears

from an additional centre-stabilizing term breaking SU(N)

to U(1)

N−1

and from quantum loop corrections, as discussed e.g.

in [5,6]. In our paper we focus on the derivation of kinetic terms

in the low-energy limit of pure Yang–Mills theory. Therefore, for

the time being, we ignore a possible symmetry-breaking potential.

Corresponding author.

E-mail

addresses: ita @theor.jinr.ru (T.A. Ivanova),

olaf .lechtenfeld @itp .uni -hannover.de (O. Lechtenfeld),

alexander.popov @itp .uni -hannover.de (A.D. Popov).

The main message of the paper is that there is important free-

dom

in the choice of the framing

of the gauge bundle, and that

this leads to the option of enlarging the sigma-model target space

from T ×T

∨

/W to a non-Abelian group, up to the maximal space

M = SU(N)×SU(N)/Z

. In other words, we shall show how the

classical Yang–Mills model on R

2,1

× S

can be reduced to a

sigma model on R

2,1

with non-Abelian target space M or a sub-

group

thereof including the torus T ×T

∨

∼

U(1)

2(N−1)

. For a gen-

eral

gauge group G with weight lattice 

, the sigma-model target

space will be M = G×G

∨

, where G

∨

denotes the Langlands dual

group, whose weight lattice 

∨

is dual to 

. Thus, the target-

space

geometry M of our sigma models obtained from Yang–Mills

theory on R

2,1

× S

in the small-R limit essentially depends on

conditions imposed on the gauge potential A and the gauge trans-

formations

along S

The

Yang–Mills reduction to the Abelian sigma model on R

2,1

(where M is toroidal) points at an Abelian conﬁnement mech-

anism

based on Dirac monopoles, Abelian vortices and the dual

Meissner effect. The Abelian dual superconductor approach has

various limitations, like any other conﬁnement mechanism (see

e.g. [1,10]). For this reason there have been efforts to extend

the dual superconductor mechanism to models with non-Abelian

A bundle over a manifold M is called framed over a submanifold N ⊂ M if its

ﬁbres over N are ﬁxed. Framed bundles are often used in discussions of instantons

and monopoles as well as on manifolds with boundaries, marked points, punctures

etc. (see e.g. [7–9]).

https://doi.org/10.1016/j.physletb.2018.04.013

0370-2693/

SCOAP

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