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
a
, Olaf Lechtenfeld
b,c,∗
, Alexander D. Popov
b
a
Bogoliubov Laboratory of Theoretical Physics, JINR, 141980 Dubna, Moscow Region, Russia
b
Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany
c
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
ˇ
c
We consider SU(N) Yang–Mills theory on R
2,1
× S
1
, where S
1
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
N
. The latter is the
direct product of SU(N) and its Langlands dual SU(N)/Z
N
, and it contains the above-mentioned torus as
its maximal Abelian subgroup. An analogous result is obtained for any non-Abelian gauge group.
© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
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
1
R
with small radius R (see e.g. [1,2]and
references therein). In the adiabatic limit, when the metric on S
1
R
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
1
R
, and T
∨
is the Cartan torus in the Langlands dual SU(N)/Z
N
. The torus T
∨
parametrizes the dual (magnetic) photons on R
2.1
and corresponds
to ’t Hooft loops around S
1
R
[3,4]. Finally, W is the Weyl group,
which for SU(N) is the finite permutation group S
N
.
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
1
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
N
. In other words, we shall show how the
classical Yang–Mills model on R
2,1
× S
1
R
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
w
, the sigma-model target
space will be M = G×G
∨
, where G
∨
denotes the Langlands dual
group, whose weight lattice
∨
w
is dual to
w
. Thus, the target-
space
geometry M of our sigma models obtained from Yang–Mills
theory on R
2,1
× S
1
R
in the small-R limit essentially depends on
conditions imposed on the gauge potential A and the gauge trans-
formations
along S
1
R
.
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
1
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/
© 2018 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
下载后可阅读完整内容，剩余4页未读，立即下载
weixin_38672940
- 粉丝: 5
- 资源: 970
我的内容管理
展开
最新资源
- JSP+SSM科研管理系统响应式网站设计案例
- 推荐一款超级好用的嵌入式串口调试工具
- PHP域名多维查询平台：高效精准的域名搜索工具
- Citypersons目标检测数据集：Yolo格式下载指南
- 掌握MySQL面试必备：程序员面试题解析集锦
- C++软件开发培训：核心技术资料深度解读
- SmartSoftHelp二维码工具：生成与解析条形码
- Android Spinner控件自定义字体大小的方法
- Ubuntu Server on Orangepi3 LTS 官方镜像发布
- CP2102 USB驱动程序的安装与更新指南
- ST-link固件升级指南：轻松更新程序步骤
- Java实现的质量管理系统Demo功能分析与操作
- Everything高效文件搜索工具：快速精确定位文件
- 基于B/S架构的酒店预订系统开发实践
- RF_Setting(E22-E90(SL)) V1.0中性版功能解析
- 高效转换M3U8到MP4：免费下载工具发布
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
