SO(2)规范Skyrmions的拓扑电荷研究：2+1与3+1维度对比

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"关于（2）个规范的Skyrmions在2 + 1和3 + 1维度上的拓扑电荷" 这篇学术论文深入探讨了拓扑电荷在不同维度的Skyrmions中的表现，特别是在2 + 1和3 + 1维度的SO(2)规范Skyrmions上下文中。Skyrmions是凝聚态物理、粒子物理和高能物理中的重要拓扑对象，它们在磁性材料、超导体和宇宙学等领域有着广泛的应用。拓扑电荷q是衡量这些拓扑物体的重要特性，它与物体的稳定性和动力学行为紧密相关。在2 + 1维的O(3)模型中，研究者发现拓扑电荷q的值会受到Chern-Simons (CS) 项的影响。Chern-Simons项是一种在量子场论和弦理论中常见的项，它引入了规范势与拓扑密度的耦合。在存在CS项的情况下，Skyrmion的拓扑电荷可以改变符号甚至归零。这种变化可以通过分析理论中解的参数来追踪，这揭示了CS项如何影响Skyrmion的拓扑性质。另一方面，在3 + 1维的O(4)模型中，由于没有无尺度的CS密度，因此在该维度下未观察到拓扑电荷q的变化。这意味着在3 + 1维度的Skyrmions中，拓扑电荷可能更为稳定，不会因为CS项而发生显著的量子修正。作者们进行了定量研究，比较了这两个模型在不同条件下的差异，提供了对Skyrmions拓扑性质理解的新视角。这些研究对于深入理解Skyrmions的动态行为、其在物质中的形成和演化过程以及它们可能的实验观测具有重要意义。关键词包括：孤子、Skyrmions、拓扑电荷和瞬子，表明这篇论文涵盖了这些领域的重要概念和技术。文章经过了多次修订，最终在2019年2月被接受并在线发表。编辑是N. Lambert，这表明了研究的质量和严谨性得到了同行评审的认可。通过这项工作，科研人员可以进一步探索Skyrmions的数学结构和物理特性，为开发新的技术和应用提供理论基础。

Physics Letters B 791 (2019) 287–292

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

On the topological charge of SO(2) gauged Skyrmions in 2 + 1and

3 + 1dimensions

Francisco Navarro-Lérida

a,∗

, Eugen Radu

b,c

, D.H. Tchrakian

c,d

Dept. de Física Teórica and IPARCOS, Ciencias Físicas, Universidad Complutense de Madrid, E-28040 Madrid, Spain

Center for Research and Development in Mathematics and Applications, (CIDMA), Campus de Santiago, 3810-183 Aveiro, Portugal

School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4, Ireland

Department of Computer Science, NUI Maynooth, Maynooth, Ireland

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

Article history:

Received

11 December 2018

Received

in revised form 5 February 2019

Accepted

11 February 2019

Available

online 13 February 2019

Editor:

N. Lambert

Keywords:

Solitons

Skyrmions

Topological

charge

Instantons

The question of the dependence of the topological charge q of a gauged Skyrmion, on the gauge ﬁeld, is

studied quantitatively. Two examples, both gauged with SO(2) are studied and contrasted: i) The O (3)

model in 2 + 1 dimensions, and ii) The O (4) model in 3 + 1 dimensions. In case i), where the (usual)

Chern–Simons (CS) term is present, the value of q changes sign, going through zero. This evolution is

tracked by a parameter characterising the solutions in the given theory. In case ii), in which dimensions

no CS density is available, the evolution of q is not observed.

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

1. Introduction

The topological charge q of gauged Skyrmions presents pecu-

liarities

that are absent in gauged Higgs solitons. While in the

latter case this quantity always equals the winding number of the

Higgs ﬁeld, in the case of Skyrmions the value of q in general may

depart from the winding number, or the “baryon number”.

The

winding number of a Skyrme system is a topological

charge [1], which provides the static energy with a lower bound.

What we refer to as the topological charge of a gauged Skyrme

system, is merely the number which gives the lower bound on the

static energy. This contrasts with (gauged) Higgs systems where

the topological charge encoded in the Higgs (and gauge) ﬁeld, also

supplies the energy lower bound [2,3]. While in Higgs models

this topological charge is the magnetic monopole charge [4], by

contrast the magnetic ﬂux of gauged Skyrme models in space di-

mensions

D ≥ 3 vanishes [5], except in D = 2,

and that only in

the presence of Chern–Simons dynamics.

The

topological charge for the SO(2) gauged O(4) (Skyrme)

model in 3 + 1dimensions is deﬁned in Ref. [7] and in [8,9], and,

for the SO(2) gauged O (3) (planar Skyrme) model in 2 + 1dimen-

Corresponding author.

E-mail

address: fnavarro@ﬁs.ucm.es (F. Navarro-Lérida).

See for example [6].

sions in Ref. [10]. The topological charges for the SO(D) gauged

O (D + 1) sigma model on R

(D = 2, 3, 4) were deﬁned in [11],

and the soliton of the SO(3) gauged O (4) sigma model on R

was ﬁrst reported in [5]. More detailed studies of the latter soli-

tons

were given in [12], while solitons of the SO(4) gauged O (5)

sigma model on R

were reported in [13]. The most comprehen-

sive

and recent deﬁnitions of the generic topological charges of

gauged Skyrmions are given in Appendix B of Ref. [14].

With

the exception of the work in Ref. [7], where attention was

focused on the decay of the baryon number,

the works of [8–11,

5,13]are

concerned with the construction of topologically stable

solitons. Here, we are concerne d with the SO(2) gauged O (3) and

O (4) sigma models in 2 + 1and 3 + 1 dimensions, respectively.

But unlike in the works of Refs. [10,8,9], our focus here will be the

evolution of the topological charge, e.g., its possible decay.

In odd dimensional spacetimes, where the Chern–Simons (CS)

term is deﬁned, one ﬁnds that the effect of this CS dynamics re-

sults

in the mass/energy of the static solitons both to increase

and to decrease with increasing global charge (electric charge and

Well known works in this direction are, e.g., [16,17], employ instead SU(2)

gauging that does not result in topologically stable solitons.

SO(2) gauged Skyrmions in 2 + 1dimensions have been studied extensively

(see Refs. [10,6,18–20]and more recently in [21,22]). The SO(2) gauged Skyrmions

in 3 + 1dimensions have been quantitatively studied mainly in Refs. [8,9](see also

the recent analytical results in Ref. [23]).

https://doi.org/10.1016/j.physletb.2019.02.011

0370-2693/

SCOAP

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