New soliton solutions of anti-self-dual Yang-Mills equations
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Springer
Received: April 21, Revised: August 6, Accepted: September 15, Published: October 15,
2020 2020 2020 2020
Masashi Hamanaka and Shan-Chi Huang Department of Mathematics, University of Nagoya, Nagoya, 464-8602, Japan
E-mail: [email protected], [email protected] Abstract: We study exact soliton solutions of anti-self-dual Yang-Mills equations for G = GL(2) in four-dimensional spaces with the Euclidean, Minkowski and Ultrahyperbolic signatures and construct special kinds of one-soliton solutions whose action density TrFµν F µν can be real-valued. These solitons are shown to be new type of domain walls in four dimension by explicit calculation of the real-valued action density. Our results are successful applications of the Darboux transformation developed by Nimmo, Gilson and Ohta. More surprisingly, integration of these action densities over the four-dimensional spaces are suggested to be not infinity but zero. Furthermore, whether gauge group G = U(2) can be realized on our solition solutions or not is also discussed on each real space. Keywords: Integrable Field Theories, Solitons Monopoles and Instantons, Integrable Hierarchies ArXiv ePrint: 2004.09248 Dedicated to the memory of Jon Nimmo
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP10(2020)101
JHEP10(2020)101
New soliton solutions of anti-self-dual Yang-Mills equations
Contents 1 Introduction
1
2 Soliton solutions on four-dimensional complex spaces 2.1 Anti-self-dual Yang-Mills equations 2.2 Soliton solutions and action densities for G = GL(2)
2 2 4 6 7 9 10 11
4 Comparison to already known soliton solutions 4.1 Atiyah-Ward ansatz solutions (G = GL(2)) 4.2 ’t Hooft ansatz solutions (G = SU(2))
12 12 13
5 Conclusion and discussion
13
A Calculation of action density (2.12)
14
1
Introduction
Yang-Mills theories are at the center of elementary particle physics to describe fundamental laws of interactions. Topological solitons in these theories, such as instantons, monopoles, vortices, calorons, merons, played central roles in the study of non-perturbative aspects, duality structures, quark confinements and so on. (See e.g. [1, 5, 8, 10, 13, 16, 24, 26, 29].) To study these topological solitons, the anti-self-dual (ASD) Yang-Mills equation would be in the most important position. For instance, the instantons are global solutions of this equation with a special boundary condition such that the action is finite. For mathematical aspects, the instantons are described very elegantly by the ADHM construction [2]. On the other hand, the anti-self-dual Yang-Mills equation has a very close relationship with lower-dimensional integrable equations, such as the KdV equation, the Toda equations, the Painlev´e equations and so on [18, 30]. Energy densities of some soliton solutions to these equations are localized on hyperplanes in the whole space-time dimensions and hence they can be interpreted as domain walls in the space-times. Existence of these solitons solutions also relate to their int
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