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.
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References
A. Actor, Classical Solutions of SU(2) Yang-Mills Theories, Rev. Mod. Phys. 51 (1979) 461 [INSPIRE].
M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Y. Manin, Construction of Instantons, Phys. Lett. A 65 (1978) 185 [INSPIRE].
M.F. Atiyah and R.S. Ward, Instantons and Algebraic Geometry, Commun. Math. Phys. 55 (1977) 117 [INSPIRE].
S.R. Coleman, Nonabelian Plane Waves, Phys. Lett. B 70 (1977) 59 [INSPIRE].
S. Coleman, Aspects of Symmetry, Cambridge University Press, (1985).
E. Corrigan and D.B. Fairlie, Scalar Field Theory and Exact Solutions to a Classical SU(2) Gauge Theory, Phys. Lett. B 67 (1977) 69 [INSPIRE].
E. Corrigan, D.B. Fairlie, R.G. Yates and P. Goddard, The Construction of Selfdual Solutions to SU(2) Gauge Theory, Commun. Math. Phys. 58 (1978) 223 [INSPIRE].
N.S. Craigie, P. Goddard and W. Nahm, Monopoles in Quantum Field Theory, World Scientific, (1982).
H.J. de Vega, Nonlinear Multiplane Wave Solutions of Selfdual {Yang-Mills} Theory, Commun. Math. Phys. 116 (1988) 659 [INSPIRE].
N. Dorey, T.J. Hollowood, V.V. Khoze and M.P. Mattis, The Calculus of many instantons, Phys. Rept. 371 (2002) 231 [hep-th/0206063] [INSPIRE].
C.R. Gilson, M. Hamanaka, S.-C. Huang and J.J.C. Nimmo, Soliton Solutions of Noncommutative Anti-Self-Dual Yang-Mills Equations, J. Phys. A 53 (2020) 404002 [arXiv:2004.01718] [INSPIRE].
C.R. Gilson, M. Hamanaka and J.J.C. Nimmo, B¨acklund Transformations and the Atiyah-Ward ansatz for Noncommutative Anti-Self-Dual Yang-Mills Equations, Proc. Roy. Soc. Lond. A 465 (2009) 2613 [arXiv:0812.1222] [INSPIRE].
D.J. Gross, R.D. Pisarski and L.G. Yaffe, QCD and Instantons at Finite Temperature, Rev. Mod. Phys. 53 (1981) 43 [INSPIRE].
M. Hamanaka and H. Okabe, Soliton Scattering in Noncommutative Spaces, Theor. Math. Phys. 197 (2018) 1451 [Teor. Mat. Fiz. 197 (2018) 68] [arXiv:1806.05188] [INSPIRE].
Y. Kodama, KP Solitons and the Grassmannians, Springer, (2017).
N.S. Manton and P. Sutcliffe, Topological solitons, Cambridge University Press, (2004).
N. Marcus, The N = 2 open string, Nucl. Phys. B 387 (1992) 263 [hep-th/9207024] [INSPIRE].
L.J. Mason and N.M. Woodhouse, Integrability, Self-Duality, and Twistor Theory, Oxford University Press, (1996).
V.B. Matveev and M.A. Salle, Darboux Transformations and Solitons, Springer-Verlag, (1991).
Y. Nakamura, Transformation group acting on a self-dual Yang-Mills hierarchy, J. Math. Phys. 29 (1988) 244 [INSPIRE].
J.J.C. Nimmo, C.R. Gilson and Y. Ohta, Applications of Darboux transformations to the selfdual Yang-Mills equations, Theor. Math. Phys. 122 (2000) 239 [Teor. Mat. Fiz. 122 (2000) 284] [INSPIRE].
H. Ooguri and C. Vafa, Geometry of N = 2 strings, Nucl. Phys. B 361 (1991) 469 [INSPIRE].
H. Ooguri and C. Vafa, N = 2 heterotic strings, Nucl. Phys. B 367 (1991) 83 [INSPIRE].
A.M. Polyakov, Gauge Fields and Strings, Harwood Academic, (1987).
N. Sasa, Y. Ohta and J. Matsukidaira, Bilinear Form Approach to the Self-Dual Yang-Mills Equation and Integrable System in (2+1)-Dimension, J. Phys. Soc. Jap. 67 (1998) 83.
M.A. Shifman, Instantons in Gauge Theories, World Scientific, (1994).
K. Takasaki, A New Approach To The Selfdual Yang-Mills Equations, Commun. Math. Phys. 94 (1984) 35 [INSPIRE].
G. ’t Hooft, unpublished.
G. ’t Hooft, 50 Years of Yang-Mills Theory, World Scientific, (2005).
R.S. Ward, Integrable and solvable systems, and relations among them, Phil. Trans. Roy. Soc. Lond. A 315 (1985) 451.
F. Wilczek, Geometry and interactions of instantons, in Quark Confinement and Field Theory, Wiley, (1977), pg. 211.
C.N. Yang, Condition of Selfduality for SU(2) Gauge Fields on Euclidean Four-Dimensional Space, Phys. Rev. Lett. 38 (1977) 1377 [INSPIRE].
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Dedicated to the memory of Jon Nimmo
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ArXiv ePrint: 2004.09248
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Hamanaka, M., Huang, SC. New soliton solutions of anti-self-dual Yang-Mills equations. J. High Energ. Phys. 2020, 101 (2020). https://doi.org/10.1007/JHEP10(2020)101
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DOI: https://doi.org/10.1007/JHEP10(2020)101