Liouville and Painlevé equations and Yang–Mills strings
Liouville and Painlevé equations and Yang–Mills strings
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Liouville 和 Painlevé 方程以及 Yang–Mills 弦
DOI：
10.1063/1.526066
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发表时间：
1984
影响因子：
1.3
通讯作者：
C. Saçlıoğlu
中科院分区：
文献类型：
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作者：
C. Saçlıoğlu
求出了自对偶杨-米尔斯方程(降维为R2)的弦形解。多弦Anatz得到了Sinh-Gordon和Liouville方程。根据一般定理，解必须是实的和奇异的并且有无限作用，或者复数和非奇异的零作用。在Liouville情形下，给出了这两类问题的显式任意分离n-弦解。这些解的磁通量是Kaehler流形的陈氏磁通，因此它假定量子化值为4πn/e。轴对称的Sinh-Gordon版本用第三Painleve超越P3求解，使用Wu等人在P3上的结果。[物理。Rev.B 13,316(1976)]和McCoy等人。[J.Math.太棒了。18、10(1977)]。轴对称的情况可以投射到Ernst方程框架中，以产生进一步的解。在附录中，欧几里德化的Ernst方程被证明给出了自对偶Gibbons-Hawking引力瞬子。
Stringlike solutions of the self‐dual Yang–Mills equations (dimensionally reduced to R2) are sought. A multistring Ansatz results in the sinh–Gordon and Liouville equations. According to a general theorem, the solutions must be either real and singular and have infinite action, or complex and nonsingular, with zero action. In the Liouville case, explicit arbitrarily separated n‐string solutions of both classes are given. The magnetic flux for these solutions is found to be the Chern class of a Kaehler manifold, and it consequently assumes quantized values 4πn/e. The axisymmetric version of the sinh–Gordon is solved by the third Painleve transcendent P3, using the results on P3 by Wu et al. [Phys. Rev. B 13, 316 (1976)] and McCoy et al. [J. Math. Phys. 18, 10 (1977)]. The axisymmetric case can be cast into the Ernst equation framework for the generation of further solutions. In the Appendix, the Euclideanized Ernst equation is shown to give self‐dual Gibbons–Hawking gravitational instantons.