辛杨-米尔斯理论
Symplectic Yang-Mills Theory
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中文总结 AI 辅助
本文在辛流形上通过分解曲率二形式，将杨-米尔斯泛函分裂为原始与迹两部分，并研究其临界解性质，证明PYM泛函的椭圆性、平坦解分类及单调性公式，为刻画解模空间奠定基础。
中文摘要 AI 辅助
在辛流形上，任何微分二形式都有自然的分解为两个分量：一个原始部分和一个非原始部分。将此分解应用于辛流形上主丛的曲率二形式，我们得到了杨-米尔斯（YM）泛函的一个自然分裂，即分解为两个本质上依赖于辛结构的泛函：原始杨-米尔斯（PYM）泛函和迹杨-米尔斯（TYM）泛函。我们研究了这两个泛函的临界解的基本性质。特别是，PYM泛函展现出YM泛函的许多理想性质，包括其欧拉-拉格朗日方程的椭圆性以及G-丛上平坦解的代数分类。我们还证明了PYM泛函的单调性公式，作为刻画其解模空间的第一步。
英文摘要
On a symplectic manifold, any differential two-form has a natural decomposition into two components: a primitive part and a non-primitive one. Applying this decomposition to the curvature two-form of a principal bundle over a symplectic manifold, we obtain a natural splitting of the Yang-Mills (YM) functional into two functionals that intrinsically depend on the symplectic structure: the primitive Yang-Mills (PYM) functional and the trace Yang-Mills (TYM) functional. We work out the basic properties of the critical solutions of these two functionals. The PYM functional in particular exhibits many of the desirable properties of the YM functional, including the ellipticity of its Euler-Lagrange equations and an algebraic classification of its flat solutions on G-bundles. We also prove a monotonicity formula for the PYM functional as a first step towards characterizing its moduli space of solutions.
