AI 中文总结
研究通过莫尔斯理论构建紧致黎曼曲面上联络空间的杨 - 米尔斯测度，利用莫尔斯规范、随机上同调方程求解及梯度流指数收敛改进等方法，定义并计算相关测度及全纯性定律，恢复前人公式。
AI 中文摘要
我们引入一种莫尔斯理论方法来构建紧致黎曼曲面上联络空间的杨 - 米尔斯测度。这提供了该测度的直接连续版本，此前在平坦环面情形下由切维列夫通过格点近似得到，一般紧致黎曼曲面情形下由作者之一与诺拉得到。起点是莫尔斯规范的新定义以及与莫尔斯 - 斯马尔向量场相关的随机上同调方程的求解。通过改进莫尔斯 - 斯马尔梯度流到平衡结果的指数收敛性来实现，这在研究吕埃勒谱时由两位作者得到，在流体力学简化模型中由贾、斯图尔特和斯韦拉克得到。将这些随机解与莫尔斯复形给出的数据相结合，在联络空间引入自由杨 - 米尔斯测度，并利用随机微分方程的经典工具说明如何理解沿一大类曲线的随机联络的全纯性。最后，通过这些随机全纯性对该自由测度进行适当条件设定，定义杨 - 米尔斯测度并计算其配分函数以及关于此测度的随机全纯性定律，恢复了米格尔、威滕和莱维作品中的公式。
英文摘要
We introduce a Morse theoretical approach to the construction of the Yang--Mills measure on the space of connections of a compact Riemannian surface. This provides a direct continuous version of this measure which was previously obtained through lattice approximations by Chevyrev in the case of the flat torus and by one of the authors and Nohra for general compact Riemannian surfaces. The starting point is the new notion of a Morse gauge together with the resolution of random cohomological equations associated to Morse--Smale vector fields. This is achieved by improving exponential convergence to equilibrium results for Morse--Smale gradient flows that were obtained by two of the authors in the context of the study of Ruelle spectra and by Jia, Stewart and Sverak in the context of simplified models from fluid mechanics. Combining these random solutions with the data given by the Morse complex, we introduce a free Yang-Mills measure on space of connections and, using classical tools from stochastic differential equations, we show how to make sense of holonomies for random connections along a large class of curves. Finally, by setting a proper conditioning of this free measure through these random holonomies, we define the Yang--Mills measure and we compute its partition function together with the law of random holonomies with respect to this measure, recovering the formulas from the works of Migdal, Witten and Lévy.
Comments112 pages, 14 figures. v1: Preliminary version. All comments are welcome
