AI 中文总结
研究将经典杨 - 米尔斯理论从主丛推广到李代数胚，用乘法埃雷斯曼联络取代主丛联络，构建作用泛函并推导方程，定义自对偶解，证明解空间规范不变性，还给出丛 gerbe 上联络的杨 - 米尔斯理论这一例子。
AI 中文摘要
我们对经典杨 - 米尔斯理论进行了双重推广，将其从主丛扩展到可能非传递且不可积的李代数胚情形。当考虑主丛的阿蒂亚代数胚时可恢复经典理论。在我们的框架中，主丛联络被更一般的（无穷小）乘法埃雷斯曼联络概念所取代。构建了此类联络的作用泛函，除通常的曲率2 - 形式项外还包括曲率3 - 形式贡献，所得变分问题自然受上同调条件约束。推导了相关的欧拉 - 拉格朗日方程，在4维和5维定义了一类自对偶解（瞬子）。还表明解空间在规范变换下不变，并计算了其在解处的切空间。作为重要例子，展示了我们的框架为丛 gerbe 上的联络产生杨 - 米尔斯理论。
英文摘要
We develop a twofold generalization of classical Yang-Mills theory, extending it from principal bundles to the setting of possibly non-transitive and non-integrable Lie algebroids. The classical theory is recovered when one considers the Atiyah algebroid of a principal bundle. In our framework, principal bundle connections are replaced by the more general notion of (infinitesimal) multiplicative Ehresmann connections. An action functional for such connections is constructed, now including a curvature 3-form contribution, alongside the usual curvature 2-form term, and the resulting variational problem is naturally constrained by a cohomological condition. We derive the associated Euler-Lagrange equations, and define a class of self-dual solutions (instantons) in both 4 and 5 dimensions. We also show that the solution space is invariant under gauge transformations, and compute its tangent space at a solution. As an important example, we show that our framework produces a Yang-Mills theory for connections on bundle gerbes.
Comments30 pages
