AI 中文总结
研究紧致曲面上杨 - 米尔斯随机联络的半经典极限，通过引入各向异性巴拿赫空间，利用莫尔斯梯度流相关联络给出平坦联络模空间新描述，证明特殊酉群下零面积极限情况，回答了相关问题。
AI 中文摘要
我们引入紧致曲面上分布 1 - 形式的各向异性巴拿赫空间，用于捕捉莫尔斯规范固定的杨 - 米尔斯随机联络的精细正则性。利用莫尔斯梯度流不稳定曲线上的奇异联络，我们给出了平坦联络模空间及其典范阿蒂亚 - 博特 - 戈德曼辛形式的新描述。当群为特殊酉群时，我们证明在零面积极限下，杨 - 米尔斯随机联络在这些各向异性空间中收敛到一个随机分布 1 - 形式，其定律与阿蒂亚 - 博特 - 戈德曼测度的莫尔斯理论代表一致。我们的方法通过在随机分布联络层面建立杨 - 米尔斯测度的零面积极限，扩展了威滕、福尔曼、刘和森古普塔的工作，回答了蒂埃里·列维关于杨 - 米尔斯随机联络半经典极限的问题。
英文摘要
We introduce anisotropic Banach spaces of distributional 1-forms on compact surfaces designed to capture the fine regularity properties of Morse gauge-fixed Yang--Mills random connections. Using singular connections supported on unstable curves of a Morse gradient flow, we provide a new description of the moduli space of flat connections and of its canonical Atiyah--Bott--Goldman symplectic form. When the group is the special unitary group, we prove that in the zero-area limit, the Yang--Mills random connection converges, within these anisotropic spaces, to a random distributional 1-form whose law coincides with our Morse-theoretic representative of the Atiyah--Bott--Goldman measure. Our approach extends the works of Witten, Forman, Liu, and Sengupta by establishing the zero-area limit of the Yang--Mills measure at the level of random distributional connections, rather than only at the level of holonomies. This answers a question of Thierry Lévy on the semiclassical limits of Yang--Mills random connections.
