标题： 平面QCD的精确解通过复矩阵系综 显示英文标题

标题： An exact solution of planar QCD by a complex matrix ensemble

摘要： 我们使用一种新颖的矩阵值动量环形式主义，在四维欧几里得空间中构造了平面QCD环方程的精确解。 这一构造的核心是一种新的环微积分，其中泛函导数作用于环速度$\dot C(\theta)$。 该框架为环空间中的点和面积导数提供了有限且定义良好的表达式，并突出了环空间Bianchi恒等式在确保与规范理论一致性的角色。 威尔逊环被表示为在由自对偶性和边界条件约束的紧复流形上追踪闭合路径的矩阵值动量环$P(\theta)$的平均值。 这些随机行走由支持在这些流形上的测度所支配。 自对偶性条件确保了环方程中经典杨-米尔斯项的消失，而额外的约束消除了平面（$N \to \infty$）极限中的接触项。 该解的一个关键结构特征是其在反射-共轭对称性下的不变性：反转环方向并取复共轭将一个解映射到另一个解。 这对于用于动量系综的任意紧李群都成立，并且独立于原始SU($N$)规范群。 我们猜想，在经典极限下，矩阵系综定义了一个广义的极小曲面，该曲面不是嵌入在$\mathbb{R}^4$中，而是嵌入在一个高维矩阵空间中。 这个（假设的）扩展曲面具有自对偶面积元素，并通过几何和Bianchi恒等式满足环方程。 显示英文摘要

摘要： We construct an exact solution to the planar QCD loop equation in four-dimensional Euclidean space using a novel matrix-valued momentum loop formalism. Central to this construction is a new loop calculus, where functional derivatives act on the loop velocity $\dot C(\theta)$. This framework yields finite, well-defined expressions for point and area derivatives in loop space and highlights the role of the loop-space Bianchi identity in ensuring consistency with gauge theory. The Wilson loop is represented as an average over matrix-valued momentum loops $P(\theta)$ tracing closed paths in a compact complex manifold constrained by self-duality and boundary conditions. These random walks are governed by a measure supported on such manifolds. The self-duality conditions ensure the vanishing of the classical Yang--Mills term in the loop equation, while additional constraints eliminate the contact terms in the planar ($N \to \infty$) limit. A key structural feature of the solution is its invariance under reflection--conjugation symmetry: reversing loop orientation and taking complex conjugation maps a solution to another. This holds for arbitrary compact Lie groups used in the momentum ensemble and is independent of the original SU($N$) gauge group. We conjecture that, in the classical limit, the matrix ensemble defines a generalized minimal surface embedded not in $\mathbb{R}^4$ but in a higher-dimensional matrix space. This (hypothetical) extended surface has self-dual area elements and satisfies the loop equation via geometry and the Bianchi identity.