Title: New perspectives in Kac-Moody algebras associated to higher dimensional manifolds Show Chinese title

Title: Kac-Moody代数在高维流形中的新视角

Abstract: In this review, we present a general framework for the construction of Kac-Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on the circle $\mathbb{S}^{1}$, we extend the approach to compact and non-compact group manifolds, coset spaces, and soft deformations thereof. After recalling the necessary geometric background on Riemannian manifolds, Hilbert bases and Killing vectors, we present the construction of generalized current algebras $\mathfrak{g}(\mathcal{M})$, their semidirect extensions with isometry algebras, and their central extensions. We show how the resulting algebras are controlled by the structure of the underlying manifold, and illustrate the framework through explicit realizations on $SU(2)$, $SU(2)/U(1)$, and higher-dimensional spheres, highlighting their relation to Virasoro-like algebras. We also discuss the compatibility conditions for cocycles, the role of harmonic analysis, and some applications in higher-dimensional field theory and supergravity compactifications. This provides a unifying perspective on KM algebras beyond one-dimensional settings, paving the way for further exploration of their mathematical and physical implications. Show Chinese abstract

Abstract: 在本综述中，我们提出了一种构建与高维流形相关的Kac-Moody（KM）代数的一般框架。从圆上的循环代数的经典情况$\mathbb{S}^{1}$出发，我们将该方法扩展到紧致和非紧致群流形、齐次空间及其软变形。在回顾了黎曼流形、Hilbert基和Killing向量的必要几何背景后，我们介绍了广义电流代数$\mathfrak{g}(\mathcal{M})$的构造，它们与等距代数的半直积扩张以及它们的中心扩张。我们展示了所得代数如何由底层流形的结构所控制，并通过在$SU(2)$、$SU(2)/U(1)$和高维球面的显式实现来说明该框架，突出它们与类似Virasoro代数的关系。我们还讨论了上同调类的相容性条件、调和分析的作用以及在高维场论和超引力紧化中的一些应用。这提供了一个超越一维设置的KM代数统一视角，为进一步探索其数学和物理意义铺平了道路。