2+1维的Maxwell群及其无限维增强
The Maxwell group in 2+1 dimensions and its infinite-dimensional enhancements
摘要：
2+1维的麦克斯韦群是由半直积的一个特殊推广给出的。这个数学结构提供了一个健全的框架，研究不同的推广麦克斯韦对称性在三个时空维度。通过给出扩展半直积的一般定义，构造了Maxwell群的无限维增强，通过引入非对易超平移扩展了${\rm ISL}(2,\mathbb{R})$ Kac-Moody群和${\rm BMS}_3$群.定义了每种情况下的余伴随表示，并给出了余伴随轨道上相应的几何作用。这些作用导致新的Wess-Zumino条款，自然实现上述无限维对称性。我们简要地阐述了在三维引力，高自旋对称性和量子霍尔系统的背景下的潜在应用。
英文摘要：
The Maxwell group in 2+1 dimensions is given by a particular extension of a semi-direct product. This mathematical structure provides a sound framework to study different generalizations of the Maxwell symmetry in three space-time dimensions. By giving a general definition of extended semi-direct products, we construct infinite-dimensional enhancements of the Maxwell group that enlarge the ${\rm ISL}(2,\mathbb{R})$ Kac-Moody group and the ${\rm BMS}_3$ group by including non-commutative supertranslations. The coadjoint representation in each case is defined, and the corresponding geometric actions on coadjoint orbits are presented. These actions lead to novel Wess-Zumino terms that naturally realize the aforementioned infinite-dimensional symmetries. We briefly elaborate on potential applications in the contexts of three-dimensional gravity, higher-spin symmetries, and quantum Hall systems.
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