收敛到余伴随轨道的矩阵代数的Dirac算子
Dirac operators for matrix algebras converging to coadjoint orbits
AI总结：
该研究针对高能物理中矩阵代数逼近空间结构的需求，提出余伴随轨道及收敛矩阵代数的统一Dirac算子构造，证明对应量子度量空间在强量子Gromov-Hausdorff距离下收敛。
摘要：
在高能物理文献中，我们能看到诸如“矩阵代数收敛到球面”之类的表述。此前笔者提出了一个通用的精确框架来理解这类表述：将矩阵代数视为量子度量空间，收敛性则通过量子Gromov-Hausdorff型距离来衡量。\n 但物理学家还希望处理球面（及其他空间）上的结构，例如向量丛、Yang-Mills泛函、Dirac算子等，并且想要用矩阵代数上的对应结构来逼近这些对象。本文中，我们针对余伴随轨道以及收敛到这些轨道的矩阵代数，提出了一种颇具统一性的Dirac算子构造方法。借此我们证明了核心定理：对于由我们构造的Dirac算子所确定的量子度量空间结构，矩阵代数确实会在一种相当强的量子Gromov-Hausdorff距离意义下收敛到余伴随轨道。
英文摘要：
In the high-energy physics literature one finds statements such as ``matrix algebras converge to the sphere''. Earlier I provided a general precise setting for understanding such statements, in which the matrix algebras are viewed as quantum metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. But physicists want even more to treat structures on spheres (and other spaces), such as vector bundles, Yang-Mills functionals, Dirac operators, etc., and they want to approximate these by corresponding structures on matrix algebras. In the present paper we provide a somewhat unified construction of Dirac operators on coadjoint orbits and on the matrix algebras that converge to them. This enables us to prove our main theorem, whose content is that, for the quantum metric-space structures determined by the Dirac operators that we construct, the matrix algebras do indeed converge to the coadjoint orbits, for a quite strong version of quantum Gromov-Hausdorff distance.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
