In the high-energy quantum-physics literature one finds statements such as “matrix algebras converge to the sphere”. Earlier I provided a general setting for understanding such statements, in which the matrix algebras are viewed as compact quantum metric spaces, and convergence is with respect to a quantum Gromov–Hausdorff-type distance. More recently I have dealt with corresponding statements in the literature about vector bundles on spheres and matrix algebras. But physicists want, even more, to treat structures on spheres (and other spaces) such as Dirac operators, Yang–Mills functionals, etc., and they want to approximate these by corresponding structures on matrix algebras. In preparation for understanding what the Dirac operators should be, we determine here what the corresponding “cotangent bundles” should be for the matrix algebras, since it is on them that a “Riemannian metric” must be defined, which is then the information needed to determine a Dirac operator. (In the physics literature there are at least 3 inequivalent suggestions for the Dirac operators.)

在高能量子物理学文献中，人们发现诸如“矩阵代数收敛于球体”之类的陈述。早些时候，我为理解这样的陈述提供了一个通用设置，其中矩阵代数被视为紧凑的量子度量空间，并且收敛是关于量子Gromov–Hausdorff型距离的。最近，我处理了有关球体和矩阵代数上的向量束的文献中的相应陈述。但是，物理学家们甚至还希望处理球体（和其他空间）上的结构，例如狄拉克算子，Yang-Mills泛函等，并且他们希望通过矩阵代数上的相应结构来近似这些结构。为了准备了解Dirac算子应该是什么，我们在这里确定矩阵代数应该对应的“余切束”是什么，由于必须定义“黎曼度量”，因此这是确定Dirac算子所需的信息。（在物理学文献中，对于Dirac算子至少存在3个不等价的建议。）