# Cotangent bundles for "matrix algebras converge to the sphere''. (arXiv:1907.06263v1 [math.OA])

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.)