Further studies on the notion of differentiable maps from Azumaya/matrix manifolds, I. The smooth case

Chien-Hao Liu Shing-Tung Yau

Differential Geometry mathscidoc:1912.43672

arXiv preprint arXiv:1508.02347, 2015.8
In this follow-up of our earlier two works D (11.1)(arXiv: 1406.0929 [math. DG]) and D (11.2)(arXiv: 1412.0771 [hep-th]) in the D-project, we study further the notion of adifferentiable map from an Azumaya/matrix manifold to a real manifold'. A conjecture is made that the notion of differentiable maps from Azumaya/matrix manifolds as defined in D (11.1) is equivalent to one defined through the contravariant ring-homomorphisms alone. A proof of this conjecture for the smooth (ie C^{\infty} ) case is given in this note. Thus, at least in the smooth case, our setting for D-branes in the realm of differential geometry is completely parallel to that in the realm of algebraic geometry, cf.\arXiv: 0709.1515 [math. AG] and arXiv: 0809.2121 [math. AG]. A related conjecture on such maps to C^{\infty} , as a C^{\infty} -manifold, and its proof in the C^{\infty} case is also given. As a by-product, a conjecture on a division lemma in the finitely differentiable case that generalizes the division lemma in the smooth case from Malgrange is given in the end, as well as other comments on the conjectures in the general C^{\infty} case. We remark that there are similar conjectures in general and theorems in the smooth case for the fermionic/super generalization of the notion.
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@inproceedings{chien-hao2015further,
  title={Further studies on the notion of differentiable maps from Azumaya/matrix manifolds, I. The smooth case},
  author={Chien-Hao Liu, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204903725408236},
  booktitle={arXiv preprint arXiv:1508.02347},
  year={2015},
}
Chien-Hao Liu, and Shing-Tung Yau. Further studies on the notion of differentiable maps from Azumaya/matrix manifolds, I. The smooth case. 2015. In arXiv preprint arXiv:1508.02347. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204903725408236.
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