Nontrivial Azumaya noncommutative schemes, morphisms therefrom, and their extension by the sheaf of algebras of differential operators: D-branes in a B-field

Chien-Hao Liu Shing-Tung Yau

Mathematical Physics mathscidoc:1912.43661

arXiv preprint arXiv:0909.2291, 2009.9
In this continuation of [L-Y1],[LLSY],[L-Y2], and [L-Y3](arXiv: 0709.1515 [math. AG], arXiv: 0809.2121 [math. AG], arXiv: 0901.0342 [math. AG], arXiv: 0907.0268 [math. AG]), we study D-branes in a target-space with a fixed B -field background B along the line of the Polchinski-Grothendieck Ansatz, explained in [L-Y1] and further extended in the current work. We focus first on the gauge-field-twist effect of B -field to the Chan-Paton module on D-branes. Basic properties of the moduli space of D-branes, as morphisms from Azumaya schemes with a twisted fundamental module to B , are given. For holomorphic D-strings, we prove a valuation-criterion property of this moduli space. The setting is then extended to take into account also the deformation-quantization-type noncommutative geometry effect of B -field to both the D-brane world-volume and the superstring target-space (-time) B . This brings the notion of twisted B -modules that are realizable as twisted locally-free coherent modules with a flat connection into the study. We use this to realize the notion of both the classical and the quantum spectral covers as morphisms from Azumaya schemes with a fundamental module (with a flat connection in the latter case) in a very special situation. The 3rd theme (subtitled" Sharp vs. Polchinski-Grothendieck") of Sec. 2.2 is to be read with the work [Sh3](arXiv: hep-th/0102197) of Sharp while Sec. 5.2 (subtitled less appropriately" Dijkgraaf-Holland-Sukowski-Vafa vs. Polchinski-Grothendieck") is to be read with the related sections in [DHSV](arXiv: 0709.4446 [hep-th]) and [DHS](arXiv: 0810.4157 [hep-th]) of Dijkgraaf, Hollands, Sukowski, and Vafa.
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@inproceedings{chien-hao2009nontrivial,
  title={Nontrivial Azumaya noncommutative schemes, morphisms therefrom, and their extension by the sheaf of algebras of differential operators: D-branes in a B-field},
  author={Chien-Hao Liu, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204813592534225},
  booktitle={arXiv preprint arXiv:0909.2291},
  year={2009},
}
Chien-Hao Liu, and Shing-Tung Yau. Nontrivial Azumaya noncommutative schemes, morphisms therefrom, and their extension by the sheaf of algebras of differential operators: D-branes in a B-field. 2009. In arXiv preprint arXiv:0909.2291. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224204813592534225.
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