2009/09/12 by Chien‐Hao Liu, Chien-Hao Liu, Shing-Tung Yau +3 · 1 citation
Mathematics · Physics and Astronomy · #14A22 #81T30 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Symplectic Geometry (math.SG) #hep-th #math.AG #math.SG #msc:14A22 #msc:81T30
paper · pdf · doi:10.48550/arxiv.0909.2291
36+2 pages
arxiv created 2009/09/12 · openalex publication_date 2009/09/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this continuation of [L-Y1], [L-L-S-Y], [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 (Y,αB) 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 (Y,αB), 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) Y. This brings the notion of twisted \cal D-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-Sułkowski-Vafa vs. Polchinski-Grothendieck") is to be read with the related sections in [D-H-S-V] (arXiv:0709.4446 [hep-th]) and [D-H-S] (arXiv:0810.4157 [hep-th]) of Dijkgraaf, Hollands, Sułkowski, and Vafa.