2018/08/15 by Chien-Hao Liu, Shing-Tung Yau, Liu, Chien-Hao +1 · 3 citations
Mathematics · Physics and Astronomy · #14A22 #16S50 #46L87 #58C25 #81Q60 #81T30 #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Symplectic Geometry (math.SG) #hep-th #math.AG #math.DG #math.SG #msc:14A22 #msc:16S50 #msc:46L87 #msc:58C25 #msc:81Q60 #msc:81T30
paper · pdf · doi:10.48550/arxiv.1808.05011
85+2 pages, 5 figures
arxiv created 2018/08/15 · arxiv updated 2018/08/16
As the necessary background to construct from the aspect of Grothendieck's Algebraic Geometry dynamical fermionic D3-branes along the line of Ramond-Neveu-Schwarz superstrings in string theory, three pieces of the building blocks are given in the current notes: (1) basic C^∞-algebrogeometric foundations of d=4, N=1 supersymmetry and d=4, N=1 superspace in physics, with emphasis on the partial C^∞-ring structure on the function ring of the superspace, (2) the notion of SUSY-rep compatible hybrid connections on bundles over the superspace to address connections on the Chan-Paton bundle on the world-volume of a fermionic D3-brane, (3) the notion of \widehatD-chiral maps \widehatφ from a d=4 N=1 Azumaya/matrix superspace with a fundamental module with a SUSY-rep compatible hybrid connection \widehat∇ to a complex manifold Y as a model for a dynamical D3-branes moving in a target space(-time). Some test computations related to the construction of a supersymmetric action functional for SUSY-rep compatible (\widehat∇, \widehatφ) are given in the end, whose further study is the focus of a separate work. The current work is a sequel to D(11.4.1) (arXiv:1709.08927 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]) and is the first step in the supersymmetric generalization, in the case of D3-branes, of the standard action functional for D-branes constructed in D(13.3) (arXiv:1704.03237 [hep-th]).