2020/02/27 by Chien‐Hao Liu, Shing–Tung Yau, Liu, Chien-Hao +1
Mathematics · Physics and Astronomy · #16S38 #58A50 #81T60 #83E50 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG) #primary: 14A22 #secondary: 14M30
paper · pdf · doi:10.48550/arxiv.2002.11868
openalex publication_date 2020/02/27 · openalex created_date 2020/03/06 · openalex updated_date 2026/07/28
Forty-six years after the birth of supersymmetry in 1973 from works of Julius Wess and Bruno Zumino, the standard quantum-field-theorists and particle physicists' language of `superspaces', `supersymmetry', and `supersymmetric action functionals in superspace formulation' as given in Chapters IV, V, VI, VII, XXII of the classic on supersymmetry and supergravity: Julius Wess & Jonathan Bagger: Supersymmetry and Supergravity (2nd ed.), is finally polished, with only minimal mathematical patches added for consistency and accuracy in dealing with nilpotent objects from the Grassmann algebra involved, to a precise setting in the language of complexified \Bbb Z/2-graded C^∞-Algebraic Geometry. This is completed after the lesson learned from D(14.1) (arXiv:1808.05011 [math.DG]) and the notion of `d=3+1, N=1 towered superspaces' as complexified \Bbb Z/2-graded C^∞-schemes, their distinguished sectors, and purge-evaluation maps first developed in SUSY(1) (= D(14.1.Supp.1)) (arXiv:1902.06246 [hep-th]) and further polished in the current work. While the construction depends on a choice of a trivialization of the spinor bundle by covariantly constant sections, as long as the transformation law and the induced isomorphism under a change of trivialization of the spinor bundle by covariantly constant sections are understood, any object or structure thus defined or constructed is mathematically well-defined. The construction can be generalized to all other space-time dimensions with simple or extended supersymmetries. This is part of the mathematical foundation required to study fermionic D-branes in the Ramond-Neveu-Schwarz formulation.