Grothendieck meeting [Wess & Bagger]: [Supersymmetry and supergravity: IV, V, VI, VII, XXII] (2nd ed.) reconstructed in complexified $\mathbb{Z}/2$-graded $C^{\infty }$-Algebraic Geometry, I. Construction under trivialization of spinor bundle

Abstract

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 $\mathbb{Z}/2$-graded $C^{\infty }$-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 $\mathbb{Z}/2$-graded $C^{\infty }$-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.

83+2 pages. Some overlaps of footnotes with SUSY(1) are kept due to the intention to make SUSY(2.1) conceptually more self-contained

Cited by in corpus (2)

• The geometry and DSZ quantization of four-dimensional supergravity
• Soft noncommutative schemes via toric geometry and morphisms from an Azumaya scheme with a fundamental module thereto -- (Dynamical, complex algebraic) D-branes on a soft noncommutative space