Further studies of the notion of differentiable maps from Azumaya/matrix supermanifolds I. The smooth case: Ramond-Neveu-Schwarz and Green-Schwarz meeting Grothendieck

2017/09/26 by Chien‐Hao Liu, Shing–Tung Yau, Liu, Chien-Hao +1 · 1 citation
Mathematics · #14A22 #16S50 #46L87 #51K10 #58A40 #81T30 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #High Energy Physics - Theory (hep-th) #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.1709.08927

openalex publication_date 2017/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this sequel to works D(11.1) (arXiv:1406.0929 [math.DG]), D(11.2) (arXiv:1412.0771 [hep-th]), and D(11.3.1) (arXiv:1508.02347 [math.DG]), we re-examine --- and reformulate when in need --- several basic notions in super C^∞-algebraic geometry as guided by the mathematical formulation of Ramond-Neveu-Schwarz fermionic strings and of Green-Schwarz fermionic strings from the viewpoint of Grothendieck on Algebraic Geometry. Two theorems that are the super counterpart of Theorem~3.1.1 and Theorem~3.2.1 of D(11.3.1) are proved. They unify the notion of "smooth maps from an Azumaya/matrix super smooth manifold with a fundamental module to a super smooth manifold" introduced in D(11.2), making it a complete super parallel to the setting for D-branes in the realm of algebraic geometry in D(1) (arXiv:0709.1515 [math.AG]) and D(2) (arXiv:0809.2121 [math.AG]), and in the realm of differential or C^∞-algebraic geometry in D(11.1) and D(11.3.1). A prototypical definition of dynamical fermionic stacked D-brane world-volume on a space-time in the same spirit of RNS fermionic strings or GS fermionic strings is thus laid down. Similar to D(11.3.1), which paved the path to the construction of non-Abelian Dirac-Born-Infeld action (D(13.1) (arXiv:1606.08529 [hep-th])) and the standard action (D(13.3) (arXiv:1704.03237 [hep-th])) for fundamental bosonic stacked D-branes, the current notes shall serve the same for the construction of supersymmetric action for fundamental fermionic stacked D-branes of various dimensions --- a theme of another subseries of the D-project. A notion of "noncommutative C^∞-rings" and "morphism" between them is introduced at the end as a byproduct.

Citations

• D-branes and Azumaya/matrix noncommutative differential geometry, I: D-branes as fundamental objects in string theory and differentiable maps from Azumaya/matrix manifolds with a fundamental module to real manifolds
• Further studies on the notion of differentiable maps from Azumaya/matrix manifolds, I. The smooth case
• D-branes and Azumaya/matrix noncommutative differential geometry,II: Azumaya/matrix supermanifolds and differentiable maps therefrom -- with a view toward dynamical fermionic D-branes in string theory
• Morphisms from Azumaya prestable curves with a fundamental module to a projective variety: Topological D-strings as a master object for curves
• Dynamics of D-branes I. The non-Abelian Dirac-Born-Infeld action, its first variation, and the equations of motion for D-branes --- with remarks on the non-Abelian Chern-Simons/Wess-Zumino term

Cited by

• C^∞-superrings and C^∞-superschemes

Related

• D-branes and Azumaya/matrix noncommutative differential geometry,II: Azumaya/matrix supermanifolds and differentiable maps therefrom -- with a view toward dynamical fermionic D-branes in string theory
• D-branes and Azumaya/matrix noncommutative differential geometry, I: D-branes as fundamental objects in string theory and differentiable maps from Azumaya/matrix manifolds with a fundamental module to real manifolds
• Further studies on the notion of differentiable maps from Azumaya/matrix manifolds, I. The smooth case
• Dynamics of D-branes II. The standard action --- an analogue of the Polyakov action for (fundamental, stacked) D-branes
• D-branes and Azumaya noncommutative geometry: From Polchinski to Grothendieck