D-branes and synthetic/$C^{\infty}$-algebraic symplectic/calibrated geometry, I: Lemma on a finite algebraicness property of smooth maps from Azumaya/matrix manifolds

TL;DR

This paper establishes a fundamental lemma about smooth maps from Azumaya/matrix manifolds, providing a foundation for developing synthetic symplectic and calibrated geometries inspired by D-brane phenomena in string theory.

Contribution

It introduces a key algebraic property of smooth maps from Azumaya/matrix manifolds, enabling the construction of $C^{ abla}$-algebraic geometries related to D-branes.

Findings

Proves a finite algebraicness property of smooth maps from Azumaya/matrix manifolds.

Lays groundwork for synthetic symplectic and calibrated geometries influenced by string theory.

Supports the mathematical modeling of D-branes in a $C^{ abla}$-algebraic framework.

Abstract

We lay down an elementary yet fundamental lemma concerning a finite algebraicness property of a smooth map from an Azumaya/matrix manifold with a fundamental module to a smooth manifold. This gives us a starting point to build a synthetic (synonymously, $C^{\infty}$-algebraic) symplectic geometry and calibrated geometry that are both tailored to and guided by D-brane phenomena in string theory and along the line of our previous works D(11.1) (arXiv:1406.0929 [math.DG]) and D(11.2) (arXiv:1412.0771 [hep-th]).