📝 SummarXiv

Abstract

In contrast to the world-sheet of a fundamental string, the world-volume of stacked D-branes carries an Azumaya noncommutative structure ([L-Y1: Sec.\ 2] (D(1))), allowing it to directly serve as a probe into noncommutative target-spaces. This feature leads to a D-brane fantasy: {\it Noncommutative Mirror Symmetry between noncommutative Calabi-Yau spaces may be realized as different realizations of a supersymmetric D-brane world-volume quantum field theory exactly like the string world-sheet aspect for Mirror Symmetry between (commutative) Calabi-Yau manifolds}. Driven by this fantasy, in the current notes a class of noncommutative ringed spaces shadowing over a $C^\infty$-manifold with corners are constructed from gluing local noncommutative crepant resolutions of Gorenstein isolated singularities. Dynamical D-branes on such noncommutative target-spaces are realized as maps/morphisms from an Azumaya manifold with a fundamental module with a connection $\nabla$ thereto. The notion of $\nabla$-adjusted kinetic energy for such a map is given via the basic noncommutative differential calculus developed earlier in [L-Y4] (D(11.1)). This provides an action functional for dynamical D-branes on such noncommutative spaces in parallel to the Polyakov action functional for fundamental bosonic strings on a commutative target-space. This sets up a basic stage to begin with for the realization of the D-brane fantasy on Noncommutative Mirror Symmetry. Questions beyond are sampled along the discussion.