String theory has had a profound influence on research in Calabi-Yau spaces over the past twenty-five years. We first briefly mention some of the work in Khler Calabi-Yau manifolds that was influenced by the discovery of mirror symmetry in the late 1980s. We then discuss some of the mathematical motivations behind the recent work on non-Khler Calabi-Yau manifolds, which arise in string compactifications with fluxes. After extending mirror symmetry to non-Khler Calabi-Yau manifolds, we show how this leads to new cohomologies and invariants of non-Khler symplectic manifolds.

We explain how Polchinski's work on D-branes re-read from a noncommutative version of Grothendieck's equivalence of local geometries and function rings gives rise to an intrinsic prototype definition of D-branes (of B-type) as an Azumaya-type noncommutative space. Several originally open-string induced properties of D-branes can be reproduced solely by this intrinsic definition. We study also the moduli space of D0-branes on a commutative target space in this setup. Some of its features resembles gas of D0-branes in Vafa's work.

In this Part II of D (11), we introduce new objects: super- C^ k -schemes and Azumaya super- C^ k -manifolds with a fundamental module (or, synonymously, matrix super- C^ k -manifolds with a fundamental module), and extend the study in D (11.1)([L-Y3], arXiv: 1406.0929 [math. DG]) to define the notion ofdifferentiable maps from an Azumaya/matrix supermanifold with a fundamental module to a real manifold or supermanifold'. This allows us to introduce the notion offermionic D-branes' in two different styles, one parallels Ramond-Neveu-Schwarz fermionic string and the other Green-Schwarz fermionic string. A more detailed discussion on the Higgs mechanism on dynamical D-branes in our setting, taking maps from the D-brane world-volume to the space-time in question and/or sections of the Chan-Paton bundle on the D-brane world-volume as Higgs fields, is also given for the first time in the D-project. Finally note that mathematically string theory begins with the notion of a differentiable map from a string world-sheet (a C^ k -manifold) to a target space-time (a real manifold). In comparison to this, D (11.1) and the current D (11.2) together bring us to the same starting point for studying D-branes in string theory as dynamical objects.

A D5 elliptic fibration is a fibration whose generic fiber is modeled by the complete intersection of two quadric surfaces in P3. They provide simple examples of elliptic fibrations admitting a rich spectrum of singular fibers (not all on the list of Kodaira) without introducing singularities in the total space of the fibration and therefore avoiding a discussion of their resolutions. We study systematically the fiber geometry of such fibrations using Segre symbols and compute several topological invariants.

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