We proved the existence of supersymmetric Hermitian metrics with torsion on a class of non-Kaehler manifolds.

We give close formulas for the counting functions of rational curves on complete intesection Calabi-Yau manifolds in terms of special solutions of generalized hypergeometric differential systems. For the one modulus cases we derive a differential equation for the Mirror map, which can be viewed as a generalization of the Schwarzian equation. We also derive a nonlinear seventh order differential equation which directly governs the Prepotential.

We introduce a new model for elliptic fibrations endowed with a Mordell-Weil group of rank one. We call it a Q _7 (\mathscr {L},\mathscr {S}) model.

This is the first of a series of papers in which we systematically use singularity theory to study four dimensional N= 2 superconformal field theories. Our main focus in this paper is to identify what kind of singularity is needed to define a SCFT. The constraint for a hypersurface singularity has been found by Sharpere and Vafa, and here the complete set of solutions are listed using a related mathematical result of Stephen ST Yau and Yu. We also study other type of singularities such as the complete intersection, quotient of hypersurface singularity by a finite group and non-isolated singularity. We finally conjecture that any three dimensional rational Gorenstein graded isolated singularity should define a N= 2 SCFT. We explain how to extract various interesting physical quantities such as Seiberg-Witten geometry, central charges, exact marginal deformations, BPS quiver, RG flow trajectory, etc from the properties of singularity.

We consider for <i>j</i>=, a spherically symmetric, static system of (2<i>j</i>+1) Dirac particles, each having total angular momentum <i>j</i>. The Dirac particles interact via a classical gravitational and electromagnetic field.