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 characterize the geometric moduli of non-Khler manifolds with torsion. Heterotic supersymmetric flux compactifications require that the six-dimensional internal manifold be balanced, the gauge bundle be Hermitian YangMills, and also the anomaly cancellation be satisfied. We perform the linearized variation of these constraints to derive the defining equations for the local moduli. We explicitly determine the metric deformations of the smooth flux solution corresponding to a torus bundle over <i>K</i>3.

Texture mapping and texture synthesis are two popular methods for the decoration of surfaces with visual detail. Here, an existing challenge is to preserve, or at least balance, two competing metrics: scale and angle. In this paper we present two methods for this, both based on global conformal parameterization. First, we describe a texture synthesis algorithm for surfaces with arbitrary topology. By using the conformal parameterization, the 3D surface texture synthesis problem can be converted to a 2D image synthesis problem, which is more intuitive, easier, and conceptually simpler. While the conformality of the parameterization naturally preserves the angles of the texture, in this paper we provide a multi-scale technique to also maintain a more uniform area scaling factor. A second novel contribution is to employ the global parameterization to simultaneously preserve orthogonality and size in texture

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.