On Non Commutative Calabi-Yau Hypersurfaces

Using the algebraic geometry method of Berenstein et al (hep-th/0005087), we reconsider the derivation of the non commutative quintic algebra ${\mathcal{A}}_{nc}(5)$ and derive new representations by choosing different sets of Calabi-Yau charges ${C_{i}^{a}}$. Next we extend these results to higher $d$ complex dimension non commutative Calabi-Yau hypersurface algebras ${\mathcal{A}}_{nc}(d+2)$. We derive and solve the set of constraint eqs carrying the non commutative structure in terms of Calabi-Yau charges and discrete torsion. Finally we construct the representations of ${\mathcal{A}}_{nc}(d+2)$ preserving manifestly the Calabi-Yau condition $\sum_{i}C_{i}^{a}=0$ and give comments on the non commutative subalgebras.Comment: 16 pages, Latex. One more subsection on fractional branes, one reference and minor changes are added. To appear in Phy. Let.

Mirror Symmetry and Landau Ginzburg Calabi-Yau Superpotentials in F-theory Compactifications

We study Landau Ginzburg (LG) theories mirror to 2D N=2 gauged linear sigma models on toric Calabi-Yau manifolds. We derive and solve new constraint equations for Landau Ginzburg elliptic Calabi-Yau superpotentials, depending on the physical data of dual linear sigma models. In Calabi-Yau threefolds case, we consider two examples. First, we give the mirror symmetry of the canonical line bundle over the Hirzebruch surfaces $\bf F_n$. Second, we find a special geometry with the affine so(8) Lie algebra toric data extending the geometry of elliptically fibered K3. This geometry leads to a pure N=1 six dimensional SO(8) gauge model from the F-theory compactification. For Calabi-Yau fourfolds, we give a new algebraic realization for ADE hypersurfaces.Comment: 27 pages, latex. To appear in Journal of Physics A: Mathematical and Genera

On ADE Quiver Models and F-Theory Compactification

Based on mirror symmetry, we discuss geometric engineering of N=1 ADE quiver models from F-theory compactifications on elliptic K3 surfaces fibered over certain four-dimensional base spaces. The latter are constructed as intersecting 4-cycles according to ADE Dynkin diagrams, thereby mimicking the construction of Calabi-Yau threefolds used in geometric engineering in type II superstring theory. Matter is incorporated by considering D7-branes wrapping these 4-cycles. Using a geometric procedure referred to as folding, we discuss how the corresponding physics can be converted into a scenario with D5-branes wrapping 2-cycles of ALE spaces.Comment: 21 pages, Latex, minor change

Geometric Engineering of N=2 CFT_{4}s based on Indefinite Singularities: Hyperbolic Case

Using Katz, Klemm and Vafa geometric engineering method of $\mathcal{N}=2$ supersymmetric QFT$_{4}$s and results on the classification of generalized Cartan matrices of Kac-Moody (KM) algebras, we study the un-explored class of $\mathcal{N}=2$ CFT$_{4}$s based on \textit{indefinite} singularities. We show that the vanishing condition for the general expression of holomorphic beta function of $\mathcal{N}=2$ quiver gauge QFT$_{4}$s coincides exactly with the fundamental classification theorem of KM algebras. Explicit solutions are derived for mirror geometries of CY threefolds with \textit{% hyperbolic} singularities.Comment: 23 pages, 4 figures, minor change

Classification of N=2 supersymmetric CFT_{4}s: Indefinite Series

Using geometric engineering method of 4D $\mathcal{N}=2$ quiver gauge theories and results on the classification of Kac-Moody (KM) algebras, we show on explicit examples that there exist three sectors of $\mathcal{N}=2$ infrared CFT$_{4}$s. Since the geometric engineering of these CFT$_{4}$s involve type II strings on K3 fibered CY3 singularities, we conjecture the existence of three kinds of singular complex surfaces containing, in addition to the two standard classes, a third indefinite set. To illustrate this hypothesis, we give explicit examples of K3 surfaces with H$_{3}^{4}$ and E$_{10}$ hyperbolic singularities. We also derive a hierarchy of indefinite complex algebraic geometries based on affine $A_{r}$ and T$$ algebras going beyond the hyperbolic subset. Such hierarchical surfaces have a remarkable signature that is manifested by the presence of poles.Comment: 12 pages, 2 figure