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

On Local Calabi-Yau Supermanifolds and Their Mirrors

We use local mirror symmetry to study a class of local Calabi-Yau super-manifolds with bosonic sub-variety V_b having a vanishing first Chern class. Solving the usual super- CY condition, requiring the equality of the total U(1) gauge charges of bosons \Phi_{b} and the ghost like fields \Psi_{f} one \sum_{b}q_{b}=\sum_{f}Q_{f}, as \sum_{b}q_{b}=0 and \sum_{f}Q_{f}=0, several examples are studied and explicit results are given for local A_{r} super-geometries. A comment on purely fermionic super-CY manifolds corresponding to the special case where q_{b}=0, \forall b and \sum_{f}Q_{f}=0 is also made.\bigskipComment: 17 page

On N=1 gauge models from geometric engineering in M-theory

We study geometric engineering of four-dimensional N=1 gauge models from M-theory on a seven-dimensional manifold with G_2 holonomy. The manifold is constructed as a K3 fibration over a three-dimensional base space with ADE geometry. The resulting gauge theory is discussed in the realm of (p,q) webs. We discuss how the anomaly cancellation condition translates into a condition on the associated affine ADE Lie algebras