A bstract 6d superconformal field theories (SCFTs) are the SCFTs in the highest possible dimension. They can be geometrically engineered in F-theory by compactifying on non-compact elliptic Calabi-Yau manifolds. In this paper we focus on the class of SCFTs whose base geometry is determined by 2 curves intersecting according to ADE Dynkin diagrams and derive the corresponding mirror Calabi-Yau manifold. The mirror geometry is uniquely determined in terms of the mirror curve which has also an interpretation in terms of the Seiberg-Witten curve of the four-dimensional theory arising from torus compactification. Adding the affine node of the ADE quiver to the base geometry, we connect to recent results on SYZ mirror symmetry for the A case and provide a physical interpretation in terms of little string theory. Our results, however, go beyond this case as our construction naturally covers the D and E cases as

We shall consider the following Dirichlet eigenvalue problem on a smooth bounded domain S~ eRn, I where V is a nonnegative function defined on,~. As is well-known, the eigenvalues of problem (1.1) can be interpreted as the energy levels of a particle travelling under an external force field of a potential q in Rn, where and the corresponding eigenfunctions are wave functions of the Schrodinger equation-J~+ qu= lu. Furthermore, the set of eigenvalues {A,} of (1.1) are nonnegative and can be arranged in a nondecreasing order as follows, It is a significant problem to find a lower bound for~, 1 in terms of the geometry of Q. This subject has been studied extensively by many authors. A rather precise bound in the case V== 0 was worked out not only for a bounded domain in but actually valid for a general Riemannian manifold with certain curvature conditions; we refer to [4] for these recent develop-ments. Nevertheless, very little is known about the obvious interesting question of how big the gap is between 2 and l. There are both physical

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.

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 define a notion of stability for chiral ring of four dimensional N= 1 theory by introducing test chiral rings and generalized a maximization. We conjecture that a chiral ring is the chiral ring of a superconformal field theory if and only if it is stable. We then study N= 1 field theory derived from D3 branes probing a three-fold singularity X, and show that the K stability which implies the existence of Ricci-flat conic metric on X is equivalent to the stability of chiral ring of the corresponding field theory.