Four-dimensional Superconformal Field Theory and String Duality

1.Studying the low-energy effective action of four-dimensional N=2supersymmetric gauge theory in the Coulomb branch，we find an intimate relation between the Seiberg-Witten theory and the ADE simple singularity theory。Given spectral curves of the periodic Toda system as Seiberg-Witten curves，we calculate the instanton corrections in the low-energy effective potential.The result is in agreement with that obtained by the microscopic instanton calculus。Our results imply a deep relevance of two-dimensional topological gravity to four-dimensional Seiberg-Witten theory.2。Considering four-dimensional N=1supersymmetric gauge theory with the ADE gauge group coupled to adjoint matters，we derive the low-energy effective superpotential in the confining phase of U(1)photon。As a result，we obtain the complex curves for N=2gauge theories and propose the Seiberg-Witten geometry for N=2E I D26西耶D2gauge theory with massive fundamental hypermultiplets。The relevant manifold is described as a fibration of the ALE space of E I D 26 ii D 2 type。Then we determine the Seiberg-Witten geometry for N=2SO and SU theory with massive spinor and vector matters by breaking the gauge symmetry of N=2E I D26西耶D2 theory with massive fundamental matters.3。We study mass deformations of N=2superconformal theories with ADE global symmetries on a D3-brane in detail。As an extension of this work，elliptic curves for the IIB7-brane configurations realizing the E-type affine Lie algebras are systematically derived from the cubic@equation for a rational elliptic surface。Moreover，we analyze the structure of singularities，Mordell-Weil lattices and torsions of a rational elliptic surface using string junctions in the background of 12IIB7-branes，and find complete agreement with the classification of the Mordell-Weil lattices。

1. Studying the low-energy effective action of four-dimensional N = 2 supersymmetric gauge theory in the Coulomb branch, we find an intimate relation between the Seiberg-Witten theory and the ADE simple singularity theory. Given spectral curves of the periodic Toda system as Seiberg-Witten curves, we calculate the instanton corrections in the low-energy effective potential. The result is in agreement with that obtained by the microscopic instanton calculus. Our results imply a deep relevance of two-dimensional topological gravity to four-dimensional Seiberg-Witten theory.2. Considering four-dimensional N = 1 supersymmetric gauge theory with the ADE gauge group coupled to adjoint matters, we derive the low-energy effective superpotential in the confining phase of U (1) photon. As a result, we obtain the complex curves for N = 2 gauge theories and propose the Seiberg-Witten geometry for N = 2 EィイD26ィエD2 gauge theory with massive fundamental hypermultiplets. The relevant manifold is described as a fibration of the ALE space of EィイD26ィエD2 type. Then we determine the Seiberg-Witten geometry for N = 2 SO and SU theory with massive spinor and vector matters by breaking the gauge symmetry of N = 2 EィイD26ィエD2 theory with massive fundamental matters.3. We study mass deformations of N = 2 superconformal theories with ADE global symmetries on a D3-brane in detail. As an extension of this work, elliptic curves for the IIB 7-brane configurations realizing the E-type affine Lie algebras are systematically derived from the cubic equation for a rational elliptic surface. Moreover, we analyze the structure of singularities, Mordell-Weil lattices and torsions of a rational elliptic surface using string junctions in the background of 12 IIB 7-branes, and find complete agreement with the classification of the Mordell-Weil lattices.