Four-dimensional Superconformal Field Theory and String Duality

1. Studying the low-energy effective action of four-dimensional N = 2 supersymmetric gauge theory inthe Coulomb branchwe find an intimate relation between the Seiberg-Witten theory and the ADE simple singularityGiven spectral curves of the periodic Toda system as Seiberg-Witten curves，我们calculate the instanton corrections in the低能量效应potential. the result is in与microscopic instanton calculus. Our results imply a deeprelevance of two-dimensional topological gravity to four-dimensional Seiberg-Witten theory.2Considering four-dimensional N = 1 supersymmetric gauge theory with the ADE gauge group coupled toadjoint matters,we derive the low-energy effective superpotential in the confining phase of U (1) photon. As aresult,我们obtain the complex curves for N = 2 gauge theories and propose the Seiberg-Witten geometry for N = 2 gauge theories and propose the Seiberg-Witten geometry for N= 2e D26 D2 gauge theory with massive fundamental hypermultiplets. The relevant manifold isdescribed as a fibration of the ALE space of E D26 D2 type. Then we determine the Seiberg-Wittengeometry for N = 2 SO and SU theory with massive spinor and vector matters by breaking the gaugesymmetry of N = 2e D26 D2 theory with massive fundamental matters.3. We study mass deformationsof N = 2 superconformal theories with ADE global symmetries on a D3-brane in detail. As an extensionof this work，elliptic curves for the IIB 7-brane配置realizing the E-type affine Lie algebras are从cubic equation for a rational elliptic surface. Moreover，we analyze the structure of singularitiesmordel - weil lattices and torsions of a rational elliptic surface using string junctions in thebackground of 12 IIB 7-branes，and find complete agreement with the classification of the mordel - 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.