Supersymmetric Gauge Theory and Superstring Geometry

这项研究的目的是通过各种数学和物理考虑来研究超弦理论和超对称规范理论（超弦几何）的模空间。在1998年（学年），我们导出了一些四维超对称规范理论的WDVV方程，推广了相关的Donaldson-Witten不变量，并确定了例外型规范群情形下的Seiberg-Witten几何。1999年，我们得到了具有异常型味对称的四维超对称规范理论的低能有效理论，并阐明了它与五维理论的关系。我们还找到了一个系统的方法来分析Calabi-Yau流形的轨道奇点上的D-膜，以及有理曲线个数的生成函数。 ...更多信息 在有理椭圆曲面中。2000年，我们证明了Calabi-Yau流形中某些弦结与del Pezzo曲面上的D-膜之间的对偶关系，并详细分析了与ADE型退化Calabi-Yau流形相关的超共形代数。2001年，我们在E_8对称的六维非临界弦理论中得到了四维紧化模型的前势和Seiberg-Witten曲线。我们还导出了AdS_3连续傅里叶变换的算子乘积展开式和非对易几何上的孤子散射的严格性质，从而得到了超弦几何和弦对偶的许多有用的严格结果。少

The purpose of this research was to study superstring theory and moduli spaces of supersymmetric gauge theories ('superstring geometry') by various mathematical and physical considerations. and verify the superstring duality in quantitative and rigorous manners.In 1998 (academic year), we derived the WDVV equations for some four-dimensional supersymmetric gauge theories, generalized related Donaldson-Witten invariants, and determined the Seiberg-Witten geometries in the cases with the gauge groups of the exceptional type. A rigorous description of the moduli spaces was also obtained by embedding them into M-theory.In 1999, we obtained the low-energy effective theories of the four-dimensional supersymmetric gauge theories with the exceptional-type flavor symmetries, and clarified its relation to five-dimensional theories. We also found a systematic method for analyzing D-branes on orbifold singularities of Calabi-Yau manifolds, and the generating function for the number of rational curv … More es in rational elliptic surfaces. Moreover, we systematically constructed the superconformal algebras in the AdS/CFT correspondence, which is related to the string duality.In 2000, we showed the dualities between some string junctions and the D-branes on del Pezzo surfaces in Calabi-Yau manifolds, and analyzed in detail the superconformal algebras related to the degenerated Calabi-Yau manifolds of the ADE type. We also obtained the exact correlation functions of the CFT with the AdS_3 target space and, using the AdS/CFT correspondence, clarified properties of the three-dimensional black hole and field theories on non-commutative geometries.In 2001, we found the prepotentials and Seiberg-Witten curves for the four-dimensional compactified models obtained from the six-dimensional non-critical string theory with the E_8 symmetry. We also derived rigorous properties about the operator product expansion for the AdS_3 CFT and the soliton scattering on non-commutative geometries.In this way, we obtained many useful and rigorous results about the superstring geometry and the string duality. Less