Supersymmetric Gauge Theory and Superstring Geometry

本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。并以定量和严格的方式验证了超弦对偶性。在1998学年，我们推导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况下的Seiberg-Witten几何。通过将模空间嵌入m理论，得到了模空间的严格描述。1999年，我们获得了具有异常型风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。我们还建立了一种系统的方法来分析Calabi-Yau流形的轨道奇异点上的d膜，以及在有理椭圆曲面上有理曲线个数的生成函数。此外，我们系统地构造了与弦对偶有关的AdS/CFT对应中的超共形代数。在2000年，我们证明了Calabi-Yau流形中del Pezzo曲面上一些弦结与d膜之间的对偶性，并详细分析了ADE型退化Calabi-Yau流形相关的超共形代数。我们还得到了CFT与AdS_3目标空间的精确相关函数，并利用AdS/CFT对应关系澄清了三维黑洞的性质和非交换几何的场论。2001年，我们发现了由六维非临界弦理论得到的具有E_8对称性的四维紧化模型的预势和Seiberg-Witten曲线。我们还得到了AdS_3 CFT算子积展开式和非交换几何上孤子散射的严格性质。通过这种方法，我们得到了关于超弦几何和弦对偶的许多有用而严谨的结果。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