标题： 具有八个超荷的秩二理论，第二部分：勒夫谢茨纤维化 显示英文标题

标题： On rank two theories with eight supercharges part II: Lefschetz pencils

摘要： 研究了具有八个超荷的秩二理论的全局Seiberg-Witten（SW）几何。 理论被一般地变形，使得Coulomb分支上只有最简单的$I_1$或$\tilde{I}_1$奇点，这在几何上给出了所谓的Lefschetz纤维丛。局部奇点被证明由映射类群（MCG）的共轭类决定； 然后全局研究归结为关于MCG的问题：a）将奇异纤维的MCG元素分解为Dehn扭转变换的正乘积（这给出$I_1$奇点或$\tilde{I}_1$奇点）；b）用Dehn变换表示单位元的分解。 我们解决了大多数秩二理论的上述两个MCG问题。这些结果对于确定4d SCFTs所有真空的红外物理非常有帮助。 我们的方法是组合性的，许多方面可以被直接推广到高秩理论的研究中。 显示英文摘要

摘要： The global Seiberg-Witten (SW) geometries for rank two theories with eight supercharges are studied. The theory is deformed generically so that there are only simplest $I_1$ or $\tilde{I}_1$ singularities on the Coulomb branch, which then geometrically gives the so-called Lefchetz pencils, The local singularity was shown to be determined by the conjugacy class of mapping class group (MCG); The global study is then reduced to the questions about MCG: a) Find the factorization of the MCG element of the singular fiber into positive products of Dehn twists (which gives the $I_1$ singularity or $\tilde{I}_1$ singularity); b) Find the factorization of identity element in terms of Dehn twists. We solved above two MCG problems for most rank two theories.The results are very helpful in determining IR physics for all vacua of 4d SCFTs. Our approach is combinatorial and many aspects can be straightforwardly generalized to the study of higher rank theory.