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

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.