5-brane Webs And Seiberg Witten Curves In 5d Gauge Theories

5d supersymmetric gauge theories have interesting and rich physics.Many 5d supersymmetric gauge theories become conformal at infinite coupling.There are intriguing phenomena such as global symmetry enhancement and various dualities at the UV fixed point.Such 5d superconformal theories can be engineered from M-theory or Type ?B string theory.In this thesis,we will explore 5d gauge theories using 5-brane webs constructed by 5-branes in Type ?B string theory and study M-theory configurations called Seiberg-Witten curves.The main contents are as follows:(1)Based on the application of 5-brane webs,the corresponding calculation programs are designed for two directions of 5-brane webs,SW curve calculation and partition function computation.On the one hand,these two programs simplifies the procedure of calculation,on the other hand,some novel algorithms are proposed for the program implementation of graphic calculation and symbol calculation,which solves the graphic structure analysis and symbols formula derivation and so on.(2)The properties of 5-brane webs with orientifolds are studied in this thesis,on the one hand,the monopole tensions of Sp(2),SO(6)and SO(7)theories are calculated and compared with the geometric parameters of corresponding 5-brane webs.It is concluded that the monopole tensions correspond to the combination of different areas in the 5-brane webs according to different theories.On the other hand,two kinds of 5-brane webs with orientifolds in SU(4)theory are derived by using higgsing method and the transition rules in the 5-brane webs.By calculating the geometric parameters of the two new 5-brane configurations and comparing with the monopole tension of SU(4)theory,it is verified that the two new 5-brane configurations correspond to the SU(4)theory with Chen-Simons level 0.The partition function of SU(4)theory obtained from 5-brane configuration with orientifolds is consistent with that obtained from 5-brane configuration without orientifolds.(2)We study the duality of two dual 5d gauge theories after they are compact to 4d,As a specific example,we consider two 5d gauge theories which are UV-dual to each other Sp(2)theory and SU(3)theory with Chern-Simons level 2,these theories have 6 hypermultiplets in the fundamental representation(flavors).We compute 5d Seiberg-Witten curves of Sp(2)theory from the dual toric-like diagram.According to the duality,the 5d Seiberg-Witten curve of SU(3)theory is obtained through the corresponding parameter transformation.Compacting the 5d Seiberg Witten curve into a cycle give rise to 4d Seiberg-Witten curve.The analysis of these two kinds of 4d curves shows that the 5d UV duality does not hold in 4d,we give an account why two UV-dual theories in 5d are no longer dual in 4d.