Research On Several Issues In Chern-Simons Theory

Gauge field theory plays an important role in physics,and a good feature of the theory is that it exists soliton inside.The application of soliton is widespread,including the fractional quantum Hall effect,high polymer physics,Biology,Cosmology,high temperature superconductors and so on.Therefore,it is important to study the partial differential equations derived from gauge field theory for solving practical problems.There have recently been a great amount of activities in the study of Chern-Simons models in gauge field theory.In physics,Chern-Simons models can be widely used in particle physics,high temperature superconductivity,quantum Hall effect and other fields.However,it is difficult to deal with Chern-Simons term mathematically.Since the discovery of the self-dual structure in the Abelian Chern-Simons model in 1990,which makes the theory simplified and further developed,there have been a wide range of explorations on the reduction of numerous Chern-Simons models.In this paper we establish the existence results of vortex solutions for different Chern-Simons models in gauge field theory.In Chapter 1,we mainly introduce the history of Chern-Simons gauge field theory,and present many important results of partial differential equations in Chern-Simons theory in recent years.Besides,we introduce the defintion of 'tHooft periodic bound-ary.In Chapter 2,we consider a Chern-Simons model with a generic renormalizable potential,this model becomes more complicated due to the existence of coupling pa-rameters that can be varied in the potential function.In order to simplify the problem,we study the existence of the topological vortex solution in symmetric ansatz.The first term in corresponding functionals of the model is difficult to deal with,thus we utilize the results of planar vortex minimizers for the Ginzburg-Landau energy to solve our problems.We also derive some properties of the solutions by analytical technique.In Chapter 3,we study the relativistic self-dual U(1)×U(1)Chern-Simons system.We can find the first order BPS equations by BPS method and simplify them to a nonlinear elliptic system.We consider the equations in the following cases.In the case of full plane,we utilize a minimization approach to establish the existence result and derive the decay estimates of the solutions.While in the case of doubly periodic domain,we use a constrained minimization approach to obtain the existence result under suitable 'tHooft boundary condition.The system admits a solution when the Chern-Simons coupling parameter ?>0 is sufficiently small,while no solution exists when ?>0 is not sufficiently small.In both cases we get the quantized integrals.In Chapter 4,we study the nonrelativistic self-dual Chern-Simons system with an external electromagnetic field.We simplify the first order BPS equations of the model to a coupled nonlinear elliptic system.We prove the existence and uniqueness results of multiple vortices over the full plane and a doubly periodic domain by variational methods.In the doubly periodic situation,the necessary and sufficient conditions for the existence of the unique solution are expressed in terms of the size of the domain and vortex number.We also derive the exponential decay estimates of the solutions in the full plane.Furthermore,we establish the quantized integrals in both cases.