Symmetry-Protected Topological Relationship Between Group SU(2)×U(1) And SU(3) In Two Dimension

Gauge field theory,as a theory based on the constant of physical equations under local symmetrical transformation,can establish a standard model of unified strong interaction,weak interaction and electromagnetic interaction based on the existing knowledge system.The basic group of the standard model is SU(3)×SU(2)×U(1),which can be divided into the group SU(2)×U(1)and the group SU(3)that are used to describe the unified theory of electroweak and quantum chromodynamics,respectively.This thesis uses the nonlinear? model to study the symmetric-protected topological phases of the two dimensions of the groups SU(2)×U(1)and SU(3)for the first time.Besides,it also studies the mathematical structure of the basic matrix of the two groups and gives the mapping relationship between them.The research work and methods as follows:(1)The choice of research model.Gauge field theory will encounter the breaking effect caused by the introduction of non-commutative gauge field in specific research calculations.The symmetry-protected topological phase is a short-range excitation quantum phase with a gap under the action of the symmetry group,which is very suitable for dealing with the problem of broken.The symmetric-protected topological phase model of the two dimensions can be represented by a nonlinear ? model,and then the corresponding detection field is introduced,and the resulting action equation can be processed by Chern-Simons norm theory.The physical properties of the original action equation at a fixed point can be expressed by the effective action,and then by deriving the effective action with respect to the component of the detection field,we can get the response current density equation corresponding to each component.Through the research on these equations,we can explore the physical properties in the original model.(2)Calculate the symmetric-protected topological phase of the groups SU(2)×U(1)and SU(3)in two dimensions.From a mathematical point of view,the groups SU(2)and SU(3)are both non-commutative unitary groups,and the group U(1)acts as a one-dimensional commutative group on the group SU(2),which makes SU(2)×U(1)and SU(3)can be regarded as a group with three degrees.Then we started from the perspective of algebraic geometry and calculated the group cohomology of the two groups separately,and found that their two-dimensional group cohomology are both Z,so they can be given nonlinear ? models and discussed by the non-commutative Chern-Simons gauge theory.(3)Research on the mapping relationship between groups SU(2)×U(1)and SU(3).In order to analyze the relationship between groups SU(2)×U(1)and SU(3)from the most basic level,we started with their basic composition matrix and discovered a subtle connection between them.Then we give three feasible mapping relations between the groups SU(2)and SU(3),and introduce the rotation of the group U(1)on the geometric level and the group representation level.Through reasonable calculation and analysis,we can connect the groups SU(2)×U(1)and SU(3)through this mapping relationship.The findings of the research work are as follows:(1)The calculation results obtained according to the two-dimensional cohomology of groups SU(2)×U(1)and SU(3)are same,which shows that their symmetric-protected topological phase has an action equation in nonlinear ? model and an effective action equation defined by Chern-Simons gauge theory are very similar.After deriving the action equation by probe field components,we can find that their response current dense with the same structure that can be used to describe quantum Hall effect,and the quantization parameters are equal.(2)The subsets of the fundamental matrix of group SU(3)can be regarded as the three-dimensional extension of the fundament matrix of group SU(2),so we established three mapping relations between these two matrices,and let the group U(1)be a rotating action group.Subsequently,we give the definition of group rotation and the conditions for the establishment from the geometric level and the group representation level,respectively.Finally,we found that the mapping relationship between the groups SU(2)×U(1)and SU(3)can be connected by the above method,so it can be seen that these groups have a high degree of similarity in both the mathematical structure and the physical meaning.