Topological methods in gauge theory
Abstract
We begin with an overview of the important topological methods used in gauge theory. In the first chapter, we discuss the general structure of fiber bundles and associated mathematical concepts and briefly discuss their application in gauge theory. The second chapter deals with the study of instantons in both gauge and gravity theories. These self-dual solutions are presented and their importance pointed out. This chapter is also a broad introduction to certain topics in gravitational physics. Gravity and gauge theory are unified in Kaluza-Klein theory as discussed in the third chapter. Of particular interest is the physics of the U(1) bundles over non-trivial manifolds. The radius of the fifth dimension is undetermined classically in the Kaluza-Klein theory. Here we describe a mechanism using some topological information to derive the functional form of the radius of the fifth dimension and thus show that it is possible classically to derive expressions for the radius as a consequence of topology. As desired, the behavior of the radius is dependent on the information present in the base metric. Results are computed for three gravitational instantons. Consequences of this mechanism are discussed. Finally, we study the description of instantons in terms of projector valued fields and universal bundles. The results of the previous chapter and this are connected via the study of universal bundles. Projector valued transformations are defined and their consequences touched upon. With the solutions of instantons in this formalism, we show explicitly that there can be solutions which allow for a $Sp(n)$ instanton to be transformed to a $Sp(k)$ instanton, thus showing that there can be interpolations which carry one instanton with a rank n to another characterized by rank k with different topological numbers. The meaning of such constructions is also discussed, namely the possible extension of gauge theory to allow for more complicated transformations other than Lie group transformations and the possibility of establishing a meaning for a zero lagrangian as in topological field theory and related areas.
Degree
Ph.D.
Advisors
Kuo, Purdue University.
Subject Area
Particle physics|Physics|Mathematics
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