In this thesis, we discuss the Yang-Mills functional and its critical points on quantum Heisenberg manifolds using the noncommutative geometrical method developed by Alain Connes. Quantum Heisenberg manifolds are invented by Marc Rieffel, which are the deformation quantizations of Heisenberg manifolds, denoted by [special characters omitted]. We describe Grassmannian connection and its curvature on a projective module Ξ∞ over the noncommutative C*-algebra, [special characters omitted], and produce a specific element R in this projective module that determines both a non-trivial Rieffel projection and the curvature of the corresponding Grassmannian connection. Also, we will introduce the notion of multiplication-type elements of [special characters omitted](Ξ) in order to find a set of critical points of the Yang-Mills functional on quantum Heisenberg manifolds. In our main result, we construct a certain family of connections on Ξ∞ that give rise to critical points of the Yang-Mills functional, using a multiplication-type operator. Moreover we show that this set of solutions can be described by a set of solutions to Laplace's equation on quantum Heisenberg manifolds.
|Advisor:||Packer, Judith A.|
|Commitee:||Gorokhovsky, Alexander, Grunwald, Dirk, Gustafson, Karl, Ramsay, Arlan|
|School:||University of Colorado at Boulder|
|School Location:||United States -- Colorado|
|Source:||DAI-B 70/04, Dissertation Abstracts International|
|Keywords:||Grassmannian connections, Laplace's equation, Quantum Heisenberg manifolds, Yang-Mills functional|
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