The Yang-Mills functional and Laplace's equation on quantum Heisenberg manifolds.

摘要

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 &cubl0;Dc,&plank;mn&cubr0; ℏ∈R . We describe Grassmannian connection and its curvature on a projective module xiinfinity over the noncommutative C*-algebra, &parl0;Dc,&plank;mn&parr0; infinity , 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 EndDc,&plank;mn (xi) 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 xiinfinity 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.