Noncommutative geometry and Yang-Mills theory

In this dissertation I consider some problems concerning applications of noncommutative geometry to theoretical physics. An introduction to noncommutative geometry with an emphasis on noncommutative tori is given in Chapter 2.; In a recent study of M-theory in a paper by A. Connes, M. Douglas and A. Schwarz noncommutative spaces emerged quite naturally. Later it was shown by A. Schwarz that the notion of Morita equivalence is related to physical duality in Yang-Mills theory. The original proof of invariance of Yang-Mills action functional under the duality transformations was done in the framework of C*-algebras. In Chapter 3 of the present paper we give another derivation of the duality in the broader framework of associative algebras and take care of all constant factors. This is described in sections 3.4–3.6. In section 3.7 we give the modified duality relation for Yang-Mills action functionals. In section 3.8 we give the classification of modules over noncommutative tori that admit a constant curvature connection. It is shown that such modules can be characterized by the property that the corresponding K-theory class is a generalized quadratic exponent.; It is important to study the duality induced by Morita equivalence on the level of quantum theory. In supersymmetric field theories information about quantum theory can be most easily extracted from Bogomolny, Prasad, Sommerfeld (BPS) spectrum of the theory. In Sections 4.1–4.3 we derive the supersymmetry algebra of the theory using the Hamiltonian formalism. We calculate the central charges of the supersymmetry algebra taking into account all possible topological terms in the theory. In Section 4.5 we derive the quantization conditions for the central charges. A general formula for BPS energies is obtained in Section 4.6 and the explicit expressions for the cases of 2, 3 and 4-dimensional tori are given in Section 4.7. In some cases these expressions were obtained (partially at the level of conjectures) by others. In this paper we give a detailed derivation of the BPS spectra in full generality. We check in Section 4.8 that BPS energies are invariant under the duality induced by Morita equivalence.; To calculate the degeneracies of BPS states we study in detail the space of classical BPS fields. An explicit description of general classical BPS solutions is given in Section 4.9. We show that the space of solutions preserved by the same set of supersymmetry transformations can be described by means of a special version of an orbifold sigma model. Section 4.10 is devoted to the quantization of this model. Upon quantization we are able to calculate the multiplicities of BPS states on basic modules.