Topological Properties of Gauge Fields on a Lattice

Euclidean solutions to the classical Yang-Mills equations (instantons, merons, etc.) are important for the non-perturbative description of gauge theories. lt is believed that these (topological in nature) solutions should provide a hint of confinement, chiral synmmetry breaking etc. It was realised by 't-Hooft that there are several types of boundary conditions(Twisted Boundary Conditions) for the gauge field which preserve the periodicity of non-local gauge invariant quantities. In this work we present numerical evidence that on a lattice and using Twisted Boundary Conditions we can have topological objects with non-integer topological charge (second Chern class). The theory of fibre bundles (Chapter 2) is discussed and emphasis is given on its connection with the topological properties of the Gauge Theories. We introduce the gauge invariants twists etamunu and we define non-Abelian fluxes. Since the Z(N) Group does not act identically on the matter field we have to introduce the flavor twist in addition to the color twist on the Psi-field. We express(Chapter 3) the winding number in terms of the twist transition functions in a finite Euclidean box in the continuum and its implications on the lower bounds of the action are discussed. Various combinations of the etamunus were constructed and applied on the lattice. The only assumption for the incorporation of the twist on the lattice is the periodicity of the plaquette. We evaluated (Chapter 4) the eigenvalues of Gamma5M (M is the Fermion Matrix) both for Wilson and Kogut-Susskind Fermions. The spectrum of the eigenvalues showed that under the introduction of color and flavor twist for the fermion field there is evidence for the restoration of a "form" of the Index Theorem on a Lattice. The eigenvalue problem for specific twisted configurations was studied and arguments are presented for the discrepancy between our numerical results and the number of zero modes obtained via the Index Theorem