Title:

Spacetime topology, conserved charges and the splitting of classical multiplets

The starting point of our work is the examination of topological extensions of Noether charge algebras carried by Dpbranes in a flat superMinkowski space, this latter regarded as a solution to D = 10 type IIA supergravity. After examining under which circumstances the algebra of extended charges actually closes, we consider the problem when the Lagrangian, in addition to the field multiplet living on the target space, also contains an Abelian gauge field on the worldvolume which is forced to transform under the symmetry group of the target space in order to keep the kinetic part of the Lagrangian invariant. We show that algebra extensions arise both from these gauge fields, as well as from WessZumino terms which exhibit semiinvariance under the symmetry group. The extended pieces due to the gauge field contain the canonical momenta of the gauge potential. We then apply this general reasoning to the superMinkowski background mentioned above. Consistency checks with supergravity lead to generalized Γmatrix identities which are shown to be valid. We finally present the precise form of the extended algebra in this background. The above analysis leads to the question of how supergravity vacua with nontrivial topology can be constructed from simply connected spacetimes, and how the precise form of their isometry group looks like. We study this problem for the case of the compactification of the bosonic part of a superMinkowski spacetime over a lightlike lattice. We work out the precise form of the isometry group of the compactified space, and find that this group can be extended to a semigroup by discrete transformations which are invertible on the original spacetime, but have no inverse on the compactification. The associated Lie algebra is shown to be a direct sum of an Abelian Lie algebra with a centrally extended Galilean algebra. The third chapter of the thesis is inspired by the question on which kind of space an extended algebra such as the one encountered in the first chapter might be possibly realized. We argue that, in a classical context, such a space can be constructed using a certain partition of a phase space of a physical system which inherits the nontrivial topology from the underlying compactified spacetime. This partition is accomplished by employing a socalled moment map, which does not exist as a globally defined function on the phase space, unless the latter is simply connected. The third chapter therefore deals with the question of how one can generalize the notion of a global moment map to what we term a "local moment map". We show that the latter can be defined using a global moment map on a simple connected symplectic covering space. The local moment map then becomes multivalued on account of the nontrivial topology of the phase space, which gives rise to a phenomenon which generalizes the usual KaluzaKlein tower of states, and which we term the "Splitting of Classical Multiplets".
