On the spectrum of some gravitational instantons
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In this thesis we study Dirac operators on the Euclidean Taub-NUT and Schwarzschild spaces coupled to abelian gauge fields, with the aim of computing the zero-modes and bound states. The work is motivated by recently proposed Geometric Models of Matter, where single particles are modelled by 4-manifolds and their quantum numbers realised as topological invariants of the model manifolds. In these models, the spin degrees of freedom are given by the zero-modes of the Dirac operator. In the case of the Taub-NUT manifold coupled to an U(1) gauged eld with selfdual curvature, which is the model for the electron, we are able to obtain explicit expressions for the zero modes of the Dirac operator. We show that they decompose into an irreducible representation of SU(2) and use this to interpret a known index theorem in this geometry first deduced by Pope. We also study the dynamical symmetry of this space in the classical and quantum cases, and show that the gauge eld allows the existence of classical bounded orbits and quantum bound states. We compute scattering cross sections and find a surprising electric-magnetic duality. Using twistor formalism we are able to show that the dynamical symmetry is preserved in the gauged case and that this makes possible to deduce the energy of the quantum bound states in an algebraic manner. We consider the Euclidean Schwarzschild manifold coupled to an U(1) gauge field as a neutron candidate. In this case the zero-modes of the Dirac operator also decompose into an irreducible representation of SU(2). Using the open code SLEIGN2, we compute the spectrum of the Laplace-Beltrami operator acting on scalar fields.