On Supergravity on a 3-Torus, Automorphic Scalar Fields in 2-dimensional de Sitter space and Harmonics on Complex Spheres

In the first part of this thesis we study linearization stability conditions in quantum supergravity on a flat 3-torus. Solutions to linearized supergravity on this background space-time can only be extended to solutions of the non-linear theory if they satisfy additional quadratic constraints, called the linearization stability conditions. This situation is well known in linearized gravity. The novel feature in the case of supergravity is the appearance of fermionic linearization stability constraints, in addition to the kind of bosonic constraints which arise already for linearized gravity. We show how to incorporate the fermionic and bosonic linearization stability constraints in the quantum theory and construct a physical space of states by group-averaging. Unlike higher dimensional de Sitter spaces, two dimensional de Sitter space is not simply connected. This allows for the existence of fields which pick up nontrivial phases when making a full rotation of the spatial sections. In the second part of this thesis we study the quantum theory of automorphic complex scalar fields in two dimensional de Sitter space, extending the work of Epstein and Moschella. We define de Sitter invariant vacuum states when corresponding unitary irreducible representations of the universal covering group of SL(2, R) exist. By calculating the two-point functions we show that these states can only be Hadamard if the field is periodic. We also define a class of de Sitter non-invariant Hadamard states for the automorphic theories. In the final part of this thesis we study harmonics on complex spheres. Using Mackey’s tensor product theorem, the harmonics on complex spheres can be used to decompose tensor products of principal series representations of the Lorentz group.