Examples of Riemannian Functorial Quantum Field Theory

Following Atiyah, Segal, Kontsevich and others, a $d$-dimensional Riemannian Functorial Quantum Field Theory $E$ assigns to a closed $d-1$ dimensional oriented Riemannian manifold a Hilbert space $E(Y)$ and to a bordism $Sigma$ from $Y_1$ to $Y_2$(which is a compact oriented Riemannian manifold with $partialSigma=Y_2sqcup overline{Y_1}$) a Hilbert-Schmidt operator $E(Sigma):E(Y_1) o E(Y_2)$ so that gluing bordisms corresponds to composing the associated operators. If we forget the Riemannian structure on the $Y$'s and the bordisms, then there are many examples which are known has Topological Quantum Field Theories. In 2007, Douglas Pickrell constructed a family of examples of $2$-dimensional theory. In this dissertation, we construct examples of $d$-dimensional theory when $d$ is even.