A Spin TQFT Related to the Ising Categories

Most interesting 3d Topological Quantum Field Theories (TQFTs) are constructed by starting with algebraic data, usually in the form of some kind of category. This category typically comes from an area of mathematics different from 3-manifold topology, and its topological nature can be hard to understand. This dissertation reverses the process, at least in one simple example, by constructing a Spin TQFT from pure topology and then uncovering some interesting categories.The topology used to construct the Spin TQFT is entirely classical. If $(M, s)$ is a closed Spin 3-manifold, then an embedded surface $\Sigma\subset M$ inherits a $\Pin^-$ structure, $s|_{\Sigma}$, from $(M, s)$. If $\Sigma$ and $\Sigma'$ represent the same class in $H_2(M;\bbZ/2)$, then $s|_{\Sigma}$ and $s|_{\Sigma'}$ are isomorphic. If $t$ is a $\Pin^-$ structure on $\Sigma$, there is a classical invariant of the isomorphism type of $(\Sigma, t)$, denoted $\beta(\Sigma, t)$, that is an eight root of unity. One can therefore form the Spin 3-manifold invariant\[ Z_3(M, s) := \frac{1}{2^{b_0(M)}} \sum_{[\Sigma]\in H_2(M;\bbZ/2)} \beta(\Sigma, s|_{\Sigma}).\]It turns out that this invariant fits into a Spin TQFT. The detailed construction of this TQFT is the subject of this dissertation.By a theorem of Kirby and Melvin, $Z_3$ is very much related to the Ising categories. In extending the TQFT for $Z_3$, one encounters a category (associated with the bounding Spin circle) which has most of the same properties as the Ising categories. One also encounters a category (associated with the interval) from which $Z_3$ can be reconstructed in the style of Turaev-Viro and Barrett-Westbury. Because $Z_3$ is a Spin TQFT, these categories are linear over super vector spaces. In fact, they are realized as the module categories of certain explicit super algebras