Classifying Stable Invertible Topological Field Theories with Spin Structure

An invertible topological field theory is a symmetric monoidal functor from the category of bordisms with a given tangential structure to the Picard subcategory of a target symmetric monoidal category. Generalizing the celebrated Galatius-Madsen-Tillman-Weiss theorem, a theorem of C. Schommer-Pries shows such field theories may be classified via computations in the stable homotopy category, namely the cohomology of connective covers of the Madsen-Tillman spectra associated with the choice of tangential structure. We classify all such extended and partially extended stable field theories with spin structure in dimensions up to n=6. By taking values in a certain universal target spectrum and its connective covers, we broaden the choice of target categories from categories of n-vector spaces to include categories of n-super vector spaces and n-superalgebras, among others. The computations involve use of the Steenrod algebra, as well as what seems to be a novel use of the octahedral axiom of triangulated categories.