Invertible Topological Field Theories

A $d$-dimensional invertible topological field theory is a functor from the symmetric monoidal $(\infty,n)$-category of $d$-bordisms (embedded into $\mathbb{R}^\infty$ and equipped with a tangential $(X,\xi)$-structure) which lands in the Picard subcategory of the target symmetric monoidal $(\infty,n)$-category. We classify these field theories in terms of the cohomology of the $(n-d)$-connective cover of the Madsen-Tillmann spectrum. This is accomplished by identifying the classifying space of the $(\infty,n)$-category of bordisms with $\Omega^{\infty-n}MT\xi$ as an $E_\infty$-spaces. This generalizes the celebrated result of Galatius-Madsen-Tillmann-Weiss in the case $n=1$, and of Bokstedt-Madsen in the $n$-uple case. We also obtain results for the $(\infty,n)$-category of $d$-bordisms embedding into a fixed ambient manifold $M$, generalizing results of Randal-Williams in the case $n=1$. We give two applications: (1) We completely compute all extended and partially extended invertible TFTs of dimension $d \leq 4$ with target a certain category of $n$-vector spaces (for $n \leq 4$), and (2) we use this to give a negative answer to a question raised by Gilmer and Masbaum.