Invertible Topological Field Theories

Understand

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.