$\mathbb{P}^n$-functors and cyclic covers

I will give and explain the definition of a (non-split) $\mathbb{P}^n$-functor, prove that its P-twist is unique, and give our first ever example of such functor using the geometry of a cyclic $(n+1)$-to-$1$ cover ramified in a divisor. This is a joint work with Rina Anno (Kansas).