Group actions on cluster categories from Ginzburg morphisms

Chris Fraser has discovered a natural birational action of the extended affine braid group on $d$ strands on the Grassmannian of $k$-dimensional subspaces in $n$-dimensional space. Here, the integer $d$ is the greatest common divisor of $k$ and $n$ and the action is via cluster transformations. In joint work with Fraser, we have shown how this action lifts to Jensen–King-Su's additive categorification of the Grassmannian. We will explain how it fits into the theory of (relative) Calabi–Yau structures and (relative) Calabi–Yau completions due to Ginzburg, …, Toën, Brav–Dyckerhoff and Yeung.