Quantum cohomology and irrationality of Gushel-Mukai fourfolds

We compute the small quantum cohomology of Gushel-Mukai fourfolds. Following [13], our computations imply that the very general ones are not rational. Following [8], and thanks to a suitable deformation of the small quantum cohomology ring, we also deduce that a rational Gushel-Mukai fourfold has the same rational cohomology as some K3 surface.