Quasi-morphisms on contactomorphism groups and Grassmannians of 2-planes

We construct a natural prequantization space over a monotone product of a toric manifold and an arbitrary number of complex Grassmannians of 2-planes in even-dimensional complex spaces, and prove that the universal cover of the identity component of the contactomorphism group of its total space carries a nonzero homogeneous quasi-morphism. The construction uses Givental’s nonlinear Maslov index and a reduction theorem for quasi-morphisms on contactomorphism groups previously established together with M. S. Borman. We explore applications to metrics on this group and to symplectic and contact rigidity. In particular we obtain a new proof that the quaternionic projective space HP^n-1, naturally embedded in the Grassmannian G2(C2n) as a Lagrangian, cannot be displaced from the real part G2(R2n) by a Hamiltonian isotopy.