  We use Bott periodicity to relate previously defined quantum classes to
certain "exotic Chern classes" on $BU$. This provides an interesting
computational and theoretical framework for some Gromov-Witten invariants
connected with cohomological field theories. This framework has applications to
study of higher dimensional, Hamiltonian rigidity aspects of Hofer geometry of
$ \mathbb{CP} ^{n}$, one of which we discuss here.
