Let $(M,\omega)$ be a symplectic manifold, and $Mp^c(n)=Mp(n)\times\_{\mathbb Z\_2}U(1)$ which $Mp(n)$ here is Metaplectic group which is the double cover of symplectic group. I am looking for a nesessary and sufficient condition which $M$ has [$Mp^c(n)$-structure.](http://arxiv.org/abs/1307.1634)



All symplectic manifolds have $Mp^c$-structures?
There is no obstruction to the existence of $\mathrm{Mp}^c$ structures on $(TM,\omega)$ or more generally any symplectic vector bundle $(E,\omega)\to M$. The set of their equivalence classes identifies with $H^2(M,\mathbf Z)$. This is all in [Rawnsley-Robinson 1989](http://www.ams.org/mathscinet-getitem?mr=1015418), pp. 3, 46, 55 who quote also [Forger-Hess 1979](https://projecteuclid.org/euclid.cmp/1103904723), p. 272.


