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.
1 Answer
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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, pp. 3, 46, 55 who quote also Forger-Hess 1979, p. 272.
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$\begingroup$ Thanks, are there any other such Mpc structures , except $spin^c$, $pin$ or, $Mp$, $Ml$, metalinear structures, which $(M,\omega)$ always admit it ? $\endgroup$– user21574May 5, 2014 at 7:21