Spin structures on Sasakian manifolds and the Kähler analogy

Question

A Sasakian manifold is often said to be the odd dimensional analogue of a Kähler manifold.

Now for a $2n$-dimensional Kähler manifold we know from Atiyah that it is spin exactly if the line bundle $\Omega^{(0,n)}$ admits a square root ${\cal S}$, and a choice of spin structure is equivalent to a choice of holomorphic structure ${\cal S}$. In this case the associated spinor bundle is the tensor product of ${\cal S}$ with the anti-holomorphic complex. Moreover, the associated Dirac operator is the tensor product of $\overline{\partial} + \overline{\partial}^*$ with the $\overline{\partial}$-operator corresponding to the choice of holomorphic structure.

So does the above analogy present any Sasakian versions of these spin geometry to complex geometry dictionary?

Answer 1

Every Sasakian manifold $M$ (of dimension $2k+1$) has a canonical $\mathrm{Spin}^c$ structure, because the cone $\overline{M}$ over $M$ is Kähler and thus has a canonical $\mathrm{Spin}^c$ structure which restricts to $\mathrm{Spin}^c$ structure on $M$.

If $M$ is Einstein, then the cone $\overline{M}$ is Ricci flat and that in turn implies that the auxiliary bundle $\Lambda^{k+1, 0} \overline{M}$ of the canonical $\mathrm{Spin}^c$ structure on the cone $\overline{M}$ is flat. Now $\mathrm{Spin}^c$ structures on simply connected manifolds with trivial auxiliary bundle is canonically identified with a spin strucutre.

For more details see Moroianu, Andrei: Parallel and Killing spinors on $Spin^c$ manifolds, Comm. Math. Phys. 187 (1997), no. 2, 417–427