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?