Let $\mathcal{M}\_g$ be the moduli space of curves of genus $g$. Consider the holomorphic bundle $\mathcal{H}^k\rightarrow\mathcal{M}\_g$ whose fiber over a curve $C\in\mathcal{M}\_g$ is the space of holomorphic $k$-differentials $H^0(C,K\_C^k)$. This is often called the Hodge bundle.


My question is, **is there a natural (flat or not) connection on $\mathcal{H}^k$?**


Even for $k=1$, I don't find any reference by internet search. We do have a Gauss-Mannin connection on the bundle $E\rightarrow\mathcal{M}\_g$ with fibers $E\,\big|\_C=H^1(C,\mathbb{C})=H^{1,0}(C)\oplus H^{0,1}(C)=H^0(C,K\_C)\oplus H^0(C,K\_C)^\*$, which however does not preserve the splitting.



Connections on the Hodge bundle?
The fourth power of the Hodge bundle is isomorphic to the $E\_8$ conformal block bundle at level one, and for that bundle (or rather its projectivization) you have the Hitchin/KZ/WZW connection, which is projectively flat.


