Let $M$ be a compact symplectic manifold equipped with a line bundle $\mathcal L$ with curvature $\omega$. Denote by $H^0(M,\mathcal L)$ the space of smooth global sections of $\mathcal L$ (this space is infinite-dimensional). Let $N$ parameterize complex structures on $M$ compatible with $\omega$. For $J\in N$, we may consider the space $\mathcal H^0_J(M,\mathcal L)\subseteq H^0(M,\mathcal L)$, the finite-dimensional subspace of $J$-holomorphic sections of $\mathcal L$. The Hitchin connection is a natural flat connection on $\mathcal H^0_J(M,\mathcal L)$ considered as a bundle over $N$.

The same construction of course applies to $\mathcal L^{\otimes k}$ for any $k\geq 1$.

Suppose we have (over some small open set in $N$) two flat sections of $\mathcal H^0_J(M,\mathcal L^{\otimes a})$ and $\mathcal H^0_J(M,\mathcal L^{\otimes b})$. Clearly their tensor product is a section of $\mathcal H^0_J(M,\mathcal L^{\otimes(a+b)})$. Is it flat?

I am mainly interested in the case where $M$ is the character variety of a compact Riemann surface.


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