dub}
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Let $X$ be a Fano manifold. B. Dubrovin predicted some relations between the derived category $D^b(X)$ of bounded complexes of coherent sheaves on $X$ and the quantum cohomology ring of $X$. More precisely, he conjectured that $D^b(X)$ has a full exceptional collection if and only if the quantum cohomology ring of $X$ is (generically) semisimple. This conjecture is proved for many examples [@baysem], [@chaqua3], [@cioont], [@galdub], [@herupd], [@tabfro], etc.

In the case where these two conjecturally equivalent conditions are hold, he also predicted a relationship between full exceptional collections of $D^b(X)$ and the quantum connection associated to the quantum cup product. To be more precise, let us roughly recall the definition of the quantum connection. For simplicity, we take the quantum parameter $\tau$ to be $0$.

Let $\mathcal{H}_X:={H^\bullet(X)}\otimes \mathcal{O}_{{\mathbb{C}}_z}$ be the trivial (${\mathbb{Z}}/2{\mathbb{Z}}$-graded) $\mathcal{O}_{{\mathbb{C}}_z}$-module. Here, ${\mathbb{C}}_z$ denotes the complex plane whose coordinate function is $z$, and $\mathcal{O}_{{\mathbb{C}}_z}$ denotes the sheaf of algebraic functions on ${\mathbb{C}}_z$. The quantum connection $\nabla:\mathcal{H}_X\to\mathcal{H}_X\otimes\Omega^1_{{\mathbb{C}}_z}(\log \{0\})\otimes\mathcal{O}_{{\mathbb{C}}_z}(\{0\})$ is defined by $$\begin{aligned}
              \nabla:={d}-\Big{(}\frac{1}{z}(c_1(X)*_0)-{\mu}\Big{)}\frac{dz}{z},
             \end{aligned}$$ where $c_1(X)$ is the first Chern class of $X$, $*_0$ is the quantum cup product with respect to the quantum parameter $\tau=0$, and $\mu$ is the grading operator (see Definition \[def of 