$\mathrm{HH}_\bullet(X):=\bigoplus_{k\in{\mathbb{Z}}}\mathrm{HH}_k(X)$. We have a pairing $[\cdot,\cdot\rangle_X$ defined by $$[\alpha,\beta\rangle_X:=\frac{1}{(2\pi\i)^{\dim X}}\int_X e^{\pi\i \rho_{\!_X}}W(\alpha)\cup\beta,$$ where $W(\alpha)=\i^{p+q}\alpha$ for $\alpha\in H^q(X,\Omega_X^p)$ (cf. [@calmuk1], [@pollef], [@ramrel], [@rammuk], [@shkhir]). Let $D^b(X)=\langle \mathcal{A}_1,\dots,\mathcal{A}_m \rangle$ be a semiorthogonal decomposition. Then, by using a theorem of Kuznetsov [@kuzhoc], [@kuzbas], we can define the subspace $\mathrm{HH}_\bullet(\mathcal{A}_i)$ of $\mathrm{HH}_\bullet(X)$, which we call the Hochschild homology of $\mathcal{A}_i$ (cf. [@calmuk1], [@kuzhoc], [@polche], [@shkhir]). They satisfies $\mathrm{HH}_\bullet(X)=\bigoplus_i\mathrm{HH}_\bullet (\mathcal{A}_i)$. Roughly speaking, this decomposition defines a mutation system, which we call a B-mutation system. The braid group action on the set of semiorthogonal decompositions of $D^b(X)$ is compatible with the mutation on the B-mutation systems.

The formulation of the analogue and the main theorem {#intro main theorem}
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Using the Gamma class, we define an isomorphism $\Gamma:\mathrm{HH}_\bullet(X){\xrightarrow{\sim}}H^\bullet(X)$ with $[\alpha,\beta\rangle_X=[\Gamma(\alpha),\Gamma(\beta))_X$. We define an analogue of Dubrovin’s conjecture by the existence of a semiorthogonal decomposition such that the B-mutation system for the semiorthogonal decomposition is isomorphic to the A-mutation system via $\Gamma$. We call the analogue \`\`Dubrovin type conjecture”. For more precise, see Definition \[dubconj\]. In the case where the quantum cohomology ring is semisimple, Gamma conjecture II implies this conjecture.

The main result of this paper is to give a class of examples such that this 