system from the quantum connection of $X$ under an assumption (Assumption \[assumption of exponential type\]). The construction has the following steps.

1.  \[step 1\] Apply the Riemann-Hilbert correspondence [@mallac], [@sabint] for the quantum connection (around $z=0$), we get a local system on $S^1$ with a filtration called a Stokes filtration. We use the assumption here.

2.  \[step 2\] A reformulation of Stokes data by Hertling and Sabbah [@herexa] gives a mutation system (we also use the Poincaré pairing on ${H^\bullet(X)}$). Here, we need to fix a real number $\theta_\circ$ with a genericity condition. We give this reformulation in §2.

3.  \[step 3\] By using the fundamental solution at $z=\infty$, the vector space $V$ with a pairing ${[\cdot,\cdot\rangle}$ underlying the mutation system constructed in (\[step 2\]) is identified with $(H^\bullet(X),[\cdot,\cdot)_X)$.

The resulting mutation system is called a A-mutation system. Each step is closely related to the construction of asymptotic classes in the semisimple case. Although the resulting mutation system depends on the choice of $\theta_\circ$, they are all equivalent by mutations.

B-mutation systems
------------------

We can construct a mutation system for a semiorthogonal decomposition of $D^b(X)$ (see, for example, [@kuzhoc Definition 2.3] for the definition of semiorthogonal decompositions). Put $\mathrm{HH}_k(X):=\bigoplus_{p-q=k}H^q(X,\Omega_X^p)$ and 