the space of solutions on the sector $ \{z\mid |\arg z-\theta_\circ|<\pi/2+\varepsilon \}$ with sufficiently small $\varepsilon>0$ have a basis $(y_{i})_{i=1}^m$. The basis $(y_i)_{i=1}^m$ is characterized by their asymptotic growth as $z\to 0$, and $y_i$ corresponds to $\exp(c_i/z)$ where $c_i:=\tau_{\theta_\circ}^{-1}(i)$. Since the fundamental solution at infinity identifies this space of solutions with ${H^\bullet(X)}$ (see Proposition \[fundamental solution\] or [@ggi Proposition 2.3.1]), the basis $(y_i)_i$ gives a basis $(A_{ i})_{i=1}^m$ of ${H^\bullet(X)}$. The classes $A_{ i}$ are called asymptotic classes (see [@ggi §4.5] for more precise).

The Gamma class [@iriint], [@kkp1], [@libche] of $X$ is defined by $$\hat\Gamma_X:=\prod_{j=1}^{\dim X}\Gamma(1+\delta_j)$$ where $\delta_1,\dots,\delta_{\dim X}$ are the Chern roots of the tangent bundle of $X$ and $\Gamma(x)$ is the Gamma function. Gamma conjecture II states that there exists a full exceptional collection $E_1,\dots, E_m$ of $D^b(X)$ such that $$A_i=\hat\Gamma_X \mathrm{Ch}(E_i)$$ for all $i$ (under the semisimplicity of the quantum cohomology ring and the existence of at least one full exceptional correction of $D^b(X)$).

This conjecture refines Conjecture \[conject\] in the following sense. Let $[\cdot,\cdot)_X:{H^\bullet(X)}\otimes{H^\bullet(X)}\to{\mathbb{C}}$ be a non-symmetric linear map defined by $$\begin{aligned}
              [\alpha,\beta)_X:=\frac{1}{(2\pi)^{\dim X}}\int_Xe^{\pi\i\mu}e^{-\pi\i\rho_{\!{}_X}}\alpha \cup \beta
             \end{aligned}$$ where $\rho_{\!{}_X}$ is the cup product $c_1(X)\cup$. Then, on the one hand, the Stokes matrix coincides with $([A_i,A_j)_X)_{i, 