Noncommutative minimal model conjecture for Fano varieties

Let $X$ be a smooth Fano variety. Write $\tau\in {\mathrm H}^\bullet(X)$ for a quantum-cohomology parameter, let $\mathscr{S}\subset\mathbf{C}^*$ be a sector, and let $\operatorname{Stab}(X)$ denote the space of stability conditions on $X$. For the Euler operator $\cE_\tau\star_\tau(-)$, write $\sigma(\cE_\tau)$ for its multiset of eigenvalues and $\lvert\sigma(\cE_\tau)\rvert$ for the underlying set. A path $\sigma_{t,\varphi}^\tau=(\cZ_{te^{\mathtt{i}\varphi}}^\tau,\cP_{t,\varphi})$ is called quasi-convergent when it has the limiting behavior specified in the source.

Noncommutative minimal model conjecture. For any smooth Fano variety $X$, there exist $\tau\in {\mathrm H}^\bullet(X)$, a sector $\mathscr{S}\subset\mathbf{C}^*$, $\epsilon>0$, and a map

$$\mathscr{S}\cap\{z\in\mathbf{C}^*:\lvert z\rvert<\epsilon\}\longrightarrow\operatorname{Stab}(X)$$

such that, for an open dense set of $\{\varphi\in\mathbf{R}:\mathbf{R}_{>0}\cdot e^{\mathtt{i}\varphi}\subset\mathscr{S}\}$, there is a quasi-convergent path $\sigma_{t,\varphi}^\tau$ in $\operatorname{Stab}(X)$ defined as $t\to0$. Moreover, the induced semiorthogonal decomposition is

$$\operatorname{D^b_{coh}}(X)=\langle\cD_\lambda:\lambda\in\lvert\sigma(\cE_\tau)\rvert\rangle.$$

The ordering on $\lvert\sigma(\cE_\tau)\rvert$ is given by $\lambda<\mu$ if $\operatorname{Im}(-e^{-\mathtt{i}\varphi}\mu)>\operatorname{Im}(-e^{-\mathtt{i}\varphi}\lambda)$, and limit semistable objects $E\in\cD_\lambda$ satisfy the asymptotic estimates specified in the source.

This conjecture proposes a categorical lift of the A-model mutation system from smooth Fano complete intersections to all smooth Fano varieties, with the semiorthogonal components indexed by the eigenvalues of the quantum Euler operator and arising from quasi-convergent paths in the stability manifold.