## 4. Modular interpretation of $\mathrm{(Q)ALX}^{(*)}$ spaces

\( \newcommand{\Sha}{\text{III}} \)-
### Modularity problem for Sha spaces

Almost all gravitational instantons are known to be realized as gauge-theoretic moduli spaces. The missing case of the so-called "modularity conjecture" corresponds to the $\Sha$ (read as "Sha") spaces [Cherkis19].#### Problem 4.1.

[Sergey A. Cherkis] Prove that the $\Sha_{*}$ ALG-gravitational instantons are mirrors of E-type ALG-gravitational instantons. More concretely,

$\Sha_0$ is a mirror of $E_8$,

$\Sha_1$ is a mirror of $E_7$, and

$\Sha_2$ is a mirror of $E_6$.

Moreover, what are they moduli spaces of? -
### Relation to Torelli classification

#### Problem 4.2.

[Jacques Hurtubise] How does the moduli theoretic classification of $\mathrm{ALG}/{ALG}^{*}$-gravitational instantons fit into their Torelli classification? Also build the dictionary for the missing cases of the modularity conjecture.

Cite this as: *AimPL: Geometry and physics of ALX metrics in gauge theory, available at http://aimpl.org/geomphysalx.
*