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

\( \newcommand{\Sha}{\text{III}} \)
    1. 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.