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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 [1].

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      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


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          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.