Gravitational instantons are complete hyperkähler four-manifolds with finite topological type — Einstein manifolds with self-dual curvature that serve as the gravitational analogues of Yang-Mills instantons. They come in families classified by their asymptotic geometry: ALE, ALF, ALG, and ALH, depending on how the metric behaves at infinity.
Fredrickson, Mazzeo, Swoboda, and Weiss (arXiv:2603.17020) prove that a 12-parameter family of parabolic SU(2)-Hitchin moduli spaces on the four-punctured sphere are ALG gravitational instantons of type D4. Moreover, they realize all allowable Torelli parameters — the geometric data that uniquely specify the instanton within its deformation class.
This is the first verification of any case of the Modularity Conjecture, which predicts that every ALG gravitational instanton arises as a Hitchin moduli space. The conjecture connects two independently developed mathematical structures: gravitational instantons from differential geometry and gauge theory, and Hitchin systems from algebraic geometry and integrable systems. That one structure produces the other is not forced by any known general principle — it is a deep and largely unexplained correspondence.
The four-punctured sphere is the simplest Riemann surface where the parabolic Hitchin system has enough parameters to produce a rich moduli space. The 12 parameters correspond to the parabolic weights at the four punctures, and the Torelli parameters of the resulting gravitational instanton are determined explicitly in terms of these weights.
The result is both a construction and a classification. Every ALG D4 instanton with the right Torelli data is isometric to one of these Hitchin moduli spaces. The gauge-theoretic object is exactly the gravitational object — not an approximation or an analogue, but an isometry.