Persistence of automorphisms under the II* specialization

Prove that the automorphisms constructed for the dP9 surface with an I8 elliptic fiber persist when the complex structure is tuned back to the singular geometry with a II* fiber.

Background

The geometric construction uses a special complex-structure deformation in which the II* fiber associated with the E8 flavor symmetry is deformed to an I8 fiber. In this deformed setting, the required automorphisms can be explicitly constructed using Mordell–Weil translations and Picard–Lefschetz transformations.

The validity of the construction for the original E-string geometry requires these automorphisms to survive the specialization back to the II* configuration. The paper presents this persistence as an assumption motivated by physics and calls for a mathematical theory of automorphisms of partially resolved elliptic fibrations.

References

We will here derive the automorphisms in a particular choice of complex structure, where the $II*$ fiber is deformed to an $I_8$, and then conjecture that these automorphisms survive the limit back to the original geometry.

— $G_2$-Manifolds from 4d $\mathcal{N}=1$ Quivers  (2608.21238 - Braun et al., 21 Aug 2026) in Section 3, subsection 'The dP9 Surface S'; Section 'Discussion and Outlook', paragraph 'Future questions'