The Geometric Schwinger Effect I: Pair Creation from Modular Curvature of the Bosonic Vacuum

This work presents a geometric reformulation of the Schwinger effect in which non-perturbative pair creation arises from modular curvature rather than external fields. Using the genus-one Legendre family and the framework of Eynard–Orantin topological recursion, the vacuum is equipped with a compact phase and a periodic potential determined entirely by modular data. The resulting barrier scale, wall tension, and thin-wall tunnelling exponents are obtained in closed analytic form from elliptic and modular functions. This paper establishes the geometric foundations of the mechanism and forms Part I of the “Geometric Schwinger Effect’’ programme, setting the stage for the development of the effective bias and vacuum structure explored in subsequent parts.