Resurgence of Tritronqu\'ees Solutions of the Deformed Painlev\'e I Equation
Abstract
We prove that the formal -power series solution of the deformed Painlev\'e I equation is resurgent, which means it is generically Borel summable and its Borel transform admits endless analytic continuation. In particular, we find that the Borel transform defines a global multivalued holomorphic function on a singular algebraic surface isomorphic to the Fermat quintic surface x5 + y5 + z5 = 0 modulo an involution. This surface is an algebraic fibration over the complex plane of the differential equation with generic fibre a smooth quintic curve. Each fibre is equipped with a fivefold covering map over another complex plane (the Borel plane) with ten ramification points (the Borel singularities) spread equally over two branch points giving two opposite Stokes rays.
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