Resurgence and Riemann--Hilbert problems for orientifolded conifolds

Abstract

We study the crosscap part of the large-$N$ SO/Sp orientifold conifold free energies. The unprojected crosscap block is a single $q$-Pochhammer tower. Its rank-one shift equation matches the functional equation for Faddeev's quantum dilogarithm after a change of variables. Using the known Borel-resurgence theorem for this quantum dilogarithm, we compute the Borel transform, pole structure, sectorial sums, Stokes jumps, and limiting sectors of the primitive block and of its odd projection. Combining the odd-projected crosscap calculation with the resolved-conifold summation theorem gives the corresponding resurgence statement for the large-$N$ orientifold free energy. We also formulate an axiomatic classical self-dual Riemann--Hilbert problem whose local wall-crossing factors are extracted from the crosscap Stokes jumps. The construction uses a doubled charge lattice and a chosen O-plane incidence function. Within this axiomatic setting, normalized scalar crosscap sectorial functions give $τ$-functions whose elementary shift-ratios solve the coordinate problem.