Crosscap resurgence and Riemann--Hilbert problems for orientifolded conifolds

Abstract

We study the crosscap contribution to the large-$N$ SO/Sp free energy of the orientifolded conifold by comparing its formal expansion with Faddeev's quantum dilogarithm. This determines its Borel transform, sectorial sums, Stokes jumps, and accumulation limits; in the convergent chamber, these limits combine with resolved-conifold summation to recover the full orientifold free energy. We also construct a convergent ray-finite uncoupled rank-five rational BPS structure supported on the finite crosscap rays, and show that its Stokes tails violate both unrestricted half-plane normalization and two-sided polynomial growth at the non-active accumulation rays. Restricting normalization to closed subsectors and dropping the growth condition at those rays yields a unique solution expressed in terms of normalized double-sine functions. Finally, finite differences of the scalar Stokes transition recover the two BPS character multipliers and determine the supported active rational invariants.

40 pages, 1 figure; v3: Riemann--Hilbert formulation corrected in Sec. 4; BPS reconstruction and tau-functions added in Sec. 5. Comments are welcome