Mirror Symmetry and the Type II String

TL;DR

The paper investigates whether mirror symmetry between Calabi–Yau threefolds X and Y extends to a nonperturbative equivalence between type II string theories compactified on these manifolds, focusing on the structure of their semiclassical moduli spaces. It shows that while the IIA moduli space is naturally organized by $H^3(X,\mathbb{Z})$-valued RR data and ${\cal M}_X\times{\cal M}_Y$ structures, the IIB moduli space requires a $B$-field–dependent refinement of the RR-configuration, encoded in a nontrivial interplay between $e^{B}$ and $H^{\rm even}(Y,\mathbb{Z})$. Monodromy in the CFT moduli space maps under mirror symmetry to $B$-field shifts on the IIB side, constraining how discrete RR data can vary; these insights are tied to the intermediate Jacobian geometry and the $C$-map framework, suggesting a coherent, locally valid dual picture at weak coupling. The work presents three key conjectures tying RR discretization to integral cohomology, preserving monodromy under mirror symmetry, and enforcing a B-dependent structure in IIB, with implications for extending the duality to nonperturbative regimes. Overall, the results illuminate how mirror symmetry constrains the global geometry of type II moduli spaces and point toward a deeper, nonperturbative equivalence between IIA and IIB compactifications.

Abstract

If $X$ and $Y$ are a mirror pair of Calabi--Yau threefolds, mirror symmetry should extend to an isomorphism between the type IIA string theory compactified on $X$ and the type IIB string theory compactified on $Y$, with all nonperturbative effects included. We study the implications which this proposal has for the structure of the semiclassical moduli spaces of the compactified type II theories. For the type IIB theory, the form taken by discrete shifts in the Ramond-Ramond scalars exhibits an unexpected dependence on the $B$-field. (Based on a talk at the Trieste Workshop on S-Duality and Mirror Symmetry.)