Topological strings on genus one fibered Calabi-Yau 3-folds and string dualities

Understand

We calculate the generating functions of BPS indices using their modular properties in Type II and M-theory compactifications on compact genus one fibered CY 3-folds with singular fibers and additional rational sections or just $N$-sections, in order to study string dualities in four and five dimensions as well as rigid limits in which gravity decouples.

• The generating functions are Jacobi-forms of $\Gamma_1(N)$ with the complexified fiber volume as modular parameter.
• The string coupling $\lambda$, or the $\epsilon_\pm$ parameters in the rigid limit, as well as the masses of charged hypermultiplets and non-Abelian gauge bosons are elliptic parameters.
• To understand this structure, we show that specific auto-equivalences act on the category of topological B-branes on these geometries and generate an action of $\Gamma_1(N)$ on the stringy K\"ahler moduli space.