  We study BPS excitations in M5-M2-brane configurations with a compact
transverse direction, which are also relevant for type IIa and IIb little
string theories. These configurations are dual to a class of toric elliptically
fibered Calabi-Yau manifolds $X_N$ with manifest $SL(2, \mathbb{Z}) \times
SL(2,\mathbb{Z})$ modular symmetry. They admit two dual gauge theory
descriptions. For both, the non-perturbative partition function can be written
as an expansion of the topological string partition function of $X_N$ with
respect to either of the two modular parameters. We analyze the resulting BPS
counting functions in detail and find that they can be fully constructed as
linear combinations of the BPS counting functions of M5-M2-brane configurations
with non-compact transverse directions. For certain M2-brane configurations, we
also find that the free energies in the two dual theories agree with each
other, which points to a new correspondence between instanton and monopole
configurations. These results are also a manifestation of T-duality between
type IIa and IIb little string theories.
