  We study three dimensional $\mathcal{N}=2$ supersymmetric abelian gauge
theories with various matter contents living on a squashed sphere. In
particular we focus on two problems: firstly we perform a Picard-Lefschetz
decomposition of the localised path integral but, due to the absence of a
topological theta angle in three dimensions, we find that steepest descent
cycles do not permit us to distinguish between contributions to the
path-integral coming from (would-be) different topological sectors, for example
a vortex from a vortex/anti-vortex. The second problem we analyse is the
truncation of all perturbative expansions. Although the partition function can
be written as a transseries expansion of perturbative plus non-perturbative
terms, due to the supersymmetric nature of the observable studied we have that
each perturbative expansion around trivial and non-trivial saddles truncates
suggesting that normal resurgence analysis cannot be directly applied. The
first problem is solved by complexifying the squashing parameter, which can be
thought of as introducing a chemical potential for the global $U(1)$ rotation
symmetry, or equivalently an omega deformation. This effectively introduces a
hidden "topological angle" into the theory and the path integral can be now
decomposed into a sum over different topological sectors via Picard-Lefschetz
theory. The second problem is solved by deforming the matter content making
manifest the Cheshire Cat resurgence structure of the supersymmetric theory,
allowing us to reconstruct non-perturbative information from perturbative data
even when these do truncate.
