  When placed on four-manifolds, $ \mathcal{N} = 2 $ gauge theories couple to
topological invariants of the background via two functions $ A $ and $ B $.
General considerations allow for these functions to be fixed in terms of the
Coulomb moduli and other parameters in the theory, but only up to
multiplicative factors about which little is known. We extend earlier work on
the microscopic study of these functions in the $ \Omega $-background to $
\mathcal{N} = 2 ^{\star } $ gauge theories with higher-rank $ \mathrm{U}(N) $
gauge groups. We complement this analysis by carrying out a perturbative study
of these functions. This allows us to determine the manner in which these
multiplicative factors scale with the rank of the gauge group and the mass of
the adjoint hypermultiplet.
