Effective gravitational couplings of four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories

The low energy effective couplings of a four-dimensional $\mathcal{N}=2$ supersymmetric gauge theory to topological invariants of the background gravitational field are described by two functions $A$ and $B$. These two functions play an important role in the study of topologically twisted four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and in the computation of central charges of $\mathcal{N}=2$ superconformal theories. In this paper, we compute $A$ and $B$ from the partition function in the $\Omega$-background for $\mathrm{SU}(2)$ gauge theories. Our results not only confirm the predicted expressions of the effective gravitational couplings, but also give the previously undetermined overall multiplicative factors. We also analyze $A$ and $B$ for the $\mathrm{SU}(N)$ super-Yang-Mills theory, and confirm all the previous predictions.