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

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