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.

中文翻译：

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四维的有效引力耦合 $$ \mathcal{N} $$ = 2 超对称规范理论

四维 $\mathcal{N}=2$ 超对称规范理论与背景引力场的拓扑不变量的低能量有效耦合由两个函数 $A$ 和 $B$ 描述。这两个函数在拓扑扭曲的四维 $\mathcal{N}=2$ 超对称规范理论的研究和 $\mathcal{N}=2$ 超共形理论的中心电荷计算中起着重要作用。在本文中，我们从 $\Omega$ 背景中的分区函数计算 $A$ 和 $B$，用于 $\mathrm{SU}(2)$ 规范理论。我们的结果不仅证实了有效引力耦合的预测表达式，而且还给出了先前未确定的整体乘法因子。我们还分析了 $\mathrm{SU}(N)$ 超级杨米尔斯理论的 $A$ 和 $B$，并确认了之前的所有预测。