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

TL;DR

This paper computes the effective gravitational couplings in four-dimensional $ =2$ supersymmetric gauge theories using the partition function in the $ ext{Omega}$-background, confirming predictions and determining previously unknown factors.

Contribution

It provides explicit calculations of the functions $A$ and $B$ for $ ext{SU}(2)$ and $ ext{SU}(N)$ gauge theories, validating theoretical predictions and refining understanding of gravitational couplings.

Findings

Confirmed the predicted expressions of $A$ and $B$ functions.

Determined the overall multiplicative factors previously unknown.

Validated predictions for $ ext{SU}(N)$ super-Yang-Mills theories.

Abstract

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.