Seiberg-Witten theory and modular lambda function

In this paper, we will apply the tools from number theory and modular forms to the study of the Seiberg-Witten theory. We will express the holomorphic functions $a, a_D$, which generate the lattice $Z=n_e a+n_m a_D, (n_e, n_m) \in \mathbb{Z}^2$ of central charges, in terms of the periods of the Legendre family of elliptic curves. Thus we will be able to compute the transformations of the quotient $a_D/a$ under the action of the modular group $\text{PSL}(2,\mathbb{Z})$. We will show the Schwarzian derivative of the quotient $a_D/a$ with respect to the complexified coupling constant is given by the theta functions. We will also compute the scalar curvature of the moduli space of the $N=2$ supersymmetric Yang-Mills theory, which is shown to be asymptotically flat near the perturbative limit.