The signature of the Seiberg-Witten surface

Abstract

The Seiberg-Witten family of elliptic curves defines a Jacobian rational elliptic surface $\Z$ over $\mathbb{C}\mathrm{P}^1$. We show that for the $\bar{\partial}$-operator along the fiber the logarithm of the regularized determinant $-1/2 \log \det' (\bar\partial^* \bar\partial)$ satisfies the anomaly equation of the one-loop topological string amplitude derived in Kodaira-Spencer theory. We also show that not only the determinant line bundle with the Quillen metric but also the $\bar{\partial}$-operator itself extends across the nodal fibers of $\mathrm{Z}$. The extension introduces current contributions to the curvature of the determinant line bundle at the points where the fibration develops nodal fibers. The global anomaly of the determinant line bundle then determines the signature of $\mathrm{Z}$ which equals minus the number of hypermultiplets.