The signature of the Seiberg-Witten surface

Understand

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.