# The signature of the Seiberg-Witten surface - Mathematics > Differential Geometry

Abstract: The Seiberg-Witten family of elliptic curves defines a Jacobian rationalelliptic surface $\Z$ over $\mathbb{C}\mathrm{P}^1$. We show that for the$\bar{\partial}$-operator along the fiber the logarithm of the regularizeddeterminant $-1-2 \log \det- \bar\partial^* \bar\partial$ satisfies theanomaly equation of the one-loop topological string amplitude derived inKodaira-Spencer theory. We also show that not only the determinant line bundlewith the Quillen metric but also the $\bar{\partial}$-operator itself extendsacross the nodal fibers of $\mathrm{Z}$. The extension introduces currentcontributions to the curvature of the determinant line bundle at the pointswhere the fibration develops nodal fibers. The global anomaly of thedeterminant line bundle then determines the signature of $\mathrm{Z}$ whichequals minus the number of hypermultiplets.

Author: Andreas Malmendier

Source: https://arxiv.org/