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The signature of the Seiberg-Witten surface. (English) Zbl 1243.14035

Mrowka, Tomasz (ed.) et al., Perspectives in mathematics and physics: Essays dedicated to Isadore Singer’s 85th birthday. Somerville, MA: International Press (ISBN 978-1-57146-145-2/hbk). Surveys in Differential Geometry 15, 255-277 (2011).
As shown by N. Seiberg and E. Witten [Nucl. Phys., B 431, No. 3, 484–550 (1994; Zbl 1020.81911)], the moduli space of the topological \(SU(2)\) Yang-Mills theory on a four-manifold decomposes into two branches. One of them, the Coulomb branch, is the moduli of a topological \(U(1)\) gauge theory and can be described by a family of elliptic curves called the Seiberg-Witten family. The paper explores geometry and topology of this family.
The Seiberg-Witten family together with \(N_f\) additional fields called hypermultiplets defines a Jacobian rational elliptic surface \(\overline{Z}\to\mathbb{C}\mathbb{P}^1\) with singular fibers. Assuming that masses of the hypermultiplets are generic all singular fibers are nodes except for a cusp over the north pole. Removing the cusp one obtains a smooth elliptic surface \(Z\to\mathrm{UP}\) over the complex plane \(\mathrm{UP}\subset\mathbb{C}\mathbb{P}^1\). For the \(\bar{\partial}\) operator along the fiber of \(Z\) the regularized determinant of the Laplacian \(\det'(\bar{\partial}^*\bar{\partial})\) becomes a smooth function on \(\mathrm{UP}\), and is used in a Quillen construction to define metric and connection on the determinant line bundle \(\mathrm{DET}\bar{\partial}\). Moreover, \(-\frac12\ln\det'(\bar{\partial}^*\bar{\partial})\) satisfies the anomaly equation for the one-loop amplitude of the Kodaira-Spencer theory, for \(\mathrm{DET}\bar{\partial}\) the local anomaly vanishes.
It follows from the results of J.-M. Bismut and J.-B. Bost [Acta Math. 165, No. 1–2, 1–103 (1990; Zbl 0709.32019)] that \(\mathrm{DET}\bar{\partial}\) with the Quillen metric extends smoothly across the nodes. Even the \(\bar{\partial}\) operator itself extends smoothly as well by the results of R. Seeley and I. M. Singer [J. Geom. Phys. 5, No. 1, 121–136 (1988; Zbl 0692.30038)], who used a different Laplacian shown by the author to be related to the one of Bismut-Bost by a change in the conformal gauge.
The author then considers the signature operator \(\mathcal{D}\) along the fiber of \(Z\to\mathrm{UP}\) and computes its global anomaly. He shows that well-defined logarithmic monodromies of the canonical section of \(\mathrm{DET}\mathcal{D}\) determine the signature of the elliptic surface \(Z\to\mathrm{UP}\). The elliptic signature operator on \(Z\) is also considered. The author shows that its analytic index is equal to minus the number of hypermultiplets \(-N_f\) if one imposes the Atiyah-Patodi-Singer boundary conditions on \(\partial Z\).
For the entire collection see [Zbl 1226.00040].

MSC:

14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
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