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Seiberg-Witten equations and pseudoholomorphic curves. (English) Zbl 1157.53048

Gilligan, Bruce (ed.) et al., Symmetries in complex analysis. Workshop on several complex variables, analysis on complex Lie groups and homogeneous spaces, Hangzhou, China, October 17–29, 2005. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4459-5/pbk). Contemporary Mathematics 468, 191-223 (2008).
From the introduction: The so-called Taubes “equation”
\[ Gr = SW \]
is just a mnemonic formula, involving the Seiberg-Witten invariant of a 4-dimensional symplectic manifold \(M\) on the right hand side and the Gromov invariant of \(M\) on the left hand side.
The Taubes equation is based on a certain limiting procedure, due to Taubes, which associates to a pseudoholomorphic curve a family of solutions of Seiberg-Witten equations, depending on a scale parameter. It turned out that this construction has a natural 3-dimensional analogue, where the role of Seiberg-Witten equations is played by the hyperbolic Ginzburg-Landau equations and pseudoholomorphic curves are replaced by the so-called adiabatic paths. This construction, called the adiabatic limit, associates to a parameter-dependent family of solutions of Ginzburg-Landau equations on \(\mathbb R^{2+1}\) an adiabatic path in the moduli space of static solutions.
We show that the Taubes 4-dimensional construction can be considered as a complex analogue of the 3-dimensional adiabatic limit so that pseudoholomorphic curves come out as complex adiabatic paths, depending on the complex time parameter.
Even the 2-dimensional analogue of Seiberg-Witten equations is non-trivial. In this case we obtain a correspondence between solutions of the so-called vortex equations on a compact Riemann surface and effective divisors on this surface, asserted by the Kazdan-Warner Theorem.
For the entire collection see [Zbl 1147.32001].

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
32Q65 Pseudoholomorphic curves
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
57R57 Applications of global analysis to structures on manifolds
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