Adiabatic paths and pseudoholomorphic curves (Q2460723)

The paper under review is to interpret a correspondence between the static Ginzburg-Landau moduli space and the adiabatic limit solutions of Ginzburg-Landau equation with slow-time \(\tau = \varepsilon \cdot t\) in \(\varepsilon \to 0\), and to show that this method applies to the complexified 4-manifold case to recapture the Taubes correspondence between the Seiberg-Wittern solutions and pseudoholomorphic curves. In section 1, the Ginzburg-Landau equations are described as the Euler-Lagrangian equations of the Ginzburg-Landau actions on \(\mathbb R \times\mathbb R^2\) (two-dimensional space and time \(2 + 1\)). Results of Jaffe and Taubes show that there is a one-to-one correspondence between the moduli space of static Ginzburg-Landau equations with finite potential energy and the moduli space of effective divisors of degree \(d\) on \(\mathbb C \cong\mathbb R^2\) the complex line. Then the author explains that the time-dependent Ginzburg-Landau solutions can be regarded as paths in the configuration space of \(d\)-vortices in section 1.4. By taking the adiabatic limit with \(\tau = \varepsilon \cdot t\) in \(\varepsilon \to 0\), every point of the path is a static solution, but the limiting path is no longer the time-dependent Ginzburg-Landau solution, and the limiting path is an extremal of the kinetic energy function. In section 2, the author applies the method of section 1 to the 4-dimensional Seiberg-Witten equations, and the adiabatic limit is replaced by the scale limit. Thus the Taubes correspondence between the solution of the Seiberg-Witten equations and the pseudo-holomorphic curves can be obtained in the scale limiting process. The paper is basically written as an interpretation of the adiabatic limit procedure in both \(2+1\) and \(2+2\) cases without any mathematical proof.