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6. Dynamics of doubly-periodic instantons

The understanding of the asymptotic hyperkähler geometry of moduli spaces of monopoles is an interesting and outstanding problem at the interface of geometry and physics of gauge theories. In this regard, the work of Kottke-Singer [1512.02979] suggests that the model metric for widely separated monopoles is exponentially close to the $L^{2}$-metric as one goes to infinity in the moduli space.

A possible strategy is to relate this problem to the dynamics of doubly-periodic instantons (i.e., instantons on the product of a 1-dimensional complex torus $\mathbb{T}^{2}$ with a complex line $\mathbb{C}$. Biquard–Jardim [MR1866161] studied the asymptotic behaviour of moduli spaces of doubly-periodic instantons and in particular showed that a certain Nahm transform to a moduli space of meromorphic Higgs bundles on the dual torus is a hyperkähler isometry.
    1. Problem 6.1.

      [Benoit Charbonneau, Akos Nagy] $(i)$ Provide a rigorous mathematical foundation to the work of Gibbons–Manton [MR1346718] on the moduli space metric for well-separated BPS monopoles. Verify the validity of the Atiyah–Hitchin adiabatic approximation of monopole dynamics by geodesics.

      $(ii)$ The same question, for doubly-periodic instantons and their approximation by vortices.
        • Compactifications of moduli spaces of monopoles

          Problem 6.2.

          [Sergey A. Cherkis] What is the right compactification of the monopole moduli spaces, such that the resulting intersection cohomology equals the $L^{2}$-cohomology? The special cases of the Taub–NUT space and the Atiyah–Hitchin manifold are particular interesting.

              Cite this as: AimPL: Geometry and physics of ALX metrics in gauge theory, available at http://aimpl.org/geomphysalx.