Guido Franchetti (Pisa) will speak in the differential geometry seminar on 25th November. Guido’s title is “L^2 geometry of hyperbolic monopoles” and his abstract is below.
Hyperbolic monopoles, that is solutions of the Bogomolny equations on hyperbolic space, are similar in many respects to their Euclidean counterparts but also exhibit significant differences. In particular, the L^2 metric on the moduli space of monopoles, defined using the Coulomb gauge-fixing condition, results in a hyperkähler structure in the case of Euclidean monopoles but diverges for hyperbolic ones. In this talk, based on joint work with D. Harland, I will show that taking a different gauge-fixing condition, inspired by supersymmetry, cures this divergence yielding a finite symmetric bilinear form on the moduli space of hyperbolic monopoles which can be shown to be real on the space of inversion symmetric ones.