The L^2 geometry of spaces of harmonic maps S^2 -> S^2 and RP^2 -> RP^2

Abstract

Harmonic maps from S^2 to S^2 are all weakly conformal, and so are represented by rational maps. This paper presents a study of the L^2 metric gamma on M_n, the space of degree n harmonic maps S^2 -> S^2, or equivalently, the space of rational maps of degree n. It is proved that gamma is Kaehler with respect to a certain natural complex structure on M_n. The case n=1 is considered in detail: explicit formulae for gamma and its holomorphic sectional, Ricci and scalar curvatures are obtained, it is shown that the space has finite volume and diameter and codimension 2 boundary at infinity, and a certain class of Hamiltonian flows on M_1 is analyzed. It is proved that \tilde{M}_n, the space of absolute degree n (an odd positive integer) harmonic maps RP^2 -> RP^2, is a totally geodesic Lagrangian submanifold of M_n, and that for all n>1, \tilde{M}_n is geodesically incomplete. Possible generalizations and the relevance of these results to theoretical physics are briefly discussed.

27 pages, 2 figures

References in corpus (1)

• Quantum Mechanics on Moduli Spaces

Cited by in corpus (13)

• Reduced dynamics of Ward solitons
• Dynamics of CP^1 lumps on a cylinder
• Slow equivariant lump dynamics on the two sphere
• The volume of the space of holomorphic maps from S^2 to CP^k
• The geodesic approximation for lump dynamics and coercivity of the Hessian for harmonic maps
• Some special Kahler metrics on SL(2,C) and their holomorphic quantization
• Ricci magnetic geodesic motion of vortices and lumps
• The volume of a vortex and the Bradlow bound
• The deformed conifold as a geometry on the space of unit charge CP^1 lumps
• Generalized Maxwell-Higgs vortices in models with enhanced symmetry
• Compactifications of Deformed Conifolds, Branes and the Geometry of Qubits
• Quantum lump dynamics on the two-sphere
• Moduli Spaces of Lumps on Real Projective Space