The Sen conjecture, made in 1994, makes precise predictions about the existence of L^2 harmonic forms on the monopole moduli spaces. For each positive integer k, the moduli space M_k of monopoles of charge k is a non-compact smooth manifold of dimension 4k, carrying a natural hyperkaehler metric. Thus studying Sen’s conjectures requires a good understanding of the asymptotic structure of M_k and its metric. This is a challenging analytical problem, because of the non-compactness of M_k and because its asymptotic structure is at least as complicated as the partitions of k. For k=2, the metric was written down explicitly by Atiyah and Hitchin, and partial results are known in other cases. In this talk, I shall introduce the main characters in this story and describe recent work aimed at proving Sen’s conjecture.
- Geometry and Analysis Seminar