Morse Theory for the Space of Higgs Bundles

Details Morse Theory for the Space of Higgs Bundles Morse Theory for the Space of Higgs Bundles

2006-11-05

arxiv.org/abs/math/0611113v2

Mathematics

Differential Geometry

45 pages, to appear Communications in Analysis and Geometry

Scientific paper

Here we prove the necessary analytic results to construct a Morse theory for the Yang-Mills-Higgs functional on the space of Higgs bundles over a compact Riemann surface. The main result is that the gradient flow with initial conditions $(A'', \phi)$ converges to a critical point of this functional, the isomorphism class of which is given by the graded object associated to the Harder-Narasimhan-Seshadri filtration of $(A'', \phi)$. In particular, the results of this paper show that the failure of hyperk\"ahler Kirwan surjectivity for rank 2 fixed determinant Higgs bundles does not occur because of a failure of the existence of a Morse theory.

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