Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles
Abstract
The moduli space of stable vector bundles on a Riemann surface is smooth when the rank and degree are coprime, and is diffeomorphic to the space of unitary connections of central constant curvature. A classic result of Newstead and AtiyahBott asserts that its rational cohomology ring is generated by the universal classes, that is, by the Kunneth components of the Chern classes of the universal bundle. This paper studies the larger, noncompact moduli space of Higgs bundles, as introduced by Hitchin and Simpson, with values in the canonical bundle K. This is diffeomorphic to the space of all connections of central constant curvature, whether unitary or not. The main result of the paper is that, in the rank 2 case, the rational cohomology ring of this space is again generated by universal classes. The spaces of Higgs bundles with values in K(n) for n > 0 turn out to be essential to the story. Indeed, we show that their direct limit has the homotopy type of the classifying space of the gauge group, and hence has cohomology generated by universal classes. A companion paper treats the problem of finding relations between these generators in the rank 2 case.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 March 2000
 arXiv:
 arXiv:math/0003093
 Bibcode:
 2000math......3093H
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematical Physics;
 Mathematics  Differential Geometry;
 Mathematics  Mathematical Physics;
 Mathematics  Symplectic Geometry;
 14H60 (Primary) 14D20;
 14H81;
 32Q55;
 58D27 (Secondary)
 EPrint:
 29 pages, LaTeX. Correction of an erroneous lemma in section 10 requires the addition, in Theorem 1.1, of the hypothesis that the rank is 2. Title changed accordingly