ul 2 00 2 Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles

A central object of study in gauge theory is the moduli space of unitary flat connections on a compact surface. Thanks to the efforts of many people, a great deal is understood about the ring structure of its cohomology. In particular, the ring is known to be generated by the so-called universal classes [2, 36], and, in rank 2, all the relations between these classes are also known [3, 27, 40, 52]. If instead of just unitary connections one allows all flat connections, one obtains larger moduli spaces of equal importance and interest. However, these spaces are not compact and so very little was known about the ring structure of their cohomology. This paper will show that, in the rank 2 case, the cohomology ring of this noncompact space is again generated by universal classes. A companion paper [23] gives a complete set of explicit relations between these generators. The noncompact spaces studied here have significance extending well beyond gauge theory. They play an important role in 3-manifold topology: see for example Culler-Shalen [10]. And they are the setting for much of the geometric Langlands program: see for example Beilinson-Drinfeld [5]. But they have received perhaps the most attention from algebraic geometers, in the guise of moduli spaces of Higgs bundles. A Higgs bundle is a holomor-phic object, related to a flat connection by a correspondence theorem similar to that of Narasimhan-Seshadri in the unitary case. This point of view has been exploited to great effect, notably by The Higgs point of view predominates in this paper also. Indeed, it is strongly influenced by, and occasionally parallel to, the works of Hitchin [24] and Atiyah-Bott [2]. It is true that the moduli spaces of Higgs bundles and of flat connections carry different complex structures, but thanks to the correspondence theorem, they are diffeomorphic, and hence interchangeable topologically. What is important for our purposes is that the Higgs moduli space carries a holomorphic action of the complex torus C ×. This allows the topology of the moduli space, in some sense, to be determined from that of the fixed-point set. Most of the results in this paper are valid for bundles of arbitrary rank. We state them this way in the hope that they may hold independent interest. We especially have in mind the construction in §9 of the classifying space of the gauge group as a direct limit of Higgs …