SL(∞,ℝ), Higgs Bundles and Quantization

Nigel Hitchin recently proposed a theory of SL(∞, R)-Higgs bundles which should parametrize a Hitchin component of representations of surface groups into SL(∞, R). This chapter discusses some properties and propose a formal approximation of SL(∞, R) representations by SL(n, R) representations in the large N limit, where n goes to infinity.

REPRESENTATIONS OF SURFACE GROUPS IN THE PROJECTIVE GENERAL LINEAR GROUP

Given a closed, oriented surface X of genus g ≥ 2, and a semisimple Lie group G, let [Formula: see text] be the moduli space of reductive representations of π1X in G. We determine the number of connected components of [Formula: see text], for n ≥ 4 even. In order to have a first division of connected components, we first classify real projective bundles over such a surface. Then we achieve our goal, using holomorphic methods through the theory of Higgs bundles over compact Riemann surfaces. We also show that the complement of the Hitchin component in [Formula: see text] is homotopically equivalent to [Formula: see text].