Yang-Mills connections on nonorientable surfaces

Abstract

Let G be a compact, connected Lie group. In [1], Atiyah and Bott identified the affine space A of connections on a principal G-bundle P over a Riemann surface with the affine space C of holomorphic structures on P = P ×G G, where G is the complexification of G. The identification A ∼= C is an isomorphism of affine spaces, thus a diffeomorphism. It was conjectured in [1] that under this identification the Morse stratification of the Yang–Mills functional on A exists and coincides with the stratification of C from algebraic geometry [14, 25]. The conjecture was proved by Daskalopoulos in [6] (see also [24] by Rade). The top stratum Css of C consists of semi-stable holomorphic structures on P. Atiyah and Bott showed that the stratification of C is G-perfect, where G = Aut(P). It has strong implications on the topology of the moduli space M(P ) of S-equivalence classes of semi-stable holomorphic structures on P. When M(P ) is smooth, Atiyah and Bott found a complete set of generators of the cohomology groups H∗(M(P );Q) and recursive relations which determine the Poincare polynomial Pt(M(P );Q). WhenM(P ) is singular, their results give generators of the equivariant cohomology groups H∗ GC(Css;Q) and formula for the equivariant Poincare series P G C t (Css;Q). Under the isomorphism A ∼= C, the top stratum Css corresponds to Ass which is the stable manifold of Nss, the set of central Yang–Mills connections, where the Yang–Mills functional achieves its absolute minimum [1,6].