Yang-Mills Connections On Orientable and Nonorientable Surfaces

TL;DR

This paper extends the analysis of Yang-Mills functional on surfaces to non-unitary classical groups, providing explicit descriptions of Morse strata and formulas for equivariant Poincaré series, highlighting differences from the unitary case.

Contribution

It generalizes the description of Morse strata and computes explicit formulas for non-unitary classical groups on orientable and nonorientable surfaces.

Findings

Explicit descriptions of Morse strata for SO(n) and Sp(n) groups.

Formulas for rational equivariant Poincaré series of semistable strata.

Differences identified between unitary and non-unitary cases.

Abstract

In math.SG/0605587, we studied Yang-Mills functional on the space of connections on a principal G_R-bundle over a closed, connected, nonorientable surface, where G_R is any compact connected Lie group. In this sequel, we generalize the discussion in "The Yang-Mills equations over Riemann surfaces" by Atiyah and Bott, and math.SG/0605587. We obtain explicit descriptions (as representation varieties) of Morse strata of Yang-Mills functional on orientable and nonorientable surfaces for non-unitary classical groups SO(n) and Sp(n). It turns out to be quite different from the unitary case. we use Laumon and Rapoport's method in "The Langlands lemma and the Betti numbers of stacks of G-bundles on a curve" to invert the Atiyah-Bott recursion relation, and write down explicit formulas of rational equivariant Poincar\'{e} series of the semistable stratum of the space of holomorphic structures on a principal $SO(n,\bC)$-bundle or a principal $Sp(n,\bC)$-bundle.