User jack evans - MathOverflow

Answer by Jack Evans for Representations of surface groups via holomorphic connections

2009-12-15T22:36:18Z

I'm too new to add this to my previous comment so apologies.

Dmitri, the trivial bundle can be part of a stable Hitchin Pair (specifically if A and B are two matrices without common eigenspaces tensored with independent sections of the canonical bundle).

The construction above generates maps from the Hitchin moduli space to itself if we start with the flat $SU(2)$ connections and use the Higgs field to define a holomorphic connection and then map the flat connection induced by the holomorphic connection to the one defined by the self-duality equations. Is this not likely to be holomorphic or well behaved under any of the complex structures?

Likewise for a fixed $SU(2)$ representation there will be a map from the space constructed as suggested above to the Hitchin moduli space for any family of curves (or $SL(2,\mathbb{R})$ representation) and holomorphic connections.

Answer by Jack Evans for Why is it useful to study vector bundles?

2009-12-05T19:44:59Z

They are tractable and naturally occuring yet encode lots of information. They also provide a link between different mathematical techniques. One good comparison is between solving Yang Mills equations on a vector bundle and the Einstein equations on a Riemannian manifold.

K-Theory shows that there is lots of topological information contained in them.

Hitchin Kobayashi correspondence linking differential and algebraic techniques, the Atiyah Singer index theorem linking analysis and topology. Flat connections and curvature link geometry and representation theory. The Torelli theorem and Donaldson's work use them to reveal information about finer structures (algebraic and differentiable respectively). They occur naturally, tangent and normal bundles obviously but also projective embeddings.

Hitchin's paper The Self Duality Equations on a Riemann Surface combines many of them beautifully.