Opers and Non-Abelian Hodge: Numerical Studies

Opers versus Nonabelian Hodge

For a complex simple simply connected Lie group G, and a compact Riemann surface C, we consider two sorts of families of flat G-connections over C. Each family is determined by a point u of the base of Hitchin’s integrable system for (G, C). One family ∇h̄,u consists of G-opers, and depends on h̄ ∈ C×. The other family ∇R,ζ,u is built from solutions of Hitchin’s equations, and depends on ζ ∈ C×, ...

Higgs Bundles and Non-abelian Hodge Theory

The Non-abelian Hodge Correspondence for Non-Compact Curves

In this talk I will describe the non-abelian Hodge theory of a non-compact curve. This was worked out by Simpson in the paper Harmonic Bundles on Non-Compact Curves, and as such almost everything I say here can be found in that paper in more detail. Let X be a smooth non-compact curve, with smooth completion j : X ↪→ X, and S = X \X a finite set of punctures. For simplicity, I’ll assume we have...

General solution of the non-abelian Gauss law and non-abelian analogs of the Hodge decomposition

General solution of the non-abelian Gauss law in terms of covariant curls and gradients is presented. Also two non-abelian analogs of the Hodge decomposition in three dimensions are addressed. i) Decomposition of an isotriplet vector field V a i (x) as sum of covariant curl and gradient with respect to an arbitrary background Yang-Mills potential is obtained. ii) A decomposition of the form V a...

Hodge cycles on abelian varieties

This is a TeXed copy of – Hodge cycles on abelian varieties (the notes of most of the seminar “Périodes des Intégrales Abéliennes” given by P. Deligne at I.H.E.S., 1978–79; pp9– 100 of Deligne et al. 1982). somewhat revised and updated. See the endnotes1 for more details.