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Register a new userHiggs bundles and diffeomorphism groups
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Nigel Hitchin
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By studying the Higgs bundle equations with the gauge group replaced by the group of symplectic diffeomorphisms of the 2-sphere we encounter the notion of a folded hyperkaehler 4-manifold and conjecture the existence of a family of such metrics parametrised by an infinite-dimensional analogue of Teichmueller space.
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Let $X$ be a compact orientable non-Haken 3-manifold modeled on the Thurston geometry $text{Nil}$. We show that the diffeomorphism group $text{Diff}(X)$ deformation retracts to the isometry group $text{Isom}(X)$. Combining this with earlier work by many authors, this completes the determination the homotopy type of $text{Diff}(X)$ for any compact, orientable, prime 3-manifold $X$.
We complete the proof of the Generalized Smale Conjecture, apart from the case of $RP^3$, and give a new proof of Gabais theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than $S^3$ and $RP^3$ and hyperbolic manifolds, to prove that the moduli space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3-manifold $X$, the inclusion $text{Isom} (X,g)to text{Diff}(X)$ is a homotopy equivalence for any Riemannian metric $g$ of constant sectional curvature.
We give an overview of the work of Corlette, Donaldson, Hitchin and Simpson leading to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface and the moduli space of Higgs bundles. We then explain how this can be generalized to a correspondence between character varieties for representations of surface groups in real Lie groups G and the moduli space of G-Higgs bundles. Finally we survey recent joint work with Bradlow, Garcia-Prada and Mundet i Riera on the moduli space of maximal Sp(2n,R)-Higgs bundles.
We formulate a correspondence between SU(2) monopole chains and ``spectral data, consisting of curves in $mathbb{CP}^1timesmathbb{CP}^1$ equipped with parabolic line bundles. This is the analogue for monopole chains of Donaldsons association of monopoles with rational maps. The construction is based on the Nahm transform, which relates monopole chains to Higgs bundles on the cylinder. As an application, we classify charge $k$ monopole chains which are invariant under actions of $mathbb{Z}_{2k}$. We present images of these symmetric monopole chains that were constructed using a numerical Nahm transform.
Given a compact Riemann surface $Sigma$ of genus $g_Sigma, geq, 2$, and an effective divisor $D, =, sum_i n_i x_i$ on $Sigma$ with $text{degree}(D), <, 2(g_Sigma -1)$, there is a unique cone metric on $Sigma$ of constant negative curvature $-4$ such that the cone angle at each $x_i$ is $2pi n_i$ (see McOwen and Troyanov [McO,Tr]). We describe the Higgs bundle corresponding to this uniformization associated to the above conical metric. We also give a family of Higgs bundles on $Sigma$ parametrized by a nonempty open subset of $H^0(Sigma,,K_Sigma^{otimes 2}otimes{mathcal O}_Sigma(-2D))$ that correspond to conical metrics of the above type on moving Riemann surfaces. These are inspired by Hitchins results in [Hi1], for the case $D,=, 0$.
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