Publications

This is the first of two papers which together prove that the 12-parameter family of parabolic $\mathrm{SU}(2)$-Hitchin moduli spaces on the four-punctured sphere are all ALG gravitational instantons of type D4, and hence are asymptotic to $(\mathbb{C}×T^2_\tau)/\mathbb{Z}_2$ at infinity. The elliptic modulus $\tau$ is determined by the cross-ratio of the four points. In this first paper, we consider each Hitchin moduli space corresponding to an allowable set of parabolic data and compute its Torelli parameters. There is a 12-parameter family of Hitchin moduli spaces corresponding to different parabolic data, and we show that these realize all possible allowable Torelli parameters. In the companion paper, we we will show there that all of the Hitchin moduli spaces studied here are indeed ALG of type D4, and consequently that every ALG-D4 gravitational instanton can be realized as a Hitchin moduli space. Altogether, this will give the first verification of any case of the Modularity Conjecture: that all ALG gravitational instantons with tangent cone $\mathbb{C}/\mathbb{Z}_2$ can be realized as Hitchin moduli spaces with their natural associated $L^2$-metrics.

Related project(s):
77Asymptotic geometry of the Higgs bundle moduli space II

We establish a duality between harmonic maps from Riemann surfaces to hyperbolic 3-space $\mathbb{H}^3$ and harmonic maps from Riemann surfaces to de Sitter three-space $\mathrm{dS}_3$, best viewed as a generalized Gauß map. On the gauge theoretic side, it matches $\mathrm{SU}(2)$ and $\mathrm{SU}(1,1)$ solutions of Hitchin's self-duality equations via a signature flip along an eigenline of the Higgs field. Reversing this operation typically produces singular solutions, occurring where the eigenline becomes lightlike. Motivated by explicit model examples and this singular behavior, we extend this duality to a class of \emph{transgressive} harmonic maps $f:M\to \mathbb S^3$: these are harmonic on the hemispheres equipped with the hyperbolic metric, intersect the equator orthogonally, and have vanishing Hopf differential along the crossing set. We construct large families by gluing and analyze their regularity, and as an application obtain $\tau$-real negative sections of the Deligne--Hitchin moduli space of arbitrarily large energy that are not twistor lines.

Related project(s):
55New hyperkähler spaces from the the self-duality equations77Asymptotic geometry of the Higgs bundle moduli space II

We use the theory of Gaiotto, Moore and Neitzke to construct a set of Darboux coordinates on the moduli space \(\mathcal{M}\) of weakly parabolic \(SL(2,\mathbb{C})\)-Higgs bundles. For generic Higgs bundles\((\mathcal{E},R\Phi)\) with \(R\gg 0\) the coordinates are shown to be dominated by a leading term that is given by the coordinates for a corresponding simpler space of limiting configurations and we prove that the deviation from the limiting term is given by a remainder that is exponentially suppressed in \(R\).

    We then use this result to solve an associated Riemann-Hilbert problem and construct a twistorial hyperkähler metric \(g_{\text{twist}}\) on \(\mathcal{M}\). Comparing this metric to the simpler semiflat metric \(g_{\text{sf}}\), we show that their difference is \(g_{\text{twist}}-g_{\text{sf}}=O\left(e^{-\mu R}\right)\), where \(\mu\) is a minimal period of the determinant of the Higgs field.

Related project(s):
77Asymptotic geometry of the Higgs bundle moduli space II