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New hyperkähler spaces from the the self-duality equations

This project concerns the investigation of new hyper-Kähler manifolds which are given by spaces of singular solutions to Hitchin's  equation.

Hitchin's self-duality equations over compact Riemann surfaces are two-dimensional reductions of the Yang-Mills equations. These equations and their solutions provide deep links between various mathematical subjects like algebraic geometry and integrable systems, topology and representations of fundamental groups of surfaces, and differential geometric structures and harmonic maps. The moduli space of solutions of the self-duality equations carries a variety of interesting geometric structures, e.g., a completely integrable system and a hyper-Kähler structure induced by a natural $L^2$-metric. (Irreducible) solutions are parametrized by means of the Hitchin-Kobayashi correspondence either by (stable) Higgs pairs or by conjugacy classes of (irreducible) representations, and yield harmonic maps in corresponding non-compact symmetrical spaces.

The twistor space of the moduli space of solutions of the Hitchin equations has a complex analytical reincarnation as the Deligne-Hitchin moduli space of $\lambda$-connections. In this project we investigate special classes of singular solutions of Hitchin's equations over compact Riemannian surfaces which are given by real holomorphic sections of the Deligne-Hitchin moduli space. Global solutions of the Hitchin equations correspond to so-called  twistor lines. It was shown recently that other  components of real holomorphic sections besides twistor lines do exists, and that some of these components  of real holomorphic sections  carry at least locally an induced hyper-Kähler metric. We will examine questions regarding the completeness of these metrics and investigate new methods for constructing real holomorphic sections.

In particular, we will answer the following questions and problems:

• detecting different components of real holomorphic sections;
• construction of new real holomorphic sections via the analytic construction of  special singular harmonic mappings;
• gluing of  local prototypes of singular solutions of the Hitchin equations;
• a first investigation of the hyper-Kähler metric near infinity;
• generalisation of previous results to the higher rank $\geq2$ case;
• applications in the AdS / CFT correspondence: Construction of examples of minimal surfaces with non-trivial topology in anti-deSitter space-time; examination of the renormalised area in terms of a natural energy functional on the space of holomorphic sections.

## Publications

The non-abelian Hodge correspondence  is a real analytic map between the moduli space of stable Higgs bundles and the deRham moduli space of irreducible flat connections mediated by solutions of the self-duality equation. In this paper we construct such solutions for strongly parabolic $\mathfrak{sl}(2,\mathbb C)$ Higgs fields on a $4$-punctured sphere with parabolic weights $t \sim 0$ using loop groups methods through an implicit function theorem argument. We identify the rescaled limit hyper-K\"ahler moduli space at $t=0$ to be (the completion of) the nilpotent orbit in $\mathfrak{sl}(2, \mathbb C)$ equipped the Eguchi-Hanson metric. Our methods and computations are based on the twistor approach to the self-duality equations using Deligne and Simpson's  $\lambda$-connections interpretation. Due to the implicit function theorem, Taylor expansions of these quantities can be computed at $t=0$. By construction they have closed form expressions in terms of Multiple-Polylogarithms and their geometric properties lead to some identities of $\Omega$-values which we believe deserve further investigations.

Using non-Abelian Hodge theory for parabolic Higgs bundles,

we construct infinitely many non-congruent  hyperbolic affine spheres modeled on a thrice-punctured sphere with monodromy in ${\bf SL}_3(\Z)$. These give rise to non-isometric semi-flat Calabi--Yau metrics on special Lagrangian torus bundles over an open ball in $\R^{3}$  with a Y-vertex deleted, thereby answering a question raised by Loftin, Yau, and Zaslow.

Related project(s):
55New hyperkähler spaces from the the self-duality equations

The Lawson surfaces $\xi_{1,g}$ of genus $g$ are constructed by rotating and reflecting the Plateau solution $f_t$ with respect to a particular geodesic $4$-gon $\Gamma_t$ along its boundary, where $t= \tfrac{1}{2g+2}$ is an angle of  $\Gamma_t$.  In this paper we combine the existence and regularity of the Plateau solution $f_t$ in $t \in (0, \tfrac{1}{4})$ with topological information about the moduli space of Fuchsian systems on the 4-puncture sphere to obtain existence of a Fuchsian DPW potential $\eta_t$ for every $f_t$ with $t\in(0, \tfrac{1}{4}]$. Moreover, the coefficients of $\eta_t$ are shown to depend real analytically on $t$. This implies that the Taylor approximation of the DPW potential $\eta_t$ and of the area obtained at $t=0$ found in \cite{HHT2} determines these quantities for  all $\xi_{1,g}$. In particular, this leads to an algorithm to conformally parametrize all Lawson surfaces $\xi_{1,g}$.

We prove the existence of a pair $(\Sigma ,\, \Gamma)$, where $\Sigma$ is a compact Riemann surface with $\text{genus}(\Sigma)\, \geq\, 2$, and $\Gamma\, \subset\, {\rm SL}(2, \C)$ is a cocompact lattice, such that there is a generically injective holomorphic map $\Sigma \, \longrightarrow\, {\rm SL}(2, \C)/\Gamma$. This gives an affirmative answer to a question raised by Huckleberry and Winkelmann \cite{HW} and by Ghys \cite{Gh}.

For every $g \gg 1$, we show the existence of a complete and smooth family  of closed constant mean curvature surfaces $f_\varphi^g,$ $\varphi \in [0, \tfrac{\pi}{2}],$ in the round $3$-sphere deforming the Lawson surface $\xi_{1, g}$ to a doubly covered geodesic 2-sphere with monotonically increasing Willmore energy. To do so we use an implicit function theorem argument in the parameter $s= \tfrac{1}{2(g+1)}$. This allows us to give an iterative algorithm to compute the power series expansion of the DPW potential and area of $f_\varphi^g$  at $s= 0$ explicitly.  In particular,  we obtain for large genus Lawson surfaces $\xi_{1,g}$, due to the real analytic dependence of its area  and DPW potential on $s,$  a scheme to explicitly compute the coefficients of the power series in $s$ in terms of multilogarithms. Remarkably, the third order coefficient of the area expansion coincides numerically with $\tfrac{9}{4}\zeta(3),$ where $\zeta$ is the Riemann $\zeta$ function (while the first and second order term were shown to be $\log(2)$  and $0$ respectively in \cite{HHT}).

For every integer $g \,\geq\, 2$ we show the existence of a compact Riemann surface $\Sigma$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${\mathcal O}^{\oplus 2}_{\Sigma}$ admits holomorphic connections with $\text{SL}(2,{\mathbb R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. This also answers a question of \cite{CDHL}. The construction carries over to all very stable and compatible real holomorphic structures over the topologically trivial rank two bundle on $\Sigma$, and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.

We study the holomorphic symplectic geometry of (the smooth locus of) the space of holomorphic sections of a twistor space with rotating circle action. The twistor space has a meromorphic connection constructed by Hitchin. We give an interpretation of Hitchin's meromorphic connection in the context of the Atiyah--Ward transform of the corresponding hyperholomorphic line bundle. It is shown that the residue of the meromorphic connection serves as a moment map for the induced circle action, and its critical points are studied. Particular emphasis is given to the example of Deligne--Hitchin moduli spaces.

Related project(s):
55New hyperkähler spaces from the the self-duality equations

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## Team Members

Dr. Sebastian Heller