A Hitchin component for the group $\mathrm{SL}(\infty, \mathbb{R})$, after Hitchin, Biquard

Following the works of Hitchin and Biquard, we will study in this talk the solutions to the self-duality equations for the group $\mathrm{SU}(\infty)$. This group can be interpreted as the group of Hamiltonian diffeomorphisms of the 2-sphere $\mathbb{S}^2$, and its Lie algebra consists of the smooth functions on $S^2$ modulo the constant functions. Then, a solution to the self-duality equations is equivalent to a folded hyperk\"{a}hler metric, that is, a hyperk\"{a}hler metric on the total space of an $S^2$-bundle with a fold singularity on the equator of each $\mathbb{S}^2$-fiber. A conjectural picture suggests that an appropriate solution of these equations can provide a flat symplectic connection on a symplectic $\mathbb{S}^1 \times \mathbb{R}$-bundle over a negatively curved surface whose holonomy lies in the proper subgroup $\mathrm{Sp}(\infty, \mathbb{R})$ of Hamiltonian diffeomorphisms commuting with the involution given by reversing the orientation of an oriented geodesic.