The Conformal Limit and Projective Structures

The non-abelian Hodge correspondence maps a polystable $\textrm{SL}(2, {\mathbb{R}})$-Higgs bundle on a compact Riemann surface $X$ of genus $g \geq 2$ to a connection that, in some cases, is the holonomy of a branched hyperbolic structure. Gaiotto's conformal limit maps the same bundle to a partial oper, that is, to a connection whose holonomy is that of a branched complex projective structure compatible with $X$. In this article, we show how these are both instances of the same phenomenon: the family of connections appearing in the conformal limit can be understood as a family of complex projective structures, deforming the hyperbolic ones into the ones compatible with $X$. We also show that, for zero Toledo invariant, this deformation is optimal, inducing a geodesic on Teichm & uuml;ller's space.