The Conformal Limit and Projective Structures

The non-abelian Hodge correspondence maps a polystable SL(2, R)-Higgs bundle on a compact Riemann surface X of genus g ≥ 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 Teichmüller’s space.