Janne Junnila: Real opers and curves with a geodesic property

Abstract:

I will discuss complex projective structures with real holonomy on Riemann surfaces, in particular the punctured sphere. Such structures can be identified with certain differential operators known as real opers that appear in the semiclassical limit of Liouville quantum gravity and also play a prominent role in the analytic Langlands correspondence. A canonical example is given by the Fuchsian projective structure on a hyperbolic Riemann surface, and a result of Goldman says that on closed surfaces all other real opers can be obtained via a grafting procedure applied to the Fuchsian structure. With Bonk, Rohde and Wang we showed a similar grafting result in a certain non-compact setting, where the projective structure is induced by a piecewise geodesic Jordan curve on the punctured sphere. We also proved that the associated Schwarzian derivative is related to the derivative of the Loewner energy of the curve, a result analogous to a famous formula conjectued by Polyakov and proven by Takhtajan and Zograf in the Fuchsian case. In addition to explaining these results and giving some general background on complex projective structures, I will also showcase some explicit examples of real opers with reflection symmetry about the unit circle.