Integrability of SLE and Liouville Conformal Field Theory through Conformal Welding

here are two frameworks to study quantum surfaces in Liouville quantum gravity: random planar geometry and Liouville conformal field theory (LCFT). A key feature of the first framework is the coupling with Schramm-Loewner evolution (SLE). A key feature of the second framework is the application of conformal field theoretical techniques. We report an ongoing program that unifies these two frameworks using conformal welding of quantum surfaces, which gives strong links between the integrability of SLE and LCFT. We obtain exact formulae for both SLE and LCFT which are hard to obtain from methods within their own framework. Based on a joint work with Ang and Holden, a joint work with Ang and Remy; and a series of ongoing projects with Ang.