Random surfaces, planar lattice models, and conformal field theory

Liouville quantum gravity (LQG) is a theory of random surfaces that originated from string theory. Schramm Loewner evolution (SLE) is a family of random planar curves describing scaling limits of many 2D lattice models at their criticality. Before the rigorous study via LQG and SLE in probability, random surfaces and scaling limits of lattice models have been studied via  another approach in theoretical physics called conformal field theory (CFT) since the 1980s. In this talk, I will demonstrate how a combination of ideas from LQG/SLE and CFT can be used to rigorously prove several long standing predictions in physics on random surfaces and planar lattice models, including the law of the random modulus of the scaling limit of uniform triangulation of the annular topology, and the crossing formula for critical planar percolation on an annulus. I will then present some conjectures which further illustrate the deep and rich interaction between LQG/SLE and CFT.  Based on joint works with Ang, Holden, Remy, Xu, and Zhuang.