Week 05.02.2023 – 11.02.2023

These lectures aim to provide a self-contained introduction to the modern conformal bootstrap method. The study of conformal field theory (CFT) will first be motivated and the “old” way of studying CFTs as endpoints of RG flows will be explained. The set of ideas necessary to understand the conformal bootstrap method will then be introduced, and both analytic and numerical implementations of the conformal bootstrap method will be discussed.

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Cone spherical metrics on compact Riemann surfaces are conformal metrics of constant curvature +1 with finitely many conical singularities. They are called irreducible if any developing maps of such metrics don't have monodromy in U(1). By using projective structures and indigenous bundles on compact Riemann surfaces, we construct a canonical surjective map from the moduli space of stable extensions of two line bundles to that of irreducible cone spherical metrics with cone angles in 2\pi Z. We also prove that the map is generically injective in algebro-geometric sense if the Riemann surface has genus >1. As an application, we obtain some new existence results about irreducible cone spherical metrics. This is a joint work with Lingguang Li and Bin Xu.