Information Theory and the Conformal Bootstrap
Two dimensional conformal field theories are among the most important quantum field theories: they describe important statistical and condensed matter systems near criticality, and -- while not exactly solvable -- many exact techniques can be used which are not available in higher dimensions. I will consider these theories defined on surfaces of genus g>1, where the partition function of the theory can be interpreted as an entanglement Renyi entropy. This leads to a fascinating relationship between the geometry of conformal field theory and information theory. In particular, statements about information theory lead to novel constraints on the structure constants of the theory which are invisible in the usual conformal bootstrap approach to the classification of CFTs. I will also discuss the relationship with holography, where the phase structure of black holes teaches us about the entanglement structure of strongly coupled field theory.