Conformal Measure Spaces

The conformal bootstrap equations in general dimension are an infinite set of coupled non-linear equations in infinitely many variables. According to the lore, the solutions of the full set of equations correspond to physical CFTs. At the same time, the only solutions truly known to exist above two dimensions are the mean field theories. In this talk, I will define conformal measure spaces, which are mathematical objects guaranteed to produce solutions of the conformal bootstrap in any dimension. I will review why hyperbolic manifolds give rise to a particular class of conformal measure spaces, and use the bootstrap equations to prove new bounds on their spectra. Finally, I will discuss the similarities and differences between these solutions, and those that are believed to arise in physical CFTs.