I will discuss a connection between harmonic analysis on hyperbolic n-manifolds and conformal field theory in n-1 dimensions. Used in one direction, this connection leads to new spectral bounds on hyperbolic manifolds. Used in the other direction, it offers a new viewpoint on the spectra data of conformal field theories.

It has been a long-standing conjecture that any CFT with a large central charge and a large gap M in the spectrum of single-trace operators must be dual to a local effective field theory in AdS. In my talk, I will discuss a proof of a sharp form of this conjecture. In particular, I will explain how to derive numerical bounds on bulk Wilson coefficients in terms of M using the conformal bootstrap. The bounds exhibit scaling in M expected from dimensional analysis in the bulk. The main technical tools are dispersive CFT sum rules. These sum rules provide a dictionary between CFT dispersion relations and S-matrix dispersion relations in appropriate limits. This dictionary allows one to apply recently-developed flat-space methods to construct positive CFT functionals. My talk will be based on https://arxiv.org/pdf/2106.10274.pdf, which is joint work with S. Caron-Huot, L. Rastelli, and D. Simmons-Duffin.

I will discuss conformal bootstrap for SCFTs with four supercharges (eight superconformal charges) between two and four dimensions in a unified language. The special cases of interest are (2,2) SCFTs in d=2, N=2 SCFTs in d=3, and N=1 SCFTs in d=4. I will show how a large class of superconformal blocks can be found from the Casimir differential equation. I will describe the numerical bounds arising from the two independent bootstrap equations of the four-point function involving a chiral field and its conjugate. The bound involves three kinks, one of which corresponds to the IR fixed point of the Wess-Zumino model, and the other two remain mysterious.