Aspects of Non-perturbative Conformal Bootstrap

Conformal field theories (CFTs) are special landmarks in the space of Quantum field theories. They sit at the fixed points of renormalization group flow and describe the physics of systems at critical points. Furthermore, CFTs provide a UV-complete definition of quantum gravity via the AdS/CFT correspondence. For example, high energy states in CFT capture the physics of black holes in AdS. I will introduce the notion of conformal bootstrap, a non-perturbative tool to study CFTs, leveraging various fundamental consistency conditions such as locality, unitarity, and crossing symmetry. In this talk, using analytical conformal bootstrap, I will derive rigorous universal results about various observables in CFT, e.g., the tightest bound on the number of energy eigenstates in some window of order-one size centered at high energy. As a corollary, I will derive an optimal bound on the asymptotic spacing between Virasoro primaries in any unitary 1+1D CFT. This helps us to probe the universality of high-energy spectra beyond the coarse-grained smooth approximation and carve out the landscape of (chaotic and/or holographic) CFTs. I will mention how one can further extend this idea to study other observables in CFTs in various dimensions. These results have applications in black hole physics and the study of entanglement entropy in statistical mechanics. Finally, I will describe a surprising and novel connection between hyperbolic manifolds and conformal bootstrap, elucidating how it produces the best, nearly saturating rigorous   bounds on the spectrum of the Laplacian on compact hyperbolic manifolds.