Week 05.02.2024 – 11.02.2024

In these lectures we will discuss various aspects of conformal field theories in Lorentzian signature. First, we will study the general properties of Lorentzian correlation functions, including their global conformal structure and the relation to Euclidean correlators. We will then consider the Regge limit of correlation functions and how this limit requires the introduction of complex spin. We will define complex spin using the Lorentzian inversion formula, and interpret it in terms of non-local light-ray operators. Finally, we will discuss applications of light-ray operators to even shape observables.

Abstract: On general grounds, the spectrum of a Conformal Field Theory (CFT) is expected to be extremely complicated away from special limits. One of such limits consists of the lowest energy states for every given spin, which are also known as the leading-twist states. In this talk I will discuss the structures that emerge in the spectrum as one departs from this limit towards the higher-twist states, focusing especially on the Regge trajectories. In particular, I will describe a general mechanism that reconciles the idea of Regge trajectories with the growing number of local operators at large spin. I will then demonstrate how this mechanism works in the case of one-loop phi^4 Wilson-Fisher theory in epsilon-expansion. This example will also clarify the properties of double-twist trajectories and light-cone bootstrap at higher-twist. Finally, I will briefly discuss a work in progress on going beyond the double-twist limit.

Double-scaled SYK (DSSYK) is a model with interesting dynamics, and many known exact results. Yet, the gravitational behavior of the system is not fully understood. I will discuss the reasoning behind a suggested connection between DSSYK and de Sitter holography. We will mention the general form of the correspondence as well as the mapping of parameters between the two sides. On the SYK side we consider two copies of DSSYK at infinite temperature with an equal energy constraint. We will discuss explicitly the two-point function in double-scaled SYK and compare it to de Sitter.

In holographic duality, a higher dimensional quantum gravity system is equivalent to a lower dimensional conformal field theory (CFT) with a large number of degrees of freedom. In this talk, I will introduce a framework to describe using the CFT how geometric notions in the gravity system, such as spacetime subregions, different notions of times, causal structure, and spacetime connectivity, emerge in the semi-classical limit.