Extended operators in quantum field theory Light-ray operators and generalized symmetries

Over the last century, it has become clear that Quantum Field Theory (QFT) is the language of theoretical physics. Vastly different phenomena permit a field-theoretic description. Despite the successes of QFT, there remain crucial systems such as strongly-interacting ones, where our standard methods fail. The theoretical tools necessary to understand their dynamics are still in development. In this thesis, we study such strongly-coupled theories by combining three tools: holography, the conformal bootstrap, and generalized symmetries.
In the first part of this thesis, we combined holography with the conformal bootstrap to probe four-dimensional Conformal Field Theories (CFTs). Holography relates a non-gravitational CFT to a higher-dimensional quantum gravitational theory living in Anti-de Sitter (AdS) space, while the conformal bootstrap is a program that uses the full power of conformal symmetry to derive powerful constraints on CFT observables. We focused on extended operators known as light-ray operators, and we analyzed the space of holographic four-dimensional CFTs. In particular, in chapter two, we used the ANEC operator to understand the necessary and sufficient condition that a CFT has to satisfy to reproduce weakly-coupled Einstein gravity in the bulk. In chapter three, we investigated a possible infinite-dimensional algebra that resembles the two-dimensional Virasoro algebra and that is spanned by light-ray operators. Uncovering such an algebra would have far-reaching consequences, as an infinite number of consistency conditions would follow.
In chapter four, we investigated the interplay between higher-form symmetries and the mechanism of spontaneous symmetry breaking. Generalized symmetries are a generalization of regular symmetries that act on, and help to classify extended objects in QFT. In particular, they are ubiquitous in very well-known systems. We showed that anomalous higher-form symmetries can be used as a starting point to prove Goldstone’s theorem, and as a consequence, there is a massless mode in the spectrum. This massless mode is protected by the anomaly. We also rephrased superfluid hydrodynamics in this language, thus streamlining the usual construction. We found perfect agreement with already available results in the literature.