Two-dimensional conformal field theories: the irrational, the rational and the celestial

Thesis

The aim of this thesis is to explore the world of two-dimensional Conformal Field Theories via bootstrap techniques. The thesis is divided in three central chapters, where we will study Irrational, Rational and Celestial CFTs.

The holographic dual to a theory of pure higher-spin gravity in AdS3 is an Irrational CFT with WN algebra, where N corresponds to the highest-spin field present in the theory. Our aim in chapter 3 will be to study the spectrum of such a theory of gravity by constructing the density of states via the Poincar ́e and Rademacher constructions, which both exploit the modular invariance of the torus partition function. When studying the densities, we encounter exponentially large amounts of negative-norm states in the near-extremal region, which would render the theory non-unitary. We propose to cure these negativities by adding operators of low twist, which can be interpreted naturally from the gravity perspective as orbifolds with conical defects and can be added without losing unitarity.

The classification of Rational CFTs will occupy our attention in chapter 4. Our goal is to understand the spectral decomposition of the torus partition function of several RCFTs, and see how it can aid in the efforts to explore the space of such theories. We begin by introducing the basics of harmonic analysis on the fundamental domain of the modular group. Afterwards, we study the Rankin-Selberg transform of one-character RCFTs’ partition function and find an interesting relation between different central charges. From there, we start investigating multi-character RCFTs, and make a proposal to build families of multi-character RCFTs from one-character theories via their Rankin-Selberg transforms.

The duality between scattering amplitudes of massless particles in R(2,2) space and the correlation functions of a Celestial CFT living on the celestial torus will be the subject of Chapter 5. The goal of this chapter is to provide a prescription that allows us to go from momentum-space amplitudes to the ambidextrous basis, where the celestial amplitudes take the form of a two-dimensional CFT correlation function, via an improved light transform. We explicitly build 2-, 3- and 4-point amplitudes of massless external particles, and show how the ambidextrous prescription maintains conformal covariance in all scattering channels whilst ridding the amplitude of its distributional factors. We then move onto an explicit tree-level four-gluon, where we will use alpha space decomposition, an inversion formula method specific to SL(2, R) primaries, to obtain data from the Celestial CFT’s spectrum.

Authors: Jorge Díaz, C

• DOI: 10.5287/ora-j5axngpjd
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: string theory; holographic correspondence; celestial holography; rational CFTs; quantum gravity; ADS/CFT correspondence; conformal bootstrap; conformal field theories
• Subjects: String Theory; Mathematical physics; Theoretical Physics; High Energy Physics