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Lecture

Conformal Field Theories in AdS: Energy Spectra and Unitarity

In course

PHYS-739: Conformal Field theory and Gravity

This course is an introduction to holography, the modern approach to quantum gravity. ...

Description

This lecture delves into the energy spectra of free scalar fields in Anti-de Sitter space (AdS), revealing a correspondence with conformal field theories (CFTs). The instructor discusses the primary and descendant states, emphasizing the unitarity constraints on the energy levels. By exploring the solutions to the Klein-Gordon equation and the role of isometries in AdS, the lecture uncovers the quantization of energies and the significance of the BF bound. The presentation concludes with a detailed analysis of the hypergeometric functions and the construction of the full representations of the conformal group. The ultimate goal is to establish a comprehensive dictionary between AdS and CFTs, shedding light on the intricate relationship between gravitational theories and quantum field theories.

Instructors (2)

Kyriakos Papadodimas

Grégoire Olivier Mathys

Official source

Related concepts (31)

Quantum field theory

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles.

Conformal field theory

A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.

Two-dimensional conformal field theory

A two-dimensional conformal field theory is a quantum field theory on a Euclidean two-dimensional space, that is invariant under local conformal transformations. In contrast to other types of conformal field theories, two-dimensional conformal field theories have infinite-dimensional symmetry algebras. In some cases, this allows them to be solved exactly, using the conformal bootstrap method. Notable two-dimensional conformal field theories include minimal models, Liouville theory, massless free bosonic theories, Wess–Zumino–Witten models, and certain sigma models.

Conformal symmetry

In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group, known as the conformal group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry has 15 degrees of freedom: ten for the Poincaré group, four for special conformal transformations, and one for a dilation. Harry Bateman and Ebenezer Cunningham were the first to study the conformal symmetry of Maxwell's equations.

Liouville field theory

In physics, Liouville field theory (or simply Liouville theory) is a two-dimensional conformal field theory whose classical equation of motion is a generalization of Liouville's equation. Liouville theory is defined for all complex values of the central charge of its Virasoro symmetry algebra, but it is unitary only if and its classical limit is Although it is an interacting theory with a continuous spectrum, Liouville theory has been solved. In particular, its three-point function on the sphere has been determined analytically.

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