Lecture

Conformal Symmetries in Euclidean and AdS Spaces

Graph Chat

Chat with Graph Search

Ask any question about EPFL courses, lectures, exercises, research, news, etc. or try the example questions below.

DISCLAIMER: The Graph Chatbot is not programmed to provide explicit or categorical answers to your questions. Rather, it transforms your questions into API requests that are distributed across the various IT services officially administered by EPFL. Its purpose is solely to collect and recommend relevant references to content that you can explore to help you answer your questions.

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 covers the concept of conformal symmetries in Euclidean and Anti-de Sitter (AdS) spaces, exploring the embedding of lower-dimensional manifolds in higher-dimensional spacetimes, the isometries, and the induced metric. It also discusses the Poincaré coordinates, the conformal structure, and the boundary of AdS spaces.

Instructors (2)

Grégoire Olivier Mathys

Kyriakos Papadodimas

Official source

Related concepts (32)

Euclidean space

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes.

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.

Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.

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.

Euclidean distance

In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 18th century.

Related lectures (64)

Manifolds and Tangent Space

Introduces manifolds, charts, atlases, tangent space, tensors, and the metric in curved spaces.

Conformal Transformations: Part 1

Covers the topic of conformal transformations, including translations, dilations, rotations, and the conformal algebra.

Embedding Space: AdS and Euclidean Geometry

Covers the concept of embedding space for AdS and Euclidean geometry, discussing AdS isometries and Euclidean Ads.

Riemannian metrics and gradients: Examples and Riemannian submanifolds MOOC: Introduction to optimization on smooth manifolds: first order methods

Explores Riemannian metrics on manifolds and the concept of Riemannian submanifolds in Euclidean spaces.

Hands on with Manopt: Optimization on Manifolds MOOC: Introduction to optimization on smooth manifolds: first order methods