Birapport

Résumé

Le birapport, ou rapport anharmonique selon la dénomination de Michel Chasles est un outil puissant de la géométrie, en particulier la géométrie projective. La notion remonte à Pappus d'Alexandrie, mais son étude systématique est réalisée en 1827 par Möbius. Birapport de quatre points alignés Définition élémentaire thumb|Les divisions sont supposées régulières. Le birapport de C, D par rapport à A, B est : \frac{\frac{-2}{-1}}{\frac{-3}{-2}}=\frac{-4}{-3}. thumb|Les divisions sont supposées régulières. Le birapport de C, D par rapport à A, B est : {\frac{-1}{3}}=1-{\frac{4}{3}}. Si A, B, C et D sont quatre points distincts d'une droite (d), le birapport, ou rapport anharmonique de ces quatre points est formé à partir des rapports des mesures algébriques des segments qu'ils découpent de la manière suivante : : r=(A,B,C,D)=\frac{\frac{\overline{\mathrm{CA}}}{\overline{\mathrm{CB

Source officielle

https://fr.wikipedia.org/wiki/Birapport

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Publications associées (7)

Publications associées

Chargement

Personnes associées (1)

Personnes associées

Chargement

Unités associées (1)

Unités associées

Chargement

Concepts associés (29)

Concepts associés

Chargement

Cours associés (3)

This course is an introduction to the non-perturbative bootstrap approach to Conformal Field Theory and to the Gauge/Gravity duality, emphasizing the fruitful interplay between these two ideas.

Ce cours traite des 3 sujets suivants : la perspective, la géométrie descriptive, et une initiation à la géométrie projective.

Introduction to local and global behavior of nonlinear dynamical systems arising from maps and ordinary differential equations. Theoretical and computational aspects studied.

Cours associés

Chargement

Séances de cours associées (6)

Séances de cours associées

Chargement

Loops in AdS: from the spectral representation to position space

Din Carmi

We compute a family of scalar loop diagrams in AdS. We use the spectral representation to derive various bulk vertex/propagator identities, and these identities enable to reduce certain loop bubble diagrams to lower loop diagrams, and often to tree- level exchange or contact diagrams. An important example is the computation of the finite coupling 4-point function of the large-N conformal O(N ) model on AdS(3). Remarkably, the re-summation of bubble diagrams is equal to a certain contact diagram: the D1,1,32,32zz function. Another example is a scalar with phi (4) or phi (3) coupling in AdS(3): we compute various 4-point (and higher point) loop bubble diagrams with alternating integer and half- integer scaling dimensions in terms of a finite sum of contact diagrams and tree-level exchange diagrams. The 4-point function with external scaling dimensions differences obeying (12) = 0 and (34) = 1 enjoys significant simplicity which enables us to compute in quite generality. For integer or half-integer scaling dimensions, we show that the M -loop bubble diagram can be written in terms of Lerch transcendent functions of the cross- ratios z and z. Finally, we compute 2-point bulk bubble diagrams with endpoints in the bulk, and the result can be written in terms of Lerch transcendent functions of the AdS chordal distance. We show that the similarity of the latter two computations is not a coincidence, but arises from a vertex identity between the bulk 2-point function and the double-discontinuity of the boundary 4-point function.

2020

Chargement

A conformal dispersion relation: correlations from absorption

Din Carmi

We introduce the analog of Kramers-Kronig dispersion relations for correlators of four scalar operators in an arbitrary conformal field theory. The correlator is expressed as an integral over its "absorptive part", defined as a double discontinuity, times a theory-independent kernel which we compute explicitly. The kernel is found by resumming the data obtained by the Lorentzian inversion formula. For scalars of equal scaling dimensions, it is a remarkably simple function (elliptic integral function) of two pairs of cross-ratios. We perform various checks of the dispersion relation (generalized free fields, holographic theories at tree-level, 3D Ising model), and get perfect matching. Finally, we derive an integral relation that relates the "inverted" conformal block with the ordinary conformal block.

2020

Chargement

Radial coordinates for defect CFTs

Marco Meineri

We study the two-point function of local operators in the presence of a defect in a generic conformal field theory. We define two pairs of cross ratios, which are convenient in the analysis of the OPE in the bulk and defect channel respectively. The new coordinates have a simple geometric interpretation, which can be exploited to efficiently compute conformal blocks in a power expansion. We illustrate this fact in the case of scalar external operators. We also elucidate the convergence properties of the bulk and defect OPE decompositions of the two-point function. In particular, we remark that the expansion of the two-point function in powers of the new cross ratios converges everywhere, a property not shared by the cross ratios customarily used in defect CFT. We comment on the crucial relevance of this fact for the numerical bootstrap.

SPRINGER2018

Chargement