Cette these examine deux aspects des theories conformes des champs (TCC) en d dimensions.Sa premiere parti est dediee aux blocs conformes, des fonctions speciales qui contribuent au developpement en ondes partielles des fonctions a quatre points dans les TCC. On montre que ces blocs admettent un developpement en coordonnees polaires dont les coecients se calculent par une recurrence. Les blocs conformes sont naturellement denis sur le plan complexe : on considere alors leur restriction a l'axe r eel, an de montrer qu'ils obeissent une equation dierentielle sur ce domaine, ce qui mene a un algorithme ecace pour calculer les blocs conformes et leurs derivees pour tout d. Quelques applications au programme de bootstrap sont developpees. La seconde partie de cette these examine les perturbations d'une TCC par des operateurs pertinents. On etudie de tels ots du groupe de renormalisation en utilisant la Methode de Troncature Conforme (MTC) de Yurov et Zamolodchikov, une methode numerique qui permet de faire des calculs non-perturbatifs en theorie quantique des champs. Deux theories derentes sont considerees : le boson libre avec un terme de masse, et la theorie 4. Pour le dernier cas, les resultats de la MTC mettent en evidence la brisure de symetrie Z2. Finalement, on developpe une methode pour reduire les erreurs de troncature en ajoutant des contre-termes a l'action \nue" de la MTC, suivant des travaux anterieurs en d = 2 dimensions. / This thesis investigates two aspects of Conformal Field Theories (CFTs) in d dimensions. Its rst part is devoted to conformal blocks, special functions that arise in the partial wave expansion of CFT four-point functions. We prove that these conformal blocks admit an expansion in terms of polar coordinates and show that the expansion coecients are determined by recursion relations. Conformal blocks are naturally dened on the complex plane: we study their restriction to the real line, and show that they obey a fourth-order dierential equation there. This ODE can be used to eciently compute conformal blocks and their derivatives in general d. Several applications to the conformal bootstrap program are mentioned. The second half of this thesis investigates RG ows that are dened by perturbing a CFT by a number of relevant operators. We study such ows using the Truncated Conformal Space Approach (TCSA) of Yurov and Zamolodchikov, a numerical method that allows for controlled computations in strongly coupled QFTs. Two dierent RG ows are considered: the free scalar feld deformed by a mass term, and 4 theory. The former is used as a benchmark, in order to compare numerical TCSA results to exact predictions. TCSA results for 4 theory display spontaneous Z2 symmetry breaking at strong coupling: we study the spectrum of this theory both in the Z2-broken and preserved phase, and we compare the critical exponents governing the phase transition to known values. In a separate chapter, we show how truncation errors can be reduced by adding suitable counterterms to the bare TCSA action, following earlier work in d = 2 dimensions.
| Identifer | oai:union.ndltd.org:theses.fr/2015ENSU0013 |
| Date | 29 June 2015 |
| Creators | Hogervorst, Matthijs |
| Contributors | Paris, Ecole normale supérieure, Rychkov, Vyacheslav |
| Source Sets | Dépôt national des thèses électroniques françaises |
| Language | English |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation, Text |
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