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Space-dependent RG, anomalous dimensions in a hierarchical model for 3d CFT and connections to the AdS/CFT correspondence

Banff International Research Station for Mathematical Innovation and Discovery

(Joint work with Ajay Chandra and Gianluca Guadagni)

An outstanding problem in the area of rigorous renormalization group theory is to develop a Wilsonian formalism which can handle space-dependent couplings in the Euclidean setting. I will present such a method in the simpler hierarchical model case and explain how this allowed us to prove a 46 years old prediction by Wilson regarding the anomalous scaling of the square/energy field in a hierarchical ferromagnet.

The model we studied is a hierarchical version of a 3d ferromagnet with long-range interactions in a range of parameters which puts it slightly below the upper critical dimension. We constructed the scaling limit for the joint law of the elementary/spin field together with the square/energy field as well as all mixed correlation functions. The Euclidean version of this scaling limit is conjectured to be a conformal field theory in three dimensions according to recent work by physicists in the conformal bootstrap program. There is a natural analogue of conformal invariance in the hierarchical model which thus provides an ideal testing ground for renormalization group-based attempts at rigorously proving conformal invariance. The idea is to feed the space-dependent renormalization group space-dependent ultraviolet cutoffs. I will also mention emerging connections to the AdS/CFT correspondence.

Mathematics; Quantum theory; Functional analysis; Mathematical physics

video/mp4

Author affiliation: University of Virginia

2019-03-14

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/68715

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/