55. Heidelberger Physik Graduiertentage

THE FABULOUS WORLD OF LARGE-N FIELD THEORIES

Dario Benedetti

Quantum field theory is a beautiful formalism underlying and unifying many different branches of physics. At the same time, it is a hard subject, very few models being exactly solvable. For this reason, several approximation methods have been developed: the most standard one is perturbation theory around a Gaussian model, but other popular ones include lattice field theory, truncations of the functional renormalization group, and the bootstrap method.

In these lectures, I will cover the large-N expansion, where N is the number of fields in the theory. When such fields belong to a representation of a large-enough symmetry group, such as O(N), in the large-N limit the theory often becomes solvable, and a perturbative expansion in 1/N can be constructed.
I will present the basic combinatorial aspects of the large-N limits in theories where the fields are in vector, matrix, or higher-order tensor representations.
I will then focus on what we can do and learn with the large-N limit (particularly in the vector and tensor case) in the context of the renormalization group, and of the nontrivial conformal field theories that arise at its fixed points.
In particular, I will dedicate some time to introducing some conformal field theory methods (such as the conformal partial wave expansion), some functional methods (the 2PI effective action), and explaining their powerful interplay with the simplified diagrammatics provided by the large-N limit.
Lastly, I will discuss the limitations of the large-N limit, with some examples examples providing cautionary tales.