RG Flows and Dynamical Systems

TL;DR

This paper explores the mathematical structure of renormalization group flows using dynamical systems theory, highlighting thermodynamics-like properties, stability, bifurcations, and applications to holographic theories.

Contribution

It applies dynamical systems concepts to RG flows, analyzing thermodynamics analogies, stability, bifurcations, and holographic applications, providing new insights into their global behavior.

Findings

RG flows exhibit thermodynamics-like properties such as monotonic functions.

Bifurcation theory explains transitions in RG flow behaviors.

Holographic RG flows share structural features with dual field theories.

Abstract

In the context of Wilsonian Renormalization, renormalization group (RG) flows are a set of differential equations that defines how the coupling constants of a theory depend on an energy scale. These equations closely resemble thermodynamical equations and how thermodynamical systems are related to temperature. In this sense, it is natural to look for structures in the flows that show a thermodynamics-like behaviour. The mathematical theory to study these equations is called Dynamical Systems, and applications of that have been used to study RG flows. For example, the classical Zamolodchikov's C-Theorem and its higher-dimensional counterparts, that show that there is a monotonically decreasing function along the flow and it is a property that resembles the second-law of thermodynamics, is related to the Lyapunov function in the context of Dynamical Systems. It can be used to rule out exotic asymptotic behaviours like periodic flows (also known as limit cycles). We also study bifurcation theory and index theories, which have been proposed to be useful in the study of RG flows, the former can be used to explain couplings crossing through marginality and the latter to extract global information about the space the flows lives in. In this dissertation, we also look for applications in holographic RG flows and we try to see if the structural behaviours in holographic theories are the same as the ones in the dual field theory side.