Counting RG flows

TL;DR

This paper develops a topological framework for analyzing renormalization group (RG) flows using Morse theory, providing new insights into counting flows, their topological sectors, and phase transitions, with applications to conformal manifolds and AdS/CFT.

Contribution

It introduces a unified topological approach to RG flows based on Morse theory, linking conformal manifold topology to RG flow properties and predicting phenomena like phase transitions.

Findings

Relation between conformal manifold topology and RG flow properties

Identification of topological obstructions to the strong C-theorem

Discovery of new conformal manifolds in 4d CFTs

Abstract

Interpreting renormalization group flows as solitons interpolating between different fixed points, we ask various questions that are normally asked in soliton physics but not in renormalization theory. Can one count RG flows? Are there different "topological sectors" for RG flows? What is the moduli space of an RG flow, and how does it compare to familiar moduli spaces of (supersymmetric) dowain walls? Analyzing these questions in a wide variety of contexts --- from counting RG walls to AdS/CFT correspondence --- will not only provide favorable answers, but will also lead us to a unified general framework that is powerful enough to account for peculiar RG flows and predict new physical phenomena. Namely, using Bott's version of Morse theory we relate the topology of conformal manifolds to certain properties of RG flows that can be used as precise diagnostics and "topological obstructions" for the strong form of the C-theorem in any dimension. Moreover, this framework suggests a precise mechanism for how the violation of the strong C-theorem happens and predicts "phase transitions" along the RG flow when the topological obstruction is non-trivial. Along the way, we also find new conformal manifolds in well-known 4d CFT's and point out connections with the superconformal index and classifying spaces of global symmetry groups.