Counting RG flows

TL;DR

The paper introduces a unified topological framework for renormalization-group (RG) flows by treating flows as trajectories in theory space ${\mathcal{T}}$ and employing Bott's Morse-Bott theory to relate endpoint data to the global topology of ${\mathcal{T}}$. Central to the approach is the mu-index $\mu(T_*)$, the number of relevant operators at a fixed point, which governs the dimension of the RG moduli space ${\mathcal{M}}(T_{UV},T_{IR})$ via $\dim {\mathcal{M}}=\mu(T_{UV})-\mu(T_{IR})$ in transverse cases, and generalizes to conformal manifolds through Morse-Bott structure and symmetry refinements. The framework yields a set of diagnostic conjectures (the ${\mu}$-theorem variants) that predict when RG flows exist, how many distinct flows or walls interpolate between fixed points, and where phase transitions can occur along flows, with explicit tests across 2d, 3d, and 4d supersymmetric theories, including holographic RG flows, RG walls, and the role of the superconformal index as a practical computational tool. The work identifies new conformal manifolds, connects with classifying spaces of global symmetries, and provides a topological lens to understand violations of the strongest form of the C-theorem, offering a robust, cross-dimensional framework for RG-flow topology and its physical consequences.

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.

Figures (4)

• Figure 1: A schematic representation of a renormalization group (RG) flow from a UV conformal theory $T_{\text{UV}}$ to an IR conformal theory $T_{\text{IR}}$. The flow is parametrized by $t \in (- \infty, + \infty)$.
• Figure 2: The "spectral flow" of scaling dimensions in CFT$_d$ and a cartoon illustrating how the space of flows fails to be a manifold at the "phase transition" point.
• Figure 3: Stable and unstable manifolds, ${\mathcal{I}}(T_*)$ and ${\mathcal{R}}(T_*)$, are spanned by RG flow trajectories in and out of the CFT $T_*$. The intersection of ${\mathcal{R}}(T_{\text{UV}})$ and ${\mathcal{I}}(T_{\text{IR}})$ defines the set of RG flows from $T_{\text{UV}}$ to $T_{\text{IR}}$ in the theory space ${\mathcal{T}}$.
• Figure 4: The spectrum of ${\mathcal{N}}=1$ adjoint SQCD, the theory $\widehat{A}$, as a function of $x = \tfrac{N_c}{N_f}$. R-charges of generalized mesons $Q X^n \widetilde{Q}$ for $n=0,1,2,3$ (Left) and R-charges of ${\rm Tr \,} (X^n)$ for $n=2,3,4,5$ (Right). The dashed lines represent $R=2$ and $R=\tfrac{2}{3}$.