Refined half-integer condition on RG flows

TL;DR

The work sharpens the constraints on RG flows between two-dimensional CFTs with braided symmetry by refining the half-integer condition on conformal dimensions, relating it to the existence of $\mathbb{Z}_2$-odd disorder objects and the universal grading of the surviving symmetry category. It shows that RG defects imply a necessary half-integer relation $h_c^{UV}+h_{F(c)}^{IR} \in \tfrac{1}{2}\mathbb{Z}$ and provides a concrete criterion to determine when the sum is half-integer through universal grading data. Through four explicit diagonal RCFT flows, the authors verify the refined condition and extract the corresponding IR theories, highlighting where emergent symmetries are required to satisfy the $c$-theorem and modular consistency. The results yield a coherent picture of when a massless IR theory can accommodate the original Braided MFC, and when a massive phase is forced to realize a TQFT with specific GSD, thus tightening the landscape of admissible IR endpoints for braided-symmetry RG flows.

Abstract

Renormalization group flows are constrained by symmetries. Traditionally, we have made the most of 't Hooft anomalies associated to the symmetries. The anomaly is mathematically part of the data for the monoidal structure on symmetry categories. The symmetry categories sometimes admit additional structures such as braiding. It was found that the additional structures give further constraints on renormalization group flows. One of these constraints is the half-integer condition. The condition claims the following. Braidings are characterized by conformal dimensions. A symmetry object $c$ in a braided symmetry category surviving all along the flow thus has two conformal dimensions, one in ultraviolet $h_c^\text{UV}$ and the other in infrared $h_c^\text{IR}$. In a renormalization group flow with a renormalization group defect, they add up to a half-integer $h_c^\text{UV}+h_c^\text{IR}\in\frac12\mathbb Z$. We find a necessary condition for the sum to be half-integer. We solve some flows with the refined half-integer condition.