Refined half-integer condition on RG flows

TL;DR

This paper refines the half-integer condition on RG flows by analyzing symmetry categories with additional structures, providing a necessary condition and solving some flows using this refined constraint.

Contribution

It introduces a refined half-integer condition involving conformal dimensions in braided symmetry categories for RG flows, enhancing previous anomaly-based constraints.

Findings

Derived a necessary condition for the half-integer sum of conformal dimensions.

Solved specific RG flows satisfying the refined half-integer condition.

Extended the understanding of symmetry constraints in renormalization group analysis.

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.