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International Journal of Modern Physics A

Rational RG flow and half-integer condition

Ken Kikuchi

Published Apr 10, 2025• 0 citations
Publisher

Abstract

In this paper, we propose a mathematical formulation of renormalization group flows to (two-dimensional) rational conformal field theories. It is given in terms of symmetry categories. Surprisingly, the formulation constrains infrared conformal dimensions mod 1/2. This claims discrete braiding structures jump at conformal fixed points under renormalization group flows. We apply the formulation to solve a flow from the E-type minimal model.

DOI
10.1142/s0217751x2548001x

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In this paper, we propose a mathematical formulation of renormalization group flows to (two-dimensional) rational conformal field theories. It is given in terms of symmetry categories. Surprisingly, the formulation constrains infrared conformal dimensions mod 1/2. This claims discrete braiding structures jump at conformal fixed points under renormalization group flows. We apply the formulation to solve a flow from the E-type minimal model.
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 ccc in a braided symmetry category surviving all along the flow thus has two conformal dimensions, one in ultraviolet hcUVh_c^\text{UV}hcUV​ and the other in infrared hcIRh_c^\text{IR}hcIR​. In a renormalization group flow with a renormalization group defect, they add up to a half-integer hcUV+hcIR∈12Zh_c^\text{UV}+h_c^\text{IR}\in\frac12\mathbb ZhcUV​+hcIR​∈21​Z. We find a necessary condition for the sum to be half-integer. We solve some flows with the refined half-integer condition.
We explain when and why symmetries enhance in fermionic rational conformal field theories. In order to achieve the goal, we first clarify invariants under renormalization group flows. In particular, we find the Ocneanu rigidity is not enough to protect some quantities. Concretely, while (double) braidings are subject to the rigidity, they jump at conformal fixed points. The jump happens in a specific way, so the double braiding relation further constrains renormalization group flows. The new constraints enable us three things; 1) to predict infrared conformal dimensions in massless flow, 2) to reveal some structures of the theory space, and 3) to obtain a necessary condition for a flow to be massless. We also find scaling dimensions ``monotonically'' decrease along massless flows. Combining the discovery with predictions, sometimes, we can uniquely fix infrared conformal dimensions.
Recently, Gaiotto (2012 arXiv:1201.0767) considered conformal defects which produce an expansion of infrared local fields in terms of the ultraviolet (UV) ones for a given renormalization group flow. In this paper, we propose that for a boundary RG flow in two dimensions there exist boundary condition changing fields (RG defect fields) linking the UV and the infrared (IR) conformal boundary conditions which carry similar information on the expansion of boundary fields at the fixed points. We propose an expression for a pairing between IR and UV operators in terms of a four-point function with two insertions of the RG defect fields. For the boundary flows in minimal models triggered by ψ13 perturbation, we make an explicit proposal for the RG defect fields. We check our conjecture by a number of calculations done for the example of (p, 2) → (p − 1, 1)⊕(p + 1, 1) flows.
A bstract We consider topological defect lines (TDLs) in two-dimensional conformal field theories. Generalizing and encompassing both global symmetries and Verlinde lines, TDLs together with their attached defect operators provide models of fusion categories without braiding. We study the crossing relations of TDLs, discuss their relation to the ’t Hooft anomaly, and use them to constrain renormalization group flows to either conformal critical points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-degenerate gapped state. For various massive flows, we determine the infrared TQFTs completely from the consideration of TDLs together with modular invariance.
Recently, Gaiotto (2012 arXiv:1201.0767) considered conformal defects which produce an expansion of infrared local fields in terms of the ultraviolet (UV) ones for a given renormalization group flow. In this paper, we propose that for a boundary RG flow in two dimensions there exist boundary condition changing fields (RG defect fields) linking the UV and the infrared (IR) conformal boundary conditions which carry similar information on the expansion of boundary fields at the fixed points. We propose an expression for a pairing between IR and UV operators in terms of a four-point function with two insertions of the RG defect fields. For the boundary flows in minimal models triggered by ψ 13 perturbation, we make an explicit proposal for the RG defect fields. We check our conjecture by a number of calculations done for the example of ( p , 2) → ( p − 1, 1)⊕( p + 1, 1) flows.