Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2602.12085

Refined half-integer condition on RG flows

Ken Kikuchi

Published Feb 12, 2026• 0 citations
arXivHTMLPDF

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.

DOI
10.48550/arxiv.2602.12085

Related papers

Rational RG flow and half-integer condition

Ken Kikuchi

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.

International Journal of Modern Physics A•Apr 10, 2025•0 citations

Renormalization group defects for boundary flows

Anatoly Konechny

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.

•Aug 17, 2016•4 citations

Symmetry enhancement in RCFT II

Kikuchi, Ken

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.

arXiv:2207.06433•Jul 13, 2022•5 citations

Renormalization group defects for boundary flows

Anatoly Konechny

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.

arXiv:1211.3665•Journal of Physics A Mathematical and Theoretical•Mar 20, 2013•6 citations

RG flows from WZW models

Ken Kikuchi

We constrain renormalization group flows from $ABCDE$ type Wess-Zumino-Witten models triggered by adjoint primaries. We propose positive Lagrangian coupling leads to massless flow and negative to massive. In the conformal phase, we prove an interface with the half-integral condition obeys the double braiding relations. Distinguishing simple and non-simple flows, we conjecture the former satisfies the half-integral condition. If the conjecture is true, some previously allowed massless flows are ruled out. For $A$ type, known mixed anomalies fix the ambiguity in identifications of Verlinde lines; an object is identified with its charge conjugate. In the massive phase, we compute ground state degeneracies.

arXiv:2212.13851•Dec 28, 2022•1 citation

Fusion category symmetry. Part II. Categoriosities at c = 1 and beyond

Ryan Thorngren, Yifan Wang

A bstract We study generalized symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. This paper follows our companion paper on gapped phases and anomalies associated with these symmetries. In the present work we focus on identifying fusion category symmetries, using both specialized 1+1D methods such as the modular bootstrap and (rational) conformal field theory (CFT), as well as general methods based on gauging finite symmetries, that extend to all dimensions. We apply these methods to c = 1 CFTs and uncover a rich structure. We find that even those c = 1 CFTs with only finite group-like symmetries can have continuous fusion category symmetries, and prove a Noether theorem that relates such symmetries in general to non-local conserved currents. We also use these symmetries to derive new constraints on RG flows between 1+1D CFTs.

Journal of High Energy Physics•Jul 5, 2024•99 citations
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.
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.
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 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.
We constrain renormalization group flows from ABCDEABCDEABCDE type Wess-Zumino-Witten models triggered by adjoint primaries. We propose positive Lagrangian coupling leads to massless flow and negative to massive. In the conformal phase, we prove an interface with the half-integral condition obeys the double braiding relations. Distinguishing simple and non-simple flows, we conjecture the former satisfies the half-integral condition. If the conjecture is true, some previously allowed massless flows are ruled out. For AAA type, known mixed anomalies fix the ambiguity in identifications of Verlinde lines; an object is identified with its charge conjugate. In the massive phase, we compute ground state degeneracies.
A bstract We study generalized symmetries of quantum field theories in 1+1D generated by topological defect lines with no inverse. This paper follows our companion paper on gapped phases and anomalies associated with these symmetries. In the present work we focus on identifying fusion category symmetries, using both specialized 1+1D methods such as the modular bootstrap and (rational) conformal field theory (CFT), as well as general methods based on gauging finite symmetries, that extend to all dimensions. We apply these methods to c = 1 CFTs and uncover a rich structure. We find that even those c = 1 CFTs with only finite group-like symmetries can have continuous fusion category symmetries, and prove a Noether theorem that relates such symmetries in general to non-local conserved currents. We also use these symmetries to derive new constraints on RG flows between 1+1D CFTs.