⚛️ high-energy theory

Defect flows from functional relations in the asymptotic semiring

This paper investigates how perturbations of topological defects by relevant holomorphic fields in two-dimensional conformal field theory lead to renormalization group flows that define a semiring homomorphism preserving functional relations between translation invariant defects and an asymptotic semiring constructed from topological defects, thereby constraining the infrared endpoints of these flows in models such as Virasoro minimal models.

Original authors: Federico Ambrosino, Anatoly Konechny, Ingo Runkel, Gérard M. T. Watts

Published 2026-10-23

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CC BY 4.0

Original authors: Federico Ambrosino, Anatoly Konechny, Ingo Runkel, Gérard M. T. Watts

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of theoretical physics, researchers often study the fundamental rules that govern how energy and matter behave. One powerful way to explore these rules is by looking at "defects." Imagine a piece of fabric that is perfectly smooth everywhere, except for a single, sharp line running through it. In the quantum world, these lines are not just imperfections; they are active features that can carry energy, interact with other lines, and even change the nature of the space they inhabit. Some of these lines are "topological," meaning they are incredibly robust and can be stretched or twisted without changing their essential character, much like a knot in a rope that stays a knot no matter how you pull the ends. Others are more rigid, able to move only in straight lines without bending.

When scientists introduce a specific type of disturbance to these lines, something fascinating happens. If the disturbance is strong enough, the line does not just wiggle; it flows. It changes its identity, evolving from its starting form into a completely different, stable form as it settles into a new state. This process is known as a "renormalization group flow." For decades, physicists have known that these lines eventually settle down into topological forms, but predicting exactly which form they will become has been a difficult puzzle. The challenge lies in the fact that the path the line takes is complex, and the rules governing its journey are hidden within layers of mathematical abstraction. Understanding where these lines end up is crucial because it reveals the deep, hidden symmetries of the universe, offering a glimpse into how different physical theories are connected.

A team of researchers has now developed a new method to solve this puzzle, turning a complex physical problem into a structured algebraic game. Instead of trying to track the messy, continuous motion of the defect line as it changes, the authors focused on the "functional relations" that govern how these lines combine. They realized that if you take two lines and fuse them together, the result follows strict rules, similar to how numbers add or multiply. By treating these lines as elements in a special kind of mathematical system called a "semiring," they could map the entire journey of a defect from its starting point to its final destination. The key insight was to look at the behavior of the line when the disturbance becomes extremely large. In this extreme limit, the line's behavior simplifies into a predictable pattern that depends on the direction from which the disturbance is applied.

The researchers constructed a new mathematical framework, which they call an "asymptotic semiring," to capture this final, simplified behavior. They proved that the flow of the defect acts like a translator, taking the complex rules of the starting point and converting them into the simpler rules of the endpoint. This translation preserves the essential relationships between the lines. If the starting lines follow a certain equation, the ending lines must follow a corresponding equation in the new framework. By solving these equations, the team could determine exactly which topological line a defect would become, without needing to simulate the entire flow step-by-step.

To test their theory, the team applied it to a specific family of quantum models known as minimal models, which are simplified versions of the theories used to describe real-world materials and particle interactions. They focused on a particular type of disturbance that is well-understood in these models. Using their new algebraic method, they derived a set of equations that the final states must satisfy. When they solved these equations, they found a unique solution that predicted exactly where each defect would end up. For the simplest case, known as the Lee-Yang model, their method predicted that the defect would always settle into the most basic, identity-like state, regardless of the direction of the flow. This prediction matched perfectly with previous, highly detailed numerical simulations, confirming that their algebraic approach was correct.

The power of this method becomes even more apparent when looking at more complex models. In these cases, the final state of the defect depends on the angle at which the disturbance is applied. The researchers found that the defect could end up in one of several different topological forms, switching between them as the angle changed. Their algebraic framework successfully mapped out these transitions, identifying the precise angles where the switch occurs. They discovered that for certain models, the defect flows to a specific set of stable lines that form a predictable pattern, while for others, the pattern is more intricate. Crucially, they showed that this flow cannot be described by a simple, direct mapping of the starting rules to the ending rules; the angle dependence is essential. This finding rules out the idea that the final state is determined solely by the initial properties of the defect, highlighting the dynamic role of the environment in shaping the outcome.

The study also addressed the limits of their approach. The method relies on the assumption that the mathematical integrals describing the flow converge without needing special adjustments. This condition holds true for some models but fails for others, particularly those that are "unitary," which are models that describe physical systems with positive energy. The authors noted that for these unitary models, the standard mathematical tools break down, and the flow might behave differently. They did not claim to have solved the problem for these cases, but rather identified the boundary where their current method works and where it does not. This distinction is vital for future research, as it points to where new mathematical tools will be needed.

By framing the problem in terms of algebraic structures rather than continuous motion, the researchers have provided a powerful new tool for understanding quantum field theories. Their work demonstrates that the complex, fluid behavior of quantum defects can be captured by rigid, logical relationships. This approach not only confirms existing numerical results but also offers a way to predict the behavior of defects in models where numerical simulations are too difficult to perform. The ability to constrain the possible endpoints of these flows using pure algebra suggests that there is a deeper, more fundamental order underlying the chaotic appearance of quantum interactions. As the authors look forward, they see opportunities to extend this method to more complex systems, including those with boundaries and combined disturbances, potentially revealing even more of the hidden symmetries that govern our universe.

Title: Defect flows from functional relations in the asymptotic semiring

Problem Statement
The paper addresses the challenge of predicting the endpoints of renormalisation group (RG) flows for translation-invariant defects (TIDs) in two-dimensional conformal field theories (CFTs). Specifically, it investigates TIDs obtained by perturbing a topological defect $T$ with a relevant holomorphic defect field $\psi$ of conformal weight $(h, 0)$ with $h < 1$ . While it is known that such flows start at a topological defect and end at another topological defect in the infrared (IR), determining the specific IR endpoint is non-trivial. The authors ask whether the additional categorical structure available on TIDs can be used to constrain or predict these flow endpoints.

Methodology
The authors develop a framework combining category theory, representation theory, and asymptotic analysis:

Categorical Description of Perturbed Defects:
- They utilize the chiral 3D topological field theory (TFT) presentation of 2D CFT correlators.
- Holomorphically perturbed defects are described by a monoidal category $\mathcal{C}_X$ , where objects are pairs $(T, \eta)$ consisting of a topological defect $T$ and a morphism $\eta: X \otimes T \to T$ (where $X$ is the representation of the perturbing field).
- To facilitate computation, $\mathcal{C}_X$ is equivalently described as the category of modules over a specific quiver $Q_X$ . The tensor product (fusion) of perturbed defects corresponds to a specific tensor product of quiver modules.
- Functional relations between defect operators are derived from short exact sequences in this category, which translate to relations in the Grothendieck semiring $\text{Gr}(\mathcal{C}_X)_{\ge 0}$ .
Asymptotic Semiring and RG Flow:
- The authors assume that for large coupling $\lambda = re^{i\theta}$ , the perturbed defect operators exhibit " $\alpha$ -asymptotics," where $\alpha = 1/(1-h)$ . The leading behavior is of the form $D_\infty(\theta) \exp(u(\theta) r^\alpha)$ .
- They construct an "asymptotic semiring" $\mathcal{A}$ , whose elements are pairs $[D_\infty(\theta), a(\theta)]$ representing the IR topological defect and the real part of the asymptotic exponent, modulo finite sets of angles (anti-Stokes rays).
- The addition in $\mathcal{A}$ is defined by the "max-plus" rule (dominance of the largest exponent), and multiplication is the standard product of defects with additive exponents.
The Homomorphism Theorem:
- The core theoretical result is that under the assumption of $\alpha$ -asymptotics, the RG flow defines a semiring homomorphism $\Lambda: \text{Gr}(\mathcal{C}_X)_{\ge 0} \to \mathcal{A}$ .
- This homomorphism preserves the functional relations derived in the UV (perturbed) theory. Consequently, the image of these relations in the asymptotic semiring must hold, imposing strong constraints on the possible IR endpoints $D_\infty$ .

Key Results
The authors apply this formalism to Virasoro minimal models $M(p, q)$ perturbed by the holomorphic $(1,3)$ -field.

Lee-Yang Model ( $M(2,5)$ ):
- Using numerical inputs from Truncated Conformal Space Approach (TCSA) regarding the location of anti-Stokes rays and the sign of the exponent, they solve the functional relations in the asymptotic semiring.
- They prove that the flow endpoint for the perturbed defect $D_\tau(\lambda)$ is uniquely the identity defect ( $D_\infty = \text{id}$ ) for all angles except the anti-Stokes ray. This confirms previous numerical findings.
General Minimal Models ( $M(p, q)$ ):
- Case $q > 2p$ : Assuming a single Stokes sector (one anti-Stokes ray along the negative real axis) and a positivity condition, they prove that $p$ must be 2. In this case, the solution is unique, and all flow endpoints are the identity defect.
- General $p, q$ ( $q > \frac{4}{3}p$ ): They construct an explicit solution to the Hirota system of functional relations in the asymptotic semiring for general $p, q$ $p, q$ .
  - The solution involves a "tower" of defects $X_s$ corresponding to the perturbed defects $D_{(1,s)}$ .
  - The IR endpoints $D_\infty$ depend on the angle $\theta$ of the coupling. As $\theta$ varies, the flow endpoints cycle through the topological defects $D_{(r,1)}$ (the first column of the Kac table).
  - For $p > 2$ , the perturbed defects necessarily possess multiple anti-Stokes rays, and the flow endpoint is not constant but switches between different topological defects depending on the direction of the RG flow.
- Non-existence of Direct Homomorphisms: The paper demonstrates that there is no semiring homomorphism from the semiring of the UV defect labels $D_{(1,s)}$ directly to the semiring of the IR labels $D_{(r,1)}$ . The angle dependence and the max-plus structure of the asymptotic semiring are essential for the relations to hold.

Significance and Claims
The paper claims to provide a new algebraic tool for analyzing RG flows of defects without solving integral equations (as in Thermodynamic Bethe Ansatz) or relying solely on numerical simulations.

Constraint Mechanism: The primary significance is the demonstration that functional relations in the UV, when mapped to the asymptotic semiring, strictly constrain the possible IR endpoints. This allows for the prediction of flow endpoints based purely on algebraic consistency.
Algebraic Structure: The work identifies that the correct mathematical structure to describe these flows is a semiring homomorphism involving an angle-dependent asymptotic semiring, rather than a simple map between fusion rings of topological defects.
Validation: The analytical solutions derived in the asymptotic semiring are shown to match numerical results from TCSA (detailed in the companion paper [1]), validating the approach.
Scope: The authors note that their current formalism relies on the convergence of perturbative integrals without regularization, which currently restricts the rigorous application to non-unitary models or specific regimes of unitary models where counterterms are not required. They view the construction of a "categorified" version of the homomorphism (a monoidal functor) as a natural and important future direction.

The paper does not claim to solve the general problem of defect flows for all CFTs or all perturbations, but rather establishes a rigorous method for a specific class of holomorphic perturbations in rational CFTs, highlighting the power of asymptotic algebraic structures in constraining RG dynamics.

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