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On general background of quantum non-invertible symmetry in 2D

This paper establishes a general framework for computing partition functions of a two-dimensional theory with dual quantum symmetry $\mathrm{Rep}(G)$ in arbitrary non-invertible symmetry backgrounds by relating them to the partition functions of the original theory with non-abelian symmetry $G$ , a result validated through examples involving both finite and compact Lie groups.

Original authors: Jin Chen, Qiang Jia

Published 2026-09-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jin Chen, Qiang Jia

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 quantum world, symmetry is not just a pretty pattern; it is a fundamental rulebook that dictates how particles behave and how the universe organizes itself. For decades, physicists have understood symmetry through the lens of groups, mathematical structures that describe how things can be rotated, flipped, or swapped without changing the underlying laws of nature. When a system possesses such a symmetry, it leaves behind a signature: a set of constraints on what particles can exist and how they interact. However, a newer, more exotic form of symmetry has recently emerged in theoretical physics. Unlike the familiar kind, which can be undone by reversing a move, these "non-invertible" symmetries cannot simply be flipped back. They are more like a knot that, once tied, cannot be untied by a simple reversal of the knotting process. These symmetries are described by complex networks of invisible lines that weave through space, and understanding how to measure them has been a significant challenge.

The core question researchers have been grappling with is how to translate between two different ways of looking at a physical system. On one side, you have a theory with a standard, ordinary symmetry, where you can turn on a background field to probe its behavior. On the other side, you have the "dual" version of that theory, created by a process called gauging, where the original symmetry is promoted to a dynamic force. In this dual world, the ordinary symmetry is replaced by the mysterious non-invertible kind. The difficulty lies in the fact that while we know how to calculate the behavior of the original system, we have lacked a clear, direct method to calculate the behavior of the dual system when it is subjected to these complex, non-invertible backgrounds. It is as if we had a map of a city but no way to translate it into the language of a different culture that lives in the same space, leaving us unable to predict how the new culture would react to specific events.

In a new study, researchers have constructed a precise mathematical bridge to solve this problem. They developed a general framework that acts as a translator, allowing physicists to compute the behavior of a theory with non-invertible symmetry directly from the known behavior of its ordinary counterpart. The team focused on two-dimensional theories, which serve as simplified models for understanding more complex quantum systems. They started with a theory that has a standard, non-abelian symmetry—a type of symmetry where the order of operations matters, much like putting on socks before shoes is different from shoes before socks. By applying a specific procedure known as flat gauging, which involves summing over all possible ways the symmetry can be arranged in a flat, non-curving space, they generated a new theory. This new theory possesses a dual symmetry described by the representation category of the original group, a structure that contains the non-invertible lines.

The breakthrough lies in the method used to connect the two theories. The researchers derived a set of rules, or kernels, that function like a conversion formula. To find the behavior of the new, dual theory in a specific non-invertible background, one does not need to start from scratch. Instead, they can take the known results from the original theory, which are calculated based on pairs of commuting elements in the symmetry group, and apply a specific transformation. This transformation involves tracing the path of the invisible symmetry lines through the background, multiplying the mathematical matrices that represent the symmetry operations, and then combining them at the points where the lines meet. Crucially, this process keeps track of every detail, including the specific channels through which the lines fuse and the different ways they can join, ensuring that no information is lost in the translation.

The authors tested this framework on several concrete examples using finite groups, which are mathematical structures with a limited number of elements. They examined groups like the symmetries of a triangle, a square, and a quaternion group. In cases where the fusion of symmetry lines was simple, with no overlapping possibilities, the method worked perfectly, matching previous results derived through more indirect means. However, the true power of their approach was demonstrated in more complex scenarios where multiple fusion channels existed simultaneously. In these cases, the researchers showed that their method could distinguish between different ways the lines could combine, a level of detail that previous techniques struggled to resolve. They also extended their formalism to infinite groups, specifically the special unitary group of two dimensions, showing that the same logic holds even when the number of possible symmetry operations is infinite, provided one uses integrals instead of simple sums.

The study confirms that the relationship between an ordinary symmetry and its non-invertible dual is not just a theoretical curiosity but a calculable reality. By establishing a direct link between the partition functions—the mathematical quantities that summarize the physical state of a system—of the original and dual theories, the researchers have provided a tool that can be used to explore a wide variety of quantum phases. They demonstrated that if one takes the dual theory and applies the reverse transformation, the original theory is recovered exactly, proving that the two descriptions are two sides of the same coin. This work does not just offer a new calculation technique; it provides a deeper conceptual understanding of how non-invertible symmetries operate, showing that they are deeply rooted in the representation theory of the original groups. The findings suggest that the complex, knotted nature of non-invertible symmetry can be unraveled and understood through the familiar language of standard group theory, opening the door to exploring new phases of matter and quantum field theories that were previously out of reach.

Technical Summary: On General Background of Quantum Non-Invertible Symmetry in 2D

Problem Statement
The paper addresses the challenge of explicitly relating the partition functions of a two-dimensional quantum field theory $T_G$ with an ordinary finite symmetry group $G$ to those of its orbifold theory $\tilde{T}_{\text{Rep}(G)} = T_G/G$ , which possesses a dual, generally non-invertible symmetry described by the fusion category $\text{Rep}(G)$ . While the categorical definition of gauging is conceptually complete, it lacks a direct, computable formula for partition functions in arbitrary non-invertible symmetry backgrounds. Existing approaches, such as modular bootstrap methods, often rely on ansätze and modular covariance to constrain coefficients, leaving open the question of whether these coefficients can be derived directly from the microscopic representation theory of $G$ without invoking modular transformations. Specifically, there is a need to resolve the partition functions in $\text{Rep}(G)$ backgrounds, which are labeled not only by the lines wrapping cycles but also by fusion channels and junction multiplicities, in terms of the flat $G$ -backgrounds (commuting holonomies).

Methodology
The authors develop a general framework based on the representation theory of the finite group $G$ to construct a direct Fourier transform between the two sets of partition functions.

Forward Transform ( $G \to \text{Rep}(G)$ ): The partition function $\tilde{Z}$ of the dual theory in a resolved $\text{Rep}(G)$ defect network (specifically on a torus with trivalent junctions) is expressed as a sum over the flat $G$ -backgrounds $Z_{g,h}$ (where $g,h$ are commuting holonomies). The kernel of this transform, $W$ , is derived by transporting Wilson lines (realizing the $\text{Rep}(G)$ defects) through the fixed $G$ -background. This involves contracting representation matrices with intertwiners and co-intertwiners at the junctions of the defect network. The method explicitly tracks fusion channels and multiplicity indices, ensuring the result is independent of the specific position of junctions due to the intertwining properties of the operators.
Inverse Transform ( $\text{Rep}(G) \to G$ ): To recover the original $G$ -background partition functions from the dual theory, the authors utilize the regular Frobenius algebra $A = \text{Fun}(G) \cong \bigoplus_{\sigma \in \hat{G}} d_\sigma \sigma$ in $\text{Rep}(G)$ . They construct the inverse kernel by "dressing" the multiplication and comultiplication maps of this algebra with commuting right-regular actions. This dressing isolates the simultaneous-conjugacy orbits of general commuting pairs $(g,h)$ . The authors derive a specific gauge choice (based on Schur orthogonality) that simplifies the inverse kernel, showing it is proportional to the forward kernel with swapped indices and arguments.
Generalization to Lie Groups: The framework is formally extended to compact Lie groups, specifically $\text{Rep}(SU(2))$ . In this case, finite sums are replaced by Peter-Weyl expansions and integrals over flat connections. The authors address the regularization of the divergent dimension of the Frobenius algebra and the measure on the moduli space of commuting pairs, focusing on the regular stratum where the centralizer is a maximal torus.

Key Contributions and Results

Explicit Kernel Construction: The paper provides explicit formulas for the Fourier kernel $W_{\rho,\sigma}^{\lambda;i,j}(g,h)$ and the inverse kernel $C_{\rho,\sigma}^{\lambda;i,j}(g,h)$ for arbitrary $\text{Rep}(G)$ defect networks. These kernels are computed directly from the representation matrices, intertwiners, and co-intertwiners of $G$ .
Resolution of Multiplicities: Unlike previous modular bootstrap approaches that might compress multiplicity spaces, this framework retains full resolution of the multiplicity indices ( $i, j$ ) at junctions. This is demonstrated explicitly in the $\text{Rep}(A_4)$ example, where the fusion channel $Z \otimes Z \to Z$ has multiplicity 2.
Verification via Examples: The framework is tested on several finite groups:
- Multiplicity-Free Cases: $\text{Rep}(S_3)$ , $\text{Rep}(D_4)$ , and $\text{Rep}(Q_8)$ . The results are compared with existing modular bootstrap literature, showing exact agreement for $S_3$ and agreement up to a discrete torsion phase for $D_4$ and $Q_8$ .
- Higher-Multiplicity Case: $\text{Rep}(A_4)$ , demonstrating the handling of non-trivial multiplicity spaces in the fusion rules.
Inverse Relation: The authors prove that the composition of the inverse and forward transforms yields the identity on the physical space of $G$ -backgrounds. Conversely, the composition of the forward and inverse transforms acts as a projector on the space of resolved defect partition functions, which may form a redundant set.
Lie Group Extension: For $\text{Rep}(SU(2))$ , the authors derive the kernels using Clebsch-Gordan coefficients and Wigner D-matrices. They show that "regauging" the $\text{Rep}(SU(2))$ symmetry recovers the original partition function $Z_{g,h}$ on the regular stratum, consistent with flat gauging over the Cartan torus.

Significance
The paper claims to provide a "controlled setting" where ordinary flat backgrounds and fusion-category defect networks are related explicitly through a direct microscopic calculation. By constructing the transform directly from the representation theory of $G$ , the authors bypass the need for modular covariance to propagate ansätze between partial traces. This offers a more fundamental derivation of the coefficients appearing in generalized orbifold partition functions. The work establishes a concrete dictionary between the data of a theory with a non-invertible symmetry and its parent theory with an ordinary symmetry, applicable to both finite groups and compact Lie groups (at least on the regular stratum). The results verify that the categorical condensation of the regular Frobenius algebra correctly implements the inverse gauging procedure, recovering the original theory sector by sector.

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