Candidate Gaugings of Categorical Continuous Symmetry

Imagine you are trying to understand the different "flavors" or phases of a complex physical system, like a strange new type of liquid or a quantum material. For a long time, scientists used a standard rulebook (the Landau paradigm) to explain how these systems change from one state to another. But recently, they discovered some exotic materials—like certain quantum liquids—that don't follow these old rules. To understand them, physicists need a new kind of map.

This paper is about drawing a new map for systems that have continuous symmetries (think of a perfect sphere that looks the same no matter how you rotate it) and some hidden "glitches" or anomalies (like a secret rule that breaks the symmetry in a specific way).

Here is a breakdown of what the authors did, using simple analogies:

1. The Big Picture: The "Shadow" Theory

The authors are working with a concept called a SymTFT (Symmetry Topological Field Theory).

The Analogy: Imagine you have a 2D movie playing on a screen (the physical system you are studying). The authors suggest that this movie is actually the "shadow" cast by a 3D object floating behind it (the SymTFT).
The Goal: By studying the 3D object, you can figure out all the possible phases and rules of the 2D movie. If you know the shape of the 3D object, you know everything about the 2D shadow.

2. The "Glitch" and the "Kernel"

The systems they are studying have a specific "glitch" labeled by a number, $k$ .

The Analogy: Think of $k$ as a specific type of twist or knot in the fabric of the system.
The Tool: To study this, the authors use a mathematical tool called a Kernel.
- Imagine you have a giant, fuzzy photo of a crowd (the continuous symmetry). It's too blurry to see individual faces.
- The "Kernel" is like a special filter or a lens. When you look through this lens, the blur clears up just enough to see specific patterns and connections between people.
- The authors built a specific "lens" (based on a mix of two theories: BF theory and Chern-Simons theory) to look at these continuous symmetries.

3. The "Hopf Link" Test

To make their lens work, they needed to test it. They used a specific shape called a Hopf Link.

The Analogy: Imagine two rings of string linked together like a chain. In their mathematical world, they "thread" these rings through their 3D shadow object.
The Result: By calculating how these linked rings interact, they derived a set of numbers (matrices called S and T). These numbers act like a codebook.
- S-matrix: Tells you how different parts of the system swap places.
- T-matrix: Tells you how the system twists on itself.

4. Finding the "Safe" Symmetries (Gauging)

The main goal of the paper is to find which symmetries can be "gauged."

The Analogy: Imagine you have a group of people holding hands in a circle (the symmetry). "Gauging" is like asking, "Can we lock this circle in place and make it a rigid rule for the whole system?"
The Problem: Sometimes, if you try to lock the circle, the "glitch" ( $k$ ) causes the whole thing to fall apart.
The Solution: The authors used their new "lens" (the S and T matrices) to find the specific patterns that remain stable even with the glitch. They looked for a special "common eigenvector"—a pattern that stays exactly the same when you apply the S and T rules.
- If a pattern survives this test, it's a candidate for a stable phase.
- They found that for simple cases (like a circle, $U(1)$ ), their method perfectly matched what scientists already knew.
- For more complex shapes (like a sphere, $SU(2)$), their method produced new, specific formulas that suggest how these complex systems might behave.

5. The "Working Assumption" Caveat

It is important to note the authors' honesty about their method.

The Analogy: They are like architects who say, "If we assume this specific type of foundation exists, then here is the blueprint for the house."
They admit they haven't proven why the foundation (the specific 3D theory they chose) is the only correct one for all continuous symmetries. They are saying, "If we accept this model, here are the concrete results we get."
They treat their results as candidates. They are strong hints and consistent with known facts, but they are presented as a working model to be tested further, not as a final, unchangeable law of the universe.

Summary

In short, the authors built a new mathematical "lens" to look at complex, continuous quantum systems with hidden glitches. By threading linked rings through their theoretical 3D model, they created a codebook (matrices) that helps identify which symmetries can be safely "locked in" to create new phases of matter. Their method works perfectly for known simple cases and offers a promising new way to explore complex, unknown systems.

1. Problem Statement

The paper addresses the classification of phases and gaugings of quantum field theories (QFTs) possessing continuous global symmetries (specifically compact Lie groups) and potential 't Hooft anomalies.

Context: While the classification of phases for finite group symmetries is well-understood via Modular Tensor Categories (MTCs) and their Drinfeld centers, the extension to continuous symmetries remains an open problem.
Challenge: The symmetry category $\mathcal{C}_k(G)$ for a continuous group $G$ with anomaly $k$ is not uniquely defined in the literature (candidates include categories of quasi-coherent sheaves, measurable fields of Hilbert spaces, etc.). Furthermore, the standard algebraic criteria for identifying "Lagrangian algebra objects" (which correspond to consistent gaugings) in MTCs rely on semisimplicity, a property often lost in continuous settings.
Goal: To propose a concrete, calculable framework for identifying candidate gaugings and modular invariants for continuous symmetries without requiring a complete, rigorous definition of the underlying symmetry category.

2. Methodology

The authors adopt a kernel-theoretic approach based on two explicit working assumptions:

SymTFT Identification: The Symmetry Topological Field Theory (SymTFT) associated with a QFT with continuous $G$ $G$ -symmetry and anomaly $k$ $k$ is the 3D BF theory coupled to level- $k$ Chern-Simons (CS) theory with gauge group $G$ $G$ .
- Action: $S = \int \text{tr}(B \wedge F) + \frac{k}{4\pi} CS[A]$ .
Modular Criterion: Candidate Lagrangian algebra data (gaugings) in the Drinfeld center $\mathcal{Z}(\mathcal{C}_k(G))$ correspond to common $+1$ eigenvectors of the modular $S$ and $T$ kernels, generalizing the known theorem for finite groups/MTCs.

Technical Execution:

Semiclassical Calculation: Instead of solving the full path integral for the non-abelian continuous theory, the authors perform a semiclassical evaluation of topological correlators on $S^3$ .
Hopf Link Correlators: They compute the expectation values of loop operators (Wilson and 't Hooft/Gukov-Witten operators) linked as a Hopf link.
- The $S$ -kernel is derived from the correlator of two linked loops $W_{[g],\sigma}(\ell_1) W_{[h],\rho}(\ell_2)$ .
- The $T$ -kernel is derived from the self-linking (framing) of a single loop.
Reduction to Holonomies: The path integral is reduced to an integral over flat connections with prescribed holonomies. The authors utilize the Weyl group and the structure of the centralizer groups $C_G(g)$ to evaluate the integrals, focusing on the "regular sector" (where centralizers are maximal tori) and handling singular sectors via limits.

3. Key Contributions

A. Derivation of Modular Kernels

The paper derives explicit integral kernel formulas for the modular $S$ and $T$ matrices for a general compact Lie group $G$ with anomaly $k$ .

$S$ -Kernel: For regular elements $g, h$ (conjugacy classes) and representations $\sigma, \rho$ of their centralizers, the kernel is given by a sum over the Weyl group $W$ :
$S^{(k)}_{([g],\sigma),([h],\rho)} \propto \sum_{w \in W} e^{-i \frac{k}{2\pi} \text{tr}(\alpha w(\beta))} \chi^*_\sigma(w(\beta)) \chi^*_\rho(w^{-1}(\alpha))$
where $g=e^{i\alpha}, h=e^{i\beta}$ . This formula includes Jacobian factors related to the Weyl denominator.
$T$ -Kernel: Derived as a diagonal kernel involving the quadratic Casimir (trace of $\alpha^2$ ) and the character:
$T^{(k)}_{([g],\sigma),([h],\rho)} \propto \delta_{[g],[h]} \delta_{\sigma,\rho} e^{i \frac{k}{4\pi} \text{tr}(\alpha^2)} \frac{\chi_\sigma(g)}{d_\sigma}$

B. Recovery of Known Cases

The authors verify that their semiclassical formulas reproduce established results in specific limits:

$U(1)$ Symmetry: The derived kernels match the known modular transformations of the $\widehat{u(1)}_k$ algebra and correctly identify the modular invariants for compact bosons.
$SU(2)$ at Level $k=0$ : The formulas recover the Kac-Peterson $S$ -matrix for the $SU(2)_k$ WZW model when restricted to the Verlinde lattice (singular sector), demonstrating that the continuous kernel correctly descends to the discrete lattice theory.

C. Identification of Candidate Gaugings

Using the derived kernels, the authors solve the eigenvalue equations $S \cdot V = V$ and $T \cdot V = V$ to find candidate Lagrangian algebra objects (which correspond to consistent gaugings). They identify several types of boundary data:

Dirichlet Type ( $L_{Dir}$ ): Corresponds to the un-gauged global symmetry.
Neumann Type ( $L_{Neu}$ ): Corresponds to gauging the entire global symmetry group.
$SO(3)$ Type: Corresponds to gauging the center $\mathbb{Z}_2$ of $SU(2)$.
$A_q$ Type: Corresponds to gauging a discrete $\mathbb{Z}_q$ subgroup of the symmetry.

4. Results

Consistency: The derived kernels satisfy the modular relations $SS^\dagger = TT^\dagger = 1$ and $(ST)^3 = C$ (charge conjugation) in the regular sector, validating the internal consistency of the semiclassical model.
Singular Sector Extension: The paper demonstrates that the singular sectors (where the centralizer is larger than the maximal torus) are not pathological but are essential extensions of the regular sector. Specifically, restricting the continuous $SU(2)$ kernel to the Verlinde lattice recovers the exact discrete $S$ -matrix of the $SU(2)_k$ theory.
Modular Invariants: For the $U(1)$ case, the candidate Lagrangian algebras are shown to be in one-to-one correspondence with the known modular invariants of the $\widehat{u(1)}_k$ algebra.

5. Significance

Bridging Algebra and Physics: The work provides a concrete "semiclassical kernel" bridge between the abstract algebraic data of continuous symmetry categories (which are mathematically subtle) and physical observables (partition functions and modular invariants).
Practical Tool: It offers a calculable method to determine candidate gaugings for continuous symmetries without needing to resolve the deep mathematical questions regarding the precise analytic realization of the symmetry category (e.g., quasi-coherent vs. measurable sheaves).
Generalization: It extends the framework of SymTFT and Drinfeld centers from finite groups to compact Lie groups, suggesting a unified picture where the "kernel-theoretic" data is the primary physical object.
Future Directions: The paper sets the stage for future work to rigorously prove the categorical equivalence and to extend these results to non-semisimple continuous categories and more general Lie groups.

In summary, the paper successfully constructs a working model where semiclassical path integrals in BF+kCS theory yield modular kernels that correctly predict candidate gaugings for continuous symmetries, recovering known results and providing new predictions for general compact Lie groups.