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Anomalies of TQFTs coupled to parameter spaces and constraints on ground-state degeneracies

This paper formulates anomalies in $d$ -dimensional topological quantum field theories coupled to parameter spaces as obstructions to constructing parametrized families, deriving a general constraint that relates ground-state degeneracy to torsion anomalies and applying it to analyze (2+1)-dimensional TQFTs with framing and 1-form symmetry anomalies.

Original authors: Takamasa Ando, Ryan Thorngren

Published 2026-11-02

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Takamasa Ando, Ryan Thorngren

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

Deep within the fabric of the universe, there exist strange states of matter that do not behave like ordinary solids, liquids, or gases. These are quantum systems where the particles are so deeply linked to one another that they form a single, vast, entangled whole. In these materials, the behavior of the whole cannot be understood by looking at the parts in isolation. Scientists call this "topological order." A defining feature of such systems is that their lowest energy state, the ground state, is not unique. Instead, the system can settle into several different configurations that have the exact same energy. The number of these possible configurations, known as the ground-state degeneracy, is not random; it is a rigid topological invariant, a number that depends only on the shape of the space the material occupies, not on the specific details of the material itself.

When these systems possess global symmetries—rules that remain unchanged no matter how you rotate or shift the system—they can carry subtle inconsistencies known as anomalies. In physics, an anomaly is not a mistake, but rather a fundamental obstruction. It means that the symmetry of the system cannot be perfectly preserved when the system is coupled to a background field, much like a background magnetic or gravitational field. These anomalies act as powerful constraints, dictating what kinds of quantum behavior are possible and what kinds are strictly forbidden. For decades, physicists have used these constraints to understand the limits of quantum matter, but a complete picture of how these anomalies interact with the geometry of space and the specific number of ground states has remained elusive.

In a recent study, researchers Takamasa Ando and Ryan Thorngren have developed a new framework to map these obstructions and derive precise rules for the number of ground states in such systems. They approached the problem by treating the quantum system not as a static object, but as a family of theories that can vary continuously as they are coupled to different parameter spaces. Imagine a quantum material whose properties change smoothly as you dial a knob; the researchers asked whether it is possible to define the theory consistently for every setting of that knob. They found that in many cases, it is not possible. The inability to construct a smooth, consistent family of theories across all settings is the anomaly. This obstruction is not merely a mathematical curiosity; it is a physical reality that leaves a measurable imprint on the system.

The core of their work involves a clever geometric trick. They considered what happens when you take a quantum system living in a certain number of dimensions and wrap it around a closed surface, effectively compressing it into a lower-dimensional system. This process, known as compactification, turns the complex quantum field theory into a simpler quantum mechanical system. The researchers discovered that the anomaly of the original system manifests as a specific phase factor in this lower-dimensional version. This phase factor is directly tied to the ground-state degeneracy. By analyzing how this phase behaves when the system is placed on different shapes of space, they derived a universal constraint: the product of the ground-state degeneracy and a specific value related to the anomaly must always be a whole number.

This finding provides a powerful new tool for physicists. It means that if you know the anomaly of a system, you can immediately calculate strict limits on how many ground states it can have. Conversely, if you measure the ground-state degeneracy, you can infer the nature of the anomaly. The researchers applied this rule to specific types of quantum systems in three dimensions, including those with fractional quantum Hall responses and those with framing anomalies, which arise from the way the system is oriented in space. In these cases, their formula successfully predicted the allowed ground-state degeneracies, confirming that the anomaly acts as a strict gatekeeper.

The study also addressed a long-standing question about symmetry-preserving topological phases. Some theories suggest that certain symmetries can be preserved in a quantum system even if the system has a unique ground state. However, Ando and Thorngren showed that for many types of anomalies, this is impossible. If a system carries a specific kind of anomaly, it cannot have a unique ground state while preserving that symmetry. The anomaly forces the system to either break the symmetry or to have multiple ground states. This rules out the existence of certain hypothetical quantum phases that were previously thought to be possible, narrowing the landscape of what nature can actually build.

To reach these conclusions, the authors had to navigate complex mathematical structures involving the classification of symmetries and the topology of the spaces in which the theories live. They distinguished between two ways of viewing the symmetries involved: one where the symmetries are treated as continuous deformations, and another where they are treated as discrete, rigid categories. They found that the choice of how to view these symmetries changes the classification of the anomalies. By focusing on the discrete perspective, they were able to uncover obstructions that were invisible in the continuous view. This distinction allowed them to formulate a precise condition for when a symmetry can be preserved and when it must be broken.

The researchers also proposed a conjecture regarding how these anomalies can be detected. They suggested that by wrapping the quantum system around specific geometric shapes, such as a bundle of surfaces, one can reveal the presence of an anomaly. If the anomaly is non-trivial, the system will exhibit a specific phase shift that cannot be removed. This provides a concrete experimental or numerical protocol for identifying anomalies in complex quantum materials. While the full mathematical proof of this conjecture remains a subject for future work, the authors argue that it is physically reasonable and consistent with known examples.

Ultimately, this work bridges the gap between abstract mathematical classifications of quantum anomalies and the tangible physical properties of quantum matter. By showing that anomalies impose strict, calculable limits on the number of ground states, the researchers have provided a new lens through which to view the quantum world. Their findings suggest that the universe is far more constrained than it might appear, with the very possibility of certain quantum states being dictated by deep topological rules. For scientists studying quantum materials, this offers a new way to predict the behavior of exotic systems and to understand why certain phases of matter exist while others do not. The study does not claim to have solved every mystery of quantum topology, but it has laid down a firm foundation, turning vague obstructions into precise, testable constraints.

Technical Summary: Anomalies of TQFTs Coupled to Parameter Spaces and Constraints on Ground-State Degeneracies

Problem Statement
Global symmetry anomalies in long-range entangled quantum many-body systems impose constraints on their dynamics, particularly regarding ground-state degeneracies (GSD). While 't Hooft anomalies are well-understood as obstructions to gauging global symmetries, this paper addresses a broader class of anomalies arising when Topological Quantum Field Theories (TQFTs) are coupled to parameter spaces. These anomalies, often referred to as sigma model anomalies or anomalies in spaces of coupling constants, manifest as obstructions to constructing continuous families of TQFTs. The central problem is to formulate these anomalies rigorously within the framework of parametrized TQFTs and to derive general, non-perturbative constraints on the GSD of gapped systems, specifically relating the degeneracy to torsion anomalies.

Methodology
The authors employ the cobordism hypothesis and the homotopy theory of moduli spaces to formulate anomalies.

Parametrized TQFTs and Moduli Spaces: The paper utilizes the correspondence between TQFTs coupled to a parameter space $X$ and parametrized families of TQFTs over $X$ . The moduli space $\mathcal{M}$ of a TQFT serves as the classifying space for these families.
Postnikov Tower Analysis: Anomalies are defined as obstructions to lifting maps from the parameter space $X$ to the moduli space $\mathcal{M}$ through its Postnikov tower. The $k$ -th anomaly corresponds to an obstruction to lifting to the $(k+1)$ -th stage of the tower, classified by cohomology classes in $H^{k+2}(X, \pi_{k+1}\mathcal{M})$ .
Topology of the Universal Symmetry: A critical methodological distinction is made regarding the topology assigned to the universal $(d-1)$ $(d - 1)$ -form $U(1)$ $U (1)$ symmetry (which acts by an overall phase on quantum states). The authors analyze two choices:
- Standard Topology: Corresponds to deformation classes of invertible bulk theories.
- Discrete Topology: Corresponds to isomorphism classes. This choice is essential for capturing torsion anomalies and constraints on GSD that are invisible under deformation equivalence (e.g., in systems with fractional filling).
Compactification and Fiber Bundles: To derive constraints on GSD, the authors consider the compactification of a $d$ -dimensional TQFT on a spatial manifold $\Sigma^{d-1}$ , resulting in a $(0+1)$ -dimensional quantum mechanical system. They analyze the projective ground-state bundle over a parameter space and relate its fractional phase to the bulk anomaly via fiber bundles of the form $\Sigma^{d-1} \to W^{d+1} \to Y^2$ , where $Y^2$ is a closed surface.

Key Contributions and Results

Formulation of Anomalies as Obstructions: The paper formalizes anomalies of TQFTs coupled to parameter spaces as obstructions to the existence of genuine parametrized families. It clarifies that the classification of these anomalies depends on the choice of topology for the universal $U(1)$ symmetry. The discrete topology is shown to be necessary for detecting torsion anomalies relevant to ground-state constraints.
The Anomaly-GSD Formula: The primary result is a general constraint relating the ground-state degeneracy $N(\Sigma^{d-1}, \hat{g})$ to the torsion anomaly $\omega$ of the system. For a $d$ -dimensional TQFT coupled to a parameter space $X$ , and for any oriented closed $(d+1)$ -manifold $W^{d+1}$ fitting into a fibration $\Sigma^{d-1} \to W^{d+1} \to Y^2$ with parameter map $g: W^{d+1} \to X$ , the following constraint holds:
$N(\Sigma^{d-1}, \hat{g}) \int_{W^{d+1}} g^*\omega \in \mathbb{Z}$
Here, $\hat{g}$ is the restriction of $g$ to a fiber, and the integral represents the evaluation of the anomaly on the bordism class. This formula generalizes previous results on framing anomalies (e.g., Kong and Wen) to arbitrary flux insertions and parameter spaces.
Applications to (2+1)D TQFTs:
- Framing and 1-Form Anomalies: The constraint is applied to (2+1)D TQFTs with framing anomalies (chiral central charge $c_-$ ) and 1-form symmetry anomalies. For a surface $\Sigma^2_g$ of genus $g \geq 3$ , the ground-state degeneracy $N$ satisfies $N \cdot c_- / 2 \in \mathbb{Z}$ .
- Abelian Anyons: For abelian anyons with topological spin $\theta_a$ , the degeneracy on a surface of genus $g \geq 1$ satisfies $N \cdot 2\theta_a \in \mathbb{Z}$ . This recovers known commutation relations of Wilson loop operators via the anomaly framework.
Obstructions to Symmetry-Preserving TQFTs: The paper establishes conditions under which a symmetry-preserving TQFT (one with a unique ground state on the sphere under twisted boundary conditions) cannot exist. If a torsion anomaly $\omega$ admits a symmetry-preserving TQFT, then $\int_{W^{d+1}} g^*\omega = 0 \pmod 1$ for all relevant bundles. The authors demonstrate that nontrivial anomalies for certain 1-form and mixed symmetries (e.g., $\mathbb{Z}_p$ 1-form symmetry with specific anomaly coefficients) obstruct the existence of such TQFTs.
Bordism Generators and Detectability: The authors propose a conjecture (Conjecture 3.5) that any nontrivial internal torsion anomaly of a finite $(d-1)$ -group symmetry in $d > 2$ dimensions can be detected by a fiber bundle $\Sigma^{d-1} \to W^{d+1} \to Y^2$ . This suggests that fibered manifolds serve as generators for detecting anomalies via the pushforward of the anomaly class.

Significance and Claims
The paper claims to provide a unified framework for understanding anomalies in gapped systems through the lens of parametrized TQFTs. By distinguishing between deformation classes (standard topology) and isomorphism classes (discrete topology), the authors resolve why certain constraints on ground-state degeneracies (like those in Lieb-Schultz-Mattis type systems) appear as anomalies only when considering specific sectors or discrete topologies.

The significance lies in:

Generalization: Extending anomaly constraints beyond 't Hooft anomalies of global symmetries to include anomalies of parameter spaces and higher-form symmetries.
Quantitative Constraints: Providing a concrete, calculable formula linking the topological invariant of the anomaly (the integral over $W^{d+1}$ ) directly to the physical observable of ground-state degeneracy.
Non-Existence Proofs: Offering a mechanism to prove the non-existence of symmetry-preserving gapped phases for specific anomaly classes, thereby constraining the possible infrared phases of quantum many-body systems.

The authors note that while the framework is rigorous for invertible symmetries in bosonic TQFTs, generalizations to non-invertible symmetries and spin TQFTs remain open questions. They also acknowledge that the conjecture regarding the detectability of all anomalies by fibered manifolds is physically motivated but not yet mathematically proven.

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