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Topological interfaces between universes and generalized families of 2d QFTs

This paper establishes a framework for decomposing two-dimensional quantum field theories with topological point operators into distinct "universes" governed by multi-fusion categories, classifying the topological interfaces between them into three types and connecting this structure to generalized families of quantum field theories.

Original authors: Pinak Banerjee, Yuhan Gai, Alonso Perez-Lona, Daniel Robbins, Subham Roy, Eric Sharpe, with an appendix by Xingyang Yu

Published 2026-10-30

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CC BY 4.0

Original authors: Pinak Banerjee, Yuhan Gai, Alonso Perez-Lona, Daniel Robbins, Subham Roy, Eric Sharpe, with an appendix by Xingyang Yu

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 modern physics, quantum field theories serve as the fundamental rulebooks for how particles and forces interact. For decades, physicists have relied on the concept of symmetry to make sense of these rules. A symmetry is a transformation that leaves a system looking exactly the same, like rotating a perfect sphere or shifting a pattern in time. These symmetries act as powerful constraints, dictating what particles can exist and how they can interact. However, in recent years, the definition of symmetry has expanded dramatically. Physicists discovered that symmetries are not just about smooth transformations but can also be encoded in topological operators—special kinds of defects or lines that can be moved around without changing the energy of the system. These new, generalized symmetries have revealed that some quantum theories are not single, unified entities. Instead, they can split apart into distinct, independent pieces, a phenomenon known as decomposition. When this happens, the theory effectively becomes a collection of separate "universes" that coexist side by side, each with its own internal physics, yet all arising from the same underlying mathematical structure.

A team of researchers has now mapped out the intricate architecture of these split theories, revealing how the different universes within a single quantum system can be related to one another. By treating the symmetries of these theories as complex mathematical structures called multi-fusion categories, the authors have developed a precise classification system. They found that the relationship between any two universes in a decomposed theory falls into one of four distinct types. In the most connected scenario, universes are identical twins, linked by invertible interfaces that allow them to be swapped perfectly. In a second type, the universes are related by a process of "gauging," where one is a modified version of the other, but they remain distinct and cannot be swapped. A third type involves universes that are related by a more complex, non-invertible connection, where the symmetry of one is fundamentally different from the other, yet they still share a deep structural bond. Finally, there is a fourth type where universes exist in the same theory but are completely isolated from one another; no topological interface connects them, and they share no common symmetry boundary.

The researchers demonstrated that the way a theory is constructed determines which of these relationships is possible. For instance, when a theory is created by "orbifolding"—a process involving a symmetry that acts trivially or does nothing on certain parts of the system—the resulting universes are always connected. They found that such constructions never produce the isolated, Type III pairs. Instead, these methods always yield universes that are either identical or related by some form of gauging. To find the truly isolated pairs, the team had to look elsewhere. They constructed examples using specific extensions of symmetry groups and by tracing the path of theories as they evolve from high-energy states to low-energy states. In these cases, the universes end up in completely separate mathematical compartments, with no topological bridge between them. This distinction is crucial because it changes the nature of the "Symmetry Topological Field Theory," a higher-dimensional framework that encodes all the symmetry data of the system. For connected universes, this framework is a single, unified space. For the isolated ones, the framework itself splits into disconnected components.

Beyond classifying these relationships, the paper connects this new understanding to a broader concept in physics known as families of theories. Physicists often study how a theory changes as they tweak its parameters, such as coupling constants, viewing these variations as a family of related theories. The authors propose that the different types of universe pairs correspond to different ways these families can be organized. The identical, swap-able universes correspond to families where the theories are related by simple, reversible operations. The more complex, non-invertible connections correspond to families where the relationship involves a deeper, categorical change that cannot be undone. The isolated universes, meanwhile, represent families that exist over a discrete set of parameters, where the theory jumps from one state to another without a continuous path. This work provides a rigorous dictionary for translating between the abstract language of category theory and the physical reality of quantum fields, offering a clearer picture of how the building blocks of the universe can be both separate and deeply intertwined.

Technical Summary: Topological Interfaces between Universes and Generalized Families of 2d QFTs

Problem Statement
Two-dimensional quantum field theories (QFTs) possessing a $(d-1)$ -form symmetry (generated by topological point operators) exhibit a phenomenon known as decomposition, where the theory splits into a direct sum of distinct sectors, referred to as "universes." While the existence of these universes is well-established, particularly in the context of trivially acting symmetries, a systematic classification of the relationships between these universes and the nature of the topological interfaces connecting them has been lacking. Furthermore, there is a need to connect this decomposition framework to the recently proposed notion of "generalized families" of QFTs by Brennan and Intriligator (BI), which organize theories into families via symmetry actions that may be explicitly broken by couplings. The central problem addressed is how to rigorously characterize the symmetry structure of decomposing theories and classify the topological interfaces between their constituent universes.

Methodology
The authors employ the mathematical framework of multi-fusion categories to describe the symmetries of decomposing theories. Their methodology proceeds as follows:

Symmetry as Multi-fusion Categories with Action Functors: The authors define a finite symmetry of a decomposing 2d QFT $T$ as a pair $(\mathcal{C}, \mathcal{F})$ , where $\mathcal{C}$ is a multi-fusion category and $\mathcal{F}: \mathcal{C} \to \text{Top}(T)$ is a tensor functor (the action functor) mapping to the topological defects of the theory. This setup allows for the tracking of trivially acting symmetries and the specific interfaces between universes.
Decomposition of the Identity: In a decomposing theory, the identity line splits into a sum of identity lines of the individual universes ( $1 = \bigoplus 1_i$ ). The multi-fusion category $\mathcal{C}$ reflects this structure, decomposing into a matrix of abelian categories $\mathcal{C}_{ij}$ , where diagonal blocks $\mathcal{C}_{ii}$ represent the symmetry lines of universe $i$ , and off-diagonal blocks $\mathcal{C}_{ij}$ represent topological interfaces between universe $i$ and universe $j$ .
Classification of Universe Pairs: The authors classify pairs of universes $(i, j)$ $(i, j)$ based on the properties of the block $\mathcal{C}_{ij}$ $C_{ij}$ and the relationship between the diagonal blocks $\mathcal{C}_{ii}$ $C_{ii}$ and $\mathcal{C}_{jj}$ $C_{j j}$ . They define four distinct types:
- Type I: $\mathcal{C}_{ij}$ contains an invertible object. The universes are isomorphic via a topological interface in the image of the symmetry.
- Type IIa: $\mathcal{C}_{ij}$ is non-zero but contains no invertible objects; however, $\mathcal{C}_{ii} \simeq \mathcal{C}_{jj}$ (tensor equivalent). One universe is a gauging of the other that returns a tensor-equivalent symmetry category.
- Type IIb: $\mathcal{C}_{ij}$ is non-zero but contains no invertible objects, and $\mathcal{C}_{ii} \not\simeq \mathcal{C}_{jj}$ (inequivalent). The universes are related by gauging that changes the symmetry, though the categories are Morita equivalent.
- Type III: $\mathcal{C}_{ij} = 0$ . No topological interface exists between the universes within the image of the chosen symmetry.
Derivation via Orbifolds and Extensions: The authors derive these structures by analyzing orbifolds of theories with trivially acting normal subgroups (using cohomological base change and exact sequences of tensor categories) and by constructing specific examples involving 2-group extensions and Renormalization Group (RG) flows.

Key Contributions and Results

Formalization of Symmetries in Decomposing Theories: The paper establishes that the finite symmetry categories of decomposing theories are multi-fusion categories equipped with an action functor. This formalism precisely captures how trivially acting symmetries manifest and how they partition the theory into sectors.
Classification of Interfaces (Types I–III): The primary result is the rigorous classification of pairs of universes into Types I, IIa, IIb, and III.
- Type I and II pairs arise naturally from gauging trivially acting symmetries in fusion categories. Specifically, the authors prove that for orbifolds with a trivially acting abelian normal subgroup, the resulting universes always form Type I or Type II pairs; Type III pairs never occur in this specific construction.
- Type III pairs, where universes are disconnected by topological interfaces, are shown to arise from extensions of 2-group type (where the associator is twisted by a 3-cocycle) and from RG flows out of a common ultraviolet theory.
SymTFT Interpretation: The authors map these types to the structure of the Symmetry Topological Field Theory (SymTFT).
- Type I and II pairs correspond to boundaries within a single indecomposable bulk SymTFT component.
- Type IIa pairs correspond to symmetry boundaries in the same orbit of bulk symmetries (related by invertible bulk surfaces).
- Type IIb pairs correspond to boundaries in different orbits.
- Type III pairs correspond to a decomposable bulk SymTFT where no topological wall connects the components.
Connection to Brennan-Intriligator Families: The paper proposes a correspondence between these decomposition types and the generalized families of QFTs defined by Brennan and Intriligator:
- Type I pairs correspond to equivariant families.
- Type II pairs correspond to categorical families.
- Family anomalies appear as universes carrying different symmetry-protected topological (SPT) phases.
- Type III pairs realize families over a discrete parameter.
Explicit Examples: The paper provides concrete calculations for various cases, including:
- $n$ identical universes (Type I) arising from gauging trivially acting $\mathbb{Z}_n$ symmetries, yielding the multi-fusion category $\text{Mat}_n(\text{Vec})$ .
- Orbifolds by $S_3$ with a trivially acting $\mathbb{Z}_3$ subgroup, yielding a Type IIa pair.
- Orbifolds involving $Q_8$ and $D_4$ , illustrating both Type I and Type IIa pairs.
- Extensions of 2-groups (e.g., $\text{Rep}(D_4) \oplus \text{Rep}(Q_8)$ ) and cohomological base changes that generate Type III pairs.

Significance and Claims
The paper claims to provide a unified framework for understanding the "universes" of decomposing 2d QFTs. By treating symmetries as multi-fusion categories with action functors, the authors offer a precise dictionary between categorical data (blocks, invertibility, Morita equivalence) and physical phenomena (isomorphism, gauging, interface existence).

The significance lies in:

Clarifying the Landscape of Decomposition: It moves beyond the existence of decomposition to a detailed taxonomy of how universes relate, distinguishing between isomorphic universes, universes related by non-invertible gauging, and completely disconnected universes.
Bridging Symmetry and Families: It establishes a concrete link between the modern understanding of generalized symmetries (topological operators, multi-fusion categories) and the concept of generalized families of theories, suggesting that the "universes" of a decomposing theory are the physical realization of members of a BI family.
Identifying the Origin of Disconnected Universes: The paper demonstrates that while gauging trivially acting symmetries in fusion categories always yields connected universes (Types I/II), Type III pairs require more complex origins, such as 2-group extensions or specific RG flows, thereby clarifying the conditions under which a theory splits into truly disjoint sectors.

The authors maintain a modest stance, noting that the classification is relative to a chosen symmetry and that the multi-fusion category describing the symmetry is not unique; however, the types of pairs provide a robust invariant for the relationship between universes within that specific symmetry context. The work is presented as a theoretical classification and derivation, supported by Lagrangian examples (such as the higher-charge Schwinger model in the appendix) and categorical constructions, without proposing new experimental tests.

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