⚛️ high-energy theory

Non-unitary Haagerup-like TQFTs and RCFTs from generalized S-fold SCFTs

This paper extends the class of 3-dimensional non-unitary TQFTs and 2-dimensional non-unitary RCFTs by deriving them from generalized S-fold SCFTs via topological twist, providing explicit modular data and identifying specific instances with generalized Haagerup-Izumi modular matrices.

Original authors: Kibok Jeong, Huijoon Sohn

Published 2026-08-13

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CC BY 4.0

Original authors: Kibok Jeong, Huijoon Sohn

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

Imagine the universe as a giant, invisible Lego set. Physicists have long been trying to figure out the rules for how these Lego bricks snap together to build everything from stars to subatomic particles. Usually, they focus on "unitary" rules—rules where energy is always conserved, and if you build a tower, it stays standing. But there's a whole other side of the Lego box: "non-unitary" rules. These are the weird, wobbly instructions where towers might collapse, or where the math allows for things that seem impossible in our everyday world. For a long time, these non-unitary theories were just abstract math puzzles, studied by people who love axioms and equations but had no physical way to test them.

Recently, however, scientists discovered that these wobbly, non-unitary rules actually show up in real physics, specifically in a special corner of the universe called "supersymmetry." Think of supersymmetry as a magical mirror that pairs every particle with a heavier, invisible twin. In certain 3-dimensional worlds made of these particles, if you twist the rules just right (a process called a "topological twist"), the messy, wobbly physics simplifies into a clean, topological game. This game is called a Topological Quantum Field Theory, or TQFT. It's like a video game where the graphics don't matter, only the connections between objects. Even cooler, these 3D games are linked to 2D "rational conformal field theories" (RCFTs), which are like the soundtrack or the rulebook for the game. The big question has been: what do these weird, non-unitary soundtracks and rulebooks actually look like?

This paper is like a detective story where the authors, Kibok Jeong and Huijoon Sohn, go hunting for new, strange rulebooks. They start with a known class of theories called "S-fold SCFTs," which are like a specific, well-understood type of Lego set. But instead of sticking to the standard instructions, they decide to build a "Generalized S-fold." Imagine taking several copies of that standard Lego set and gluing them together in a more complex, twisted way using different "Chern-Simons levels" (which you can think of as different types of glue that change how the pieces interact). By doing this, they create a much broader family of these 3D topological worlds.

The authors then perform a "topological twist" on these new worlds to turn them into the TQFTs mentioned earlier. Once twisted, they use a mathematical tool called the "Bethe vacua analysis" to peek inside the worlds and count the possible states, much like counting the different ways a Rubik's cube can be solved. Based on this counting, they propose the "modular matrices" for these new theories. In plain English, these matrices are the instruction manuals that tell you how the different pieces of the theory swap places or transform when you look at them from different angles.

For some specific families of these generalized theories, the authors go even deeper. They identify specific "simple lines" (which are like special strings or threads running through the 3D world) and use them to write down the exact "characters" of the 2D RCFTs. Characters are essentially the musical notes or the unique signatures of the theory. They found that for certain members of this new family, the resulting instruction manuals look very similar to a famous set of matrices known as "Haagerup-Izumi," but with a twist: they are "Galois conjugated," which is a fancy way of saying they are related by a specific kind of mathematical reshuffling, like looking at a reflection in a funhouse mirror.

The paper doesn't claim to have solved the entire mystery of non-unitary physics. Instead, it suggests that by generalizing the way we build these S-fold theories, we can unlock a whole new zoo of these strange, non-unitary TQFTs and their associated RCFTs. They provide the specific formulas (the S and T matrices) for these new theories and show that they fit the expected patterns. While they can't list every single possible line for every single theory yet, they have successfully mapped out a significant new territory, showing that the "Haagerup-like" family of theories is much larger and more varied than anyone previously realized. It's a step forward in understanding the weird, wobbly side of the universe's Lego set.

Technical Summary: Non-unitary Haagerup-like TQFTs and RCFTs from generalized S-fold SCFTs

Problem Statement
The paper addresses the classification and characterization of 3-dimensional non-unitary topological quantum field theories (TQFTs) and their associated 2-dimensional non-unitary rational conformal field theories (RCFTs). While recent developments have provided physical realizations of such theories via topological twists of specific $\mathcal{N}=4$ rank-0 superconformal field theories (SCFTs), the existing class of known theories—specifically those arising from "S-fold" SCFTs—remains limited. The authors aim to extend this known class by investigating a broader family of theories, termed "generalized S-fold SCFTs," to uncover new families of non-unitary TQFTs and RCFTs, particularly those related to generalized Haagerup-Izumi modular data.

Methodology
The authors employ a dual-description approach to analyze the theories labeled by once-punctured torus bundles $M_3 = \Sigma_{1,1} \times_\phi S^1$ , where $\phi \in SL(2, \mathbb{Z})$ is a monodromy with $|\text{tr}\phi| > 2$ .

Dual UV Descriptions:
- Abelian Chern-Simons-Matter Theory ( $T_{DGG}[\phi]$ ): Utilizing the Dimofte-Gaiotto-Gukov (DGG) construction, the authors describe the theories as $\mathcal{N}=2$ abelian Chern-Simons-matter theories derived from ideal triangulations of the 3-manifold. They focus on specific monodromies decomposable into products of $L$ and $R$ matrices (e.g., $\phi = LR^n$ and $\phi = L^2R^n$ ).
- Generalized S-fold SCFTs ( $S(\vec{k})$ ): Theories are also described as 3-dimensional $\mathcal{N}=4$ rank-0 SCFTs obtained by gauging diagonal $SU(2)$ subgroups of multiple copies of the $T[SU(2)]$ theory at non-vanishing Chern-Simons levels. This corresponds to a linear quiver diagram where the gauging data $\vec{k}$ is derived from the decomposition of the monodromy $\phi$ .
Topological Twisting:
The authors apply a topological A-twist ( $\nu = -1$ ) to these $\mathcal{N}=4$ rank-0 SCFTs. This procedure renders the theories independent of the background metric, yielding non-unitary TQFTs.
Extraction of Data:
- For Specific Families ( $\phi = LR^n, L^2R^n$ ): The authors utilize BPS partition functions, specifically the superconformal index and half-indices on $S^2 \times_q S^1$ and $D^2 \times_q S^1$ , to identify "simple lines" in the TQFT. These lines are mapped to chiral primaries in the boundary RCFT. Using the Nahm sum formula, they derive closed-form expressions for RCFT characters and propose modular $S$ matrices.
- For Broader Families: For general monodromies where the full set of simple lines is unknown, the authors perform a Bethe vacua analysis on the squashed 3-sphere partition function ( $S^3_b$ ) in the limit $b^2 \to 0$ . This saddle-point approximation allows them to extract "Handle-gluing" ( $H$ ) and "Fibering" ( $F$ ) data. These HF data are used to determine the absolute values of the modular $S$ and $T$ matrices and to propose the full matrices for broader classes of monodromies.

Key Contributions and Results

Extension of Non-Unitary TQFTs: The paper constructs a broader class of non-unitary TQFTs by generalizing the S-fold construction. These theories are identified as the IR fixed points of generalized S-fold SCFTs.
RCFT Characters and Modular Matrices:
- For the families $\phi = LR^n$ ( $n \ge 2$ ) and $\phi = L^2R^n$ ( $n \ge 1$ ), the authors explicitly propose a complete set of simple lines.
- They derive the conformal weights ( $h$ ) and central charges ( $c$ ) for the associated boundary RCFTs. Notably, for $\phi = L^2R^n$ , the resulting RCFTs are identified as fermionic due to half-integer powers in their $q$ -series expansions.
- Explicit modular $S$ matrices are proposed for these families, which satisfy the required modularity conditions.
Connection to Haagerup-Izumi Data: The authors demonstrate that for specific members of the generalized family, the resulting modular matrices agree with the "generalized Haagerup-Izumi modular matrices" up to Galois conjugation. This establishes a physical realization of these mathematical structures.
Bethe Vacua Analysis: The paper provides a detailed Bethe vacua analysis for the generalized S-fold theories. It confirms that the sum of inverse Handle-gluing factors equals 1, supporting the claim that these theories are rank-0 (lacking local operators in the topological sector).
Modular Matrix Proposals: For a broader family of monodromies defined by $\phi = ST^{k_1}ST^{k_2}\dots$ (with odd integers $k_i$ ), the authors propose modular $S$ and $T$ matrices based on the HF data extracted from the Bethe vacua.

Significance and Claims
The paper claims to significantly expand the landscape of physically realized non-unitary TQFTs and RCFTs. By moving beyond the specific S-fold constructions previously studied, the authors provide a systematic framework to generate new families of these theories. The work bridges the gap between axiomatic studies of non-unitary TQFTs and their physical realization via supersymmetric field theories.

The authors modestly note that while they can determine the complete set of characters and simple lines for specific monodromy families, the general case remains partially understood. However, the Bethe vacua analysis allows for the extraction of partial modular data (specifically the absolute values of matrix entries) for a much wider class of theories. The identification of these theories with generalized Haagerup-Izumi modular data (up to Galois conjugation) suggests a deep connection between the geometry of punctured torus bundles and the algebraic structure of non-unitary RCFTs. The results serve as a foundation for further exploration of bulk-boundary correspondences in non-unitary settings.

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