Hypergroup Symmetry in Relative Quantum Field Theories and Chiral Algebras

Abstract: A QFT is said to be relative if it lives at the boundary of a topological QFT in one higher dimension. We develop a general framework for working with noninvertible symmetries of relative theories in two spacetime dimensions, extending several well-known results for absolute QFTs. We emphasize various new features which arise in the relative setting, including the role of topological surfaces of the bulk, and the appearance of hypergroups and certain generalizations of tube algebras known as dome algebras. Our formalism is particularly well-suited for studying rational chiral algebras, where it predicts that finite symmetries are in explicit one-to-one correspondence with conformal embeddings of finite index. We describe several implications of our framework for absolute theories. First, we explain how to "glue" together symmetries of the left- and right-moving chiral algebras of a 2D CFT to produce topological line defects of the full theory. Second, we derive a precise correspondence between boundary conditions of a 2D CFT and symmetries of its chiral algebra. This correspondence has several structural corollaries: in diagonal rational CFTs, we demonstrate that the topological line defects of the theory act transitively on its boundary conditions, and further that the identity Cardy state has the smallest $g$-function amongst all boundary conditions, including those which only preserve Virasoro symmetry. We conclude by illustrating our results in a variety of examples. For instance, we show that, if there exists a rational chiral algebra with central charge $c=8$ whose modular tensor category is the Drinfeld center of the Haagerup fusion category, then it must arise as the fixed points of a rank-2 hypergroup acting on the $SU(3)1\otimes (E{6})_1$ chiral algebra.

• The paper introduces hypergroup symmetry as a novel replacement for traditional group symmetries, enabling the study of noninvertible structures in relative QFTs and chiral algebras.
• It employs modular tensor and fusion categories to construct precise hypergroup gradings and dome algebras that govern the fusion of quantum operators.
• The results establish clear links between boundary conditions, defect classifications, and absolute CFT constructions, offering new tools for spectral analysis and orbifold operations.

Hypergroup Symmetry in Relative QFTs and Chiral Algebras: An Expert Analysis

Context and Motivation

The paper establishes a comprehensive structure for understanding symmetries, both invertible and noninvertible, in two-dimensional relative quantum field theories (QFTs) and chiral algebras, particularly rational vertex operator algebras (VOAs). By extending the absolute QFT symmetry paradigm to relative theories—those living at the boundary of a higher-dimensional topological QFT—it incorporates generalized algebraic concepts such as hypergroups and dome algebras. The work is technically grounded in modular tensor categories (MTCs), fusion categories, and their correspondence to rational chiral algebras and conformal field theories (CFTs). The framework is developed with an explicit eye toward applications in rational chiral algebras and rational CFTs, with implications for boundary conditions, defect theory, and generalizations of existing Moore-Seiberg theory.

Main Formalism: Hypergroups and Symmetry Categories

The central formalism replaces ordinary symmetry groups with hypergroups—a structure that generalizes fusion rings by accommodating non-integer structure constants. The core result establishes:

• Given a rational chiral algebra $V$ as a boundary of a semi-simple bulk 3D TQFT, any fusion category $\mathcal{C}$ of boundary topological line operators containing the bulk anyons (i.e., $B = \mathrm{Rep}(V)$) is a $K$-graded extension of $B$, with $K = K_{\mathcal{C}} // K_B$ the double-coset hypergroup associated to the fusion rings of $\mathcal{C}$ and $B$.

The hypergroup grading is realized via bulk topological surfaces and line junctions anchoring to the boundary, and the action is encoded by noninteger structure constants governing the fusion of quantum dimensions.

This generalizes the group-crossed braided extension paradigm (e.g., $G$-actions in orbifolds) to noninvertible symmetries, laying the algebraic groundwork for a "hypergroup-crossed braided tensor category" (a precise construction deferred to future work).

Symmetry–Subalgebra Duality and Dome Algebras

A symmetry/subalgebra duality emerges: fusion categories of boundary topological lines correspond bijectively to rational conformal subalgebras of $V$. Given a conformal embedding $\mathcal{C}$0, the symmetry category is the fusion category $\mathcal{C}$1 of topological lines neutral under $\mathcal{C}$2, realized via gauging interfaces in the SymTFT construction.

Twisted modules and sectors associated to these symmetries are acted upon by dome algebras, generalizations of tube algebras suitable for relative QFTs. The dome algebra acts faithfully on the extended Hilbert space and encodes the action of hypergroup symmetries on local operators, leading to a Schur–Weyl-type bimodule decomposition involving the dome algebra and transparent subalgebras.

Implications for Absolute CFTs: Gluing and Boundary Correspondences

The framework yields explicit constructions for absolute CFTs. Symmetries of left- and right-moving chiral algebras can be "glued" via the relative Deligne product over the MTC $\mathcal{C}$3, yielding the symmetry category of the full CFT. The paper derives:

• Unitary boundary topological line operators are in one-to-one correspondence with unitary boundary conditions preserving conformal subalgebras, and the topological line defects act transitively on boundary conditions.
• The identity Cardy boundary has the smallest $\mathcal{C}$4-function among all boundaries (including those preserving only Virasoro), establishing a rigorous structural bound observed numerically in bootstrap studies.

This establishes a powerful paradigm for boundary classification, interface manifold construction, and defect mediation in rational CFTs.

Numerical and Structural Highlights

• Symmetry categories constructed via this mechanism are explicitly computed in examples, including cases with cleft group automorphism actions, leading to pointed fusion categories with Verlinde lines.
• The symmetry category for the $\mathcal{C}$5-even sector of the Ising CFT is demonstrated as a $\mathcal{C}$6-graded extension of the toric code, with explicit fusion rules for the full symmetry category.
• Construction of noninvertible hypergroup actions, notably related to the Haagerup fusion category, shows the realization of hypergroup symmetries in chiral algebras whose representation categories are Drinfeld centers of exotic fusion categories.
• Hypergroup gradings are rigorously computed in explicit examples; certain structure constants are non-integerizable, providing strong evidence for genuinely non-group-like symmetry structures.

Practical and Theoretical Implications

This formalism unifies the treatment of noninvertible symmetries in boundary QFTs, chiral algebras, and rational CFTs, providing new computational tools and invariants for classification:

• Bulk–Boundary Correspondence: The categorical bulk is determined entirely by the choice of chiral algebra boundary, and symmetry enrichment is functorially inherited from boundary automorphism actions.
• Boundary Conditions and Defects: Structural correspondences enable systematic classification of boundary and defect conditions via symmetry categories, with robust implications for bootstrap bounds and spectral properties.
• Orbifold and Gauging Operations: The machinery gives universal results for how topological manipulations (conformal extension, orbifolding, equivariantization) mutate symmetry categories in rational and irrational settings.
• The framework extends to infinite settings (e.g., the full symmetry category of the $\mathcal{C}$7 Kac-Moody algebra), foreshadowing generalizations to continuous categorical structures.

Future development points include:

• Rigorous formulation of hypergroup-crossed braided tensor categories,
• Removal of finiteness/rationality/unitarity restrictions,
• New orbifold constructions and generalized VOAs characterized primarily by hypergroup symmetries.

Conclusion

The paper supplies a technically rich, categorical framework for analyzing noninvertible symmetries, hypergroup gradings, and their manifestations in relative quantum field theories, chiral algebras, and rational CFTs. By connecting fusion category symmetries, dome algebras, and explicit boundary constructions, it elevates the classification, structure, and spectral analysis of quantum field theories beyond group symmetries, opening new avenues in the study of exotic symmetry categories, boundary conditions, and algebraic invariants in mathematical and theoretical physics.