Dome Algebras in Relative QFT

• Dome algebras are quotient structures derived from tube algebras that mod out bulk-trivial configurations in relative 2D quantum field theories.
• They encode effective boundary actions and hypergroup symmetries by faithfully representing twisted-sector operators and defect interactions.
• Comparative examples demonstrate that dome algebras isolate the boundary effective operators, clarifying defect structures in relative QFTs.

Searching arXiv for papers on "dome algebras" and closely related context. Dome algebras are algebraic structures introduced in the study of relative two-dimensional quantum field theories, especially rational chiral algebras that occur as boundary conditions for three-dimensional topological field theories. In that setting, a dome algebra is defined as the faithful quotient of the tube algebra acting on twisted-sector operators, obtained by modding out those tube configurations that become trivial because bulk lines can be absorbed into the ambient three-dimensional bulk (Gannon et al., 3 Jun 2026). The notion therefore belongs to the interface of noninvertible symmetry, fusion-categorical defect theory, and chiral-algebraic constructions in relative QFT. The term does not occur in the classification of two-dimensional algebras by structure constants (Ahmed et al., 2017), nor in diagrammatic algebraic treatments of polyhedral cones and polyhedra (Bonchi et al., 2021); in those contexts, any apparent similarity in terminology is accidental. By contrast, dome algebras arise specifically as a relative-theory generalization of the Ocneanu tube algebra and are tied to hypergroup symmetry in boundary and chiral settings (Gannon et al., 3 Jun 2026).

1. Definition and conceptual placement

The formal definition appears in a framework where $V$ is a rational chiral algebra viewed as a boundary condition of a $3$D topological field theory $B(V)$, and $C$ is a fusion category of boundary topological line operators satisfying

$B(V)\subset C.$

For each simple object $X\in C$, there is a twisted-sector Hilbert space $V_X$, and the direct sum

$V_C=\bigoplus_{X\in \mathrm{Irr}(C)} V_X$

carries an action of the ordinary tube algebra $\mathrm{Tube}(C)$ by the usual lasso construction around operators (Gannon et al., 3 Jun 2026).

In a relative theory, however, this action fails to be faithful. The reason is that some lines in the bulk category $B(V)$ can be pulled off the boundary into the bulk and collapsed, so certain annular configurations act trivially on genuine boundary observables. The dome algebra is defined precisely to remove that redundancy: $3$0 where $3$1 identifies tube configurations differing by insertions or motions of bulk lines $3$2 that are bulk-trivial in the relative theory (Gannon et al., 3 Jun 2026).

This construction makes dome algebras the relative-theory analogue of tube algebras. The paper characterizes the geometric picture behind the name: instead of a full tube surrounding an operator, the relevant topological action is represented by a dome-shaped surface operator anchored to the boundary (Gannon et al., 3 Jun 2026). A plausible implication is that the terminology emphasizes the dimensional asymmetry of relative theories: bulk-topological data modifies the familiar annular algebra into a boundary-supported quotient structure.

2. Relative QFT origin and twisted-sector action

The central physical setting is a relative $3$3D QFT living on the boundary of a $3$4D TQFT. In this framework, boundary operators may interact with bulk topological surfaces, and the algebraic action of symmetry defects must record which topological manipulations are physically nontrivial on the boundary. Dome algebras encode exactly that effective action (Gannon et al., 3 Jun 2026).

The paper describes a decomposition in which a boundary line $3$5 belonging to a component $3$6 is represented by a surface $3$7 together with a boundary junction $3$8. Acting on a local operator $3$9 is then depicted by collapsing the corresponding dome onto $B(V)$0, producing an operator

$B(V)$1

These operators compose according to the hypergroup multiplication law, after normalization, and the dome algebra is the algebraic structure that controls this action on local and twisted-sector data (Gannon et al., 3 Jun 2026).

The extended Hilbert space carries more refined structure. The paper gives the decomposition

$B(V)$2

where $B(V)$3 is the subalgebra transparent to $B(V)$4. The dome algebra acts on each $B(V)$5, hence on each multiplicity space $B(V)$6, and Claim 2.13 identifies the resulting structure as a Schur–Weyl decomposition: $B(V)$7 This places dome algebras squarely in the algebraic control of twisted sectors, multiplicity spaces, and transparent subalgebras (Gannon et al., 3 Jun 2026).

3. Relation to tube algebras, centers, and representation theory

The dome algebra is best understood by comparison with the ordinary tube algebra. In an absolute $B(V)$8D QFT, the tube algebra acts faithfully on the full extended Hilbert space of twisted sectors, and its representation category recovers the Drinfeld center: $B(V)$9 In the relative setting, bulk-trivial configurations generate a kernel, so one passes to the quotient $C$0 (Gannon et al., 3 Jun 2026).

The fundamental representation-theoretic statement is Claim 2.12: $C$1 Here $C$2 is the centralizer of $C$3 in the center of $C$4. The paper also compares this directly with the unquotiented tube algebra: $C$5 Thus the dome algebra removes the extra bulk factor $C$6, leaving precisely the algebra that governs the “non-bulk” part of the defect action (Gannon et al., 3 Jun 2026).

This comparison is the cleanest categorical characterization of dome algebras. They are not merely smaller tube algebras in an ad hoc sense; they are quotient algebras whose representation category captures the centralizer of bulk degrees of freedom. This suggests that dome algebras are the natural algebraic object for relative theories in the same way that tube algebras are natural for absolute theories.

4. Hypergroup symmetry and dome-algebra multiplication

A major reason dome algebras matter is that the effective symmetry acting on local boundary operators is often a hypergroup rather than a group or fusion ring. The paper defines a hypergroup $C$7 with multiplication

$C$8

where the structure constants are nonnegative real numbers rather than necessarily integers (Gannon et al., 3 Jun 2026).

For a boundary line category $C$9, the effective symmetry is obtained as a double-coset hypergroup quotient

$B(V)\subset C.$0

and this hypergroup acts faithfully on genuine local operators: $B(V)\subset C.$1 The dome algebra is the algebraic mechanism underlying this action (Gannon et al., 3 Jun 2026).

The multiplication law is represented by normalized dome operators satisfying

$B(V)\subset C.$2

with

$B(V)\subset C.$3

The coefficients $B(V)\subset C.$4 arise from point operators on the relevant surface junctions, and the $B(V)\subset C.$5 are normalization factors. In this way, hypergroup symmetry is encoded both geometrically, through composition of dome-shaped surface defects, and algebraically, through the quotient structure of the dome algebra (Gannon et al., 3 Jun 2026).

The paper further states that any boundary line category $B(V)\subset C.$6 is a hypergroup-graded extension of $B(V)\subset C.$7: $B(V)\subset C.$8 This identifies hypergroup grading as an organizing principle for the relative symmetry structure and situates dome algebras as the operator-algebraic realization of that grading (Gannon et al., 3 Jun 2026).

5. Examples and model cases

The paper develops several examples illustrating how dome algebras behave across finite, noninvertible, and even infinite settings (Gannon et al., 3 Jun 2026).

Example	Structural feature	Reported conclusion
Finite-group action $B(V)\subset C.$9 on $X\in C$0	Effective hypergroup is $X\in C$1	Dome algebra reduces to a quotient of the twisted quantum double algebra
Ising $X\in C$2-even sector	Six simple objects in boundary-line category	Effective hypergroup is $X\in C$3
$X\in C$4	$X\in C$5 fusion-category realization	Hypergroup has non-integerizable structure constants
$X\in C$6 Kac-Moody / Heisenberg VOA	Infinite symmetry category	Effective hypergroup is $X\in C$7

In the finite-group case, the paper conjectures in the pointed setting that

$X\in C$8

and notes the expected representation category

$X\in C$9

This situates dome algebras close to twisted quantum double constructions in familiar orbifold-type settings (Gannon et al., 3 Jun 2026).

For the $V_X$0-even sector $V_X$1 of the Ising CFT, the boundary-line category has

$V_X$2

while the effective hypergroup is only $V_X$3. The example is used to show that the boundary line content can be noninvertible even when the effective action on local operators simplifies considerably; $V_X$4 is interpreted as a dome of the electric-magnetic duality surface in the toric code bulk (Gannon et al., 3 Jun 2026).

The $V_X$5 example is singled out because the associated hypergroup has non-integerizable structure constants. This demonstrates that dome algebras naturally encode symmetries beyond ordinary finite-group or integral fusion data (Gannon et al., 3 Jun 2026).

The $V_X$6 Kac-Moody / Heisenberg VOA example extends the framework beyond finite settings. The symmetry category has simple lines

$V_X$7

and the effective hypergroup is

$V_X$8

This shows that the dome-algebra formalism is not intrinsically finite and can accommodate continuous hypergroup symmetry (Gannon et al., 3 Jun 2026).

6. Mathematical significance and relation to neighboring usages of “algebra”

Within the paper that introduces them, dome algebras are central because they provide a faithful algebraic handle on symmetries of relative $V_X$9D QFTs, especially when ordinary tube algebras retain bulk-trivial information and therefore overcount physically effective operators (Gannon et al., 3 Jun 2026). Their significance is distributed across several closely related tasks: classifying symmetries, organizing twisted sectors, relating chiral data to boundary conditions, and encoding noninvertible symmetry through hypergroups rather than groups.

The paper also connects dome-algebra data to a generalized Galois-type picture in which intermediate conformal subalgebras correspond to subhypergroups. This suggests a structural role for dome algebras analogous to the role of group actions in classical fixed-point theory, but in a genuinely noninvertible and relative setting (Gannon et al., 3 Jun 2026).

The term should not be conflated with unrelated algebraic classifications. The paper “Complete Classification of Two-Dimensional Algebras” classifies non-trivial two-dimensional algebras over algebraically closed fields up to $V_C=\bigoplus_{X\in \mathrm{Irr}(C)} V_X$0-equivalence using matrices of structure constants and trace vectors $V_C=\bigoplus_{X\in \mathrm{Irr}(C)} V_X$1 and $V_C=\bigoplus_{X\in \mathrm{Irr}(C)} V_X$2, but it does not define or mention dome algebras (Ahmed et al., 2017). Likewise, “Diagrammatic Polyhedral Algebra” develops complete ordered-prop axiomatizations for polyhedral cones and polyhedra via $V_C=\bigoplus_{X\in \mathrm{Irr}(C)} V_X$3 and $V_C=\bigoplus_{X\in \mathrm{Irr}(C)} V_X$4, yet it also does not introduce any class called dome algebra (Bonchi et al., 2021). These distinctions matter because the word “algebra” in these works refers to very different objects: coordinate algebras classified by structure constants in one case, ordered props for polyhedral geometry in another, and quotient algebras of defect-tube operators in relative QFT for dome algebras (Ahmed et al., 2017, Bonchi et al., 2021, Gannon et al., 3 Jun 2026).

A further potential confusion arises with “algebras of generalized dihedral type,” which are symmetric tame indecomposable algebras classified via triangulated surfaces and quivers with relations (Erdmann et al., 2017). Despite the superficial phonetic resemblance between “dome” and “dihedral,” these are unrelated notions. Generalized dihedral type concerns finite-dimensional representation theory, stable Auslander–Reiten structure, and biserial weighted surface algebras, not the defect-algebraic apparatus of relative quantum field theory (Erdmann et al., 2017).

7. Scope, limitations, and current status

At present, dome algebras are a specialized notion tied to the formalism of relative $V_C=\bigoplus_{X\in \mathrm{Irr}(C)} V_X$5D QFTs and rational chiral algebras developed in “Hypergroup Symmetry in Relative Quantum Field Theories and Chiral Algebras” (Gannon et al., 3 Jun 2026). The available arXiv record indicates that this work is the source in which dome algebras are explicitly introduced and motivated. The concept is not part of the standard nomenclature of low-dimensional algebra classification (Ahmed et al., 2017), polyhedral diagrammatics (Bonchi et al., 2021), or generalized dihedral representation theory (Erdmann et al., 2017).

The core definition is stable and precise: a dome algebra is the quotient

$V_C=\bigoplus_{X\in \mathrm{Irr}(C)} V_X$6

that acts faithfully on twisted sectors after modding out bulk-trivial relations. Its irreducible representations are identified with

$V_C=\bigoplus_{X\in \mathrm{Irr}(C)} V_X$7

and its operator product realizes hypergroup multiplication on local boundary observables (Gannon et al., 3 Jun 2026).

This suggests that dome algebras should be regarded as the boundary-effective, bulk-reduced operator algebras of relative noninvertible symmetry. Their novelty lies not in providing another finite-dimensional algebra classification, but in isolating the correct algebraic object for the defect and symmetry structure seen by observables on the boundary of a higher-dimensional topological phase (Gannon et al., 3 Jun 2026).