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arxiv: 2606.05279 · v1 · pith:MJ5SVSMQnew · submitted 2026-06-03 · ✦ hep-th · cond-mat.str-el· math.QA

Hypergroup Symmetry in Relative Quantum Field Theories and Chiral Algebras

Terry Gannon , Brandon C. Rayhaun This is my paper

Pith reviewed 2026-06-28 05:09 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elmath.QA
keywords relative quantum field theorynoninvertible symmetrieshypergroupsconformal embeddingschiral algebrasrational conformal field theorytopological line defectsboundary conditions
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The pith

Finite symmetries in relative 2D QFTs stand in explicit one-to-one correspondence with finite-index conformal embeddings.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a framework for noninvertible symmetries in relative quantum field theories that live at the boundary of a higher-dimensional topological theory. The relative setup brings in bulk topological surfaces that organize the symmetries into hypergroups and replace tube algebras with dome algebras. For rational chiral algebras the framework delivers a direct matching between finite symmetries and conformal embeddings of finite index. A reader would care because the matching supplies a concrete classification tool that links abstract symmetries to standard constructions in conformal field theory. The same framework also yields concrete results for ordinary absolute theories, such as gluing left- and right-moving symmetries into line defects and relating boundary conditions to chiral-algebra symmetries.

Core claim

In relative quantum field theories in two spacetime dimensions, finite symmetries are in explicit one-to-one correspondence with conformal embeddings of finite index. The formalism incorporates the role of topological surfaces of the bulk and introduces hypergroups together with dome algebras that generalize tube algebras, extending earlier results known for absolute theories.

What carries the argument

Hypergroups induced by bulk topological surfaces, which replace ordinary groups as the symmetry structure in the relative setting.

If this is right

  • Symmetries of the left- and right-moving chiral algebras can be glued to produce topological line defects of the full 2D CFT.
  • Boundary conditions of a 2D CFT stand in precise correspondence with symmetries of its chiral algebra.
  • In diagonal rational CFTs the topological line defects act transitively on the set of boundary conditions.
  • The identity Cardy state has the smallest g-function among all boundary conditions, including those that preserve only Virasoro symmetry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The explicit correspondence supplies a practical search method for new rational chiral algebras by enumerating finite-index embeddings that admit compatible hypergroup actions.
  • The Haagerup example in the paper indicates that any c=8 rational chiral algebra whose modular tensor category is the Drinfeld center of the Haagerup fusion category must arise as fixed points under a rank-2 hypergroup action on SU(3)_1 ⊗ (E6)_1.
  • The hypergroup and dome-algebra structures may extend to relative theories in higher dimensions or to other boundary setups where a bulk topological theory is present.

Load-bearing premise

The theory must be relative, living at the boundary of a topological QFT in one higher dimension so that bulk surfaces can generate the hypergroup symmetry structure.

What would settle it

A single rational chiral algebra that is relative and possesses a finite symmetry with no corresponding finite-index conformal embedding would disprove the claimed correspondence.

Figures

Figures reproduced from arXiv: 2606.05279 by Brandon C. Rayhaun, Terry Gannon.

Figure 1
Figure 1. Figure 1: The Hilbert space Va consists of boundary local operators O(z) on which the bulk anyon a can terminate. By the state/operator correspondence, it can also be viewed as the Hilbert space that the bulk TQFT assigns to a disk which is punctured at the center by the anyon a. One can also extract a category B from the bulk TQFT. Its objects by definition are anyons, or topological line operators, and its morphis… view at source ↗
Figure 2
Figure 2. Figure 2: Left: the representation of an automorphism [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The twisted module Va consists of (non-genuine) boundary local operators O(z) which live at the junction of a g-twisted anyon a and the boundary topological line junction Lg. By the state/operator correspondence, Va can also be thought of as the Hilbert space that the bulk TQFT assigns to a disk with boundary condition described by V and with insertions of topological defects as shown on the right. (i.e. t… view at source ↗
Figure 4
Figure 4. Figure 4: The bulk to boundary functor FB→∂. As anticipated earlier, we advocate that the correct generalization arises when one replaces groups with hypergroups, which we will define shortly. (See also [22,24] for closely related ideas.) In particular, in the next subsection, we will define a hypergroup-graded extension of the Moore￾Seiberg data of a rational chiral algebra V whenever V possesses a noninvertible sy… view at source ↗
Figure 5
Figure 5. Figure 5: A (non-genuine) local operator O(z) ∈ Va can be thought of either as living at the junction of a and Lg as in the left, or at the end of the boundary line FB→∂(a) as on the right. category C of topological line operators supported on a rational chiral algebra boundary condition V of a bulk TQFT. When can one think of C as a symmetry-enrichment of the Moore-Seiberg data of V ? Our answer is always, provided… view at source ↗
Figure 6
Figure 6. Figure 6: The SymTFT picture of a conformal embedding [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The definition of the 2D theory CFTV,S. 1.3 Implications for absolute CFTs So far, the discussion has been about generalized symmetries of relative QFTs, especially chiral algebras. Do our results have any utility for someone who is mainly interested in absolute QFTs? Indeed, we will now sketch how symmetries of chiral algebras distill invaluable information about symmetries, and even boundary conditions, … view at source ↗
Figure 8
Figure 8. Figure 8: Left: the bulk 3D TQFT on a solid ball B(3) with a chiral algebra V imposed as a boundary condition on ∂B(2) = S 2 and a boundary topological line operator X wrapped on the equator. Right: after squashing, this produces a boundary condition ∂X of the diagonal theory CFTV built on V . that the “phantom symmetry” mechanism described in [44] can always be used to explain the existence of an interface conforma… view at source ↗
Figure 9
Figure 9. Figure 9: A vertex operator algebra V and a ribbon equivalence Φ : B(V ) → Rep(V ) together define a gapless chiral boundary condition of the 3D topological field theory (B(V ), c(V )). The local operators O(z) living on the boundary and attached to a bulk line a ∈ B(V ) belong to the V -module Va := Φ(a). By the state/operator correspondence, Va can equivalently be described as the Hilbert space that the 3D TQFT (B… view at source ↗
Figure 10
Figure 10. Figure 10: The action of a bulk topological surface defined by ribbon auto-equivalence [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: An isomorphism f : V → V ′ of chiral algebras is interpreted as a bulk surface Sf terminating on a boundary topological line interface f, as on the left. Bending the bulk surface so that it fuses onto half of the boundary produces an invertible topological line interface I between (V, f∗ ◦ Φ ′ ) and (V ′ , Φ ′ ). To explain why the boundaries (V, Φ) and (V ′ , Φ ′ ) are isomorphic when f∗ ◦Φ ′ = Φ, note t… view at source ↗
Figure 12
Figure 12. Figure 12: The category B(V ) of topological line operators in the bulk is a fusion subcategory of the category Sym(V ) of topological line operators on the boundary defined by (V, Φ). 2.2 Boundary topological line operators We are interested in probing the category of topological line operators supported on a gapless chiral boundary condition (V, ΦV ). Unlike gapped boundary conditions, which are home to only finit… view at source ↗
Figure 13
Figure 13. Figure 13: The action of a boundary topological line operator [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Operators O(z) in V C are by definition transparent to every topological line b in C. or a transparent subalgebra. In the other direction, if one has a unitary conformal subalgebra W ⊂ V , one can consider the category Ver(V/W) of topological lines in Sym† (V ) which commute with the local operators in W, that is Ver(V/W) = {X ∈ Sym† (V ) | X commutes w/ O, ∀O ∈ W}. (2.7) While symmetry/subalgebra duality… view at source ↗
Figure 15
Figure 15. Figure 15: A surface in a 3D TQFT B is defined by 1-gauging a gaugeable algebra B of lines. The dark green lines are described by B, and the purple junctions are the multiplication and comultiplication morphisms m : B ⊗ B → B and ◦m : B → B ⊗ B of the algebra. W B(V ) B(W) IA V B(V ) X ⇋ X S S [PITH_FULL_IMAGE:figures/full_fig_p026_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Every object X of the category VerS(V/W) is a B-module (M, µ) in Ver(V/W), i.e. a topological line M on the V boundary on which the bulk surface S can consistently end via a junction µ : M ⊗ B → M. The SymTFT shows that the line junctions X commute with local operators in W [PITH_FULL_IMAGE:figures/full_fig_p026_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The SymTFT interpretation of the decompositions in Equation ( [PITH_FULL_IMAGE:figures/full_fig_p027_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: By the state/operator correspondence, the Hilbert space [PITH_FULL_IMAGE:figures/full_fig_p028_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The physical interpretation of the restriction functor [PITH_FULL_IMAGE:figures/full_fig_p030_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: In a relative theory, lasso actions of Ver(V/W) generate a quotient of the tube algebra, rather than the tube algebra itself, due to the fact that certain lines a ∈ B(V ) can be dragged into the bulk. because the lassos appear in the SymTFT supported on the topological interface IA in [PITH_FULL_IMAGE:figures/full_fig_p031_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Pinching the interface I near the location of the line X produces a line X at the boundary of the surface S = I ∗ ⊗ I in B(V ). Then, instead of using the surface S in [PITH_FULL_IMAGE:figures/full_fig_p037_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Left: The action of an element ri of the hypergroup K on a local operator O(z) in V can be represented via a dome, up to a ri-dependent proportionality constant 1/d(ri). Right: More generally, the action of the dome algebra on twisted local operators can also be represented via domes. To make contact with the effective hypergroup, we work in a slightly unusual normalization in which the action of topologi… view at source ↗
Figure 23
Figure 23. Figure 23: The definition of the T-junction line operator. Example 2.22 Consider again the situation of a cleft group G of automorphisms acting on a chiral algebra V appearing at the boundary of an Abelian TQFT, B(V ) ∼= Vecσ,ω D . Take W = V G so that Ver(V/V G) ∼= Vecω˜ Γ . (Cf. Example 2.9.) We conjectured in Example 2.14 that the dome algebra in this case is given by a Mason-Ng algebra Dω˜ (Γ, D) [75, Definition… view at source ↗
Figure 24
Figure 24. Figure 24: The T-junction defines a tensor product ⊗T on the category of lines which appear on the boundary of S. This tensor product agrees with the fusion of lines on I. The T-junction is useful for a variety of purposes. For example, it can be used to define a tensor product ⊗T on the category of line operators which live on the boundary of the surface S, as illustrated on the top row of [PITH_FULL_IMAGE:figures… view at source ↗
Figure 25
Figure 25. Figure 25: Replacing the outgoing surface S = L rk Srk with one of its components Srk has the effect of projecting X ⊗Y onto its part πk(X ⊗Y ) which resides in the graded component B(V )rk . B(V ) B(V ) IA = S S L L L B(V ) IA B(W) IA B(W) B(W) S S S T = [PITH_FULL_IMAGE:figures/full_fig_p042_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Two S surfaces terminating on the interface IA can be fused together using the T￾junction. recombination rule of [PITH_FULL_IMAGE:figures/full_fig_p042_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: The composition law of the effective hypergroup. [PITH_FULL_IMAGE:figures/full_fig_p044_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: The boundary topological line operators after gauging a condensable algebra [PITH_FULL_IMAGE:figures/full_fig_p045_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: For each h ∈ H, there is a functor h∗ : C → C, with C the category of boundary line operators. B(V ) V B(V ) X V = Sh Sg h∗(X) g∗(h∗(X)) X (gh)∗(X) Sg Sh Lg Lh Lgh ωX(g, h) [PITH_FULL_IMAGE:figures/full_fig_p046_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: The physical interpretation of the associativity natural isomorphisms [PITH_FULL_IMAGE:figures/full_fig_p046_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: The physical interpretation of the tensor structure natural isomorphisms [PITH_FULL_IMAGE:figures/full_fig_p047_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: The character [χµ] k,v X can be represented as the S 2 × S 1 partition function obtained by gluing together the two solid tori on the left along their common boundary (making sure to glue X to X and Lk to Lk). Alternatively, [χµ] k,v X can be reduced to evaluating configurations like those on the right (i.e. determining the multiple of the identity operator obtained when one shrinks the configuration down… view at source ↗
Figure 33
Figure 33. Figure 33: The induction functor I : B(W) → Ver(V/W) is a bulk-to-wall map which describes the result of bringing a bulk topological line in B(W) to the interface IA. To gain more detailed information about Ver(V/W), we note that there is an induction functor, defined as I : B(W) → B(W)A = Ver(V/W) µ 7→ (µ ⊗ A, idµ ⊗ m), (2.99) where m : A ⊗ A → A is the multiplication morphism on A. That is, I(µ) is the A-module wh… view at source ↗
Figure 34
Figure 34. Figure 34: The physical derivation of Frobenius reciprocity, Equation ( [PITH_FULL_IMAGE:figures/full_fig_p057_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: The generalized induction functor U I V W describes the result of bringing a line µ on the condensation interface between B(U) and B(W) to the condensation interface between B(W) and B(V ) to produce a topological line U I V W (µ) on the condensation interface between B(U) and B(V ). This can be easily seen because the lines a ∈ B(V ) can be pulled off the interface IA and into the bulk, and then pushed b… view at source ↗
Figure 36
Figure 36. Figure 36: Top left: The Kapustin-Saulina representation of a CFT built on [PITH_FULL_IMAGE:figures/full_fig_p060_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: The category L(I) of topological lines supported on the interface I possesses the structure of a (C, D)-bimodule category. of [PITH_FULL_IMAGE:figures/full_fig_p062_37.png] view at source ↗
Figure 38
Figure 38. Figure 38: The physical interpretation of the functor [PITH_FULL_IMAGE:figures/full_fig_p064_38.png] view at source ↗
Figure 39
Figure 39. Figure 39: The physical interpretation of the natural isomorphisms [PITH_FULL_IMAGE:figures/full_fig_p064_39.png] view at source ↗
Figure 40
Figure 40. Figure 40: The physical interpretation of the equivalence [PITH_FULL_IMAGE:figures/full_fig_p067_40.png] view at source ↗
Figure 41
Figure 41. Figure 41: The physical interpretation of the D-balanced functor B : L(I) ⊠ L(J ) → FunD(L(I) ∨ ,L(J )). As a category of modules for an algebra Another useful perspective on the relative Deligne product is obtained by folding. By folding the left of [PITH_FULL_IMAGE:figures/full_fig_p068_41.png] view at source ↗
Figure 42
Figure 42. Figure 42: The physical interpretation of the equivalence [PITH_FULL_IMAGE:figures/full_fig_p069_42.png] view at source ↗
Figure 43
Figure 43. Figure 43: The functor ∂ between (non-genuine) lines X on a chiral algebra boundary V and boundary conditions of CFTV,S. algebra, then what we learn is that every boundary condition of CFTV,S corresponds to some line on V on which S can terminate. Claim 3.11 Let S be a topological surface defined by a gaugeable algebra B in Rep(V ). Then there is the following equivalence of categories: ∂ : Sym† S (V ) → Bdy† (CFTV,… view at source ↗
Figure 44
Figure 44. Figure 44: The pictorial interpretation of the relationship Equation ( [PITH_FULL_IMAGE:figures/full_fig_p074_44.png] view at source ↗
Figure 45
Figure 45. Figure 45: The topological line N+ (middle) can be obtained by pushing the line (σ, 1) in Ising ⊠ Ising (left) onto the interface IA. Pulling N+ further into the bulk Z2 gauge theory TC leads to a line N+ attached to a Z2 surface which implements electric-magnetic duality (right). We can also determine the action of the lines N± on the Hilbert spaces A1, Ae, Am, Af . This can be done using Equation (2.109), where we… view at source ↗
Figure 46
Figure 46. Figure 46: A rational chiral algebra V Hg with Rep(V Hg) = Z(Hg) exists if and only if there is a purely chiral 2D CFT V with Hg symmetry. Claim 4.3 There exists a purely chiral 2D CFT V with central charge c = 8n and C symmetry if and only if there exists a rational chiral algebra V C with central charge c = 8n and Rep(V C ) = Z(C). See [39, 30, 40] for related discussions. We remark that there could of course exis… view at source ↗
Figure 47
Figure 47. Figure 47: The existence of a Haagerup chiral algebra [PITH_FULL_IMAGE:figures/full_fig_p086_47.png] view at source ↗
Figure 48
Figure 48. Figure 48: The duality line D supported on the sub (3)1 chiral algebra is trailed by a charge conju￾gation surface C when it is dragged into the bulk. Let us start with the famous conformal embedding sub (2)4 ⊂ sub (3)1. (4.110) This is a Z2 simple current extension, which means in particular that SU(3)1 Chern-Simons theory can be obtained from SU(2)4 Chern-Simons theory by gauging a Z (1) 2 one-form symmetry. Thus,… view at source ↗
Figure 49
Figure 49. Figure 49: The topological lines Z of the SU(3)1 WZW model which preserve the left- and right￾moving sub (2)4 subalgebras can be identified with topological lines on the surface S = 1 ⊕ C. Let us start by calculating how many simple objects there are in this category. We do this using the physical interpretation of the relative Deligne product afforded by [PITH_FULL_IMAGE:figures/full_fig_p098_49.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript develops a framework for noninvertible symmetries of relative 2D QFTs, emphasizing bulk topological surfaces, hypergroups, and dome algebras as generalizations of tube algebras. For rational chiral algebras it predicts an explicit one-to-one correspondence between finite symmetries and conformal embeddings of finite index. It further derives a gluing procedure for left- and right-moving chiral symmetries into topological line defects of the full CFT, a correspondence between boundary conditions and chiral-algebra symmetries, and structural corollaries for diagonal rational CFTs (transitive action of topological lines on boundary conditions; identity Cardy state minimizes the g-function). A conditional illustration is given involving a putative c=8 chiral algebra whose MTC is the Drinfeld center of the Haagerup fusion category.

Significance. If the claimed bijective correspondence can be established independently of auxiliary bulk choices, the framework would supply a concrete classification tool linking symmetries of rational chiral algebras to finite-index embeddings and would extend several standard results on absolute QFTs to the relative setting. The derived corollaries on boundary conditions and g-functions would also be of structural interest for 2D CFTs.

major comments (1)
  1. [Abstract and the section presenting the one-to-one correspondence] Abstract and the section presenting the one-to-one correspondence: the asserted bijectivity between finite hypergroup symmetries (arising from bulk surfaces via dome algebras) and finite-index conformal embeddings is presented as a general prediction, yet the sole concrete illustration is conditional on the existence of a chiral algebra with MTC equal to the Drinfeld center of the Haagerup category. The manuscript must demonstrate that the map remains bijective and canonical when the bulk TQFT or the precise dome-algebra action is varied; otherwise the correspondence is not shown to be independent of additional data and the central claim is not fully supported.
minor comments (1)
  1. [Abstract] The abstract introduces 'dome algebras' and 'hypergroups' without a brief parenthetical reminder of their relation to tube algebras; a short clarifying sentence would improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comment. We respond to it below and will incorporate revisions to strengthen the presentation of the central claim.

read point-by-point responses

  1. Referee: Abstract and the section presenting the one-to-one correspondence: the asserted bijectivity between finite hypergroup symmetries (arising from bulk surfaces via dome algebras) and finite-index conformal embeddings is presented as a general prediction, yet the sole concrete illustration is conditional on the existence of a chiral algebra with MTC equal to the Drinfeld center of the Haagerup category. The manuscript must demonstrate that the map remains bijective and canonical when the bulk TQFT or the precise dome-algebra action is varied; otherwise the correspondence is not shown to be independent of additional data and the central claim is not fully supported.

    Authors: We thank the referee for this comment. The bijective correspondence is established as a general result in the framework developed for rational chiral algebras. The map is constructed by associating to each hypergroup symmetry its dome algebra, which determines the embedding as the fixed points under the symmetry action, and the inverse map is given by the symmetries induced by the embedding. This construction is independent of the choice of bulk TQFT because the dome algebra is defined directly from the topological surfaces in the bulk acting on the boundary theory; varying the bulk would correspond to a different relative QFT. The Haagerup illustration is conditional only because the existence of the c=8 chiral algebra is not established, but the general correspondence does not rely on it. We will update the abstract and the section to make the canonicity and independence explicit, thereby addressing the concern. revision: yes

Circularity Check

0 steps flagged

No significant circularity; correspondence presented as derived prediction of new relative formalism.

full rationale

The paper introduces a framework for relative 2D QFTs emphasizing bulk topological surfaces, hypergroups, and dome algebras as extensions of absolute QFT results. The central claim of an explicit one-to-one correspondence between finite symmetries and finite-index conformal embeddings is stated as a prediction of this formalism for rational chiral algebras, with a conditional illustration involving the Drinfeld center of the Haagerup category. No quoted equations or self-citations reduce the bijectivity or the prediction to a fitted input, self-definition, or prior author result by construction. The derivation chain for gluing symmetries, boundary correspondences, and transitivity on Cardy states is framed as independent structural corollaries. This qualifies as a self-contained development against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

Framework rests on standard category theory and TQFT axioms from prior literature; introduces dome algebras as new but without independent evidence beyond the formalism.

axioms (2)
  • standard math Standard assumptions of modular tensor categories and fusion categories for chiral algebras
    Invoked for rational chiral algebras and Drinfeld centers in the examples.
  • domain assumption Existence of a bulk topological QFT in one higher dimension for relative theories
    Central to defining the relative setting and bulk surfaces.
invented entities (1)
  • Dome algebras no independent evidence
    purpose: Generalizations of tube algebras suited to relative theories with bulk surfaces
    New structure introduced to handle noninvertible symmetries in the relative case.

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