Let \mathcal A be a completely rational local Möbius covariant net on S^1 , which describes a set of chiral observables. We show that local Möbius covariant nets \mathcal B_2 on 2D Minkowski space which contains \mathcal A as chiral left-right symmetry are in one-to-one correspondence with Morita equivalence classes of Q-systems in the unitary modular tensor category \mathrm{DHR}(\mathcal A) . The Möbius covariant boundary conditions with symmetry \mathcal A of such a net \mathcal B_2 are given by the Q-systems in the Morita equivalence class or by simple objects in the module category modulo automorphisms of the dual category. We generalize to reducible boundary conditions. To establish this result we define the notion of Morita equivalence for Q-systems (special symmetric \ast -Frobenius algebra objects) and non-degenerately braided subfactors. We prove a conjecture by Kong and Runkel, namely that Rehren's construction (generalized Longo–Rehren construction, \alpha -induction construction) coincides with the categorical full center. This gives a new view and new results for the study of braided subfactors.
Let \mathcal A be a completely rational local Möbius covariant net on S^1 , which describes a set of chiral observables. We show that local Möbius covariant nets \mathcal B_2 on 2D Minkowski space which contains \mathcal A as chiral left-right symmetry are in one-to-one correspondence with Morita equivalence classes of Q-systems in the unitary modular tensor category \mathrm{DHR}(\mathcal A) . The Möbius covariant boundary conditions with symmetry \mathcal A of such a net \mathcal B_2 are given by the Q-systems in the Morita equivalence class or by simple objects in the module category modulo automorphisms of the dual category. We generalize to reducible boundary conditions. To establish this result we define the notion of Morita equivalence for Q-systems (special symmetric \ast -Frobenius algebra objects) and non-degenerately braided subfactors. We prove a conjecture by Kong and Runkel, namely that Rehren's construction (generalized Longo–Rehren construction, \alpha -induction construction) coincides with the categorical full center. This gives a new view and new results for the study of braided subfactors.
Let A be a completely rational local Möbius covariant net on S1, which describes a set of chiral observables. We show that local Möbius covariant nets B2 on 2D Minkowski space which contains A as chiral left-right symmetry are in one-to-one correspondence with Morita equivalence classes of Q-systems in the unitary modular tensor category DHR(A). The Möbius covariant boundary conditions with symmetry A of such a net B2 are given by the Q-systems in the Morita equivalence class or by simple objects in the module category modulo automorphisms of the dual category. We generalize to reducible boundary conditions. To establish this result we define the notion of Morita equivalence for Q-systems (special symmetric ∗-Frobenius algebra objects) and non-degenerately braided subfactors. We prove a conjecture by Kong and Runkel, namely that Rehren's construction (generalized Longo-Rehren construction, α-induction construction) coincides with the categorical full center. This gives a new view and new results for the study of braided subfactors.
We classify E2 condensable algebras in a modular tensor category C up to 2-Morita equivalence. From a physical perspective, this is equivalent to providing a criterion for when different E2 condensable algebras result in the same condensed topological phase in a 2d anyon condensation process. By considering the left and right centers of E1 condensable algebras in C, we exhaust all 2-Morita equivalent E2 condensable algebras in C and provide a method to recover E1 condensable algebras from 2-Morita equivalent E2 condensable algebras. We also prove that intersecting Lagrangian algebras in C⊠C with its left and right components generates all 2-Morita equivalent E2 condensable algebras in C. This paper establishes a complete interplay between E1 condensable algebras in C, 2-Morita equivalent E2 condensable algebras in C, and Lagrangian algebras in C⊠C. The relations between different condensable algebras can be translated into their module categories, which correspond to domain walls in topological orders. We introduce a two-step condensation process and study the fusion of domain walls. We also show that an automorphism of an E2 condensable algebra may lead to a nontrivial braided autoequivalence in the condensed phase. As concrete examples, we interpret the categories of quantum doubles of finite groups. We also discuss examples beyond group symmetries. Moreover, our results can be generalized to Witt-equivalent modular tensor categories.
A (1 + 1)D unitary bosonic rational conformal field theory (RCFT) may be organized according to its genus, a tuple (c,C) consisting of its central charge c and a unitary modular tensor category C which describes the (2 + 1)D topological quantum field theory for which its maximally extended chiral algebra forms a holomorphic boundary condition. We establish a number of results pertaining to RCFTs in “small” genera, by which we informally mean genera with the central charge c and the number of primary operators rank(C) both not too large. We start by completely solving the modular bootstrap problem for theories with at most four primary operators. In particular, we characterize, and provide an algorithm which efficiently computes, the function spaces to which the partition function of any bosonic RCFT with rank(C)≤4 must belong. Using this result, and leveraging relationships between RCFTs and holomorphic vertex operator algebras which come from “gluing” and cosets, we rigorously enumerate all bosonic theories in 95 of the 105 genera (c,C) with c ≤ 24 and rank(C)≤4. This includes as (new) special cases the classification of chiral algebras with three primaries and c < 120/7 ∼ 17.14, and the classification of chiral algebras with four primaries and c < 62/3 ∼ 20.67. We then study two applications of our classification. First, by making use of chiral versions of bosonization and fermionization, we obtain the complete list of purely left-moving fermionic RCFTs with c < 23 as a corollary of the results of the previous paragraph. Second, using a (conjectural) concept which we call “symmetry/subalgebra duality,” we precisely relate our bosonic classification to the problem of determining certain generalized global symmetries of holomorphic vertex operator algebras.
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⊗(E6)1 chiral algebra.
Boundaries of Walker-Wang models have been used to construct commuting projector models which realize chiral unitary modular tensor categories (UMTCs) as boundary excitations. Given a UMTC A representing the Witt class of an anomaly, the article \cite{MR4640433} gave a commuting projector model associated to an A -enriched unitary fusion category X on a 2D boundary of the 3D Walker-Wang model associated to A . That article claimed that the boundary excitations were given by the enriched center/Müger centralizer ZA(X) of A in Z(X) .In this article, we give a rigorous treatment of this 2D boundary model, and we verify this assertion using topological quantum field theory (TQFT) techniques, including skein modules and a certain semisimple algebra whose representation category describes boundary excitations. We also use TQFT techniques to show the 3D bulk point excitations of the Walker-Wang bulk are given by the Müger center Z2(A) , and we construct bulk-to-boundary hopping operators Z2(A)→ZA(X) reflecting how the UMTC of boundary excitations ZA(X) is symmetric-braided enriched in Z2(A) .This article also includes a self-contained comprehensive review of the Levin-Wen string net model from a unitary tensor category viewpoint, as opposed to the skeletal 6j symbol viewpoint.
We introduce and study (fermionic and bosonic) invertible quasi-local algebras over uniformly locally finite metric spaces X with infinite-dimensional local von Neumann algebras. We show that the group of Brauer equivalence classes of such algebras is isomorphic to both the group of phases of invertible states and the group of stable equivalence classes of quantum cellular automata over X×Z. Using K-theory of the symmetric monoidal category of invertible quasi-local algebras and bounded spread isomorphisms, we propose a definition of an Ω-spectrum of invertible phases as conjectured by Kitaev. We then show that the c=21 chiral Majorana fermion net and the (E8)1 conformal net provide Brauer non-trivial invertible quasi-local algebras, thus providing explicit constructions of non-trivial invertible states and quantum cellular automata on Z2. In addition, we show that the time-slice nets of rational diagonal conformal field theories admit lattice degrees of freedom, which implies the discretization of any holomorphic conformal net is invertible.