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Topological sectors and a dichotomy in conformal field theory

Roberto Longo, Feng Xu

Published Nov 5, 2012• 60 citations
OpenAlex

Abstract

Let A be a local conformal net of factors on S 1 with the split property. We provide a topological construction of soliton representations of the n-fold tensor product A ⊗ · · · ⊗ A, that restrict to true representations of the cyclic orbifold (A ⊗ · · · ⊗ A) Zn. We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then A is not completely rational iff the symmetrized tensor product (A ⊗ A) flip has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of A have a conjugate sector then either A is completely rational or A has uncountably many different irreducible sectors. Thus A is rational iff A is completely rational. In particular, if the µ-index of A is finite then A turns out to be strongly additive. By [30], if A is rational then the tensor category of representations of A is automatically modular, namely the braiding symmetry is non-degenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.

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A comprehensive symmetry resolution of the entanglement entropy in ( 1 + 1 )-d rational conformal field theories with categorical noninvertible symmetries is presented. This amounts to symmetry resolving the entanglement with respect to the generalized twisted and anyonic charged sectors of the theory. The anyonic sectors label the irreducible representations of a modular fusion category defining the symmetry and can be understood through the ( 2 + 1 )-d symmetry topological field theory that encodes the symmetry features of the conformal field theory. Using this, we define the corresponding generalized boundary-dependent charged moments necessary for the symmetry resolution of the entanglement entropy and find that, contrary to the case of invertible symmetries, the entanglement equipartition between different charged sectors at the next-to-leading order in the ultraviolet cutoff, is broken. Published by the American Physical Society 2024

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Braid Group Representations and Modular Data: Non-Uniqueness, Finite Images, and the Ising Test Case

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A central question in the theory of topological order asks whether the braid group representation $B_N$ carried by $N$ non-Abelian anyons uniquely determines the modular data — the $S$ and $T$ matrices — of the underlying topological quantum field theory. We reconcile three lines of analysis into a single assessment. Structurally, we formalize the question as a reduction map $\Phi$ from modular tensor category data to projective braid group representations, whose kernel is the locus of potential counterexamples. Two rigorous finiteness results constrain the problem: braid representations from twisted quantum doubles of finite groups factor through finite groups, and the $SO(N)_2$/$O(N)_2$ braid representations are Gaussian with finite image. We then perform an explicit computation on the Ising anyon model — the canonical non-Abelian order of Majorana zero modes — reconstructing the topological spins $\theta_\sigma = e^{i\pi/8}$, $\theta_\psi = -1$, the total quantum dimension $\mathcal{D}=2$, and the full $S$ matrix from the braid eigenvalues, with all arithmetic shown. We exhibit a Galois-conjugate deformation $\mathrm{Ising}(7)$ with identical fusion rules but distinct $S$ matrix, and verify that its braid representation is inequivalent to the Ising representation on $B_3$ by an explicit trace computation, while remaining indistinguishable at the level of fusion rules. We conclude that braid representations constrain but do not provably uniquely determine modular data in general: uniqueness can hold only under additional hypotheses, and the Majorana fusion-rule classification is not closed by braid data alone. The affirmative claim that braiding is a complete invariant is restricted to a narrower regime than sometimes asserted, and we document this divergence explicitly. ## 1. Introduction Non-Abelian anyons in two spatial dimensions are characterized by two layers of algebraic structure. The first is combinatorial: fusion rules $N_{ab}^c$ specifying how anyon c Full text and updates: papers.qnfo.org/papers/braid-group-representations-and-modular-data-non-uniqueness-finite-images-and-th/

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Twisted representations of conformal nets and crossed balanced tensor categories

Adrià Marín-Salvador

Let $\mathcal{A}$ be a (not necessarily rational) conformal net with an action of a discrete group $G$. We show that the category $\text{Rep}^G(\mathcal{A})$ of $G$-twisted representations of $\mathcal{A}$ is canonically a $G$-crossed balanced $\mathrm{W}^*$-tensor category. This extends the results of Müger arXiv:math/0403322, in the language of localized endomorphisms, that $\text{Rep}^G(\mathcal{A})$ is a $G$-crossed braided tensor category.

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Let A be a local conformal net of factors on S 1 with the split property. We provide a topological construction of soliton representations of the n-fold tensor product A ⊗ · · · ⊗ A, that restrict to true representations of the cyclic orbifold (A ⊗ · · · ⊗ A) Zn. We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then A is not completely rational iff the symmetrized tensor product (A ⊗ A) flip has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of A have a conjugate sector then either A is completely rational or A has uncountably many different irreducible sectors. Thus A is rational iff A is completely rational. In particular, if the µ-index of A is finite then A turns out to be strongly additive. By [30], if A is rational then the tensor category of representations of A is automatically modular, namely the braiding symmetry is non-degenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.
In the framework of algebraic quantum field theory, we study the category ΔBFA{\Delta_{{\rm BF}}^{\mathfrak{A}}}ΔBFA​ of stringlike localised representations of a net of observables O↦A(O){\mathcal{O} \mapsto \mathfrak{A}(\mathcal{O})}O↦A(O) in three dimensions. It is shown that compactly localised (DHR) representations give rise to a non-trivial centre of ΔBFA{\Delta_{{\rm BF}}^{\mathfrak{A}}}ΔBFA​ with respect to the braiding. This implies that ΔBFA{\Delta_{{\rm BF}}^{\mathfrak{A}}}ΔBFA​ cannot be modular when non-trivial DHR sectors exist. Modular tensor categories, however, are important for topological quantum computing. For this reason, we discuss a method to remove this obstruction to modularity.Indeed, the obstruction can be removed by passing from the observable net A(O){\mathfrak{A}(\mathcal{O})}A(O) to the Doplicher-Roberts field net F(O){\mathfrak{F}(\mathcal{O})}F(O) . It is then shown that sectors of A{\mathfrak{A}}A can be extended to sectors of the field net that commute with the action of the corresponding symmetry group. Moreover, all such sectors are extensions of sectors of A{\mathfrak{A}}A . Finally, the category ΔBFF{\Delta_{{\rm BF}}^{\mathfrak{F}}}ΔBFF​ of sectors of F{\mathfrak{F}}F is studied by investigating the relation with the categorical crossed product of ΔBFA{\Delta_{{\rm BF}}^{\mathfrak{A}}}ΔBFA​ by the subcategory of DHR representations. Under appropriate conditions, this completely determines the category ΔBFF{\Delta_{{\rm BF}}^{\mathfrak{F}}}ΔBFF​ .
Abstract We identify the maximal chiral algebra of conformal cyclic orbifolds. In terms of this extended algebra, the orbifold is a rational and diagonal conformal field theory, provided the mother theory itself is also rational and diagonal. The operator content and operator product expansion of the cyclic orbifolds are revisited in terms of this algebra. The fusion rules and fusion numbers are computed via the Verlinde formula. This allows one to predict which conformal blocks appear in a given four-point function of twisted or untwisted operators, which is relevant for the computation of various entanglement measures in one-dimensional critical systems.
A comprehensive symmetry resolution of the entanglement entropy in ( 1 + 1 )-d rational conformal field theories with categorical noninvertible symmetries is presented. This amounts to symmetry resolving the entanglement with respect to the generalized twisted and anyonic charged sectors of the theory. The anyonic sectors label the irreducible representations of a modular fusion category defining the symmetry and can be understood through the ( 2 + 1 )-d symmetry topological field theory that encodes the symmetry features of the conformal field theory. Using this, we define the corresponding generalized boundary-dependent charged moments necessary for the symmetry resolution of the entanglement entropy and find that, contrary to the case of invertible symmetries, the entanglement equipartition between different charged sectors at the next-to-leading order in the ultraviolet cutoff, is broken. Published by the American Physical Society 2024
A central question in the theory of topological order asks whether the braid group representation BNB_NBN​ carried by NNN non-Abelian anyons uniquely determines the modular data — the SSS and TTT matrices — of the underlying topological quantum field theory. We reconcile three lines of analysis into a single assessment. Structurally, we formalize the question as a reduction map Φ\PhiΦ from modular tensor category data to projective braid group representations, whose kernel is the locus of potential counterexamples. Two rigorous finiteness results constrain the problem: braid representations from twisted quantum doubles of finite groups factor through finite groups, and the SO(N)2SO(N)_2SO(N)2​/O(N)2O(N)_2O(N)2​ braid representations are Gaussian with finite image. We then perform an explicit computation on the Ising anyon model — the canonical non-Abelian order of Majorana zero modes — reconstructing the topological spins θσ=eiπ/8\theta_\sigma = e^{i\pi/8}θσ​=eiπ/8, θψ=−1\theta_\psi = -1θψ​=−1, the total quantum dimension D=2\mathcal{D}=2D=2, and the full SSS matrix from the braid eigenvalues, with all arithmetic shown. We exhibit a Galois-conjugate deformation Ising(7)\mathrm{Ising}(7)Ising(7) with identical fusion rules but distinct SSS matrix, and verify that its braid representation is inequivalent to the Ising representation on B3B_3B3​ by an explicit trace computation, while remaining indistinguishable at the level of fusion rules. We conclude that braid representations constrain but do not provably uniquely determine modular data in general: uniqueness can hold only under additional hypotheses, and the Majorana fusion-rule classification is not closed by braid data alone. The affirmative claim that braiding is a complete invariant is restricted to a narrower regime than sometimes asserted, and we document this divergence explicitly. ## 1. Introduction Non-Abelian anyons in two spatial dimensions are characterized by two layers of algebraic structure. The first is combinatorial: fusion rules NabcN_{ab}^cNabc​ specifying how anyon c Full text and updates: papers.qnfo.org/papers/braid-group-representations-and-modular-data-non-uniqueness-finite-images-and-th/
It has been a long-standing open problem to construct a general framework for relating the spectra of dual theories to each other. Here, we solve this problem for the case of one-dimensional quantum lattice models with symmetry-twisted boundary conditions. In Ref. [PRX Quantum 4, 020357], dualities are defined between (categorically) symmetric models that only differ in a choice of module category. Using matrix product operators, we construct from the data of module functors explicit symmetry operators preserving boundary conditions as well as intertwiners mapping topological sectors of dual models onto one another. We illustrate our construction with a family of examples that are in the duality class of the spin- 12 Heisenberg XXZ model. One model has symmetry operators forming the fusion category Rep(S3) of representations of the group S3 . We find that the mapping between its topological sectors and those of the XXZ model is associated with the nontrivial braided autoequivalence of the Drinfel’d center of Rep(S3) . Published by the American Physical Society 2024
Let A\mathcal{A}A be a (not necessarily rational) conformal net with an action of a discrete group GGG. We show that the category RepG(A)\text{Rep}^G(\mathcal{A})RepG(A) of GGG-twisted representations of A\mathcal{A}A is canonically a GGG-crossed balanced W∗\mathrm{W}^*W∗-tensor category. This extends the results of Müger arXiv:math/0403322, in the language of localized endomorphisms, that RepG(A)\text{Rep}^G(\mathcal{A})RepG(A) is a GGG-crossed braided tensor category.