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arXiv:2606.03623

Twisted representations of conformal nets and crossed balanced tensor categories

Adrià Marín-Salvador

Published Jun 2, 2026• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2606.03623

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Let A\mathcal{A}A be a completely rational local Möbius covariant net on S1S^1S1, which describes a set of chiral observables. We show that local Möbius covariant nets B2\mathcal{B}_2B2​ on 2D Minkowski space which contains A\mathcal{A}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)\mathrm{DHR}(\mathcal{A})DHR(A). The Möbius covariant boundary conditions with symmetry A\mathcal{A}A of such a net B2\mathcal{B}_2B2​ 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, ααα-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.
Q-systems describe "extensions" of an infinite von Neumann factor NNN, i.e., finite-index unital inclusions of NNN into another von Neumann algebra MMM. They are (special cases of) Frobenius algebras in the C* tensor category of endomorphisms of NNN. We review the relation between Q-systems, their modules and bimodules as structures in a category on one side, and homomorphisms between von Neumann algebras on the other side. We then elaborate basic operations with Q-systems (various decompositions in the general case, and the centre, the full centre, and the braided product in braided categories), and illuminate their meaning in the von Neumann algebra setting. The main applications are in local quantum field theory, where Q-systems in the subcategory of DHR endomorphisms of a local algebra encode extensions A(O)⊂B(O)A(O)\subset B(O)A(O)⊂B(O) of local nets. These applications, notably in conformal quantum field theories with boundaries, are briefly exposed, and are discussed in more detail in two separate papers [arXiv:1405.7863, 1410.8848].
The classification of mixed-state topological order requires indices that behave monotonically under finite-depth quantum channels. In two dimensions, a braided C∗C^*C∗-tensor category, which corresponds to strong symmetry, arises from a state satisfying approximate Haag duality. In this note, we show that the SSS-matrix and topological twists of the braided C∗C^*C∗-tensor category are quantities that are monotone under finite-depth quantum channels.