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1. Superselection Sectors for Posets of von Neumann Algebras

   https://doi.org/10.1007/s00220-025-05315-4  

   Bhardwaj, Anupama; Brisky, Tristen; Chuah, Chian_Yeong; Kawagoe, Kyle; Keslin, Joseph; Penneys, David; Wallick, Daniel (July 2025, Communications in Mathematical Physics)

   Abstract We study a commutant-closed collection of von Neumann algebras acting on a common Hilbert space indexed by a poset with an order-reversing involution. We give simple geometric axioms for the poset which allow us to construct a braided tensor category of superselection sectors analogous to the construction of Gabbiani and Fröhlich for conformal nets. For cones in$$\mathbb {R}^2$$ $R^2$, we weaken our conditions to a bounded spread version of Haag duality and obtain similar results. We show that intertwined nets of algebras have isomorphic braided tensor categories of superselection sectors. Finally, we show that the categories constructed here are equivalent to those constructed by Naaijkens and Ogata for certain 2D quantum spin systems. 

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2. Synchronizing dynamical systems: Their groupoids and C*-algebras

   https://doi.org/10.1090/tran/8969  

   Deeley, Robin J; Stocker, Andrew (March 2024, Transactions of the American Mathematical Society)

   Building on work of Ruelle and Putnam in the Smale space case, Thomsen defined the homoclinic and heteroclinic -algebras for an expansive dynamical system. In this paper we define a class of expansive dynamical systems, called synchronizing dynamical systems, that exhibit hyperbolic behavior almost everywhere. Synchronizing dynamical systems generalize Smale spaces (and even finitely presented systems). Yet they still have desirable dynamical properties such as having a dense set of periodic points. We study various -algebras associated with a synchronizing dynamical system. Among other results, we show that the homoclinic algebra of a synchronizing system contains an ideal which behaves like the homoclinic algebra of a Smale space. 

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3. Planar Algebras in Braided Tensor Categories

   https://doi.org/10.1090/memo/1392  

   Henriques, André; Penneys, David; Tener, James (February 2023, Memoirs of the American Mathematical Society)

   We generalize Jones’ planar algebras by internalising the notion to a pivotal braided tensor category $\mathcal{C}$. To formulate the notion, the planar tangles are now equipped with additional ‘anchor lines’ which connect the inner circles to the outer circle. We call the resulting notion ananchored planar algebra. If we restrict to the case when $\mathcal{C}$is the category of vector spaces, then we recover the usual notion of a planar algebra. Building on our previous work on categorified traces, we prove that there is an equivalence of categories between anchored planar algebras in $\mathcal{C}$and pivotal module tensor categories over $\mathcal{C}$equipped with a chosen self-dual generator. Even in the case of usual planar algebras, the precise formulation of this theorem, as an equivalence of categories, has not appeared in the literature. Using our theorem, we describe many examples of anchored planar algebras. 

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4. Unitary Anchored Planar Algebras

   https://doi.org/10.1007/s00220-024-04985-w  

   Henriques, André; Penneys, David; Tener, James (May 2024, Communications in Mathematical Physics)

   Abstract In our previous article (http://arxiv.org/abs/1607.06041), we established an equivalence between pointed pivotal module tensor categories and anchored planar algebras. This article introduces the notion of unitarity for both module tensor categories and anchored planar algebras, and establishes the unitary analog of the above equivalence. Our constructions use Baez’s 2-Hilbert spaces (i.e., semisimple$$\textrm{C}^*$$ ${\text{C}}^{\ast }$-categories equipped with unitary traces), the unitary Yoneda embedding, and the notion of unitary adjunction for dagger functors between 2-Hilbert spaces. 

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5. Symmetries as Ground States of Local Superoperators: Hydrodynamic Implications

   https://doi.org/10.1103/PRXQuantum.5.040330  

   Moudgalya, Sanjay; Motrunich, Olexei I (November 2024, PRX Quantum)

   Symmetry algebras of quantum many-body systems with locality can be understood using commutant algebras, which are defined as algebras of operators that commute with a given set of local operators. In this work, we show that these symmetry algebras can be expressed as frustration-free ground states of a local superoperator, which we refer to as a “super-Hamiltonian.” We demonstrate this for conventional symmetries such as ${\mathbb{Z}}_2$, $U\left(1\right)$, and $SU\left(2\right)$, where the symmetry algebras map to various kinds of ferromagnetic ground states, as well as for unconventional ones that lead to weak ergodicity-breaking phenomena of Hilbert-space fragmentation (HSF) and quantum many-body scars. In addition, we show that the low-energy excitations of this super-Hamiltonian can be understood as approximate symmetries, which in turn are related to slowly relaxing hydrodynamic modes in symmetric systems. This connection is made precise by relating the super-Hamiltonian to the superoperator that governs the operator relaxation in noisy symmetric Brownian circuits and this physical interpretation also provides a novel interpretation for Mazur bounds for autocorrelation functions. We find examples of gapped (gapless) super-Hamiltonians indicating the absence (presence) of slow modes, which happens in the presence of discrete (continuous) symmetries. In the gapless cases, we recover hydrodynamic modes such as diffusion, tracer diffusion, and asymptotic scars in the presence of $U\left(1\right)$symmetry, HSF, and a tower of quantum scars, respectively. In all, this demonstrates the power of the commutant-algebra framework in obtaining a comprehensive understanding of exact symmetries and associated approximate symmetries and hydrodynamic modes, and their dynamical consequences in systems with locality. Published by the American Physical Society2024 

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