arXiv:2412.07198

Tube Category, Tensor Renormalization and Topological Holography

Tian Lan

Published Dec 10, 2024• 0 citations

Abstract

Ocneanu's tube algebra provides a finite algorithm to compute the Drinfeld center of a fusion category. In this work we reveal the universal property underlying the tube algebra. Take a base category $\mathcal V$ which is strongly concrete, bicomplete, and closed symmetric monoidal. For physical applications one takes $\mathcal V=\mathbf{Vect}$ the category of vector spaces. Given a $\mathcal V$-enriched rigid monoidal category $\mathcal C$ (not necessarily finite or semisimple) we define the tube category $\mathbb X \mathcal C$ using coends valued in $\mathcal V$. Our main theorem established the relation between (the category of representations of) the tube category $\mathbb X \mathcal C$ and the Drinfeld center $Z(\mathcal C)$: $Z(\mathcal C)\hookrightarrow \mathrm{Fun}(\mathbb X \mathcal C^{\mathrm{op}},\mathcal V)\cong Z(\mathcal C\hookrightarrow\mathrm{Fun}(\mathcal C^{\mathrm{op}},\mathcal V))\hookrightarrow Z(\mathrm{Fun}(\mathcal C^{\mathrm{op}},\mathcal V))$. Physically, besides viewing the tube category as a version of TFT with domain being the tube, we emphasize the "Wick-rotated" perspective, that the morphisms in $\mathbb X \mathcal C$ are the local tensors of fixed-point matrix product operators which preserves the symmetry $\mathcal C$ in one spatial dimension. We provide a first-principle flavored construction, from microscopic quantum degrees of freedom and operators preserving the symmetry, to the macroscopic universal properties of the symmetry which form the Drinfeld center. Our work is thus a proof to the 1+1D topological holography in a very general setting.

DOI: 10.48550/arxiv.2412.07198

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arXiv:2412.07198

Tube Category, Tensor Renormalization and Topological Holography

Tian Lan

Published Dec 10, 2024• 0 citations

arXiv HTML PDF

Abstract

Ocneanu's tube algebra provides a finite algorithm to compute the Drinfeld center of a fusion category. In this work we reveal the universal property underlying the tube algebra. Take a base category

\mathcal V

which is strongly concrete, bicomplete, and closed symmetric monoidal. For physical applications one takes

\mathcal V=\mathbf{Vect}

the category of vector spaces. Given a

\mathcal V

-enriched rigid monoidal category

\mathcal C

(not necessarily finite or semisimple) we define the tube category

\mathbb X \mathcal C

using coends valued in

\mathcal V

. Our main theorem established the relation between (the category of representations of) the tube category

\mathbb X \mathcal C

and the Drinfeld center

Z(\mathcal C)

Z(\mathcal C)\hookrightarrow \mathrm{Fun}(\mathbb X \mathcal C^{\mathrm{op}},\mathcal V)\cong Z(\mathcal C\hookrightarrow\mathrm{Fun}(\mathcal C^{\mathrm{op}},\mathcal V))\hookrightarrow Z(\mathrm{Fun}(\mathcal C^{\mathrm{op}},\mathcal V))

. Physically, besides viewing the tube category as a version of TFT with domain being the tube, we emphasize the "Wick-rotated" perspective, that the morphisms in

\mathbb X \mathcal C

are the local tensors of fixed-point matrix product operators which preserves the symmetry

\mathcal C

in one spatial dimension. We provide a first-principle flavored construction, from microscopic quantum degrees of freedom and operators preserving the symmetry, to the macroscopic universal properties of the symmetry which form the Drinfeld center. Our work is thus a proof to the 1+1D topological holography in a very general setting.

DOI: 10.48550/arxiv.2412.07198

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A variational principle for quantum-state distinguishability is established in many-body systems with generalized symmetries, including noninvertible cases described by fusion categories. Standard fidelity and symmetry-resolved diagnostics emerge as coarse-grained limits of a more refined operational structure. When symmetry actions terminate at entanglement cuts, distinguishability is governed by boundary tube algebras within a symmetry-constrained measurement resource theory. The physically admissible instruments are characterized by complete positivity, entanglement-cut locality, boundary-module covariance, and sequential stability. The resulting optimal measurement structure is uniquely fixed by the center of the boundary tube algebra, $\mathcal{A}_{\mathrm{phys}} = Z\!\left(\mathrm{Tube}_{\mathcal{C}}(\mathcal{M}_A)\right)$ , whose primitive dempotents define tube-sector probabilities that refine fidelity-based and symmetry-resolved descriptions. The associated tube positive-operator-valued measures (POVMs) are extremal and yield optimal one-shot hypothesis-testing distinguishability under symmetry constraints. The categorical selection rule is universal across fusion categories and independent of microscopic realization; microscopic models enter by choosing the boundary module and the protocol family used to witness a nontrivial tube fiber.A variational principle for quantum-state distinguishability is established in many-body systems with generalized symmetries, including noninvertible cases described by fusion categories. Standard fidelity and symmetry-resolved diagnostics emerge as coarse-grained limits of a more refined operational structure. When symmetry actions terminate at entanglement cuts, distinguishability is governed by boundary tube algebras within a symmetry-constrained measurement resource theory. The physically admissible instruments are characterized by complete positivity, entanglement-cut locality, boundary-module covariance, and sequential stability. The resulting optimal measurement structure is uniquely fixed by the center of the boundary tube algebra, $\mathcal{A}_{\mathrm{phys}} = Z\!\left(\mathrm{Tube}_{\mathcal{C}}(\mathcal{M}_A)\right)$, whose primitive dempotents define tube-sector probabilities that refine fidelity-based and symmetry-resolved descriptions. The associated tube positive-operator-valued measures (POVMs) are extremal and yield optimal one-shot hypothesis-testing distinguishability under symmetry constraints. The categorical selection rule is universal across fusion categories and independent of microscopic realization; microscopic models enter by choosing the boundary module and the protocol family used to witness a nontrivial tube fiber.

hep-thCross-listedarXiv:2606.19678Jun 19, 20260 citations

Ocneanu's tube algebra provides a finite algorithm to compute the Drinfeld center of a fusion category. In this work we reveal the universal property underlying the tube algebra. Take a base category

\mathcal V

which is strongly concrete, bicomplete, and closed symmetric monoidal. For physical applications one takes

\mathcal V=\mathbf{Vect}

the category of vector spaces. Given a

\mathcal V

-enriched rigid monoidal category

\mathcal C

(not necessarily finite or semisimple) we define the tube category

\mathbb X \mathcal C

using coends valued in

\mathcal V

. Our main theorem established the relation between (the category of representations of) the tube category

\mathbb X \mathcal C

and the Drinfeld center

Z(\mathcal C)

Z(\mathcal C)\hookrightarrow \mathrm{Fun}(\mathbb X \mathcal C^{\mathrm{op}},\mathcal V)\cong Z(\mathcal C\hookrightarrow\mathrm{Fun}(\mathcal C^{\mathrm{op}},\mathcal V))\hookrightarrow Z(\mathrm{Fun}(\mathcal C^{\mathrm{op}},\mathcal V))

. Physically, besides viewing the tube category as a version of TFT with domain being the tube, we emphasize the "Wick-rotated" perspective, that the morphisms in

\mathbb X \mathcal C

are the local tensors of fixed-point matrix product operators which preserves the symmetry

\mathcal C

This is the continuation of our study of the Levin-Wen model based on an arbitrary unitary fusion category

\mathcal{C}

on the infinite plane. The ground state of the Levin-Wen model hosts anyonic excitations whose fusion and braiding properties are captured by the associated braided

\rm C^*

-tensor category of superselection sectors

\mathsf{SSS}

. By constructing explicit isomorphisms between the fusion spaces of

\mathsf{SSS}

and those of the Drinfeld center

Z(\mathcal{C})

, we show that these two categories have isomorphic

F

- and

R

-symbols. It follows that the full subcategory of finite sectors is unitarily braided monoidally equivalent to the Drinfeld center,

\,\mathsf{SSS}_f \simeq Z(\mathcal{C}).

This provides the first complete characterisation of the category of superselection sectors for a class of two-dimensional lattice models supporting anyons with non-integer quantum dimensions.

B

of a quasi-local algebra

A

, there is a canonical fusion category

\mathcal{C}

that acts on

A

by bimodules and whose fusion ring acts by locality preserving quantum channels on the quasi-local algebra such that

B

is recovered as the fixed point operators. We show that a fusion category can be realized as symmetries on a tensor product quasi-local algebra if and only if all of its objects have integer dimensions, and that it admits an ``on-site" action on a tensor product spin chain if and only if it admits a fiber functor. We give a formal definition of a topological symmetric state, and prove two anomaly enforced gaplessness theorems, one for internal categorical symmetries and one for anomalous duality channels. Using the first, we show that for any fusion category

\mathcal{C}

with no fiber functor there always exist gapless pure symmetric states on an anyon chain.

\mathcal{A}_{\mathrm{phys}} = Z\!\left(\mathrm{Tube}_{\mathcal{C}}(\mathcal{M}_A)\right)

, whose primitive dempotents define tube-sector probabilities that refine fidelity-based and symmetry-resolved descriptions. The associated tube positive-operator-valued measures (POVMs) are extremal and yield optimal one-shot hypothesis-testing distinguishability under symmetry constraints. The categorical selection rule is universal across fusion categories and independent of microscopic realization; microscopic models enter by choosing the boundary module and the protocol family used to witness a nontrivial tube fiber.