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On Reducibility of Mapping Class Group Representations: The SU(N) Case

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J{o}rgen Ellegaard Andersen, Jens Fjelstad

Quantum Algebra · Submitted 25 Feb 2009

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim As a result we show that the quantum representations are reducible in the SU(N) case, N>2, for all levels k\in \mathbb{N}.

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arXiv ID0902.4375 (version 1)
Pith ID3FVRBFBN, Pith's permanent record of this paper and its review
CitationJens Fjelstad, J{\o}rgen Ellegaard Andersen. "On Reducibility of Mapping Class Group Representations: The SU(N) Case." arXiv:0902.4375 (2009). https://pith.science/paper/3FVRBFBN
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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.21 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.50 · compares with prior results 0.93 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; .

We review and extend the results of [1] that gives a condition for reducibility of quantum representations of mapping class groups constructed from Reshetikhin-Turaev type topological quantum field theories based on modular categories. This criterion is derived using methods developed to describe rational conformal field theories, making use of Frobenius algebras and their representations in modular categories. Given a modular category C, a rational conformal field theory can be constructed from a Frobenius algebra A in C. We show that if C contains a symmetric special Frobenius algebra A such that the torus partition function Z(A) of the corresponding conformal field theory is non-trivial, implying reducibility of the genus 1 representation of the modular group, then the representation of the genus g mapping class group constructed from C is reducible for every g\geq 1. We also extend the number of examples where we can show reducibility significantly by establishing the existence of algebras with the required properties using methods developed by Fuchs, Runkel and Schweigert. As a result we show that the quantum representations are reducible in the SU(N) case, N>2, for all levels k\in \mathbb{N}. The SU(2) case was treated explicitly in [1], showing reducibility for even levels k\geq 4.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We review and extend the results of [1] that gives a condition for reducibility of quantum representations of mapping class groups constructed from Reshetikhin-Turaev type topological quantum field theories based on modular categories. contribution 0.85 0.75 0.09 asserted 0.96
s1 This criterion is derived using methods developed to describe rational conformal field theories, making use of Frobenius algebras and their representations in modular categories. method 0.87 0.21 0.03 asserted 1.00
s2 Given a modular category C, a rational conformal field theory can be constructed from a Frobenius algebra A in C. background 0.98 0.09 0.02 asserted 0.97
s3 We show that if C contains a symmetric special Frobenius algebra A such that the torus partition function Z(A) of the corresponding conformal field theory is non-trivial, implying reducibility of the genus 1 representation of the modular group, then the representation of the genus g mapping class group constructed from C is reducible for every g\geq 1. result 0.94 0.92 0.68 asserted 1.00
s4 We also extend the number of examples where we can show reducibility significantly by establishing the existence of algebras with the required properties using methods developed by Fuchs, Runkel and Schweigert. contribution 0.80 0.92 0.06 asserted 1.00
s5 As a result we show that the quantum representations are reducible in the SU(N) case, N>2, for all levels k\in \mathbb{N}. result 0.99 0.94 0.75 asserted 1.00
s6 The SU(2) case was treated explicitly in [1], showing reducibility for even levels k\geq 4. background 0.95 0.06 0.04 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We review and extend the results of [1] that gives a condition for reducibility of quantum representations of mapping class groups constructed from Reshetikhin-Turaev type topological quantum field theories based on modular categories. This criterion is derived using methods developed to describe rational conformal field theories, making use of Frobenius algebras and their representations in modular categories. Given a modular category C, a rational conformal field theory can be constructed from a Frobenius algebra A in C. We show that if C contains a symmetric special Frobenius algebra A such that the torus partition function Z(A) of the corresponding conformal field theory is non-trivial, implying reducibility of the genus 1 representation of the modular group, then the representation of the genus g mapping class group constructed from C is reducible for every g\geq 1. We also extend the number of examples where we can show reducibility significantly by establishing the existence of algebras with the required properties using methods developed by Fuchs, Runkel and Schweigert. As a result we show that the quantum representations are reducible in the SU(N) case, N>2, for all levels k\in \mathbb{N}. The SU(2) case was treated explicitly in [1], showing reducibility for even levels k\geq 4.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 31 sentences without a confident label

Builds on 2 papers
  • redcrit We first review Theorem 1 of [ref] , that gives a condition for reducibility of quantum representations, and then show that the conditions of the theorem is satisfied for the modular categories of integrable highest weight representations of $𝔰𝔲(N)$ for $N>2$ and any level $k∈ℕ$.

    Reducibility of quantum representations of mapping class groups · A Reducibility Criterion · 0.87

  • This generalises a structure in the $𝔰𝔲(2)$ case that was used in [ref] to construct an action of a Heisenberg group on $ℋ_g$, which in turn is responsible for the reducibility of the corresponding quantum representations shown in [ref] .

    Involutions on Moduli Spaces and Refinements of the Verlinde Formula · TQFT and Quantum Representations · 0.81

Uses the method or tool of 8 papers
  • The techniques we employ were developed in the series of papers [ref] , and the proof presented here for the $𝔰𝔲(N)$ case relies in particular on results from [ref] concerning Frobenius algebras constructed from direct sums of invertible objects in modular categories.

    TFT construction of RCFT correlators I: Partition functions · Introduction · 0.85

  • The techniques we employ were developed in the series of papers [ref] , and the proof presented here for the $𝔰𝔲(N)$ case relies in particular on results from [ref] concerning Frobenius algebras constructed from direct sums of invertible objects in modular categories.

    TFT construction of RCFT correlators II: Unoriented world sheets · Introduction · 0.85

  • The techniques we employ were developed in the series of papers [ref] , and the proof presented here for the $𝔰𝔲(N)$ case relies in particular on results from [ref] concerning Frobenius algebras constructed from direct sums of invertible objects in modular categories.

    TFT construction of RCFT correlators III: Simple currents · Introduction · 0.85

Compares with 2 papers
  • For $k=10$ an algebra can be constructed on the object $(0)⊕(6)$, for $k=16$ there is an algebra on the object $(0)⊕(8)⊕(16)$, and for $k=28$ there is an algebra on the object $(0)⊕(10)⊕(18)⊕(28)$, all of which belong to different Morita classes than the trivial algebra or the Schellekens algebra.This classification coincides precisely with the ADE classification of modular invariants for the same categories [ref] , and algebras not Morita equivalent to $1$ give non-trivial modular invariants $Z(A)$ [ref] .

    Modular Invariants, Graphs and $\alpha$-Induction for Nets of Subfactors II · The $SU(N)$ Case · 0.85

  • When $N=3$ one easily compares the torus partition functions of the Schellekens algebras with the complete list of modular invariants from [ref] , with the expected result.

    The Classification of Affine SU(3) Modular Invariant Partition Functions · The $SU(N)$ Case · 0.99

Cites as background 12 papers

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