## ON PICTURE (2+1)-TQFTS (2008)

### BibTeX

@MISC{Freedman08onpicture,

author = {Michael Freedman and Chetan Nayak and Kevin Walker and Zhenghan Wang},

title = {ON PICTURE (2+1)-TQFTS},

year = {2008}

}

### OpenURL

### Abstract

The goal of the paper is an exposition of the simplest (2 + 1)-TQFTs in a sense following a pictorial approach. In the end, we fell short on details in the later sections where new results are stated and proofs are outlined. Comments are welcome and should be sent to the 4th author.

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Citation Context ...ones polynomial evaluation Z(S 3 , L) = JL(e 2πi/r ), r = 3, 4, 5, . . ., as the “closed 3−manifold” invariants, which mathematically are the ReshetikhinTuraev invariants based on quantum groups [Jo1]=-=[Witt]-=-[RT]. This is the best known example. Note that physicists tend to index the same family by the levels k = r − 2. The shift 2 is the dual Coxeter number of SU(2). We will use both indices. Most of the... |

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Citation Context ...olynomial evaluation Z(S 3 , L) = JL(e 2πi/r ), r = 3, 4, 5, . . ., as the “closed 3−manifold” invariants, which mathematically are the ReshetikhinTuraev invariants based on quantum groups [Jo1][Witt]=-=[RT]-=-. This is the best known example. Note that physicists tend to index the same family by the levels k = r − 2. The shift 2 is the dual Coxeter number of SU(2). We will use both indices. Most of the “qu... |

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Citation Context ... a complete analysis of the possible local relations. Experts have long been troubled by certain sign discrepancies between the S−matrix arising from representations (or loop groups or quantum groups)=-=[MS]-=-[Witt][KM] on the one hand and from the Kauffman bracket on the other [Li][Tu][KL]. The source of the discrepancy is that the fundamental representation of SU(2) is antisymmetrically self dual whereas... |

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Citation Context ...s a Jones polynomial evaluation Z(S 3 , L) = JL(e 2πi/r ), r = 3, 4, 5, . . ., as the “closed 3−manifold” invariants, which mathematically are the ReshetikhinTuraev invariants based on quantum groups =-=[Jo1]-=-[Witt][RT]. This is the best known example. Note that physicists tend to index the same family by the levels k = r − 2. The shift 2 is the dual Coxeter number of SU(2). We will use both indices. Most ... |

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Citation Context ...r the braid group representations. All above theories can be doubled to get picture TQFTs: the doubled JonesKauffman TQFTs are the diagram TQFTs, while the doubled WRT TQFTs are the Turaev-Viro TQFTs =-=[TV]-=-. The doubles of even sub-categories for r odd form part of the Black-White TQFTs in Theorem 9.3, while for r even this is still a conjecture. When A is a primitive 2rth or rth root of unity and r odd... |

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Citation Context ...led by certain sign discrepancies between the S−matrix arising from representations (or loop groups or quantum groups)[MS][Witt][KM] on the one hand and from the Kauffman bracket on the other [Li][Tu]=-=[KL]-=-. The source of the discrepancy is that the fundamental representation of SU(2) is antisymmetrically self dual whereas there is no room in Kauffman’s spin-network notation to record the antisymmetry. ... |

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Citation Context ... to “skeletonizing” the larger category and replacing some “strict” associations by “weak” ones. Apparently a theorem of S. MacLane guarantees that no harm follows, so either viewpoint can be adopted =-=[Ma]-=-. We will work with the continuously many objects version. Recall the following definition from Appendix B: Definition 4.1. A representation of a linear category Λ is a functor ρ : Λ → V, where V is t... |

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Citation Context ... course (2). We do recommend to some interested brave reader that she produce her own article hewing to course (1).4 MICHAEL FREEDMAN, CHETAN NAYAK, KEVIN WALKER, AND ZHENGHAN WANG In the literature =-=[BHMV]-=- comes closest to the goals of the notes, and [Wal2] exploits deeply the picture theories in many directions. Actually, a large part of the notes will follow from a finished [Wal2]. If one applies the... |

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Citation Context ...rt is the mapping class group action. This is explained at the end of Section 3.14. 8. WRT and Turaev-Viro SU(2)-TQFTs The pictorial approach to the Witten-Reshetikhin-Turaev SU(2) TQFTs was based on =-=[KM]-=-. The paper [KM] finished with 3-manifold invariants, just as [KL] for the Jones-Kauffman theories. The paper [BHMV] took the picture approach in [KL] one step further to TQFTs, but the same for WRT T... |

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Citation Context ...his form or slightly more subtle quantum doubles or Drinfeld centers in which the original theory V violates some axiom (the nonsingularity of the S−matrix) but this deficiency is “cured” by doubling =-=[K]-=-[Mu]. Although those notes focus on picture TQFTs based on variations of the Jones-Wenzl projectors, the approach can be generalized to an arbitrary spherical tensor category. The Temperley-Lieb categ... |

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Citation Context ...ussion, but it is instructive to see how things work in TLJ R d=1 and TLJ A d=1, the TLJ rectangular and picture categories for d = 1. These simple examples include the celebrated toric codes TQFT in =-=[Ki1]-=- or Z2 gauge theory, and illustrate the general techniques. Since it is almost no extra work, we will include the corresponding calculation for TLJ R d=−1 and TLJ A d=−1 where A = e 2πi 12 , d = −1 an... |

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Citation Context ...ants but there has been a growing awareness that a deeper understanding is locked up in the representation spaces V (Y ) and the “higher algebras” associated to boundary (Y ) (circles) and points [FQ]=-=[Fd]-=-. Let us explain this last statement. While invariants of 3−manifolds may be fascinating in their interrelations there is something of a shortage of work for them within topology. Reidemeister was pro... |

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Citation Context ...variants but there has been a growing awareness that a deeper understanding is locked up in the representation spaces V (Y ) and the “higher algebras” associated to boundary (Y ) (circles) and points =-=[FQ]-=-[Fd]. Let us explain this last statement. While invariants of 3−manifolds may be fascinating in their interrelations there is something of a shortage of work for them within topology. Reidemeister was... |

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Citation Context ...bra. But when d is a root of some Chebyshev polynomial, TLn(A) is in general not semi-simple. Jones discovered a semi-simple quotient by introducing local relations, called the Jones-Wenzl projectors =-=[Jo4]-=-[We][KL]. Jones-Wenzl projectors have certain rigidity. Represented by formal diagrams in TL algebras, Jones-Wenzl projectors make it possible to describe two families of TQFTs labelled by integers. C... |

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Citation Context ...ors. Our version of the Turaev-Viro SU(2)-TQFTs have non-trivial FS indicators, but no anomaly; while the WRT TQFTs have both anomaly, and non-trivial FS indicators. Our treatment essentially follows =-=[Wal1]-=- with two variations: first the axioms in [Wal1] apply only to TQFTs with trivial FS indicators, so we extend the label set to cover the non-trivial FS indicators; secondly we choose to resolve the an... |

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Citation Context ...roubled by certain sign discrepancies between the S−matrix arising from representations (or loop groups or quantum groups)[MS][Witt][KM] on the one hand and from the Kauffman bracket on the other [Li]=-=[Tu]-=-[KL]. The source of the discrepancy is that the fundamental representation of SU(2) is antisymmetrically self dual whereas there is no room in Kauffman’s spin-network notation to record the antisymmet... |

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Citation Context ...ze was for the prediction of an anyon carrying change e/3 and with braiding statistics e 2πi/3 . In the FQHE central charge c ̸= 0 is enforced by a symmetry breaking magnetic field B. It is argued in =-=[Fn]-=- that solid state realizations of doubled or “picture” TQFTs may - if found - be more stable (larger spectral gap above the degenerate ground state manifold) because no symmetry breaking is required. ... |

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Citation Context ...e. mij = tr(DiDj), then we have: n∏ (2.13) Det(Mcn×cn) = ∆i(d) i=1 an,i , where an,i = ( ) 2n + n−i−2 ( ) 2n − 2 n−i ( ) 2n . n−i−1PICTURE TQFTS 9 Figure 5. Jones Wenzl projectors This is derived in =-=[DGG]-=-. As a quick consequence of this formula, we have: Lemma 2.14. The dimension of TLn(d) as a vector space over C(d) is cn if d is not a root of the Chebyshev polynomials ∆i, 1 ≤ i ≤ n, where cn = 1 ) .... |

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Citation Context ...aces. In Section 6.3, we will give the definition of a TQFT, and later verify all axioms for diagram TQFTs. The irreps of the non-semi-simple TL annular categories at roots of unity were contained in =-=[GL]-=-, but we need the irreps of the semi-simple quotients of TL annular categories, i.e., the TLJ annular categories. The irreps of the TLJ annular will be analyzed in Sections 5.5 5.6. In the following, ... |

10 | The n-eigenvalue problem and two - Larsen, Rowell, et al. - 2005 |

8 |
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Citation Context ...en troubled by certain sign discrepancies between the S−matrix arising from representations (or loop groups or quantum groups)[MS][Witt][KM] on the one hand and from the Kauffman bracket on the other =-=[Li]-=-[Tu][KL]. The source of the discrepancy is that the fundamental representation of SU(2) is antisymmetrically self dual whereas there is no room in Kauffman’s spin-network notation to record the antisy... |

8 |
From subfactors to categories and topology
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(Show Context)
Citation Context ... form or slightly more subtle quantum doubles or Drinfeld centers in which the original theory V violates some axiom (the nonsingularity of the S−matrix) but this deficiency is “cured” by doubling [K]=-=[Mu]-=-. Although those notes focus on picture TQFTs based on variations of the Jones-Wenzl projectors, the approach can be generalized to an arbitrary spherical tensor category. The Temperley-Lieb categorie... |

4 |
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Citation Context ...on which might throw off readers from physics. When different terminologies prevail within mathematics and physics we will try to note both. Within physics, TQFTs are referred to as “anyonic systems” =-=[Wil]-=-[DFNSS]. These are 2-dimensional quantum mechanical systems with point like excitations (variously called “quas-particle” or just “particle”, anyon, or perhaps “nonabelion”) which under exchange exhib... |

2 |
On classification of modular tensor categories. arXiv:0712.1377
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Citation Context ...eje, 0 ≤ i, j ≤ k can be defined just as in last TQFT they are β∗ ieieβjeje. Also each Li,j section. As operators on the SU(2) even k 2 These TQFTs are called SO(3)-TQFTs by many authors. As noted in =-=[RSW]-=-, there is some mystery about those TQFTs as SO(3)-Witten-Chern-Simons TQFTs. Since they are the same TQFTs as SU(2)-Witten-Reshetikhin-Turaev TQFTs restricted to integral spins, therefore we adopt th... |

1 |
TQFTs, 2006 notes at http://canyon23.net/math
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(Show Context)
Citation Context ...e reader that she produce her own article hewing to course (1).4 MICHAEL FREEDMAN, CHETAN NAYAK, KEVIN WALKER, AND ZHENGHAN WANG In the literature [BHMV] comes closest to the goals of the notes, and =-=[Wal2]-=- exploits deeply the picture theories in many directions. Actually, a large part of the notes will follow from a finished [Wal2]. If one applies the set up of [BHMV] to skeins in surface cross interva... |