Reshetikhin–Turaev invariant

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In the mathematical field of quantum topology, the Reshetikhin–Turaev invariants (RT-invariants) are a family of quantum invariants of framed links. Such invariants of framed links also give rise to invariants of 3-manifolds via the Dehn surgery construction. These invariants were discovered by Nicolai Reshetikhin and Vladimir Turaev in 1991, [1] and were meant to be a mathematical realization of Witten's proposed invariants of links and 3-manifolds using quantum field theory. [2]

Contents

Overview

To obtain an RT-invariant, one must first have a -linear ribbon category at hand. Each -linear ribbon category comes equipped with a diagrammatic calculus in which morphisms are represented by certain decorated framed tangle diagrams, where the initial and terminal objects are represented by the boundary components of the tangle. In this calculus, a (decorated framed) link diagram , being a (decorated framed) tangle without boundary, represents an endomorphism of the monoidal identity (the empty set in this calculus), or in other words, an element of . This element of is the RT-invariant associated to . Given any closed oriented 3-manifold , there exists a framed link in the 3-sphere so that is homeomorphic to the manifold obtained by surgering along . Two such manifolds and are homeomorphic if and only if and are related by a sequence of Kirby moves. Reshetikhin and Turaev [1] used this idea to construct invariants of 3-manifolds by combining certain RT-invariants into an expression which is invariant under Kirby moves. Such invariants of 3-manifolds are known as Witten–Reshetikhin–Turaev invariants (WRT-invariants).

Examples

Let be a ribbon Hopf algebra over a field (one can take, for example, any quantum group over ). Then the category , of finite dimensional representations of , is a -linear ribbon category. [3] There is a diagrammatic calculus in which morphisms in are represented by framed tangle diagrams with each connected component decorated by a finite dimensional representation of . That is, is a -linear ribbon category. In this way, each ribbon Hopf algebra gives rise to an invariant of framed links colored by representations of (an RT-invariant).

For the quantum group over the field , the corresponding RT-invariant for links and 3-manifolds gives rise to the following family of link invariants, appearing in skein theory. Let be a framed link in with components. For each , let denote the RT-invariant obtained by decorating each component of by the unique -dimensional representation of . Then

where the -tuple, denotes the Kauffman polynomial of the link , where each of the components is cabled by the Jones–Wenzl idempotent , a special element of the Temperley–Lieb algebra.

To define the corresponding WRT-invariant for 3-manifolds, first of all we choose to be either a -th root of unity or an -th root o–unity with odd . Assume that is obtained by doing Dehn surgery on a framed link . Then the RT-invariant for the 3-manifold is defined to be

where is the Kirby coloring, are the unknot with framing, and are the numbers of positive and negative eigenvalues for the linking matrix of respectively. Roughly speaking, the first and second bracket ensure that is invariant under blowing up/down (first Kirby move) and the third bracket ensures that is invariant under handle sliding (second Kirby move).

Properties

The Witten-Reshetikhin–Turaev invariants for 3-manifolds satisfy the following properties:

  1. where denotes the connected sum of and
  2. where is the manifold with opposite orientation, and denotes the complex conjugate of

These three properties coincide with the properties satisfied by the 3-manifold invariants defined by Witten using Chern–Simons theory (under certain normalization) [4]

Open problems

Witten's asymptotic expansion conjecture [5]

Pick . Witten's Asymptotic Expansion Conjecture suggests that for every 3-manifold , the large -th asymptotics of is governed by the contributions of flat connections.

Conjecture: There exists constants and (depending on ) for and for such that the asymptotic expansion of in the limit is given by

where are the finitely many different values of the Chern–Simons functional on the space of flat -connections on .

Volume conjecture for the Reshetikhin–Turaev invariant [6]

The Witten's Asymptotic Expansion Conjecture suggests that at , the RT-invariants grow polynomially in . On the contrary, at with odd , in 2015 Q. Chen and T. Yang suggested the volume conjecture for the RT-invariants, which essentially says that the RT-invariants for hyperbolic 3-manifolds grow exponentially in and the growth rate gives the hyperbolic volume and Chern–Simons invariants for the 3-manifold.

Conjecture: Let be a closed oriented hyperbolic 3-manifold. Then for a suitable choice of arguments,

where is odd positive integer.

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References

  1. 1 2 Reshetikhin, Nicolai, and Vladimir G. Turaev. "Invariants of 3-manifolds via link polynomials and quantum groups." Inventiones mathematicae 103.1 (1991): 547–597.
  2. Witten, Edward. "Quantum field theory and the Jones polynomial." Communications in Mathematical Physics 121.3 (1989): 351-399.
  3. Turaev, Vladimir G. Quantum invariants of knots and 3-manifolds. Vol. 18. Walter de Gruyter GmbH & Co KG, 2016.
  4. Witten, Edward. "Quantum field theory and the Jones polynomial." Communications in Mathematical Physics 121.3 (1989): 351–399.
  5. Andersen, Jørgen Ellegaard, and Søren Kold Hansen. "Asymptotics of the quantum invariants for surgeries on the figure 8 knot." Journal of Knot theory and its Ramifications 15.04 (2006): 479–548.
  6. Chen, Qingtao, and Tian Yang. "Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants." arXiv preprint arXiv:1503.02547 (2015).