View source for Reshetikhin–Turaev invariant

{{Short description|Family of quantum invariants}}
In the mathematical field of [[quantum topology]], the '''Reshetikhin–Turaev invariants''' ('''RT-invariants''') are a family of [[quantum invariants]] of [[framed link]]s.
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,<ref name=RT1991>{{cite journal |last1=Reshetikhin |first1=Nicolai |first2=Vladimir G. |last2=Turaev |title=Invariants of 3-manifolds via link polynomials and quantum groups |journal=Inventiones Mathematicae |volume=103 |issue=1 |year=1991 |pages=547–597 |doi=10.1007/BF01239527 |bibcode=1991InMat.103..547R |s2cid=123376541 }}</ref> and were meant to be a mathematical realization of Witten's proposed invariants of links and 3-manifolds using [[quantum field theory]].<ref name="Witten1989">{{cite journal |last=Witten |first=Edward |title=Quantum field theory and the Jones polynomial |journal=Communications in Mathematical Physics |volume=121 |issue=3 |year=1989 |pages=351–399 |doi=10.1007/BF01217730 |bibcode=1989CMaPh.121..351W |s2cid=14951363 |url=http://projecteuclid.org/euclid.cmp/1104178138 }}</ref>

==Overview==

To obtain an RT-invariant, one must first have a <math>\Bbbk</math>-linear [[ribbon category]] at hand. Each <math>\Bbbk</math>-linear ribbon category comes equipped with a diagrammatic calculus in which morphisms are represented by certain decorated framed [[tangle diagram]]s, where the initial and terminal objects are represented by the boundary components of the tangle. In this calculus, a (decorated framed) link diagram <math>L</math>, 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 <math>\Bbbk</math>. This element of <math>\Bbbk</math> is the RT-invariant associated to <math>L</math>. Given any closed oriented 3-manifold <math>M</math>, there exists a framed link <math>L</math> in the 3-sphere <math>S^3</math> so that <math>M</math> is homeomorphic to the manifold <math>M_L</math> obtained by surgering <math>S^3</math> along <math>L</math>. Two such manifolds <math>M_L</math> and <math>M_{L^\prime}</math> are homeomorphic if and only if <math>L</math> and <math>L^\prime</math> are related by a sequence of [[Kirby moves]]. Reshetikhin and Turaev <ref name=RT1991 /> 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 <math>A</math> be a [[ribbon Hopf algebra]] over a field <math>\Bbbk</math> (one can take, for example, any [[quantum group]] over <math>\mathbb{C}</math>). Consider the category <math>\textbf{Rep}^{\text{f.d.}}(A)</math>, of finite dimensional representations of <math>A</math>. There is a diagrammatic calculus in which morphisms in <math>\textbf{Rep}^{\text{f.d.}}(A)</math> are represented by framed tangle diagrams with each connected component decorated by a finite dimensional representation of <math>A</math>.<ref>{{cite book |last=Turaev |first=Vladimir G. |title=Quantum invariants of knots and 3-manifolds |series=De Gruyter Studies in Mathematics |volume=18 |location=Berlin |publisher=Walter de Gruyter |year=2016 |isbn=978-3-11-044266-3 }}</ref> That is, <math>\textbf{Rep}^{\text{f.d.}}(A)</math> is a <math>\Bbbk</math>-linear ribbon category. In this way, each ribbon [[Hopf algebra]] <math>A</math> gives rise to an invariant of framed links colored by representations of <math>A</math> (an RT-invariant).

For the quantum group  <math>A=U_q(\mathfrak{sl}_2(\mathbb{C}))</math> over the field <math>\mathbb{C}(q)</math>, the corresponding RT-invariant for links and 3-manifolds gives rise to the following family of link invariants, appearing in [[skein theory]]. Let <math>L</math> be a framed link in <math>S^3</math> with <math>m</math> components. For each <math>r\in\mathbb{N}</math>, let <math>\text{RT}_r(S^3, L)</math> denote the RT-invariant obtained by decorating each component of <math>L</math> by the unique <math>N+1</math>-dimensional representation of <math>A</math>. Then 
:<math>\operatorname{RT}_r(S^3,L) = \langle e_n, e_n, \dots, e_n \rangle_L \in\mathbb{C}(q)</math>
where the <math>m</math>-tuple, <math>\langle e_n, e_n, \dots, e_n \rangle_L</math> denotes the Kauffman polynomial of the link <math>L</math>, where each of the <math>m</math> components is cabled by the Jones–Wenzl idempotent <math>e_n</math>, a special element of the [[Temperley–Lieb algebra]].

To define the corresponding WRT-invariant for 3-manifolds, first of all we choose <math>t</math> to be either a <math>2r</math>-th root of unity or an <math>r</math>-th root of unity with odd <math>r</math>. Assume that <math>M_L</math> is obtained by doing Dehn surgery on a framed link <math>L</math>. Then the RT-invariant for the 3-manifold <math>M</math> is defined to be 
:<math>\operatorname{RT}_r(M_L)  = \langle \omega_r \rangle_{O^+}^{b_-} \langle \omega_r \rangle_{O^-}^{b_+} \langle \omega_r, \omega_r, \dots, \omega_r \rangle_L (t)\in \mathbb{C},</math>
where <math>\omega_r = \sum_{n=0}^{r-2} \langle e_n \rangle_{O} e_n</math> is the Kirby coloring, <math>O^\pm</math> are the unknot with <math>\pm 1</math> framing, and <math>b_\pm</math> are the numbers of positive and negative eigenvalues for the linking matrix of <math>L</math> respectively. Roughly speaking, the first and second bracket ensure that <math>\text{RT}_r(M_L)</math> is invariant under blowing up/down (first Kirby move) and the third bracket ensures that <math>\text{RT}_r(M_L)</math> is invariant under handle sliding (second Kirby move).

==Properties==

The Witten–Reshetikhin–Turaev invariants for 3-manifolds satisfy the following properties:
# <math>\text{RT}_r(M\#N) = \text{RT}_r(M)\text{RT}_r(N),</math> where <math>M\# N</math> denotes the [[connected sum]] of <math>M</math> and <math>N</math>
# <math>\operatorname{RT}_r(-M)=\overline{\text{RT}_r(M)},</math> where <math>-M</math> is the manifold <math>M</math> with opposite orientation, and <math>\overline{\text{RT}_r(M)}</math> denotes the complex conjugate of <math>\operatorname{RT}_r(M)</math>
# <math>\operatorname{RT}_r(S^3)=1</math>
These three properties coincide with the properties satisfied by the 3-manifold invariants defined by Witten using Chern–Simons theory (under certain normalization)<ref name="Witten1989" />

==Open problems==
===Witten's asymptotic expansion conjecture===
Pick <math>t = e^{\frac{\pi i}{r}}</math>. Witten's asymptotic expansion conjecture suggests that for every 3-manifold <math>M</math>, the large <math>r</math>-th asymptotics of <math>\text{RT}_r(M)</math> is governed by the contributions of flat connections.<ref>{{cite journal |last1=Andersen |first1=Jørgen Ellegaard |first2=Søren Kold |last2=Hansen |title=Asymptotics of the quantum invariants for surgeries on the figure 8 knot |journal=Journal of Knot Theory and Its Ramifications |volume=15 |issue=4 |year=2006 |pages=479–548 |doi=10.1142/S0218216506004555 |arxiv=math/0506456 |s2cid=8713259 }}</ref>

'''Conjecture:''' 
There exists constants <math>d_j \in \mathbb{Q}</math> and <math>b_j \in \mathbb{C}</math> (depending on <math>M</math>) for <math>j = 0,1, \dots, n</math> and <math>a^l_j \in \mathbb{C}</math> for <math>j=0,1,\dots, n, l=1,2,\dots</math> such that the asymptotic expansion of <math>\text{RT}_r(M)</math> in the limit <math>r \to \infty</math> is given by
:<math> \operatorname{RT}_r(M) \sim \sum_{j=0}^n e^{2\pi i r q_j} r^{d_j} b_j \left( 1 + \sum_{\ell=1}^\infty a^\ell_j r^{-\ell} \right)</math>
where <math>q_0 = 0, q_1,\dots q_n</math> are the finitely many different values of the Chern–Simons functional on the space of flat <math>\text{SU}(2)</math>-connections on <math>M</math>.

===Volume conjecture for the Reshetikhin–Turaev invariant ===
The Witten's asymptotic expansion conjecture suggests that at <math>t =e^{{\pi i}/{r}} </math>, the RT-invariants grow polynomially in <math>r</math>. On the contrary, at <math>t=e^{{2\pi i}/{r}}</math> with odd <math>r</math>, in 2018 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 <math>r</math> and the growth rate gives the hyperbolic volume and Chern–Simons invariants for the 3-manifold.<ref>{{cite journal |last1=Chen |first1=Qingtao |first2=Tian |last2=Yang |title=Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants |journal=Quantum Topology |volume=9 |issue=3 |pages=419–460 |year=2018 |doi=10.4171/QT/111 |arxiv=1503.02547 |s2cid=18870964 }}</ref>

'''Conjecture:'''
Let <math>M</math> be a closed oriented hyperbolic 3-manifold. Then for a suitable choice of arguments, 
:<math>\lim_{r\to \infty} \frac{4\pi}{r} \log \left(\operatorname{RT}_r \big(M,e^{{2\pi i}/{r}}\big)\right) = \operatorname{Vol}(M) - i \operatorname{CS}(M) \mod \pi^2 i\mathbb{Z}</math>
where <math>r</math> is odd positive integer.

== References ==
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{{reflist}}

== External links ==
*https://ncatlab.org/nlab/show/Reshetikhin-Turaev+construction

{{DEFAULTSORT:Reshetikhin-Turaev invariant}}
[[Category:Quantum groups]]
[[Category:Quantum field theory]]