Reshetikhin–Turaev invariant explained

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]

Overview

To obtain an RT-invariant, one must first have a

\Bbbk

-linear ribbon category at hand. Each

\Bbbk

-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

\Bbbk

. This element of

\Bbbk

is the RT-invariant associated to

. Given any closed oriented 3-manifold

, there exists a framed link

in the 3-sphere

S³

so that

is homeomorphic to the manifold

M_L

obtained by surgering

S³

along

. Two such manifolds

M_L

and

M
	L^\prime

are homeomorphic if and only if

and

L^\prime

are related by a sequence of Kirby moves. Reshetikhin and Turaev 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

\Bbbk

(one can take, for example, any quantum group over

). Consider the category

bf{Rep}^f.d.(A)

, of finite dimensional representations of

. There is a diagrammatic calculus in which morphisms in

bf{Rep}^f.d.(A)

are represented by framed tangle diagrams with each connected component decorated by a finite dimensional representation of

.^[3] That is,

bf{Rep}^f.d.(A)

is a

\Bbbk

-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

A=U_q(ak{sl}_2(C))

over the field

C(q)

, 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

S³

with

components. For each

r\inN

, let

	3,
RT
	r(S

denote the RT-invariant obtained by decorating each component of

by the unique

N+1

-dimensional representation of

. Then

	3,L)
\operatorname{RT}
	r(S

=\langlee_n,e_n,...,e_n\rangle_L\inC(q)

where the

-tuple,

\langlee_n,e_n,...,e_n\rangle_L

denotes the Kauffman polynomial of the link

, where each of the

components is cabled by the Jones–Wenzl idempotent

e_n

, 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 of unity with odd

. Assume that

M_L

is obtained by doing Dehn surgery on a framed link

. Then the RT-invariant for the 3-manifold

is defined to be

\operatorname{RT}_r(M_L)=\langle\omega_r

	b_-
\rangle
	O⁺

\langle\omega_r

	b₊
\rangle
	O^-

\langle\omega_r,\omega_r,...,\omega_r\rangle_L(t)\inC,

where

\omega_r=

	r-2
\sum
	n=0

\langlee_n\rangle_Oe_n

is the Kirby coloring,

O^\pm

are the unknot with

\pm1

framing, and

b_\pm

are the numbers of positive and negative eigenvalues for the linking matrix of

respectively. Roughly speaking, the first and second bracket ensure that

RT_r(M_L)

is invariant under blowing up/down (first Kirby move) and the third bracket ensures that

RT_r(M_L)

is invariant under handle sliding (second Kirby move).

Properties

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

RT_r(M\#N)=RT_r(M)RT_r(N),

where

M\#N

denotes the connected sum of

and

\operatorname{RT}_{r(-M)=\overline{RT}_r(M)},

where

-M

is the manifold

with opposite orientation, and

\overline{RT_r(M)}

denotes the complex conjugate of

\operatorname{RT}_r(M)

	3)=1
\operatorname{RT}
	r(S

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

Open problems

Witten's asymptotic expansion conjecture

Pick

	\pii
	r

. Witten's asymptotic expansion conjecture suggests that for every 3-manifold

, the large

-th asymptotics of

RT_r(M)

is governed by the contributions of flat connections.^[4]

Conjecture: There exists constants

d_j\inQ

and

b_j\inC

(depending on

) for

j=0,1,...,n

and

	l
a
	j

\inC

for

j=0,1,...,n,l=1,2,...

such that the asymptotic expansion of

RT_r(M)

in the limit

r\toinfty

is given by

\operatorname{RT}_r(M)\sim

	n
\sum
	j=0

	2\piirq_j
e

	d_j
r

b_j\left(1+

	infty
\sum
	\ell=1

	\ell
a
	j

r^-\ell\right)

where

q₀=0,q_1,...q_n

are the finitely many different values of the Chern–Simons functional on the space of flat

SU(2)

-connections on

Volume conjecture for the Reshetikhin–Turaev invariant

The Witten's asymptotic expansion conjecture suggests that at

t=e^{\pi/{r}}

, the RT-invariants grow polynomially in

. On the contrary, at

t=e^{2\pi/{r}}

with odd

, 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

and the growth rate gives the hyperbolic volume and Chern–Simons invariants for the 3-manifold.^[5]

Conjecture:Let

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

\lim_r\to

	4\pi
	r

log\left(\operatorname{RT}_r(M,e^{2\pi/{r}})\right)=\operatorname{Vol}(M)-i\operatorname{CS}(M)\mod\pi²iZ

where

is odd positive integer.

External links

https://ncatlab.org/nlab/show/Reshetikhin-Turaev+construction

Notes and References

Reshetikhin . Nicolai . Vladimir G. . Turaev . Invariants of 3-manifolds via link polynomials and quantum groups . Inventiones Mathematicae . 103 . 1 . 1991 . 547–597 . 10.1007/BF01239527 . 1991InMat.103..547R . 123376541 .
Witten . Edward . Quantum field theory and the Jones polynomial . Communications in Mathematical Physics . 121 . 3 . 1989 . 351–399 . 10.1007/BF01217730 . 1989CMaPh.121..351W . 14951363 .
Book: Turaev, Vladimir G. . Quantum invariants of knots and 3-manifolds . De Gruyter Studies in Mathematics . 18 . Berlin . Walter de Gruyter . 2016 . 978-3-11-044266-3 .
Andersen . Jørgen Ellegaard . Søren Kold . Hansen . Asymptotics of the quantum invariants for surgeries on the figure 8 knot . Journal of Knot Theory and Its Ramifications . 15 . 4 . 2006 . 479–548 . 10.1142/S0218216506004555 . math/0506456 . 8713259 .
Chen . Qingtao . Tian . Yang . Volume conjectures for the Reshetikhin–Turaev and the Turaev–Viro invariants . Quantum Topology . 9 . 3 . 419–460 . 2018 . 10.4171/QT/111 . 1503.02547 . 18870964 .