научная статья по теме MATRIX INTEGRAL EXPANSION OF COLOURED JONES POLYNOMIALS FOR FIGURE-EIGHT KNOT Физика

Текст научной статьи на тему «MATRIX INTEGRAL EXPANSION OF COLOURED JONES POLYNOMIALS FOR FIGURE-EIGHT KNOT»

Pis'ma v ZhETF, vol. 101, iss. 1, pp. 54-58 © 2015 January 10

Matrix integral expansion of coloured Jones polynomials for figure-eight

knot

A. Alexandrov+*, D. Melnikov*x + Mathematics Institute, University of Freiburg, D-79104 Freiburg, Germany * Institute for Theoretical and Experimental Physics, 117218 Moscow, Russia x International Institute of Physics, UFRN, Capim Macio, 59078-400 Natal-RN, Brazil Submitted 20 November 2014

In this note we examine a possible extension of the matrix integral representation of knot invariants beyond the class of torus knots. In particular, we study a representation of the SU{2) quantum Racah coefficients by double matrix integrals. We find that the Racah coefficients are mapped to expansion coefficients in some basis of double integrals. The transformed coefficients have a number of interesting algebraic properties.

DOI: 10.7868/S0370274X15010117

Classification of knots is a central problem of knot theory. One way to solve it is to derive most general formulae for the topological invariants, which would distinguish any pair of knots. For torus knots a general formula, computing the corresponding HOMFLY polynomials, was derived by Rosso and Jones [1]. Since the work of Rosso and Jones a number of new general formulae was derived (e.g. [2-8] and references therein), which yield (coloured) Jones or HOMFLY polynomials of certain knot series.

One possible way to extend the known results would be to construct a matrix integral representation of the knot invariants. The relevant matrix integral for the Rosso-Jones formula was derived in [9, 10]. Per se, the restriction to the Rosso-Jones formula seems to be solely a technical issue and the authors are unaware of any conceptual obstacles to generalize this result to a more general class of knots or links. Indeed, the existence of the matrix model formulation is advocated by the relation of the knot invariants to the generic Hurwitz r-functions, as observed in [11].

In a recent work [12] a step towards a generalization of the matrix model beyond the torus knots was taken. For a series of twisted knots it was demonstrated that it is natural to consider matrix integrals with the integration measure made of the Laplace evolution of the corresponding Jones polynomial. Unfortunately, this approach requires some unknown ingredient, and the naive matrix integrals give a discrepancy in the perturbative expansion of HOMFLY polynomials starting from gzs terms.

-'-'e-mail: dmitry@iip.ufrn.br

The topological quantum field theory (TQFT) approach [13] allows one to write generic formulae for knots invariants. However, in this case the final result depends on the quantum extensions of the Racah coefficients (Wigner 6j-symbols). These coefficients are known in the case of SU(2) [6, 7], and for symmetric and antisymmetric representations of SU(N) [7, 14], but remain largely unknown beyond those results (see [5,15-18] for some exceptions), especially for the cases of non-trivial multiplicity.

Based on the TQFT approach, in this work we study a matrix integral representation of the quantum Racah coefficients, which naturally arises in the analysis of the invariants of the figure-eight (4i) knot. The main idea behind our approach is to consider these invariants as deformations of those of the product of a Hopf link and its mirror image. This will lead to an expansion of the invariants in terms of a basis of double matrix integrals. We found that the coefficients of the expansion can be thought as of a transform of the original Racah coefficients. To explain this idea we will rely on an explicit realization of Witten's proposal developed in [19] and subsequent papers, e.g. see [20, 15, 8]. A nice refined review of the method was recently given in [18].

In the TQFT approach the invariant (HOMFLY polynomial of variables q and A = qN) of the Hopf link 2\ is given by the formula

HRuR2{2\) <x (Vs| dim, i?s &r2 I > oc

RseRi0R2

oc ]T dim qRsq2C^\ (1)

RseRi0R2

where i?i and Ro are representations associated to the components of the link, dim9 R and C2(R) are the quantum dimension and quadratic Casimir of the representation R. Fig. 1 illustrates the above formula in the TQFT approach.

Fig. 1. The topological invariant of the Hopf link 2\ can be obtained by sandwiching the braid group element b±2 between the initial and final state

The formula for the Hopf link does not involve the Racah coefficients. This is also true for any torus knot of the series (2m + 1,2) or torus links (2m, 2) with m > 0 in the four-strand case. The simplest non-trivial example is the figure-eight knot 4i (Fig. 2). The invariant of this knot can be computed clS du double sum

Hr{4i)= dimq(Ri)q~2ci x

ri,rjeror

R R Rj

R R R<;

dim q(Rj)q

(2)

where the expression in curly brackets denotes the Racah matrix.

For example, in the fundamental representation of SU(2) (2) gives the following Jones polynomial

J[ i](4i)

1 + q2

(3)

where we used the following expression for the Racah coefficients, cf. [15],

R R Ri R R Rn

i

dim9 R

1

(1 - [Nf

(4)

Here, we use the following convention for the quantum version of a number

1 ~ <1

q-q

-l

(5)

Eq. (2) can be seen as a product of a Hopf link and its mirror image twisted by the Racah matrix, inserted in between. Pictorially we would like to illustrate the deformation of the product of two Hopf links as an operation of cutting and gluing together the links as shown in Fig. 2. The operation is valid if the representations of the Hopf links coincide. An important difference between formulae (1) and (2) is in the multiplicity of summation, single in the first example and double in the second. Now we would like to construct a similar generalization for the matrix integral.

For the torus link (m, n) coloured with representations i?i, ..., Rl of U(N) the HOMFLY polynomial is given by the following matrix (eigenvalue) integral [9, 10] representation:

H,

Ri ,...,RL

l

n

z„

du J^J (

-ujL2/4gs

n

X

J^J 4 sinh

i<j

where

Ui — Sj \ , ( Ui — u - si

^/t^Wln^c*).«)

gs = mngs, q = e9',

(7)

Sr(€u) is the character of U(N), which is given by the Schur polynomial and ZmjU is the normalization factor given by the same integral with Sr —> 1. The integral is taken over the eigenvalues in, i = 1,..., N of the U(N) matrices. The U(l) part can be factorized out and we will be interested only in the SU(N) part, what amounts to an appropriate renormalization of characters. In the following we use the "correlator" notation

n

(0(x))m,„,L

n

x J^4sinh

i<j

dx J^J (

sinh

and

2m/L J V 2n/L

{0(x))m^L

(l)m,n,I

0(x) (8)

(9)

56

A. Alexandrov, D. Melnikov

Fig. 2. Construction of the figure-eight in the TQFT approach (left). Figure-eight knot obtained by a connect-sum of two Hopf links (right)

for the normalized version. In particular, for the Hopf link coloured with the spin k/2 representation of S'U(2) one finds the following Jones polynomial

Jfc,fc( 2?) = ((Sk(ex)2))2,2,2 =

There is a simple, mirror-symmetric (q —> 1 /q) basis for the coloured Jones polynomials of the knot. This is given by double matrix integrals of the form

«Sfc(e*)Sfc(e«)» =

1

r ~ 1

/CO

dxeC

-OO

e(fc+l)x/2 _ e-(fc+ l)x/2

k +2 k

( 1 )2,2,2 { 1 }2, — 2,2 J dxdy exp i - Sk(ex) Sfc(e") x

E

(10)

j=o

X

Going from the Hopf link to the figure-eight knot we propose the following generalization of the matrix integral formula, where the product of two matrix integrals is twisted by some (non-local) kernel insertion:

1

( 1 }2,2,2 { 1 ) 2,-2,2 n

x 4 sinh

i<j

dx dy e

sinh

-x2/4gi-y2/4g2

x SR(e*)SR(ex)GR(x, y) SR(e«)SR(e«),

(11)

where function GR(x,y) glues and twists two braids, representing 2\ links, in such a way that we get the 4i knot. Here we introduce two coupling constants g\ and g2 to distinguish the link and its mirror. After integration one sets e91 = q = e~92. If GR(x, y) = 1 we recover the result for the product of two links given by (10).

. Xl — xn \ ( y\ — yn 4 sinh -1 sinh

(12)

2 J \ 2 Here the "correlator" of the two characters is normalized

by two normalization factors of the 2\ Hopf link and of its mirror image. The integral yields

((Sk{ex)Sk{ev))) = [k + l]2 = q2k + 2 q2k-2 + ...+ kq2 + (A- + 1) + kq-2 + . . . + 2q2-2k + q-2k.

(13)

This basis can be related to the standard monomial basis using the Jones polynomial of 4i in the representation p/2 as an example. We have2'

n

dimjp] Jp(4i) = o0 + J2a,i (q2i + '

¿=1

n = p{p + 2). (14)

Then the correlator, which reproduces this result is

2'With a slight abuse of notation we label [p] the spin-p/2 representation of SU{2). Otherwise [•] stand for numbers.

0 200 400 600 800 0 600 1000 1500

Fig. 3. Value of the coefficients bk as a function of k in representations [10], [20], [30] and [40]

J2bkSk(ex)Sk(ey:

k=0

where bk = (A_1)fciOj, (15)

and A is a non-degenerate (n + 1) x (n + 1) matrix / n + 1 n n — 1 ... 1 \

A =

0 0

V o

n ii — 1 0 ii - 1

0

0

1 1

1 /

(16)

It is interesting to note the following property of the correlators, which immediately follows from the comparison of (13) and (10):

<<S'fc(e*)2S'fc(^)2)) = ((Sk{k+2)(e*)Sk{k+2)(ey))). (17)

The lhs of this expression is the invariant of the product of a link and its mirror image. The rhs is the highest order term in expansion (15) of the Jones polynomial (14). The above property shows us how the invariant of the knot (14) can be understood clS cl deformation of the formula for the product of two links. One can also notice that the number of terms (p + l)2 in the sum (14) coincides with the number of terms in the sum (2).

Thus, the Jones polynomials can be found as an expansion

p(p+2)

dim9[p]J[p](4i)= ]T bk(p)[k+l]2 (18)

fc=0

with some integer coefficients bk, which depend on the representation [p]:

bp2+2p — 1, b.p2 + 2p-l — —1,

°P2+2p-2

"I,

(19)

Since the general formula for coloured Jones polynomials of the figure-eight can be found in [3], this allows us to find the coefficients bk in

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