SU(N) quantum Racah coefficients & non-torus links

0.3 MB·

by  Zodinmawia

CQUEST 2011-0459 July 2011
1 SU(N) quantum Racah coefficients & non-torus links
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0
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g Zodinmawia1 and P. Ramadevi1,2
u
A 1 Department of Physics,
Indian Institute of Technology Bombay,
8
1
Mumbai 400 076, India

] and
h
t 2 Center for Quantum Spacetime,
-
p
e Sogang University, Seoul, S.Korea
h
[

4
v
8
1
9 Abstract
3
.
7 It is well-known that the SU(2) quantum Racah coefficients or the
0
1 Wigner 6j symbols have a closed form expression which enables the
1
evaluation of any knot or link polynomials in SU(2) Chern-Simons
:
v
field theory. Using isotopy equivalence of SU(N) Chern-Simons func-
i
X
tional integrals over three balls with one or more S2 boundaries with
r
a punctures,we obtain identities to be satisfied by the SU(N) quantum
Racah coefficients. This enables evaluation of the coefficients for a
class of SU(N) representations. Using these coefficients, we can com-
pute the polynomials for some non-torus knots and two-component
links. These results are useful for verifying conjectures in topological
string theory.
1 Introduction
Following the seminal work of Witten[1] on Chern-Simons theory as a theory
of knots and links, generalised invariants[2, 3] for any knot or link can be
directly obtained without going through the recursive procedure. For torus
links, which can bewrapped ona two-torusT2, using thetorus link operators
[4], explicit polynomial form of these invariants could be obtained. However
for non-torus links, the generalised invariants [2] in SU(N) Chern-Simons
theory involves SU(N) quantum Racah coefficients which are not known in
closed form as known for SU(2)[5, 16]. This prevents in writing the polyno-
mial form for the non-torus links.
For simple SU(N) representations placed on knots, whose Young Tableau
are , and , we had obtained some Racah coefficients from isotopy
equivalence of knots and links which was useful to obtain polynomial for
few non-torus knots[2, 3]. Going beyond these simple representations and
finding the quantum Racah coefficients appeared to be a formidable task.
In fact, determining these coefficients would help in verifying the topological
string conjectures for a general non-torus knot or link observable proposed
by Ooguri-Vafa[6, 7]. Using the few Racah coefficients data[2], Ooguri-Vafa
conjecturefor4 ,6 non-torusknotsasindicatedinFig.1wereverified[8]. For
1 1
verifying Ooguri-Vafa conjecture for the non-torus two-component links[7],
weneed toevaluatethenon-toruslinkwhose component knotscarrydifferent
representations.
These non-torus links invariants will also be useful to generalise some of
the results of recent papers[9, 10, 11, 12] where torus knots and links are
studied. So, it is very important to determine the SU(N) quantum Racah
coefficients.
Using the correspondence between Chern-Simons functional integral and
thecorrelatorconformalblocksstatesinthecorrespondingWess-Zuminocon-
formal field theory[1], we can derive identities to be obeyed by the SU(N)
quantum Racah coefficients. Using the identities and the the properties of
quantum dimensions for any N, we could determine the form of these coeffi-
1
cients for some class of SU(N) representations. These coefficients are needed
to obtain polynomial invariants of some non-torus knots and non-torus two-
component links.
The plan of the paper is as follows: In section 2, we briefly review Chern-
Simons functional integrals and properties of the Racah coefficients. In sec-
tion 3, we systematically study the equivalence of states and obtain identities
to be obeyed by the SU(N) quantum Racah coefficients. In section 4, we
tabulate the SU(N) quantum Racah coefficients. In section 5, we use these
coefficients to obtain the generalised polynomial invariant for a non-torus
knot and a non-torus link and verify Ooguri-Vafa conjecture. Finally, we
conclude with summary and discussions.
2 Chern-Simons Field Theory
Chern-Simons fields on S3 with U(1) SU(N) gauge group with levels k ,k
1
×
respectively is given by by the following action:
k k 2
S = 1 B dB + Tr A dA+ A A A , (2.1)
4π ∧ 4π ∧ 3 ∧ ∧
ZS3 ZS3 (cid:18) (cid:19)
where B is the U(1) gauge connection and A is the SU(N) matrix valued
gauge connection. The observables in this theory are Wilson loop operators:
r
W [ ] = Tr UA[ ]Tr UB[ ] , (2.2)
(R1,n1),(R2,n2),...(Rr,nr) L Rβ Kβ nβ Kβ
β=1
Y
where the holonomy of the gauge field A around a component knot ,
β
K
carrying a representation R , of a r-component link is denoted by UA[ ] =
β β
K
P[exp A] and n is the U(1) charge carried by the component knot .
Kβ β Kβ
The expectation value of these Wilson loopoperators arethe link invariants:
H
[ B][ A]eiSW [ ]
V{U(N)} [ ](q,λ) = W [ ] = D D (R1,n1),...(Rr,nr) L
(R1,n1),... L h (R1,n1),... L i [ B][ A]eiS
R
D D
= V{SU(N)} [ ]V{U(1} [ ] .R (2.3)
R1,R2,...Rr L n1,n2,...,nr L
2
(b(2+))(−1)(b(1−))(−1)(b(2−))2 (b(2+))−1(b1(−))−1(b3(−))−1(b(2+))−2 (b(2+))−1(b1(−))−3(b2(−))2
41 52 61
(b(2−))−1(b(1+))−1(b(3+))−1(b(2−))−1(b(3−))b(2+) b2(−)b(1+)b2(−)(b3(−))−1(b(2+))−2 (b(2+))−1(b1(−))−2(b3(−))−2(b(2+))−2
62 63 72
b(2−)(b(1+))2b(3+)(b(2−))3 b(2+)b1(−)b3(−)b(2+)b1(−)b3(−)b(2+) (b2(−))−1(b(1+))−1(b(3+))−1(b2(−))−2(b(3+))−1(b(2−))−1
74 75
73
(b(2−))−1(b(1+))−1(b(2−))−1(b(3−))2(b(2−))−2 b(1+)b1(−)(b2(−))−1(b(3+))−1(b2(−))−1b1(−)b(2+)
76 77
Figure 1: Plat representation of non-torus knots upto 7-crossings
3
R2
R2
R1 R2 R1
R1
(b(2−))−1b(4+)(b(3+))−1(b(2−))−1b(4+) (b(2+))−1(b1(−))−1(b3(−))−1(b(2+))−3 b(2+)b1(−)(b2(−))−2b3(−)b(2+)
51 62 63
R2 R2
R1
R1 R2 R1
(b(2−))−1(b(1+))−3(b(2−))−1b(3−)b(2+) b2(−)b(1+)b(3+)b2(−)(b3(−))−1(b(2+))−2 (b2(−))−1(b(1+))−1(b2(−))−3b(3−)b(2+)
71 72 73
Figure 2: Plat representation of non-torus links upto 7-crossings
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We make a specific choice of the U(1) charges and the coupling k so that the
1
above invariant is a polynomial in two variables q = exp 2πi , λ =qN[13, 14]:
k+N
(cid:16) (cid:17)
l
β
n = ; k = k+N , (2.4)
β 1
√N
where l is the total number of boxes in the Young Tableau representation R .
β β
The U(1) link invariant for the link with this substitution gives
V{l1U(,1..}., lr [L] = (−1)Pβlβpβ expk+iπN r lβ2Npβexpk+iπN lαlβNlkαβ ,
√N √N β=1 α=6 β
X X
   (2.5)
wherep is the framingnumber of the component knot and lk is the linking
β β αβ
K
number between the component knots and . In order to directly evaluate
α β
K K
SU(N) link invariants, we need to use the following two ingredients:
1. Therelation between SU(N)Chern-Simonsfunctionalintegral on thethree-
dimensional ball to the two-dimensional SU(N) Wess-Zumino conformal
k
field theory on the boundary of the three-ball[1].
2. Any knot or link can be drawn as a platting of braids[15].
In Fig. 1 and Fig. 2 we have drawn some non-torus knots and non-torus links
as a plat representation of braids. We have labelled them in the Thistlewaithe
notation and written their braid words. We have indicated the orientation and
(−)
labelled the representation R on the component knots in the link. Note that b
i i
( b(−) −1) in the braid word denotes right-handed crossing (left-handed crossing)
{ i }
between i-strand and i+1-th strand which are anti-parallelly oriented. Similarly
b(+) ( b(+) −1)denotesright-handedcrossing(left-handedcrossing)between j and
j { j }
j+1-th strand which are parallelly oriented. Theplat representation of these non-
torus knots and non-torus links except link 5 involves braids with four-strands.
4
Hencewecanviewtheseknotsandlinksexceptlink5 inS3 asgluingofthree-balls
4
with4-puncturedboundaryasshowninFig.3. Therearetwothree-ballsB andB
1 3
withoneS2 boundary. Athree-balldenotedasB inFig.3withtwoS2 boundaries
2
with a braid which can represent any of the braid words corresponding to the
B
non-torus knots and links in Figs. 1&2. The gluing of the three-balls are along
the oppositely oriented S2 boundaries. The functional integral over the ball B is
3
5
S3 oppositely oriented S2 boundary
1 2
1 2
R¯1 R2
R1 R¯1 R2
R2 B ≡ R1 B R¯2
R¯2 R1 R2 R¯1
R¯2 R1
B1 B2 B3
hΨ(1)| Bν(1),(2) |χ(2)i
0 0
Figure 3: Link in S3 viewed as gluing of three-balls with four-punctured S2
boundaries
given by a state χ(2) where the superscriptdenotes the label of the S2 boundary.
| 0 i
The representation R indicate that the lines are going into the S2 boundary of
i
the three-ball and the conjugate representation denotes the lines going out of the
S2 boundary. Thestate correspondingto a functionalintegral on a three-ball with
an oppositely oriented boundary is written in a dual space along with conjugating
all the representations as illustrated for the ball B . The expectation value of the
1
Wilson-loop operator gives the link invariant for a nontorus link L
VSU(N)[L] = Ψ(1) ν(1),(2) χ(2) . (2.6)
R1,R2 h 0 |B | 0 i
These invariants multiplied with the U(1) invariant (2.5) are polynomials in two
variables q = exp(2πi/(k + N)) and λ = qN. In order to see the polynomial
form, we write these states on a four-punctured boundary in a suitable basis of
four-point conformal block of the SU(N) Wess-Zumino conformal field theory.
k
There are two different four-point conformal block bases as shown in Fig. 4 where
t (R R ) (R R ) and s (R R¯ ) (R¯ R ). Using these bases, the
1 2 3 4 2 3 1 4
∈ ⊗ ∩ × ∈ ⊗ ∩ ×
states corresponding to three balls B ,B and B in Fig. 3, can be expanded as
1 2 3
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R1 R2 R¯3 R¯4 R2 R¯3
R1
R¯4
s
t
φ (R ,R .R¯ ,R¯ ) φˆ (R ,R .R¯ ,R¯ )
t 1 2 3 4 s 1 2 3 4
Figure 4: four-point conformal block bases states
[2, 3]
Ψ(1) = dim R dim R φ (R¯ ,R ,R ,R¯ )(1) (2.7)
h 0 | q 1 q 2h 0 1 1 2 2 |
= p ǫs(R1,R2) dimqshφˆs(R¯1,R1,R2,R¯2)(1)|
s∈RX1⊗R2 p
ν(1),(2) = φ (R ,R¯ ,R¯ ,R )(1) φ (R¯ ,R ,R ,R¯ )(2)
t 2 2 1 1 t 2 2 1 1
B B| ih |
= φˆ (R ,R¯ ,R¯ ,R )(1) φˆ (R¯ ,R ,R ,R¯ )(2) (2.8)
s 2 2 1 1 s 2 2 1 1
B| ih |
χ(2) = dim R dim R φ (R ,R¯ ,R¯ ,R )(2) (2.9)
| 0 i q 1 q 2| 0 2 2 1 1 i
= p ǫs(R1,R2) dimqs|φˆs(R2,R¯2,R¯1,R1)(2)i , (2.10)
s∈RX1⊗R2 p
where the subscript ‘0’ in the basis state in eqns.(2.7, 2.9) denotes the singlet
state. The states are written such that the invariant of a simple circle (also called
unknot) carrying representation R is normalised as dim R, which is the quantum
q
dimension of the representation R, defined as
[α.(ρ+Λ )]
R
dim R = , (2.11)
q
[α.ρ]
α>0
Y
where Λ denotes the highest weight of the representation R, α’s are the positive
R
rootsandρisequaltothesumofthefundamentalweightsoftheLiegroupSU(N).
The square bracket refers to quantum number defined as
qn2 q−n2
[n]= − . (2.12)
q21 q−21
−
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The symbol ǫ(R1,R2) = 1 which we will fix from equivalence of states in the next
s
±
section. To operate the braid word in eqn.(2.8), we need to find the eigenbasis
B
(±)
of the braiding generators b ’s.
i
The conformal block φ (R ,R ,R¯ ,R¯ ) is suitable for the braiding operator
t 1 2 3 4
| i
(±) (±)
b and b . Similarly braiding in the middle two strands involving the operator
1 3
b(±) requires the conformal block φˆ (R ,R ,R¯ ,R¯ ) . That is,
2 | s 1 2 3 4 i
b(±) φˆ (R ,R ,R¯ ,R¯ ) = λ(±)(R ,R¯ )φˆ (R ,R¯ ,R ,R¯ ) ,
2 | s 1 2 3 4 i s 2 3 | s 1 3 2 4 i
b(±) φ (R ,R ,R¯ ,R¯ ) = λ(±)(R ,R )φˆ (R ,R ,R¯ ,R¯ ) ,
1 | t 1 2 3 4 i t 1 2 | s 2 1 3 4 i
b(±) φˆ (R ,R ,R¯ ,R¯ ) = λ(±)(R¯ ,R¯ )φˆ (R ,R ,R¯ ,R¯ ) , (2.13)
3 | s 1 2 3 4 i t 3 4 | s 1 2 4 3 i
(±)
where braiding eigenvalues λ (R ,R ) in vertical framing are
t 1 2
±1
λt(±)(R1,R2)= ǫ(t;±R)1,R2 qCR1+CR22−CRt . (2.14)
(cid:18) (cid:19)
In this framing, framingnumberp for the componentknot is equal to writhew of
β
that component knot which is the difference between total number of left-handed
crossings and total number of right-handed crossing. For example, torus knots
(±)
4 ,5 in Fig. 1 have writhe w equal to 0 and 5 respectively. The symbol ǫ is
1 2 t;R1,R2
a sign which can be fixed by studying equivalence of states or equivalence of links
which we shall elaborate for a class of representations in the next section and C
R
denotes the quadratic casimir for the representation R given by
l2 1
C = κ , κ = [Nl+l+ (l2 2il )] , (2.15)
R R− 2N R 2 i − i
i
X
wherel isthenumberofboxesinthei-throwoftheYoungTableaurepresentation
i
R and l is the total number of boxes. The two bases in Fig. 4 are related by a
duality matrix a as follows:
R R
φ (R ,R ,R¯ ,R¯ ) = a 1 2 φˆ (R ,R ,R¯ ,R¯ ) . (2.16)
| t 1 2 3 4 i ts"R¯3 R¯4#| s 1 2 3 4 i
From the definition of t,s, we can see that the duality matrix a obeys the following
properties:
R R R¯ R R R¯ R¯ R¯ R R
1 2 3 2 1 4 3 4 3 4
a = a = a = a = a .
ts"R¯3 R¯4# st"R1 R¯4# s¯t¯"R¯3 R2# t¯s¯"R1 R2# ts"R¯1 R¯2#
(2.17)
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If one of the representation is singlet (denoted by 0), we see that the matrix
elements are
R 0 0 R
1 2
ats"R¯3 R¯4# = δtR1δsR¯3 , ats"R¯3 R¯4# = δtR2δsR4 ,
R R R R
1 2 1 2
ats" 0 R¯4# = δtR4δsR2 , ats"R¯3 0 # = δtR3δsR¯1 . (2.18)
From the procedure presented in this section, we can write the U(N) invariants of
non-torus knots and links as a product of U(1) invariant times SU(N) invariant
(2.6). For example, non-torus knot 5 with framing p = 5, the invariant will be
2
VR{U(N)}[52;p] = (−1)5lq52lN2 ǫRs,R dimqs ǫRs′,R dimqs′(λ(s+)(R,R))−1
s,t,s
X′ p p
R¯ R R R¯
a (λ(−)(R¯,R))−2a (λ(+)(R,R))−2,(2.19)
ts"R R¯# t ts′"R¯ R# s′
where l is the total number of boxes in the Young Tableau representing R and we
have indicated the framing number of the knot. We could add additional framing
p by multiplying these invariants by a framing factor as follows:
1
VR{U(N)}[K;(p+p1)] = (−1)lp1qp21Nl2(λ(0−)(R,R¯))−p1VR{U(N)}[K;p]
= ( 1)lp1qp1κRV(U(N)[K;p] . (2.20)
− R
So, to obtain 5 invariant with zero framing, we have to take p = 5. Similarly
2 1
−
the invariant for link 6 with linking number lk = 3 with framing p ,p on the
2 1 2
component knots will be
2
V({RU1(,RN2))}[62;p1,p2] = {(−1)lRipiqpiκRi}q3lRN1lR2 ǫRs1,R2 dimqs (2.21)
i=1 s,t,s
Y X′ p
ǫR¯1,R¯2 dim s′(λ(+)(R ,R ))−3a R2 R¯1
s′ q s 1 2 ts" R¯2 R1 #
p
R¯ R
(λ(−)(R¯ ,R ))−2a 1 2 (λ(+)(R¯ ,R¯ ))−1 ,
t 1 2 ts′" R1 R¯2 # s′ 1 2
where l is the total number of boxes in the Young diagram representation R
Ri i
placed on the component knots in link 6 . However to see the polynomial form of
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