A Contribution of the Trivial Connection to Jones Polynomial and Witten's Invariant of 3d Manifolds I

29 Pages·0.26 MB·English

by  Lev Rozansky

UMTG–172
UTTG–30–93
hep-th/9401061
4
9 A Contribution of the Trivial Connection to Jones Polynomial
9
1 and Witten’s Invariant of 3d Manifolds. I

n
a
J L. Rozansky1

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1 Physics Department, University of Miami

1 P. O. Box 248046, Coral Gables, FL 33124, U.S.A.
v
1
6
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1
0 Abstract
4
9
/
h
We use a path integral formulation of the Chern-Simons quantum field theory
t
-
p
in order to give a simple “semi-rigorous” proof of a recently conjectured limita-
e
h
: tion on the 1/K expansion of the Jones polynomial of a knot and its relation
v
i
X to the Alexander polynomial. A combination of this limitation with the finite
r
a version of the Poisson resummation allows us to derive a surgery formula for the
contribution of the trivial connection to Witten’s invariant of rational homology
spheres. The 2-loop part of this formula coincides with Walker’s surgery formula
forCasson-Walker invariant. This proves aconjecture thatCasson-Walker invari-
ant is proportionalto the 2-loopcorrection to the trivial connection contribution.
A contribution of the trivial connection to Witten’s invariant of a manifold with
nontrivial rational homology is calculated for the case of Seifert manifolds.
1
Work supported in part by the National Science Foundation under Grants No. PHY-92 09978 and
9009850and by the R. A. Welch Foundation.
1 Introduction
In his paper [1], Witten defined a topological invariant of a 3d manifold M with an n-
component link inside it as a partition function of a quantum Chern-Simons theory. Let
L
us attach representations Vα , 1 i n of a simple Lie group G to the components of (in
i ≤ ≤ L
our notations α are the highest weights shifted by ρ = 1 λ , ∆ is a set of positive
i 2 λi∈∆+ i +
roots of G). Then Witten’s invariant is equal to the path iPntegral over all gauge equivalence
classes of G connection on M:
i n
Zα α (M, ;k) = [ A ]exp S Trα Pexp A dxµ , (1.1)
1,..., n L D µ h¯ CS i µ
Z (cid:18) (cid:19)i=1 (cid:18)ILi (cid:19)
Y
here A is a connection, S is its Chern-Simons action
µ CS
1 2
S = Trǫµνρ dx(A ∂ A + A A A ), (1.2)
CS µ ν ρ µ ν ρ
2 3
ZM
Tr is a trace in the fundamental representation (so that Trλ2 = 2 for long roots of G), h¯ is
i
a Planck’s constant:
2π
h¯ = , k ZZ. (1.3)
k ∈
Trα Pexp A dxµ arethetraces ofholonomies alongthelinkcomponents L taken inthe
i Li µ i
representat(cid:16)iHons Vα .(cid:17)Witten showed that for a link in S3 his invariant was proportional to
i
the Jones polynomial of that link. In what follows we will refer to eq. (1.1) as the definition
of the Jones polynomial and its normalization.
Witten derived a surgery algorithmfor anexact calculation of thepath integral (1.1). We
review it briefly in order to set our notations. Consider a manifold M with a knot inside
K
it. Let us choose a basis of cycles on the boundary of its tubular neighborhood Tub( ).
K
C is a cycle contractible through the tubular neighborhood (i.e. C is the meridian of ).
1 1
K
C is a cycle which has a unit intersection number with C (C is defined only modulo C ).
2 1 2 1
Cut out the tubular neighborhood Tub( ) and glue it back in such a way that the cycles
K
pC +qC and rC +sC on the boundary of the complement of are identified with the
1 2 1 2
K
cycles C and C on the boundary of Tub( ). As a result of this surgery, a new manifold
1 2
K
M′ is constructed.
1
The integer numbers p,q,r,s form a unimodular matrix
p r
U(p,q) = SL(2,ZZ), ps qr = 1. (1.4)
 
∈ −
q s
 
 
The group SL(2,ZZ) has a unitary representation in the space of affine characters of G which
is in fact a Hilbert space of the Chern-Simons theory corresponding to T2 = ∂Tub( ). The
K
basis vectors of this space α,1 (α ∆ = Λw/(W KΛR) walls, K = k +c , c is a
G V V
| i ∈ × \
dual Coxeter number of G, c = N for SU(N)) are the eigenstates of the holonomy operator
V
along the cycle C :
1
2πi
Pexp Aˆµdxµ α,1 = exp α α,1 , (1.5)
| i K | i
(cid:18)IC1 (cid:19) (cid:18) (cid:19)
here Aˆµ is an operator corresponding to the classical field Aµ. The matrix elements of U(p,q)
represented in this basis are (for a simply laced group)
1
[isign(q)]|∆+| iπ VolΛw 2
U˜(p,q) = exp dimGΦ(U(p,q)) (1.6)
αβ (K|q|)rankG/2 (cid:20)−12 (cid:21) VolΛR!
iπ
( 1)|w|exp pα2 2α (Kn+w(β))+s(Kn+w(β))2 ,
× − Kq − ·
n∈ΛXR/qΛRwX∈W h i
here ∆ is a number of positive roots in G, W is the Weyl group and Φ(U(p,q)) is the
+
| |
Rademacher function defined as follows:
p r p+s
Φ = 12s(s,q), (1.7)
 
q −
q s
 
 
s(s,q) is a Dedekind sum:
1 n−1 πj πmj
s(m,n) = cot cot . (1.8)
4n n n
j=1 (cid:18) (cid:19) (cid:18) (cid:19)
X
The formula (1.6) was derived by L. Jeffrey [3] for G = SU(2):
sign(q) q−1 iπ
U˜(p,q) = i e−i4πΦ(U(p,q)) µexp pα2 2α(2Kn+µβ)+s(2Kn+µβ)2 ,
αβ 2K q 2Kq −
| | µX=±1nX=0 h i
q ∆ : 1 α,β K 1. (1.9)
SU(2)
≤ ≤ −
2
According to Witten [1], the invariant of the manifold M′ constructed by a U(p,q) surgery
on a knot in a manifold M can be expressed through the Jones polynomial of that knot
K
and the representation (1.8) of the surgery matrix:
Z(M′;k) = eiφfr Zα(M, ;k)U˜α(p,ρq) (1.10)
K
α
X∈∆G
(recall that ρ is a shifted highest weight of the trivial representation). The phase φ is a
fr
framing correction. If both invariants are reduced to canonical framing, then
π K c p
φ = − V dimG Φ(U(p,q)) 3sign +ν , (1.11)
fr
12 K " − q !#
here ν is a self-linking number of defined as a linking number between C and .
2
K K
For a more general case when a surgery is performed on a link in M Witten concluded
L
that
Z(M′;k) = eiφfr Zα1,...,αn(M,L;k)U˜α(p11,ρq1)...U˜α(pnn,ρqn). (1.12)
α α
1,...Xn∈∆G
Reshetikhin and Turaev showed in [2] that eq. (1.12) is invariant under Kirby moves. There-
fore they proved that Z(M;k) is a topological invariant of the manifold without invoking the
path integral representation (1.1) which still lacks mathematical rigor. They also established
a general set of conditions on the components of the r.h.s. of eq. (1.12) which guarantee its
topological invariance.
The disadvantage of eqs. (1.10) and (1.12) is that they do not make the relation between
Witten’s invariant and classical topological invariants of 3d manifolds quite transparent
(Alexander polynomial was the only quantum invariant which had a clear topological nature
since it was originally constructed from the fundamental group of the knot complement).
A possible way to deal with this problem is to consider a large k asymptotics of the path
integral (1.1) by applying a stationary phase approximation. The stationary phase points
are flat connections. Therefore the invariant is presented as a sum over connected pieces
c
M
of the moduli space of flat connections on M:
M
Zα1,...,αn(M,L;k) = Zα(M1,c..).,αn(M,L;k),
XMc
i ∞
Zα(Mc) α (M, ;k) = exp S(c) + S(c)h¯n , (1.13)
1,..., n L h¯ CS n !
n=1
X
3
here S is a Chern-Simons action of flat connections of and S(c) are the quantum n-
CS Mc n
loop corrections to the contribution of . The 1-loop correction is a determinant of the
c
M
quadratic formdescribing the small fluctuations of S (A ) around a stationary phase point.
CS µ
Its major features were determined by Witten [1], Freed and Gompf [4], and Jeffrey [3] (some
further details were added in [5]):
(2πh¯)dimHc0−dimHc1 i iπ
eiS1(c) = 2 exp cVSCS Nph (1.14)
Vol(H ) 2π − 4
c (cid:18) (cid:19)
n
τ Trα Pexp A dxµ ,
×ZMc "q| R|iY=1 i (cid:18)ILi µ (cid:19)#
hereH isanisotropygroupof (i.e. asubgroupofGwhichcommuteswiththeholonomies
c c
M
of connections A(c) of ), N is expressed [4] as
µ Mc ph
N = 2I +dimH0 +dimH1 +(1+b1 )dimG, (1.15)
ph c c c M
I is a spectral flow of the operator L = ⋆D + D⋆ acting on 1- and 3-forms, D being a
c −
covariant derivative, H0 and H1 are cohomologies of D, and b1 is the first Betti number of
c c M
M. τ is a Reidemeister-Ray-Singer torsion. It was observed in [3] that √τ defines a ratio
R R
of volume forms on and H .
c c
M
The higher loop corrections S(c) are calculated by Feynman rules. They are expressed
n
as multiple integrals of the products of propagators taken over the manifold M and the
link . Such representation might make the nature of invariants S(c) more transparent.
L n
Bar-Natan [6] and Kontsevich [7] studied the Feynman diagrams related to the link. These
diagrams produce Vassiliev invariants. In particular, Bar-Natan observed that the 2-loop
correction to the SU(2) invariant of the knot in S3 is proportional to the second derivative
of its Alexander polynomial.
In their recent paper [8] Melvin and Morton conjectured2 a rather strict limitation on the
possible powers of α in the K−1 expansion of the SU(2) Jones polynomial Z (S3, ;k) as
α
K
well asa relationbetween the dominant part of this expansion andthe Alexander polynomial
which generalizes the result of [6].
2
This conjecture was proven recently by D. Bar-Natan and S. Garoufalidis [9] at the level of weight
systems.
4
The properties of Feynman diagrams related to the manifold were studied in early papers
[10],[11] and then by Axelrod and Singer [12] and Kontsevich [13]. A convergence of those
diagrams was proven, however no multiloop diagrams were explicitly calculated. An “exper-
imental” approach to their study was initiated in [4] and [3]. Freed and Gompf checked the
1-loop formula (1.14) by comparing it numerically to the surgery formula (1.12) applied to
some lens spaces and Seifert homology spheres. L. Jeffrey transformed the surgery formula
for lens spaces and some mapping tori into the asymptotic form (1.13) thus obtaining all the
loop corrections for those manifolds. This program was further extended to Seifert manifolds
in [5]. It was observed there among other things that the 2-loop correction to the contribu-
tion of the trivial connection was proportional to Casson-Walker invariant as calculated by
C. Lescop [14].
In this paper we study the trivial connection contribution to Witten’s invariant of a
knot, a link and a manifold. In Section 2 we prove the relation between the Jones and
Alexander polynomials of a knot (Proposition 2.1) conjectured in [8] by relating the former
to the Reidemeister-Ray-Singer torsion of the knot complement. We also generalize this
result to the case of an arbitrary rational homology sphere (RHS ). In Section 3 we derive
a knot surgery formula for the trivial connection contribution to Witten’s invariant of a
RHS (Proposition 3.1). We show that at the 2-loop level this formula coincides with
Walker’s formula [15] for Casson-Walker invariant. This proves the relation between the 2-
loop correction to the contribution of the trivial connection and the Casson-Walker invariant
(Proposition 3.2) conjectured in [5]. In Section 4 we try to go beyond RHS by considering
a Seifert manifold with nontrivial rational homology. We derive a formula for the trivial
connectioncontributiontoitsWitten’sinvariant(Proposition4.3)andcompareitsproperties
tothepartitionfunctionofa2dgaugetheorystudied byWitten [16]. The results ofSection2
are illustrated in Appendix, where a large k asymptotics of the Jones polynomial of a torus
knot is calculated. The contributions of reducible and irreducible connections in the knot
complement are identified. Similarly to the results of [5], the contribution of the irreducible
connections appears to be 2-loop exact.
5
2 Jones Polynomial and Reidemeister-Ray-Singer Tor-
sion
We are going to study a Jones polynomial of a knot in a rational homology sphere M (i.e.
K
b1 = 0). We start with the case of M = S3. Then the SU(2) Jones polynomial (in Witten’s
M
normalization (1.1)) can be expanded in K−1:
Z (S3, ;k) = C αmK−n. (2.1)
α m,n
K
m,n≥0
X
Melvin and Morton [8] suggested3 the following
Proposition 2.1 If the knot is canonically framed (i.e. the linking number ν between the
K
cycle C which determines the framing and is zero), then
2
K
C = 0 if m > n. (2.2)
m,n
Moreover,
2 sin(πa)
C an = , 0 a 1, (2.3)
n,n sK ∆ (S3, ;exp(2πia)) ≤ ≤
nX≥0 A K
here ∆ (S3, ;exp(2πia)) is the Alexander polynomial of normalized in such a way that
A
K K
∆(S3,unknot;exp(2πia)) = 1, ∆ (M, ;exp(2πia)) is real.
A
K
It was established by Milnor [17] and Turaev [18] that in this normalization ∆ is related to
A
the Reidemeister torsion of the knot complement:
2sin(πa)
∆ (M, ;exp(2πia)) = (2.4)
A
K τ (M Tub( );exp(2πia))
R
\ K
Some simple quantum field theory arguments were used in [19] to show that the Alexander
polynomial was related by eq. (2.4) to the Ray-Singer torsion of the knot complement. Here
we will apply the same arguments to the Jones polynomial Z (S3, ;k).
α
K
3I am thankful to D. Bar-Natan and S. Garoufalidis for drawing my attention to the paper [8].
6
Consider the values of α of order K. We introduce a new variable
α
a = , 0 a 1. (2.5)
K ≤ ≤
Let us split the path integral (1.1) for a knot into an integral over the connection A
µ
K
inside the tubular neighborhood Tub( ) and inside its complement S3 Tub( ) with certain
K \ K
boundary conditions on the boundary T2 = ∂Tub( ), as well as an integral over these
K
boundary conditions. According to [20], one possible set of boundary conditions requires
that the gauge fields A on T2 should belong to the Cartan subalgebra, the curvature F
1,2 1,2
should be zero and the integral I = A dxµ should be fixed. In fact, it was established
1 C1 µ
in [20] that in accordance with eq. (1H.5), the path integral over connections on Tub( ) is
K
proportional to δ(I 2πia). Therefore the Jones polynomial Z (S3, ;k) is equal to the
1 α
− K
path integral over connections on S3 Tub( )
\ K
i
Z (S3, ;k) = [ A ]exp S′ (2.6)
α K Z[S3\Tub(K)] D µ (cid:18)h¯ CS(cid:19)
taken with the boundary condition
Pexp A dxµ = exp(2πia). (2.7)
µ
(cid:18)IC1 (cid:19)
The Chern-Simons action is modified [20] by the boundary term
1
S′ = S + Tr A A d2x, (2.8)
CS CS 2 ZT2 1 2
which is necessary for the choice (2.7) of boundary conditions.
Letuscalculatethepathintegral(2.6)bythestationaryphaseapproximationmethod(1.13).
First of all, we look for stationary phase points, i.e. flat connections satisfying the boundary
condition (2.7). There is only one such connection for a < a (a > 0 being a critical value
0 0
depending on ). This connection is reducible: all the holonomies belong to the maximal
K
torus U(1) SU(2). For this connection S = 0. Since the linking number ν of C and
CS 2
⊂
is zero, the homology class of C in S3 Tub( ) is trivial. Therefore A = 0 and the
2 2
K \ K
boundary term in eq. (2.8) is also zero. Thus the whole classical Chern-Simons action S′
CS
is zero.
7
We will estimate the 1-loop correction (1.14) up to a phase factor exp iπN . The
− 4 ph
(cid:16) (cid:17)
flat U(1) connection on S3 Tub( ) satisfying eq. (2.7) has no moduli, so dimH1 = 0.
\ K c
The isotropy group is H = U(1), so VolH = 2√2π (recall that the radius of U(1) is
c c
√2), while dimH0 = 1. The determinants in the SU(2) Ray-Singer torsion τ split into
c R
three factors for three Lie algebra components of A which have the definite U(1) charge.
µ
The chargeless Cartan subalgebra (i.e. diagonal) component of A contributes 1, while
µ
each of the two off-diagonal components contribute the square root of the U(1) torsion
τ (S3 Tub( );exp(2πia)). As a result of all this and eq. (2.4) we conclude that
R
\ K
Proposition 2.2 The loop expansion formula (1.13) for the Jones polynomial of a knot in
S3 can be presented in the form
2 sin(πa) ∞ 2π
Z (S3, ;k) = exp i S (a) , (2.9)
α K sK ∆A(S3,K;exp(2πia)) " nX=1(cid:18)K (cid:19) n+1 #
here S (a) are the higher loop corrections for the path integral (2.6), they depend on the
n
boundary holonomy exp(2πia).
We will show later that e−iNph = 1.
The substitution (2.5) turns the r.h.s. of this equation into the expansion (2.1) with
limitations (2.2) and property (2.3). We also learn that the sum of the terms
C αmK−m−n (2.10)
m,m+n
m≥0
X
comes from the n-loop Feynman diagrams (including disconnected ones) in the knot com-
plement S3 Tub( ).
\ K
Consider now a general RHS M with a knot inside it. This time there may be
K
many flat connections (both reducible and irreducible) with a given holonomy (2.7) even
if a is very small. Each of them will contribute to the stationary phase approximation of
the path integral (2.6) turning it into the sum (1.13). We will concentrate on the reducible
U(1) connections because their 1-loop contributions can again be related to the Alexander
polynomial of .
K
8
Some changes have to be made to eq. (2.9). Let b define the holonomy along C for a
2
reducible flat connection on M Tub( ):
\ K
Pexp A dxµ = exp(2πib). (2.11)
µ
(cid:18)IC2 (cid:19)
The holonomies (2.7) and (2.11) are related by the fact that the homomorphism
H (∂Tub( ),ZZ) H (M Tub( ),ZZ) (2.12)
1 1
K → \ K
has a kernel. Let the cycle
C = d(m C +m C ), d,m ,m ZZ, m ,m coprime (2.13)
0 1 1 2 2 1 2 1 2
∈ −
be its generator. Then
Pexp A dxµ exp[2πid(m a+m b)] = 1, (2.14)
µ 1 2
≡
(cid:18)IC0 (cid:19)
so that
1 n
b = m a+ , n ZZ, 0 n < d. (2.15)
1
−m d ∈ ≤
2 (cid:18) (cid:19)
If we smoothly reduce a to zero, then the flat connection on M Tub( ) will transform into
\ K
a flat connection on M. Let S be its Chern-Simons invariant. Then according to [21]
CS,0
and eq. (2.8), the Chern-Simons action of the original connection is
m na
S′ = π2 1a2 +2 +S . (2.16)
CS − m m b CS,0
(cid:18) 2 2 (cid:19)
In particular, if the flat connection on M at a = 0 is trivial, then S = 0 and n = 0, so
CS
that
m
S′ = π2 1a2. (2.17)
CS − m
2
The Reidemeister-Ray-Singer torsion for the Cartan subalgebra part of A is known to be
µ
equal to ordH (M,ZZ). As for the off-diagonal Lie algebra components of A , we can use
1 µ
again eq. (2.4). However the argument of the torsion is related to the holonomy along the
9