arXiv:hep-th/0306165v1 17 Jun 2003
hep-th/0306165
HUTP-03/A003
ITEP-TH-50/02
Three-Dimensional Quantum Gravity,
Chern-Simons Theory, And
The A-Polynomial
Sergei Gukov
Jefferson Physical Laboratory, Harvard Univ ersity,
Cambridge, MA 02138, USA
Abstract
We study three-dimensional Chern-Simons theory with complex g auge group SL(2,
C),
which has many interesting connections with three-dimensional quantum gravi ty and ge-
ometry of hyperbolic 3 -manifolds. We show that, in the presence of a single knotted Wilson
loop in an infinite-dimensional representation of the gauge group, the classical and quan-
tum properties of such theory a re described by an algebraic curve called the A-polynomial
of a knot. Using this approach, we find some new and rather surprising relations be-
tween the A-polynomial, the colored Jones polynomial, and other inva riants of hyperbolic
3-manifolds. These relations generalize the volume conjecture and t he Melvin-Morton-
Rozansky conjecture, and suggest an intriguing connection between the SL(2,
C) partition
function and the colored Jones polynomial.
June 2003
1. Introduction and Motivation
In this paper we study three-dimensional Chern-Simons theory with complex g auge
group. Of particular interest is a Chern-Simons theory with gauge group G
C
= SL(2,
C)
(viewed as a complexification of G = SU(2)), which has many interesting connections with
three-dimensional quantum g ravity and geometry o f hyperbolic three-manifolds. In this
introductory section we review some aspects o f these relations, formulate the problem, and
describe various applications.
1.1. Chern-Simons Theory
Consider an orient ed three-dimensional space M . We wish to formulate a Chern-
Simons gauge theory on M with complex gauge group G
C
, whose real form we denote
by G. Let g
C
and g be the corresponding Lie algebras. In these notations, the gauge
connection A is a o ne-form on M valued in the complex Lie algebra g
C
. Explicitly, we
can write A =
P
a
A
a
· T
a
where T
a
denote the g enerators of g, which are assumed to be
orthonormal, Tr(T
a
T
b
) = δ
ab
. Then, the Chern-Simons action can be written as a sum of
the holomorphic and anti-holomorphic terms,
I =
t
8π
Z
M
Tr
A∧ dA +
2
3
A ∧ A ∧A
+
+
t
8π
Z
M
Tr
A∧ dA +
2
3
A∧ A ∧A
(1.1)
where t = k + is and
t = k − is are the corresponding coupling constants. Consistency of
the quantum theory requires the “level” k to be an integer, k ∈ ZZ. The other parameter,
s, is not quantized. However, s must obey certain constraints i mposed by unitarity [1].
In Euclidean space, unitarity implies that the argument of the Fey nman path integral
Z(M) =
Z
DA e
iI
(1.2)
must be complex conjugated under a reversal of the orientation on M . In the Chern-Simons
theory defined by the action (1.1), there are two possibilities to achieve this, corresponding
to either purely real or purely imaginary values of s. In the first case, A is invariant under
the reversal of the orientation and
t is the usual complex conjugate of t. On the other
hand, the second possibility, s ∈ iIR, is realized when the gauge connection transforms non-
triviall y under the reversal of the orientation, A 7→
A. In the rest of the paper we mainly
consider the case of imaginary s and G
C
= SL(2,
C), which is related to the Euclidean
quantum gravi ty i n three dimensions (see discussion below).
1
γ
Fig. 1: A knotted Wilson loop in the IR
3
.
Now let us consider observables in this theory. If M is a manifold without boundary,
holonomies of the gauge connection prov ide a complete set of observables, also known as
“Wilson lines” in the context of gauge theory or “loop variables” in the context of gravity
[2]. Specifically, gi ven a closed oriented curve γ ⊂ M and a representation R of G
C
, one
can define a gauge invariant observable as
W
R
(γ) = Tr
R
Hol
γ
(A) = Tr
R
P exp
I
γ
A
(1.3)
The Wilson loop observables W
R
(γ) are naturally associated with knots in M. Indeed,
even t hough intrinsically the curve γ is simply a circle, it s embedding in M may be highly
non-trivial, represent ed by a knot, as in fig. 1. More generally, an embedding of a collection
of circles into M is called a link, and the image of each circle is called a component of
the link. Thus, g iven a link with (non-int ersecting) components γ
i
, i = 1, . . . , r, and a
set of representations R
i
assigned to each component of the li nk , one can study a natural
generalization of the Feynman path integral (1.2):
Z(M; γ
i
, R
i
) =
Z
DA exp(iI)
r
Y
i=1
W
R
i
(γ
i
) (1.4)
which, following [3], we call the (unnormalized) expectation value of the link. By construc-
tion, it is a function of t and
t, which also depends on t he topology of the three-manifold
M, on the choice of the Wilson lines γ
i
, and on t he corresp onding representations R
i
.
A large class of representations R
i
can be naturally obtained by complexification from
the correspo nding representations of the real Lie algebra g. However, no new information
can be gained by studying such representations since the evaluation of (1.4) essentially
reduces to the Chern-Simons theory with t he real form of the gauge group, G, at least
2
in perturbation theory
1
. On the other hand, of particular interest are certain infinite
dimensional representations R
i
that we describe explicitly in the next subsection, after
explaining the connection with three-dimensional quantum gravity.
1.2. Three-Dimensional Quantum Grav ity
As we already mentioned earlier, there are several int riguing connections between
three-dimensional gravity and Chern-Simons theory with complex gauge group G
C
=
SL(2,
C) and imaginary values of the parameter s (for convenience, in what follows we
shall use a real parameter σ = is). Thus, SL(2,C) appears as the Poincare group in
three-dimensional Euclidean gravity with negative cosmological constant
2
. Moreover, by
writing the complex gauge field A in terms of the real and imaginary components one can
relate the Chern-Simons acti o n (1.1) to the usual form of the E instein-Hilbert action of
three-dimensional gravity with negative cosmological constant [6,7]. Specifically, writing
A = w + ie and
A = w − ie one finds
I =
k
4π
Z
M
Tr
w ∧ dw − e ∧ de +
2
3
w ∧ w ∧ w − 2w ∧e ∧ e
+
+
iσ
2π
Z
M
Tr
w ∧ de + w ∧ w ∧ e −
1
3
e ∧ e ∧e
(1.5)
The second term in this expression is indeed equivalent to the Einstein-Hilbert action
3
with negative cosmolo gical constant , Λ = −1, written in terms of the vielbein e and the
spin connection w. We can also w rite it in t he standard form:
I
grav
= −
1
4π
Z
M
d
3
x
√
g
R + 2
(1.6)
As will be shown b el ow, the first term in (1.5) also has a nice interpretation. It is
related to the Chern-Simons invariant of the three-manifold M. Therefo re, it is convenient
to denote this term as I
CS
. In the new not ations, we can w rite the original actio n (1.1 ) as
I(k, σ) = kI
CS
+ iσI
grav
(1.7)
1
See e.g. exercise 6.32 in [4].
2
More precisely, Isom
+
(IH
3
) = P SL(2,
C) = SL(2,C)/{±1}, but according to [5], a repre-
sentation of P SL(2,C) corresponding to a complete hyperbolic structure can always be lifted to
SL(2,
C), and it is SL(2,C) with which we shall work.
3
In our notations, the length scale ℓ = 1 and the Newton constant G
N
= 1/(4σ).
3
Summarizing, following Witten [7], we conclude that the real and imaginary compo-
nents of the SL(2,
C) Chern-Simons action (1.1) have a nice physical interpretation. In
particular, a theory with k = 0 represents, at least (semi-)classically, a three-dimensional
Euclidean quantum grav ity with negative cosmological constant
4
. However, this equiv-
alence does not readily extend to q uantum theories due to a number of subtle issues,
typical ly related to degenerate vielbeins etc. (see e.g. [10] for a recent discussion of these
questions). For example, in the Chern-Simons theory, it is natural to expand around a
trivial vacuum, A = 0, which corresponds to a very degenerate metric, g
ij
= 0 . Also, in
the Chern-Simons path integral (1.2) one integrates over all (equivalence classes of) gauge
connections, w hereas in quantum g ravity one takes only a subset of those corresponding to
positive-definite volume elements. Nevertheless, o ne would hope that, for certain questions,
the relation to gravity can still be helpful even beyond t he classical limit. Thus, in order to
avoid the above problems t hroughout the paper we shall mainly consider the semi-classical
expansion around an isolated critical point, corresponding to a non-degenerate metric on
M. Then, the quantum fluctuations are small, and both theories are expected to agree.
So far we discussed a relation between SL(2,
C) Chern-Simons theory and pure gravity.
Now let us add sources representing point particles. Assuming that particles don’t have
any internal structure, they can be characterized by two numbers: a mass and a spin. As
we will see later, it is natural to combine these numbers into a single complex quantity,
which labels an infinite dimensional representation of SL(2,C), see [11,12]. Interacti ng with
gravity, matter particles produce conical defects in the geometry of the space manifold M
[13,14,15 ]. In particular, light -l ike particles correspond to cusps in M. We shall say more
about this aspect later, when we will be talking about the relation to hyperboli c geometry.
The coupling of point-like sources to gravity can be described by Wilson lines in the
Feynman path integral (1.4), see e.g. [16,17,18,19]. For example, if we int roduce extra
variables p
a
and x
a
, which represent momentum and coordinate of a particle in space M ,
then the Wilson line operator for a spinless particle can be explicitly written as [1 6]:
W
R
(γ) =
Z
Dx
a
(s)Dp
a
(s)Dλ(s) exp (iI
W
) (1.8)
4
Chern-Simons theory with SL( 2,
C) gauge group and real values of the parameter s is also
related t o three-dimensional gravity, namely to de Sitter gravi ty in 2+1 dimensions [6,7]. This
theory can be treated simil arly, and many of the arguments below easily extend to this case. For
work on quantization of this theory see [1,8,9].
4
