Baez J.C. (Ed.) Knots and Quantum Gravity (Oxford Lecture Series in Mathematics and Its Applications)

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Geometric Struct ures and Loop Variables 109

equations of mo tion,

dˆτ

= i

H, ˆτ

dˆp

= i

H, ˆp

, (4.3)

with a Hamiltonian

H[ˆτ , ˆp

, T ] that can again be calculated ex plicitly.

Let me stres s that despite the familiar appearance of (4.3), the pa-

rameter T is not a time coordinate in the o rdinary sense; the operators

ˆτ(T ) and ˆp

(T ) ar e fully diﬀeomorphism invariant. We thus have a kind

of ‘time dependence without time dependence’, an expr ession of dynamics

in terms of operators that are individually constants of motion. For more

complicated topologies, such an explicit construction seems quite diﬃcult,

although Unr uh and Newbury have taken some interesting steps in that

direction [40]. Ideally, one would like to ﬁnd some kind of perturbation

theory for geometrical variables like

Q, but little progress has yet been

made in this direction.

The speciﬁc c onstructions I have described here are unique to 2+1 di-

mensions, of course. But I believe that some of the basic features are

likely to extend to realistic (3+1)- dimensional gravity. The quantization

of holonomies is a form of ‘covariant canonical quantization’ [41, 42], or

quantization of the space of classical solutions . We do not yet understand

the classical solutions of the (3+1)-dimensional ﬁeld equations well enough

to duplicate such a strategy, but a similar approach may be useful in mini-

supe rspace models. The invariant but T -dependent operators ˆτ(T ) and

ˆp

(T ) are examples of Rovelli’s ‘evolving constants of motion’ [43], whose

use has also been suggested in (3+1)-dimensional g ravity. Finally, our sim-

ple model has strikingly conﬁrmed the power of the Rovelli–Smolin loop

va riables. A full extension to 3+1 dimensions undoubtedly remains a dis-

tant goal, but for the ﬁrst time in years, there seems to be some real cause

for o ptimism.

Acknowledgements

This work was supported in part by US Department of Energy grant DE-

FG03-91ER40674.

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110 Steven Carlip

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Geometric Struct ures and Loop Variables 111

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112 Steven Carlip

From Chern–Simons to WZW via Path

Integrals

Dana S. Fine

Department of Mathematics, University of Massachusetts,

Dartmouth, Massachusetts 02747, USA

(email: dﬁne@cis.umassd.edu)

Abstract

Direct analysis of the Chern–Simons path integral reduces quantum

expectations of unknotted Wilson lines in Chern–Simons the ory on

with group G to n-point functions in the WZW model of maps

from S

to G. The reduction hinges on the characterization of A/G

the space of connections modulo those gauge transformations which

are the identity at a point n, as itself a principal ﬁber bundle with

aﬃne-linear ﬁber.

1 Introduction

In this chapter, I will describe how to reduce the path integral for Chern–

Simons theory on S

to the path integral for the Wess– Zumino–Witten

(WZW) model for maps from S

to G, where G is the symmetry group of

the Chern–Simons theory. The po ints I would like to emphasize are, ﬁrst,

that the Chern–Simons path integral o n S

is tractable, and, second, that

it provides an unusually direct link between the Chern–Simons theory and

the WZW model. The details I omit appear in [4].

That the two theories are related has been known at least since Witten

ﬁrst described the geometric quantization of the Chern–Simons theory in

the axial gauge on manifolds of the form R × Σ, where Σ is a Riemann

surface [9]. Although this is frequently described as a path integral ap-

proach, it does not use a direct evaluation of the path integral; rather , the

formal properties of the path integral provide axioms crucial to extending

the theory from R×Σ to compact 3-manifolds. In this approach, the WZW

model appears in a highly reﬁned form, as the source of a certain modular

functor. That is, one must already know how to solve the WZW model (or

at least know that its solution deﬁnes this modular functor) to reco gnize

its appearance in the Chern–Simons theory.

Treatments of the path integral fo r Chern–Simons in the axial gauge on

R ×Σ tend to make contact with the WZW model at earlier stages. Moore

and Seiber g [6] proceed with the evaluation of the path integral far enough

113

114 Dana S. Fine

to obtain a path integral over the phase space LG/G. This links the two

theories as theories of the representations of LG/G. Fr¨ohlich and King [5]

obtain the Knizhnik–Zamolodchikov equation of the WZW model from the

same g auge-ﬁxed Chern–Simons path integral.

The approach I will describe is novel in that it treats the path integral

on S

directly, it does not require a choice of gauge, and one nee d only

know the action of the WZW model, not its solution nor even its current

algebra, to recognize its appearance in the Chern–Simons theory. The key

ingredient is a technique for integrating directly on the space of connections

modulo ga uge transformatio ns which I developed to evaluate path integrals

for Yang–Mills on Riemann surfaces [3].

2 The main result

The Chern–Simons path integral to evaluate is

hfi

A/G

f([A])e

iS(A)

∗

(DA),

where A is a connection on the principal bundle P = S

× G, and S(A) is

the Chern–Simons action:

4π



A ∧ dA +

A ∧ [A ∧ A]



Here A is the space of smooth connections on P , G

is the space of gauge

transformations which are the identity at the north pole n of S

, ρ : A →

A/G

is the standard map to the quotient, DA means the standard measure

on A, and ρ

∗

(DA) is the push-forward to A/G

. Note that even the formal

deﬁnition of DA requires a metric. Take this to be the standard metr ic on

. The WZW path integral for Ω

G (bas e d maps from S

to G) is

hfi

WZW

Ω

f(X)e

iS(X)

DX,

where the WZW action is

S(X) =

4π



Tr(∂X ∧

∂X) +

Tr(

−1

X ∧

−1

X ∧

−1

)



Here M is any 3-manifold with ∂M = S

, and

X is any extension of X

from S

to M .

The main result is

hfi

f(X)e

4π

Tr(∂X∧

∂X)+i

12π



−1



DX,

where X ∈ Ω

G, and f explicitly determines

f. In particular, if f is

a Wilson line, then

f is the trace of a product of evaluation-at-a-point

From Chern–Simons to WZW 115

Fig. 1. The path γ(e)

maps. That is, the Chern–Simons path integral directly reduces to the

path integral for a WZW model of based maps from S

to G.

The technique is as follows:

1. Characterize A/G

as a bundle over Ω

G with aﬃne-linear ﬁbers.

2. Integrate along the ﬁber over each point X ∈ Ω

3. Recognize the resulting ‘measure’ on Ω

G as a WZW path integral.

3 A/G

as a bundle over Ω

Fo r each connection A, and each point e ∈ S

, deﬁne ξ

(e) by holonomy

about paths γ(e) of the form pictured in Fig . 1. That is, γ(e) follows a

longitude from the north pole n to the south pole s through a ﬁx e d point e

of the equator and then returns along the longitude through the (variable)

point e. Relative to a basepoint in the ﬁber over n, ξ

(e) lies in G. As e

va ries throughout the e quator (identiﬁed with S

), ξ

(e) deﬁnes, for e ach

A, a map from S

to G. Since ξ

) = 1, it is a based ma p, s o ξ

∈ Ω

Moreover, since a gauge transformation aﬀects holonomy about γ(e) by

conjugation by the value of the gauge transformation at n, ξ

φ•A

= ξ

for

φ ∈ G

. Thus the assignment [A] 7→ ξ

deﬁnes a mapping ξ : A/G

→ Ω

which will serve as our projection.

Turn now to the ﬁber of ξ. Clearly, ξ

A+τ

= ξ

if τ vanishes along the

longitudes of Fig. 1. If P

, g) →

, g) denotes projection to

the long itudinal part, the space of such τ is ker P

. In fact, the ﬁber over

116 Dana S. Fine

is the set of orbits

{[A + τ] : τ ∈ ker P

}. (3.1)

Thus the ﬁber is the aﬃne version of the linear space ker P

, and the

connection A in the above presentation represents a choice of origin in the

ﬁber.

The image of ξ is all of Ω

G. To see this, use a homotopy from X

to the identity to determine lifts to P of the family of paths γ(e) in S

which deﬁned ξ. (Such a homotopy exists, bec ause π

(G) = 0 for any

Lie group G). This determines P

A such that ξ(A) = X. This is a slight

reﬁnement of [7] and [1], which describe a homotopy equivalence between

A/G

for bundles over S

and Ω

G. The reason to redo this is to describe

the (topologically trivial) ﬁber in an explicit ma nner.

According to eqn 3.1, the integral over the ﬁber looks like

ﬁber

(X) = det

1/2

∗

)

f([A + τ])e

iS(A+τ )

Dτ.

The determinant factor relates ρ

∗

(DA) on the ﬁber to the metric-compatible

measure Dτ. The action along the ﬁber is, after an integration by parts,

S(A + τ) = S(A) +

4π



Tr (τ ∧ D

τ + 2 F

∧ τ) +

∂M

Tr(A ∧ τ )



where, for reasons which will soon be clear, the boundary contribution is

retained. To integrate over the ﬁbe r, choose a speciﬁc origin [

A] to remove

the linear term.

One such choice is determined by

A =

−1

where

X : S

→ G is parallel transport by A along the longitudinal paths

γ(e) used in deﬁning ξ. Note that

X is not well deﬁned at n. In fact

∂

= X, where |

∂

is the re striction to a small 2-sphere appr oaching n.

Thus

A is not continuous at n (nor is any other such choice), so to keep

A + τ smooth requires

τ|

∂

= −



∂

= −X

−1

dX.

With this choice, the boundary term vanishes, and

A + τ) = S(

A) +

4π

Tr(τ ∧ D

τ).

The integral over the ﬁber becomes

ﬁber

(X) = det

1/2

∗

iS(

ker P

A + τ )e

4π

Tr(τ ∧D

τ )

Dτ,

From Chern–Simons to WZW 117

where

ker P

denotes elements of ker P

with the prescrib e d discontinuity

at n. Now assume f is independent of τ and consider

4π

Tr(τ ∧D

τ )

Dτ .

But for the discontinuity at n, this would be standard. The result would be

a determinant times the exponential of the eta -invariant as in [8] and [9].

Repeating these arguments, which refer to a ﬁnite-dimensional analog, with

a minor variation to account for the discontinuity, will evaluate I

ﬁber

. The

metric on

, g) restricts to ker P

to deﬁne a decomposition (of its

complexiﬁcation) ker P

⊕

−

, in terms of which

Tr(τ ∧ D

τ) = 2

Tr(τ

∧ D

−

) −

Tr(τ

−

∧ τ

)

= 2

Tr(τ

∧ D

−

) +

Tr(∂X ∧

∂X).

Model this by a vector space V = V

⊕V

−

with a linear mapping T : V

→

∓

)

∗

. Add a small quadratic term, and integrate to ﬁnd

2ihT v

−

= det

−1/2



∗

−



The determinant analogous to the right-hand side is that of P

−

∗

−

acting on elements of ker P

which vanish at n. Now, regularize and ac c ount

for neg ative eigenvalues using the eta-invariant; that is, take

det

−

∗

−

−i

η(0)

where η(0) is the eta-invariant, as the analogue for det

−1/2



∗

−



. Then

ﬁber

(X) =

det

∗

)

det

−

∗

−

−i

η(0)

iS(

4π

Tr(∂X∧

∂X)

Performing the integration over the ﬁbers in the path integral for hfi,

assuming f constant along the ﬁbers, thus gives

hfi =

A)e

iS(

4π

Tr(∂X∧

∂X)

det

∗

)

det

−

∗

−

−i

η(0)

base

where µ

base

is the measure on Ω

G induced by the metric on A/G

. To

interpret this in terms of Ω

G requires four obser vations:

1. S





12π



−1



2. µ

base

= constant × DX.

3. Each determina nt is constant along the base in A/G

4. f can be expressed as a function

f of X.

118 Dana S. Fine

Fig. 2. The path whose holonomy is X(e

)

−1

X(e

)

Absorbing everything independent of X into Z

, then,

hfi

f(X)e

4π

Tr(∂X∧

∂X)+i

12π



−1



DX,

as claimed.

4 An explicit reduction from Chern–Simons to WZW

Let R be a repres e ntation of G, and let W

denote the Wilson line co rre-

sp onding to the longitudinal path indicated in Fig. 2. Then



X(e

)

−1

X(e

)



4π

Tr(∂X∧

∂X)+i

12π



−1







X(e

)

−1

X(e

)



WZW

As a description of the relation be tween Chern–Simons theory on S

and the WZW for Ω

G, this work is nearly complete. However, as an eval-

uation of the Chern–Simons path integral it lacks an essential component;

namely, an evaluation of the WZW path integral appe aring on the right-

hand side of the above equation. For ex pecta tions of evaluation-at-a-point

maps, the WZW path integral is tractable, at least formally, but this is

work in progress. The basic idea is to reinterpret X ∈ Ω

G as an element

of Path(ΩG) with some constraints on its endpoint values. Then, the WZW