Quantum holonomy in three-dimensional general covariant field theory and link invariant

Abstract

We consider quantum holonomy of some three-dimensional general covariant non-Abelian field theory in the Landau gauge and confirm a previous result partially proven. We show that quantum holonomy retains metric independence after explicit gauge fixing and hence possesses the topological property of a link invariant. We examine the generalized quantum holonomy defined on a multicomponent link and discuss its relation to a polynomial for the link.

Full text

Quantum holonomy in three-dimensional general covariant field theory and the link invariant
W. F. Chen*
Helsinki Institute of Physics, P.O. Box 9 (Siltavuorenpenger 20 C), FIN-00014, Helsinki, Finland
H. C. Lee
Department of Physics, National Central University, Chungli, Taiwan 320, Republic of China
Z. Y. Zhu
Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing, 100080, China
~Received 14 February 1997!
We consider quantum holonomy of some three-dimensional general covariant non-Abelian field theory in
the Landau gauge and confirm a previous result partially proven. We show that quantum holonomy retains
metric independence after explicit gauge fixing and hence possesses the topological property of a link invari-
ant. We examine the generalized quantum holonomy defined on a multicomponent link and discuss its relation
to a polynomial for the link. @S0556-2821~97!02314-X#
PACS number~s!: 11.15.Tk, 02.40.Pc, 03.65.Fd
Some time ago it was shown that quantum holonomy in a
three-dimensional general covariant non-Abelian gauge field
theory possesses topological information of the link on
which the holonomy operator is defined @1#. The quantum
holonomy operator was shown to be a central element of the
gauge group so that, in a given representation of the gauge
group, it is a matrix that commutes with the matrix represen-
tations of all other operators in the group. In an irreducible
representation, it is proportional to the identity matrix. Quan-
tum holonomy should therefore in general have more infor-
mation on the link invariant than the quantum Wilson loop
which, for the SU~2!Chern-Simons quantum field theory,
was shown by Witten @2#to yield the Jones polynomial @3#.
Horne @4#extended Witten’s result to some other Lie groups.
The difference between quantum holonomy and the Wilson
loop becomes apparent in the SU(N
u
N) Chern-Simons
theory, where the quantum Wilson loop vanishes identically
for any link owing to the property of supertrace, but the
quantum holonomy @1#yields the important Alexander-
Conway polynomial @5–8#.
However, the argument used in Ref. @1#was based only
on the formal properties of the functional integral and
complications that may arise from the necessity for gauge
fixing in any actual computation were not taken into consid-
eration. In addition, in a case when a metric is needed
for gauge fixing, the metric independence of quantum ho-
lonomy may be violated. Furthermore, in the standard
Faddeev-Popov technique used for gauge fixing, ghost fields
and auxiliary fields that are introduced reduce the original
local gauge symmetry to Becchi-Rouet-Stora-Tyutin ~BRST!
symmetry, and it is no longer certain that the formal argu-
ments and manipulations used in Ref. @1#to derive its results
are still valid. As well, the case of the quantum holonomy
defined on multicomponent links was not explicitly consid-
ered.
The main aim of the present work is to clarify these prob-
lems. We explicitly work in the Landau gauge1and confirm
the results obtained in Ref. @1#for the case of a one-
component contour. We then show that a quantum holonomy
operator defined on a n-component link, which by construc-
tion is a tensor product of those operators defined on the
components, is a central element of the universal enveloping
algebra of the Lie algebra of the gauge group and, when
evaluated on a set of nirreducible representations of the
gauge group, has a uniquely defined eigenvalue that is a
polynomial invariant of the link.
The quantum holonomy is defined as
Z@C#[1
V
E
DAexp~iS@A#!f@A,C#,
f@A,C#[Pexp
S
i
R
CA
D
,~1!
where Cis a contour in the three-dimensional manifold M,
S@A#is the action of some three-dimensional general cova-
riant non-Abelian gauge field theory, Pmeans path ordering,
and V5
*
Dgis gauge-invariant group volume. Now we
choose the Lorentz gauge condition
F@A#5
]
m
~
A
2GG
m
n
A
n
!50, G5det~G
m
n
!,~2!
where G
m
n
is the metric of the space-time manifold. Accord-
ing to standard Faddeev-Popov procedure, we insert the
identity
1[DF@A#
E
DgPx
d
„F@Ag~x!#…~3!
into Eq. ~1!and obtain
*Also at ICSC-World Laboratory, Lausanne, Switzerland.
1The reason why we prefer this gauge is because it allows one to
avoid the infrared divergence in low-dimensional gauge theories;
for a discussion on Chern-Simons theory, see @15#.
PHYSICAL REVIEW D 15 JULY 1997VOLUME 56, NUMBER 2
56
0556-2821/97/56~2!/1170~5!/$10.00 1170 © 1997 The American Physical Society

Z@C#51
V
E
DADF@A#
E
DgPx
d
„F@Ag~x!#…exp~iS@A#!f@A,C#.~4!
Denoting Agas Aand replacing the original Aby Ag21, we rewrite Eq. ~4!as
Z@C#51
V
E
DgDAg21DF@Ag21#Px
d
~F@A#!exp~iS@Ag21#!f@Ag21,C#
51
V
E
DgDADF@A#Px
d
~F@A#!exp~iS@A#!f@Ag21,C#
51
V
E
DgDADBDc
¯
Dc
H
exp
F
iS@A#2i
E
d3x
A
2GG
m
n
~
]
m
BaA
n
a2
]
m
c
¯
aD
n
ca!
G
f@Ag21,C#
J
~5!
where we have used the gauge invariance of S@A#and Ba(x),c
¯
a(x), and ca(x) are, respectively, auxiliary fields, ghost, and
antighost fields.
We perform the following maneuver on Eq. ~5!. Suppose g8is a global group element, write Aas Ag8g821, and rename
Ag821as Aand hence the original Ais replaced by Ag8. We thus obtain
Z@C#51
V
E
DgDADBDc
¯
Dc
H
exp
F
iS@A#1i
E
d3x
A
2G~Ba
]
m
A
m
a2c
¯
a
]
m
D
m
ca!
G
f@~Ag21!g8,C#
J
51
V
E
DgDADBDc
¯
Dc
H
exp
F
iS@A#1i
E
d3x
A
2G~Ba
]
m
A
m
a2c
¯
a
]
m
D
m
ca!
G
Vg8
21f@Ag21,C#Vg8
J
5Vg8
21Z@C#Vg8,
~6!
where we have used the fact that all the fields are in the adjoint representation of global gauge group. Since Eq. ~6!is true for
every global gauge transformation, according to Schur’s lemma, we conclude that when Z@C#is valued in an irreducible
representation
r
it has the form
r
~Z@C#!5F@C#1
r
,~7!
where 1
r
is the matrix representation of the identity element in
r
and F@C#, the eigenvalue of Z@C#in
r
, is a scalar function
depending on the contour C. Equation ~7!is one of the results obtained in Ref. @1#without explicit gauge fixing.
In the following, we shall show explicitly the metric independence of quantum holonomy. All fields are valued in the
adjoint representation of the gauge group. As a first step, Eq. ~5!can be rewritten as
Z@C#51
V
E
DgDADBDc
¯
Dcexp
F
i
S
S@A#1
E
d3x
A
2GG
m
n
d
B@c
¯
a
]
m
A
n
a#
D
G
f@Ag21,C#,~8!
where the BRST transformations are
d
BAa5D
m
ca,
d
Bca51
2fabccbcc,
d
Bc
¯
a5Ba,
d
BBa50. ~9!
Assuming that the functional measures of fields have no dependence on the metric, we have that
22i
A
2G
d
Z@C#
d
G
m
n
52 2i
A
2G
d
F@C#
d
G
m
n
1
r
52 2i
A
2G
d
d
G
m
n
H
1
V
E
DgDADBDc
¯
Dcexp
F
i
S
S@A#1
E
d3x
A
2GG
ab
d
B~c
¯
a
]
a
A
b
a!
D
G
f@Ag21,C#
J
51
V
E
DgDADBDc
¯
Dcexp
F
iS@A#1i
E
d3x
A
2GG
ab
d
B~c
¯
a
]
a
A
b
a!
G
T
m
n
f@Ag21,C#,~10!
where T
m
n
is the canonical symmetric energy momentum,
56 1171QUANTUM HOLONOMY IN THREE-DIMENSIONAL . . .

T
m
n
52
A
2G
d
Seff
d
G
m
n
5A
m
a
]
n
Ba1A
n
a
]
m
Ba2
]
m
c
¯
a~D
n
c!a2
]
n
c
¯
a~D
m
c!a2G
m
n
@A
a
a
]
a
Ba2
]
a
c
¯
a~D
a
c!a#,
Seff5S@A#1
E
d3x
A
2GG
ab
d
B~c
¯
a
]
a
A
b
a!.~11!
It can be written as a BRST trivial form from a careful observation,
T
m
n
5
d
BQ
m
n
,Q
m
n
52
]
m
c
¯
aA
n
a2
]
n
c
¯
aA
m
a2G
m
n
]
a
c
¯
aA
a
a.~12!
So we can obtain that
22i
A
2G
d
Z@C#
d
G
m
n
52 2i
A
2G
d
F~C!
d
G
m
n
1
r
5
E
DADBDc
¯
Dcexp
H
iS@A#1i
E
d3x
A
2GG
ab
d
B~c
¯
a
]
a
A
b
a!
J
d
BQ
m
n
1
V
E
Dgf@Ag21,C#
5
^
0
u
@Q
ˆB,Q
ˆ
m
n
#1
V
E
Dgf@A
ˆg21,C#
u
0
&
5
^
0
u
F
Q
ˆB,Q
ˆ
m
n
1
V
E
Dgf@A
ˆg21,C#
G
u
0
&
50, ~13!
where QBis the BRST charge corresponding to the BRST
transformation, Eq. ~9!, and the caret denotes an operator. In
the above, we have used the physical state condition in
BRST quantization, Q
ˆB
u
phys
&
50@9#, and the fact that
(1/V)
*
Dgf@Ag21,C#is gauge invariant,
F
Q
ˆB,1
V
E
Dgf@A
ˆg21,C#
G
50, ~14!
as well as the explicit corollary observed by Witten @10#: For
two operators A
ˆand B
ˆ,if@
Q
ˆ
B
,A
ˆ
#
50, then A
ˆ@Q
ˆB,B
ˆ#
5@Q
ˆB,A
ˆB
ˆ#. Therefore, from Eq. ~13!,Z@C#and F@C#are
generally covariant.
We shall now justify the assumption that there is no met-
ric independence for the path integral measure. This assumed
property was very crucial in the proof of the general covari-
ance of quantum holonomy. The justification can be made
from two aspects. First, a metric dependence in the path in-
tegral measure means that under metric variation the path
integral measure has a nontrivial Jacobian factor. According
to Fujikawa @11#, this implies that the theory would have a
conformal anomaly; i.e., the trace of the energy momentum
^
Q
m
m
&
does not vanish. Since the trace of the energy momen-
tum is proportional to the
b
function of the theory @12#, the
existence of a conformal anomaly would mean that the
theory is not finite. On the other hand, it has been proved that
three-dimensional topological field theories such as Chern-
Simons @13#and BF @14#theories are finite to any order; that
is, the
b
function and anomalous dimension vanish identi-
cally. This has also been verified by explicit computation in
concrete regularization schemes to two loops @15–17#.
Therefore the conformal anomaly and hence the metric de-
pendence of the path integral measure should not exist.
We could also approach the issue from the opposite direc-
tion and impose the conformal anomaly-free condition. Then
the Jacobian associated with the metric variation would be
trivial, which means that the correct path integration vari-
ables would not be the original fields F5(A,B,c,c
¯
) but
would be the appropriate tensorial densities F
˜
given by
F
˜
i5~2G!2wi/2Fi,~15!
where Gis the determinant of metric tensor G
m
n
, the sub-
script ilabels a specific field in F, and wiis the weight
associated with the field Fiwhose value depends on the
tensorial character of the corresponding field. Replacing Fi
by F
˜
ias the path integral variables and at the same time
rewriting the action in terms of these new variables, we
could make the path integral measure metric independent.
This procedure would transfer the metric dependence of the
path integral measure to the effective action, and this would
affect all the symmetries of the effective action such as
BRST symmetry, etc. In Ref. @18#, it was shown that for a
cohomological topological field theory, whose action can al-
ways be written as a BRST-trivial form, this line of reason-
ing can be used to define an invariant path integral measure
so that the topological character of the theory is preserved.
However, for topological field theories of the Chern-Simons
type, whose action cannot be written as a BRST commutator,
it is not clear how this technique can be applied. We intend
to explore this problem in detail elsewhere.
Now we try to understand the result, Eq. ~7!, from the
operator viewpoint. Since the global gauge symmetry is not
affected by gauge fixing, the Noether current and charge cor-
responding to global gauge transformation are, respectively,
j
m
a52iTr
]
L
]]
m
F@Ta,F#,Qa5
E
d3xj0
a,~16!
where F[(A,B,c,c
¯
). After quantization, since there is no
anomaly in three-dimensional gauge theory, we have
1172 56W. F. CHEN, H. C. LEE, AND Z. Y. ZHU

]
m
^
j
ˆ
m
a
&
50, Q
ˆa
u
0
&
50. ~17!
The Q
ˆaconstitute the operator realization of the generators
of the global gauge group,
@Q
ˆa,Q
ˆb#5ifabcQ
ˆc.~18!
Correspondingly, Ug85exp@2i
j
aQ
ˆa#are global gauge
group elements and
j
aare group parameters. Under a global
gauge transformation, the holonomy operator f@A
ˆ,C#trans-
forms as
f@A
ˆ,C#→f@A
ˆ,C#g8
5Ug8
21f@A
ˆ,C#Ug85Vg8
21f@A
ˆ,C#Vg8,
~19!
where Vg85exp@2i
j
aTa#are the matrix representations of
group elements. So we have
Z@C#5
^
0
u
f@A
ˆ,C#
u
0
&
5
^
0
u
Ug8
21f@A
ˆ,C#Ug8
u
0
&
5
^
0
u
Vg8
21f@A
ˆ,C#Vg8
u
0
&
5Vg8
21Z@C#Vg8.~20!
Z@C#commutes with every global gauge transformation, and
from Schur’s lemma we obtain Eq. ~7!.
Finally let us consider the generalized quantum holonomy
defined on a multicomponent link L, the disjoint union of
nsimple knotted contours Cj,j51,2...n. The holonomy
operator is the tensor product of those defined on each com-
ponent Cj:
f@A,L#[Pexp
S
i
R
LA
D
5^j51
nPexp
S
i
R
CjA
D
.~21!
Using the same reasoning that was used to derive Eq. ~7!,
we can see that quantum holonomy defined a multicompo-
nent link commutes with any appropriately tensored genera-
tors of the gauge group,
@^i51
nT~i!
a,Z@L##50,
Z
@
L
#
5
^
f~
A
ˆ,L!
&
5
K
Pexp
S
i
R
LA
ˆ
D
L
.~22!
This shows that Z@L#is a commutant of the universal envel-
oping algebra ~of the Lie algebra of gauge group!. The ma-
trix representation of Z@L#is not as simple as that of the
quantum holonomy of a simple ~one-component!knotted
contour, however. If we now evaluate Z@L#in the represen-
tation ^i51
n
r
i, the result will not be a polynomial times the
N-dimensional matrix representation of the unity element,
where Nis equal to the product of the dimensions of
r
i,i
51,...,n:
N5
)
i51
n
N
~
r
i!
.~23!
This is because ^i51
n
r
iis reducible. Suppose this tensored
representation decomposes as
^i51
n
r
i5%j51
m
t
j,N5(
j51
m
N~
t
j!,~24!
where each
t
jis irreducible. Then it follows from Eq. ~22!
that ^i51
n
r
i(Z@L#) will be a diagonal matrix of dimension
N, with its Ndiagonal matrix elements being composed of
possibly mdistinct polynomials, each polynomial repeating
N(
t
j) times.
The above is a general algebraic property of Z@L#, regard-
less of whether the theory is topological or not. For a topo-
logical field theory, the polynomials in ^i51
n
r
i(Z@L#) carry
additional topological information. In the case of the Chern-
Simons theory in three dimensions, the polynomials will
carry information pertaining to Lbeing a member of an iso-
topy class. Let us take the traces of all the representations in
^i51
n
r
iexcept one, say, that of
r
k. Then we obtain the
counterpart of Eq. ~7!:
^
jÞk
Tr
r
j„^i51
n
r
i~Z@L#!…5F~L!1
r
k,~25!
where F(L) is a polynomial of the mpolynomials in
^i51
n
r
i(Z@L#); it is a polynomial for L. One might think that
F(L) would be labeled by
r
k, but it has been shown @19#
that Eqs. ~22!and ~25!are sufficient to prove that F(L)is
independent of the choice of
r
k. Thus F(L) is a uniquely
defined eigenvalue of Z@L#and is a link polynomial for Lon
the set of representations
$
r
1,
r
2,...,
r
n
%
.
An explicit evaluation of Z@L#by the right-hand side of
Eq. ~22!is nontrivial. This contrasts with the rather straight-
forward evaluation of the link polynomial F(L) by algebraic
means. This method, based on long-standing theorems by
Alexander @20#and Reidemeister @21#, begins by making a
planar projection of a link, called a link diagram, in such a
way that the projection is a network whose only nodes are
either one of two kinds of crossings — overcrossing or un-
dercrossing — each being a four-valent planar diagram. To-
pologically equivalent classes of links are classified accord-
ing to isotopic classes of link diagrams. The connection to a
gauge group either through a skein relation @3#, the braid
group @22#, or directly @19#is made by mapping the over-
crossing ~undercrossing!to ~the inverse of!an invertible uni-
versal Rmatrix, which is a construct ~more specifically, a
second rank tensor product!that exists in universal envelop-
ing algebras of Lie algebras such as the Lie algebra of
SU(N).
In Ref. @23#, the emergence of an Rmatrix in Wilson
loops of the three-dimensional Chern-Simons theory was in-
vestigated. It was shown that by choosing a special gauge —
the almost axial gauge — and working in the space-time
manifold S13R2, one can use the technique of standard per-
turbation theory to reveal the assignment of an Rmatrix to
the crossing on a link diagram. Since the trace of the Wilson
loop was actually not taken in Ref. @23#, its conclusions more
appropriately apply to quantum holonomy. However, be-
cause the object of investigation, the link invariant, is a non-
perturbative property of the Chern-Simons theory, the con-
clusion is somewhat clouded through a lack of accuracy
owing to the nature of the perturbation method. Perhaps a
direct nonperturbative evaluation of Eq. ~22!, such as by lat-
tice gauge theory, is called for.
H.C.L. was partially supported by Grant No. 85-2112-M-
008-011 from the National Science Council, ROC, and
W.F.C. is grateful to ICSC-World Laboratory, Lausanne,
Switzerland for financial support.
56 1173QUANTUM HOLONOMY IN THREE-DIMENSIONAL . . .

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W. F. CHEN, H. C. LEE, AND Z. Y. ZHU