1. Introduction
This is an unplanned sequel to [
1,
2]. Let
M be a 3-manifold and
G be a compact connected semisimple Lie group. Without loss of generality we will assume that
G is a Lie subgroup of
,
. We will identify the Lie algebra
of
G with a Lie subalgebra of the Lie algebra
of
throughout this article. Suppose we write
. Then we can define a positive, non-degenerate bilinear form by
for
.
Let be a subspace. The vector space of all smooth -valued 1-forms on a manifold Σ (need not be a 3-manifold) will be denoted by . We will identify the space of connection 1-forms on the trivial principal fiber bundle with group G and base manifold M with .
Denote the group of all smooth
G-valued mappings on
M by
, called the gauge group. The gauge group induces a gauge transformation on
,
given by
for
,
. The orbit of an element
under this operation will be denoted by
and the set of all orbits by
.
For
, the Chern-Simons action is given by
Note that so that is invariant under gauge transformation even though is not.
The interest in Chern-Simons path integrals is the evaluation of Wilson Loop observables, that is we want to compute
where
is a normalising constant.
Here,
is a link in
M with non-intersecting (closed) curves
and
is the Wilson loop associated to
. And,
is some heuristic Lebesgue measure on
,
is the matrix trace for some representation
,
, and
is the time ordering operator.
Note that is the holonomy operator of A, computed along the loop . The integral in Equation (2) will be known as the Wilson Loop observable associated to the link L and q will be called the charge of the link. When L consists of only one curve, the link is termed as a knot.
It was argued in [
3] that if one can make sense of the RHS of Equation (2), then one can define a suitable generalization of the Jones polynomial of the link
L in
M. The objective of this article is to compute the right hand side of Equation (2) for the case of
in the non-abelian case. The case when the manifold is
is also singled out in [
3] and is the next simple case to consider after
.
The main purpose of this article is to define a Chern-Simons path integral in using torus gauge fixing and non-abelian gauge group. We will further show how link invariants appear from these path integrals in the second half of this article.
The case
was worked out by [
1,
2] for the abelian and non-abelian gauge group
G respectively. Using axial gauge fixing, it suffices to only consider connections which are zero in the
z-direction.
Unfortunately, in the case of
, it is not possible to make the connection disappear in the
direction. On our compact Lie group, fix a maximal torus
T and let
be the Lie algebra of
T. Under torus gauge fixing, we can choose the connection such that it takes values in
in the
direction. This was accomplished by Hahn in [
4] and he wrote down an expression for the Chern-Simons path integral in Expression 6. We will try to make sense of this expression instead.
Using local coordinates, we will work on , which we will call it the classical space. The link L is mapped inside , called a truncated link. Now, consider , whereby is a complexification of . We will refer as a quantum space. After `scaling’ the truncated link and embed it inside , the Wilson Loop observable will then be defined on this quantum space. Details to be given later.
Over this quantum space, we will explain how to construct two Wiener spaces. The first Wiener space will be the space of analytic 2-tuple -valued functions over the quantum space. The second Wiener space will be space of analytic 4-tuple -valued functions over the quantum space. The Chern-Simons path integral is defined as an integral over the product space of these 2 Wiener spaces. For the Wilson Loop observable, we will explicitly work out this integral for the truncated link embedded inside the quantum space.
The link invariants that we are interested in will only appear when we take the limit of the Wilson Loop observable as κ goes to infinity. This limit can be computed easily from a truncated link diagram, by projecting L on . By assigning to crossings on this link diagram, we can write down a formula for the Wilson Loop observable directly from this link diagram. Furthermore, we will show that the Wilson Loop observable is equal to a state model for links when the representation is the same for all curves in L.
Two diagrams represent the same link up to ambient isotopy if the 2 diagrams can be obtained from each other by applying Reidemeister moves. It is not true that the state model defines a link invariant. The state model for links has to satisfy certain algebraic equations to be a link invariant, including the Yang Baxter Equation (34). This will impose quantization conditions on the charge q of the link.
As an application, we will work out explicitly for the gauge groups and . We will show that using gauge group , the Wilson Loop observable will satisfy a Homfly skein relation Equation (38), with and . For gauge group , the Wilson Loop observable will satisfy a Conway-type skein relation, with . For both cases, is quantized to take only a discrete number of values.
This article is organized as follows. In
Section 2, we will explain Hahn’s heuristic expression for the Chern-Simons path integral using torus gauge fixing. This section will contain mainly definitions. In
Section 3, we will give a heuristic but equivalent definition, whereby the path integral will be defined on. In
Section 4, we will compute some simple functional integrals, which motivates the definition of the Chern-Simons path integral. This is an extension to the path integral considered in [
1]. In
Section 5, we need to introduce some important linear operators which are necessary in defining the Chern-Simons path integral. In
Section 6, we will give our definition of the Chern-Simons path integral. As an application, we will define the Wilson Loop observable given in Equation (2) and compute it.
The second half of this article concentrates on taking the limit as
κ goes to infinity of the Wilson Loop observable. In
Section 7, we will define a link diagram for a framed link
L. In
Section 8, we will compute the limit of the Wilson Loop observable. In
Section 9, we will obtain framed link invariants in the case of gauge group
and
. We will make some ending remarks in
Section 10.
We end this section by stating some notations which will be assumed throughout this article.
Notation 1. Suppose we have two Hilbert spaces, and .
We consider the tensor product .
The inner product on the tensor product is given by This definition of the inner product on the tensor product of Hilbert spaces will be assumed throughout this article.
Now consider the direct product .
The inner product on is defined by This definition of the inner product on the direct product of Hilbert spaces will also be assumed throughout this article.
If , we abbreviate by writing .
Finally, we always use to denote an inner product.
2. Some Definitions and Notations
From this point onwards, we only consider the 3-manifold . On , fix a north pole and let the south pole sit on the origin of . We use the stereographic projection as local coordinates. Let be local coordinates on .
On , let denote the mapping and we set . The restriction of onto , which is a bijective mapping , will also be denoted by and its inverse will be denoted by . The tangent vector of at the point , induced by the curve , will be denoted by , for . Finally, will denote the vector field on given by for and , the real-valued 1-form on is dual to .
For the rest of this article, instead of working in , we work in local coordinates . All the formulas in the sequel will be written using these local coordinates.
2.1. Quasi-Axial and Torus Gauge Fixing
Let
be the vector space of (smooth)
-valued 1-forms on
. We further impose the condition that it vanishes at infinity. Now, we write
, where
For every , and will denote the unique elements of , respectively such that holds. For a given , we set , i.e., is the element of given by .
Let
T be a maximal torus of
G and denote the Lie algebra of
T by
. An element
will be called “quasi-axial” (respectively “in the
T-torus gauge”) if the functions
,
are constant (respectively constant and
-valued). We will denote the set of all quasi-axial elements (respectively all elements in the
T-torus gauge) of
by
(respectively
). Thus, we have
The following proposition is Proposition 5.2 taken from [
4], the proof is omitted. We present the proposition using local coordinates
X.
Proposition 1. Let and let and be given by .
Then we have Definition 1. (Regular elements) Let denote the set of regular elements of G, i.e., the set of all which are contained in a unique maximal torus of G. Similarly, let denote the set of regular elements of , i.e., the set of all which are contained in a unique maximal Abelian Lie subalgebra of . We set .
It is not difficult to see that (resp. ) if and only if the set of fixed points of (resp. the kernel of ) is a maximal Abelian Lie subalgebra of . Thus, .
Hahn in [
4] was able to write 2 expressions for Expression 2 on the subspace
and
. Let
be a link. Using quasi-axial gauge fixing, we have the following expression taken from Equation (6.3) in [
4], (∼ means up to a constant.)
where
is the informal “Lebesgue measure” on
and
is the informal Lebesgue measure on
. For
,
and from Proposition 1, we will write
with
denotes the bilinear form on the vector space of smooth
-valued 1-forms on
,
, given by
for
A,
. Similar definition for
, with
A,
. Here,
is the vector space of
-valued 1-forms on
. Finally, for
, we have
Here, is viewed as an operator on .
Definition 2. (Maximal Torus)Let T be a fixed maximal torus of G. The Lie algebra of T will be denoted by . Moreover, we set and . Note that .
Let denote the scalar product on and let be the Lie algebra of T. Let be the orthogonal complement of in .
Suppose we write
, whereby
Here, is the projection operator onto the second term in the direct sum . And, (respectively )denotes the vector space of -valued (-valued) smooth 1-forms on .
Let
,
. Note that
. For
, we have the following torus gauge analogue of Equation (4), taken from Equation (6.6) in [
4],
where
and
is a link. In this case, do note that
with
denoting the “Lebesgue measure" on
. Now,
is dense in
T and since
is a local homeomorphism, we can conclude immediately that
is dense in the vector space,
. Thus, we will in the rest of the article, replace
with
in Equations (6) and (7).
And with ∼ denoting equality up to a multiplicative constant independent of
B,
where the operator
in the numerator is defined on
. For
,
whereby a direct calculation using Equation (5) gives
We refer the reader to [
4] for the derivation of these expressions as our main focus in this article is to make sense of Expression 6. For ease of notations, we omit
κ on the RHS of Expression Equation (6), but the reader should note its dependence on
κ.
2.2. Infinite Dimensional Determinant
Let us first digress a little and discuss the function
Y, which is defined as
Note that is skew symmetric and thus the operator is unitary on , thus it is not a compact operator and hence is not trace class. Therefore we cannot define as a Fredholm determinant.
Alternatively, we can interpret
as a product form,
i.e.,
where
is a suitable measure on
. More precisely, we should have
, where
. This suggests the following heuristic formula
whereby
is some orthonormal basis in
and
is Lebesgue measure on the subspace spanned by
. However, the term
is still ill-defined and we need to resolve this.
Note that
is a skew symmetric operator,
i.e.,
. Let
N be the dimension of
and
be an orthonormal basis in
, and
is simultaneously diagonalizable. Suppose that
are the complex eigenvalues of
and let
be an orthonormal basis in
. Then we write for
,
where
,
is Lebesgue measure. That is, we interpret the determinant as an infinite product.
Unfortunately, the infinite product given in Equation (9) converges to 0. Furthermore, if we use Definition 8, we observe that the normalizing constant in Equation (6) can be shown to be 0. See Remark 4.
As such, we will drop the term in future for reasons cited above. Another reason for dropping this term is that we really do not need this term to define the link invariants in the second half of this article.
3. Heuristic Argument
Notation 2. Throughout the rest of this article, we adopt the following notation. For , we let , which is a Gaussian measure with variance . And let .
We let denote the 2-tuple ,
are integers with .
And we write .
For ,
.
Let denote the set of all such 2-tuples, i.e., Let .
Consider the Schwartz space , with the Gaussian function , . The inner product is given by , λ is Lebesgue measure on . Let be the smallest Hilbert space containing , using this inner product.
The Hermite polynomials form an orthogonal set on with the Gaussian measure . Let , with , be a product of Hermite polynomials and .
We have the normalized Hermite polynomials
with respect to the Gaussian measure
. Then
is an orthonormal basis for
.
Definition 3. Define a transformation , and . Thus and . Observe that for any κ and that .
For each Hermite polynomial
, we will define a function
on
. Define
Note that approaches 0 as or .
Now define a real subspace
, spanned by
. We make
into an inner product space by defining an inner product,
whereby
f and
g are polynomials. Complete the inner product space
into a Hilbert space, denoted by
. Clearly,
is an orthonormal basis.
Remark 1. We remark that the constant function 1 is not inside . Furthermore, it is not necessary to consider all the functions on . To obtain the link invariants later, is good enough for our consideration.
Definition 4. Let be the smallest Hilbert space containing .
We define and We will write and .
Observe that .
With this definitions, we are going replace Equation (6) with
Henceforth, we will try to make sense of the RHS of Equation (10).
Remark 2. Note that we replace and to be -valued forms instead of -valued forms.
Let
be the
q exterior power of the cotangent bundle over
. Let
denote the space of
sections in
. We use local coordinates
. To define an
space on the space of
q-forms, we have to introduce a metric
g on
. We pick the standard metric
. This metric defines an inner product on
which we denote by
and we can define a volume form
. (See [
5] for details.) Therefore, we can define a Hodge star operator * acting on
k-forms,
, such that for
,
Note that because
, we have that
if
;
if
. We define an
inner product on sections of real-valued q-forms,
by
By the choice of the metric, note that
is a sub Hilbert space inside
. Let
Then, is a Hilbert space.
Now, there are 2 Hilbert spaces and that we need to consider for the Chern-Simons integral, which we will each make into a direct product , for .
The first Hilbert space
is
. Take the direct product
. This is similar to the construction used in [
1].
The second Hilbert space that we need to consider is . Now we need to take the direct product , which is isomorphic to the direct product of 4 copies of .
3.1. Heuristic Argument
Lemma 1. Now write .
Then, Proof. Because
is an anti-symmetric operator, we have
☐
Thus, from Equation (8), we can write
Here and what follows, will always denote an inner product in a Hilbert space.
Definition 5. (Orthonormal basis )
The orthonormal basis in ,
will be fixed throughout this article. Let ,
and .
We let denote a -
valued inner product, i.e., Refer to Definition 4. We will now give a heuristic argument for Expression 10. Let
δ denote the Dirac delta function and for
, we write
Let
,
and
, with
. Define
We will also write
. Denote
For simplicity, we want to make sense of
with
Z is a normalizing constant. Note that
is defined by Equation (12). As discussed in Subsection 2.2, we drop the term
Y in Equation (10).
Write
. Then
, so
Note that we make use of the fact that is a skew symmetric operator, so .
Now we make the following substitution
. The Jacobian factor is
, thus Expression 13 becomes,
where
Now,
and
. With this new notation, we can write
using Stokes’ Theorem and Equation (11). So if we make the substitution
, then
and
up to some constant.
Thus, the path integral, up to a constant, can written in the form
We wish to point out that this integral in Expression 15 is of the form
where
is a linear operator that maps
,
and
,
.
Thus, our goal is to give a sensible definition for Expression 16. From Expression 15, we also need to define
Unfortunately, the Dirac delta function δ is not inside . Therefore, the term is ill defined. Furthermore, the operators and will be shown later, to be only defined on a dense subspace of and respectively. Hence these operators do not operate on the Dirac delta function.
We would like to end this section by saying that to define the path integral, we will need the following inputs, namely , and . And in Expression refe.x.1 is given by , which will be defined later.
The reader should think of
, and similarly,
The path integral is simply an integral over the product space , which we will define in the next section.
4. Functional Integral
Consider the real Hilbert space spanned by
, integrable with respect to the Gaussian measure, equipped with a sesquilinear complex inner product, given by
Note that
means complex conjugate. Denote this Hilbert space over
, by
. An orthonormal basis is given by
Let be the smallest Hilbert space containing .
It is well-known that there is no sensible notion of Lebesgue measure on an infinite dimensional space. Our next strategy will be to define a Gaussian type of measure on . Unfortunately, this space is too small to support a Gaussian measure.
Let
,
. Introduce a norm by setting
Here, is the ball with radius , center 0 in . Note this norm is weaker than the norm in .
Using this weaker norm, complete
into a Banach space
B. In [
1], it was shown that one can equip
B with a Gauss measure
, with variance
. Identify
and denote the pairing
.
The space
B can be described explicitly. Let
and
. In [
1], it was shown that
Proposition 2. The support of in the Banach space B is the space of holomorphic -valued functions on .
Let and define an evaluation map, . Then is in .
Remark 3. Note that χ will play the role of the Dirac delta function discussed earlier. The advantage of this is that now .
Notation 3. We denote the Abstract Wiener space containing H by B, with Gauss measure , variance . If is an Abstract Wiener space, then .
Definition 6. Recall in Section 3, we said that there are 2 Hilbert spaces that need to be considered for the path integral. Instead of considering the space of Schwartz functions, we will replace it by considering the Hilbert space and complete it into an Abstract Wiener space, denoted by ,
with Wiener measure .
Consider the Hilbert space ,
which is the smallest Hilbert space containing .
In a similar way, we can construct an Abstract Wiener space containing it, denoted by ,
with Wiener measure .
There are two Abstract Wiener spaces that we will consider in this article, necessary for the definition of the path integral; Consider the tensor product and complete it into an Abstract Wiener space, denoted by . The Abstract Wiener measure will be the product measure , N copies in total, N is the dimension of .
Consider the direct product , and complete it into an Abstract Wiener space, denoted by . The Abstract Wiener measure will be the product measure , copies in total.
Let
H be any Hilbert space and
. The following expression,
with
is the basis whereby the Chern-Simons path integral is build upon. We define Expression 19 as
with
Suppose
B is an Abstract Wiener space containing
H. Now, one can show that there exists a complex measure
on
, such that
is a probability measure on
and we can define for
,
Furthermore, it can be shown directly that
Thus, using analytic continuation, we will define
The reader may refer to [
1] for details.
We can now give a definition to the heuristic Expression 19.
Definition 7. Let and .
Then we define We remark that is not taking expectation, but rather it should be viewed as a linear functional acting on functions of the form .
Let
. One can show that for any polynomials
, we have
Thus, we can extend Definition 7 to include polynomials.
However, given a general (continuous and bounded) function
F on one variable, then it is not clear that
admits an analytic continuation. However, from the above calculations, it is possible to extend Definition 7 to include
F.
Definition 8. Let ,
with for .
Let be continuous functions on and Y be continuous on ,
with for any ,
such that .
Then we define if Remark 4. Recall that Y in Subsection 2.2 is defined as an infinite dimensional determinant. If we use Definition 8, notice that . Hence the normalizing constant is 0. As such, we have to remove the term Y in order to obtain non trivial results for the path integral.
Notation 4. Let B be an Abstract Wiener space containing the Hilbert space H, equipped with inner product .
We will also write for and is an orthonormal basis in .
Let , be 2 Hilbert spaces and , be Abstract Wiener spaces containing them respectively. For any , let be a linear operator that maps to . Recall the path integral we want to make sense of is given by Expression 16.
Proposition 3. Refer to Notation 4. Let ,
.
For any ,
let be a bounded linear operator. Using Definition 8, we define Expression 16 as with
,
.
Using the definition again, we have
☐
The 2 Abstract Wiener spaces,
and
we have in mind, are defined as follows:
5. Linear Operators
Refer back to Expression 15. If we wish to apply Proposition 3, then we have to define linear operators and , . But does not map into . And for any , . Thus it seems that we are not able to apply Proposition 3.
However, as long as we can make sense of the RHS of Equation (20), we can define the Chern-Simons path integral. If one goes back to Expression 15 and compare with Expression 16, what we really need to define are the terms
Once we can define these terms, we can proceed to define our Chern-Simons path integral.
Now, we define the path integral as a linear functional, on the direct product of 2 Abstract Wiener spaces, . The operators and act on a dense subspace in and respectively. We need to transfer these operators to act on and . To do this, we need to construct an isometry between these Hilbert spaces.
Fortunately, there is a natural map, the Segal Bargmann transform
, that sends
In the sequel, we will extend this definition to tensor products or direct products of hermite polynomials.
For example, on the tensor product space
, we have
, by
Recall , so we have for .
Definition 9. Let .
We define an operator acting on by We leave to the reader to check that For ,
we define an operator by Recall we have the exterior derivative d and the Hodge star operator acting on .
We define a linear operator by For ,
we define an operator by
Remark 5.
When ,
then .
The operator and the operator which appeared in [1] differs by a factor 2, i.e., .
By their definitions, it is clear that
and
. However, to define the path integral in Proposition 3, what we really need to define is
where
.
Definition 10. We define for and ,
Remark 6. It is possible that the integrals might not be defined. However, as we will show later, for our choice of α and β, the integrals are well-defined.
Recall we define the evaluation map
in Proposition 2. The linear functionals
,
and
, we have in mind are of the form
where
,
and
. Next, we need to know how to compute
.
Proposition 4. For each ,
Proof. We will leave to the reader to check that
Now
maps
to
for
,
s real, which upon simplification gives
☐
Here, is the usual scalar product in .
Notation 5. Define for ,
Note that and ,
so we define .
We will also write Finally, for ,
,
define Corollary 1. Under the isometry ,
and whereby and .
Lemma 2. Suppose whereby ,
and .
For ,
we have Proof. Using Definition 10 and Corollary 1,
☐
Lemma 3. Suppose for ,
whereby and .
Then, we have for ,
Proof. Using Definition 10 and Corollary 1,
☐
6. Definition of the Chern-Simons Path Integral
Here, we have
,
,
, where
for
,
. We want to give a definition for Expression 13, with
and
defined above. In Subsection 3.1, we also showed that Expression 13 can be written heuristically as Expression 15. By replacing the Dirac delta function
by
in Expression 15 and applying Proposition 3, we have the following definition.
Definition 11. (Chern-Simons Path Integral)
Refer to Definition 10. Write .
Applying Proposition 3 to Expression 15, we define Expression 13, with and defined as above, as Using Lemmas 2 and 3, the exponent can be explicitly computed as
with
Definition 12. (Time ordering operator)
For any permutation ,
Suppose now our matrices are indexed by the curves k and time s. Extend the definition of the time ordering operator, first ordering in decreasing values of k, followed by the time s.
Definition 13. ()
Define a linear functional as follows. Suppose a matrix A is index by time s and representation ,
.
In other words,
.
Let be a finite set of matrices. Let and write .
For any ,
define a linear operator, such that for each ,
and for .
Let us apply Definition 11 to the Wilson Loop observable, given by Equation (2).
Notation 6. Suppose is a link in such that the projected link on does not pass through . Using local coordinates , we map L into . Let be a parametrization for , such that , hence giving an orientation to each curve. In components with respect to the local coordinates , we have . We will also write . Without loss of generality, we also assume that for only finite number of points s in .
Next, we map inside ,
by As a result of this scaling, we represent our original link as a set of (possibly open ended) curves.
For each curve , let be a representation for .
Remark 8. The scaling of the curves by was carried out in [1]. We will now define the Wilson Loop observable on the set of curves
, in
. We will scale the integrand by
, which was also carried out in the case of
and hence interpret the line integral in Equation (3) as (See also Subsection 10.1.)
whereby
,
, and
Note that k tracks the representation used and s tracks the ordering of .
Corollary 2. Refer to Notation 6. Consider the 3 manifold .
Let ,
and denote Also denote Apply Definition 11, the Wilson Loop observable, with a charge q, is defined as where Note that is dependent on κ, but we omit κ to ease the notation.
Proof. Observe that we can commute
and
because the time ordering only acts on the matrices
and
. Note that the time ordering operator
arranges the matrices according to
j, followed by
s. Now apply Definition 11, by replacing the finite sum in the definition by an integral, using a Riemannian sum type of argument. See [
1] for such an argument. To obtain the RHS of Equation (24), we apply Lemmas 2 and 3. For more details, we refer the reader to [
2]. ☐
Equation (24) will not give us the link invariants we desire, as the path integral depends on the parametrization used. And the path integral depends on the parameter κ as we used the parameter κ in constructing the isometry .
To obtain the link invariants, the rest of this article will focus on computing the limit as κ goes to infinity, of the RHS of Equation (24). It is only by taking the limit as κ goes to infinity that we will obtain the desired link invariants, independent of the parametrization used.
7. Planar Diagrams
Definition 14. (Framed link)
Let be a link in . Define a continuous normal vector along each closed curve such that is nowhere tangent to . Let be a new closed curve obtained by shifting in the direction , is some small number. Now, forms a closed thin band or ribbon, whereby a finite number of twists can be added. We will write to denote a framed link, .
The Wilson Loop observable can be computed from a link diagram in . Up to isotopy, we insist that the truncated link, is embedded “nicely” inside and thus projected “nicely” onto , so that we get a nice planar diagram . The following definition makes this `nice’ embedding and projection more precise.
Definition 15. (Planar Diagrams)
Assume that a link is made up of individual closed curves that do not intersect one another, i.e., for any j, k and when projected onto , the curve does not pass through the north pole . Using local coordinates , we map the link into , denoted by . We will refer it as a truncated link. Project the truncated link onto the plane using the projection map .
Parametrise each curve by such that , hence giving an orientation to each curve. Without loss of generality, we also assume that for only finite number of points s in .
Note that in the following definitions, it applies if L is just a knot, i.e., .
We define a truncated link diagram for on the plane if the following conditions are met.Define a standard projection of the truncated link onto if the following conditions are satisfied: for any , intersects at most 2 distinct arcs in L. We say p is called a crossing if intersects exactly 2 distinct arcs in .
at each crossing , there exists an such that for all and , the tangent vectors and are linearly independent at p. Furthermore, we also assume that and in a small neighborhood containing and respectively.
Denote the set of crossings between curves and by .
And will denote the set of crossings in .
We will write to denote the set of crossings of the standard projection of the truncated link onto .
For each curve , write the interval as a union of intervals , where in each interval , is a bijection. Write be the image of the interval under . Without loss of generality, further assume the image contains at most one crossing which is an interior point in .
Given 2 arcs which intersect at p, define to be the sign of the determinant of the Jacobian at the crossing . Otherwise, define it to be zero if the 2 arcs do not intersect at all. We will also write , and call this the orientation of p.
Using the same notation as the previous item, for each crossing ,
define If the 2 arcs do not intersect, set it to be 0. We will also write and call this the height of p.
For each crossing ,
the algebraic crossing number is defined by This is actually well defined on an oriented truncated link diagram, independent of the parametrization used.
Remark 9. The sets and only make sense for a truncated link diagram in . Different link diagrams on will give different sets of crossings.
We can also represent a truncated link diagram with a graph, which would be more convenient to use in computing the Wilson Loop observables in the next section. The vertices will represent crossings on a link diagram.
Definition 16. (Edges.)
Let be a link in .
Let be a parametrization for and project it down onto ,
forming a planar graph. Refer to Definition 15. The vertex set will be the set of crossings in . The terms vertices and crossings are used interchangeably. The set of edges is simply the set of lines in the planar diagram of L joining each vertex. Each edge , . The end points will be a vertex or crossing in . Each vertex has 4 edges incident onto it.
Fix a j. For each crossing p in , , let be the set of all such points p.
Suppose . Let be the set of all such p’s on a planar diagram of .
Define , which defines the vertex set of the graph . The set of edges , is a subset of , joining only vertices in . Note that .
Suppose e and belong to . We say that an edge precedes another edge if .
Each crossing is denoted by 4 edges, labeled by , whereby and are edges belonging to with the bigger index j and is the edge that precedes at the vertex p. When all 4 edges belong to the same curve, then and are the edges that precede and .
Now suppose we define a frame on and project the framed oriented truncated link onto . The crossings in the planar diagram will define the set of vertices as in the case of an oriented link. A half twist q will be represented by a vertex with only 2 edges incident onto it, labeled . Thus, a full twist, given by 2 consecutive half twists, twisted in the same direction, will be represented in the planar graph of the curve by 2 vertices, joined together by a common edge.
Remark 10. For a half twist q, we can define an algebraic number associated to it. A positive half twist is given an algebraic number +1; a negative half twist is given an algebraic number –1. We refer the reader to [2], whereby there is a discussion on how to define the algebraic number of a half twist in a framed link. In the next section, we will show how to calculate the Wilson Loop observable using the graph of a framed truncated link diagram.
8. Wilson Loop Observables
Let L be a link in and using local coordinates, we represent each component of the link L by .
Recall from Corollary 2, we have the double sum in the exponent,
where
was defined in Equation (25).
Lemma 4. We have as .
Furthermore, Proof. Using Item 2 from Lemma 5,
as
. From Definition 9,
By Item 1 of Lemma 5, we have
as
and this completes the proof. ☐
As a result of Lemma 4, the limit of the RHS of Equation (24) is equivalent to compute the limit of
Notation 7. For , let × denote the cross product and let denote the a-th component of , .
Using the fact that
,
are skew symmetric operators, it is straightforward to show that
and
whereby
,
is the Kronecker delta function.
Our next task is to compute the limit of Equations (26) and (27), as κ goes to infinity. We break up the computations into 2 simple lemmas.
Lemma 5. We have Let .
Then, and
Proof. We will prove (1) first. Make a substitution
Note that for any
,
as
. By definition of
in Equation (21),
And by definition of
in Definition 9,
The last step requires the following explanation. Note that
is bounded and there exists a small neighborhood
,
δ small enough, such that for all
,
for any given
ϵ.
Using Notation 5, we have
. We will only prove (2) for +, the other case is similar. Note that
A direct computation will give
and
To prove (3) and (4), a direct computation gives
and
☐
The following lemma is is similar to Lemma 4.5 found in [
1]. Note that there should not be a negative sign in Lemma 4.5.
Lemma 6. Refer to Definition 15. For ,
The proof is similar to the proof for Lemma 4.5 in [
2], so the proof is omitted.
When
, we have a problem with the following expression,
i.e.,
do not exist.
The solution as explained in [
3], would be to consider a framing
whereby
is a normal vector field along the curve
that is nowhere tangent to
. Define
,
ϵ is some small number,
i.e.,
is a parametrization of the shifted curve
in the direction
. The limit in Expression 28 is now defined as
The framing on the curve
will give rise to half twists. Using Lemma 6, one can show that the limit of Expression 29 can be written as a sum of the algebraic numbers of crossings and half twists, which form on the curve
. We refer the reader to [
2] for the details.
We now focus on Expression 27. Unfortunately, the limit
is not well-defined as
κ goes to infinity. The limit, if it exists, will depend on the ambient isotopy of the link. This is similar to the self-linking problem.
Using Lemma 5, it is straightforward to show that this limit is equal to 0. Thus, the Expression 27 is defined as 0.
Definition 18. Given ,
an oriented framed link in ,
map it into using .
Let be any parametrization of ,
whose image is then projected down onto the plane to define a graph as in Definition 16. And let be the dimension of each representation and be the maximum of all the ’s.
Let N, be any positive integers. Define with , with . For , the components are given by with respect to the basis of .
Denote a map such that for each k,
Let denote all such mappings.
We are now ready to state our formula for the Wilson Loop observable in Equation (24), in the limit as κ goes to infinity.
Theorem 1. Let be an oriented link in which when projected down on , does not pass through . Choose a framing for L. Map it into using and project it onto .
Suppose for each curve , we assign a representation to it. Refer to Definitions 5, 15, 16 and 18. For , , the usual matrix multiplication.
Given any gauge group G with its complex Lie algebra ,
the Wilson Loop observable in Equation (2), as κ goes to infinity, is given by If ,
with and ,
then If ,
with ,
then Note that ε is the algebraic crossing number and was defined in Definition 15. See also Remark 10.
Notation 8. Suppose for all l,
for some representation. Denote When the representation is clear, we will drop the subscript ρ.
Proof. Because of Lemma 4 and because Expression 27 is defined as 0, it suffices to compute the limit of
To compute Expression 31, we note that it suffices to consider a framed truncated link diagram which is projected on plane. Using Lemma 6, the exponent will be given by a sum of terms, each involving a crossing or half-twists. We also note that we will have a problem when , whereby we have to consider Expressions 29 instead.
The rest of the proof now follows similarly to the argument used in
Section 2.1 in [
2].
9. Σ-Model
Equation (30) defines a -valued map on a framed link diagram . From the definition of , it is clear that it is invariant under ambient isotopies.
Notation 9. Let . Fix a N. Recall that given , the components . The upper indices a and b refer to the rows, the lower indices c and d refer to the columns.
Definition 19. (State model for framed truncated links.)
Fix a natural number N. A state model of type is given by ,
with ,
.
For every state model of type ,
there is a unique -
valued mapping on the set of framed truncated link diagrams in for every framed link L in ,
such that holds.
When for some representation ρ, Theorem 1 says that the Wilson Loop observable defines a state model on the set of framed link diagrams.
Two framed truncated link diagrams in are equivalent if they can be obtained from the other by the Reidemeister Moves. Fortunately, there are algebraic conditions on and which tells us when is invariant under the 3 Reidemeister moves.
Proposition 5. (Reidemeister Move I’.)
is invariant under Reidemeister Move if for all a,
c.
Proposition 6. (Reidemeister Move II.)
is invariant under Reidemeister Move II if for all a, b, c, d. There are 2 Reidemeister Moves II, the other obtained by reversing the orientation of one strand, keeping the orientation of the other fixed. In this case, the equation becomes .
Proposition 7. (Reidemeister Move III.)
is invariant under Reidemeister Move III if for all i,
j,
k,
l,
m,
n.
Finally,
satisfies the skein relation with parameters
α,
β and
γ,
for all
,
and
as in [
6] if
for all
a,
b,
c,
d. Note that this is a correction to Equation 14 in [
2].
Definition 20. (Special elements in .)
Define I,
J,
K in ,
K commutes with J. Note that if is the identity matrix in ,
then .
Furthermore,
,
and .
We will now present the corrected version of 2 examples taken from [
2].
Example 1 (SU(N))
Suppose our Lie group is
. Considering its standard representation, one shows that
. Hence,
Let , thus satisfy the Reidemeister Equations (33) and satisfy Reidemeister Equation (32) for any values of λ. It will satisfy Equation (34) if λ is an integer or half integer.
If we solve Equation (36), we get
and hence
Therefore,
satisfy a Homfly polynomial skein relation
with parameters
and
. Compare with the Jones polynomial skein relation, given by
Let us summarize the result as a theorem.
Theorem 2. Consider the standard representation of and let q be the charge of the link. Then the Wilson Loop observable in Equation (30) can be written as a state model of a framed link. If is an integer or half integer, then defines a framed link invariant. Furthermore, satisfy a skein relation Equation (38).
Example 2 (SO(N))
Now consider
. Considering its standard representation, then
. Hence,
Write
. Note that
satisfy the Reidemeister Equation (33) and
satisfy Reidemeister Equation (32) for any values of
λ . Equation (34) will be satisfied in any of the following 3 cases:
Now, solve Equation (36), we get
Compare this with the skein relation for the Conway polynomial,
The only interesting case would be when N is a multiple of 4 and is an odd integer. Then the state model would satisfy the skein relation Equation (39) for the Conway polynomial, with .
10. Final Comments
We would like to end this article with a few comments.
10.1. Normalizing Constants
In the definition of the line integral in the Wilson Loop observable, we scale
by
and
,
B with
. The factor
ψ is required to obtain non trivial results when we take the limit as
κ goes to infinity and this scaling was also done in [
1]. But in that article, we notice that the constant was
.
Now, the factor
κ is put there for technical reasons. The thing we want to address is the discrepancy in the constants
and
. In fact, there is no discrepancy as the operator
used in [
1] is actually twice of the operator
defined in this article. If we had used the operator
instead in this article, then we would use the normalizing constant
, instead of
.
Finally, we would like to point out that the normalizing constants were specially chosen so that the
R-matrices obtained in Theorem 1 will be consistent with the
R-matrices obtained in [
2].
10.2. The Solid Torus
Consider the solid torus , where is the open disc of radius 1 in . Given any link embedded inside T or on the surface of a torus, we may as well assume that it is embedded inside . Now the open disc is homeomorphic to , so we can map the link into .
However, we wish to point out that quasi axial gauge or torus gauge fixing may not apply to . We will not address this issue here. Instead we will use the RHS of Equation (10) as the heuristic expression for the Wilson Loop observable, for a manifold of the form , where Σ is any simply connected Riemann surface.
Hence we can apply the results in this article, define and compute the Wilson Loop observable and obtain link invariants for a link embedded inside . In particular, we can define the link invariants for a link embedded inside the solid torus T.
10.3. The W Polynomial
The Wilson Loop observable in the case of , was meant to give us the Jones Polynomial of a link. However, the Wilson Loop observable gives us a number. So how does one even obtain a polynomial invariant out of it?
Firstly, we have shown that the Wilson Loop observable is an invariant for a framed link. Secondly, the Wilson Loop observable yields for the unlink with n number of components.
Let us go back to our
example. Now, when
is an integer,
and thus the state model just yield us
Here, is the sum of the algebraic numbers of all the crossings, and is the sum of the algebraic numbers of all the half twists in L.
The more interesting case is when
is a half integer. In this case,
and the state model will yield a polynomial
, whereby
Thus, the Wilson Loop observable defines a polynomial for a framed link L.