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Some geometric and topological implications of noncommutative Wilson loops are explored via the Seiberg-Witten map. In the abelian Chern-Simons theory on a three-dimensional manifold, it is shown that the effect of noncommutativity is the appearance of

Noncommutativity of spacetime has strongly attracted the attention in the last two decades (see, e.g., [

Under such a map gauge fields are written as an infinite series on the noncommutative parameter

It is well known that the Wilson lines and loops are very useful in the description and computation of some nonperturbative aspects of gauge theory just as confinement [

Noncommutative gauge theories have very striking topological and geometrical features. For instance, in string theory, in the absence of

The Wilson loops possess by themselves some interesting geometrical and topological properties; for example, in the abelian case, they can be regarded in terms of the linking number [

In the present work we explore some geometrical and topological aspects based on the previous ideas but immersed in a noncommutative space using the Wilson lines constructed by means of the gauge field provided by the Seiberg-Witten map. As our main result, we will see that in this context it is possible to establish a correspondence between the terms of a power series (in the noncommutative parameter

The linking number is ordinarily a topological invariant; now the noncommutative linking numbers considered here will represent a topological invariant of the corresponding more general noncommutative topology. Thus, the arising extra terms and their involved mathematical structures deserve a detailed mathematical and physical interpretation and further analysis. In the present paper we will restrict ourselves to compute the first order noncommutative corrections to the linking numbers. This should be considered a first step of a description of the subject.

To explore how the modifications to topology of knots are immersed on the noncommutative space, we consider the abelian Jones-Witten polynomials in the path integral formalism which are given in terms of Wilson loops. We will show explicitly that the polynomials are changed due to noncommutativity already at the first order; there will be a nonvanishing and nontrivial modification of the linking numbers due the noncommutative generalization of the notion of topology.

In order to explore our proposal in a more detailed way we consider the application of the noncommutative Wilson loops to the Aharonov-Bohm effect. Some literature on the noncommutative Aharonov-Bohm effect and its relation to the Landau levels can be found in [

It is worth mentioning that the Wilson loops have been used in the description of some quantum theories of gravity. Some of these results can be found in [

This paper is structured as follows. In Section

Our aim is not to provide an extensive review on the Seiberg-Witten map [

We are interested in noncommutativity utilizing the Seiberg-Witten map [

The central idea is to deform the algebraic structure of continuous spaces in particular the polynomials in

Explicitly this modifies the way we multiply polynomials and in general functions over the noncommutative variables in terms of the commutative variables through the Moyal

In this context the gauge transformations of a matter field

The covariant derivative is defined by

For example, up to first order in

For the gauge field we expand again in orders of

The usual Wilson loop is given by the following expression:

The

For the abelian case the Wilson loop can be written as follows:

As an example let us compute

It is well known that on a three-manifold

Since

In the previous integral we integrate out by parts and in this way we can regard

Now we will consider the noncommutative case; then the phase in the noncommutative Wilson loop assuming the expansion on

The figure accounts the computation at the first order in

Let us introduce the following notation:

Now consider the second order term and reordering the expression

Finally in general for the

The Jones polynomials in Witten’s path integral formulation are given by the correlation functions of Wilson loops

We will consider the simplest case when the gauge group is

First of all we change in the path integral

Now we consider the second integral which is rewritten as

In order to integrate out the path integral measure is decomposed into

Moreover, it is necessary to introduce some extra conventions. Let

Now consider a basis of transverse

In a similar spirit we compute the first order

Therefore, (

In order to integrate out expression (

Finally the third contribution in (

This section is devoted to explore some physical applications of the noncommutative Wilson loops and linking numbers; in particular we consider the Aharonov-Bohm effect which is a very good arena to test the physical ideas and extract visible effects. We are aware that some noncommutative extensions of the Aharonov-Bohm effect are present in the literature; see, for instance, [

In the usual Aharonov-Bohm effect it is assumed that the wave function is of the form

In the usual Aharonov-Bohm effect the potential outside the solenoid is given by

Now we proceed to show that the phase of (

Let us construct the creation-annihilation operators as

To estimate the order of

In this paper we proposed to use the gauge field provided by the Seiberg-Witten map to study noncommutative Wilson loops. After a brief account on noncommutative Wilson loops, we study abelian Chern-Simons theory on a three-dimensional manifold. It was shown that the effect of noncommutativity is the appearance of

Furthermore as a topological application of the noncommutative gauge theories and Wilson loops in the abelian case, by using the path integral formalism and the Chern-Simons theory we computed the first order and nonvanishing correction due to the noncommutativity of the abelian Jones-like polynomials (

Furthermore as a physical application we compute explicitly the abelian Aharonov-Bohm effect in

It could be interesting to explore some geometrical aspects; we might extend the linking number between knots in the three-dimensional Euclidean space in a noncommutative sense and explore the different orders of the gauge potential and how they could give new information about linking numbers.

Moreover we would like to extend the present noncommutative ideas to higher-dimensional theories, through a BF theory since it is a higher-dimensional generalization of Chern-Simons theory and the Wilson line will be interpreted in terms of linking numbers between higher-dimensional objects [

Wilson loops for the spin connection are very important in some theories of quantum gravity [

As a further step we are interested in the natural extension to the nonabelian case, where we will deal with two expansions: the first one focusing on the noncommutative parameter and the second one being due to the nonabelianity of the Chern-Simons theory. Also we will study the physical implications using nonabelian Aharonov-Bohm effect [

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work of H. García-Compeán was partially supported by the CONACyT research Grant: 128761. The work of O. Obregón is supported by PROMEP, CONACyT, and UG grants. In addition the work of R. Santos-Silva was partially supported by a PROMEP and CONACyT postdoctoral fellowship.

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^{∗}-products, Wilson lines and the solution of the Seiberg-Witten equations