^{1}

^{2}

^{1}

^{2}

^{3}.

We advocate and develop the use of the dreibein (and the metric) as prepotential for three-dimensional SO(

In this paper we advocate and develop the use of the dreibein or triad (the 3D version of the vielbein) as the basic variable or prepotential for Yang-Mills theory. The first clear proposal to use the dreibein came from Haagensen and Johnson [

Our formulation is useful in several ways:

The metric arising out of the dreibein is gauge invariant, and so the topological properties of the field configurations which we link to it are also gauge invariant. The Yang-Mills action is like

Our formulation can be useful for a nonperturbative understanding of Yang-Mills theory. We find that the topological, monopole configurations (which may drive confinement) correspond to conformally flat metrics. In our earlier works [

There has been extensive interest in obtaining an ansatz for the Yang-Mills potential that exhibits the topological aspects [

Our techniques can be extended in a straightforward way to SO(

Ever since the proposal of nonabelian gauge theory, the similarity with Einstein gravity has led to extensive work exploring the relationship between the two. Examples of this on the gravity side include the Chern-Simons formulation of 2 + 1 gravity [

The paper is arranged as follows. In Section

Consider the set of nine first-order partial differential equations (see, e.g., [

It may be helpful to link the above with Cartan’s structure equations of general relativity. Cartan’s first equation,

Equation (

In 3 space dimensions, the Riemann curvature tensor can be completely expressed in terms of the Einstein tensor:

Equation (

When the matrix

To sum up, (

When

In [

Now

First we note that operating on (

We have demonstrated that, on fixing the boundary condition, a unique torsion-free dreibein

It is expected that topological degrees of freedom like magnetic monopoles are responsible for nonperturbative properties such as confinement [

We now summarize the idea of

For the ’t Hooft-Polyakov monopole,

Let us then consider the gauge field part of the ’t Hooft-Polyakov monopole configuration

The configurations (

Now since conditions 2 and 3 are equivalent to

For the special, spherical symmetric case given by (

It is interesting to apply our formalism to the Wu-Yang monopole [

We now consider the polar decomposition of the most general

Now the symmetric matrix

Thus (

In this work, we have used the dreibein and the metric as basic variables for the SO(

By showing that a

We have applied this formalism to nonperturbative aspects of the theory. The conformal mode of the metric is found to contain the topological aspects of the gauge potential. We have demonstrated that topological centres are located at certain points at which this conformal mode, a scalar field, is an extremum. This criterion is thus an alternative to that developed by us in [

We have also proposed an ansatz for the gauge potential, which incorporates the topological aspects contained in the conformal mode after separating the gauge variant and the gauge invariant degrees of freedom.

When our formalism is applied to the Hamiltonian formulation of 3 + 1-dimensional Yang-Mills theory, the canonical variables are the metric and its conjugate variable. The physical wave functionals are simply functionals of the metric. However, due to the change of variables, the inner product of the wave functionals will involve the Jacobian of the transformation. For 2 + 1-dimensional SU(

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Indrajit Mitra thanks UGC (DRS) for support.