Dreibein as prepotential for three-dimensional Yang-Mills theory

We advocate and develop the use of the dreibein (and the metric) as prepotential for three-dimensional SO(3) Yang-Mills theory. Since the dreibein transforms homogeneously under gauge transformation, the metric is gauge invariant. For a generic gauge potential, there is a unique dreibein on fixing the boundary condition. Topologically non-trivial monopole configurations are given by conformally flat metrics, with scalar fields capturing the monopole centres. Our approach also provides an ansatz for the gauge potential covering the topological aspects.


Introduction
In this paper we advocate and develop the use of the dreibein (and the metric) as the basic variable or prepotential for Yang-Mills theory. The first clear proposal to use the dreibein came from Haagensen and Johnson [1] in the context of Hamiltonian formalism in 3+1 dimensions. (See also Refs. [2]- [7] for other works involving various constructions of the metric.) However, the authors of Ref. [1] claimed there are too many zero modes associated with the defining equation for the dreibein (the condition for the dreibein to be torsion-free with respect to a connection one-form). Such zero modes would make the Jacobian of the transformation singular, and so Haagensen, Johnson and Lam [8] followed it up with a deformation of the defining equation to avoid the problem. Here we analyse this issue in some detail, considering a generic Yang-Mills potential. We conclude that the transformation from the gauge potential to the torsion-free dreibein is possible, and so the latter is a useful prepotential.
Our formulation is useful in several ways: 1. We rewrite the functional integral of Yang-Mills theory in 3-Euclidean dimensions exactly as a local theory of the metric.The action is quadratic in curvature (without the diffeomorphism invariance of gravity theories). In addition there is a local action involving anticommuting tensor fields which takes care of the Jacobian of the transformation from the gauge potential to the metric. Thus Yang-Mills theory is completely expressed as a local theory of gauge-invariant degrees of freedom contrary to common beliefs.
2. There have been many proposals of an ansatz for the gauge potential that exposes its topological properties ( [9]- [14]). We propose a complete ansatz which has a scalar and a spin-two field in addition to a pure gauge. The scalar field is the conformal mode. It captures the gauge field part of the 't Hooft-Polyakov monopole configuration. Indeed the monopoles can be located at the points at which this scalar field is an extremum and satisfies some additional conditions. At such points the 'pure gauge' part is also singular.
3. Our formulation can be useful for a non-perturbative understanding of Yang-Mills theory.
It is important to understand confinement from the R 2 action. Keeping only the scalar (conformal part of the metric) gives a self-interacting scalar theory that is quartic in derivatives This theory might by itself exhibit confinement. This could be tested by simulations of this action (in contrast to lattice gauge theories).
Our techniques can be extended in a straightforward way to SO(4) Yang-Mills theory in 4-Euclidean dimensions. Haagensen and Johnson [1] have also addressed generalization to other gauge groups. Our approach can also be applied to the 3+1-dimensional Yang-Mills theory in the Hamiltonian formulation, with the physical states described as wavefunctionals of the metric.
Ever since the proposal of non-abelian gauge theory, the similarity with the Cartan formulation of Einstein gravity has led to extensive work over decades exploring the relationship between the two. This has enriched both disciplines. Examples of this are gauge theories of gravity including Chern-Simons formulation of 2+1 gravity, Ashtekar and loop gravity formulation, ADS/CFT correspondence, and more recently the connection between the scattering amplitudes of gauge theories and Einstein gravity. There is also a set of approaches which uses the non-abelian electric field or other combination of gauge fields to construct the metric. In spite of all this, the connection between the two as we do it here has not been exploited. Our approach gives a new way to attack the exotic features of Yang-Mills theory.
The paper is arranged as follows. In Sec. 2, the dreibein is defined through the torsion-free condition, and the Yang-Mills theory expressed as R 2 theory. In Sec. 3, we determine the Jacobian of the transformation from A a i to e a i . Sec. 4 deals with topological field configurations, the resulting scalar field action and the gauge potential ansatz. In Sec. 5, we discuss our results along with outlook on 3+1-dimensional Yang-Mills theory. The Appendix analyses the existence and uniqueness of e a i for a given A a i .

Dreibein as prepotential and R theory
Consider the set of nine first order partial differential equations, which are the 'torsion-free condition' for the dreibein e a i (x) (i = 1, 2, 3, are the space indices and a = 1, 2, 3, are the group indices) with respect to a connection one-form A ab i (x). If the determinant |e a i (x)| = 0, we can write expanding the LHS in the e a i basis. Imposing the symmetry is equivalent to the torsion-free condition (1). Define Then equation (2) is D j e a k = 0, and so [D i , D j ]e a k = 0. This gives ab is the Yang-Mills field strength (see, for example, [15]). Using the inverse matrix {e bk }: e a k e bk = δ ab , we can write The dreibein e a i is viewed as the square root of a metric Eq. (1) may be written as where D j is the gauge-covariant derivative: When the Yang-Mills potential, as given by transforms inhomogeneously under an SO(3) gauge transformation, eq. (9) ensures that the dreibein e a i transforms homogeneously: Here O ab is an SO(3) matrix. The metric g ij (x) is therefore gauge-invariant. All gauge invariant objects can be rewritten in terms of g ij . For instance, where R (ij) is the matrix with the elements (R (ij) ) lk = R klij and g −1 has the elements (g −1 ) km = g km . This may be obtained from F ab ij = R klij g km e a m e bl . In 3 space dimensions, the Riemann curvature tensor can be completely expressed in terms of the Einstein tensor: where g = detg ij . Thus the action of 3-dimensional Yang-Mills theory is This is like R 2 gravity, but without the diffeomorphism invariance.
Eq. (1) or (9) comprise a set of linear equations for the variables A a i : When the matrix {e a i } is non-singular, (16) can be uniquely solved for A a i . Indeed, multiplying by e a l and summing over a we get ǫ ijk e a l ∂ j e a k = a il − δ il a mm (17) where a jm = |e|A b j e bm . Now set i = l in (17) to get a mm and put it back in (17). Therefore [1] In the Appendix, we consider (16) as equations for the variables e a i for a given gauge potential A a i , and argue that for any A a i there is a unique dreibein e a i on fixing the boundary condition, at least in the generic case. So the transformation A a i → e a i is also well-defined.
3 The Jacobian |δA a i (x)/δe b j (y)| For a change δe a i (x) of the dreibein, the corresponding change δA a i (x) of the Yang-Mills potential is given by (We performed infinitesimal variation of (16) and followed the steps leading from (16) to (18).) This equation is covariant under local gauge rotation (the inhomogeneous term in the local gauge transformation of A a i being absent in δA a i (x)). It is also covariant under diffeomorphism when e a i and A a i transform as covariant vectors (the determinant |∂x ′ /∂x| resulting from ǫ ijk cancels against one from 1/|e|). Define which transform as conventional tensors: ε ijk g ip g jq g kr = ε pqr . Then we have the manifestly covariant expression Therefore the Jacobian |δA a i (x)/δe b j (y)| will be invariant under local gauge rotation and diffeomorphisms, and so it can be fully expressed in terms of curvature tensors and covariant derivatives built out of the metric g ij .
Using the standard trick of using anti-commuting bosonic fields, where we have simply replaced δe a i by η a i . Note that we have √ g with d 3 x. This is necessary to get manifestly diffeomorphism invariant entity on the RHS of (22).
We now use e a i to construct tensors fromη cl and η b k : Then using (2). Therefore where ∇ j is the geometric covariant derivative involving the affine connection. The last two terms on the RHS of (25) cancel against each other, while the second term on the RHS drops out because of ε ijk in (22) and symmetry of Γ m jk in j and k. Hence in the integrand in the exponent in (22), the first term is simply √ gε ijkη il ∇ j η l k . The second term involves g im e am D ab j η b k . Using (25), this is g im ∇ j η m k , which equals ∇ j η ik . Thus we finally have

Non-perturbative configurations
We now point out that the dreibein and the metric are very useful for characterizing nonperturbative configurations. Ever since the Wu-Yang monopole ansatz [16] there has been extensive interest in obtaining an ansatz for the Yang-Mills potential that exhibits the topological aspects ([9]- [14]). This issue is all the more important because the non-perturbative aspects like confinement are expected to be driven by the topologically non-trivial configurations. The Faddeev-Niemi ansatz has led to extensive work in recent years. Here we show that the relation of the Yang-Mills potential to the dreibein gives a complete ansatz which brings out the topological property.

Conformal mode of metric and monopole configuration
First consider the gauge field part of the 't Hooft-Polyakov monopole configuration which possesses spherical symmetry. Using eq. (18), we find that with leads to the configuration (27).
Next consider a generalization of (28), with φ(x), an arbitrary scalar function of x 1 , x 2 and x 3 , in the place of g(r): This corresponds to a conformally flat metric, and φ is conformal degree of freedom. Using (30) in (18) leads to Then Now consider a point at which the following hold.
Then at that point, B a i ∼ δ ia and so B a i B a j ∼ δ ij . Thus the matrix S ij ≡ B a i B a j is triply degenerate at the point, which is the criterion for locating the centre of a monopole configuration in a gauge-invariant description of monopoles [5,17,18]. (We recall that the topological properties of a monopole can be characterised using only the gauge field even in the interior by the eigenvector fields of S ij . The points at which these vector fields become singular, or equivalently, the eigenvalues of S ij become degenerate, locate the monopoles. These points are called the 'centres' of the monopole configurations. The details of such characterisation are to be found in Refs. [5] and [17]).
For the special, spherical symmetric case given by (27) and (28), using K(r) = 1 + O(r 2 ) for r → 0, it can be checked that the function g(r) indeed satisfies the above conditions on φ(x) at the monopole centre r = 0. On the other hand, the conditions on φ(x 1 , x 2 , x 3 ) stated above show that in the more general form given by (30), the function φ needs to be symmetric only upto the second order terms in the Taylor expansion about the centre. Thus, taking the topological centre to be at r = 0, the forms A + Br 2 + Cx 3 1 + · · · and A + Br 2 + Dx 2 1 x 2 + · · · (where A, B, C, D are constants) are two examples of possible Taylor expansion for the function φ satisfying the conditions. Using (33), the Yang-Mills action becomes on integration by parts. A multi-monopole configuration corresponds to a function φ(x) satisfying the conditions given after equation (33) at isolated points. This simple massless scalar theory with quartic derivatives is likely to be confining. It is a good toy model for understanding confinement and this can be checked in simulations.
To evaluate the Jacobian |δA a i (x)/δe b j (y)| for the case of the ansatz (30), we put (31) in the affine connections in (26). This gives

Ansatz for gauge field
We now consider the polar decomposition of the most general 3×3 matrix e a i into an orthogonal matrix R and a symmetric matrix E: Under a gauge transformation, while the symmetric matrix E ji is gauge-invariant and is the symmetric square-root of the metric g ij . The decomposition (36) corresponds to where and ω d l is formally a pure gauge: (To obtain (38)-(40), one puts (36) in (18). The terms containing derivative of E immediately give the first term on the RHS of (38). In the terms containing derivative of R, we put ∂ j R a p = −ǫ abc ω b j R c p , which is the same as (40). Then using ǫ abc R a n R c p = ǫ nqp R b q (since |R| = 1) and ǫ ijk E in E kp = ǫ nrp |E|(E −1 ) jr , we get the ω d l term in (38).) Now the symmetric matrix E can be further decomposed into a spin-two traceless part and a scalar trace part: ) Since δ ij e −φ corresponds to the metric (31), we see that the topological configurations are contained in this part, when φ satisfies the three conditions given after (33) at isolated points. At such points, ω d l is not strictly a pure gauge. For example, the non-Abelian magnetic field corresponding to ω d l has a Dirac string contribution when R is the singular gauge in which the configuration (27) becomes a Dirac monopole [17].
Thus equations (38), (39),(40) and (41) comprise a complete ansatz for the gauge potential, which contains the topological aspects in a natural way. The three gauge degrees of freedom reside in R and the six gauge-invariant degrees of freedom reside in E. The conformal mode in E captures the topological configurations.

Discussion
In this work, we have used the dreibein and the metric as basic variables for the SO(3) Yang-Mills theory in three Euclidean dimensions, and arrived at a local theory of only gauge-invariant degrees of freedom. The theory has been rewritten as R 2 theory (without diffeomorphism invariance) and a local action involving anticommuting tensor fields.
We have applied this formalism to non-perturbative aspects of the theory. The conformal mode of the metric is found to contain the topological aspects of the gauge potential. This leads to a scalar field action, with the topological centres located at the points at which this scalar field is an extremum and satisfies some additional conditions. We have proposed that this scalar theory itself may be confining.
We have also proposed a complete ansatz for the gauge potential, incorporating the topological aspects contained in the conformal mode in a natural way. The criterion of points of triple degeneracy of B a i B a j as centres of monopole configurations, which was earlier developed in Refs. [5] and [17], has now been expressed in terms of the gauge potential.
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. We can do away with the Gauss law constraint, and the physical wavefunctionals are simply functionals of the metric. However, due to the change of variables, the inner product of the wavefunctionals will involve the Jacobian of the transformation, i.e., the local action of the anticommuting bosonic variables given in eq. (26). For 2+1-dimensional SU(N) gauge theory, Nair et al ( [19]- [22]) proposed an N × N complex matrix as a prepotential. They obtained an inner product for physical states which involves the WZW action, a consequence of the change of variables. They arrived at a trial wave functional which exhibits confinement and gives string tension in striking agreement with lattice gauge theory simulations. Our proposal here of using the metric is a natural generalization to 3+1-dimensions. It will be interesting to guess the wave functional which gives confinement and see the role played by the anticommuting Bose fields. Acknowledgement I.M. thanks UGC (DRS) for support.
A On existence and uniqueness of e a i for given A a i When A a i (x) = 0, equation (16) implies e a i is curl-free for each a = 1, 2, 3 and we have a general solution where ϕ a (x) are arbitrary functions. Thus the equation (16) has a large set of zero modes. This is precisely the situation where the curvature F ab ij (or R klij ) vanishes and corresponds to a flat space. Equation (42) just corresponds to a set of curvilinear coordinates φ a (x) of the flat space: g ij (x) = ∂ i ϕ a ∂ j ϕ a . We would like to know whether there is such a large class of solutions e a i for a generic A a i . Note that the equation (16) (42) as accidental to the case A a i = 0. We next consider eq. (1) or (9) or (16) for a generic A a i . These equations have been earlier analysed in Ref. [23]. For i = 1, 2 in eq. (9), we have These being linear equations in x 3 for e a 1 and e a 2 , we get unique solutions for any given e a 3 . But the e a 1 and e a 2 so obtained have to further satisfy for any x 3 . So one has to address whether this is always possible with some choice of e a 3 , and if yes, whether the choice of e a 3 is unique. For the case A i (x) = 0, the solutions of equations (43) are These automatically satisfy (44). But we will now explain that the situation for generic A a i is very different.
We presume that the initial data on x 3 = x 0 3 satisfies (44). Then the entire content of (44) is contained in the equation obtained by applying ∂/∂x 3 on it. So we may equivalently consider an equation obtained by applying ∂/∂x 3 on (44) and using equations (43). Such an equation is which is obtained by operating on (9) by D i . The original equations are now equivalent to the set of equations comprising (43) and (46). We decompose e 3 in directions parallel and perpendicular to B 3 : is an arbitrary function. Let us consider the generic non-Abelian configurations where the 3 × 3 matrix {B a i } is non-singular in the region of interest. Putting e 3 as decomposed above into (46) and taking cross-product with B 3 , we get Substituting this into (43), we obtain e 1 and e 2 as functions of α(x). However this e 1 and e 2 have to satisfy (obtained by taking the dot product of (46) with B 3 ). So one has to address whether this condition can be satisfied. An analysis on these lines was done in Ref. [23].
This has to be satisfied at all x 3 . The variable α 33 enters in this consistency equation. If the coefficient of α 33 , that is, we can solve for α 33 in favour of the other five variables α 11 , α 12 , α 13 , α 22 , α 23 . Substituting this for α 33 in the (five) independent equations (54)-(59), we get evolution equations for these five independent variables. The solution is unique with initial choice on x 3 = x 0 3 surface. (If β 123 − β 213 = 0, we can consider the evolution equation in x 1 (or x 2 ) instead of x 3 and obtain unique solution with initial choice on x 1 = x 0 1 (or x 2 = x 0 2 ) surface.) Thus in this Appendix, we have demonstrated that with appropriate boundary condition, a unique torsion-free dreibein e a i exists for a given generic Yang-Mills potential A a i . We disagree with Ref. [1] where it is claimed that the zero modes for the torsion-free condition (1) make the Jacobian for the transformation A a i → e a i singular. Fixing the boundary condition, this transformation is indeed possible. Even though our arguments are mainly in the generic case with |B a i | = 0, we expect the transformation to be useful for any Yang-Mills potential.