# Dreibein as Prepotential for Three-Dimensional Yang-Mills Theory

Authors: Indrajit Mitra; H. S. Sharatchandra
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

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 nontrivial 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.


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## Body

## 1. Introduction

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 [1] in the context of Hamiltonian formalism in 3 + 1 dimensions. (References [2–8] are some other works on Yang-Mills theory involving various constructions of the metric.) The defining equation for the dreibein is the condition for the dreibein to be torsion-free with respect to a connection one-form. The authors of [1] claimed that there are zero modes associated with this equation due to nonuniqueness of the dreibein eai corresponding to a given gauge potential Aia but also gave arguments that the zero modes would not affect their discussion. Then Haagensen et al. [9] followed it up with a deformation of the defining equation for the dreibein, removing the deformation at the end. References [5–8] have used the original defining equation for the dreibein as in [1] and not the deformation of it. In this work also, we use the original defining equation of [1]. We analyse the case of ageneric Yang-Mills potential, satisfying det⁡Bia≠0 (Bia being the nonabelian magnetic field) and argue that a unique dreibein is obtained by fixing the boundary condition. Consequently, the torsion-free dreibein is a useful prepotential. The calculation of the Jacobian will be affected by the zero modes present for vanishing nonabelian magnetic field, but the Jacobian will not be needed in the present work.Our formulation is useful in several ways:(1)
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 likeR2 theory, but without the diffeomorphism invariance. It should be noted thatour interest in the present work is about Yang-Mills theory, not gravity.(2)
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 [5, 10], we showed how to locate such a configuration to an internal point (the “centre”) in a gauge invariant way. In this work, the “centres” are identified as certain points at which the conformal mode (a scalar field) is extremum.(3)
There has been extensive interest in obtaining an ansatz for the Yang-Mills potential that exhibits the topological aspects [11–16]. We propose an ansatz which has a scalar and a spin-two part in addition to a pure gauge. The scalar part comes from the conformally flat metric which captures the monopole configurations.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 wave functionals of the metric.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 [17], Ashtekar and loop gravity formulation [18], and the pure connection formulation of general relativity [19]. We have already cited several works which attack gauge theory using gravity [1–9]; a recent work is [20]. In spite of all this, our approach gives a new way to attack the exotic features of Yang-Mills theory.The paper is arranged as follows. In Section2, the dreibein eia is defined through the torsion-free condition, and the gauge invariance of the metric is emphasized. In Section 3, we analyse the existence and uniqueness of eia for a given gauge potential Aia. Section 4 deals with topological (monopole) field configurations and Section 5 presents the gauge potential ansatz. In Section 6, we discuss our results.


## 2. Dreibein as Prepotential

Consider the set of nine first-order partial differential equations (see, e.g., [1])(1)ϵijk∂jeka+Ajabekb=0,which constitute the “torsion-free condition” for the dreibein eia(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 Aiab(x). The dreibein eia is viewed as the square root of a metric(2)gijx=eiaxejax.If det⁡eia(x)≠0, we can expand the LHS of (1) in the eia basis and write(3)∂jeka+Ajabekb=Γjklela.Imposing the symmetry(4)Γjkl=Γkjlis equivalent to the torsion-free condition (1). Define(5)Dieja=∂ieja+Aiabejb-Γijlela.Then (3) is Djeka=0, and so [Di,Dj]eka=0. This gives(6)Rlkijela-Fijabekb=0,where Rlkij is the Riemann tensor and(7)Fijab=∂iAjab-∂jAiab+Ai,Ajab(see, e.g., [21]). Using the inverse matrix {ebk},(8)ekaebk=δab,we can write(9)Fijab=Rlkijelaebk.It may be helpful to link the above with Cartan’s structure equations of general relativity. Cartan’s first equation,(10)dea+ωab∧eb=0, is our equation (1); we have used the symbol A in the place of ω and and put all group indices as upper indices. (Since we have SO(3) and not the Lorentz group, there is no need to distinguish between upper and lower indices.) Cartan’s other equation (11)Rab=dωab+ωac∧ωcbin components is (on using the dreibein to convert the group indices of Rabij to space indices) (12)Rlkijelaebk=∂iωjab-∂jωiab+ωi,ωjab.Equation (12) is (7) and (9) combined (in our notation).Equation (1) can be written as (13)ϵijkDjeka=0, where Dj is the gauge-covariant derivative: (14)Djab=δab∂j+Ajab. Now write (15)Aiab=-ϵabcAic,where Aic is the Yang-Mills potential. Then (7) gives the Yang-Mills field strength. Aia transformsinhomogeneously under an SO(3) gauge transformation, but (13) ensures that the dreibein eia transformshomogeneously:(16)eia′x=Oabxeibx.Here Oab is an SO(3) matrix. The metric gij(x) as given by (2) is thereforegauge-invariant. All gauge invariant objects can be rewritten in terms of gij. For instance,(17)trFi1j1Fi2j2⋯=trRi1j1g-1Ri2j2g-1⋯,where R(ij) is the matrix with the elements Rijkl=Rklij and g-1 has the elements g-1km=gkm. This may be obtained from Fijab=Rklijgkmemaebl.In 3 space dimensions, the Riemann curvature tensor can be completely expressed in terms of the Einstein tensor:(18)Rklij=-gϵklmϵijnGmn,where g=det⁡gij. This allows us to express the Yang-Mills action in 3D (19)S=12gYM2∫d3xBiaBia,(where Bia≡(1/2)ϵijkFjka) also in terms of the Einstein tensor, as follows. In (6), we use (18) and also Fijab=-ϵabcFijc. This gives [1] (20)Bia=det⁡eeajGij, where det⁡e=det⁡eia=g. Then the Yang-Mills action in 3D is (21)S=12gYM2∫d3xggjkGijGik. This is like R2 gravity, butwithout the diffeomorphism invariance.Equation (1) or (13) comprises a set of linear equations for the variables Aia:(22)ϵijk∂jeka+ϵabcAjbekc=0.Equation (22) has diffeomorphism covariance provided both Aia and eia transform as covariant GL(3) vectors (the Christoffel symbol does not contribute due to antisymmetry of ϵijk). Note that ϵijk in (22) can be replaced by the Levi-Civita tensor εijk since the two differ by just a factor of g.When the matrixeia is nonsingular, (22) can be uniquely solved for Aia. Indeed, multiplying by ela and summing over a we get(23)ϵijkela∂jeka=ail-δilamm,where ajm=(det⁡e)Ajbebm. Now set i=l in (23) to get amm and put it back in (23). Therefore [1, 5](24)Alb=1det⁡eϵijkelaeib∂jeka-12δabeic∂jekc.To sum up, (1) or (13) or (22) defines eia. In our context, the dreibein does not arise from transformation between coordinate basis and orthonormal basis. The gauge transformation of Aia ensures that eia transforms homogeneously and so gij as defined by (2) is gauge invariant. Equation (22) is diffeomorphism covariant. But the action is not diffeomorphism invariant, since we are dealing with Yang-Mills theory in flat space and not in curved space.


## 3. On Existence and Uniqueness ofeia for Given Aia

WhenAia(x)=0, (22) implies eia is curl-free for each a=1,2,3 and we have a general solution(25)eiax=∂iφax,where φa(x) are arbitrary functions. Thus (22) has a large set of zero modes. In this situation, the curvature Fijab (or Rklij) vanishes and we have a flat space. From (2), we see that (25) corresponds to gij(x)=∂iφa∂jφa, that is, a set of curvilinear coordinates ϕa(x) of the flat space. Now (22) is covariant under diffeomorphisms. If we start with Aia(x)=0 and equation (25) and apply the diffeomorphism x→x′(x) using the transformation property of covariant vectors, Aia stays zero but eia changes to eia′(x)=∂iθa(x) (where θax′=ϕa(x)); that is, we have a new dreibein for the same Aia. Thuswe may view the set of zero modes of (22) given by (25) as accidental to the case ofAia=0and as a result of diffeomorphism covariance.In [1], it is claimed that all pure gauges, and not just Aia=0, give zero modes. But a pure gauge potential Ai(x)=OT(x)∂iO(x) (where Aiab=-ϵabcAic and O is an SO(3) matrix) changes under both diffeomorphism and gauge transformation. If Aia were invariant but the dreibein changed under a transformation, we would have had nonunique eia for a given pure gauge Aia. This does not appear to be the case. So, while we do not exclude zero modes for nonvanishing pure gauges, such zero modes do not follow from any symmetry (either gauge covariance or diffeomorphism covariance) of (22), unlike the zero modes for vanishing gauge potential.Nowboth vanishing gauge potential and pure gaugeAiacorrespond toBia=0. The point we make in this section is that, fora generic gauge potential withdet⁡Bia≠0, we do not have such large set of nonunique eia.This issue was earlier addressed in Section2 of [22]. Here we present a simpler and neater analysis, arriving at the same conclusion.First we note that operating on (13) by Di gives aconsistency condition, as follows. We have ϵijkDicaDjabekb=0. Using antisymmetry of ϵijk and [Di,Dj]cb=Fijcb=-ϵcbaFija, we arrive at a condition to be satisfied by any solution to (22):(26)B→ix×e→ix=0.In the case
det
⁡
B
i
a
≠
0, without any loss of generality, we may expand e→i(x) in the basis provided by B→i(i=1,2,3):(27)eiax=Bjaαji.Putting (27) in the consistency condition (26) gives us(28)det⁡BϵijkB-1kaαji=0.Multiplying (28) by Bla, we get ϵijlαji=0. Therefore αij(x) is a symmetric matrix:(29)αijx=αjix.In terms of these variables αij, (22) becomes(30)ϵijk∂jαmk+βmjlαlk=0with i,j,k,l,m=1,2,3. Here(31)βmjl=B-1maDjBla.Equation (30) gives the following six equations for evolution of αij in x3:(32)∂3α11-∂1α13+β13lαl1-β11lαl3=0,(33)∂3α21-∂1α23+β23lαl1-β21lαl3=0,(34)∂3α31-∂1α33+β33lαl1-β31lαl3=0,(35)∂3α12-∂2α13+β13lαl2-β12lαl3=0,(36)∂3α22-∂2α23+β23lαl2-β22lαl3=0,(37)∂3α32-∂2α33+β33lαl2-β32lαl3=0.Note that there is no evolution equation for the variable α33. Also, since there are two evolution equations (33) and (35) for the same variable α12=α21, we get the consistency requirement(38)∂3α12=∂2α13+β12lαl3-β13lαl2=∂1α23+β21lαl3-β23lαl1.This has to be satisfied at all x3. The variable α33 enters in this consistency equation. The coefficient of α33 is β123-β213. If(39)β123-β213≠0we can solve for α33 in favour of the other five variables α11,α12,α13,α22,α23. Substituting this for α33 in the (five) independent equations (32)–(37), we get evolution equations for these five independent variables. The solution is unique with initial choice on x3=x30 surface. This is ensured by the Cauchy-Kowalevski theorem (see, e.g., [22], which contains further reference, for a statement of this theorem). In the generic case, at least one of (β123-β213), (β231-β321), and (β312-β132) will be nonzero. If β231-β321≠0 (or β312-β132≠0), we can consider the evolution equation in x1 (or x2) instead of x3 and obtain unique solution with initial choice on x1=x10 (or x2=x20) surface.We have demonstrated that, on fixing the boundary condition, a unique torsion-free dreibeineia exists for a generic Yang-Mills potential Aia. So, after taking into account the zero modes discussed in the first two paragraphs of this section, it should be possible to calculate the Jacobian of the transformation. Thus the transformation from Aia to eia as given by (1) is indeed possible. However, the Jacobian will not be needed in the calculations of this paper. Only the defining equation of the dreibein will be used.


## 4. Nonperturbative Monopole Configurations and Conformal Mode of Metric

It is expected that topological degrees of freedom like magnetic monopoles are responsible for nonperturbative properties such as confinement [23]. In the Georgi-Glashow model, confinement is due to the ’t Hooft-Polyakov monopole solution [24]. In contrast to this model, there is no scalar (Higgs) field in the SO(3) Yang-Mills theory in three dimensions. But the gauge field part of the ’t Hooft-Polyakov monopole with appropriate boundary conditions (see below) constitutes a finite-action field configuration and contributes to the functional integral. Moreover, it has strong qualitative effects on the Wilson loop.We now summarize the idea of“centres” of topological field configurations as developed and used in our earlier works [5, 10, 25]. (This framework is in the spirit of, but different from, the Abelian projection procedure of ’t Hooft [26].) The topological properties of the ’t Hooft-Polyakov monopole can be characterised using only the gauge field and in the interior (in contrast to the usual characterisation by the triplet scalar field at infinity). This is therefore useful in pure gauge theory. This characterisation is done using the eigenvector fields of the gauge invariant quantity Sij≡BiaBja, where Bia≡1/2ϵijkFjka is the non-Abelian magnetic field.For the ’t Hooft-Polyakov monopole,Sij=α(r)δij+β(r)xixj, where α and β are functions of the distance r from the origin. At r=0, Sij~δij and so any direction is an eigenvector. Thus the eigenvector xi with unit winding number is singular at the origin. In general, the points at which the eigenvalues of Sij become triply degenerate or, equivalently, the eigenvector fields of Sij become singular, are called the “centres” of the monopoles and other topological objects in Yang-Mills theory.Let us then consider the gauge field part of the ’t Hooft-Polyakov monopole configuration(40)Aiax=ϵaijxj1-Krr2. Here K(r)=1+O(r2) for r→0 and K(r)→0 for r→∞. (These conditions ensure the finiteness of the action.) Using (41)eiax=δiae-grin (24), we find that the resulting gauge potential corresponds to (40) with (42)dgdr=1-Krr.The configurations (40) and (41) possess spherical symmetry. We next consider a generalization of (41) with ϕ(x), an arbitrary scalar function of x1, x2, and x3, in the place of g(r): (43)eiax=δiae-ϕx.This corresponds to (44)gij=δije-2ϕ,a conformally flat metric, and we call ϕ theconformal mode. Using (43) in (24) now leads to (45)Aiax=ϵaij∂jϕ.This gives (46)Bia=∂i∂aϕ-δia∂2ϕ+∂iϕ∂aϕ.Nowconsider a point at which the following hold: (1)
∂iϕ=0.(2)
∂i∂aϕ=0 for i≠a.(3)
∂2ϕ/∂x12=∂2ϕ/∂x22=∂2ϕ/∂x32.Note that the first condition says that we have a critical point and the next two conditions say that the Hessian matrix is diagonal and its three diagonal elements (eigenvalues) are equal. These are sufficient conditions for ϕ to be an extremum at the point. (ϕ attains a maximum or a minimum depending on whether the eigenvalues are all positive or all negative.)Now since conditions 2 and 3 are equivalent to∂i∂aϕ=1/3δia∂2ϕ, it follows from (46) that, at the point under consideration, Bia~δia and so BiaBja~δij. So the matrix Sij is triply degenerate at the point, which, as explained at the beginning of this section, is the criterion for locating the “centre” of a topological configuration in a gauge-invariant description. Thusat the topological centre, the fieldϕis an extremum, satisfying the three conditions given above.For the special, spherical symmetric case given by (40) and (41), using K(r)=1+O(r2) 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, in the more general form given by (43), the conditions on ϕ(x1,x2,x3) stated above show that ϕ needs to be symmetric only up to the second-order terms in the Taylor expansion about the centre. Thus, taking the topological centre to be at r=0, the forms A+Br2+Cx13+⋯ and A+Br2+Dx12x2+⋯ (where A, B, C, and D are constants) are two examples of possible Taylor expansion for the function ϕ satisfying the conditions.It is interesting to apply our formalism to the Wu-Yang monopole [27]. This is a solution to the Yang-Mills field equations. However, it is a point-like magnetic monopole. The gauge potential for it is obtained on putting K(r)=0 in (40). Then (42) gives g(r)=ln⁡r (up to an additive constant, which sets the scale). So the conformal mode goes to negative infinity as r goes to zero. Such a singularity, instead of a smooth minimum, in the conformal mode is just what is expected since the Wu-Yang monopole is point-like. The metric in this case is gij=δij/r2 (up to a multiplicative constant).


## 5. Ansatz for Gauge Field

We now consider the polar decomposition of the most general3×3 matrix eia into an orthogonal matrix R and a symmetric matrix E: (47)eiax=RjaxEjix∀x. Under a gauge transformation, (48)Rjax⟶OabxRjbx, while the symmetric matrix Eji is gauge-invariant and is the symmetric square-root of the metric gij. The decomposition (47) corresponds to (49)Aldx=Rmdxamlx+ωldx, where (50)amlx=1det⁡EϵijkEimElp-12EipElm∂jEkp and ωld is formally a pure gauge: (51)ωldx=-12ϵbcdRmbx∂lRmcx.(To obtain (49), one puts (47) in (24). The terms containing derivative of E immediately give the first term on the RHS of (49). In the terms containing derivative of R, we put ∂jRpa=-ϵabcωjbRpc, which is the same as (51). Then using ϵabcRnaRpc=ϵnqpRqb (since det⁡R=1) and ϵijkEinEkp=ϵnrp(det⁡E)E-1jr, we get the ωld term in (49).)Now the symmetric matrixE can be further decomposed into a spin-two traceless part and a scalar trace part: (52)Eij=E~ij+δije-ϕ.(Here E~ij=Eij-(1/3)δijΣkEkk and we define (1/3)ΣkEkk to be e-ϕ.)Sinceδije-ϕcorresponds to the metric (44), we propose that the topological configurations are contained in this part, withϕsatisfying the three conditions given after (46) at isolated points. At such points, ωld is not strictly a pure gauge. For example, the non-Abelian magnetic field corresponding to ωld has a Dirac string contribution when R is the singular gauge in which the configuration (40) becomes a Dirac monopole [10].Thus (49), (50), (51), and (52) comprise our 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 contained in E captures the topological configurations.


## 6. 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. The usefulness of this lies in the fact that the metric is gauge invariant.By showing that agenericAia corresponds to a unique torsion-free eia on fixing the boundary condition, we have provided justification for the change of variables from Aia to eia.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 [5, 10] (also applied by us in [25]), namely, that the points of triple degeneracy of BiaBja are the centres of topological configurations. Both of these criteria are gauge invariant. The Yang-Mills theory expressed in terms of this conformal mode would be a theory of a scalar field, and lattice simulation of thisscalar theory (as opposed to latticegauge theory) with monopoles located at extrema of the scalar field would serve as a model for confinement.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(N) gauge theory, Nair et al. [28–31] proposed an N×N complex matrix as a prepotential and, as a consequence of the change of variables, obtained an inner product for physical states which involves the Wess-Zumino-Witten action. They arrived at a trial wave functional which exhibits confinement and gives string tension in striking agreement with lattice gauge theory simulations. Viewing our proposal of using the dreibein as prepotential as a generalization to 3 + 1-dimensions, it will therefore be interesting to calculate the Jacobian of the transformation in our case.



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