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University of Maryland, College Park, MD 20742

arXiv:gr-qc/9303032v1 26 Mar 1993February 1993UMDGR-93-129GEOMETRODYNAMICSVS.CONNECTION DYNAMICSJoseph D. Romano1Department of PhysicsUniversity of Maryland, College Park, MD 20742ABSTRACTThe purpose of this review is to describe in some detail the mathematicalrelationship between geometrodynamics and connection dynamics in the con-text of the classical theories of 2+1 and 3+1 gravity. I analyze the standardEinstein-Hilbert theory (in any spacetime dimension), the Palatini and Chern-Simons theories in 2+1 dimensions, and the Palatini and self-dual theories in3+1 dimensions.

I also couple various matter fields to these theories and brieflydescribe a pure spin-connection formulation of 3+1 gravity. I derive the Euler-Lagrange equations of motion from an action principle and perform a Legendretransform to obtain a Hamiltonian formulation of each theory.

Since constraintsare present in all these theories, I construct constraint functions and analyze theirPoisson bracket algebra. I demonstrate, whenever possible, equivalences betweenthe theories.PACS: 04.20, 04.501romano@umdhep.umd.edu

Contents:1.Overview2.Einstein-Hilbert theory2.1Euler-Lagrange equations of motion2.2Legendre transform2.3Constraint algebra3.2+1 Palatini theory3.1Euler-Lagrange equations of motion3.2Legendre transform3.3Constraint algebra4.Chern-Simons theory4.1Euler-Lagrange equations of motion4.2Legendre transform4.3Constraint algebra4.4Relationship to the 2+1 Palatini theory5.2+1 matter couplings5.12+1 Palatini theory coupled to a cosmological constant5.2Relationship to Chern-Simons theory5.32+1 Palatini theory coupled to a massless scalar field6.3+1 Palatini theory6.1Euler-Lagrange equations of motion6.2Legendre transform6.3Relationship to the Einstein-Hilbert theory7.Self-dual theory7.1Euler-Lagrange equations of motion7.2Legendre transform7.3Constraint algebra8.3+1 matter couplings8.1Self-dual theory coupled to a cosmological constant8.2Self-dual theory coupled to a Yang-Mills field9.General relativity without-the-metric9.1A pure spin-connection formulation of 3+1 gravity9.2Solution of the diffeomorphism constraints10.DiscussionReferences1

1. OverviewEinstein’s theory of general relativity is by far the most attractive classical theory ofgravity today.

By describing the gravitational field in terms of the structure of spacetime,Einstein effectively equated the study of gravity with the study of geometry. In generalrelativity, spacetime is a 4-dimensional manifold M with a Lorentz metric gab whose curvaturemeasures the strength of the gravitational field.

Given a matter distribution described bya stress-energy tensor Tab, the curvature of the metric is determined by Einstein’s equationGab = 8π Tab. This equation completely describes the classical theory.As written, Einstein’s equation is spacetime covariant.

There is no preferred time vari-able, and, as such, no evolution. However, as we shall see in Section 2, general relativityadmits a Hamiltonian formulation.

The canonically conjugate variables consist of a positive-definite metric qab and a density-weighted, symmetric, second-rank tensor field epab—bothdefined on a 3-manifold Σ. These fields are not free, but satisfy certain constraint equations.Evolution is defined by a Hamiltonian, which (if we ignore boundary terms) is simply a sumof the constraints.Now it turns out that the time evolved data defines a solution, (M, gab), of the fullfield equations which is unique up to spacetime diffeomorphisms.

In a solution, Σ can beinterpreted as a spacelike submanifold of M corresponding to an initial instant of time, whileqab and epab are related to the induced metric and extrinsic curvature of Σ in M.2 Thus, theHamiltonian formulation of general relativity can be thought of as describing the dynamics of3-geometries. Following Wheeler, I will use the phrase “geometrodynamics” when discussinggeneral relativity in this form.On the other hand, all of the other basic interactions in physics—the strong, weak, andelectromagnetic interactions—are described in terms of connection 1-forms.

For example,the Hamiltonian formulation of Yang-Mills theory has a connection 1-form Aa (which takesvalues in the Lie algebra of some gauge group G) as its basic configuration variable. Thecanonically conjugate momentum (or “electric field”) eEa is a density-weighted vector fieldwhich takes values in the dual to the Lie algebra of G. As in general relativity, these variablesare not free, but satisfy constraint equations:The Gauss constraint Da eEa = 0 (where Dais the generalized derivative operator associated with Aa) tells us to restrict attention todivergence-free electric fields.

Thus, just as we can think of the Hamiltonian formulation ofgeneral relativity as describing the dynamics of 3-geometries, we can think of the Hamiltonianformulation of Yang-Mills theory as describing the dynamics of connection 1-forms. I will2More precisely, qab is the induced metric on Σ, while epab is related to the extrinsic curvature Kab viaepab = √q(Kab −Kqab).2

often use the phrase “connection dynamics” when discussing Yang-Mills theory in this form.Despite the apparent differences between geometrodynamics and connection dynamics,many researchers have tried to recast the theory of general relativity in terms of a connection1-form. Afterall, if the strong, weak, and electromagnetic interactions admit a connectiondynamic description, why shouldn’t gravity?

Early attempts in this direction used Yang-Mills type actions, but these actions gave rise, however, to new theories of gravity.Aconnection dynamic theory was gained, but Einstein’s theory of general relativity was lostin the process. Later attempts (like the ones I will concentrate on in this review) left generalrelativity alone, but tried to reinterpret Einstein’s equation in terms of the dynamics of aconnection 1-form.

The most familiar of these approaches is due to Palatini who rewrotethe standard Einstein-Hilbert action (which is a functional of just the spacetime metric gab)in such a way that the spacetime metric and an arbitrary Lorentz connection 1-form areindependent basic variables. However, as we shall see in Section 6, the 3+1 Palatini theorydoes not succeed in recasting general relativity as a connection dynamical theory.

The 3+1Palatini theory collapses back to the standard geometrical description of general relativitywhen one writes it in Hamiltonian form.More recently, Ashtekar [1, 2, 3] has proposed a reformulation of general relativity inwhich a real (densitized) triad eEai and a connection 1-form Aia (which takes values in thecomplexified Lie algebra of SO(3)) are the basic canonical variables.He obtained thesenew variables for the real theory by performing a canonical transformation on the standardphase space of real general relativity.For the complex theory, Jacobson and Smolin [4]and Samuel [5] independently found a covariant action that yields Ashtekar’s new variableswhen one performs a 3+1 decomposition. This action is the Palatini action for complexgeneral relativity viewed as a functional of a complex co-tetrad and a self-dual connection1-form.3 In one sense, it is somewhat surprising that these new variables could capture thefull content of Einstein’s equation since they involve only half (i.e., the self-dual part) of aLorentz connection 1-form.

On the other hand, the special role that self-dual fields play inthe theory of general relativity was already evident in the work of Newman, Penrose, andPlebanski on self-dual solutions to Einstein’s equation. In fact, much of this earlier workprovided the motivation for Ashtekar’s search for the new variables.Not only did the new variables give general relativity a connection dynamic descrip-tion; they also simplified the field equations of the theory—particularly the constraints.

Interms of the standard geometrodynamical variables (qab, epab), the constraint equations arenon-polynomial. However, in terms of the new variables, the constraint equations become3To recover the phase space variables for the real theory, one must impose reality conditions to select areal section of the complex phase space.3

polynomial. This result has rekindled interest in the canonical quantization program for 3+1gravity.

Due to the simplicity of the constraint equations in terms of these new variables,Jacobson, Rovelli, and Smolin [6, 7] and a number of other researchers have been able tosolve the quantum constraints exactly. Although the quantization program has not yet beencompleted, the above results constitute promising first steps in that direction.The Palatini and self-dual theories described above were attempts to give general rel-ativity in 3+1 dimensions a connection dynamic description.A few years later, Witten[8] considered the 2+1 theory of gravity.

He was able to show that this theory simplifiesconsiderably when expressed in Palatini form. In fact, Witten demonstrated that the 2+1Palatini theory for vacuum 2+1 gravity was equivalent to Chern-Simons theory based on theinhomogeneous Lie group ISO(2, 1).4 He then used this fact to quantize the theory.

Thisresult startled both relativists and field theorists alike: relativists, since the Wheeler-DeWittequation in geometrodynamics is as hard to solve in 2+1 dimensions as it is in 3+1 dimen-sions; field theorists, since a simple power counting argument had shown that perturbationtheory for 2+1 gravity around a flat background metric is non-renormalizable—just as it isfor the 3+1 theory. The success of canonical quantization and failure of perturbation theoryin 2+1 dimensions came as a welcome surprise.

Despite key differences between 2+1 and3+1 gravity (in particular, the lack of local degrees of freedom for 2+1 vacuum solutions),Witten’s result has proven to be useful to non-perturbative approaches to 3+1 quantumgravity. In particular, since the overall structure of 2+1 and 3+1 gravity are the same (e.g.,they are both diffeomorphism invariant theories, there is no background time, and the dy-namics is generated in both cases by 1st class constraints), researchers have been able to use2+1 gravity as a “toy model” for the 3+1 theory [9].Finally, the most recent developments relating geometrodynamics and connection dy-namics involve formulations of general relativity that are independent of any metric variable.This idea for 3+1 gravity dates back to Plebanski [10], and was recently developed fullyby Capovilla, Dell, and Jacobson (CDJ) [11, 12, 13, 14].

Shortly thereafter, Peld´an [15]provided a similar formulation for 2+1 gravity. These pure spin-connection formulationsof general relativity are defined by actions that do not involve the spacetime metric gab inany way whatsoever—the action for the complex 3+1 theory depends only on a connection1-form (which takes values in the complexified Lie algebra of SO(3)) and a scalar density ofweight −1.

Moreover, the Hamiltonian formulation of this theory is the same as that of theself-dual theory, and by using their approach, CDJ have been able to write down the most4Chern-Simons theory, like Yang-Mills theory, is a theory of a connection 1-form. However, unlike Yang-Mills theory, it is defined only in odd dimensions and does not require the introduction of a spacetimemetric.4

general solution to the 4 diffeomorphism constraint equations. Whether or not these resultswill lead to new insights for the quantization of the 3+1 theory remains to be seen.With this brief history of geometrodynamics and connection dynamics as background,the purpose of this review can be stated as follows: It is to describe in detail the theoriesmentioned above, and, in the process, clarify the mathematical relationship between ge-ometrodynamics and connection dynamics in the context of the classical theories of 2+1 and3+1 gravity.

While preparing the text, I made a conscious effort to make the presentation asself-contained and internally consistent as possible. The calculations are somewhat technicaland rather detailed, but I have included many footnotes, parenthetical remarks, and math-ematical digressions to fill various gaps.

I felt that this style of presentation (as opposed torelegating the necessary mathematics to appendices at the end of the paper) was more inkeeping with the natural interplay between mathematics and physics that occurs when oneworks on an actual research problem. Also, I felt that the added details would be of valueto anyone interested in working in this area.In Section 2, I recall the standard Einstein-Hilbert theory and take some time to introducethe notation and mathematical techniques that I will use repeatedly throughout the text.Although this section is a review of fairly standard material, readers are encouraged to at leastskim through the pages to acquaint themselves with my style of presentation.

In Sections 3and 4, I restrict attention to 2+1 dimensions and describe the 2+1 Palatini and Chern-Simonstheories and demonstrate the relationship between them. In Section 5, I couple a cosmologicalconstant and a massless scalar field to the 2+1 Palatini theory.

2+1 Palatini theory coupledto a cosmological constant Λ is of interest since we shall see that the equivalence betweenthe 2+1 Palatini and Chern-Simons theories continues to hold even if Λ ̸= 0; 2+1 Palatinitheory coupled to a massless scalar field is of interest since it is the dimensional reductionof 3+1 vacuum general relativity with a spacelike, hypersurface-orthogonal Killing vectorfield (see, e.g., Chapter 16 of [16]).In fact, recent work in progress (by Ashtekar andVaradarajan) in the hamiltonian formulation of this reduced theory indicates that its non-perturbative quantization is likely to be successful. In Sections 6 and 7, I turn my attentionto 3+1 dimensions and describe the 3+1 Palatini and self-dual theories.

In Section 8, Icouple a cosmological constant and a Yang-Mills field to 3+1 gravity. Section 9 describesa pure spin-connection formulation of 3+1 gravity, and Section 10 concludes with a briefsummary and discussion of the results.

All of the above theories are specified by an action. Iobtain the Euler-Lagrange equations of motion by varying the action and perform a Legendretransform to put each theory in Hamiltonian form.

I emphasize the similarities, differences,and equivalences of the various theories whenever possible. While this paper is primarily areview, some of the material is in fact new, or at least has not appeared in the literature in5

the form given here. Much of Sections 3, 4, and 5 on the 2+1 theory fall in this category.I should also list a few of the topics that are not covered in this review.

First, I haverestricted attention to the more “standard” theories of 2+1 and 3+1 gravity. I have madeno attempt to treat higher-derivative theories of gravity, supersymmetric theories, or any oftheir equivalents.

Second, I have chosen to omit any discussion of quantum theory, althoughit is here, in quantum theory, that the change in emphasis from geometrodynamics to con-nection dynamics has had the most success. All of the theories described in this paper aretreated at a purely classical level; issues related, for instance, to quantum cosmology and thenon-perturbative canonical quantization program for 3+1 gravity are not dealt with.

This re-view serves, instead, as a thorough pre-requisite for addressing the above issues. Moreover,many books and review articles already exist which discuss the quantum theory in greatdetail.

Interested readers should see, in addition to the text books [2, 3], review articles[17, 18, 19, 20] and references mentioned therein. Third, in 2+1 dimensions, I have chosento concentrate on the relationship between the 2+1 Palatini and Chern-Simons theories, andhave all but ignored the equally interesting relationships between these formulations and thestandard 2+1 dimensional Einstein-Hilbert theory.

Fortunately, other researchers have al-ready addressed these issues, so interested readers can find details in [21, 22, 23]. Also, sinceChern-Simons theory is not available in 3+1 dimensions, the equivalence of the 2+1 Palatiniand Chern-Simons theories does not have a direct 3+1 dimensional analog.

However, recentwork by Carlip [24, 25] and Anderson [26] on the problem of time in 2+1 quantum gravitymay shed some light on the corresponding issue facing the 3+1 theory. Finally, Section 9 ongeneral relativity without-the metric deals exclusively with 3+1 gravity.

Readers interestedin a pure spin-connection formulation of 2+1 gravity should see [15].Penrose’s abstract index notation will be used throughout. Spacetime and spatial tensorindices are denoted by latin letters from the beginning of the alphabet a, b, c, · · · , whileinternal indices are denoted by latin letters from the middle of the alphabet i, j, k, · · · orI, J, K, · · · .

The signature of the spacetime metric gab is taken to be (−+ +) or (−+ ++),depending on whether we are working in 2+1 or 3+1 dimensions. If ∇a denotes the unique,torsion-free spacetime derivative operator compatible with the spacetime metric gab, thenRabcdkd := 2∇[a∇b]kc, Rab := Racbc, and R := Rabgab define the Riemann tensor, Ricci tensor,and scalar curvature of ∇a.Finally, since I eventually want to obtain a Hamiltonian formulation for each theory, Iwill assume from the beginning that the spacetime manifold M is topologically Σ×R.

If thetheory depends on a spacetime metric, I assume Σ to be spacelike; if the theory does notdepend on a spacetime metric, I assume Σ to be any (co-dimension 1) submanifold of M. Ineither case, I ignore all surface integrals and avoid any discussion of boundary conditions.6

In this sense, the results I obtain are rigorous only for the case when Σ is compact. Readersinterested in a detailed discussion of the technically more difficult asymptotically flat case(in the context of the standard Einstein-Hilbert or self-dual theories) should see ChaptersII.2 and III.2 of [2].2.

Einstein-Hilbert theoryIn this section, we will describe the standard Einstein-Hilbert theory. We obtain the vac-uum Einstein’s equation starting from an action principle and perform a Legendre transformto put the theory in Hamiltonian form.

We shall see that the phase space variables con-sist of a positive-definite metric qab and a density-weighted, symmetric, second-rank tensorfield epab. These are the standard geometrodynamical variables of general relativity.

We willalso analyze the motions on phase space generated by the constraint functions and evaluatetheir Poisson bracket algebra. This section is basically a review of standard material.

Ourtreatment will follow that given, for example, in Appendix E of [27] or Chapter II.2 of [2].The standard Einstein-Hilbert theory is, of course, valid in n+1 dimensions. Everythingwe do in this section will be independent of the dimension of the spacetime manifold M.This is an important feature which will allow us to compare the standard Einstein-Hilberttheory with the Chern-Simons and self-dual theories.

Unlike the standard Einstein-Hilberttheory, Chern-Simons theory is defined only in odd dimensions, while the self-dual theory isdefined only in 3+1 dimensions.2.1 Euler-Lagrange equations of motionLet us begin with the well-known Einstein-Hilbert actionSEH(gab) :=ZM√−gR. (2.1)Here g denotes the determinant of the covariant metric gab, and R denotes the scalar cur-vature of the unique, torsion-free spacetime derivative operator ∇a compatible with gab.

Ihave taken the basic variable to be the contravariant spacetime metric gab for conveniencewhen performing variations of the action. The Einstein-Hilbert action is second-order sinceR contains second derivatives of gab.To obtain the Euler-Lagrange equations of motion, we vary the action with respect tothe field variable gab.

If we write the integrand as √−gRabgab and use the fact that δg =−g gabδgab, we getδSEH =ZM√−g(Rab −12Rgab)δgab +ZM√−gδRabgab. (2.2)7

The first integral is of the desired form, while the second integral requires us to evaluate thevariation of the Ricci tensor Rab. Since one can show that5δRabgab = ∇ava(2.3)(where va = ∇a(gbcδgbc) −∇bδgab), we see that modulo a surface integral, δSEH = 0 if andonly ifGab := Rab −12Rgab = 0.

(2.4)This is the desired result:The vacuum Einstein’s equation can be obtained starting froman action principle.I should note that, strictly speaking, the variation of (2.1) with respect to gab does notyield the vacuum Einstein’s equation Gab = 0. The surface integral does not vanish sinceva involves derivatives of the variation δgab.

Even though δgab is required to vanish on theboundary, these derivatives need not vanish. This seems to pose a potential problem, but itcan handled by simply adding to (2.1) a boundary term which will (upon variation) exactlycancel the surface integral.

As shown in Appendix E of [27], this boundary term involves thetrace of the extrinsic curvature of the boundary of M. For the sake of simplicity, however,we will continue to use the unmodified Einstein-Hilbert action (2.1) and ignore all surfaceintegrals as mentioned at the end of Section 1.2.2 Legendre transformTo put the standard Einstein-Hilbert theory in Hamiltonian form, we will follow theusual procedure: We assume that M = Σ×R for some spacelike submanifold Σ and assumethat there exists a time function t (with nowhere vanishing gradient (dt)a) such that eacht = const surface Σt is diffeomorphic to Σ. To talk about evolution from one t = const surfaceto the next, we introduce a future-pointing timelike vector field ta satisfying ta(dt)a = 1. tais the “time flow” vector field that defines the same point in space at different instants oftime.

We will treat ta and the foliation of M by the t = const surfaces as kinematical (i.e.,non-dynamical) structure. Evolution will be given by the Lie derivative with respect to ta.Since we have a spacetime metric gab as one of our field variables, we can also introduce aunit covariant normal na and its associated future-pointing timelike vector field na = gabnb.5To obtain this result, consider a 1-parameter family of spacetime metrics gab(λ) and their associatedspacetime derivative operators λ∇a.

Define Cabc by λ∇akb =: ∇akb +λCabckc and differentiate λ∇agbc(λ) = 0with respect to λ. Evaluating this expression at λ = 0 gives Cabc = −12gcd(∇aδgbd +∇bδgad −∇dδgab), wheregab := gab(0) and δgab :=ddλλ=0gab(λ).

Since Rabcd(λ) = Rabcd + λ 2∇[aCb]cd + λ2 [Ca, Cb]cd, it followsthat δRac :=ddλλ=0Rabcb(λ) = 2∇[aCb]cb. Contracting with gac (using δgab = −gacgbdδgcd) yields the aboveresult.8

Note that since nana = −1,qab := δab + nanb is a projection operator into the t = constsurfaces. We will construct the configuration variables associated with the field variable gabby contracting with na and qab.

We define the induced metric qab, the lapse N, and shift Naviaqab := qma qnb gmn (= gab + nanb),N := −natb gab,andNa := qab tb. (2.5)Note that in terms of N and Na, we can write ta = Nna +Na.

Furthermore, since Nana = 0and qabna = 0, Na and qab are (in 1-1 correspondence with) tensor fields defined intrinsicallyon Σ.The next step in constructing a Hamiltonian formulation of the Einstein-Hilbert theoryis to decompose the Einstein-Hilbert action and write it in the formSEH(gab) =Zdt LEH(q, ˙q). (2.6)LEH will be the Einstein-Hilbert Lagrangian provided it depends only on (qab, N, Na) andtheir first time derivatives.

But, as written, (2.1) is not convenient for such a decomposition.The integrand √−gR contains second time derivatives of the configuration variable qab.However, as we will now show, these terms can be removed from the integrand by subtractingoffa total divergence.To see this, let us write the scalar curvature R as R = 2(Gab −Rab)nanb. Then thedifferential geometric identitiesGabnanb = 12(R −KabKab + K2)andRabnanb = −KabKab + K2 + ∇b(na∇anb −nb∇ana)(2.7)(where Kab := qma qnb ∇mnn is the extrinsic curvature of the t = const surfaces and R is thescalar curvature of the unique, torsion-free spatial derivative operator Da compatible withthe induced metric qab) implyR = (R + KabKab −K2) + (total divergence term).

(2.8)Using the fact that √−g = N√q dt (where q denotes the determinant of qab), the Einstein-Hilbert action (2.1) becomesSEH(gab) =ZdtZΣ√qN(R + KabKab −K2) + (surface integral). (2.9)If we ignore the surface integral, we getLEH =ZΣ√qN(R + KabKab −K2).

(2.10)9

This is the desired Einstein-Hilbert Lagrangian first proposed by Arnowitt, Deser, and Misner(ADM) [28]. The identity Kab =12N (L⃗t qab −2D(aNb)) allows us to express LEH in terms ofonly (qab, N, Na) and their first time derivatives.Given the Einstein-Hilbert Lagrangian, we are now ready to perform the Legendre trans-form.

But before we do this, it is probably worthwhile to make a detour and first review thestandard Dirac constraint analysis for a theory with constraints and recall some basic ideasof symplectic geometry. I propose to examine, in detail, a simple finite-dimensional systemdescribed by a LagrangianL(q, ˙q) := 12 ˙q12 + q3 ˙q2 −q4f(q2, q3).

(2.11)Here (q1, · · · , q4) ∈C0 are the configuration variables and ( ˙q1, · · · , ˙q4) are their associated timederivatives (or velocities). f(q2, q3) can be any (smooth) real-valued function of (q2, q3).

Thetechniques that arise when analyzing this simple system will apply not only to the standardEinstein-Hilbert theory but to many other constrained theories as well. Readers interested ina more detailed description of the general Dirac constraint analysis and symplectic geometryshould see [29] and Appendix B of [3], respectively.Readers already familiar with thestandard Dirac constraint analysis may skip to the paragraph immediately following equation(2.25).To perform the Legendre transform for our simple system, we first define the momentumvariables (p1, · · ·, p4) viapi := δLδ ˙qi(i = 1, · · · , 4).

(2.12)For the special form of the Lagrangian given above, they becomep1 = ˙q1,p2 = q3,p3 = 0,andp4 = 0. (2.13)Since only the first equation can be inverted to give ˙q1 as a function of (q, p), there areconstraints: Not all points in the phase space Γ0 = T ∗C0 = {(qi, pi)| i = 1, · · ·, 4} areaccessible to the system.

Only those (q, p) ∈Γ0 which satisfyφ1 := p2 −q3 = 0,φ2 := p3 = 0,andφ3 := p4 = 0(2.14)are physically allowed. The φi’s are called primary constraints and the vanishing of thesefunctions define a constraint surface in Γ0.It is the presence of these constraints thatcomplicates the standard Legendre transform.Following the Dirac constraint analysis, we now must now write down a Hamiltonian forthe theory.

But due to (2.14), the Hamiltonian will not be unique. The usual definitionH0(q, p) := P4i=1 pi ˙qi −L(q, ˙q) does not work, since there exist ˙qi’s which cannot be written10

as functions of q and p. If, however, we restrict ourselves to the constraint surface definedby (2.14), we haveH0(q, p) = 12p12 + q4f(q2, q3). (2.15)Since the right hand side of (2.15) makes sense on all of Γ0, H0(q, p) actually defines onepossible choice of Hamiltonian.

However, as we will show below, this Hamiltonian is definitelynot the only one.For suppose λ1, λ2, and λ3 are three arbitrary functions on Γ0. ThenHT(q, p) : = H0(q, p) + λ1φ1 + λ2φ2 + λ3φ3= 12p12 + q4f(q2, q3) + λ1(p2 −q3) + λ2p3 + λ3p4(2.16)is another function (defined on all of Γ0) that agrees with H0(q, p) on the constraint surface.HT(q, p) is called the total Hamiltonian, and it differs from H0(q, p) by terms that vanish onthe constraint surface.

This non-uniqueness of the total Hamiltonian exists for any theorythat has constraints.Given HT(q, p), the next step in the Dirac constraint analysis is to require that theprimary constraints (2.14) be preserved under time evolution—i.e., that˙φi := {φi, HT}0 ≈0(i = 1, 2, 3). (2.17)Here ≈means equality on the constraint surface defined by (2.14) and { , }0 denotes thePoisson bracket defined by the natural symplectic structure6Ω0 = dp1 ∧dq1 + dp2 ∧dq2 + dp3 ∧dq3 + dp4 ∧dq4(2.18)on Γ0.

Equation (2.17) is equivalent to the requirement that the evolution of the systemtake place on the constraint surface.Evaluating (2.17) for the primary constraints (2.14), we find that {φ3, HT}0 ≈0 impliesφ4 := f(q2, q3) ≈0. (2.19)6A symplectic manifold (or phase space) consists of a pair (Γ0, Ω0), where Γ0 is an even dimensionalmanifold and Ω0 is a closed and non-degenerate 2-form.

(i.e., dΩ0 = 0 and Ω0(v, w) = 0 for all w impliesv = 0. )Ω0 is called the symplectic structure and it allows us to define Hamiltonian vector fields andPoisson brackets: Given any real-valued function f : Γ0 →R, the Hamiltonian vector field Xf is defined by−iXf Ω0 := df.

Given any two real-valued functions f, g : Γ0 →R, the Poisson bracket {f, g}0 is definedby {f, g}0 := −Ω(Xf, Xg) = −Xf(g). As a special case, if Γ0 = T ∗C0 is the cotangent bundle over somen-dimensional configuration space C0, then Ω0 = dp1∧dq1+· · ·+dpn∧dqn is the natural symplectic structureon Γ0 associated with the chart (q, p).

It follows that {f, g}0 = Pni=1( ∂f∂qi∂g∂pi −∂f∂pi∂g∂qi ), which is the standardtextbook expression for the Poisson bracket of f and g.11

The other Poisson brackets yield conditions on λ1 and λ2. φ4 is called a secondary constraint,and for consistency we must also require that˙φ4 := {φ4, HT}0 ≈0.

(2.20)Here ≈now means equality on the constraint surface defined by (2.14) and (2.19). Sinceone can show that (2.20) follows from the earlier conditions on λ1 and λ2, (2.14) and (2.19)constitute all the constraints of the theory.The final step in the Dirac constraint analysis is to take all the constraints φ1, · · ·, φ4and evaluate their Poisson brackets.

If a constraint φi satisfies {φi, φj}0 ≈0 for all φj, thenφi is said to be 1st class. If, however, {φi, φj}0 ̸≈0 for some φj, then φi and φj are saidto form a 2nd class pair.

(In terms of symplectic structures and Hamiltonian vector fields,a constraint φi is 1st class with respect to the symplectic structure Ω0 if and only if theHamiltonian vector field Xφi defined by Ω0 is tangent to the constraint surface defined bythe vanishing of all the constraints.) The goal is to solve all the 2nd class constraints (andpossibly some 1st class constraints) and obtain a new phase space (Γ, Ω) where the remainingconstraints (pulled-back to Γ) are all 1st class with respect to the Poisson bracket defined byΩ.

Evaluating {φi, φj}0 for our simple system, we find that φ3 is the only 1st class constraintwith respect to Ω0. By solving the second class pair φ1 = 0 and φ2 = 0, we getΩ0φ1=0, φ2=0 = dp1 ∧dq1 + dq3 ∧dq2 + dp4 ∧dq4(2.21)andHTφ1=0, φ2=0 = 12p12 + q4f(q2, q3) + λ3p4.

(2.22)The remaining constraints φ3 and φ4 are now both 1st class with respect to this new sym-plectic structure.Although we have successively eliminated all the 2nd class constraints, we can go onestep further. We can solve the 1st class constraint φ3 := p4 = 0 by gauge fixing the configu-ration variable q4.

Even though this step is not required by the Dirac constraint analysis, itsimplifies the final phase space structure somewhat. Solving φ3 = 0 and pulling-back (2.21)and (2.22) to this new constraint surface Γ (coordinatized by (q1, q2, q3, p1)), we obtainΩ:= dp1 ∧dq1 + dq3 ∧dq2(2.23)andH(q1, q2, q3, p1) := 12p12 + q4f(q2, q3).

(2.24)Here q4 is no longer thought of as a dynamical variable—it is a Lagrange multiplier of thetheory associated with the 1st class constraint f(q2, q3) = 0.12

To summarize: Given a Lagrangian of the formL(q, ˙q) := 12 ˙q12 + q3 ˙q2 −q4f(q2, q3),(2.25)the Dirac constraint analysis says that the momentum p1 is unconstrained, while p2 = q3and p3 = p4 = 0. Demanding that the constraints be preserved under evolution, we obtain asecondary constraint f(q2, q3) = 0.

The constraints p2 −q3 = 0 and p3 = 0 form a 2nd classpair and are easily solved; the remaining constraints p4 = 0 and f(q2, q3) = 0 now form a 1stclass set. By gauge fixing q4 we can solve p4 = 0, and thus obtain a new phase space (Γ, Ω)coordinatized by (q1, q2, q3, p1) with symplectic structure (2.23) and Hamiltonian (2.24).

Weare left with a single 1st class constraint, f(q2, q3) = 0.Let us now return to our analysis of the standard Einstein-Hilbert theory. Given LEH,we find that the momentum epab canonically conjugate to qab is given byepab := δLEHδL⃗t qab= √q(Kab −Kqab),(2.26)while the momenta canonically conjugate to N and Na are zero.

Since equation (2.26) canbe inverted to giveL⃗t qab = 2Nq−1/2(epab −12epqab) + 2D(aNb),(2.27)it does not define a constraint. However, N and Na play the role of Lagrange multipliers.Thus, by following the Dirac constraint analysis we find that the phase space (ΓEH, ΩEH)of the standard Einstein-Hilbert theory is coordinatized by the pair (qab, epab) and has sym-plectic structure7ΩEH =ZΣ dIepab ∧∧dIqab.

(2.28)The Hamiltonian is given byHEH(q, ep) =ZΣ N−q1/2R + q−1/2(epab epab −12ep2)−2NaqabDc epbc. (2.29)As we shall see in the next subsection, this is just a sum of 1st class constraint functionsassociated witheC(q, ep) := −q1/2R + q−1/2(epab epab −12ep2) ≈0and(2.30a)eCa(q, ep) := −2qabDc epbc ≈0.

(2.30b)7I use dI and ∧to denote the infinite-dimensional exterior derivative and infinite-dimensional wedgeproduct of forms on ΓEH. They are to be distinguished from d and ∧which are the finite-dimensionalexterior derivative and finite-dimensional wedge product of forms on Σ.

Note that in terms of the Poissonbracket { , } defined by ΩEH, we have {qab(x), epcd(y)} = δc(aδdb)δ(x, y).13

Note that constraint equation (2.30a) is non-polynomial in the canonically conjugate vari-ables due to the dependence of R on the inverse of qab. This is a major stumbling blockfor the canonical quantization program in terms of (qab, epab).

To date, there exist no ex-act solutions to the quantum version of this constraint in full (i.e., non-truncated) generalrelativity.2.3 Constraint algebraTo evaluate the Poisson brackets of the constraints and to determine the motions theygenerate on phase space, we must first construct constraint functions (i.e., mappings ΓEH →R) associated with the constraint equations (2.30a) and (2.30b). To do this, we smear eC(q, ep)and eCa(q, ep) with test fields N and Na on Σ—i.e., we defineC(N) :=ZΣ N−q1/2R + q−1/2(epab epab −12ep2)and(2.31a)C( ⃗N) :=ZΣ −2NaqabDc epbc.

(2.31b)They are called the scalar and vector constraint functions of the standard Einstein-Hilberttheory.The next step is to evaluate the functional derivatives of C(N) and C( ⃗N). For recallthat if f, g : ΓEH →R are any two real-valued functions on phase space, the Hamiltonianvector field Xf (defined by the symplectic structure (2.28)) is given byXf =ZΣδfδ epabδδqab−δfδqabδδ epab(2.32)and the Poisson bracket {f, g} (defined by {f, g} := −Xf(g)) is given by{f, g} =ZΣδfδqabδgδ epab −δfδ epabδgδqab.

(2.33)Note that under the 1-parameter family of diffeomorphisms on ΓEH associated with Xf,qab 7→qab + ǫ δfδ epab + O(ǫ2)and(2.34a)epab 7→epab −ǫ δfδqab+ O(ǫ2). (2.34b)We will use (2.33) to determine the various Poisson brackets between C(N) and C( ⃗N); wewill use (2.34a) and (2.34b) to determine the motions that they generate on phase space.Let us begin with the vector constraint C( ⃗N).

Integrating (2.31b) by parts and notingthat 2D(aNb) = L ⃗Nqab, we getC( ⃗N) =ZΣ(L ⃗Nqab)epab= −ZΣ qab(L ⃗N epab). (2.35)14

By inspection,δC( ⃗N)δqab= −L ⃗N epabandδC( ⃗N)δ epab= L ⃗Nqab. (2.36)Thus, we see thatqab 7→qab + ǫL ⃗Nqab + O(ǫ2)and(2.37a)epab 7→epab + ǫL ⃗N epab + O(ǫ2)(2.37b)is the motion on ΓEH generated by C( ⃗N).

Note that (2.37a) and (2.37b) are the maps onthe tensor fields qab and epab induced by the 1-parameter family of diffeomorphisms on Σassociated with the vector field Na. In other words, the Hamiltonian vector field XC( ⃗N) onΓEH is the lift of the vector field Na on Σ.Let us now consider the scalar constraint C(N).

Due to the non-polynomial dependenceof R on qab, the functional derivative δC(N)/δqab is much harder to evaluate. After a fairlylong calculation, one finds that8δC(N)δqab= −12N eC(q, ep)qab + 2Nq−1/2(epac epbc −12epepab)+ Nq1/2(Rab −Rqab) −q1/2(DaDbN −qabDcDcN).

(2.38)A much simpler calculation givesδC(N)δ epab= 2Nq−1/2(epab −12epqab). (2.39)Recall that for the vector constraint function C( ⃗N), the motion on ΓEH along XC( ⃗N) corre-sponded to the Lie derivative of qab and epab with respect to Na.

Thus, one might expect themotion on ΓEH along XC(N) to correspond to the Lie derivative with respect to ta := Nna.We will now show that if we restrict ourselves to the constraint surface ΓEH ⊂ΓEH (definedby (2.30a) and (2.30b)), then this is actually the case.Comparing (2.39) with equation (2.27) (setting Na = 0), we see that δC(N)/δ epab =L⃗t qab, soqab 7→qab + ǫL⃗t qab + O(ǫ2)(2.40)as conjectured. Similarly, writing epab = √q(Kab −Kqab) and using the differential geometricidentityLN⃗nKab = −NRab + 2NKacKbc −NKKab + DaDbN(2.41)8To obtain this result we used the facts that δq = q qabδqab and δRabqab = Dava for va = −Da(qbcδqbc)+Db(qacδqbc).

These are just the spatial analogs of the results used in subsection 2.1 when we varied theEinstein-Hilbert action with respect to gab.15

(which holds in this form when Rab = 0), we see thatδC(N)δqab= −12N eC(q, ep)qab + L⃗t epab. (2.42)Thus,epab 7→epab + ǫ12N eC(q, ep)qab + L⃗t epab+ O(ǫ2).

(2.43)If we now restrict ourselves to ΓEH (so that eC(q, ep) = 0), we getepab 7→epab + ǫL⃗t epab + O(ǫ2). (2.44)This is the desired result:When restricted to ΓEH ⊂ΓEH, the Hamiltonian vector fieldXC(N) on ΓEH is the lift of the vector field ta := Nna on Σ.We are now ready to evaluate the Poisson brackets between the constraint functions.

Butfirst note that if f(M) : ΓEH →R is any real-valued function on phase space of the formf(M) :=ZΣ Ma···bc···d efa···bc···d(q, ep)(2.45)(were Ma···bc···d is any tensor field on Σ which is independent of qab and epab), then{C( ⃗N), f(M)} =ZΣ −L ⃗N epabδf(M)δ epab−L ⃗Nqabδf(M)δqab=ZΣ −Ma···bc···d L ⃗N efa···bc···d(q, ep). (2.46)Integrating the last line of (2.46) by parts and throwing away the surface integral, we get{C( ⃗N), f(M)} = f(L ⃗NM).

(2.47)Thus, the Poisson bracket of C( ⃗N) with any other constraint function is easy to evaluate.We have{C( ⃗N), C( ⃗M)} = C([ ⃗N, ⃗M])and(2.48a){C( ⃗N), C(M)} = C(L ⃗NM),(2.48b)where [ ⃗N, ⃗M] := L ⃗NMa is the commutator of the vector fields Na and Ma on Σ. Notethat (2.48a) tells us that the subset of vector constraint functions is closed under Poissonbrackets. In fact, Na 7→C( ⃗N) is a representation of the Lie algebra of vector fields on Σ.The commutator of vector fields on Σ is mapped to the Poisson bracket of the correspondingvector constraint functions.16

We are left with only the Poisson bracket {C(N), C(M)} of two scalar constraint func-tions to evaluate. Using (2.38) and (2.39) (and eliminating all terms symmetric in M andN), we get{C(N), C(M)} =ZΣ −2M(DaDbN −qabDcDcN)(epab −12epqab) −(N ↔M)=ZΣ −2(N∂aM −M∂aN)qabDc epbc= C( ⃗K),(2.49)where Ka := (N∂aM −M∂aN) = qab(N∂bM −M∂bN).

Thus, the Poisson bracket of twoscalar constraints is a vector constraint. Although this implies that the subset of scalarconstraint functions is not closed under Poisson bracket, the totality of constraint func-tions (scalar and vector) is—i.e., the constraint functions form a 1st class set as claimed insubsection 2.2.

Note, however, that since the vector field Ka depends on the phase spacevariable qab (through its inverse), the Poisson bracket (2.49) involves structure functions.The constraint functions do not form a Lie algebra.3. 2+1 Palatini theoryIn this section, we will describe the 2+1 Palatini theory which, as we shall see at the endof subsection 3.2, is defined for any Lie group G. We will discuss the relationship betweenthe Palatini and Einstein-Hilbert actions, and show how the 2+1 Palatini theory basedon SO(2, 1) reproduces the standard results of 2+1 gravity.

After performing a Legendretransform to put this theory in Hamiltonian form, we shall see that the phase space variablesconsist of a connection 1-form AIa (which takes values in the Lie algebra of G) and itscanonically conjugate momentum (or “electric field”) eEaI . Thus, for G = SO(2, 1), the 2+1Palatini theory gives us a connection dynamic description of 2+1 gravity.

The constraintequations are polynomial in the basic variables and the constraint functions form a Lie algebrawith respect to Poisson bracket.Once we write the 2+1 Palatini action in its generalized form, we will let G be an arbitraryLie group. To reproduce the results of 2+1 gravity, we simply take G to be SO(2, 1).

Notethat much of the material in subsections 3.2 and 3.3 can also be found in [30].3.1 Euler-Lagrange equations of motionRecall the standard Einstein-Hilbert action of Section 2,SEH(gab) =ZΣ√−gR. (3.1)17

To define the 2+1 Palatini action, it is convenient to first rewrite the integrand √−gR intriad notation. But in order to do this, we will have to make a short mathematical digression.Readers interested in a more detailed discussion of what follows should see [31].

Readersalready familiar the method of orthonormal bases may skip to the paragraph immediatelyfollowing equation (3.9).Consider an n-dimensional manifold M, and let V be a fixed n-dimensional vector spacewith Minkowski metric ηIJ having signature (−+ · · · +). A soldering form at p ∈M is anisomorphism eIa(p) : TpM →V.

(Here TpM denotes the tangent space to M at p.) Althoughan n-manifold does not in general admit a globally defined soldering form eIa, we can use eIato define tensor fields locally on M. For instance,gab := eIaeJb ηIJ(3.2a)is a (locally defined) spacetime metric having the same signature as ηIJ. The inverse of eIawill be denoted by eaI; it satisfiesgabeaIebJ = ηIJ.

(3.2b)Spacetime tensor fields with additional internal indices I, J, K, · · · will be called generalizedtensor fields on M. Spacetime indices are raised and lowered with the spacetime metric gab;internal indices are raised and lowered with the Minkowski metric ηIJ.If one introduces a standard basis {bII |I = 1, · · · , n} in V , then the vector fields eaI := eaIbIIform an orthonormal basis of gab. These n-vector fields will be called a triad when n = 3and a tetrad when n = 4.

The dual co-vector fields, eIa := gabηIJebJ, will be called a co-triadand a co-tetrad when n = 3 and 4, respectively. I should note, however, that from now onI will ignore the distinction between a soldering form eIa and the co-vector fields eIa.

I willcall a 3-dimensional soldering form 3eIa a co-triad and a 4-dimensional soldering form 4eIa aco-tetrad in what follows.To do calculus with these generalized tensor fields, it is necessary to extend the definitionof spacetime derivative operators so that they also “act” on internal indices. We require(in addition to the usual properties that a spacetime derivative operator satisfies) that ageneralized derivative operator obey the linearity, Leibnitz, and commutativity with contrac-tion rules with respect to the internal indices.

Furthermore, we require that all generalizedderivative operators be compatible with ηIJ. Given these properties, it is straightforward toshow that the set of all generalized derivative operators has the structure of an affine space.In other words, if ∂a is some fiducial generalized derivative operator (which we treat as anorigin in the space of generalized derivative operators), then any other generalized derivativeoperator Da is completely characterized by a pair of generalized tensor fields Aabc and AaIJ18

defined byDakbI =: ∂akbI + AabckcI + AaIJkbJ. (3.3)We will call Aabc and AaIJ the spacetime connection 1-form and internal connection 1-formof Da.

It is easy to show thatAaIJ = Aa[IJ]andAabc = A(ab)c.(3.4)These conditions follow from the requirements that all generalized derivative operators becompatible with ηIJ and that they be torsion-free. Later in this section, we will considerwhat happens if we allow derivative operators to have non-zero torsion—i.e., if A[ab]c ̸= 0.Finally, note that Aabc need not equal AaIJeIbecJ, in general.As usual, given a generalized derivative operator Da, we can construct curvature tensorsby commuting derivatives.

The internal curvature tensor FabI J and the spacetime curvaturetensor Fabcd are defined by2D[aDb]kI =: FabIJkJand(3.5a)2D[aDb]kc =: Fabcdkd. (3.5b)If our fiducial generalized derivative operator is chosen to be flat on both spacetime andinternal indices, thenFabIJ = 2∂[aAb]IJ + [Aa, Ab]IJand(3.6a)Fabcd = 2∂[aAb]cd + [Aa, Ab]cd.

(3.6b)Here [Aa, Ab]IJ := (AaI KAbKJ −AbIKAaKJ) and [Aa, Ab]cd := (AaceAbed −AbceAaed) are thecommutators of linear operators.Just as a compatibility with a spacetime metric gab defines a unique, torsion-free spacetimederivative operator ∇a, compatibility with an orthonormal basis eaI (and thus with gab) definesa unique torsion-free generalized derivative operator, which we also denote by ∇a.TheChristoffel symbols ΓaIJ and Γabc are defined by∇akbI =: ∂akbI + ΓabckcI + ΓaIJkbJ,(3.7)and satisfyΓaIJ = −ebJ(∂aebI + ΓabcecI)and(3.8a)Γabc = −12gcd(∂agbd + ∂bgad −∂dgab). (3.8b)19

It also follows that internal and spacetime curvature tensors RabI J and Rabcd of ∇a are relatedbyRabIJ = RabcdecIeJd. (3.9)We will need the above result in this and later sections to show that the Palatini and self-dualactions reproduce Einstein’s equation.Now let us return to our discussion of the 2+1 Palatini theory.

Recall that we wanted towrite the integrand √−gR in triad notation. UsingRabIJ = Rabcd 3ecI3eJd(3.10)(which is equation (3.9) written in terms of a triad 3eaI) andǫabc =3eIa3eJb3eKc ǫIJK(3.11)(which relates the volume element ǫabc of gab = 3eIa3eJb ηIJ to the volume element ǫIJK of ηIJ),we find that√−gR = √−g δb[dδce] Rbcde= 12eηabcǫadeRbcde= 12eηabc 3eIa3eJd3eKe ǫIJKRbcde= 12eηabcǫIJK3eIa RbcJK.

(3.12)Thus, viewed as a functional of a co-triad 3eIa, the standard Einstein-Hilbert action is givenbySEH(3e) = 12ZΣeηabcǫIJK3eIa RbcJK. (3.13)To obtain the 2+1 Palatini action, we simply replace RabI J in (3.13) with the internalcurvature tensor 3FabI J of an arbitrary generalized derivative operator 3Da defined by3DakI := ∂akI + 3AaIJkJ.

(3.14)We define the 2+1 Palatini action based on SO(2, 1) to beSP(3e, 3A) := 14ZMeηabcǫIJK3eIa3FbcJK,(3.15)where 3FabI J = 2∂[a3Ab]IJ + [3Aa, 3Ab]IJ. Note that I have included an additional factor of1/2 in definition (3.15).

This overall factor will not affect the Euler-Lagrange equations ofmotion in any way, but it will change the canonically conjugate variables. I have chosen20

to use this action so that the expressions for our canonically conjugate variables agree withthose used in the literature (see, e.g., [30]).As defined above, SP(3e, 3A) is a functional of both a co-triad 3eIa and a connection 1-form3AaIJ which takes values in the defining representation of the Lie algebra of SO(2, 1). Notealso that 3Da as defined by (3.14) knows how to act only on internal indices.

We do not requirethat 3Da know how to act on spacetime indices. However, when performing calculations, wewill find that it is often convenient to consider a torsion-free extension of 3Da to spacetimetensor fields.

It turns out that all calculations and all results will be independent of ourchoice of torsion-free extension. In fact, we will see that these results hold for extensions of3Da that have non-zero torsion as well.Since the 2+1 Palatini action is a functional of both a co-triad and a connection 1-form,we will obtain two Euler-Lagrange equations of motion.

When we vary 3eIa, we geteηabcǫIJK3FbcJK = 0. (3.16)When we vary 3AaIJ, we get3Db(eηabcǫIJK3eKc ) = 0.

(3.17)To arrive at (3.17), we considered a torsion-free extension of 3Da to spacetime tensor fields(so that δ3FbcJK = 2 3D[bδ3Ac]JK) and then integrated by parts. The surface integral vanishedsince δ3AcJK = 0 on the boundary, while the remaining term gave (3.17).

Note that sincethe left hand side of (3.17) is the divergence of a skew spacetime tensor density of weight +1on M, it is independent of the choice of torsion-free extension of 3Da. Since eηabcǫIJK 3eKc =2(3e) 3e[aI3eb]J (where (3e) := √−g), we can rewrite (3.17) as3Db(3e) 3e[aI3eb]J= 0.

(3.18)We shall see in Section 6 that the form of equation (3.18) holds for the 3+1 Palatini theoryas well.To determine the content of equation (3.18), let us express 3Da in terms of the unique,torsion-free generalized derivative operator ∇a compatible with 3eIa, and 3CaIJ defined by3DakI =: ∇akI + 3CaIJkJ. (3.19)Since (3.18) is the divergence of a skew spacetime tensor density of weight +1 on M, andsince ∇a is compatible with 3eIa, we get3CbIK 3e[aK3eb]J + 3CbJK 3e[aI3eb]K = 0.

(3.20)This is equivalent to the statement that the (internal) commutator of 3CbIJ and 3e[aI3eb]Jvanishes. We will now show that (3.20) implies that 3CaIJ = 0.21

To see this, define a spacetime tensor field 3Sabc via3Sabc := 3CaIJ3eIb3eJc . (3.21)(Note, incidently, that 3Sabc is not the spacetime connection of 3Da relative to ∇a.) Then thecondition 3CaIJ = 3Ca[IJ] is equivalent to 3Sabc = 3Sa[bc].

Now contract equation (3.20) with3eIa3eJc . This yields 3Sbcb = 0, so 3Sabc is trace-free on its first and last indices.

Using thisresult, (3.20) reduces to3CbIK 3eaK3ebJ −3CbJK 3ebI3eaK = 0. (3.22)If we now contract (3.22) with 3eIc3eJd, we get3Scda = 3S(cd)a.

(3.23)Thus, 3Sabc is symmetric in its first two indices. Since 3Sabc = 3Sa[bc] and 3Sabc = 3S(ab)c, wecan successively interchange the first two and last two indices (with the appropriate signchanges) to show 3Sabc = 0.

Futhermore, since eIa are invertible, we get 3CaI J = 0. This isthe desired result.9Since 3CaIJ = 0, we can conclude that the generalized derivative operator 3Da must agreewith ∇a when acting on internal indices.

Thus, although the Palatini action started as afunctional of a co-triad and an arbitrary generalized derivative operator 3Da, we find thatone equation of motion implies that 3Da = ∇a. In terms of connection 1-forms, 3CaIJ = 0implies that 3AaIJ = ΓaIJ, where ΓaI J is the internal Christoffel symbol of ∇a.

Using thisresult, the remaining Euler-Lagrange equation of motion (3.16) becomeseηabcǫIJKRbcJK = 0. (3.24)When (3.24) is contracted with 3edI, we get Gad = 0.

Thus, the Palatini action based onSO(2, 1) reproduces the standard 2+1 vacuum Einstein’s equation.It is interesting to note that to show that the Palatini action reduces to the standardEinstein-Hilbert action in 2+1 dimensions, we need only vary the connection 1-form 3AaI J.Since we found that (3.17) could be solved uniquely for 3AaI J in terms of the remaining basicvariables 3eIa, we can pull-back SP(3e, 3A) to the solution space 3AaI J = ΓaIJ and obtain a newaction SP(3e), which depends only on a co-triad. This pulled-back action is just 1/2 timesthe standard Einstein-Hilbert action SEH(3e) given by (3.13).

But what about the boundaryterm that one should strictly include in the standard Einstein-Hilbert action? It looks as ifSP(3e) is missing this needed term.9This method of proving 3CaIJ = 0—which generalizes to the 3+1 Palatini and self-dual actions—wasshown to me by J. Samuel and A. Ashtekar.22

The answer to this question is the following: Whereas the standard Einstein-Hilbertaction is a second-order action, the 2+1 Palatini action is first-order. As mentioned at thebeginning of Section 2, varying the standard Einstein-Hilbert action (3.1) with respect to gabproduces a surface integral involving derivatives of the variation δgab.

Since we are allowedonly to keep gab fixed on the boundary, this surface integral is non-vanishing and must becompensated for by adding a boundary term to (3.1). This is also the case if we vary SEH(3e)given by (3.13) with respect to 3eIa.

On the other hand, when we vary the Palatini action(3.15) with respect to 3AaIJ, we hold 3AaI J fixed on the boundary and 3eIa fixed throughout.Then by solving (3.17) uniquely for 3AaIJ, we can pull-back SP(3e, 3A) to the solution space3AaIJ = ΓaI J. But now when we vary SP(3e) with respect to 3eIa which lie entirely in thesolution space, fixing 3eIa on the boundary also fixes certain derivatives of 3eIa on the boundary.This is a reflection of the fact that the reduction procedure comes with a prescription on howto do variations.

It is precisely the vanishing of these derivatives of δ3eIa which eliminatesthe need of a boundary term for SP(3e).It is also interesting to note that we could obtain the same result (3AaI J = ΓaIJ) byconsidering an extension of 3Da to spacetime tensor fields with non-zero torsion 3Tabc. (Recallthat if 3Aabc denotes the spacetime connection 1-form of the extension of 3Da, then the torsiontensor 3Tabc is defined by 23D[a 3Db]f =: 3Tabc 3Dcf and satisfies 3Tabc = 2 3A[ab]c.) By varyingthe 2+1 Palatini action (3.15) with respect to 3AaIJ, we would find2 3D[a3eIb] −3Tabc 3eIc = 0.

(3.25)This is the field equation for 3AaI J which holds for any extension of 3Da to spacetime tensorfields. If we restrict ourselves to torsion-free extensions, we get back equation (3.17).

Thenby following the argument given there, we would find 3AaIJ = ΓaIJ as before.However, there exists an alternative approach to solving equation (3.25) which is oftenused by particle physicists. Namely, instead of considering a torsion-free extension of 3Da tospacetime tensor fields, one considers an extension of 3Da to spacetime tensor fields which iscompatible with the co-triad 3eIa.

This can always be done, but the price of such an extensionis in general a non-zero torsion tensor 3Tabc.But since we now have 3Da 3eIb = 0, equation(3.25) implies3Tabc 3eIc = 0. (3.26)Invertibility of 3eIc then implies that 3Tabc = 0.Since there exists only one torsion-freederivative operator compatible with 3eIa, we can conclude that 3Da = ∇a(or equivalently,3AaIJ = ΓaIJ).

This is the desired result.Finally, to conclude this section, let us write the 2+1 Palatini action (3.15) in a formwhich is valid for any Lie group G. Recall that the connection 1-form 3AaIJ—being a linear23

operator on the internal 3-dimensional vector space (equipped with the Minkowski metricηIJ) and satisfying 3AaIJ = 3Aa[IJ]—takes values in the defining representation of the Liealgebra of SO(2, 1). Since dim(SO(2, 1)) = 3 (which is the same as the dimension of theinternal vector space), we can define an SO(2, 1) Lie algebra-valued connection 1-form, 3AIa,via3AaIJ =: 3AKa ǫJIK.

(3.27)This is just the adjoint representation of the Lie algebra of SO(2, 1) with respect to thestructure constants ǫI JK := ηIMǫMJK.10 That the defining representation and adjoint repre-sentation agree is a property that holds only in 2+1 dimensions since dim(SO(n, 1)) = n+ 1if and only if n = 2. In terms of 3AIa, the generalized derivative operator 3Da satisfies3DavI = ∂avI + [3Aa, v]I,(3.28)where [3Aa, v]I := ǫI JK 3AJavK.

From (3.27), it also follows that the Lie algebra valued-curvature tensor 3F Iab (which is related to 3FabI J via 3FabI J = 3F KabǫJ IK) can be written as3F Iab = 2∂[a3AIb] + [3Aa, 3Ab]I. (3.29)Thus, in terms 3AIa and 3F Iab, the Palatini action becomesSP(3e, 3A) = 14ZMeηabcǫIJK3eIa3FbcJK= 14ZMeηabcǫIJK3eIa3F Lbc ǫKJL= 12ZMeηabc 3eaI3F Ibc.

(3.30)But now note that the last line above suggests a natural generalization. Namely, let Gbe any Lie group with Lie algebra LG, and let 3AIa and 3eaI be LG- and L∗G-valued 1-forms,respectively.

Although the action given by (3.30) was originally defined for the Lie groupSO(2, 1), it is well-defined in the above sense for any Lie group G.3F Iab is still the curvaturetensor of 3AIa, but 3eaI can no longer be thought of as a co-triad. In fact, since G is now10Given a Lie algebra L with structure constants CIJK, the adjoint representation of L by linear op-erators on L is defined by the mapping vI ∈L 7→(adv)I J := vKCJ IK.Under ad, the Lie bracket[v, w]I := CIJKvJwK ∈L maps to the commutator of linear operators [adv, adw]IJ := (adv)I K(adw)KJ −(adw)I K(adv)K J. I should note that since (adv)I JwI = −[v, w]J, the above definition of the adjoint rep-resentation differs in sign from that given in most math and physics textbooks.

The sign difference can betraced to my definition of the commutator of linear operators, which also differs in sign from the standarddefinition.24

arbitrary, the index I can take any value 1, 2, · · ·, dim(G). Nonetheless, we can still definethe Palatini action based on G viaGSP(3e, 3A) := 12ZMeηabc 3eaI3F Ibc,(3.31)which we treat it as a functional of an LG-valued connection 1-form 3AIa and an L∗G-valuedcovector field 3eaI.

The equations of motion we obtain by varying 3eaI and 3AIa areeηabc 3F Ibc = 0and3Db(eηabc 3ecI) = 0,(3.32)which are the analogs of equations (3.16) and (3.17). As before, the second equation re-quires a torsion-free extension of 3Da to spacetime tensor fields, but again, all results will beindependent of this choice.3.2 Legendre transformGiven the action (3.31), it is a straightforward exercise to put the 2+1 Palatini theorybased on G in Hamiltonian form.

We will assume that M is topologically Σ × R and thatthere exists a function t (with nowhere vanishing gradient (dt)a) such that each t = constsurface Σt is diffeomorphic to Σ. As usual, ta will denote the flow vector field satisfyingta(dt)a = 1.

Since the Lie group G is arbitrary, the 2+1 Palatini theory based on G is not atheory of a spacetime metric; it does not involve a spacetime metric in any way whatsoever.Thus, in particular, t does not necessarily have the interpretation of time. Nonetheless, wecan still define “evolution” from one t = const surface to the next using the Lie derivativewith respect to ta.To write (3.31) in 2+1 form, we decompose eηabc in terms of ta and eηab (the Levi-Civitatensor density of weight +1 on Σ).

Using eηabc = 3t[aeηbc]dt, we getGSP(3e, 3A) = 12ZMeηabc 3eaI3F Ibc= 12ZdtZΣ(ta eηbc + tbeηca + tceηab) 3eaI3F Ibc=ZdtZΣ12(3e · t)I eηbcF Ibc + eEcIL⃗t AIc −eEcIDc(3A · t)I,(3.33)where (3e · t)I := ta 3eaI,eEaI := eηab 3ebI,(3A · t)I := ta 3AIa, and AIa := tba3AIb are theconfiguration variables which specify all the information contained in the field variables 3eaIand 3AIa. Note that:1.

Since G is an arbitrary Lie group, the internal index I can take any value I =1, 2, · · ·, dim(G). Thus, eEaI cannot in general be interpreted as a dyad.

In fact, thisis true even when G = SO(2, 1), since dim(SO(2, 1)) = 3. However, for SO(2, 1) wehave eEaI eEbI = eeqab (= qqab).25

2. ta 3F Iab = L⃗t3AIb −3Db(3A · t)I, which follows from a generalization of Cartan’s identityL⃗vα = i⃗vdα + d(i⃗vα). The Lie derivative L⃗t treats fields with only internal indices asscalars.3.

L⃗t tab = 0, where tab := δab −ta(dt)b is the natural projection operator into the t = constsurfaces defined by t and ta.4. Da := tba3Db is the generalized derivative operator on Σ associated with AIa.5.

F Iab := tcatdb3F Icd is the curvature tensor of Da and satisfies F Iab = 2∂[aAIb] + [Aa, Ab]I.From (3.33), we see that (modulo a surface integral) the Lagrangian GLP of the 2+1Palatini theory based on G is given byGLP =ZΣ12(3e · t)I eηabF Iab + eEaI L⃗t AIa + (Da eEaI )(3A · t)I. (3.34)By inspection, we see that the momentum conjugate to AIa is eEaI , while (3e · t)I and (3A · t)Iboth play the role of Lagrange multipliers.

Thus, the Dirac constraint analysis says that thephase space (GΓP, GΩP) is coordinatized by the pair (AIa, eEaI ) and has symplectic structure11GΩP =ZΣ dI eEaI ∧∧dIAIa. (3.35)The Hamiltonian is given byGHP(A, eE) =ZΣ −12(3e · t)I eηabF Iab −(Da eEaI )(3A · t)I.

(3.36)As we shall see in the next subsection, this is just a sum of 1st class constraint functionsassociated witheηabF Iab ≈0andDa eEaI ≈0. (3.37)Note that these equations are the field equations (3.32) pulled-back to Σ with eηab.

Notealso that they are polynomial in the canonically conjugate variables (AIa, eEaI ). This is to becontrasted with the constraint equations for the standard Einstein-Hilbert theory.

Recallthat the scalar constraint of that theory depended non-polynomially on qab.3.3 Constraint algebraAs usual, to evaluate the Poisson brackets of the constraints and to determine the motionsthey generate on phase space, we must first construct constraint functions associated with11Note that in terms of the Poisson bracket { , } defined by GΩP , we have {AIa(x), eEbJ(y)} = δbaδIJδ(x, y).26

(3.37). Given test fields vI and αI, which take values in the Lie algebra LG and its dual L∗G,we defineF(α) := 12ZΣ αI eηabF IabandG(v) :=ZΣ vI(Da eEaI )(3.38)We will call G(v) the Gauss constraint function since it will play the same role as theGauss constraint of Yang-Mills theory.

We will see that G(v) generates the usual gaugetransformations of the connection 1-form AIa and it conjugate momentum (or “electric field”)eEaI .We are now ready to evaluate the functional derivatives of F(α) and G(v). Since F(α)is independent of the momentum eEaI , and since δF Iab = 2D[aδAIb], we findδF(α)δ eEaI= 0andδF(α)δAIa= eηabDbαI.

(3.39)Similarly, if we vary G(v) with respect to eEaI and AIa, we findδG(v)δ eEaI= −DavIandδG(v)δAIa= {v, eEa}I:= CKJIvJ eEaK. (3.40)Here CIJK denote the structure constants of the Lie algebra LG and { , } : LG × L∗G →L∗Gdenotes the co-adjoint bracket.

{ , } is defined in terms of the Lie bracket [ , ] : LG ×LG →LG via {v, α}IwI := αK[v, w]K.Given (3.39) and (3.40), we can now write down the Hamiltonian vector fields XF (α) andXG(v) associated with F(α) and G(v). They areXF (α) =ZΣ −eηab(DbαI) δδ eEaIand(3.41a)XG(v) =ZΣ −(DavI) δδAIa−{v, eEa}Iδδ eEaI.

(3.41b)Thus, under the 1-parameter family of diffeomorphisms on GΓP associated with XF (α), wehaveAIa 7→AIaand(3.42a)eEaI 7→eEaI −ǫ(eηabDbαI) + O(ǫ2). (3.42b)Similarly, under the 1-parameter family of diffeomorphisms on GΓP associated with XG(v),we haveAIa 7→AIa −ǫDavI + O(ǫ2)and(3.43a)eEaI 7→eEaI −ǫ{v, eEa}I + O(ǫ2).

(3.43b)27

Note that (3.43a) and (3.43b) are the usual gauge transformations of the connection 1-formAIa and its conjugate momentum eEaI that we find in Yang-Mills theory. Equations (3.42) and(3.43) are the motions on phase space generated by F(α) and G(v).Given (3.39) and (3.40), we can also evaluate the Poisson brackets of the constraintfunctions.

Since the G(v)’s play the same role as the Gauss constraint functions of Yang-Mills theory, we would expect their Poisson bracket algebra to be isomorphic to the Liealgebra LG. This is indeed the case.

We find{G(v), G(w)} = G([v, w]),(3.44)where [v, w]I = CIJKvJwK is the Lie bracket of vI and wI. Thus, vI 7→G(v) is a represen-tation of the Lie algebra LG.

The Lie bracket in LG is mapped to the Poisson bracket of thecorresponding Gauss constraint functions.Since F(α) is independent of eEaI , it follows trivially that{F(α), F(β)} = 0. (3.45)With only slightly more effort, we can show that{G(v), F(α)} = −F({v, α}),(3.46)where {v, α}I = CKJIvJαK is the co-adjoint bracket of vI and αI.Thus, the totalityof constraint functions (G(v) and F(α)) is closed under Poisson bracket —i.e., they forma 1st class set.

Furthermore, since (3.44), (3.45), and (3.46) do not involve any structurefunctions (unlike the constraint algebra of the Einstein-Hilbert theory discussed in subsection2.3), the set of constraint functions form a Lie algebra with respect to Poisson bracket.In fact, (α, v) ∈L∗G × LG 7→(F(α), G(v)) is a representation of the Lie algebra LIG ofthe inhomogeneous Lie group IG associated with G.12The action τv(α) := −{v, α}I ofvI ∈LG on αI ∈L∗G is mapped to the Poisson bracket {G(v), F(α)} of the correspondingconstraint functions. The F(α)’s play the role of “translations” and the G(v)’s play the roleof “rotations” in the inhomogeneous group.4.

Chern-Simons theorySo far in this review, we have written down two different actions for 2+1 gravity:Thestandard Einstein-Hilbert action, which gave us a description of 2+1 gravity in terms of a12We will discuss the construction of the inhomogeneous Lie group IG and its Lie algebra LIG in subsection4.4 where we show the equivalence between the 2+1 Palatini theory based on G and Chern-Simons theorybased on IG. When G is the Lorentz group SO(2, 1), IG is the corresponding Poincar´e group ISO(2, 1).28

spacetime metric (or equivalently, a co-triad); and the 2+1 Palatini action based on SO(2, 1),which gave us a description in terms of a co-triad and a connection 1-form. In this section,we shall see that 2+1 gravity can be described by an action that depends only on a con-nection 1-form.

We shall see that the 2+1 Palatini action based on SO(2, 1) is equal to theChern-Simons action based on ISO(2, 1), modulo a surface integral that does not affect theequations of motion. This result was first shown by A. Achucarro and P.K.

Townsend [32];it was later rediscovered and used by Witten [8] to quantize 2+1 gravity. In this section,we will follow the treatment of [30] in which the result for 2+1 gravity follows as a specialcase.

We will show that the 2+1 Palatini theory based on any Lie group G is equivalent toChern-Simons theory based on the inhomogeneous Lie group IG associated with G.The above equivalence between the 2+1 Palatini and Chern-Simons theories is at thelevel of actions. The gauge groups, G and IG, are different, but the actions are the same.It is interesting to note that the 2+1 Palatini and Chern-Simons theories based on the sameLie group G are also related, but this time at the level of their Hamiltonian formulations.

Weshall see that the reduced phase space of the Chern-Simons theory based on G is the reducedconfiguration space of the 2+1 Palatini theory based on the same G. Since Chern-Simonstheory is not available in 3+1 dimensions, the relationships that we find in this section donot, unfortunately, extend to 3+1 theories of gravity.4.1 Euler-Lagrange equations of motionUnlike the standard Einstein-Hilbert and Palatini theories which are well-defined in n+1dimensions, Chern-Simons theory is defined only in odd dimensions. In 2+1 dimensions, thebasic variable is a connection 1-form 3Aia which takes values in a Lie algebra LG equippedwith an invariant, non-degenerate bilinear form kij.13 The Chern-Simons action based on Gis defined byGSCS(3A) := 12ZMeηabckij3Aia∂b3Ajc + 133Aia[3Ab, 3Ac]j,(4.1)where [3Ab, 3Ac]j := Cjmn 3Amb3Anc denotes the Lie bracket of 3Aib and 3Aic.

It is importantto note that Chern-Simons theory is not defined for arbitrary Lie groups—we need theadditional structure provided by the invariant, non-degenerate bilinear form kij.To obtain the Euler-Lagrange equations of motion, we vary GSCS(3A) with respect to 3Aia.13We will require that kij be invariant under the adjoint action of the Lie algebra LG on itself—i.e., thatkij[x, v]iwj + kijvi[x, w]j = 0 for all vi, wi, xi ∈LG. If Cijk is defined in terms of the structure constantsCijk via Cijk := kimCmjk, then invariance of kij under the adjoint action is equivalent to Cijk = C[ijk].

Ifthe Lie group is semi-simple (i.e., if it does not admit any non-trivial abelian normal subgroup), then weare guaranteed that such a kij exists. This is just the Cartan-Killing metric defined by kij := CmniCnmj.Invariance of kij is equivalent to the invariance of Cijk—that is, the Jacobi identity Cm[ijCnk]m = 0.29

Using the fact that Cijk := kimCmjk is totally anti-symmetric, we obtaineηabckij3F jbc = 0,(4.2)where 3F iab = 2∂[a3Aib]+[3Aa, 3Ab]i is the Lie algebra-valued curvature tensor of the generalizedderivative operator 3Da defined by3Davi := ∂avi + [3Aa, v]i. (4.3)If we also use the fact that kij is non-degenerate, we get 3F iab = 0.

Thus, Chern-Simonstheory is a theory of a flat connection 1-form. We will see the role that this equation playsin the next two subsections when we put the theory in Hamiltonian form.4.2 Legendre transformJust like the 2+1 Palatini theory based on an arbitrary Lie group G, Chern-Simons theoryis not a theory of a spacetime metric.

However, we can still put this theory in Hamiltonianform if we assume that M is topologically Σ × R for some submanifold Σ and assume thatthere exists a function t (with nowhere vanishing gradient (dt)a) such that each t = constsurface Σt is diffeomorphic to Σ. As usual, we let ta denote the flow vector field satisfyingta(dt)a = 1.Given t and ta, we are now ready to write the Chern-Simons action (4.1) in 2+1 form.Using the decomposition eηabc = 3t[a eηbc]dt, we getGSCS(3A) = 12ZMeηabckij( 3Aia∂b3Ajc + 133Aia[3Ab, 3Ac]j)= 12ZdtZΣ(3A · t)ikij eηbcF jbc + eηcakijAiaL⃗t Ajc,(4.4)where the last equality holds modulo a surface integral.

Here (3A · t)i := ta 3Aia and Aia :=tba3Aib (= (δba −tb(dt)a) 3Aib) are the configuration variables which specify all the informationcontained in the field variable 3Aia. The Lie derivative treats fields with only internal indicesas scalars, and F iab = 2∂[aAib] + [Aa, Ab]i is the curvature tensor associated with Aia.From (4.4), it follows that the Lagrangian GLCS of the Chern-Simons theory based on Gis given byGLCS = 12ZΣ(3A · t)ikij eηabF jab + eηabkijAjbL⃗t Aia.

(4.5)The momentum canonically conjugate to Aia is12 eηabkijAjb, while (3A · t)i plays the roleof a Lagrange multiplier.Thus, the Dirac constraint analysis says that the phase space30

(GΓCS, GΩCS) is coordinatized by (Aia) and has symplectic structure14GΩCS = −12ZΣeηabkijdIAia ∧∧dIAjb. (4.6)The Hamiltonian is given byGHCS(A) = −12ZΣ(3A · t)ikij eηabF jab.

(4.7)As we shall see in the next subsection, this is just a 1st class constraint function associatedwithkij eηabF jab = 0. (4.8)Note that constraint equation (4.8) is the field equation (4.2) pulled-back to Σ with eηab.

Justas in the 2+1 Palatini theory, the constraint equation is polynomial in the basic variable Aia.Note also that although (4.8) may not look like the standard Gauss constraint of Yang-Millstheory, we shall see that its associated constraint function generates the same motion of Aia.4.3 Constraint algebraFollowing the same procedure that we used in Sections 2 and 3, we first construct aconstraint function associated with (4.8). Given a test field vi (which takes values in the Liealgebra LG), we defineG(v) := 12ZΣ vikij eηabF jab.

(4.9)Since the phase space is coordinatized by the single field Aia, we need to evaluate only onefunctional derivative. Varying G(v) with respect to Aia, we getδG(v)δAia= kij eηabDbvj,(4.10)where Da is any torsion-free extension of the generalized derivative operator associated withAia.

From (4.10) it then follows that the Hamiltonian vector field XG(v) is given byXG(v) =ZΣ −(Davi) δδAia,(4.11)so thatAia 7→Aia −ǫDavi + O(ǫ2)(4.12)14Note that in terms of the Poisson bracket { , } defined by GΩCS, we have {Aia(x), Ajb(y)} = ∼ηabkijδ(x, y),where ∼ηab and kij denote the inverses of eηab and kij. This result follows from the fact that for any f : GΓCS →R, the Hamiltonian vector field Xf is given by Xf =RΣ ∼ηabkijδfδAjbδδAia .

Hence the Poisson bracket of anytwo functions f, g :GΓCS →R is {f, g} =RΣ ∼ηabkijδfδAiaδgδAjb .31

under the 1-parameter family of diffeomorphisms on GΓCS associated with XG(v). This isthe usual gauge transformation of the connection 1-form that we find in Yang-Mills theory.Thus, G(v) can be appropriately called a Gauss constraint function.Given (4.10), it is also straightforward to evaluate the Poisson brackets of the constraints.We find that{G(v), G(w)} = G([v, w]),(4.13)which is the expected Poisson bracket algebra of the Gauss constraint functions.

The mapvi 7→G(v) is a representation of the Lie algebra LG. The Lie bracket in LG is mapped tothe Poisson bracket of the corresponding Gauss constraint functions.4.4 Relationship to the 2+1 Palatini theoryBefore we can show the relationship between the Chern-Simons and 2+1 Palatini theories,we will first have to recall the construction of the inhomogeneous Lie group IG associatedwith any Lie group G. This will allow us to generalize the equivalence of the 2+1 Palatiniand Chern-Simons theories (as shown in [8, 32]) to arbitrary Lie groups G. We will be ableto show that the 2+1 Palatini theory based on any G is equivalent to Chern-Simons theorybased on IG.Consider any Lie group G with Lie algebra LG, and let L∗G denote the vector space dualof LG.

If vI, wI denote typical elements of LG and αI, βI denote typical elements of L∗G,then (α, v)i := (αI, vI) and (β, w)i := (βI, wI) are typical elements of the direct sum vectorspace L∗G ⊕LG. We can define a bracket on L∗G ⊕LG via[(α, v), (β, w)]i := (−{v, β} + {w, α}, [v, w])i,(4.14)where [v, w]I := CIJKvJwK and {v, β}I := CKJIvJβK are the Lie bracket and co-adjointbracket associated with LG.

By inspection, we see that (4.14) is linear and anti-symmetric.If we use{[v, w], α}I = −{v, {w, α}}I + {w, {v, α}}I(4.15)(which follows as a consequence of the Jacobi identity for LG), we can show that (4.14)satisfies the Jacobi identity as well. Thus, the vector space LIG := L∗G ⊕LG together with(4.14) is actually a Lie algebra.

We call LIG the inhomogeneous Lie algebra associated withG; the inhomogeneous Lie group IG is obtained by exponentiating the Lie algebra LIG. Aswe shall see later in this subsection, IG is simply the cotangent bundle over G.The terminology inhomogeneous is due to the fact that LIG admits an abelian Lie idealisomorphic to L∗G, and that the quotient of LIG by this ideal is isomorphic to LG.15 Thus,15A Lie ideal I of a Lie algebra L is a vector subspace I ⊂L such that [i, x] ∈I for any i ∈I, x ∈L.32

elements of L∗G are analogous to infinitesimal “translations,” while elements of LG are analo-gous to infinitesimal “rotations.” Note, however, that the space of translations and rotationshave the same dimension. As a special case, if one chooses G to be the 2+1 dimensionalLorentz group SO(2, 1), then the above construction yields for IG the 2+1 dimensionalPoincar´e group ISO(2, 1).In addition to the above Lie algebra structure, L∗G ⊕LG is equipped with a (natural)invariant, non-degenerate bilinear form kij defined bykij(α, v)i(β, w)j := αIwI + βIvI.

(4.16)Since LIG is not semi-simple (because it admits a non-trivial abelian Lie ideal), kij is not the(degenerate) Cartan-Killing metric of LIG. Nevertheless, the existence of kij will allow usto construct Chern-Simons theory for IG.

Recall that without an invariant, non-degeneratebilinear form, Chern-Simons theory could not be defined. Note also that for G = SO(2, 1),the above construction of kij reduces to that used by Witten [8].Given these remarks, we can now show that the 2+1 Palatini theory based on any Liegroup G is equivalent to Chern-Simons theory based on IG.

To do this, recall that for anyLie group G, the 2+1 Palatini action based on G is given byGSP(3e, 3A) = 12ZMeηabc 3eaI3F Ibc,(4.17)where 3AIa and 3eaI are LG- and L∗G-valued 1-forms. We now construct the inhomogeneousLie algebra LIG associated with G and define an LIG-valued connection 1-form 3Aia via3Aia := (3eaI, 3AIa).

(4.18)By simply substituting this expression for 3Aia into the Chern-Simons action IGSCS(3A), we33

find thatIGSCS(3A) = 12ZMeηabckij3Aia∂b3Ajc + 133Aia[3Ab, 3Ac]j= 12ZMeηabc3eaI∂b3AIc + (∂b3ecI)3AIa+ 13( 3eaI[3Ab, 3Ac]I −{3Ab, 3ec}I3AIa + {3Ac, 3eb}I3AIa)= 12ZMeηabc3eaI∂b3AIc + ∂b(3ecI3AIa) −3ecI∂b3AIa+ 13( 3eaI[3Ab, 3Ac]I −3ecI[3Ab, 3Aa]I + 3ebI[3Ac, 3Aa]I)= 12ZMeηabc3eaI(2∂b3AIc) + 3eaI[3Ab, 3Ac]I + ∂b(3ecI3AIa)= 12ZMeηabc 3eaI3F Ibc + (surface integral)= GSP(3e, 3A) + (surface integral),(4.19)where we have used definitions (4.14), (4.16), and (4.18) repeatedly. Since the surface termdoes not affect the Euler-Lagrange equations of motion, we can conclude that the 2+1Palatini theory based on G is equivalent to Chern-Simons theory based on IG.

This is thedesired result. Note that as a special case, we can conclude that 2+1 gravity as describedby the 2+1 Palatini action based on SO(2, 1) is equivalent to Chern-Simons theory basedon ISO(2, 1).Up to now, we have only described the inhomogeneous Lie group IG in terms of its asso-ciated Lie algebra LIG.

We just exponentiated the Lie algebra LIG to obtain IG. However,it is also instructive to give an explicit construction of IG at the level of groups and mani-folds.

But to do this, we will need to make another short digression, this time on semi-directproducts and semi-direct sums. Readers already familiar with these definitions may skip tothe paragraph immediately following equations (4.24).Let G and H be any two groups.

To define the semi-direct product H ⃝σ G, we need ahomomorphism σ from the group G into the group of automorphisms of H—i.e., for eachg, g′ ∈G and h′, h′ ∈H, the map σg : H →H must be 1-1, onto, and satisfyσg(hh′) = σg(h)σg(h′)andσgg′(h) = σg(σg′(h))(4.20)for every g, g′ ∈G and h′, h′ ∈H. Given this structure, one can check that(h, g)(h′, g′) = (hσg(h′), gg′)(4.21)defines a group multiplication law on the set H ×G.

The identity element is (eH, eG), whereeH, eG are the identities in H and G, and the inverse (h, g)−1 of (h, g) is (σg−1(h−1), g−1).34

The set H × G together with this group multiplication law defines the semi-direct productH ⃝σ G. Note that H is homomorphic to a (not necessarily abelian) normal subgroup ofH ⃝σ G, and the quotient of H ⃝σ G by this normal subgroup is homomorphic to G. As atrivial example, if σg(h) = h for all g ∈G, h ∈H, then H ⃝σ G is the usual direct productH ⊗G of groups.Now assume that G and H are Lie groups with Lie algebras LG and LH. If H ⃝σ G is thesemi-direct product of G and H with respect to some action σ of G on H satisfying (4.20),we would now like to determine the relationship between the Lie algebras LH⃝σ G, LG, andLH.

To do this, we differentiate the action σ of G on H to obtain an action τ of LG onLH. More precisely, if g(ǫ) is a 1-parameter curve in G with g(0) = eG and tangent vectorv :=ddǫ|ǫ=0g(ǫ) and h(λ) is a 1-parameter curve in H with h(0) = eH and tangent vectorα :=ddλ|λ=0h(λ), then we defineτv(α) := ddǫǫ=0ddλλ=0 σg(ǫ)(h(λ)).

(4.22)In terms of τ, the Lie bracket of (α, v) and (β, w) in LH⃝σ G becomes16[(α, v), (β, w)] = (τv(β) −τw(α) + [α, β], [v, w]),(4.23)where [v, w] and [α, β] are the Lie brackets of v, w ∈LG and α, β ∈LH. Note that (4.23)satisfies the Jacobi identity as a consequence ofτv([α, β]) = [τv(α), β] + [α, τv(β)]and(4.24a)τ[v,w](α) = τv(τw(α)) −τw(τv(α)),(4.24b)which follow from the definition of τ and the properties (4.20) satisfied by σ.

Thus, if LGand LH are two Lie algebras and τ is an action of LG on LH satisfying (4.24), then the directsum vector space LH ⊕LG together with the bracket defined by (4.23) is a Lie algebra. ThisLie algebra, denoted LH ⃝τ LG, is called the semi-direct sum of LG and LH.

Note that LH isisomorphic to a (not necessarily abelian) Lie ideal of LH ⃝τ LG, and the quotient of LH ⃝τ LGby this ideal is isomorphic to LG. If σg(h) = h for all g ∈G, h ∈H (so that H ⃝σ G = H ⊗G),then LH ⃝τ LG is the usual direct sum LH ⊕LG of Lie algebras.Given these general remarks, let us now return to our discussion of the inhomogeneousLie group IG and its associated Lie algebra LIG.

From the above definitions, we see that16To obtain this result, consider 1-parameter curves (h(ǫ), g(ǫ)) and (h′(ǫ′), g′(ǫ′)) in H ⃝σ G with(h(0), g(0)) = (h′(0), g′(0)) = (eH, eG) and tangent vectors (α, v) :=ddǫ|ǫ=0(h(ǫ), g(ǫ)) and (β, w) :=ddǫ′ |ǫ′=0(h′(ǫ′), g′(ǫ′)). Then use the definition of the Lie bracket in terms of the group multiplication law(4.21), [(α, v), (β, w)] :=ddǫ|ǫ=0 ddǫ′ |ǫ′=0(h(ǫ), g(ǫ))(h′(ǫ′), g′(ǫ′))(h(ǫ), g(ǫ))−1(h′(ǫ′), g′(ǫ′))−1.

This leads to(4.23).35

LIG is simply the semi-direct sum L∗G ⃝τ LG.L∗G is to be thought of as a Lie algebra withthe trivial Lie bracket [α, β] = 0 for all αI, βI ∈L∗G;the action τ of LG on L∗G is givenby τv(β) = −{v, β}I. Equations (4.24) hold for this action as a consequence of the Jacobiidentity in LG:Equation (4.24a) is satisfied since [α, β] = 0 for all αI, βI ∈L∗G, whileequation (4.24b) is equivalent to equation (4.15). Furthermore, the inhomogenized Lie groupIG is simply the semi-direct product L∗G ⃝σ G.L∗G is to be thought of as an abelian groupwith respect to vector addition, and the action σ of G on L∗G is induced by the adjoint actionof G on itself.17 This implies that as a manifold IG is the cotangent bundle T ∗G.

At eachpoint g ∈G, the cotangent space T ∗g G is isomorphic to L∗G.Moreover, the above relationship between G and IG allows us to prove an interestingmathematical result involving the space of connection 1-forms on a 2-dimensional manifold.We can show that for any Lie group GT ∗(GA) =IGA,(4.25)where GA and IGA denote the space of LG- and LIG-valued connection 1-forms on a 2-dimensional manifold Σ. The map(AIa, eEaI ) ∈T ∗(GA) 7→Aia := (eaI, AIa) ∈IGA(4.26)(where eaI := −∼ηab eEbI) is a diffeomorphism from the manifold T ∗(GA) to the manifold IGAthat sends the natural symplectic structureGΩ:=ZΣ dI eEaI ∧∧dIAIa(4.27)on T ∗(GA) to the natural symplectic structureIGΩ:= −12ZΣeηabkijdIAia ∧∧dIAjb(4.28)on IGA.

(Here kij denotes the (natural) invariant, non-degenerate bilinear form on LIGdefined by (4.16).) Note that (4.27) and (4.28) are the symplectic structures of the 2+1Palatini theory based on G and the Chern-Simons theory based on IG.

However, the aboveresult (4.25) does not require any knowledge of the 2+1 Palatini or Chern-Simons actions.Finally, to conclude this subsection, I would like to verify the claim made at the start ofSection 4 that the reduced phase space of Chern-Simons theory based on any Lie group G17The adjoint action of G on itself is defined by Ag(g′) = gg′g−1 for all g, g′ ∈G. By differentiatingAg at the identity e, we obtain a map Adg : LG →LG via Adg(v) := A′g(e) · v.Ad defines the adjointrepresentation of the Lie group G by linear operators on the Lie algebra LG.

The action σ of G on L∗G isthen given by (σg(α))(v) := α(Adg(v)) for any α ∈L∗G and v ∈LG.36

is the reduced configuration space of the 2+1 Palatini theory based on the same Lie groupG. This result will be simpler to prove than the previous two results since most of thepreliminary work has already been done.Let G be any Lie group whose Lie algebra LG admits an invariant, non-degenerate bilinearform kIJ.

Then Chern-Simons theory based on G is well-defined, and, as we saw in subsection4.2, the phase space GΓCS is coordinatized by LG-valued connection 1-forms AIa on Σ. Thesymplectic structure isGΩCS = −12ZΣeηabkIJdIAIa ∧∧dIAJb . (4.29)In subsection 4.3, we then verified that the constraint functions G(v) associated with thethe constraint equationkIJ eηabF Jab = 0(4.30)formed a 1st class set and generated the usual gauge transformationsAIa 7→AIa −ǫDavI + O(ǫ2).

(4.31)To pass to the reduced phase space, we must factor-out the constraint surface (defined by(4.30)) by the orbits of the Hamiltonian vector fields XG(v).18 From (4.30) and (4.31) we seethat the reduced phase space GˆΓCS of the Chern-Simons theory based on G is coordinatizedby equivalence classes of flat LG-valued connection 1-forms on Σ, where two such connection1-forms are said to be equivalent if and only if they are related by (4.31). This space is calledthe moduli space of flat LG-valued connection 1-forms on Σ.Now recall the Hamiltonian formulation of the 2+1 Palatini theory based on the sameLie group G. In subsection 3.3, we saw that the phase space GΓP was coordinatized by pairs(AIa, eEaI ) consisting of LG-valued connection 1-forms AIa on Σ and their canonically conjugatemomentum eEaI .GΓP was the cotangent bundle T ∗(GCP) over the configuration space GCP ofLG-valued connection 1-forms AIa on Σ with symplectic structureGΩP =ZΣ dI eEaI ∧∧dIAIa.

(4.32)We also saw that the constraint equations of the 2+1 Palatini theory wereeηabF Iab = 0andDa eEaI = 0,(4.33)18Recall that given a symplectic manifold (Γ, Ω), a set of constraints φi form a 1st class set if and only ifeach Hamiltonian vector field Xφi is tangent to the constraint surface Γ ⊂Γ defined by the vanishing of allthe constraints. The pull-back, Ω, of Ωto Γ is degenerate with the degenerate directions given by the Xφi.Thus, (Γ, Ω) is not a symplectic manifold.

However, by factoring-out the constraint surface by the orbits ofthe Xφi, we obtain a reduced phase space (ˆΓ, ˆΩ) whose coordinates are precisely the true degrees of freedomof the theory. ˆΩis non-degenerate; it is the projection of Ωto ˆΓ.37

and verified that their associated constraint functions F(α) and G(v) formed a 1st class set.They generated the motionsAIa 7→AIaand(4.34a)eEaI 7→eEaI −ǫ(eηabDbαI) + O(ǫ2)(4.34b)andAIa 7→AIa −ǫDavI + O(ǫ2)and(4.35a)eEaI 7→eEaI −ǫ{v, eEa}I + O(ǫ2),(4.35b)respectively. Thus, the reduced phase space GˆΓP of the 2+1 Palatini theory based on G iscoordinatized by equivalence classes of pairs consisting of flat LG-valued connection 1-formson Σ and divergence-free L∗G-valued vector densities of weight +1 on Σ, where two such pairsare said to be equivalent if and only if they are related by (4.34) and (4.35).

Since F(α)and G(v) are independent and linear in the momentum eEaI , it follows thatGˆΓP is naturallythe cotangent bundle T ∗(GˆCP) over the reduced configuration space GˆCP of the 2+1 Palatinitheory based on G. From (4.34a) and (4.35a) we see that GˆCP is again the moduli space offlat LG-valued connection 1-forms on Σ. Thus, GˆΓCS = GˆCP as desired.

In particular, thereduced configuration space of the 2+1 Palatini theory based on G has the structure of asymplectic manifold.This last result has interesting consequences. It can be used, for example, to show therelationship between the ˆT 0[γ] and ˆT 1[γ] observables for 2+1 gravity.

These are the 2+1dimensional analogs of the classical T-observables constructed by Rovelli and Smolin [7] forthe 3+1 theory. As shown in [30],ˆT 0[γ] is the trace of the holonomy of the connectionaround a closed loop γ in Σ, while ˆT 1[γ] is the function on the reduced phase space of the2+1 Palatini theory defined by the Hamiltonian vector field associated with ˆT 0[γ].

Thus,many properties satisfied by the ˆT 1[γ]’s can be derived from similar properties satisfied bythe ˆT 0[γ]’s.5. 2+1 matter couplingsIn this section, we will couple various matter fields to 2+1 gravity via the 2+1 Palatiniaction.

We will consider the inclusion of a cosmological constant and a massless scalar field.One can couple other fundamental matter fields (e.g., Yang-Mills and Dirac fields) to 2+1gravity in a similar fashion—I have chosen to consider a massless scalar field in detail since2+1 gravity coupled to a massless scalar field is the dimensional reduction of 3+1 vacuum38

general relativity with a spacelike, hypersurface-orthogonal Killing vector field [16]. As notedin Section 1, this is an interesting case since it appears likely that the non-perturbativecanonical quantization program for 3+1 gravity can be carried through to completion forthis reduced theory.In subsection 5.1, we define the 2+1 Palatini theory based on a Lie group G with cos-mological constant Λ.

We derive the Euler-Lagrange equations of motion and perform aLegendre transform to obtain a Hamiltonian formulation of the theory. Just as in Section3 (when Λ was equal to zero), we shall find that the constraint equations of the theory arepolynomial in the canonically conjugate variables.

We shall also find that their associatedconstraint functions still form a Lie algebra with respect to Poisson bracket.In subsection 5.2, we will show that the 2+1 Palatini theory based on G with cosmologicalconstant Λ is equivalent to Chern-Simons theory based on the Λ-deformation, ΛG, of G. Thisis a generalization of Witten’s result [8] for G = SO(2, 1) (and ΛG = SO(3, 1) or SO(2, 2)depending on the sign of Λ) which holds for any Lie group G that admits an invariant, totallyanti-symmetric tensor ǫIJK.This result also generalizes the Λ = 0 equivalence of the 2+1Palatini and Chern-Simons theories given in subsection 4.4.Finally, in subsection 5.3, we define the action for a massless scalar field and couple thisfield to 2+1 gravity by adding the action to the 2+1 Palatini action based on G = SO(2, 1).It is when we wish to couple matter with local degrees of freedom to 2+1 gravity (as it isfor the case of a massless scalar field) that we are forced to take the Lie group G to besuch that the fields 3eIa have the interpretation of a co-triad. We obtain the Euler-Lagrangeequations of motion for the coupled theory and then perform a Legendre transform to obtainthe Hamiltonian formulation.

We shall find that the constraint equations remain polynomialin the canonically conjugate variables and the associated constraint functions form a 1stclass set, but they no longer form a Lie algebra with respect to Poisson bracket.The basis for much of the material in this section can be found in [8, 30, 33].5.1 2+1 Palatini theory coupled to a cosmological constantRecall that the equation of motion for gravity coupled to the cosmological constant Λ isGab + Λgab = 0,(5.1)where Gab := Rab −12Rgab is the Einstein tensor of gab. We can obtain (5.1) via an actionprinciple if we modify the standard Einstein-Hilbert action by a term proportional to thevolume of the spacetime.

DefiningSΛ(gab) :=ZΣ√−g(R −2Λ),(5.2)39

we find that the variation of (5.2) with respect to gab yields (5.1) (modulo the usual boundaryterm associated with the standard Einstein-Hilbert action). These results are valid in n+1dimensions.To write this action in 2+1 Palatini form, we proceed as in Section 3.We replacethe spacetime metric gab with a co-triad 3eaI and replace the unique, torsion-free spacetimederivative operator ∇a (compatible with gab) with an arbitrary generalized derivative operator3Da.

Recalling that √−g = 13! eηabcǫIJK 3eaI 3ebJ 3ecK, we defineSΛ(3e, 3A) : = 12ZMeηabc 3eaI3F Ibc −Λ3!ZMeηabcǫIJK 3eaI3ebJ3ecK= 12ZMeηabc 3eaI( 3F Ibc −Λ3 ǫIJK 3ebJ3ecK),(5.3)where 3F Iab = 2∂[a3AIb]+[3Aa, 3Ab]I is the internal curvature tensor of the generalized derivativeoperator 3Da defined by3DavI := ∂avI + [3Aa, v]I.

(5.4)Note that [3Aa, v]I := ǫI JK 3AJavK where ǫI JK := ǫIMNηMJηNK. Just as we did for the vacuum2+1 Palatini theory, we have included an additional overall factor of 1/2 in definition (5.3).Although the action (5.3) was originally defined for G = SO(2, 1), it is well-defined forany Lie group G that admits an invariant, totally anti-symmetric tensor ǫIJK.19This isadditional structure that does not naturally exist for an arbitrary Lie group G, so unlikethe 2+1 Palatini theory with Λ = 0, the 2+1 Palatini theory with non-zero cosmologicalconstant Λ is not defined for arbitrary G. If the Lie algebra LG admits an invariant, non-degenerate bilinear form kIJ, then we are guaranteed that such an ǫIJK exists—we cantake ǫIJK := kJMkKNCIMN.Thus, in particular, 2+1 Palatini theory with a non-zerocosmological constant is well-defined for semi-simple Lie groups.We should emphasize,however, that it is not necessary to restrict ourselves to semi-simple Lie groups.

In whatfollows, we will only assume that ǫIJK exists. Given such a Lie group G, the actionGSΛ(3e, 3A) := 12ZMeηabc 3eaI( 3F Ibc −Λ3 ǫIJK 3ebJ3ecK)(5.5)will be called the 2+1 Palatini action based on G with cosmological constant Λ.

Note that3F Iab = 2∂[a3AIb] + [3Aa, 3Ab]I,(5.6)19We will require that ǫIJK be invariant under the adjoint action of the Lie algebra LG on its dual L∗G—i.e., that ǫIJK{v, α}IβJγK + ǫIJKαI{v, β}JγK + ǫIJKαIβJ{v, γ}K = 0 for all vI ∈LG and αI, βI, γI ∈L∗G. ({v, α}I is the co-adjoint bracket of vI and αI defined in terms of the structure constants CI JK via{v, α}I := CK JIvJαK.) Invariance of ǫIJK is equivalent to ǫM[IJCK]MN = 0.40

where [3Aa, 3Ab]I := CIJK 3AJa3AKb is the Lie bracket in LG. It is only for G = SO(2, 1) thatCIJK = ǫI JK = ǫIMNηMJηNK.To obtain the Euler-Lagrange equations of motion, we vary GSΛ(3e, 3A) with respect toboth 3eaI and 3AIa.

We findeηabc( 3F Ibc −ΛǫIJK 3ebJ3ecK) = 0and3Db(eηabc 3ecI) = 0,(5.7)where the second equation, as usual, requires a torsion-free extension of the generalizedderivative operator 3Da to spacetime tensor fields, but is independent of this choice. Notefurther that if Λ ̸= 0, 3Da is not flat.

In fact, for the special case G = SO(2, 1), equations(5.7) imply that the spacetime (M, gab := 3eaI 3ebJηIJ) has constant curvature equal to 6Λ.To show that the above two equations reproduce (5.1), let us restrict ourselves to G =SO(2, 1) (with ǫIJK being the volume element of ηIJ) so that 3eaI is, in fact, a co-triad.Then by following the argument given in Section 3, we find that the second equation implies3AIa = ΓIa, where ΓIa is the (internal) Christoffel symbol of the unique, torsion-free generalizedderivative operator ∇a compatible with the co-triad 3eaI. Thus, 3Da is not arbitrary, butequals ∇a when acting on internal indices.Substituting this solution back into the firstequation, we findeηabc(RIbc −ΛǫIJK 3ebJ3ecK) = 0,(5.8)where RIab is the (internal) curvature tensor of ∇a.

Contracting (5.8) with 3edI givesGad + Λgad = 0. (5.9)This is the desired result.To put this theory in Hamiltonian form, we will assume that M is topologically Σ × R,and assume that there exists a function t (with nowhere vanishing gradient (dt)a) such thateach t = const surface Σt is diffeomorphic to Σ.

By ta we will denote the flow vector fieldsatisfying ta(dt)a = 1. Using eηabc = 3t[a eηbc]dt (and our decomposition of GSP(3e, 3A) fromSection 3), we obtainGSΛ(3e, 3A) =ZdtZΣ12(3e · t)I(eηabF Iab −ΛǫIJK∼ηab eEaJ eEbK)+ eEaI L⃗t AIa −eEaI Da(3A · t)I.

(5.10)The configuration variables are (3e · t)I := ta 3eaI,eEaI := eηab 3ebI, (3A · t)I := ta 3AIa, andAIa := tba3AIb. Thus, (modulo a surface integral) the Lagrangian GLΛ of the 2+1 Palatinitheory based on G with cosmological constant Λ is given byGLΛ =ZΣ12(3e · t)I(eηabF Iab −ΛǫIJK∼ηab eEaJ eEbK)+ eEaI L⃗t AIa + (Da eEaI )(3A · t)I.

(5.11)41

GLΛ is to be viewed as a functional of the configuration variables and their first derivatives.Following the standard Dirac constraint analysis, we find that the momentum canonicallyconjugate to AIa is eEaI , while (3e · t)I and (3A · t)I both play the role of Lagrange multipliers.Thus, the phase space and symplectic structure are the same as those found for the 2+1Palatini theory with Λ = 0, and the Hamiltonian is given byGHΛ(A, eE) =ZΣ −12(3e · t)I(eηabF Iab −ΛǫIJK∼ηab eEaJ eEbK) −(Da eEaI )(3A · t)I. (5.12)We will see that this is just a sum of 1st class constraint functions associated witheηabF Iab −ΛǫIJK∼ηab eEaJ eEbK ≈0andDa eEaI ≈0.

(5.13)By inspection, constraint equations (5.13) are polynomial in the canonically conjugate vari-ables (AIa, eEaI ). They are the field equations (5.7) pulled-back to Σ with eηab.As usual, given test fields αI and vI, which take values in L∗G and LG, we can defineconstraint functionsF(α) := 12ZΣ αI(eηabF Iab −ΛǫIJK∼ηab eEaJ eEbK)andG(v) :=ZΣ vI(Da eEaI ).

(5.14)Note that G(v) is unchanged from the 2+1 Palatini theory with Λ = 0, while F(α) has anadditional term quadratic in the momentum eEaI . There is only one new functional derivative,δF(α)δ eEaI= −ΛǫIJK∼ηab eEbJαK.

(5.15)All the others are the same as before.Under the 1-parameter family of diffeomorphisms associated with the Hamiltonian vectorfield XF (α), we haveAIa 7→AIa −ǫ(ΛǫIJK∼ηab eEbJαK) + O(ǫ2)and(5.16a)eEaI 7→eEaI −ǫ(eηabDbαI) + O(ǫ2). (5.16b)Similarly, under the 1-parameter family of diffeomorphisms associated with the Hamiltonianvector field XG(v), we haveAIa 7→AIa −ǫDavI + O(ǫ2)and(5.17a)eEaI 7→eEaI −ǫ{v, eEa}I + O(ǫ2).

(5.17b)Comparing these results with those from the 2+1 Palatini theory with Λ = 0, we see that themotion of 3AIa generated by the constraint functions no longer corresponds to the usual gauge42

transformation of Yang-Mills theory. This is due to the non-zero contribution from F(α).In fact, since F(α) depends quadratically on the momentum eEaI , the reduced phase space ofthe 2+1 Palatini theory with non-zero cosmological constant Λ is not naturally a cotangentbundle over a reduced configuration space.

Thus, the result of Section 4 that the reducedphase space of the Chern-Simons theory based on G equals the reduced configuration spaceof the 2+1 Palatini theory based on the same G does not extend in general to the case Λ ̸= 0.Nevertheless, we can still evaluate the Poisson brackets of the constraint functions F(α)and G(v). As in the Λ = 0 case, we find that{G(v), G(w)} = G([v, w]),(5.18)where [v, w]I = CIJKvJwK is the Lie bracket of vI and wI, so vI 7→G(v) is a representationof the Lie algebra LG.

Although F(α) has changed, we again find that{G(v), F(α)} = −F({v, α}),(5.19)where {v, α}I = CKJIvJαK is the co-adjoint bracket of vI and αI. However, the Poissonbracket of F(α) with F(β) is no longer zero; it equals{F(α), F(β)} = −ΛG(ǫ(α, β)),(5.20)where ǫ(α, β)I := ǫIJKαJβK.Thus, the totality of constraint functions is closed underPoisson bracket —i.e., they form a 1st class set.

In fact, since (5.18), (5.19), and (5.20) donot involve any structure functions, the constraint functions form a Lie algebra with respectto Poisson bracket. The mapping (α, v) ∈L∗G × LG 7→(F(α), G(v)) is a representation ofthe Lie algebra LΛG of the Λ-deformation, ΛG, of the Lie group G.20 The F(α)’s play therole of “boosts” while the G(v)’s play the role of “rotations” in the Λ-deformation of G.5.2 Relationship to Chern-Simons theoryIn a manner similar to that used in subsection 4.4, we will now show that if G is any Liegroup which admits an invariant, totally anti-symmetric tensor ǫIJK, then the 2+1 Palatinitheory based on G with cosmological constant Λ is equivalent to Chern-Simons theory basedon the Λ-deformation, ΛG, of the Lie group G. The actions for these two theories are thesame modulo a surface term that does not affect the equations of motion.20We will discuss the construction of the Λ-deformation of a Lie group G in the following section wherewe show the equivalence between the 2+1 Palatini theory based on G with cosmological constant Λ andChern-Simons theory based on ΛG.

When Λ = 0, ΛG is just the inhomogeneous Lie group IG constructedin subsection 4.4.43

Given a Lie group G with an invariant, totally anti-symmetric tensor ǫIJK, we firstconstruct the Λ-deformation, ΛG, of G as follows: Form the direct sum vector space L∗G⊕LG(having typical elements (α, v)i := (αI, vI) and (β, w)i := (βI, wI)) and then define a bracketon L∗G ⊕LG via[(α, v), (β, w)]i := (−{v, β} + {w, α}, [v, w] −Λǫ(α, β))i,(5.21)where [v, w]I := CIJKvJwK, {v, β}I := CKJIvJβK, and ǫ(α, β)I := ǫIJKαJβK. By inspec-tion, (5.21) is linear and anti-symmetric.

By using the Jacobi identity CM [IJCN K]M = 0on LG together with the anti-symmetry and invariance of ǫIJK, one can show that (5.21)satisfies the Jacobi identity as well. Thus, the vector space LΛG := L∗G ⊕LG together with(5.21) is actually a Lie algebra.

We call LΛG the Λ-deformed Lie algebra associated with G.The Λ-deformation, ΛG, of G is obtained by exponentiating LΛG. We can think of ΛG asan extension of the inhomogeneous Lie group IG in the sense that ΛG reduces to IG whenΛ = 0.

Note also that if G = SO(2, 1), then the above construction for ΛG yields SO(3, 1)if Λ < 0 and SO(2, 2) if Λ > 0.In addition to the above Lie algebra structure, L∗G ⊕LG is also equipped with a (natural)invariant, non-degenerate bilinear formkij(α, v)i(β, w)j := αIwI + βIvI. (5.22)This is the same kij that we had when Λ = 0.

As before, the existence of kij will allow us toconstruct Chern-Simons theory for ΛG.Given these remarks, we are now ready to verify that the 2+1 Palatini theory based onG with cosmological constant Λ is equivalent to Chern-Simons theory based on ΛG. RecallthatGSΛ(3e, 3A) = 12ZMeηabc 3eaI( 3F Ibc −Λ3 ǫIJK 3ebJ3ecK),(5.23)where 3AIa and 3eaI are LG- and L∗G-valued 1-forms.

If we now construct the Λ-deformed Liealgebra LΛG associated with G and define an LΛG-valued connection 1-form 3Aia via3Aia := (3eaI, 3AIa),(5.24)then a straightforward calculation along the lines of that used in subsection 4.4 shows thatthe Chern-Simons actionΛGSCS(3A) = 12ZMeηabckij3Aia∂b3Ajc + 133Aia[3Ab, 3Ac]j(5.25)equals GSΛ(3e, 3A) modulo a surface term which does not affect the Euler-Lagrange equationsof motion. Specializing to the case G = SO(2, 1), we see that 2+1 gravity coupled to the44

cosmological constant Λ is equivalent to Chern-Simons theory based on SO(3, 1) if Λ < 0 orSO(2, 2) if Λ > 0. This was the observation of Witten [8].5.3 2+1 Palatini theory coupled to a massless scalar fieldSo far, we have seen that the 2+1 Palatini theory (with or without a cosmological constantΛ) is well-defined for a wide class of Lie groups.

If Λ = 0, the Lie group G can be completelyarbitrary; if Λ ̸= 0, then G has to admit an invariant, totally anti-symmetric tensor ǫIJK. Weare not forced to restrict ourselves to G = SO(2, 1).

However, in order to couple fundamentalmatter fields with local degrees of freedom to 2+1 gravity via the 2+1 Palatini action, wewill need to take G = SO(2, 1). The matter actions require the existence of a spacetimemetric gab, and, as such,3eIa must have the interpretation of a co-triad.

We will only considercoupling a massless scalar field to 2+1 gravity in this section—a similar treatment wouldwork for Yang-Mills and Dirac fields as well.Let us first recall that the theory of a massless scalar field φ can be defined in n+1dimensions. If gab denotes the inverse of the spacetime metric gab, then the Klein-Gordonaction SKG(gab, φ) is defined bySKG(gab, φ) := −8πZM√−g gab∂aφ∂bφ,(5.26)where ∂aφ denotes the gradient of φ.

To couple the scalar field to gravity, we simply add theKlein-Gordon action (5.26) to the standard Einstein-Hilbert actionSEH(gab) =ZM√−gR. (5.27)The total action ST(gab, φ) is then given by the sumST(gab, φ) := SEH(gab) + SKG(gab, φ),(5.28)and the Euler-Lagrange equations of motion are obtained by varying ST(gab, φ) with respectto both gab and φ.

The variation of φ yieldsgab∇a∇bφ = 0,(5.29)while the variation of gab yieldsGab = 8πTab(KG). (5.30)∇a is the unique, torsion-free spacetime derivative operator compatible with the metric gab,and Tab(KG) is the stress-energy tensor of the massless scalar field.

In terms of gab and φ,we haveTab(KG) := ∂aφ∂bφ −12gab∂cφ∂cφ. (5.31)45

Now let us restrict ourselves to 2+1 dimensions and rewrite the above actions and equa-tions of motion in 2+1 Palatini form. As we saw in Section 3, the 2+1 Palatini action canbe written asSP(3e, 3A) := 12ZMeηabc 3eaI3F Ibc,(5.32)where 3eaI is the co-triad related to the spacetime metric gab via gab := 3eIa3eJb ηIJ and 3F Iab =2∂[a3AIb] + ǫIJK3AJa3AKb is the internal curvature tensor of the generalized derivative operator3Da defined by 3AIa.

The Klein-Gordon action, viewed as a functional of 3eIa and the scalarfield φ, is given bySKG(3e, φ) = −8πZM√−g gab∂aφ∂bφ. (5.33)Note that although the Klein-Gordon action depends on the co-triad 3eaI through its depen-dence on √−g and gab, it is independent of the connection 1-form 3AIa.

In fact, of all thefundamental matter couplings, only the action for the Dirac field would depend on 3AIa.Given (5.32) and (5.33), we define the total action as the sumST(3e, 3A, φ) := SP(3e, 3A) + 12SKG(3e, φ). (5.34)The additional factor of 1/2 is needed in front of SKG(3e, φ) so that the above definitionof the total action will be consistent with the definition of SP(3e, 3A).

The Euler-Lagrangeequations of motion are obtained by varying ST(3e, 3A, φ) with respect to each field. Varyingφ givesgab∇a∇bφ = 0,(5.35)while varying 3AIa and 3eaI imply3Db(eηabc 3ecI) = 0and(5.36)eηabc 3F Ibc −8π√−g(3eaIgbc −2 3ebIgac)∂bφ∂cφ = 0,(5.37)respectively.

Note that equation (5.35) is just the standard equation of motion for φ, whileequation (5.36) implies that 3AIa = ΓIa, as in the vacuum case. Substituting this result for3AIa back into (5.37) and contracting with 3edI givesGad = 8πT ad(KG).

(5.38)These are the desired results.To put this theory in Hamiltonian form, we will basically proceed as we have in the past,but use additional structure provided by the spacetime metric gab. We will assume thatM is topologically Σ × R for some spacelike submanifold Σ and assume that there existsa time function t (with nowhere vanishing gradient (dt)a) such that each t = const surface46

Σt is diffeomorphic to Σ.ta will denote the time flow vector field (ta(dt)a = 1), while nawill denote the unit covariant normal to the t = const surfaces.na := gabnb will be theassociated future-pointing timelike vector field (nana = −1). Given na and na, it followsthat qab := δab + nanb is a projection operator into the t = const surfaces.

We can then definethe induced metric qab, the lapse N, and shift Na as we did for the standard Einstein-Hilberttheory in Section 2. Recall that ta = Nna + Na, with Nana = 0.Now let us write the total action (5.34) in 2+1 form by decomposing each of its pieces.Using eηabc = 3t[a eηbc]dt, it follows thatSP(3e, 3A) =ZdtZΣ12(3e · t)I eηabF Iab + eEaI L⃗t AIa −eEaI Da(3A · t)I,(5.39)where (3e · t)I := ta 3eaI, eEaI := eηab 3ebI, (3A · t)I := ta 3AIa, and AIa := qba3AIb.

To obtain (39),we used the fact that L⃗t qab = 0. Note also that F Iab := qcaqdb3F Icd is the curvature tensor ofthe generalized derivative operator Da (:= qba3Db) on Σ associated with AIa.

Since12(3e · t)I eηabF Iab = −12∼NǫIJK eEaI eEbJFabK −Na eEbIF Iab(5.40)(where ∼N := q−12N), we see that (modulo a surface integral) the Lagrangian LP of the 2+1Palatini theory is given byLP =ZΣ −12∼NǫIJK eEaI eEbJFabK −Na eEbIF Iab + eEaI L⃗t AIa + (Da eEaI )(3A · t)I. (5.41)Similarly, using gab = qab −nanb and the decomposition √−g = N√qdt, it follows thatSKG(3e, φ) = −8πZdtZΣn∼Neeqab∂aφ∂bφ −∼N−1(L⃗t φ −Na∂aφ)2o,(5.42)where eeqab := qqab (= eEaI eEbI).

Thus, the Klein-Gordon Lagrangian LKG is simply given byLKG = −8πZΣn∼Neeqab∂aφ∂bφ −∼N−1(L⃗t φ −Na∂aφ)2o. (5.43)The total Lagrangian LT is the sum LT = LP + 12LKG and is to be viewed as a functional ofthe configuration variables (3A · t)I, ∼N, Na, AIa, eEaI , φ and their first time derivatives.Following the standard Dirac constraint analysis, we find thateπ :=δLTδ(L⃗t φ) = 8π∼N−1(L⃗t φ −Na∂aφ)(5.44)is the momentum canonically conjugate to φ.

Since this equation can be inverted to giveL⃗t φ = 18π∼N eπ + Na∂aφ,(5.45)47

it does not define a constraint. On the other hand, eEaI is constrained to be the momentumcanonically conjugate to AIa, while (3A· t)I, ∼N, and Na play the role of Lagrange multipliers.The resulting total phase space (ΓT, ΩT) is coordinatized by the pairs of fields (AIa, eEaI ) and(φ, eπ) with symplectic structureΩT =ZΣ dI eEaI ∧∧dIAIa + Tr(dIeπ ∧∧dIφ).

(5.46)The Hamiltonian is given byHT(A, eE, φ, eE) =ZΣ ∼N12ǫIJK eEaI eEbJFabK + (4πeeqab∂aφ∂bφ +116πeπ2)+ Na( eEbIF Iab + eπ∂aφ) −(Da eEaI )(3A · t)I,(5.47)We shall see that this is just a sum of 1st class constraint functions associated with12ǫIJK eEaI eEbJFabK + (4πeeqab∂aφ∂bφ +116πeπ2) ≈0,(5.48)eEbIF Iab + eπ∂aφ ≈0,and(5.49)Da eEaI ≈0. (5.50)These are the constraint equations associated with the Lagrange multipliers ∼N, Na, (3A· t)I,respectively.Two remarks are in order: First, note that just as we found for the 2+1 Palatini theorywith or without a cosmological constant Λ, the constraint equations for the 2+1 Palatinitheory coupled to a Klein-Gordon field are polynomial in the basic canonically conjugatevariables (AIa, eEaI ) and (φ, eπ).

Since the Hamiltonian is just a sum of these constraints, itfollows that the evolution equations will be polynomial as well. Second, since the constraintequations do not involve the inverse of eEaI , the above Hamiltonian formulation is well-definedeven if eEaI is non-invertible.

Thus, we have a slight extension of the standard 2+1 theory ofgravity coupled to a massless scalar field. It can handle those cases where the spatial metriceeqab := eEaI eEbI becomes degenerate.Our next goal is to verify the claim that the constraint functions associated with (5.48)-(5.50) form a 1st class set.

To do this, we let vI (which takes values in the Lie algebra ofSO(2, 1)), ∼N, and Na be arbitrary test fields on Σ. Then we defineC(∼N) :=ZΣ ∼N12ǫIJK eEaI eEbJFabK + (4πeeqab∂aφ∂bφ +116πeπ2),(5.51)C′( ⃗N) :=ZΣ Na( eEbIF Iab + eπ∂aφ),and(5.52)G(v) :=ZΣ vI(Da eEaI ),(5.53)48

to be the scalar, vector, and Gauss constraint functions.As we saw in subsection 3.3 for the 2+1 Palatini theory, the subset of Gauss constraintfunctions form a Lie algebra with respect to Poisson bracket. Given G(v) and G(w), we have{G(v), G(w)} = G([v, w]),(5.54)where [v, w]I is the Lie bracket in LSO(2,1).

Thus, the mapping v 7→G(v) is a representationof the Lie algebra LSO(2,1). Given its geometrical interpretation as the generator of internalrotations, it follows that{G(v), C(∼N)} = 0and(5.55){G(v), C′( ⃗N)} = 0,(5.56)as well.Since one can show that the vector constraint function does not by itself have any directgeometrical interpretation (see, e.g., [34]), we will define a new constraint function C( ⃗N) bytaking a linear combination of the vector and Gauss constraints.

We defineC( ⃗N) := C′( ⃗N) −G(N),(5.57)where NI := NaAIa. We will call C( ⃗N) the diffeomorphism constraint function since themotion it generates on phase space corresponds to the 1-parameter family of diffeomorphismson Σ associated with the vector field Na.

To see this, we can writeC( ⃗N) : = C′( ⃗N) −G(N)=ZΣ Na( eEbIF Iab + eπ∂aφ) −ZΣ NI(Da eEaI )=ZΣ Na( eEbIF Iab −AIaDb eEbI) + eπNa∂aφ=ZΣeEaI L ⃗NAIa + eπL ⃗Nφ,(5.58)where the Lie derivative with respect to Na treats fields having only internal indices asscalars. To obtain the last line of (5.58), we ignored a surface integral (which would vanishanyways for Na satisfying the appropriate boundary conditions).

By inspection, it followsthat AIa 7→AIa + ǫL ⃗NAIa + O(ǫ2), etc. Using this geometric interpretation of C( ⃗N), it followsthat{C( ⃗N), G(v)} = G(L ⃗Nv),(5.59){C( ⃗N), C(∼M)} = C(L ⃗N∼M),and(5.60){C( ⃗N), C( ⃗M)} = C([ ⃗N, ⃗M]).

(5.61)49

We are left to evaluate the Poisson bracket {C(∼N), C(∼M)} of two scalar constraints.After a fairly long but straightforward calculation, one can show that{C(∼N), C(∼M)} = C′( ⃗K)= C( ⃗K) + G(K, K),(5.62)where Ka := eeqab(∼N∂b∼M −∼M∂b∼N) and eeqab = eEaI eEbI. This result makes crucial use of thefact thatǫIJKǫIMN = (−1)(ηJMηKN −ηJNηKM).

(5.63)This is a property of the structure constants ǫI JK := ǫIMNηMJηNK of the Lie algebra ofSO(2, 1). Thus, the constraint functions are closed under Poisson bracket—i.e., they forma 1st class set.

Note, however, that since the vector field Ka depends on the phase spacevariable eEaI , the Poisson bracket (5.62) involves structure functions. The constraint functionsdo not form a Lie algebra.

This result is similar to what we found for the standard Einstein-Hilbert theory in subsection 2.3.It is interesting to note that even if we did not couple matter to the 2+1 Palatini theory,but performed the Legendre transform as we did above (i.e., using the additional structureprovided by the spacetime metric gab := 3eIa3eJb ηIJ), we would still obtain the same Poissonbracket algebra. The constraint functions would still fail to form a Lie algebra due to thestructure functions in (5.62).

At first, something seems to be wrong with this statement,since we saw in subsection 3.3 that the constraint functions G(v) and F(α) of the 2+1Palatini theory form a Lie algebra with respect to Poisson bracket. One may ask why theconstraints functions obtained via one decomposition of the 2+1 Palatini theory form a Liealgebra, while those obtained from another decomposition do not.The answer to this question is actually fairly simple.

Namely, it is easy to destroy the“Lie algebra-ness” of a set of constraint functions. If φi (i = 1, · · ·, m) denote m constraintfunctions which form a Lie algebra under Poisson bracket (i.e., {φi, φj} = Ckijφk where Ckijare constants), then a linear combination of these constraints, χi = Λijφj, will not in generalform a Lie algebra if Λij are not constants on the phase space.

In essence, this is whathappens when one passes from the G(v) and F(α) constraint functions of subsection 3.3 tothe G(v), C(∼N), and C( ⃗N) constraint functions of this subsection. The transition from F(α)to C(∼N) and C( ⃗N) involve functions of the phase space variables.6.

3+1 Palatini theoryIn this section, we will describe the 3+1 Palatini theory. In subsection 6.1, we define the3+1 Palatini action and show that the Euler-Lagrange equations of motion are equivalent50

to the standard vacuum Einstein’s equation. In subsection 6.2, we will follow the standardDirac constraint analysis to put the 3+1 Palatini theory in Hamiltonian form.

We obtain aset of constraint equations which include a 2nd class pair. By solving this pair, we find thatthe remaining (1st class) constraints become non-polynomial in the (reduced) phase spacevariables.

In essence, we are forced into using the standard geometrodynamical variables ofgeneral relativity. In fact, as we shall see in subsection 6.3, the Hamiltonian formulation ofthe 3+1 Palatini theory is just that of the standard Einstein-Hilbert theory.

Thus, the 3+1Palatini theory does not give us a connection-dynamic description of 3+1 gravity.Much of the material in subsection 6.2 is based on an an analysis of the 3+1 Palatinitheory given in Chapter 4 of [3].6.1 Euler-Lagrange equations of motionTo obtain the Palatini action for 3+1 gravity, we will first write the standard Einstein-Hilbert actionSEH(gab) =ZΣ√−gR(6.1)in tetrad notation. UsingRabIJ = Rabcd 4ecI4eJd(6.2)(which relates the internal and spacetime curvature tensors of the unique, torsion-free gen-eralized derivative operator ∇a compatible with the tetrad 4eaI) andǫabcd =4eIa4eJb4eKc4eLd ǫIJKL(6.3)(which relates the volume element ǫabcd of gab = 4eIa4eJb ηIJ to the volume element ǫIJKL ofηIJ), we find that√−gR = 14eηabcdǫIJKL4eIa4eJb RcdKL.

(6.4)Thus, viewed as a functional of the co-tetrad 4eIa, the standard Einstein-Hilbert action isgiven bySEH(4e) = 14ZΣeηabcdǫIJKL4eIa4eJb RcdKL. (6.5)To obtain the 3+1 Palatini action, we simply replace RabI J in (5) with the internalcurvature tensor 4FabI J of an arbitrary generalized derivative operator 4Da defined by4DakI := ∂akI + 4AaIJkJ.

(6.6)We define the 3+1 Palatini action to beSP(4e, 4A) := 18ZMeηabcdǫIJKL4eIa4eJb4FcdKL,(6.7)51

where 4FabI J = 2∂[a4Ab]IJ + [4Aa, 4Ab]IJ. Just as we did for the 2+1 Palatini theory, we haveincluded an additional factor of 1/2 in definition (6.7) so our canonically conjugate variableswill agree with those used in the literature (see, e.g., [3]).

Note also, that as defined above,4Da knows how to act only on internal indices. We do not require that 4Da know how toact on spacetime indices.

However, we will often find it convenient to consider a torsion-freeextension of 4Da to spacetime tensor fields. All calculations and all results will be independentof this choice of extension.Since the 3+1 Palatini action is a functional of both a co-tetrad and a connection 1-form,there will be two Euler-Lagrange equations of motion.

When we vary SP(4e, 4A) with respectto 4eIa and 4AaIJ, we findeηabcdǫIJKL4eJb4FcdKL = 0and(6.8)4Db(eηabcdǫIJKL4eKc4eLd ) = 0,(6.9)respectively. The last equation requires a torsion-free extension of 4Da to spacetime tensorfields, but since the left hand side of (6.9) is the divergence of a skew spacetime tensordensity of weight +1 on M, it is independent of this choice.

Noting that eηabcdǫIJKL 4eKc4eLd =4(4e) 4e[aI4eb]J(where (4e) := √−g), we can rewrite (6.9) as4Db(4e) 4e[aI4eb]J= 0. (6.10)This equation is identical in form to equation (3.18) obtained in Section 3 for the 2+1 Palatinitheory.By following exactly the same argument used in subsection 3.1, equation (6.10) impliesthat 4AaI J = ΓaIJ, where ΓaIJ is the internal Christoffel symbol of ∇a.

Using this result, theremaining Euler-Lagrange equation of motion (6.8) becomeseηabcdǫIJKL4eJb RcdKL = 0. (6.11)When (6.11) is contracted with 4eeI, we get Gae = 0.

Thus, we can produce the 3+1 vacuumEinstein’s equation starting from the 3+1 Palatini action given by (6.7). Note that just asin the 2+1 theory, the equation of motion (6.9) for 4AaI J can be solved uniquely for 4AaIJin terms of the remaining basic variables 4eIa.

The pulled-back action SP(4e) defined on thesolution space 4AaI J = ΓaIJ is just 1/2 times the standard Einstein-Hilbert action SEH(4e)given by (6.5).6.2 Legendre transformTo put the 3+1 Palatini theory in Hamiltonian form, we will use the additional structureprovided by the spacetime metric gab. We will assume that M is topologically Σ × R for52

some spacelike submanifold Σ and assume that there exists a time function t (with nowherevanishing gradient (dt)a) such that each t = const surface Σt is diffeomorphic to Σ.tawill denote the time flow vector field (ta(dt)a = 1), while na will denote the unit covariantnormal to the t = const surfaces. na := gabnb will be the associated future-pointing timelikevector field (nana = −1).

Given na and na, it follows that qab := δab + nanb is a projectionoperator into the t = const surfaces. We can then define the induced metric qab, the lapseN, and shift Na as we did for the standard Einstein-Hilbert theory in Section 2.

Recall thatta = Nna + Na, with Nana = 0.Now let us write (6.7) in 3+1 form. Using the decomposition eηabcd = 4t[aeηbcd]dt (whereeηabc is the Levi-Civita tensor density of weight +1 on Σ), we getSP(4e, 4A) = 18ZMeηabcdǫIJKL4eIa4eJb4FcdKL=ZdtZΣ14(4e · t)IǫIJKLeηbcdeJb FcdKL + 12eEaIJL⃗t AaIJ −12eEaIJDa(4A · t)IJ,(6.12)where (4e · t)I := ta 4eaI, eEaIJ := 12ǫIJKLeηabc 4eKb4eLc , (4A · t)IJ := ta 4AaIJ, AaIJ := qba4AbIJ,and eIa := qba4eIb.

To obtain the last line of (6.12), we used the fact that L⃗t qab = 0. Notealso that FabIJ := qcaqdb4FcdIJ is the curvature tensor of the generalized derivative operatorDa (:= qba4Db) on Σ associated with AaIJ.

Since14(4e · t)IǫIJKLeηbcdeJb FcdKL = −12∼NTr( eEa eEbFab) + 12NaTr( eEbFab)(6.13)(where ∼N := q−12N and Tr denotes the trace operation on internal indices), we see that(modulo a surface integral) the Lagrangian LP of the 3+1 Palatini theory is given byLP =ZΣ −12∼NTr( eEa eEbFab) + 12NaTr( eEbFab)+ 12eEaIJL⃗t AaIJ + 12(Da eEaIJ)(4A · t)IJ. (6.14)The configuration variables of the theory are (4A · t)IJ, ∼N, Na, AaIJ, and eEaIJ.But before we perform the Legendre transform, we should note that the configurationvariable eEaIJ is not free to take on arbitrary values.

In fact, from its definitioneEaIJ := 12ǫIJKLeηabc 4eKb4eLc ,(6.15)one can show thateeφab := ǫIJKL eEaIJ eEbKL = 0andTr( eEa eEb) > 0. (6.16)53

The second condition follows from the fact that Tr( eEa eEb) = 2eeqab (= 2qqab), where qab isthe inverse of the induced positive-definite metric qab on Σ. Thus, the starting point for theLegendre transform is LP together with the primary constraint eeφab = 0.

Since the inequality isa non-holonomic constraint, it will not reduce the number of phase space degrees of freedom.If we now follow the standard Dirac constraint analysis, we find that (4A · t)IJ, ∼N, andNa play the role of Lagrange multipliers. Their associated constraint equations (which ariseas secondary constraints in the analysis) areTr( eEa eEbFab) ≈0,(6.17)Tr( eEbFab) ≈0,and(6.18)Da eEaIJ ≈0.

(6.19)There is also a primary constraint which says that12eEaIJ is the momentum canonicallyconjugate to AaIJ. By demanding that the Poisson bracket of this constraint with the totalHamiltonian and the Poisson bracket of eeφab with the total Hamiltonian be weakly zero, wefind thatχab := ǫIJKL(Dc eEaIJ)[ eEb, eEc]KL + (a ↔b) ≈0.

(6.20)This is an additional secondary constraint which must be appended to constraint equations(6.16)-(6.19).In virtue of eeφab = 0, the expression for χab is independent of the choiceof torsion-free extension of Da to spacetime tensor fields. If we further demand that thePoisson bracket of χab with the total Hamiltonian be weakly zero, we find nothing new—i.e.,there are no tertiary constraints.Let us summarize the situation so far: Out of the original set of configuration variables{(4A · t)IJ, ∼N, Na, AaIJ,eEaIJ}, the first three are non-dynamical.

We also found that12 eEaIJ is the momentum canonically conjugate to AaIJ. Thus, the phase space (Γ′P, Ω′P) ofthe 3+1 Palatini theory is coordinatized by the pair (AaIJ, eEaIJ) and has the symplecticstructureΩ′P = 12ZΣ dI eEaIJ ∧∧dIAaIJ.

(6.21)The Hamiltonian is given byH′P(A, eE) =ZΣ12∼NTr( eEa eEbFab) −12NaTr( eEbFab) −12(Da eEaIJ)(4A · t)IJ+ ∼λabǫIJKL eEaIJ eEbKL,(6.22)where ∼λab is a Lagrange multiplier. The constraints of the theory are (6.16)-(6.20).

Note alsothat at this stage of the Dirac constraint analysis all constraint (and evolution) equationsare polynomial in the canonically conjugate variables.54

The next step in the Dirac constraint analysis is to evaluate the Poisson brackets of theconstraints and solve all 2nd class pairs. Since only{eeφab(x), χcd(y)} ̸≈0,(6.23)we just have to solve constraint equations (6.16) and (6.20).

As shown in Chapter 4 of [3],the most general solution to (6.16) iseEaIJ =: 2 eEa[InJ],(6.24)for some unit timelike covariant normal nI (nInI = −1) with eEaI invertible and eEaI nI = 0.By writing eEaIJ in this form, we see that the original 18 degrees of freedom per space pointfor eEaIJ has been reduced to 12. Note also that eEaI eEbI = eeqab, so eEaI is in fact a (densitized)triad.Given (6.24), the most convenient way to solve (6.20) is to gauge fix the internal vectornI.This will further reduce the number of degrees of freedom ofeEaIJ to 9, since nowonly eEaI will be arbitrary.

However, gauge fixing nI requires us to solve the boost part of(6.19) relative to nI as well.21 We can only keep that part of (6.19) which generates internalrotations leaving nI invariant.To solve these constraints, let us first define an internal connection 1-form KaIJ viaDakI =: DakI + KaIJkJ,(6.25)where Da is the unique, torsion-free generalized derivative operator on Σ compatible withthe (densitized) triad eEaI and the gauge fixed internal vector nI. Then constraint equation(6.20) and the boost part of (6.19) becomeχab = −4ǫIJKKcIL eE(aL eEb)J eEcK ≈0and(6.26)(Da eEaIJ)nJ = −KaMN eEaNqMI ≈0,(6.27)where qIJ := δIJ + nInJ.

By using the invertibility of the (densitized) triad eEaI , one can thenshow (again, see Chapter 4 of [3]) that (6.26) and (6.27) imply that KaIJ also be pure boostwith respect to nI—i.e., that KaIJ have the formKaIJ =: 2K[Ia nJ],(6.28)with KIanI = 0. Since Da is determined completely by eEaI and nI, the original 18 degrees offreedom for AaIJ has also been reduced to 9 degrees of freedom per space point.

The infor-mation contained in AaIJ (which is independent of eEaI and nI) is completely characterized21The boost part of any anti-symmetric tensor AIJ relative to nI is defined to be AIJnJ.55

by KIa. To emphasize the fact that eEaI nI = 0 and KIanI = 0, we will use a 3-dimensionalabstract internal index i and write eEai and Kia in what follows.Thus, after eliminating the 2nd class constraints, the phase space (ΓP, ΩP) of the 3+1Palatini theory is coordinatized by the pair ( eEai , Kia) and has the symplectic structureΩP =ZΣ dIKia ∧∧dI eEai .

(6.29)The Hamiltonian is given byHP( eE, K) =ZΣ12∼N−qR−2 eEa[i eEbj]KiaKjb−2Na eEbi D[aKib]+ (4A · t)ij eEa[iKaj],(6.30)where R denotes the scalar curvature of Da. This is just a sum of the 1st class constraintsfunctions associated witheeC( eE, K) := −qR −2 eEa[i eEbj]KiaKjb ≈0,(6.31)eCa( eE, K) := 4 eEbi D[aKib] ≈0,and(6.32)eGij( eE, K) := −eEa[iKaj] ≈0.

(6.33)(The overall numerical factors have been chosen in order to facilitate the comparison withthe standard Einstein-Hilbert theory.) Note that constraint equations (6.31), (6.32), and(6.33) are the remaining constraints (6.17), (6.18), and the rotation part of (6.19) relativeto nI expressed in terms of the phase space variables ( eEai , Kia).22 We will call (6.31), (6.32),and (6.33) the scalar, vector, and Gauss constraints for the 3+1 Palatini theory.Note that as a consequence of eliminating the 2nd class constraints by solving (6.16),(6.20), and the boost part of (6.19) relative to nI, the constraint equations (6.31)-(6.33)(and hence the evolution equations generated by the Hamiltonian) are now non-polynomialin the canonically conjugate pair ( eEai , Kia).

This is due to the dependence of R on the inverseof eEai . In fact, since eEai must be invertible, we are forced to take eEai as the configurationvariable of the theory.

We are led back to a geometrodynamical description of 3+1 gravity.Thus, the Hamiltonian formulation of the 3+1 Palatini theory has the same drawback as theHamiltonian formulation of standard Einstein-Hilbert theory. As we shall see in the nextsubsection, these theories are effectively the same.6.3 Relationship to the Einstein-Hilbert theory22The rotation part of any anti-symmetric tensor AIJ relative to nI is given by qMI qNJ AMN where qIJ :=δIJ + nInJ.56

In this subsection, we will not explicitly evaluate the Poisson bracket algebra of theconstraint functions for the 3+1 Palatini theory. Rather, we will describe the relationshipbetween the constraint equations (6.31)-(6.33) and those of the standard Einstein-Hilberttheory.

We shall see that if we solve the first class constraint (6.33) by passing to a reducedphase space, we recover the Hamiltonian formulation of the standard Einstein-Hilbert theoryin terms of the induced metric qab and its canonically conjugate momentum epab.To do this, let us first define a tensor field ∼Mab (of density weight -1 on Σ) via∼Mab := Kia∼Ebi,(6.34)where ∼Eia is the inverse of the (densitized) triad eEai . Then in terms of ∼Mab, one can showthat constraint equation (6.33) is equivalent to∼M[ab] ≈0.

(6.35)Thus, the constraint surface in ΓP defined by (6.33) will be coordinatized in part by thesymmetric part of ∼Mab—i.e., by ∼Kab := ∼M(ab).But we are not yet finished. Since (6.33) is a 1st class constraint, we must also factor-out the constraint surface by the orbits of the Hamiltonian vector field associated with theconstraint functionG(Λ) :=ZΣeGij( eE, K)Λij=ZΣ −eEai KajΛij.

(6.36)(Here Λij = Λ[ij] denotes an arbitrary anti-symmetric test field on Σ.) Since it is fairlyeasy to show that G(Λ) generates (gauge) rotations of the internal indices (i.e., eEai 7→eEai +ǫΛij eEaj +O(ǫ2) and Kia 7→Kia−ǫΛjiKja+O(ǫ2)), the factor space will be coordinatized by ∼Kaband the gauge invariant information contained in eEai .

This is precisely eeqab = eEai eEbi. Thus,the reduced phase space (ˆΓP, ˆΩP) is coordinatized by the pair (eeqab, ∼Kab) and has symplecticstructureˆΩP = 12ZΣ dI∼Kab ∧∧dIeeqab.

(6.37)All we must do now is make contact with the usual canonical variables of the standardEinstein-Hilbert theory. To do this, let us work with the undensitized fields qab and Kab,and lower and raise their indices, respectively.

Then in terms of epab defined byepab := √q(Kab −Kqab),(6.38)one can show thatˆΩP = −12ZΣ dIepab ∧∧dIqab(6.39)57

andeeC(q, ep) = −qR + (epab epab −12ep2) ≈0,(6.40)eCa(q, ep) = −2qabDc epbc ≈0. (6.41)Up to overall factors, these are just the symplectic structure ΩEH and scalar and vectorconstraint equations of the standard Einstein-Hilbert theory described in Section 2.

Thefactor of −1/2 in the symplectic structure is due to the combination of using an actionwhich is 1/2 the standard Einstein-Hilbert action and using a (densitized) triad eEai insteadof a covariant metric qab as our basic dynamical variables. Thus, we see that the Hamiltonianformulation of the 3+1 Palatini theory is nothing more than the familiar geometrodynamicaldescription of general relativity.7.

Self-dual theoryIn this section, we will describe the self-dual theory of 3+1 gravity. This theory is similarin form to the 3+1 Palatini theory described in the previous section; however, it uses aself-dual connection 1-form as one of its basic variables.

We define the self-dual action forcomplex 3+1 gravity and show that we still recover the standard vacuum Einstein’s equationeven though we are using only half of a Lorentz connection. We then perform a Legendretransform to put the theory in Hamiltonian form.

In terms of the resulting complex phasespace variables, all equations of the theory are polynomial. This simplification gives theself-dual theory a major advantage over the 3+1 Palatini theory.

As noted in the previoussection, the Hamiltonian formulation of the 3+1 Palatini theory reduces to that of thestandard Einstein-Hilbert theory with its troublesome non-polynomial constraints.As mentioned in footnote 3 in Section 1, to obtain the phase space variables for the realtheory, we must impose reality conditions to select a real section of the original complexphase space.At the end of subsection 7.2 we will describe these conditions and discusshow they are implemented. The need to use reality conditions is a necessary consequenceof using an action principle to obtain the new variables for real 3+1 gravity.

Although wemention here that it is possible to stay within the confines of the real theory by performinga canonical transformation on the standard phase space of real general relativity, we will notfollow that approach in this paper. (Interested readers should see [1] for a detailed discussionof Ashtekar’s original approach.) Rather, we will start with an action for the complex theoryand obtain the new variables as outlined above.

Henceforth, the co-tetrad 4eIa will be assumedto be complex unless explicitly stated otherwise.58

7.1 Euler-Lagrange equations of motionTo write down the self-dual action for complex 3+1 gravity, all we have to do is replacethe (Lorentz) connection 1-form 4AaI J of the 3+1 Palatini theory by a self-dual connection1-form +4AaI J and let the co-tetrad 4eIa become complex. We define the self-dual action to beSSD(4e, +4A) := 14ZMeηabcdǫIJKL4eIa4eJb+4FcdKL,(7.1)where +4FabI J = 2∂[a+4Ab]IJ + [+4Aa, +4Ab]IJ is the internal curvature tensor of the self-dualgeneralized derivative operator +4Da defined by+4DakI := ∂akI + +4AaIJkJ.

(7.2)Some remarks are in order:1. We will always take our spacetime manifold M to be a real 4-dimensional manifold.Complex tensors at a point p ∈M will take values in the appropriate tensor productof the complexified tangent and cotangent spaces to M at p. The fixed internal spacewill also be complexified; however, the internal Minkowski metric ηIJ will remain real.Since the co-tetrad 4eIa are allowed to be complex, the spacetime metric gab defined bygab := 4eIa4eJb ηIJ will also be complex.2.

Although we can no longer talk about the signature of a complex metric gab, com-patibility with a complex co-tetrad 4eIa still defines a unique, torsion-free generalizedderivative operator ∇a.Thus, the complex Einstein tensor Gab := Rab −12Rgab iswell-defined; whence the complex equation of motion Gab = 0 makes sense. It is thisequation that defines for us the complex theory of 3+1 gravity.3.

When we say that the connection 1-form +4AaI J (or any other generalized tensor field)is self-dual, we will always mean with respect to its internal indices. Thus, the notionof self-duality makes sense only in 3+1 dimensions and applies only to generalizedtensor fields with a pair of skew-symmetric internal indices, T a···bc···dIJ = T a···bc···d[IJ].The dual of T a···bc···dIJ, denoted by ∗T a···bc···dIJ, is defined to be∗T a···bc···dIJ := 12ǫIJKLT a···bc···dKL,(7.3)where the internal indices of ǫIJKL are raised with the internal metric ηIJ.

Since ηIJhas signature (−+ ++), it follows that the square of the duality operator is minusthe identity. Hence, our definition of self-duality involves the complex number i. We59

define T a···bc···dIJ to be self-dual if and only if23∗T a···bc···dIJ = iT a···bc···dIJ. (7.4)Thus, self-dual fields in Lorentzian 3+1 gravity are necessarily complex.4.

Given any generalized tensor field T a···bc···dIJ = T a···bc···d[IJ], we can always decomposeit asT a···bc···dIJ = +T a···bc···dIJ + −T a···bc···dIJ,(7.5)where+T a···bc···dIJ := 12(T a···bc···dIJ −i∗T a···bc···dIJ)and(7.6a)−T a···bc···dIJ := 12(T a···bc···dIJ + i∗T a···bc···dIJ). (7.6b)Since ∗(+T a···bc···dIJ) = i+T a···bc···dIJ and ∗(−T a···bc···dIJ) = −i−T a···bc···dIJ, it follows that+T a···bc···dIJ and −T a···bc···dIJ are the self-dualand anti self-dual parts of T a···bc···dIJ.Equations (7.6a) and (7.6b) define the self-duality and anti self-duality operators +and −.5.

The generalized derivative operator +4Da is said to be self-dual only in the sense that itis defined in terms of a self-dual connection 1-form +4AaIJ. As in many of the previoustheories, +4Da (as defined by (7.2)) knows how to act only on internal indices.

Butas usual, we will often find it convenient to consider a torsion-free extension of +4Dato spacetime tensor fields. All calculations and all results will be independent of thischoice of extension.

Note also that the internal curvature tensor of the generalizedderivative operator +4Da is given by+4FabIJ = 2∂[a+4Ab]IJ + [+4Aa, +4Ab]IJ. (7.7)Since one can show that the (internal) commutator of two self-dual fields is also self-dual, it follows that +4FabI J is self-dual with respect to its internal indices.Given these general remarks, we are now ready to return to the self-dual action (7.1)and obtain the Euler-Lagrange equations of motion.

Varying SSD(4e, +4A) with respect to 4eIagiveseηabcdǫIJKL4eJb+4FcdKL = 0,(7.8)23T a···bc···dIJ is defined to be anti self-dual if and only if ∗T a···bc···dIJ = −iT a···bc···dIJ. The choice of +ifor self-dual and −i for anti self-dual is purely convention.

I have chosen our conventions to agree with thoseof [3].60

while varying SSD(4e, +4A) with respect to +4AaIJ gives+4Db(4e) +(4e[aI4eb]J )= 0. (7.9)To obtain (7.9), we used the fact that eηabcdǫIJKL 4eKc4eLd = 4(4e) 4e[aI4eb]J (where (4e) := √−g).Note also that we are forced to take the self-dual part of 4e[aI4eb]J since the variations δ+4AaIJare required to be self-dual.

This is the distinguishing feature between the self-dual and3+1 Palatini equations of motion. Finally note that although (7.9) requires a torsion-freeextension of +4Da to spacetime tensor fields, it is independent of this choice since the lefthand side is the divergence of a skew spacetime tensor density of weight +1 on M.We would now like to show that (7.9) implies that +4Da is the self-dual part of the unique,torsion-free generalized derivative operator ∇a compatible with 4eIa.

But since we are workingwith self-dual fields, the argument used for the 2+1 and 3+1 Palatini theories does not yetapply. We will have to do some preliminary work before we can use those results.If ΓaIJ denotes the internal Christoffel symbol of ∇a, we define the self-dual part +∇a of∇a by+4∇akI := ∂akI + +ΓaIJkJ,(7.10)where +ΓaIJ is the self-dual part of ΓaIJ.The difference between +4Da and +∇a is thencharacterized by a generalized tensor field +4CaIJ defined by+4DakI =: +∇akI + +4CaIJkJ.

(7.11)Note that +4CaIJ is self-dual as the notation suggests. In fact, +4CaIJ = +4AaI J −+ΓaI J. Nowlet us write (7.9) in terms of +∇a and +4CaIJ.

Using (7.11) to expand the left hand side of(7.9), we get+∇b(4e) +(4e[aI4eb]J )+ (4e)+4CbIK +(4e[aK4eb]J ) + +4CbJK +(4e[aI4eb]K)= 0. (7.12)Since +∇akI = ∇akI −−ΓaI JkJ (where −ΓaI J is the anti self-dual part of the internal Christof-fel symbol ΓaIJ), and since the last two terms of (7.12) can be written as the (internal)commutator of +4CaIJ and +(4e[aI4eb]J ), we get−h−Γb, +(4e[a 4eb])iIJ +h+4Cb, +(4e[a 4eb])iIJ = 0.

(7.13)The first commutator vanishes since −ΓbIJ is anti self-dual while +(4e[aI4eb]J ) is self-dual; in thesecond commutator, +(4e[aI4eb]J ) can be replaced by 4e[aI4eb]J . Thus, (7.13) reduces toh+4Cb, (4e[a 4eb])iIJ = 0.

(7.14)61

This is exactly the form of the equation found in the 2+1 Palatini theory with +4CbIJ replacing3CbIJ. (See equation (3.20).) We can now follow the argument given there to conclude that+4CaIJ = 0.

Thus, +4AaI J = +ΓaI J as desired.By substituting the solution +4AaI J = +ΓaI J into the remaining equation of motion (7.8),we geteηabcdǫIJKL4eJb+RcdKL = 0(7.15)where +RcdKL is the self-dual part of the internal curvature tensor RcdKL of ∇a. Then byusing the definition of +RcdKL, we see that equation (7.15) becomes0 = 12eηabcdǫIJKL4eJb (RcdKL −i2ǫKLMNRcdMN)= 12eηabcdǫIJKL4eJb RcdKL,(7.16)where the second term on the first line vanishes by the Bianchi identity R[abc]d = 0.

When(7.16) is contracted with 4eeI, we get Gae = 0. Thus, the self-dual action (7.1) reproducesthe vacuum Einstein’s equation for complex 3+1 gravity.Since this is an important—yet somewhat suprising—result, it is perhaps worthwhile torepeat the above argument from a slightly different perspective.

First note that the self-dualaction (7.1) and the 3+1 Palatini action (6.7) differ by a term involving the dual of thecurvature tensor 4FcdKL. This extra term in the self-dual action is not a total divergence andthus gives rise to an additional equation of motion that is not present in the 3+1 Palatinitheory.

This equation of motion also involves the dual of the curvature tensor. (Compareequations (7.8) and (6.8).) However, as we showed above, if we solve (7.9) for +4AaIJ andsubstitute the solution +4AaIJ = +ΓaIJ back into (7.8), the additional equation of motion isautomatically satisfied as a consequence of the Bianchi identity R[abc]d = 0.

Hence, there areno “spurious” equations of motion. Moreover, since the self-dual and 3+1 Palatini actionsdiffer by a term that is not a total divergence, the canonically conjugate variables for the twotheories will disagree.

As we shall see in the following section, it is this difference that willallow us to construct a Hamiltonian formulation of 3+1 gravity with a connection 1-form asthe basic configuration variable.Finally, we conclude this subsection by showing the relationship between the self-dualand standard Einstein-Hilbert actions. To do this, note that since the equation of motion(7.9) for +4AaIJ could be solved uniquely for +4AaI J in terms of the remaining basic variables4eIa, we can pull-back the self-dual action SSD(4e, +4A) to the solution space +4AaI J = +ΓaIJ62

and obtain a new action SSD(4e). Doing this, we findSSD(4e) = 14ZMeηabcdǫIJKL4eIa4eJb+RcdKL= 18ZMeηabcdǫIJKL4eIa4eJb RcdKL,(7.17)where we expanded +RcdKL and used the Bianchi identity R[abc]d = 0 to get the last line of(7.17).

Thus, SSD(4e) is just 1/2 times the standard Einstein-Hilbert action SEH(4e) viewedas a functional of a complex co-tetrad 4eIa. (See equation (6.5).) In fact, SSD(4e) = SP(4e),where SP(4e) is the pull-back of the 3+1 Palatini action.

It was precisely to obtain this lastequality that we defined the self-dual action (7.1) with an overall factor of 1/4 rather than1/8.7.2 Legendre transformTo put the self-dual theory for complex 3+1 gravity in Hamiltonian form, we will basicallyproceed as we did in Section 6 for the 3+1 Palatini theory. However, since the spacetimemetric gab := 4eIa4eJb ηIJ is now complex, we can only assume that M is topologically Σ×R forsome submanifold Σ and assume that there exists a real function t whose t = const surfacesfoliate M. (We cannot assume that Σ is spacelike, since the signature of a complex metricis not defined.) We can still introduce a real flow vector field ta (satisfying ta(dt)a = 1) anda unit covariant normal na to the t = const surfaces satisfying nana = −1.

(We are free tochoose −1 for the normalization of na since na is allowed to be complex.) na := gabnb is thevector field associated to na, and is related to ta by a complex lapse N and complex shiftNa via ta = Nna + Na, with Nana = 0.

Finally, the induced metric qab on Σ is given byqab = gab + nanb.Following the same steps that we used in the previous section for the 3+1 Palatini theory,we find that (modulo a surface integral) the Lagrangian LSD of the self-dual theory is givenbyLSD =ZΣ −∼NTr(+eEa +eEb +Fab) + NaTr(+eEb +Fab)+ (+eEaIJ)L⃗t+AaIJ + (+Da+eEaIJ)(+4A · t)IJ. (7.18)Here +eEaIJ denotes the self-dual part of eEaIJ := 12ǫIJKLeηabc 4eKb4eLc .

Note that the transi-tion from the 3+1 Palatini Lagrangian to the self-dual Lagrangian can be made by simplyreplacing all of the real fields by their self-dual parts. The configuration variables of thetheory are (+4A · t)IJ, ∼N, Na, +AaIJ, and +eEaIJ.Now recall that for the real 3+1 Palatini theory, the configuration variable eEaIJ was notfree to take on arbitrary values.

From its definition in terms of the co-tetrad 4eIa, we saw thateeφab := ǫIJKL eEaIJ eEbKL = 0andTr( eEa eEb) > 0. (7.19)63

The second condition followed from the fact that Tr( eEa eEb) = 2eeqab (= 2qqab), where qab wasthe inverse of the induced positive-definite metric qab on Σ. Taking the primary constraint(7.19) together with the 3+1 Palatini Lagrangian as the starting point for the Legendretransform, we found that the standard Dirac constraint analysis gave rise to additionalconstraints—one of which was 2nd class with respect to eeφab = 0.

By solving this 2nd classpair, the remaining (1st class) constraints became non-polynomial and we were forced backto the usual geometrodynamical description of real 3+1 gravity.24Similarly, we must check to see if there are any primary constraints on the configurationvariables of the self-dual theory. It turns out that although eeφab := ǫIJKL eEaIJ eEbKL = 0still follows from the definition of eEaIJ in terms of the complex co-tetrad 4eIa, it does notimply a constraint on the self-dualfield +eEaIJ.

Equation (7.19) may be viewed, instead,as a constraint on the anti self-dual field −eEaIJ. (Recall that for complex eEaIJ,−eEaIJ isnot necessarily the complex conjugate of +eEaIJ).

Thus, +eEaIJ is free to take on arbitraryvalues, and the Legendre transform for the complex self-dual theory is actually fairly simple.By following the standard Dirac constraint analysis, we find that +eEaIJ is the momentumcanonically conjugate to +AaIJ, while (+4A · t)IJ, ∼N, and Na play the role of Lagrangemultipliers. The complex phase space (CΓSD, CΩSD) is coordinatized by the pair of complexfields (+AaIJ, +eEaIJ) and has the natural complex symplectic structure25CΩSD =ZΣ dI+eEaIJ ∧∧dI+AaIJ.

(7.20)The Hamiltonian is given byHSD(+A, +eE) =ZΣ ∼NTr(+eEa +eEb +Fab) −NaTr(+eEb +Fab)−(+Da+eEaIJ)(+4A · t)IJ. (7.21)As we shall see in the next subsection, this is just a sum of 1st class constraint functionsassociated withTr(+eEa +eEb +Fab) ≈0,(7.22)24It is fairly easy to see that all of the above statements—except for the non-holonomic constraint whichwould now say that Tr( eEa eEb) be non-degenerate—apply to the complex 3+1 Palatini theory as well.

eeφab = 0is a primary constraint on the complex configuration variable eEaIJ, and it must be included when performingthe Legendre transform. The standard Dirac constraint analysis leads to a pair of 2nd class constraints which,when solved, gives back the usual geometrodynamical description of complex 3+1 gravity.25Note that in terms of the Poisson bracket { , } defined by CΩSD, we have {+AaIJ(x), +eEbKL(y)} =12δbaδ(x, y)(δ[IKδJ]L −i2ǫKLMNδIMδJN).

The “extra” term on the right hand side is needed to make the Poissonbracket self-dual in the IJ and KL pairs of indices.64

Tr(+eEb +Fab) ≈0,and(7.23)+Da+eEaIJ ≈0. (7.24)Note that all the constraints (and hence the evolution equations) are polynomial in thecanonically conjugate pair (+AaIJ, +eEaIJ).This is a simplification that we found in the2+1 Palatini theory, but lost in the 3+1 Palatini theory when we solved the 2nd classconstraints.

In fact, since the constraint equations never involve the inverse of +eEaIJ, theabove Hamiltonian formulation is well-defined even if +eEaIJ is non-invertible. Thus, we havea slight extension of complex general relativity.

The self-dual theory makes sense even whenthe induced metric eeqab = Tr(+eEa +eEb) becomes degenerate.In order to make contact with the notation used in the literature (see, e.g., [3]), let us usethe fact that the covariant normal na to Σ defines a unit internal vector nI via nI := na 4eaI.One can then show that+bKIJ := −12ǫKIJ + iqK[I nJ](7.25)is an isomorphism from the self-dual sub-Lie algebra of the complexified Lie algebra ofSO(3, 1) to the complexified tangent space of Σ. (Here ǫJKL := nIǫIJKL, qKI := δKI + nInK,and nInI = −1.) It satisfies∗(+bKIJ) := 12(+bKIJ −i2ǫIJMN +bKMN) = i +bKIJ,(7.26)[+bI, +bJ]MN = ǫIJK+bKMN,and(7.27)Tr(+bI +bJ) := −+bIMN+bJMN = −qIJ.

(7.28)The inverse of +bKIJ will be denoted by +bKIJ, and is obtained by simply raising and loweringthe indices of +bKIJ with the internal metric ηIJ.Since nK+bKIJ = 0, we will use a 3-dimensional abstract internal index i and write +biIJ and +biIJ in what follows. From property(7.27), it follows that +biIJ can actually be thought of as an isomorphism from the self-dualsub-Lie algebra of the complexified Lie algebra of SO(3, 1) to the complexified Lie algebraof SO(3).Given this isomorphism, we can now define a CLSO(3)-valued connection 1-form Aia anda CL∗SO(3)-valued vector density eEai via+AaIJ =: Aia+biIJand+eEaIJ =: −i eEai+biIJ.

(7.29)A straightforward calculation then shows that26+FabIJ = (2∂[aAib] + ǫijkAjaAkb) +biIJ =: F iab+biIJand(7.30)26To obtain equation (7.30), I assume that the fiducial derivative operator ∂a has been extended to act onCLSO(3)-indices in such a way that ∂a+biIJ = 0.65

Tr(+eEa +eEb) = eEai eEbi = eeqab. (7.31)Thus, eEai is a complex (densitized) triad and F iab is the Lie algebra-valued curvature tensorof the generalized derivative operator Da defined by Davi := ∂avi + ǫijkAjavk.In terms of Aia and eEai , the complex symplectic structure CΩSD becomesCΩSD = −iZΣ dI eEai ∧∧dIAia,(7.32)so −i eEai is the momentum canonically conjugate to Aia.

The Hamiltonian (7.21) can bewritten asHSD(A, eE) =ZΣ12∼Nǫijk eEai eEbjFabk −iNa eEbi F iab + i(Da eEai )(4A · t)i,(7.33)while the constraint equations (7.22)-(7.24) can be written asǫijk eEai eEbjFabk ≈0,(7.34)eEbi F iab ≈0,and(7.35)Da eEai ≈0. (7.36)We will take the constraint equations in this form when we analyze the Poisson bracketalgebra of the corresponding constraint functions in the following section.So far, all of the discussion in this section has dealt with complex 3+1 gravity.

In orderto recover the real theory, we must now impose reality conditions on the complex phase spacevariables (Aia, eEai ) to select a real section of (CΓSD, CΩSD). To do this, recall that in terms ofthe standard geometrodynamical variables (qab, epab), one recovers real general relativity fromthe complex theory by requiring that qab and epab both be real.

Since equation (7.31) tells usthat eEai eEbi = eeqab (= qqab), the condition that qab be real can be conveniently expressed interms of eEai aseEai eEbibe real. (7.37)Since we will want to ensure that this reality condition be preserved under the dynamicalevolution generated by the Hamiltonian, we must also demand that( eEai eEbi)•be real.

(7.38)Since in a 4-dimensional solution of the field equations epab is effectively the time derivativeof qab, requirement (7.38) is equivalent to the condition that epab be real. In addition, sincethe Hamiltonian of the theory is just a sum of the constraints (7.34)-(7.36) (all of which arepolynomial in the canonically conjugate variables), the reality conditions (7.37) and (7.38)are also polynomial in (Aia, eEai ).66

Finally, to conclude this subsection, I should point out that the self-dual action (7.1)viewed as a functional of a self-dual connection 1-form +4AaIJ and a real co-tetrad 4eIa doesnot yield the new variables for real 3+1 gravity when one performs a 3+1 decomposition.The definition of the configuration variable +eEaIJ in terms of a real co-tetrad 4eIa gives riseto a primary constraint. Although the non-holonomic constraint can be expressed in termsof +eEaIJ asTr(+eEa +eEb) > 0,(7.39)the holonomic constraint eeφab = 0 cannot be expressed solely in terms of +eEaIJ.

For real eEaIJwe have that −eEaIJ equals the complex conjugate of +eEaIJ, soeeφab = ǫIJKL(+eEaIJ + +eEaIJ)(+eEbKL + +eEbKL) = 0. (7.40)But by writing eeφab = 0 in this way, we have destroyed the possibility of completing thestandard Dirac constraint analysis.

For nowhere in the analysis have we been told how totake Poisson brackets of the complex conjugate fields. The Legendre transform of the self-dual Lagrangian for real 3+1 gravity breaks down when we try to incorporate the primaryconstraints into the analysis.7.3 Constraint algebraGiven constraint equations (7.34)-(7.36) for the complex self-dual theory, we would nowlike to verify the claim that their associated constraint functions form a 1st class set.

Todo this, let vi (which takes values in CLSO(3)), ∼N, and Na be arbitrary complex-valued testfields on Σ and defineC(∼N) := 12ZΣ ∼Nǫijk eEai eEbjFabk,(7.41)C′( ⃗N) := −iZΣ Na eEbi F iab,and(7.42)G(v) := −iZΣ vi(Da eEai ). (7.43)These will be called the scalar, vector, and Gauss constraint functions.

As the names andnotation suggest, these constraint functions will play a similar role to the constraint functionsdefined in subsection 5.3. Many of the calculations and results found there will apply hereas well.As usual, it is fairly easy to show that the Gauss constraint functions generate thestandard gauge transformations of the connection 1-form and rotation of internal indices.SinceδG(v)δ eEai= iDaviandδG(v)δAia= −i{v, eEa}i:= −iǫkjivj eEak,(7.44)67

it follows that Aia 7→Aia −ǫDavi + O(ǫ2) and eEai 7→eEai −ǫ{v, eEa}i + O(ǫ2). It also followsthat{G(v), G(w)} = G([v, w]),(7.45)where [v, w]i := ǫijkvjwk is the Lie bracket of vi and wi.

Thus, the mapping v 7→G(v) is arepresentation of the Lie algebra CLSO(3). Furthermore, given its geometrical interpretationas the generator of internal rotations, we have{G(v), C(∼N)} = 0and(7.46){G(v), C′( ⃗N)} = 0,(7.47)as well.Since it is possible to show that the vector constraint function does not by itself have anydirect geometrical interpretation (see, e.g., [34]) we will define a new constraint function,C( ⃗N), by taking a linear combination of the vector and Gauss constraints.

We defineC( ⃗N) := C′( ⃗N) −G(N),(7.48)where Ni := NaAia. We will call C( ⃗N) the diffeomorphism constraint function since themotion it generates on phase space corresponds to the 1-parameter family of diffeomorphismson Σ associated with the vector field Na.

To see this, we can writeC( ⃗N) : = C′( ⃗N) −G(N)= −iZΣ Na eEbi F iab + iZΣ Ni(Da eEai )= −iZΣ Na( eEbi F iab −AiaDb eEbi )= −iZΣeEai L ⃗NAia,(7.49)where the Lie derivative with respect to Na treats fields having only internal indices asscalars. To obtain the last line of (7.49), we ignored a surface integral (which would vanishanyways for Na satisfying the appropriate boundary conditions).

By inspection, it followsthat Aia 7→Aia + ǫL ⃗NAia + O(ǫ2), etc. Using this geometric interpretation of C( ⃗N), it followsthat{C( ⃗N), G(v)} = G(L ⃗Nv),(7.50){C( ⃗N), C(∼M)} = C(L ⃗N∼M),and(7.51){C( ⃗N), C( ⃗M)} = C([ ⃗N, ⃗M]).

(7.52)68

We are left to evaluate the Poisson bracket {C(∼N), C(∼M)} of two scalar constraints.UsingδC(∼N)δ eEai= ∼Nǫijk eEbjFabkandδC(∼N)δAia= ǫijkDb(∼N eEaj eEbk),(7.53)it follows that{C(∼N), C(∼M)} =ZΣδC(∼N)δAiaδC(∼M)δ(−i eEai ) −(∼N ↔∼M)=ZΣ iǫimnDc(∼N eEam eEcn)∼Mǫijk eEbjFabk −(∼N ↔∼M)=ZΣ iǫijkǫimn eEam eEcn eEbj(∼M∂c∼N −∼N∂c∼M)Fabk. (7.54)If we now use the fact thatǫijkǫimn = (δjmδkn −δjnδkm)(7.55)(which is a property of the structure constants of SO(3)), we get{C(∼N), C(∼M)} = C′( ⃗K)= C( ⃗K) + G(K),(7.56)where Ka := eeqab(∼N∂b∼M −∼M∂b∼N) and eeqab =eEai eEbi.

Thus, the constraint functions areclosed under Poisson bracket—i.e., they form a 1st class set. Note, however, that since thevector field Ka depends on the phase space variable eEai , the Poisson bracket (7.56) involvesstructure functions.

The constraint functions do not form a Lie algebra.8. 3+1 matter couplingsIn this section, we will couple various matter fields to 3+1 gravity.

We will repeat much ofwhat we did in Section 5, but this time in the context of the 3+1 theory, and for a Yang-Millsfield instead of a massless scalar field. In subsections 8.1 and 8.2, we couple a cosmologicalconstant Λ and a Yang-Mills field to complex 3+1 gravity using an action principle and theself-dual action as our starting point.

We shall show that the inclusion of these matter fieldsdoes not destroy the polynomial nature of the constraint equations. This is the main result.

(As usual, reality conditions should be included to recover the real theory.) As I mentionedfor the 2+1 theory, it is possible to couple other fundamental matter fields (e.g., scalar andDirac fields) to 3+1 gravity in a similar fashion and obtain the same basic results.

For amore detailed discussion of this and related issues, interested readers should see, e.g., [33].8.1 Self-dual theory coupled to a cosmological constant69

To couple a cosmological constant Λ to complex 3+1 gravity via the self-dual action, wewill start with the actionSΛ(4e, +4A) := 14ZMeηabcdǫIJKL4eIa4eJb (+4FcdKL −Λ3!4eKc4eLd ). (8.1)Here +4FabI J = 2∂[a+4Ab]IJ + [+4Aa, +4Ab]IJ is the internal curvature tensor of the self-dualgeneralized derivative operator +4Da defined by the self-dual connection 1-form +4AaI J, and4eIa is a complex co-tetrad which defines a spacetime metric gab via gab := 4eIa4eJb ηIJ.

Notethat SΛ(4e, +4A) is just a sum of the self-dual actionSSD(4e, +4A) := 14ZMeηabcdǫIJKL4eIa4eJb+4FcdKL(8.2)and a term proportional to the volume of the spacetime. In fact,Λ4!ZMeηabcdǫIJKL4eIa4eJb4eKc4eLd = ΛZM√−g,(8.3)where g is the determinant of the covariant metric gab.To show that (8.1) reproduces the standard equation of motion,Gab + Λgab = 0,(8.4)for gravity coupled to the cosmological constant Λ, we will first vary (8.1) with respect tothe self-dual connection 1-form +4AaIJ.

Since the second term (8.3) is independent of +4AaIJ,we get+4Db(4e) +(4e[aI4eb]J )= 0,(8.5)which is exactly the equation of motion we obtained in Section 7 for the vacuum case. Thus,just as we saw in subsection 7.1,+4AaI J = +ΓaIJ where +ΓaIJ is the self-dual part of theinternal Christoffel symbol of ∇a (the unique, torsion-free generalized derivative operatorcompatible with the co-tetrad.) Since (8.5) can be solved uniquely for +4AaIJ in terms ofthe remaining basic variables 4eIa, we can pull-back SΛ(4e, +4A) to the solution space +4AaI J =+ΓaI J.

We obtain a new actionSΛ(4e) = 14ZMeηabcdǫIJKL4eIa4eJb (+RcdKL −Λ3!4eKc4eLd ),(8.6)where +RcdKL is the self-dual part of the internal curvature tensor defined by ∇a. Then byusing the Bianchi identity R[abc]d = 0 for the first term and (8.3) for the second, we getSΛ(4e) = 12ZM√−g(R −2Λ).

(8.7)70

As mentioned in subsection 5.1, this is (up to an overall factor of 1/2) the action that oneuses to obtain (8.4) starting from an action principle. This is the desired result.To put this theory in Hamiltonian form, we proceed as in subsection 7.2.

Recall that(modulo a surface integral) the Lagrangian LSD of the self-dual theory is given byLSD =ZΣ −∼NTr(+eEa +eEb +Fab) + NaTr(+eEb +Fab)+ (+eEaIJ)L⃗t+AaIJ + (+Da+eEaIJ)(+4A · t)IJ,(8.8)where +eEaIJ denotes the self-dual part of eEaIJ := 12ǫIJKLeηabc 4eKb4eLc . By using the isomor-phism between the self-dual sub-Lie algebra of the complexified Lie algebra of SO(3, 1) andthe complexified Lie algebra of SO(3), we can rewrite (8.8) asLSD =ZΣ −12∼Nǫijk eEai eEbjFabk + iNa eEbi F iab −i eEai L⃗t Aia −i(Da eEai )(4A · t)i,(8.9)where eEai is a complex (densitized) triad (i.e., eEai eEbi = eeqab (= qqab)) and Aia is a connection1-form on Σ that takes values in the complexified Lie algebra of SO(3).By using the decomposition √−g = N√q dt together with the fact that13!∼ηabcǫijk eEai eEbj eEck = q,(8.10)one can similarly show thatΛ4!ZMeηabcdǫIJKL4eIa4eJb4eKc4eLd = Λ3!ZdtZΣ ∼N∼ηabcǫijk eEai eEbj eEck.

(8.11)Thus, the Lagrangian LΛ for 3+1 gravity coupled to the cosmological constant Λ via theself-dual action is given byLΛ =ZΣ −∼N(12ǫijk eEai eEbjFabk + Λ3!∼ηabcǫijk eEai eEbj eEck)+ iNa eEbi F iab −i eEai L⃗t Aia −i(Da eEai )(4A · t)i. (8.12)The configuration variables of the theory are (4A · t)i, ∼N, Na, Aia, and eEai .By following the standard Dirac constraint analysis, we find (as in the vacuum case) that−i eEai is the momentum canonically conjugate to Aia while (4A · t)i, ∼N, and Na play the roleof Lagrange multipliers.

The complex phase space and complex symplectic structure are thesame as those found for the self-dual theory with Λ = 0, while the Hamiltonian is given byHΛ(A, eE) =ZΣ ∼N12ǫijk eEai eEbjFabk + Λ3!∼ηabcǫijk eEai eEbj eEck−iNa eEbi F iab + i(Da eEai )(4A · t)i. (8.13)71

We shall see that this is just a sum of 1st class constraint functions associated with12ǫijk eEai eEbjFabk + Λ3!∼ηabcǫijk eEai eEbj eEck ≈0,(8.14)eEbi F iab ≈0,and(8.15)Da eEai ≈0. (8.16)These are the constraint equations associated with the Lagrange multipliers ∼N, Na, and(4A · t)i, respectively.

Note that they are polynomial in the canonically conjugate variables(Aia, eEai ) even when Λ ̸= 0. In fact, only constraint equation (8.14) differs from its Λ = 0counterpart.To conclude this subsection, we will verify the claim that the constraint functions associ-ated with (8.14)-(8.16) form a 1st class set.

Since the Gauss and diffeomorphism constraintfunctions associated with (8.16) and (8.15) will be the same as in subsection 7.3, we needonly concentrate on the scalar constraint functionC(∼N) :=ZΣ ∼N12ǫijk eEai eEbjFabk + Λ3!∼ηabcǫijk eEai eEbj eEck. (8.17)SinceδC(∼N)δ eEai= ∼N(ǫijk eEbjFabk + Λ2 ∼ηabcǫijk eEbj eEck)and(8.18a)δC(∼N)δAia= ǫijkDb(∼N eEaj eEbk),(8.18b)it follows that{C(∼N), C(∼M)} =ZΣδC(∼N)δAiaδC(∼M)δ(−i eEai ) −(∼N ↔∼M)=ZΣ iǫimnDc(∼N eEam eEcn)∼M(ǫijk eEbjFabk + Λ2 ∼ηabdǫijk eEbj eEdk) −(∼N ↔∼M)=ZΣ iǫijkǫimn(∼M∂c∼N −∼N∂c∼M)( eEam eEcn eEbjFabk + Λ2 qǫmjk eEcn).

(8.19)If we again use the fact that the structure constants of SO(3) satisfyǫijkǫimn = (δjmδkn −δjnδkm),(8.20)we get{C(∼N), C(∼M)} = C′( ⃗K)= C( ⃗K) + G(K),(8.21)where Ka := eeqab(∼N∂b∼M −∼M∂b∼N) and eeqab = eEai eEbi as before. Thus, the constraint functionsare closed under Poisson bracket—i.e., they form a 1st class set.

The Poisson bracket algebra72

of the constraint functions is exactly the same as it was for the Λ = 0 case. In particular,since the vector field Ka depends on the phase space variable eEai , the constraint functionsagain do not form a Lie algebra.8.2 Self-dual theory coupled to a Yang-Mills fieldTo couple a Yang-Mills field (with gauge group G) to complex 3+1 gravity via the self-dual action, we will start with the total actionST(4e, 4A, 4A) := SSD(4e, +4A) + 12SY M(4e, 4A),(8.22)where SSD(4e, +4A) is the self-dual action (8.2) and SY M(4e, 4A) is the usual Yang-Mills actionSY M(4e, 4A) := −ZM Tr(√−g gacgbd 4Fab4Fcd).

(8.23)Here SY M(4e, 4A) is to be viewed as a functional of a co-tetrad 4eIa and a connection 1-form 4Aawhich takes values in the Lie algebra of the gauge group G.27 Tr denotes the trace operationin some representation of the Yang-Mills Lie algebra, and 4Fab = 2∂[a4Ab] + [4Aa, 4Ab] is the(internal) curvature tensor of the generalized derivative operator 4Da defined by 4Aa. Theadditional factor of 1/2 is needed in front of SY M(4e, 4A) so that the above definition ofthe total action will be consistent with the definition of SSD(4e, 4A).

The Yang-Mills actiondepends on the co-tetrad 4eIa through its dependence on √−g and gab, but is independent ofthe self-dual connection 1-form +4AaIJ. As mentioned in Section 5, out of all the fundamentalmatter couplings, only the action for the Dirac field would depend on +4AaIJ.To show that (8.22) reproduces the standard Yang-Mills coupled to gravity equations ofmotion4Db(√−g 4Fab) = 0andGad = 8πT ad(Y M),(8.24)whereTab(Y M) := 14πTr(4Fac 4Fbc −14gab4Fcd4Fcd)(8.25)is the stress-energy tensor of the Yang-Mills field, we proceed as we did in the previoussubsection.

Since SY M(4e, 4A) is independent of +4AaIJ, the variation of (8.22) with respectto +4AaIJ implies+4Db(4e) +(4e[aI4eb]J )= 0. (8.26)27Yang-Mills fields will be denoted by bold face stem letters and their (internal) Lie algebra indices willbe suppressed.

Throughout, we will assume that we have a representation of the Yang-Mills Lie algebra LGby linear operators (on some vector space V ) with the trace operation Tr playing the role of an invariant,non-degenerate bilinear form k.73

As before, this tells us that +4AaI J = +ΓaIJ. Recalling that the Bianchi identity R[abc]d = 0implies that the pull-back of SSD(4e, +4A) to the solution space +4AaI J = +ΓaIJ is just 1/2times the standard Einstein-Hilbert action SEH(4e) for complex 3+1 gravity, we obtainST(4e, 4A) = 12SEH(4e) + SY M(4e, 4A).

(8.27)This is (up to an overall factor of 1/2) the usual total action that one uses to couple a Yang-Mills field to gravity. If we now vary ST(4e, 4A) with respect to 4Aa and 4eIa, and contract thesecond equation with 4edI, we recover (8.24).

Note that to write the first equation in (8.24),we had to consider a torsion-free extension of 4Da to spacetime tensor fields. But since theleft hand side is the divergence of a skew spacetime tensor density of weight +1 on M, it isindependent of this choice.To put this theory in Hamiltonian form, we need only decompose the Yang-Mills actionSY M(4e, 4A) since the self-dual Lagrangian LSD is given by (8.9).

Using gab = qab −nanb and√−g = N√q dt it follows thatSY M(4e,4A) =ZdtZΣ Trn−∼Nq−1eeqaceeqbdFabFcd + 2eeqab∼N−1q−1×× (L⃗t Aa −Da(4A · t) + NcFac)(L⃗t Ab −Db(4A · t) + NdFbd)o,(8.28)where eeqab := qqab (= eEai eEbi), (4A · t) := ta 4Aa, and Aa := qba4Ab. Here Fab := qca qdb4Fcd isthe curvature tensor of the generalized derivative operator Da (:= qba4Db) on Σ associatedwith Aa.

If we now define the “magnetic field” of Aa to be Bab := 2Fab (= 2qca qdb4Fcd), wesee that the Yang-Mills Lagrangian LY M is given byLY M =ZΣ Trn−14∼Nq−1eeqaceeqbdBabBcd + 2eeqab∼N−1q−1×× (L⃗t Aa −Da(4A · t) + 12NcBac)(L⃗t Ab −Db(4A · t) + 12NdBbd)o. (8.29)The total Lagrangian LT is the sum LT = LSD + 12LY M and is to be viewed as a functional ofthe configuration variables (4A·t), (4A·t)i, ∼N, Na, Aia, eEai , Aa and their first time derivatives.Following the standard Dirac constraint analysis, we find thateEa :=δLTδ(L⃗t Aa) = 2eeqab∼N−1q−1(L⃗t Ab −Db(4A · t) + 12NdBbd)(8.30)is the momentum (or “electric field”) canonically conjugate to Aa.

Since this equation canbe inverted to giveL⃗t Aa = 12qab∼N eEb + Da(4A · t) −12NcBac,(8.31)74

it does not define a constraint. On the other hand, −i eEai is constrained to be the momentumcanonically conjugate to Aia, while (4A · t), (4A · t)i, ∼N, and Na play the role of Lagrangemultipliers.

The resulting complex total phase space (CΓT, CΩT) is coordinatized by thepairs of fields (Aia, eEai ) and (Aa, eEa) with symplectic structureCΩT =ZΣ −idI eEai ∧∧dIAia + Tr(dI eEa ∧∧dIAa). (8.32)The Hamiltonian is given byHT(A, eE, A, eE) =ZΣ ∼N12ǫijk eEai eEbjFabk + 18q−1eeqaceeqbdTr(BabBcd + EabEcd)+ Na−i eEbi F iab + Tr( eEbFab)+ i(Da eEai )( 4A · t)i −Tr((4A · t)Da eEa),(8.33)where Eab := ∼ηabc eEc is the dual to the Yang-Mills “electric field” eEa.

We shall see that thisis just a sum of 1st class constraint functions associated with12ǫijk eEai eEbjFabk + 18q−1eeqaceeqbdTr(BabBcd + EabEcd) ≈0,(8.34)−i eEbi F iab + Tr( eEbFab) ≈0,(8.35)Da eEai ≈0,andDa eEa ≈0. (8.36)These are the constraint equations associated with the Lagrange multipliers ∼N, Na, (4A· t)I,and (4A · t), respectively.Note that by inspection (8.35) and (8.36) are polynomial in the canonically conjugatevariables.

However, constraint equation (8.34) fails to be polynomial due to the presence ofthe non-polynomial multiplicative factor q−1. But since q = 13!∼ηabcǫijk eEai eEbj eEck is polynomialin eEai , we can multiply (8.34) by q and restore the polynomial nature of all the constraints.Thus, to couple a Yang-Mills field to 3+1 gravity via the self-dual action, we are led to ascalar constraint with density weight +4.

This implies that the associated constraint functionwill be labeled by a test field (i.e., lapse function) having density weight −3.To verify the claim that the constraint functions associated with (8.34)-(8.36) form a1st class set, let vi and v (which take values in complexified Lie algebra of SO(3) and therepresentation of the Lie algebra of the Yang-Mills gauge group G), ∼N, and Na be arbitrarycomplex-valued test fields on Σ. Then defineC(∼N) :=ZΣ ∼N12ǫijk eEai eEbjFabk + 18q−1eeqaceeqbdTr(BabBcd + EabEcd),(8.37)C′( ⃗N) :=ZΣ Na(−i eEbi F iab + Tr( eEbFab)),and(8.38)G(v, v) :=ZΣ Tr(vDa eEa) −ivi(Da eEai ),(8.39)75

to be the scalar, vector, and Gauss constraint functions.As usual, it is fairly easy to show that the Gauss constraint functions generate thestandard gauge transformations of the connection 1-forms and rotations of internal indices.Using this information, we find that{G(v, v), G(w, w)} = G([v, w], [v, w]),(8.40){G(v, v), C(∼N)} = 0,and(8.41){G(v, v), C′( ⃗N)} = 0,(8.42)where [v, w] and [v, w]i are the Lie brackets in LG and CLSO(3). Thus, the subset of Gaussconstraint functions form a Lie algebra with respect to Poisson bracket.

In fact, the mapping(v, v) 7→G(v, v) is a representation of the direct sum Lie algebra LG ⊕CLSO(3).Again, the the vector constraint function will not have any direct geometrical inter-pretation, so we define the diffeomorphism constraint function C( ⃗N) by taking a linearcombination of the vector and Gauss law constraints. SettingC( ⃗N) := C′( ⃗N) −G(N, N),(8.43)where N := NaAa and Ni := NaAia, we can show thatC( ⃗N) =ZΣ −i eEai L ⃗NAia + Tr( eEaL ⃗NAa),(8.44)where the Lie derivative with respect to Na treats fields having only internal indices asscalars.

By inspection, Aia 7→Aia + ǫL ⃗NAia + O(ǫ2),etc., so the motion on phase spacegenerated by C( ⃗N) corresponds to the 1-parameter family of diffeomorphisms on Σ associatedwith Na. From this geometric interpretation of C( ⃗N), it follows that{C( ⃗N), G(v, v)} = G(L ⃗Nv, L ⃗Nv),(8.45){C( ⃗N), C(∼M)} = C(L ⃗N∼M),and(8.46){C( ⃗N), C( ⃗M)} = C([ ⃗N, ⃗M]).

(8.47)Finally, we are left to evaluate the Poisson bracket {C(∼N), C(∼M)} of two scalar constraintfunctions. After a fairly long but straightforward calculation that uses the fact that thestructure constants of SO(3) satisfyǫijkǫimn = (δjmδkn −δjnδkm),(8.48)one can show that{C(∼N), C(∼M)} = C′( ⃗K)= C( ⃗K) + G(K, K),(8.49)76

where Ka := eeqab(∼N∂b∼M −∼M∂b∼N) and eeqab = eEai eEbi. Thus, the constraint functions are againclosed under Poisson bracket—i.e., they form a 1st class set.

Just as we saw in subsection8.1 for the cosmological constant Λ, the Poisson bracket algebra of the constraint functionsfor complex 3+1 gravity coupled to a Yang-Mills field via the self-dual action is exactly thesame as it was for the vacuum case.9. General relativity without-the-metricTo conclude this review, we will describe a theory of 3+1 gravity without a metric.This will complete the transition from geometrodynamics to connection dynamics in 3+1dimensions.

Although we saw in Section 7 that the Hamiltonian formulation of the self-dualtheory for complex 3+1 gravity could be described in terms of a connection 1-form Aia and itscanonically conjugate momentum (or “electric field”) eEai , the action for the self-dual theorydepended on both a self-dual connection 1-form +4AaIJ and a complex co-tetrad 4eIa. Sincethe co-tetrad defines a spacetime metric gab via gab := 4eIa4eJb ηIJ, the self-dual action had animplicit dependence on gab.

The purpose of this section is to show that (modulo an importantdegeneracy) complex 3+1 gravity can be described by an action which does not depend ona spacetime metric in any way whatsoever. We shall see in subsection 9.1 that this actiondepends only on a connection 1-form 4Aia (which takes values in the complexified Lie algebraof SO(3)) and a scalar density ∼Φ of weight -1 on M. Hence we obtain a pure spin-connectionformulation of gravity.

We shall also see how this pure spin-connection action is related tothe self-dual action in the non-degenerate case.In subsection 9.2, we will analyze the constraint equations for this theory. Since we willhave shown in subsection 9.1 that the self-dual action and the pure spin-connection actionare equivalent when the self-dual part of the Weyl tensor is non-degenerate, the constraintequations of this theory are the same as the the constraint equations for the self-dual theoryfound in subsection 7.2.However, we will now be able to write down the most generalsolution to the four diffeomorphism constraint equations (the scalar and vector constraintsof the self-dual theory) when the “magnetic field” eBai associated with the connection 1-formAia is non-degenerate.

This is a new result for the Hamiltonian formulation of the 3+1 theorythat was made manifest by working in the pure spin-connection formalism.I should emphasize here that all of the results in this section are taken from previouswork of Capovilla, Dell, Jacobson, Mason, and Plebanski. I am not adding anything new inthis section; rather, I am reporting their results to bring the discussion of geometrodynamicsversus connection dynamics for 3+1 gravity to it logical conclusion.

Readers interested in amore detailed discussion of the general relativity without-the-metric theory (including matter77

couplings and an extension of this theory to a class of generally covariant gauge theories)should see [10, 11, 12, 13, 14] and references mentioned therein. In addition, Peld´an hasrecently provided a similar pure spin-connection formulation of 2+1 gravity.Interestedreaders should see [15].9.1 A pure spin-connection formulation of 3+1 gravityThe pure spin-connection action for complex 3+1 gravity is defined to beS(∼Φ, 4A) := 18ZM ∼Φ(eη ·4F i ∧4F j)(eη ·4F k ∧4F l)hijkl,(9.1)where 4Aia is a connection 1-form which takes values in the complexified Lie algebra of SO(3),∼Φ is a scalar density of weight -1 on M, and hijkl and (eη · 4F i ∧4F j) are shorthand notationsforhijkl := (δikδjl + δilδjk −δijδkl)and(9.2)(eη ·4F i ∧4F j) := eηabcd 4F iab4F jcd.

(9.3)As usual, 4F iab = 2∂[a4Aib] + [4Aa, 4Ab]i is the Lie algebra-valued curvature tensor of the gener-alized derivative operator 4Da defined by4Davi := ∂avi + [4Aa, v]i,(9.4)where [4Aa, v]i := ǫijk 4Ajavk denotes the Lie bracket of 4Aia and vi in CLSO(3). Although 4Dadefined by (9.4) knows how to act only on internal indices, we will often find it convenientto consider a torsion-free extension of 4Da to spacetime tensor fields.

All results and allcalculations will be independent of this choice.To show that the pure spin-connection action reproduces the standard results of complex3+1 gravity, one could vary (9.1) with respect to ∼Φ and 4Aia and analyze the resulting Euler-Lagrange equations of motion. Instead, we will start with the self-dual actionSSD(4e, +4A) := 14ZMeηabcdǫIJKL4eIa4eJb+4FcdKL(9.5)for complex 3+1 gravity and show that (modulo an important degeneracy) the self-dualaction (9.5) is actually equivalent to (9.1).

Basically, we will eliminate from (9.5) the fieldvariables which pertain to the spacetime metric by solving their associated Euler-Lagrangeequations of motion. This will require that a certain symmetric trace-free tensor ψij beinvertible as a 3 × 3 matrix.

By substituting these solutions back into the original action(9.5), we will eventually obtain (9.1). We should point out that, in a solution, ψij corresponds78

to the self-dual part of the Weyl tensor associated with the connection 1-form 4Aia. Thus, theequivalence between the two actions breaks down whenever the self-dual part of the Weyltensor is degenerate.

Note also that the pure spin-connection action describes complex 3+1gravity. To recover the real theory, one would have to impose reality conditions similar tothose used in Section 7 for the self-dual theory.

For a detailed discussion of ψij and thereality conditions see, e.g., [11].Since the self-dual action (9.5) depends on both a self-dual connection 1-form +4AaIJ and acomplex co-tetrad 4eIa, it has an implicit dependence on the spacetime metric gab := 4eIa4eJb ηIJ.Thus, it should come as no surprise that the first step in obtaining a metric-independentaction for 3+1 gravity involves the elimination of 4eIa from (9.5). To do this, let us defineΣabIJ := ǫIJKL4eKa4eLb(9.6)and +ΣabIJ to be its self-dual part.28 Then we can write the self-dual action asSSD(4e, +4A) = 14ZMeηabcd +ΣabIJ+4FcdIJ,(9.7)where we have used the fact that ΣabIJ +4FcdIJ = +ΣabIJ +4FcdIJ.

To simplify the notationsomewhat, let us recall that the self-dual sub-Lie algebra of the complexified Lie algebraof SO(3, 1) is isomorphic to the complexified Lie algebra of SO(3). Using the isomorphismdescribed in Section 7, we can define a CLSO(3)-valued connection 1-form 4Aia and a CL∗SO(3)-valued 2-form Σabi via+4AaIJ =: 4Aia+biIJand+ΣabIJ =: Σabi+biIJ.

(9.8)ThenSSD(4e, 4A) = 14ZMeηabcdΣabi4F icd,(9.9)where 4F iab = 2∂[a 4Aib] + ǫijk 4Aja4Akb is the Lie algebra-valued curvature tensor of the gener-alized derivative operator 4Da defined by 4Aia. It is related to +4FabIJ via +4FabIJ = 4F iab+biIJ.Although the right hand side of (9.9) involves just Σabi and 4Aia, the action is still afunctional of 4Aia and 4eIa since Σabi depends on 4eIa through equation (9.6).

To eliminate 4eIafrom the action, we must use the result (see, e.g., [12]) that (9.6) holds for some 4eIa if andonly if the trace-free part of Σi ∧Σj equals zero—i.e., ΣabIJ = ǫIJKL 4eKa4eLb for some 4eIa ifand only ifeηabcd(ΣiabΣjcd −13δijΣkabΣcdk) = 0. (9.10)28Recall that the self-dual part of ΣabIJ is defined by +ΣabIJ := 12(ΣabIJ −i2ǫIJ KLΣabKL).79

Thus, the self-dual action can be viewed as a functional of Σabi instead of 4eIa provided weinclude in the action a term which gives back (9.10) as one of its Euler-Lagrange equationsof motion. More precisely, let us defineS(ψ, Σ, 4A) := 14ZMeηabcd(Σabi4F icd −12ψijΣiabΣjcd),(9.11)where ψij is a symmetric trace-free tensor which will play the role of a Lagrange multiplierof the theory.

Then the variation of S(ψ, Σ, 4A) with respect to ψij will yield (9.10). Solvingthis equation and pulling-back the action (9.11) to this solution space gives back (9.9).But instead of varying S(ψ, Σ, 4A) with respect to ψij, let us vary this action with respectto Σabi and solve the resulting Euler-Lagrange equation of motion for Σabi in terms of ψijand 4Aia.

Varying (9.11) with respect to Σabi, we find4F iab −ψijΣabj = 0,(9.12)where ψij := δikδjlψkl. This equation can be solved for Σabi in terms of the remaining fieldvariables provided the inverse (ψ−1)ij of ψij exists.

Assuming that it does, we getΣabi = (ψ−1)ij4F jab. (9.13)If we now pull-back (9.11) to the solution space defined by (9.13), the resulting actionbecomesS(ψ, 4A) = 18ZMeηabcd(ψ−1)ij4F iab4F jcd.

(9.14)This is to be viewed as a functional of only the symmetric trace-free tensor ψij and theconnection 1-form 4Aia.We are almost finished. What remains to be shown is that ψij can be eliminated fromthe action (9.14) in lieu of a scalar density ∼Φ of weight -1 on M. To do this, let us write theaction in matrix notation and introduce another Lagrange multiplier eµ to guarantee that ψijis trace-free.29 Then (9.14) can be written asS(eµ, ψ, 4A) = 18ZM Tr(ψ−1 fM) + eµTrψ,(9.15)where ψij is now assumed to be only symmetric (and invertible) and fMij is defined byfMij := eηabcd 4F iab4F jcd.

(9.16)Varying S(eµ, ψ, 4A) with respect to ψij, we find−ψ−1 fMψ−1 + eµI = 0. (9.17)29By introducing eµ, we can consider arbitrary symmetric variations of ψij rather than symmetric andtrace-free variations.80

By multiplying on the left and right by ψ, we see that (9.17) is equivalent tofM = eµψ2. (9.18)This equation can be solved for ψij in terms of fMij and eµ provided the square-root of fMijexists.

Thenψ = eµ−1/2 fM1/2,(9.19)so the action (9.15) pulled-back to this solution space equalsS(eµ, 4A) = 14ZMeµ1/2Tr fM1/2. (9.20)The variation of S(eµ, 4A) with respect to eµ now implies that Tr fM1/2 = 0.

From (9.19) wesee that this is nothing more than the tracelessness of ψij.In order to write the action in its final form (9.1), recall that the characteristic equationobeyed by any 3 × 3 matrix isB3 −(TrB)B2 + 12(TrB)2 −TrB2B −(detB)I = 0. (9.21)Multiplying by B and setting B2 = fM (i.e., B = fM1/2), we get(detB)B = fM2 −12(Tr fM) fM,(9.22)(Here we have used the fact that TrB (= Tr fM1/2) = 0.

)Using (detB)2 = det fM andassuming invertibility of B (so that detB ̸= 0), we can write this last equation asB = (det fM)−1/2( fM2 −12(Tr fM) fM). (9.23)By substituting this expression for B (= fM1/2) back into (9.20), we findS(eµ, 4A) = 14ZMeµ1/2(det fM)−1/2Tr( fM2 −12(Tr fM) fM).

(9.24)Finally, if we define∼Φ = eµ1/2(det fM)−1/2(9.25)(which is a scalar density of weight -1 on M) and use the definition (9.16) of fMij in termsof 4F iab, we see thatS(∼Φ, 4A) = 18ZM ∼Φ(eη ·4F i ∧4F j)(eη ·4F k ∧4F l)hijkl(9.26)when viewed as a functional of ∼Φ and 4Aia. Note that hijkl and (eη ·4F i ∧4F j) are given asbefore by equations (9.2) and (9.3).

This is the desired result.81

9.2 Solution of the diffeomorphism constraintsGiven that the self-dual and pure spin-connection actions are equivalent when the self-dual part of the Weyl tensor is non-degenerate, it follows that the constraint equations ofthe theory can be written asǫijk eEai eEbjFabk ≈0,(9.27)eEbi F iab ≈0,and(9.28)Da eEai ≈0. (9.29)These are just the constraint equations that we found in subsection 7.2 when we put theself-dual theory in Hamiltonian form.

As before, the canonically conjugate variables consistof a pair of complex fields (Aia, eEai ), where Aia is the pull-back of the connection 1-form 4Aiato the submanifold Σ and eEai is a complex (densitized) triad which may or may not definean invertible induced metric eeqab := eEai eEbi. However, by working in the pure spin-connectionformalism, we will obtain a new result.

We will be able to write down the most generalsolution to the four diffeomorphism constraints (9.27)-(9.28) when the “magnetic field” eBaiassociated with Aia is non-degenerate.To see this, recall that in the self-dual theory+eEaIJ =: −i eEai+biIJ,(9.30)where +eEaIJ was the self-dual part ofeEaIJ := 12ǫIJKLeηabc 4eKb4eLc . (9.31)Note that in terms of ΣabIJ defined by (9.6), we have eEaIJ = 12 eηabcΣbcIJ, so that−i eEai = 12eηabcΣbci.

(9.32)If we now use the result that an invertible symmetric trace-free tensor ψij impliesΣabi = (ψ−1)ij4F jab,(9.33)it follows thateEai = i(ψ−1)ij eBaj,(9.34)where eBai := 12 eηabc 4F ibc (= 12 eηabcF ibc) is the “magnetic field” of Aia. We will now show that bytaking eEai of this form, the four diffeomorphism constraints (9.27)-(9.28) are automaticallysatisfied.82

Substituting (9.34) into the vector constraint (9.28), we geteEbi F iab = i(ψ−1)ij eBbj∼ηabc eBci = 0,(9.35)where we have used the fact that (ψ−1)ij is symmetric in i and j while ∼ηabc is anti-symmetricin b and c. Similarly, substituting (9.34) into the scalar constraint (9.27), we getǫijk eEai eEbjFabk = ǫijk eEai eEbj∼ηabc eBck= −iǫijk eEai eEbj∼ηabcψlk eEcl= −iqǫijkǫijlψlk= −2iqψkk= 0,(9.36)where we have used the fact that ψij is trace-free. Thus, the four diffeomorphism constraints(9.27)-(9.28) are automatically satisfied for eEai having the form given by (9.34).

That thisis the most general solution follows if eBai is non-degenerate. Then for a given Aia, eEai willhave 5 degrees of freedom (per space point) corresponding to the 5 degrees of freedom of thesymmetric trace-free tensor ψij.What remains to be solved is the Gauss constraint (9.29), which in terms of eBai and(ψ−1)ij can be written as0 = Da eEai= iDa((ψ−1)ij eBaj)= i eBajDa(ψ−1)ij.

(9.37)To obtain the last line of (9.37), we used the Bianchi identity Da eBaj = 12 eηabcD[aF jbc] = 0.10. DiscussionLet me begin by briefly summarizing the main results reviewed in this paper.1.

The standard Einstein-Hilbert theory is a geometrodynamical theory of gravity inwhich a spacetime metric is the fundamental field variable. The phase space variablesconsist of a positive-definite metric qab and its canonically conjugate momentum epab.These variables are subject to a set of 1st class constraints, which are non-polynomialin qab and which have a Poisson bracket algebra involving structure functions.

Thistheory is valid in n + 1 dimensions.83

2. The 2+1 Palatini theory is a connection dynamical theory defined for any Lie groupG.

The fundamental field variables consist of a LG-valued connection 1-form and aL∗G-valued covector field. The phase space is coordinatized by a connection 1-formAIa and its canonically conjugate momentum (or “electric field”) eEaI .

These are fieldsdefined on a 2-manifold Σ, and they are subject to a set of 1st class constraints. Theconstraints are polynomial in (AIa, eEaI ) and provide a representation of the Lie algebraof the inhomogeneous Lie group associated with G. One recovers 2+1 gravity by takingG = SO(2, 1).3.

Chern-Simons theory is a connection dynamical theory defined for any Lie group thatadmits an invariant, non-degenerate bilinear form. In 2+1 dimensions, the fundamentalfield variable is a Lie algebra-valued connection 1-form, and the phase space variablesare Aia—the pull-back of the field variable to the 2-dimensional hypersurface Σ. Thereare 1st class constraints, which are polynomial in Aia and provide a representation ofthe defining Lie algebra.

Chern-Simons theory is related to 2+1 Palatini theories asfollows: (i) 2+1 Palatini theory based on any Lie group G is equivalent to Chern-Simons theory based on the inhomogeneous Lie group IG associated with G; and (ii)the reduced phase space of Chern-Simons theory based on a Lie group G is the sameas the reduced configuration space of the 2+1 Palatini theory based on the same G.As a special case of (i), 2+1 gravity is equivalent to Chern-Simons theory based onthe 2+1 dimensional Poincar´e group ISO(2, 1).4. One can couple matter to 2+1 gravity via the 2+1 Palatini action.

2+1 Palatini theorycoupled to a cosmological constant Λ is defined for any Lie group G that admitsan invariant, totally antisymmetric tensor ǫIJK.This theory is equivalent to 2+1dimensional Chern-Simons theory based on the Λ-deformation of G. As a special case,2+1 gravity coupled to a cosmological constant is equivalent to Chern-Simons theorybased on SO(3, 1) or SO(2, 2) (depending on the sign of Λ). 2+1 Palatini theory canalso be coupled to matter fields with local degrees of freedom provided G = SO(2, 1).The constraints remain polynomial in the canonically conjugate variables and form a1st class set.

However, due to the presence of structure functions, they no longer forma Lie algebra.5. The 3+1 Palatini theory is a geometrodynamical theory of 3+1 gravity in which aco-tetrad and a Lorentz connection 1-form are the fundamental field variables.

Dueto the presence of 2nd class constraints, the Hamiltonian formulation of this theoryreduces to that of the standard Einstein-Hilbert theory in 3+1 dimensions. Unlike the84

2+1 Palatini theory, the 3+1 Palatini theory does not provide a connection dynamicaltheory of 3+1 gravity.6. The self-dual theory is a connection dynamical theory of complex 3+1 gravity in whicha complex co-tetrad and a self-dual connection 1-form are the fundamental field vari-ables.

The phase space variables consist of an CLSO(3)-valued connection 1-form Aiaand its canonically conjugate momentum (or “electric field”) eEai , both defined on a3-manifold Σ. These variables are subject to a set of 1st class constraints, which arepolynomial in (Aia, eEai ) but which have a Poisson bracket algebra involving structurefunctions.

In a solution, eEai is a (densitized) spatial triad. Since none of the equationsinvolve the inverse of eEai , the self-dual theory makes sense even if eEai is non-invertible.Thus, the self-dual theory provides an extension of complex general relativity that isvalid even when the induced spatial metric eeqab (=eEai eEbi) is degenerate.

One mustimpose reality conditions to recover real general relativity.7. One can couple matter to complex 3+1 gravity via the self-dual action.

The constraintsremain polynomial in the canonically conjugate variables and form a 1st class set. Sincenone of the equations involves the inverse of eEai , the self-dual theory coupled to matterprovides an extension of complex general relativity coupled to matter that includesdegenerate spatial metrics.

Reality conditions must be imposed to recover the realtheory.8. The pure spin-connection formulation of general relativity is a connection dynamicaltheory of complex 3+1 gravity in which a CLSO(3)-valued connection 1-form and ascalar density of weight −1 are the fundamental field variables.The Hamiltonianformulation of this theory is equivalent to that of the self-dual theory provided the self-dual part of the Weyl tensor is non-degenerate.

When the “magnetic” field associatedwith the connection 1-form Aia is non-degenerate, one can write down the most generalsolution to the four diffeomorphism constraints. Reality conditions must be imposedto recover the real theory.So what can we conclude from all these results?

Is gravity a theory of a metric or aconnection? In other words, is gravity a theory of geometry, where the fundamental variableis a spacetime metric which specifies distances between nearby events, or is it a theory ofcurvature, where the fundamental variable is a connection 1-form which tells us how toparallel propagate vectors around closed loops?

The answer: Either. As far as the classicalequations of motion are concerned, both a metric and a connection describe gravity equallywell in 2+1 and 3+1 dimensions.

Neither metric nor connection is preferred.85

As we have shown in this review and have summarized above, 2+1 and 3+1 gravity admitformulations in terms of metrics and connections. But despite the apparent differences (i.e.,the different actions and field variables; the different Hamiltonian formulations and canon-ically conjugate momenta; and the possiblity of extending the theories to include arbitrarygauge groups and solutions with degenerate spatial metrics), we have seen that the classicalequations of motion for all these formulations are the same.

For instance, we saw that the2+1 Palatini theory reproduces vacuum 2+1 gravity when we choose G = SO(2, 1) and solvethe equation of motion for the connection. Similarly, we saw that Chern-Simons theory re-produces 2+1 gravity coupled to a cosmological constant Λ (> 0) when we choose the gaugegroup to be SO(2, 2).

At the level of field equations, all the theories are mathematicallyequivalent. The difference between the theories is, instead, one of emphasis.Now such a small change may not seem, at first, to be worth all the effort.

Recall thatthe shift in emphasis from metric to connection came only after we successively analyzed theEinstein-Hilbert, Palatini, and Chern-Simons theories in 2+1 dimensions, and the Einstein-Hilbert, Palatini, self-dual, and pure spin-connection theories in 3+1 dimesions. This analysisrequired a fair amount of work and, as we argued in the previous paragraph, did not leadto anything particularly new at the classical level modulo, of course, the extensions of thetheories to include arbitrary gauge groups and solutions with degenerate spatial metrics.But as soon as we turn to quantum theory and consider the recent results that have beenobtained there, the question as to whether the shift in emphasis from metric to connectionwas worth the effort has a simple affirmative answer.

Yes! Indeed, almost all of the recentadvances in quantum general relativity can be traced back to this change of emphasis.

Asmentioned in the introduction, Witten [8] used the equivalence of the 2+1 Palatini theorybased on SO(2, 1) with Chern-Simons theory based on ISO(2, 1) to quantize 2+1 gravity.Others (e.g., Carlip [24, 25] and Anderson [26]) are now using Witten’s quantization toanalyze the problem of time in the 2+1 theory. In 3+1 dimensions, Jacobson, Rovelli, andSmolin [6, 7] took advantage of the simplicity of the constraint equations in the self-dualformulation of 3+1 gravity to solve the quantum constraints exactly—something that nobodycould accomplish for the quantum version of the scalar constraint in the traditional metricvariables.

And the list goes on. (See, e.g., [2, 3] and [17, 18, 19, 20] for more details.) Wherethis list will end, and whether or not the change in emphasis from metric to connectionwill lead to a mathematically consistent and physically reasonable quantum theory of 3+1gravity, remains to be seen.ACKNOWLEDGEMENTS86

I would like to thank Abhay Ashtekar, Joseph Samuel, Charles Torre, and Ranjeet Tatefor many helpful discussions.This work was supported in part by NSF grants PHY90-16733 and PHY91-12240, and by research funds provided by Syracuse University and by theUniversity of Maryland at College Park.87

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