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Einstein Manifolds in Ashtekar Variables:

arXiv:hep-th/9207056v2 30 Jul 1992VPI-IHEP-92-5hep-th/9207056Einstein Manifolds in Ashtekar Variables:Explicit ExamplesLay Nam Chang & Chopin SooInstitute for High Energy PhysicsVirginia Polytechnic Institute and State UniversityBlacksburg, Virginia 24061-0435AbstractWe show that all solutions to the vacuum Einstein field equations may be mappedto instanton configurations of the Ashtekar variables. These solutions are characterizedby properties of the moduli space of the instantons.

We exhibit explicit forms of theseconfigurations for several well-known solutions, and indicate a systematic way to get newones. Some interesting examples of these new solutions are described.7/92

1. IntroductionWe present some solutions to the vacuum Einstein equations based upon the Ashtekarvariables[1].

These variables are convenient for implementing the canonical descriptionof the Einstein field equations.The variables are SO(3) gauge fields for Riemannianmanifolds, and we shall show that the classical solutions of the field equations correspondto the instanton sector of the gauge fields. Not every instanton configuration can be usedto define Einstein manifolds.

In this note, we present the conditions under which thisdefinition will be possible, and work out some explicit examples which demonstrate theutility of such an approach.Ashtekar’s variables[1] can be obtained from the 3 + 1 decomposition of the Einstein-Hilbert action,116πGZ4Rp4gd4x(1.1)through a series of canonical transformations[2]. The canonical pair of variables consistsof the complex Ashtekar potentialsAia = iKia −12ǫabcωbci(1.2)and the densitized triad of weight 1,˜σia =p3gσia(1.3)The canonical variables obey the Poisson bracket relations{Aia(⃗x), ˜σjb(⃗y)}P B = i2δji δbaδ3(⃗x −⃗y){Aia(⃗x), Ajb(⃗y)}P B =0{˜σia(⃗x), ˜σjb(⃗y)}P B =0(1.4)for all space-time points ⃗x, ⃗y on the constant-x0 3-dimensional hypersurface M3.

In theabove, a factor of 16πG has been suppressed on the right hand side. For concreteness, wesuppose that M3 carries the signature + + +.

Our convention will be such that, unlessotherwise stated, lower case Latin indices run from 1 to 3, while upper case and Greekindices run from 0 to 3. In the above, ω is the torsionless spin connection compatible withthe triads:dσa + ωab ∧σb = 0(1.5)1

and modulo the constraint which generates triad rotations,Kia = σjaKij(1.6)where Kij is the extrinsic curvature.In terms of the Ashtekar variables, the constraints generating local SO(3) gauge trans-formations, or triad rotations, which leave the spatial metricgij = σiaσja(1.7)invariant can be written in the form Gauss’ law:Ga ≡2iDi˜σia ≃0(1.8)Ashtekar showed [1] that, modulo Gauss law constraints, the usual “supermomentum”and “superhamiltonian” constraints of ADM[3] achieve remarkable simplifications whenexpressed in terms of the new variables. Indeed, the “supermomentum” constraint−2πji |j ≃0(1.9)is proportional toHi ≡2i ˜σiaFija ≃0(1.10)while the “superhamiltonian” constraint√g16πGtrK2 −3 R −(trK)2≃0(1.11)is equivalent toǫabc˜σia˜σjbF cij ≃0(1.12)The quantity K is the extrinsic curvature given byKij = −16πG√gπij −12πgij(1.13)The presence of a cosmological termSC =2λ16πGZ p4gd4x(1.14)2

in the action modifies the usual “superhamiltonian” constraint in that one will need toadd a new term:2λ16πGp3gto the left hand side of (1.11), and H in (1.12) becomesH = ǫabc˜σia˜σjbF cij + Λǫabcǫijk˜σia˜σjb˜σkc(1.15)with Λ = λ/3.In the case of metrics with Euclidean signature, one should drop all factors of i in(1.2) and (1.4), and may further assume that the Ashtekar variables are all real. The“superhamiltonian” constraint in the ADM formalism for Euclidean signature becomesHE =√g16πG(−trK2 −(3R) + (trK)2 + 2λ) ≃0(1.16)Modulo Gauss law constraints, HE is still proportional to ǫabc˜σjbFija.

A short computationgives−16πGHEj = 2√gKij|i −Kii|j= 2˜σiaFija(1.17)2. Classification Scheme for Solution Space of ConstraintsIn this section, we exhibit a classification scheme of the solutions of Ashtekar’s con-straints, and discuss its connection to results appearing in the literature.It is known that all solutions of the Einstein field equations in 4D can be classifiedaccording to the canonical forms of the Riemann-Christofel curvature tensor.Such ascheme was first given by Petrov[4] and then further extended by Penrose[5] in the contextof spinors and null tetrads.In the ADM formalism, the “supermomentum” and “superhamiltonian” constraintsare projections of the Einstein field equations tangentially and normally to the three-dimensional hypersurface M3, on which the initial data compatible with the constraints isspecified.

The solutions to the constraints when stacked up according to their x0-evolutionby the Hamiltonian are then the solutions to the field equations. The natural question toask is if a similar classification scheme can be set up in the phase space defined by theAshtekar variables.

What we will do in this section is present one such scheme.In the ADM formalism, the metric is assumed to be non-degenerate.Ashtekar’sformulation allows for both degenerate and non-degenerate metrics. This is because the3

relevant constraints (1.8), (1.10), and (1.15) do not involve the inverse of the momenta˜σia.For non-degenerate metrics, the magnetic field of the Ashtekar connection (1.2),Bia =12ǫijkF ajk, can be expanded in terms of the densitized triad ˜σia, and the mostgeneral solution to the three-dimensional diffeomorphism constraint (1.10) isBia = ˜σibSab(2.1)with S being a symmetric 3 × 3 ⃗x-dependent matrix. Observe that (2.1) is a solution ofthe diffeomorphism constraint even for the case of degenerate metrics.The “superhamiltonian” constraint (1.15) becomes an algebraic relation:(det ˜σ) (Saa + λ) = 0(2.2)which has the solutiontrS = Saa = −λ(2.3)for non-degenerate metrics.

(2.2) will not fix trS when the metric is degenerate. It isintriguing to note the apparent shift in the specification of the dynamical degree of freedomfrom det ˜σ to trS in the case of degenerate metrics.

Metrics which become degenerate atcertain points in space-time may well be important in topology changing situations inclassical and quantum gravity.For space-times with Lorentzian signature, the Ashtekar gauge potential A is complex,and hence so is S. Complex symmetric matrices can be classified according to the numberof independent eigenvectors and eigenvalues, according Table 1.Since Sab is gauge invariant, one may classify the matrix in terms of the roots of itscharacteristic polynomial. These can in turn be expressed by:C1 = −trS = λC2 =12C1trS + trS2C3 = −13C2trS + C1trS2 + trS3(2.4)The Bianchi identity for the magnetic field associated with A further implies theconsistency condition:Di(S · ˜σi)a = 0(2.5)or˜σia (DiS)ab = 0(2.6)4

when one takes into account (1.8).There have been attempts to obtain metric-independent gravity theories by expressing˜σi in terms of Bi [6]. However, in view of the displayed classification scheme, this is not themost natural way to proceed.

For instance, the scheme of [6] will not work for the simpleF = 0 sector, which has S = 0 for finite momenta. We shall elaborate on the significanceof the cases when S is degenerate later on.

When S is invertible, we do obtain the resultsof [6], with˜σia = (Sab )−1 Biband the constraintsBia (DiS)ab = 0S−1ab = S−1ba(det B)(trS−1)2 −tr{(S−1)2} + 2λ det S−1= 0(2.7)As noted by the authors of [6], these are seven equations on the nine complex components ofS−1, and the solutions should give the two unconstrained field degrees of freedom associatedwith general relativity in 4D. When S is degenerate, though, as we will show, there couldarise phases with fewer degrees of freedom.It should be emphasized that, as in the Petrov classification scheme, types II, III and Ndo not occur for space-times with Euclidean signatures.

This is because the correspondingAshtekar variables are all real, so that S is real and symmetric, and there are always threedistinct eigenvectors.For the case when there is only one eigenvalue, the three roots (2.4) are not indepen-dent:trS3 = (trS)(trS2) = 19(trS)3 = −λ39(2.8)When two of the eigenvalues are the same, the relationship among the roots is6trS3 + λtrS2 −2λ392=trS2 −λ233(2.9)The initial value data thus falls into distinct classes with strikingly distinct properties.For instance, type I has three ⃗x-dependent eigenvalues for S, whose sum is restricted to −λ,while for type O one has only one ⃗x-independent eigenvalue −λ/3. This mismatch in theallowable fluctuations is highly suggestive of distinct phases in the theory.

For example, we5

may show[7] that type O (Sab = −(λ/3)δab) can be identified with an unbroken topologicalquantum field theory (TQFT), describing a topological phase in quantum gravity.The classification scheme described so far becomes equivalent to the usual Petrovclassification for non-degenerate metrics. In this case,Sab = Re0a0b −R0a0b(2.10)3.

Equations of Motion and Anti-InstantonsIn this section, we exhibit the manifestly covariant equations of motion for theAshtekar variables and discuss the implications. We choose to work explicitly with met-rics of Euclidean signature and use SO(3) instead of SU(2) gauge potentials, but we willindicate the necessary modifications for metrics of Lorentzian signature.

It will becomeclear as we go along, that there are Einstein manifolds that cannot be described globallyby SU(2) Ashtekar potentials, but can be described by SO(3) connections. This has to dowith the fact that not all SO(3) connections can be lifted to be SU(2) connections withinteger second Chern class, but all SU(2) connections can be thought of as SO(3) connec-tions with the first Pontrjagin class being a multiple of four.

We will furnish examples ofsuch manifolds below.In working with metrics of Euclidean signature, we should drop all i’s, starting withEqn. (1.1).

In the spatial gauge, we haveeAµ =N0eajN jeai(3.1)Eqn. (3.1) in no way compromises the values of the lapse and shift functions (N, N j)[3],and is compatible with the ADM decomposition of the metric:ds2 = ∓e02 + e12 + e22 + e32= ∓N 2(dx0)2 + gijdx1 + N idx0 dxj + N jdx0(3.2)where eA is the 1-form eAµdxµ and the +(−) sign is to be used for metrics of Euclidean(Lorentzian) signature.

On the constant x0-hypersurface M3, e0 vanishes, and we maywriteFa = e0 ∧Tabeb + 12Sbaǫbcdec ∧ed(3.3)6

which gives Eqn. (2.1).

Tab however must be chosen carefully because Eqn. (3.3) impliesthat on M3F0ia =Tabe00ebi −e0ieb0+ Sabǫbcdec0edi=NTabebi + Sabǫbcdec0edi(3.4)For Riemannian manifolds, apart from a boundary term that does not contribute to theequations of motion, the Hamiltonian in the Ashtekar formalism is[1]:H =ZM3 d3x∼Nǫabc˜σia˜σjbF cij + (λ3 )ǫabcǫijk˜σia˜σjb˜σkc+ 2N i˜σjaFija −2A0a(Di˜σi)a(3.5)and the evolution equation for Aia on M3 gives˙Aia = {Aia, H}PB=∼Nǫabc˜σjbF cij + 12λ∼Nǫijkǫabc˜σjb˜σkc+ ∂iA0a −ǫabcA0bAic −N jFija(3.6)With the use of Eqn.

(2.1) and assuming non-degenerate metrics, we can rewrite Eqn. (3.4)asF0ia = −NSabebi + Neia (Scc + λ)+ Sabǫbcdec0edi(3.7)The second term vanishes because of the “superhamiltonian” constraint and comparingwith Eqn.

(3.4) we observe that the consistent choice for Tab is:Tab = −Sab(3.8)Thus we haveFa = Sab−e0 ∧eb + 12ǫbcdec ∧ed(3.9)Similarly, for the evolution of ˜σia, we have˙˜σia =˜σia, HPB=ǫabchDj(∼N ˜σi)ib˜σjc + ǫabc∼N ˜σic(Dj˜σj)b+Dj(N j˜σi)a −(∂jN i)˜σja−N(Dj˜σj)a + A0cǫabc˜σib(3.10)7

It is not difficult to show that the equations of motion for the Ashtekar variables can thenbe succinctly written asFa =SabΣb(3.11a)(DΣ)a =0(3.11b)withSab =Sba(3.12a)tr S = −λ(3.12b)Here,Σa ≡−e0 ∧ea + 12ǫabceb ∧ec(3.13)and D is the covariant derivative with respect to the Ashtekar connection 1-form. The ninex0-evolution equaitons for Aia are contained in Eqn.

(3.11a) while the twelve equations inEqn. (3.11b) can be split offinto the set of three equations:∗[(DΣ)a |M3] = 0(3.14)which is equivalent to the set of Gauss Law constraints, and the nine equations:[∗(DΣ)a] |M3 = 0(3.15)which, modulo the Gauss Law constraints, are equivalent to the x0- evolution equationsfor ˜σia, Eqn.

(3.10). Ashtekar’s transcription of the “supermomentum” and “superhamil-tonian” constraints of general relativity takes the simple form of (3.12a, b).

(See also [8]for an alternative derivation of the equations of motion using self-dual two-forms as funda-mental variables and a discussion of gravitational instantons as SU(2) rather than SO(3)gauge fields. )We shall now examine the meaning of the equations of motion.

Firstly, observe thatΣa is explicitly anti-self-dual:∗Σa = −Σa(3.16)Since Fa is the product of a zero form S with the 2-form Σ, Eqn. (3.16) implies that∗Fa = −Fa(3.17)8

As a result, all Einstein manifolds correspond to anti-instantons of the Ashtekar potentials.However, the converse is not always true. In general, the curvature of an arbitrary anti-instanton can be expanded in terms of Σa viaFa = YabΣb(3.18)But the quantity Y will have to satisfy Eqn.

(3.12 )and Eqn. (3.11b) before the anti-instanton can correspond to an Einstein manifold.The twelve equations in Eqn.

(3.11b) suggest that the 1-form Aa can be expressedin terms of the vierbein eA. This is indeed the case, for the solution to Eqn.

(3.11b) ispreciselyAa = ω0a −12ǫabcωbc(3.19)where ωAB can be determined uniquely from eA throughdeA + ωAB ∧eB = 0(3.20)Eqn. (3.19) says that, apart from a factor of 2, Aa is the anti-self-dual part of the spin-connection and so the curvature 2-form of Aa can be expressed asFa = R0a −12ǫabcRbc(3.21)where RAB is the curvature 2-form of the spin-connection.

It is then not difficult to showthat Eqn. (3.11b) is satisfied if and only ifSab = Re0a0b −R0a0b = R0ae0b −Re0a0b(3.22)and so the constraints Eqn.

(3.12 )imply thatRABCD = RfAB fCD(3.23)and the Ricci scalar becomesR = 4λ(3.24)These equations are completely equivalent to the pure gravity field equations definingEinstein manifolds.Dimension four is the lowest dimension for which the Riemann curvature tensor as-sumes its full complexity.It is also the dimension which has the peculiarity that the9

curvature 2-form can be decomposed into parts taking values in the (±) eigenspaces Λ±2of the Hodge duality operator. The Riemann curvature tensor, having four indices, canbe dualized on the left or on the right, so that it can be viewed as a 6 × 6 mapping ofΛ2± →Λ2±[9]:AC+C−B(3.25)where in components,Aab ≡+R0a0b + R0ae0b+Re0a0b + Re0ae0b(3.26)and B and C are defined similarly according to the signs of the following:A ∼(+, +, +, +) , B ∼(+, −, −, −) , C+ ∼(+, −, +, −) , C−∼(+, +, −, +)(3.27)It is easy to check that A(B) is self-dual (anti-self-dual) with respect to both left andright duality operations, while C+(C−) is self-dual (anti-self-dual) under left duality andanti-self-dual (self-dual) under right duality operations.

A metric is Einstein if and onlyif C± = 0, i.e. when Eqn.

(3.25) assumes a block diagonal form. In view of Eqns.

(3.17)and (3.21), Fa is the doubly anti-self-dual part of the curvature and apart from a multi-plicative factor, S, can be identified with B when the equations of motion are satisfied. Inthis context, for Einstein manifolds, the Ashtekar formulation is the realization of Propo-sition 2.2 of [9] in the canonical framework.

However, it should be emphasized that itis the remarkable simplification of the constraints provided by Ashtekar that makes thenon-perturbative quantization scheme viable. While it appears that only half of the non-vanishing components of the Riemann curvature tensor is contained in Fa, the equationsof motion are completely equivalent to Einstein’s field equations for non-degenerate met-rics.

Actually, A and B interchange under a reversal of orientation because a reversal oforientation changes the definition of self- and anti-self-duality.While not all Einstein manifolds have anti-self-dual Riemann or Weyl tensors, a man-ifold is Einstein only if the curvature tensor constructed from the anti-self-dual part ofthe spin connection is anti-self-dual.It is precisely this property which allows for thedescription of all Einstein manifolds in terms of anti-instantons of the Ashtekar variables.As a corollary, we note that for Einstein manifolds, the Weyl 2-form isWAB = RAB −λ3 eA ∧eB(3.28)10

so the anti-self-dual part of the Weyl 2-form W −a becomesW −a =R0a −12ǫabcRbc + λ3−e0 ∧ea + 12ǫabceb ∧ec=Fa + λ3 Σa(3.29)so an Einstein manifold is conformally flat or self-dual (half-flat when λ = 0) if and only ifFa = −λ3 Σa(3.30)orSab = −λ3 δab(3.31)According to our classification, this situation corresponds precisely to type O.It is possible to eliminate Sab from the equations of motion. We haveΣa ∧Σb = −2δab (∗1)(3.32)where (∗1) is the 4-volume element.

So from the equations of motion Eqn. (3.11 )Sab = −14 ∗(Fa ∧Σb + Σa ∧Fb)(3.33)and the equations of motion can be written asFa = −12 [∗(Fa ∧Σb)] Σb(3.34a)(DΣ)a =0(3.34b)ǫabcFb ∧Σc =0(3.34c)Fa ∧Σa = −2λ (∗1)(3.34d)4.

Invariants and the Ashtekar variablesUnlike other fields, the gravitational field describes the dynamics of space-time. Anyviable classical and quantum theory of the gravitational field must therefore be able totake into account not just the local description of curvature, but also the large scale globaland topological aspects of the structure of space-time.

We shall see how the Ashtekarvariables can be used to capture the global invariants in 4D, especially those associatedwith Einstein manifolds.11

As we have discussed in section 2, a specification of the initial value data is equivalentto a specification of the characteristic classes of S which is compatible with the constraints.We may take the gauge-invariant quantities on M3 to be tr S = −λ, tr S2, and tr S3, fromwhich we can reconstruct the characteristic classes of S. Their integrals over M3 shouldreflect global properties of M3.It is not difficult to show that when the equations of motion are satisfied,tr S = −λ(4.1a)tr S2 =18nRfAB fCD −RAB fCDRABCDo(4.1b)tr S3 = −116nRABCD −RAB fCDRCDEF REFABo(4.1c)Thus their integrals over compact, closed 4-manifolds M4 giveZM4 (tr S) = −λV = −λ6ZΣa ∧Σa(4.2)where V is the volume of M4, andZM4tr S2=2π2 2χ(M4) −3τ(M4)= −12ZFa ∧F a= −2π2P1(4.3)where χ(M4) and τ(M4) are the Euler characteristic and signature of M4, while P1 isthe Pontrjagin number of the SO(3) Ashtekar connection. Finally,ZM4(tr S3) = −12ZM4 SabF a ∧F b(4.4)Observe that the signature τ(M4) depends on the orientation of M4.

Indeed,τ(M4) =dim H2+ −dim H2−=b+2 −b−2(4.5)where H2± are the self-dual and anti-self-dual subspaces of the second cohomology group,and b±2 are the corresponding Betti numbers. Reversing the orientation interchanges self-dual and anti-self-dual 2-forms, so thatτ(M4) = −τ(M4)(4.6)12

where M4 has the opposite orientation relative to M4. Reversing the orientation changesthe spin connections in general, and thus the Ashtekar connections via Eqn.

(3.19). Forexample, considerdeA = −ωAB ∧eBA transformation of the form (e0, ea) →(−e0, ea) reverses the orientation, though it doesnot change the metric ds2.

The new spin connections becomeω0a →−ω0a;ωab →ωab(4.7)so that the Ashtekar connections transform asAa =ω0a −12ǫabcωbc→Aa −2ω0a(4.8)The Pontrjagin numbers of the Ashtekar connections with respect to the two differentorientations areP +1 =3τ(M4) −2χ(M4)P −1 =3τ(M4) −2χ(M4) = −3τ(M4) −2χ(M4)(4.9)Since P ±1 are the Pontrjagin numbers of the anti-self-dual Ashtekar connections,P ±1 ≤0An immediate consequence is the Hitchin bound for compact, closed Einstein manifolds[10]|τ| ≤23χ(4.10)For compact, closed Einstein manifolds with Euclidean signatures,χ(M4) =132π2ZRfAB fCDRABCD=132π2Z(RABCD)2 ≥0with the equality holding only if M4 is flat. Moreover τ(M4) and χ(M4) can be computedfrom the SO(3) Ashtekar connections throughτ(M4) =16P +1 −P −1(4.11a)χ(M4) = −14P +1 + P −1(4.11b)13

If the Einstein manifold possesses an orientation reversing diffeomorphism, then P +1 =P −1 , and τ = 0. The vanishing or non-vanishing of the signature has important physicalimplications.

For according to the index theorem for the spin complex for closed, compactRiemannian manifolds,n+ −n−= −124P1T(M4)= −18τ(M4)(4.12)where n± are the number of ±1 chirality zero-frequency solutions of the Dirac equation.P1T(M4)is the Pontrjagin number of the tangent bundle, i.e. of the SO(4) spin con-nection, and is related to the τ(M4) by the Hirzebruch signature theorem:P1T(M4)= 3τ(M4)Thusτ(M4) = 0mod 8for spin manifolds, since n+−n−must be an integer.

An orientable manifold (W1 = 0) hasa spin structure iffW2 = 0. Here W refer to the Stiefel-Whitney class.

A simply-connected,compact, closed manifold of dimension four has a spin structure iffits intersection formis even, and this spin structure is unique[11]. Actually, for the case of simply-connected,compact, closed, smooth four-manifolds, the intersection form, and hence the topology viaFreedman’s theorem, is determined by τ and χ, and whether the intersection form is even(i.e.

W2 = 0) or odd. This can be explained as follows: Indefinite intersection forms aredetermined by their rank, signature, and type (even or odd).

The rank of the intersectionform is the second Betti number. Butb2 = b+2 + b−2 = χ −2(4.13)for simply-connected, compact, closed four-manifolds.

τ is the signature of the intersectionform. Although there are many definite intersection forms of the same rank and signature,Donaldson’s theorem [12] asserts that differentiable four-manifolds with definite intersec-tion forms must be of the standard typenL±(1).

So specification of P ±1 and whether themanifold is spin (W2 = 0) or not corresponds to a complete specification of the inter-section form of a smooth, simply-connected, compact, closed four-manifold. Freedman’stheorem [13] asserts that given an even (odd) intersection form, there is exactly one (two,14

distinguished by their Z2-valued Kirby-Siebenmann invariant) simply-connected, closed,compact, topological four-manifold representing that form.Before we proceed to specific illustrations, we remark that the third invariantEqn. (4.1c), which involves the explicit form of S could provide a new differential in-variant for Einstein manifolds, since the intersection form has already been accounted forby Eqn.

(4.3), at least for the case when they are smooth, simply-connected, closed, andcompact. See also [7] for a discussion of BRST-invariants of four-dimensional gravity inAshtekar variables.15

5. Examples of Einstein manifolds in Ashtekar variablesA.

Known SolutionsThe formalism developed in the previous sections provides a coherent framework todiscuss explicit Einstein manifolds in the context of Ashtekar variables.Every knownsolution of the Einstein field equationsRµν = λgµν(5.1)can be put in the form of Eqns. (3.11).

In fact, when the field equations are satisfied, wecan use Eqns. (3.12) to obtain the Ashtekar connection, and compute S via Eqn.

(3.11a).It will be convenient to introduce the 1-forms Θa, where Φa = −2 Θa obeys theMaurer-Cartan equation for SO(3):dΦa + 12ǫabcΦb ∧Φc = 0(5.2)We can choose the four-dimensional polar coordinates as (R, θ, φ, ψ), where for fixed R,0 ≤θ ≤π, 0 ≤φ < 2π, and 0 ≤ψ < 4π. Next introducex1 + ix2 =R cos(θ/2) exp i2(ψ + φ)x3 + ix0 =R sin(θ/2) exp i2(ψ −φ)(5.3)Then Θa can be written in terms of the Euler angles θ, φ, ψ on S3 asΘ1 =12 (sin ψdθ −sin θ cos ψdφ)Θ2 =12 (−cos ψdθ −sin θ sin ψdφ)Θ3 =12 (dψ + cos θdφ)(5.4)We concentrate first on solutions with Sab = −(λ/3)δab.

As we have explained, thesesolutions correspond to the conformally self-dual sector of Einstein manifolds. It is knownthat for λ > 0, S4 and CP2 are the only compact, closed, simply-connected four-manifoldswhich are conformally self-dual[14].16

(a)S4 with the de Sitter metricThe metric for this space is given byds2 ="1 +Ra2#−2 dR2 + R2 Θ12 + Θ22 + Θ32(5.5)while the vierbein is expressed aseA =dR1 + Ra2,RΘa1 + Ra2(5.6)The corresponding Ashtekar connections then have the form:Aa =ω0a −12ǫabcωbc= −2Θa1 + Ra2(5.7)givingFa =dAa + 12ǫabcAb ∧Ac= −4a2−e0 ∧ea + 12ǫabceb ∧ec(5.8)Thus,Fa = SabΣbwithSab = −4a2 δab = −λ3 δab(5.9)so thatλ = 12a2 > 0(5.10)and the diameter of the four sphere is related to λ bya =r12λ(5.11)Suppose that we now reverse the orientation, by for example defining the vierbeinfield to beeA =−dR1 + Ra2,RΘa1 + Ra2(5.12)17

The Ashtekar connections then change to the form:Aa = −2a2R2Θa1 + Ra2(5.13)althoughF a = −4a2 Σa(5.14)so that S is unchanged. S4 has an orientation reversing diffeomorphism.

By using theexplicit form for the Ashtekar connections, one obtains that the first Pontrjagin numberequals −4, and is preserved under the reversal. Thus Eqns.

(4.11a) and (4.11b) yieldτS4=0(5.15a)χS4=2(5.15b)Note that for S4, the SO(3) Ashtekar connections give P ±1 = 0 mod 4, and so can be liftedto an SU(2) connection, with second Chern classc2 = −P14=1(5.16)Actually, the Ashtekar connections given by Eqns. (5.7) and (5.13) are precisely the BPST(anti-)instanton solutions[15].

Since the intersection form has rankrank (Q) = b2 = χ −2 = 0(5.17)S4 has Q = ∅.The dimension of the moduli space for a single anti-instanton on S4 is known tobe five[9]. The parameters correspond to the size and location of the (anti-)instanton.For the Ashtekar connections, however, diffeomorphism invariance collapses this spaceentirely, since the solution must now be translationally invariant, and the size is fixed bythe cosmological constant, according to Eqn.

(5.11). S4 is not only conformally self-dual,but it is also conformally flat.

That this is so is also evident in the Ashtekar context becauseSab = λ/3δab implies, by Eqn. (3.29) that W−a = 0.

But S is unchanged by orientationreversal, so W+a = 0 also. Hence W±a = 0, and S4 is conformally flat.18

(b)CP2 and the Fubini-Study MetricThe two dimensional complex projective space is described by the Fubini-Study metric:ds2 =dR21 + λ6 R22 +(RΘ1)21 + λ6 R2 +(RΘ2)21 + λ6 R2 +(RΘ3)21 + λ6 R22(5.18)We may choose the vierbeins aseA =dR1 + λ6 R2,RΘ11 + λ6 R2 12 ,RΘ21 + λ6 R2 12 ,RΘ31 + λ6 R2(5.19)in which case the Ashtekar variables are:A1 =−2Θ11 + λ6 R2 12A2 =−2Θ21 + λ6 R2 12 ,andA3 =−2 −λ6 R2Θ31 + λ6 R2(5.20)These equations yield Fa = SabΣab, with Sab = −(λ/3)δab. The solution is thereforeagain of Type O.

However, the Pontrjagin index is found to equalP1 = 14πZFa ∧Fa = −3(5.21)As a result, the Ashtekar connections cannot be realized in a globally well-defined manneras an SU(2) gauge potential.Like S4, CP2 is conformally flat, since S is of Type O, but unlike S4, it does nothave an orientation reversing diffeomorphism. Under a reversal, we obtain CP 2, whichis described by the same metric, but the vierbein becomes (−e0, ea).

In which case, theAshtekar potentials becomeA1 = A2 = 0whileA3 = −λR2 Θ321 + λ6 R2(5.22)givingF 1 = F 2 = 0F 3 = dA3= −λ (e0 ∧e3 + e1 ∧e2)= −λΣ3(5.23)19

Thus CP2 is described by Ashtekar potentials of a non-abelian anti-instanton, whereasCP 2 is described by those of an abelian anti-instanton. The corresponding Pontrjaginindex is found to beP 1 = −9(5.24)which is different from that of Eqn.(5.21).

Accordingly, the Euler characteristic and sig-nature are given byχ(CP2) = χ(CP 2) = 3(5.25)whileτ(CP2) = −τ(CP 2) = 1(5.26)From previous studies [11], we already know that CP2 cannot support abelian instan-tons, while CP 2 can support only one such object. The Ashtekar potential is simply thatunique abelian anti-instanton.The matrix Sab for CP 2 is of the formS = diag (0, 0, −λ)(5.27)and so the solution is of Type D.This example shows how the Ashtekar variables provide a more natural context inwhich to study the topological and differential invariants of a 4-manifold.

(c)The Schwarzschild-de Sitter solutionThe Schwarzschild-de Sitter metric in Euclidean space is given byds2 =1 −2Mr−λ3 r2dτ 2 +1 −2Mr−λ3 r2−1dr2 + r2dθ2 + r2 sin2 θ dφ2(5.28)where M is G/c2 times the mass. Taking the vierbein fields to beeA =(1 −2Mr−λ3 r2 12dτ,1 −2Mr−λ3 r2−12dr, r dθ, r sin θ dφ)(5.29)yields the Ashtekar potentialsA1 =Mr2 −λ3 rdτ + cos θ dφA2 = −1 −2Mr−λ3 r2 12sin θ dφA3 =1 −2Mr−λ3 r2 12dθ(5.30)20

and the matrix S:S = diag−2Mr3 −λ3 , Mr3 −λ3 , Mr3 −λ3(5.31)This solution is therefore of Type D generally, for M ̸= 0, and of Type O in the limit ofvanishing mass. One may view the mass term as a parameter which breaks the system outof the Type O sector.

When λ →0, we recover the usual Schwarzschild solution.Reversing the orientation givesA1 = −Mr2 −λ3 rdτ −cos θdφ = −A1(5.32)withA2 = A2andA3 = A3while the form of S is preserved.The Pontrjagin index for λ = 0 can be computed to giveP1 = −12π2ZtrS2(∗1)= −1π2Z 2πφ=0Z π0Z ∞r=2MZ 8πMτ=03M 2r4 dτ ∧dr ∧sin θdθ ∧dφ= −4= P 1(5.33)The radius 2M is the usual event horizon, and we have also used the periodicity in theEuclidean time interval of 8πM inherent in the Schwarzschild metric. From Eqn.

(5.33),we conclude that χ = 2, and τ = 0, in agreement with the standard result.Finally, note that a general Type D metric with zero cosomological constant can becharacterized by S = diag(−2α, α, α). If α > 0, S is gauge equivalent toSab = 13φ2δab −φaφb(5.34)since this form can be diagonalized todiag−23φ2, 13φ2, 13φ2For the Schwarzschild solution, φ2 = 3M/r3.

In isotropic coordinates, withr ≡1 + M2ρ2ρ21

the quantity φ above takes on the valueφa = (3M)12 (ρ)−521 + M2ρ−3ρayielding for the Ashtekar magnetic fieldBia =Mρ31 + M2ρ6δab −3ρaρbρ2˜σib(5.35)This establishes the gauge-equivalence between the Schwarzschild solution in Ashtekarvariables in our general formalism and the solution exhibited in [16]. (d)The Eguchi-Hanson metricThe Eguchi-Hanson metric [17] with a cosmological constant can be written asds2 =1 − aR4−λ6 R2−1dR2 + R2 Θ12 + Θ22+ R21 − aR4−λ6 R2Θ32(5.36)We can choose the vierbein fields to beeA =(1 − aR4−λ6 R2−12dR, RΘ1, RΘ2,1 − aR4−λ6 R2 12RΘ3)(5.37)which then implies that the Ashtekar potentials are given byA1 = −21 − aR4−λ6 R2 12Θ1A2 = −21 − aR4−λ6 R2 12Θ2A3 = −21 + aR4−λ12R2 12Θ3(5.38)The corresponding matrix Sab takes the formS = diag4a4R6 −λ3 , 4a4R6 −λ3 , −8a4R6 −λ3(5.39)The Eguchi-Hanson metric is therefore of Type D when a ̸= 0, and of Type O when a = 0,so this parameter causes the system to break out of the Type O sector.22

When we apply a reversal, we get another manifold, EH, with the Ashtekar potentialstaking the form:A1 = A2 = 0whileA3 = −λ2 R2Θ3(5.40)The field strengths are now controlled by the matrixS = diag (0, 0, −λ)(5.41)Like in the case of CP 2, this matrix is not invertible, and it is described by an abeliananti-instanton.However, the Eguchi-Hanson manifold has a boundary of real projective 3-space,RP3[17]. The abelian instanton of Eqn.

(5.40) does not depend on the parameter a, andfurthermore, it is anti-self-dual relative to EH for arbitrary λ and a. In the limit λ →0,S becomes zero, and EH becomes half-flat.

As we shall see below, the Eguchi-Hansonmetric can be obtained as limiting cases of two different classes of explicit solutions, onefrom the F = 0 sector, and the other from the abelian anti-instanton sector.B. New SolutionsThe above examples illustrate the procedure for determining the appropriate anti-instanton configuration of the Ashtekar variables once the metric is known.But, theformalism can be used to go the other way and yield new solutions to the Einstein fieldequations.

We shall illustrate the method below by examining a few explicit examples.Before we do so, recall that the matrix S for Riemannian manifolds is real-symmetric.Solutions are characterized by tr S2 and tr S3, which can be further divided into classesrelative to a sign change under orientation reversal. This distinction had been utilized inthe examples presented so far, and will continue to be significant in the solutions we willbe discussing below.F = 0 sector and hyperk¨ahler manifoldsWe first examine the case where the Ashtekar field strength vanishes.When thishappens, the metric is half-flat; i.e.

the Riemann curvature is self-dual. S vanishes also,23

and for simply-connected manifolds, we may set the connection to be zero globally as well.The equations of motion reduce tod Σa = 0(5.42)so that the anti-self-dual Σa is now also closed. As a result,b2−= 3(5.43)Since the Ashtekar curvature vanishes, we obtain0 = P1 = 3τ(M) −2χ(M)(5.44)so that τ takes on the maximal value of the Hitchin bound:τ(M) = 23χ(M)(5.45)But we also have the relationχ(M) = b2 + 2 = b2+ + b2−+ 2 = b2+ + 5so that finallyτ(M) ≡b+2 −b2−= b2+ −3(5.46)These relations may be solved to give the following characteristic numbers for simply-connected compact Einstein manifolds in the F = 0 sector:b2+ = 19b2−= 3τ = 16χ = 24(5.47)It is known that K3 manifolds and the 4-torus are the only compact manifolds withoutboundary admitting metrics of self-dual Riemann curvature[18].

The 4-torus is not simply-connected, and has τ = χ = 0, since its metric is flat. So, choosing the convention thatτ(K3) = −16, we can identify the simply-connected compact half-flat manifolds withoutboundary as K3.

They have the intersection form[11]:Q =3M 01102ME8(5.48)The Pontrjagin index for K3 can be computed to beP1 = −3τ −2χ = −96(= 0 mod4 )(5.49)24

As a result, the SO(3) Ashtekar connection can be lifted to an SU(2) connection. Themetric therefore possesses an SU(2) holonomy, and is therefore hyperk¨ahler[14].Suchmetrics have been used to formulate conditions for unbroken supersymmetry in the com-pactification of superstrings[19].

In our present context, these metrics are associated withthe unbroken topological field theory of the moduli space of flat connections[7].Note that although F = 0 and S = 0 for K3, the corresponding values for K3need not be trivial. It has been calculated that these surfaces are parametrized by 58parameters[20].According to Eqn.

(5.49), these must be associated with an Ashtekarconnection with Pontrjagin number −96.We shall now construct explicitly half-flat Einstein manifolds which are not necessarilysimply-connected, or without boundary. They will have F = 0, but F ̸= 0.We begin by supposing that the vierbein is of the form:eA = {−a(R)dR, f(R)Θ1, g(R)Θ2, h(R)Θ3}(5.50)This yieldsA1 =f ′a −(g2 + h2 −f 2)ghΘ1A2 =g′a −(h2 + f 2 −g2)fhΘ2A3 =h′a −(f 2 + g2 −h2)fgΘ3(5.51)where primes denote differentiation with respect to R.Further simplification can beachieved by assuming that f = g. Setting Aa = 0 locally, we need to solvef ′a = hfandh′a + h2f 2 = 2Combining these two equations gives(h2)′(f 2)′ + h2f 2 = 2With u ≡h2 and v ≡f 2, this equation reduces to(uv)′ = (v2)′25

which has as solutionh2 = f 2 + bf 2(5.52)with b being an integration constant. The metric is therefore given byds2 = a2dR2 + f 2 Θ12 + Θ22+ h2Θ32(5.53)where a = (f 2)′/2h, and h is given Eqn.

(5.52). The function f is an arbitrary function ofR.If we now reverse the orientation, the metric is invariant, but the vierbein changes toeA = (−e0, ei).

The Ashtekar potentials becomeA1 = −2hf Θ1A2 = −2hf Θ2A3 = −22 −h2f 2Θ3(5.54)assuming that the relations among h, a and f continue to hold. A short computation thenfixes the matrix S to be:S = diag−4bf 6 , −4bf 6 , 8bf 6(5.55)Thus the equations of motion still hold, but the solution is now of Type D when b ̸= 0.For compact manifolds without boundaries,P1 = −14π2ZF i ∧F i= 12b2Z(f −8)′dR(5.56)assuming that the variables θ, φ and ψ are the coordinates of a 3-sphere for fixed valuesof R. By choosing the appropriate function f, one can obtain self-dual Einstein manifoldswith non-trivial values of the Pontrjagin number.For the special case of f = R, andb = −a4, we recover the λ →0 limit of the Eguchi-Hanson metric discussed above.

Recallthat our convention is such that EH with λ = 0 is half-flat.26

Abelian anti-instantons and K¨ahler-Einstein manifoldsWhen S = diag (0, 0, −λ), the Ashtekar potential is described by an abelian anti-instanton. In this gauge, the only non-vanishing component of the field-strengths is F3,and the equations of motion reduce todA3 = F3 = −λ Σ3dΣ1 = A3 ∧Σ2dΣ2 = −A3 ∧Σ1(5.57)We now suppose that the manifold can support a complex structure, and defineΣ+ ≡Σ1 + i Σ2(5.58)in which case,dΣ+ + i A3 ∧Σ+ = 0(5.59)Furthermore, let us defineΩ1 ≡−e0 + i e3Ω2 ≡e1 + i e2(5.60)Then,Σ3 = i2Ωα ∧Ωαα = 1, 2(5.61)is closed, by Eqn.

(5.57). Therefore, Einstein manifolds which are endowed with a complexstructure, and are described by abelian Ashtekar anti-instantons can be identified as K¨ahlermanifolds, with Σ3 as the K¨ahler form.We now construct explicit solutions for this class of manifolds.

We shall assume thatthe vierbein fields are of the form Eqn. (5.50), and that the Ashtekar potentials satisfyEqn.

(5.51). The equations Eqn.

(5.57), it can be checked, are then satisfied. We shallnow suppose, for simplicity, thatA3 = c(R)Θ3(5.62)so thatF3 = c′ dR ∧Θ3 + 2 c Θ1 ∧Θ2(5.63)To satisfy the gauge condition on S as specified above, i.e.

diag(0, 0, −λ), we must havec′ = −λa h2c = −λ f g(5.64)27

It is easy to check that for f = g, Eqn. (5.64) implies that both A1 and A2 vanish, and weare left with the conditionh′a −(2f 2 −h2)f 2= c(5.65)Substituting for f 2 from Eqn.

(5.64) gives−λh2c′ = 23c3′ + 2c2′The solution ish2 = −23λc(c + 3) + bca2 = (c′)2λ2h2f 2 = g2 = −2cλ(5.66)The function c is an arbitrary function of R, while b is an integration constant.Upon reversal of orientation, the new Ashtekar variables areA1 = −2f ′a Θ1A2 = −2f ′a Θ2A3 =−c3 −2 + 2b3 c2Θ3(5.67)The connections above now describe a non-abelian anti-instanton, with the correspondingS matrix given byS = diag−λ31 −2bc3, −λ31 −2bc3, −λ31 + 4bc3(5.68)As a result, the solution is of Type D for b ̸= 0, and of Type 0 when b = 0.The corresponding Pontrjagin numbers areP1 =14π2ZFa ∧Fa= −Z c2′ dR(5.69)for the case of the abelian anti-instanton, andP1 =14π2ZF a ∧F a= −Z c23 + 8b9c −4b23c4′dR(5.70)28

for the non-abelian case.If we letc = −λR221 + λ6 R2,b = 0as an example, we reproduce the expressions for CP2 and CP 2. Another example, withb ̸= 0 is obtained by settingc = −λ2 R2,b = −34λ2a4This ansatz gives us the Eguchi-Hanson space and the configuration for EH discussed inthe last section.6.

Matrix S as an order parameterWe have seen how S can play an effective role as an order parameter characterizingthe Type O sector. This sector corresponds classically to conformally self-dual Einsteinmanifolds.

Actually, we can go further with this hypothesis by studying it in the abeliananti-instanton sector.We have already discussed several explicit examples of Einsteinmanifolds which belong to this sector.Suppose that S is of rank one and can be expressed asSab = ±φaφb(6.1)where φa is a triplet of phenomenological real scalar fields. It is then gauge-equivalent tothe form S = diag(0, 0, ±φ2) and we may assume that||φ|| =√∓λ(6.2)where the sign in Eqn.

(6.1)is chosen in accordance with whether λ is negative or positive.In the U-gauge, with φa = ||φ||δa3, and A1,2 = 0, we have simply the condition(Dφ)a = 0(6.3)But Eqns. (6.2)and (6.3) are gauge and diffeomorphism invariant statements, and aretherefore valid in arbitrary SO(3) gauges and coordinate systems.

The situation is there-fore identical to that of a system possessing a symmetry based on the group SO(3), whichis broken down to SO(2) by the order parameter φa acquiring a non-vanishing vacuumexpectation value equal to the cosmological constant. The matrix S is non-invertible, andin this phase the gravitational fields are ordered dynamically in such a way as to break thelocal SO(3) Ashtekar symmetry.29

7.Concluding RemarksWe have presented in this paper several examples which illustrate the methods tobe used in obtaining solutions to the Einstein equations with Ashtekar variables. Theexamples have been chosen to bring out those features which are particularly transparentwithin this context.

Among these are the properties of Einstein manifolds under orientationreversals and their relations to abelian anti-instantons, the role of the cosmological constantin fixing the the type of Einstein manifolds, and finally, a perspective on spontaneousbreaking of the local SO(3) symmetry. We hope to amplify upon some of the physicalimplications of these features, especially in a quantum context, in the near future.30

Figure CaptionFig. 1 Classification of the initial data according to S.31

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