Topological ϴ-sectors in canonically quantized gravity

Volume 106B, number 3

PHYSICS LETTERS

5 November 1981

TOPOLOGICAL 0-SECTORS IN CANONICALLY QUANTIZED GRAVITY C.J. ISHAM Imperial College, Blaekett Laboratory, London SW7 2BZ, UK

Received 8 July 1981

It is shown that in canonical quantization of general relativity on a closed, orientable three-manifold 2, the diffeomorphism group Diff 2 of ~ gives rise to n- and 0-sectors analogous tothewell-known Yang-Mills results. There is also a "Gribov effect" which precludes the choice of a non-singular gauge for Dfff 2.

One of the most interesting recent developments in quantized Yang-Mills theory has been the discovery [1,2] of the existence of topological sectors; usually discussed in the language of n- and 0-vacua. On the other hand a number of workers in quantum gravity have tackled by different means the problem of uncovering the role played by space or spacetime topology. Typical examples are the discovery of solutions to the riemannianized Einstein equations (gravitational instantons) on topologically non-trivial spacetimes [3,4] and the canonical quantization o f " t w i s t e d fields" defined as cross sections of non-trivial vector bundles over three-space [5]. The purpose of this paper is to discuss a direct extension of the Yang-Mills ideas to quantum gravity. A priori, either the diffeomorphism group or the local tangent space group (the group of triad or tetrad rotations) might be expected to take the place of the Yang-Mills gauge group. However, neither of these groups appears in conventional general relativity as a precise analogue of the Yang-Mills group and a modification of the usual technique for obtaining n- and 0-vacua is appropriate. The approach considered below emphasises the role played by the topology of certain function spaces within a canonical quantization scheme associated with a three-manifold 2;. The relevant topological properties of these spaces are determined in a fairly direct way by those of Y~. The usual canonical approach to the Yang-Mills theory employs the gauge condition A 0 = 0 on the Yang-Mills field A u and involves a study of the solu188

tions to the classical vacuum condition Fi] = 0 [1,2]. If strong boundary conditions are imposed on A i the solutions are A i = ~ 3 i ~ - 1 , where ~ may be viewed as a function to the internal symmetry group G from the S3-compactification o f euclidean (R 3) three-space. The n-vacua are associated with the h o m o t o p y classes [S 3, G] = II3(G ) of these maps ~2. I f G is simple and non-abelian then II3(G ) = Z, which leads to the concept of the "winding number". The n-vacua In) are not covariant under "large" (i.e. homotopically nontrivial) gauge transformations whereas the 0-vacua defined as 10) = 2;n=_ ~ e in° In) are invariant up to an irrelevant phase factor. Although this discussion is carried out in terms of the vacuum states, the entire theory splits in this way. This scheme is readily extended to an arbitrary compact connected, orientable three-manifold Z. It is possible for the original G-bundle to be nontrivial * a over 2; but, if we concentrate on the trivial case, the theory splits into "n-sectors" labelled by the group of homotopy classes [2;, G] of(basepoint preserving) maps of 2; into G. This group can be explicitly determined [6] as a subgroup o f Z • Horn(Ill(2;), HI(G)). The 0-sectors are then labelled by the group Horn([2;, G], U(I)) of homomorphisms of [2;, G] into U(1) (i.e. the character group of [E, G]). When 111(2; ) 4= 0 some of the v a c u u m n-states may be approximately degenerate because of the existence of solutions to Fij = 0 which #1 Such bundles axe classified by elements of the the cohomology group H2 (1£; II 1(G)).

0 0 3 1 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.75 © 1981 North-Holland

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are not of pure gauge type but which can nevertheless connect together different pure gauge solutions. This leads to the representation of 0-vacua in terms of the quotient of [E, G] by a specific subgroup [7]. The concept o f a vacuum state is unclear in the constraint dominated form o f canonically quantized gravity employed below and it is useful to rederive the Yang Mills results in a different way. Let e denote the space of c o n n e c t i o n s A i defined on ~ and consider the action of the gauge group g on C .

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a system. When III(Q) 4: O, it is shown in refs. [12,13] that there exist a number of different quantization schemes labelled by the group of homomorphisms Horn(HI(Q), U(1)). For example, in a path integral formalism, the propagator Kx(q, q') can be expressed as a sum with phase factors X: Kx(q,

q') = ~ X([p])KiPl(q, q'), [P]

(5)

t

In general these groups are non-zero. For example, when the G-bundle over Z is trivial, ~ . is the space o f C =, basepoint preserving, maps from E into G and ,2

where K[p](q, q ) is the propagator obtained by summing over all paths between q and q' that lie in the homotopy class [p] of the path p. In a Hilbert space approach one may argue as follows. Normally the state vector • would be viewed as a wavefunction ~ ( q ) i.e. a complex valued function, on Q. However, when H 1 (Q) 4:0 we can consider the possibility that 'Iffq) "twists itself around" as q varies over Q. More precisely ~I, can be a cross section of a complex line bundle over Q. In general the functional differential (in a field theory) SchrSdinger equation would need to incorporate a U(1) connection ~ on Q which would completely change the theory *3. This can be avoided by employing only flat bundles, i.e. those for which the field strength of 9d vanishes, so that ~ can be (locally) gauged away. However, such bundles and connections are classified by elements of H o m ( i i I (Q), U(1)) [15] and the results of refs. [12,13] are regained +4 To see the relationship with 0-vacua in the Yang-Mills case let Q and ~" denote the base space and fibre of either of the fibre bundles in (2) and let X be a homomorphism from II1 (Q) = rio ( ~ ) to u ( 1).The corresponding flat, complex line bundle over Q is Q X xC where 0 is the universal covering space o f Q and the equivalence relation on 0 X C is (c~, k) = (qT, X(7) - l k ) for "all 3' in I I 0 ( ~ ) , where 03' is the point in (~ reached by acting with 3' on ~ E 0 in the usual way. Cross sections of this vector bundle are in one-to-one correspondence with functions ~ x from (} into C satisfying

I l l ( Q , ) = H o ( g , ) = [G, G] .

q~x(q3,) = X ( 7 ) - 1 qJx(q),

A i ~ A ~ = ~2Ai~ 1 + ~ 8 i ~ 2 - 1 .

(1)

This action may possess fixed points but these may be removed by employing the group g/C(G) [C(G) is the centre of G] which acts freely on the space ~ r of connections with an irreducible holonomy group. Alternatively, the subgroup g , C g of gauge transformations which are the identity on the base point of E, acts freely on G itself. Singer [8], Narasimhan and Ramadas [9] and Mitter and Viallet [10] have shown that, with appropriate choices of function space topologies, there'exist locally trivial fibre bundles over infinite-dimensional manifolds Qr and Q , : ~ / C ( G ) -~ e r

Qr =- erl(qlC(a))

g , -~ e

Q, - e / q ,

(2)

These quotient spaces may be viewed as tire physical configuration spaces of the two models. Now Q is contractible and all h o m o t o p y groups of G r vanish, [8] and hence 1Ii(~) = l l i ( G r ) = 0 for all i ~> 0. Thus the h o m o t o p y exact sequences of these bundels imply HI(Qr ) =

IIO(g/C(G))

and

Ill(Q. ) = II0(g. ) . (3)

(4)

Thus we are concerned with the quantum theory of a classical system with a multiply connected configuration space Q. This problem has been investigated in some depth [12,13] and Dowker pointed out in ref. [13] that Yang-Mills theory could be viewed as such ,2 Powerful results of Palais [ 1 l ] show that many natural topologies on the function space g* (including those used in refs. [8-10]) yield the same IIo(G,).

(6)

via the relation #3 +4

In a single-particle wave mechanics 21 is just the electromagnetic potential Hermitian line bundles and compatible connections are also discussed in ref. [29]. Such bundles, defined on phase space and with non-flat connections, play an important part in the Kostant-Sourieau geometric quantization programme. 189

Volume 106B, number 3 ~(q) = [q, qtx(0)] x

PHYSICS LETTERS (q lies over q ) .

(7)

The universal covering space ~) of Q in (2) is simply (respectively e / ~ *0) (this essentially follows from the homotopy exact sequences of the bundles below), where the 0 suffix denotes the set of all gauge transformations homotopic to the identity. A functional satisfying (7) is equivalent to a functional q>x on e r (respectively e ) with qSx(Aa ) = X[FZ]-lcbx(A ) where [g2] is the homotopy class of the gauge function ~2. This, however, is precisely the transformation law of a "0-state" labelled by X. The analogue of a "n-state" is qb defined by

e z/(~/C(G))o

(I),),(.4) ~ Sx X- 1(7) qbx(A) ,

(8)

where S denotes the appropriate sum or integral. Under gauge transformations these states transform as ~bv(A~) = ~ v [ a ] (A) as expected. A useful diagram is the double bundle: 6~,0-+ e 4

63./C3.o -" e / ~ , o

- Q.

J,

e/g,--Q, with a similar picture for e r. We can now consider the canonical quantization of gravity. The canonical fields [16,17] are the threemetric gi/defined on the compact * s three-space 2; and its conjugate variable

H i/=

(det

g)l/2(Ki/- Kkkg i]) ,

(9)

where Ki] is the second fundamental form of Z embedded in four-dimensional spacetime. The four Gou = 0 Einstein equations are equivalent to the constraint equations 0 =c~0 ~ (det -

(det

0=c/~ i~-

g)- 1/2(IIiJIIi! - ½(llkx)2 )

g)I/2R/K2 , Hi~i]

(10) (11)

where R is the scalar curvature ofgi] , K is 8n/c 2 times Newton's constant and I signifies covariant differentia4:5 In this scheme there are profound conceptual and technical differences between a compact and non-compact Z. However, the purely topological aspects could readily be adapted to include the latter case. 190

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tion with respect to the Christoffel symbol ofgij. In the constraint formalism of Dirac [16] Wheeler [18] and DeWitt [19], quantization consists of Solving the constraint equations in operator form: ~O(L.s ' fikl)~ = O,

(12)

9ei(~rs, fik~),I, = 0,

(13)

where

[~,rs(X), flkto,)] = ih 8 ~kSts)6(3)(x,y).

(14)

Typically the state vector 'I, is a functional ofgi] with a heuristic representation of the canonical commutation relations (14) in the form (~rs(X)'I') (g) = g~s(x)'I'(g), (fikZ(x) q')(g) : - ~

5"~(g)/Sgkt(x).

(15)

(16)

The constraintsC~ i generate coordinate transformations in 2; and hence (13) may be interpreted as a statement that tI, is invariant under the coordinate group. Thus, formally, ~ is initially a functional on the space Riem 2; of all three-metrics on 2; and is then projected by (13) onto the quotient space ("superspace") Riem Z/Diff ~ where Diff 2; is the group of all orientation preserving*6 diffeomorphisms of Z. Some of the functional analytic aspects of this scheme are discussed in ref. [20] but we see that, from a topological viewpoint, the situation is analogous to the Yang-Mills case. This becomes especially clear when it is realised that, by (somewhat formally) exponentiating (13), invariance is only in fact guaranteed under diffeomorphisms homotopic to the identity 4:7. This is the same as the situation arising in the Hilbert space quantization of Yang-Mills theory [2] and, as there, leads directly to the possibility of n-vacua. To proceed further it is necessary to construct the analogues of the fibre bundles in eq. (2). A diffeomorphism q~ of 2; acts on a metric by pulling it back, g-+ ~b*g, but the corresponding action of Diff 2; on Riem ~ is not free due to the existence of metrics with isometries (i.e. ~*g = g). There are two natural

4:6 If orientation reversing dfffeomorphisms are permitted the number of "n-sectors" increases. *7 The exponential map is not a local diffeomorphism so that (13) only really implies invariance under some subset of this group.

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ways of overcoming this defect. The first is to consider the action of Diff 2; on Riem r 2; - the subspace of Riem 13 consisting of metrics with a trivial isometry group - whilst the other is to employ the subgroup Diff. 2; of Diff 13 consisting of orientation preserving diffeomorphisms which leave the tangent space at the base point invariant and which hence act freely on Riem £. The topological structures of Diff I3 and Riem Z / D i f f Y, have been studied by Ebin [21] and Fischer [22] and, in analogy with (2), there exist two fibre bundles over infinite-dimensional manifolds c5 r and c5. : Diff ~ -+ Riem r X, 4. CSr= Riemr 13/DiffY.

Diff, ~ -+ Riem 2;. + (17) Riem 2;/Diff, 1!; ---=c5, -

We may now proceed as in the Yang-Mills theory. State vectors are represented in the two models by cross sections of flat complex line bundles over the appropriate quotient spaces and the 0-sectors are labelled by elements of H o m ( I I l ( d r ) , U(1)) and Hom(IJl(CS,), U(1)), respectively. Riem 2; is clearly contractible and from [8] all the h o m o t o p y groups of Riem r 13 vanish. Thus I l l ( 6 r ) = Ilo(Diff 2;),

111(6. ) = IIo(Diff , 2;)

(18)

and the classification of n- and 0-states reduces to the study of the components of Diff 13 and Diff. 13. Ab initio this is a much harder problem than the corresponding one in Yang Mills theory. Fortunately, however, there have been significant advances recently in the mathematical study of the diffeomorphism groups of three-manifolds. Any two three-manifolds 2;1 and 132 can be glued together by cutting out a three-ball in each and identifying along the two-sphere boundaries to give the connected sum N1 * 2;2. A manifold 2; is said to be prime if it cannot be expressed as the connected sum o f two other manifolds unless one of them is a three-sphere (which is a trivial factor) and it is a basic result that any compact threemanifold 13 can be decomposed into an essentially unique connected sum 2;1 * 2;2 * --- £n of some finite set of prime manifolds [23]. It should be noticed in passing that Il1(231 * £ 2 ) is the free product I l l ( £ 1 ) • Il1(2;2) and hence in studying Yang-Mills theories on an arbitrary three-manifold 2;, the computation o f [2;, G] in terms o f elements o f Z • Horn(Ill(2;), Ill(G)) may be reduced to the corresponding problem for the prime factors 2; 1 , 2;2 ... 13n o f 2;.

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It is clearly desirable that an analogous result holds in the quantum gravity case and fortunately this is so. Every compact, connected, orientable, prime manifold is homeomorphic to S 3, S 1 X S 2 or a p2-irreducible [23] manifold with a non-vanishing first Betti number. The group o f orientation preserving diffeomorphisms of S 3 is homotopic to SO(4) so that II0(Diff S 3) = O. On the other hand II0(Diff S1 × S 2) = Z 2 ® Z 2 [24,25] and the zeroth and first h o m o t o p y groups of the diffeomorphism group o f the irreducible manifolds are, respectively, the group of outer automorphisms and the centre of H I (2;) [26]. Finally the h o m o t o p y groups of Diff 2; for a general E are related in ref. [25] to the corresponding expressions for the prime factors. This results is too lengthy and technical to sunnnarise here but the final outcome is that, via (18), I l l ( d r ) and I I l ( e S , ) are effectively computable. Evidently II0(Diff S 3) is trivial but it is easy to construct diffeomorphisms on other spaces which lead to a nontrivial result. For example, orientation preserving diffeomorphisms o f S 1 X S 1 X S 1 which reverse the orientations of individual circles and/or permute them in different ways, are clearly not homotopic and indeed correspond to various outer automorphisms of i l l ( S 1 X S 1 X S 1 ). Similarly the orientation preserving diffeomorphism o f S 1 X S 2 of the form ~b(X,x) = (X*, a(x)) where S 1 is viewed as the complex numbers and a is the antipodal map, is not homotopic to the identity. In general II0(Diff Z) and ll0(Diff , 2;) are non-trivial, leading to the principal conclusion that, in canonically quantized gravity, there will be topological sectors associated with the diffeomorphism group and that the classification of these sectors is in principle feasible. It also follows that the fibrations in (17) are non-trivial and hence do not possess global cross sections leading to a "Gribov phenomenon" [8,27] for the diffeomorphism group +8. (This even holds on S 3 since I I l ( D i f f $3 ) = Z 2 ~ 0.) One might anticipate that similar effects arise from the tangent space group .~ of local triad rotations but in fact this is not so. This local SO(3) gauge group acts without fixed points on the space g of triads to give a principals~-fibre bundle:

4~8 If desired the group of conformal rescalings may also be factored. This group is homotopicaUy trivial and hence does not affect the topological results. 191

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-~-+ 9 J, 9 / ~ = R i e m 2;,

(19)

where the base space is the set o f triads with the local SO(3)-aetion factored out i.e. R i e m Y,. However, R i e m Z is contractible and hence I I l ( R i e m £ ) = 0, H o m ( I I l ( R i e m I~), U(1)) = 0 and there are no topological sectors induced in this way. Neither is there any Gribov effect since the bundle in (19) is actually trivial. Thus, notwithstanding the superficial resemblance b e t w e e n the triad group and a Y a n g - M i l l s gauge group, their topological effects are very different. A n y topological effect from the triad group must arise only w h e n spinors are included. In this c o n t e x t it should be emphasized that the d i f f e o m o r p h i s m group results are as applicable to supergravity as they are to pure general relativity and in particular to the, hopefully finite and currently very popular, SO(8) extended theory. A study is under way to investigate these topological effects in detail within the f r a m e w o r k o f an inverse coupling constant ( 1 / K 2) perturbation theory [20] o f Q~0(grs , [Ikl ) e m p l o y i n g Pilati's [28] recent explicit quantization o f the ultralocal hamiltonian obtained f r o m (12) by setting 1/K 2 = 0. Such a perturbation scheme is manifestly covariant and should reflect the topological effects in each order. The results will be published elsewhere.

References [1] R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172; C.G. Callen, R.F. Dashen and D.J. Goss, Phys. Lett. 63B (1976) 34. [2] R. Jackiw, Rev. Mod. Phys. 52 (1980) 661. [3] S.W. Hawking, in: General relativity - an Einstein centenary survey, eds. S.W. Hawking and W. Israel (Cambridge U. P., London, 1979). [4] C.N. Pope, in: Quantum gravity - a second Oxford symposium, eds. C.J. Isham, R. Pen.rose and D. Sciama (Oxford U.P., London, 1981), to be published. [5 ] For an extensive list of references, see C.J. Isham, in: Quantum gravity - a second Oxford symposium, eds. C.J. Isham, R. Pen.rose and D. Sciama (Oxford U.P., London, 1981) to be published.

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[6] C.J. Isham, in: Essays in honour of Wolfgang Yourgrau, ed. A. van der Merwe (Plenum, New York, 1981), to be published. [7] C.J. Isham and G. Kunstatter, Phys. Lett 102B (1981) 417; Spatial topology and Yang-Mills vacua, Imperial College preprint ICTP/80-81/27, to be published in J. Math. Phys. [8] I. Singer, Commun. Math. Phys. 60 (1978) 7. [9} M.S. Narasimhan and T.R. Ramadas, Commun. Math. Phys. 67 (1979) 21. [10] P.K. Mitter and C. Viallet, Commun. Math. Phys. 79 (1981) 457. [11] R. Palais, Topology 5 (1966) 1; Foundations of global non-linear analysis (Benjamin, New York, 1968). [12] L.S. Schulman, J. Math. Phys. 12 (1971) 304; M. Laidlaw and C. DeWitt, Phys. Rev. D3 (1971) 1375; J.S. Dowker, J. Phys. A5 (1972) 936. [13] J.S. Dowker, Selected topics in topology and quantum field theory, Austin preprint (1980). [14] M. Asorey and P.K. Mitter, Regularized continuum Yang-Mills process and Feynman-Kac functional integral, preprint LPTHE 80/22, to be published in Commun. Math. Phys. [15] J. Milnor, Commun. Math. Helv. 32 (1957) 215; F. Kamber and P. Tondeur, Flat manifolds, Lecture notes in mathematics, Vol. 67 (Springer, New York, 1968). [16] P.A.M. Dirac, Can. J. Math. 2 (1950) 129; Proc. R. Soc. A246 (1958) 333; Phys. Rev. 114 (1959) 924. [17] R. Arnowitt, S. Deser and C. Misner, Phys. Rev. 113 (1959) 745; 116 (1959) 1322; 117 (1960) 1595. [18] J.A. Wheeler, in: Relativity groups and topology, eds. C. DeWitt and B.S. DeWitt (Blackie, London, 1965); J.A. Wheeler, in: Battelle recontres 1967, eds. C. DeWitt and J.A. Wheeler (Benjamin, New York, 1968). [19] B.S. DeWitt, Phys. Rev. 160 (1967) 1113. [20] C.J. Isham, Proc. R.Soc. A351 (1976) 209. [21 ] D. Ebin, in: Proc. Symposia in pure mathematics, Vol. 15 (Am. Math. Soc., 1970). [22] A. Fischer, in: Relativity, eds. M. Carrneh, S. Fickler and L. Witten (Plenum, New York, 1967). [23] A comprehensive reference on three-manifolds is: J. Hempel, 3-manifolds (Princeton U. P., 1976). [24] H. Gluck, Bull. Am. Math. Soc. 67 (1961) 586. [25 ] E.C. de S~ and C. Rourke, Bull. Am. Math. Soc. 1 (1979) 251. [26] F. Waldhausen, Ann. Math. 18 (1968) 56; F. Laudenbach, Asterique 12 (1974); A. Hatcher, Topology 15 (1976) 343. [27] V.N. Gribov, Nucl. Phys. B139 (1978) 1. [28] M. Pilati, Imperial College preprint ICTP/80/81-32. [29] B. Kostant, in: Lecture notes in mathematics, Vol. 170 (Springer, Berlin, 1970).