Topological gravity

Volume 206, number 4

PHYSICS LETTERS B

2 June 1988

T O P O L O G I C A L GRAVITY Edward W I T T E N 1,2 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA

Received 19 February 1988

A version of conformal gravity is formulated with a local fermionic symmetry that is reminiscent of BRST invariance. It may have mathematical applications (gravitational counterpart of Donaldson theory) or physical ones (unbroken phase of general relativity ).

In ref. [ 1 ], Donaldson initiated a program of using Yang-Mills fields to study the geometry and topology o f four-manifolds. Some o f the ingredients, notably the self-dual Yang-Mills equations, were familiar to physicists, but other crucial ideas did not have any obvious relation to physics. Floer's work [2 ] on instantons and three-manifolds has provided a crucial link between Donaldson theory and physics. For on the one hand [ 3 ], Floer theory can be interpreted in terms of a certain non-relativistic q u a n t u m field theory, and on the other hand Floer theory enters when one tries to define Donaldson invariants o f a four-dimensional manifold with boundary. The latter connection between Floer and Donaldson theory led Atiyah to suggest that there should be a relativistic q u a n t u m field theory underlying Donaldson theory. This indeed turns out to be the case; the relevant theory, formulated in ref. [4] (where more background can be found), is a sort of twisted version o f supersymmetric Yang-Mills theory. In ordinary, supersymmetric Yang-Mills theory, the supercharges are space-time spinors, conserved only on a fiat fourmanifold M. In the twisted, topological version, there is a single global supercharge Q, obeying Q 2 = 0 , and conserved on an arbitrary M. Q relates bosons and fermions all o f integer spin. The Q cohomology classes

(solutions o f Q~V=0, modulo the equivalence ku___~u+ Q2) consist purely o f global topological invariants, the Floer groups; all the local excitations decouple when one constructs the Q cohomology. The Q symmetry is so similar to BRST symmetry that it is natural to think that it arises upon BRST gauge fixing o f some underlying theory with more gauge invariance; such an underlying theory is not known at present, but if it exists it must apparently be a generally covariant theory in a strongly coupled phase in which general covariance is unbroken and confined (like color in Q C D ). The reasoning behind the latter remark is explained in the last section o f ref. [ 4 ]. It is natural to ask whether a similar BRST-like symmetry is possible in general relativity. At this point, it is helpful to recall what fermionic symmetries are known to be possible in general relativity. In conventional supergravity [ 5,6 ], the purely gravitational part of the action is the Einstein-Hilbert action I c = 16nG 1 f d4x ~f~ R. One also has (in four dimensions) conformal supergravity [ 7 ], the purely gravitational part of the action being Ic = f dax x/g W 2 ,

J On leave from Department of Physics, Princeton University, Princeton, NJ 08544, USA. 2 Researchsupported in part by NSF Grants Nos. PHY 80-19754, 86-16129, 86-20266. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

with W the Weyl tensor. Since W is proportional to the second derivative of the metric tensor, Ic is quadratic in second derivatives. It gives a theory which classically has indefinite energy and q u a n t u m me601

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chanically has an indefinite metric Hilbert space with ghosts. There has hitherto been no known way to eliminate the ghosts from conformal gravity. The ghosts are not gauge degrees of freedom in the usual sense, so they are not removed by conventional gauge fixing for BRST quantization. Conventional supersymmetrization does not remove the ghosts (it adds new fermionic ones). In this paper, we will add new bosonic and fermionic degrees of freedom to conformal gravity in such a way that a new BRST-like symmetry (not obtainable by quantization of any known theory with higher gauge invariance) will appear. In the BRST sense, there will be no ghosts and in fact no local physical degrees of freedom. This corresponds roughly to a unitary, confining version of conformal gravity. For the future, there are twin challenges of finding an underlying gauge invariant system and of finding a version in which general covariance is spontaneously broken and dynamical gravity emerges. If possible, the latter would give a new perspective on the problem of induced gravity (of which a review can be found in ref. [ 8 ] ). The purely topological nature of the model considered here (and of the related one in ref. [4] ) is in certain respects reminiscent of string theory. Before plunging into a description of the model, let us describe the conventions that we will use. The structure group of the tangent bundle of an oriented four-dimensional manifold is essentially $ 0 ( 4 ) - ~ S U ( 2 ) L × SU(2)R. SU(2)L and SU(2)R indices will be denoted A, B, C and A, B, C, respectively (A, Jl = 1, 2). A field • of spin ( n / 2 , m / 2 ) under SU (2) L× SU (2) R is denoted ~ A , A~.~. A, (symmetric in Av..A, and in .~t...A, ). Spinor indices are raised and lowered by 2A=eA828, 28=~BC2 c, with eAB=--eBA, eABeBC= +8~; and similarly for A--,A. Tangent indices to space-time are denoted a, fl, y. The metric is described by a tetrad e ~ A (and inverse .~o~BB B~B tetrad eans;• they obey e,~AA~ = xuA"~, e,~Ai~ePAi~=~ ) . The metric tensor is g , p = e ~ e ~ A . The covariant derivative of a spin ( 1/2, 0) field is defined by Da2A

= O a 2 A + t o a A a . ,],B ,

O a J. A = Oa ,~.A - - t o A a.,'],a

(

1)

(and likewise for A ~ J l ) . In (1), to is known as the 602

2 June 1988

spin connection and obeys O.)o:AB~. (.I)otBA. The covariant derivative of a more general field q~A,.~,.n,..~,, is obtained by generalizing ( 1 ) in the usual way. The spin connection (and affine connection F) are determined in the usual way by requiring 0=D,~e~p _a

.mi - - zr.a ~A~ AB ,i ab A a # ~ , ~ - - t o a eft. - - t o a e# h.

--vat#

(2)

This leads to a lengthy and standard formula for to in terms of e. Now let us turn to our main problem of finding the appropriate multiplet with fermionic symmetry. As in usual BRST quantization, we will assume the conservation (at the classical level) of a global quantum number U ("ghost number" ). First we pick the anticommuting fields. The anticommuting fields in ref. [4] consisted of the multiplet (spins (0, 0), (1/2, 1/2), and (0, 1 )) that enters in deformation theory around an anti-self-dual gauge field. Following a suggestion by Atiyah, we will in the gravitational case take the anticommuting fields to be the multiplet that arises in linearization around an anti-self-dual metric. The construction of this multiplet is clear from ref. [ 9 ]. It includes a spin ( 1, 1 ) field of conformal dimension D = 0 , and U= 1; we will call it ~UAnAS-In addition, this multiplet contains a spin (1/2, 1/2) field 2 ~ of U= - 1 and a spin (0, 2) field ZASCD of U= -- 1 and D = 2 . As regards bosonic fields, apart from the tetrad e~.~, additional bosonic fields are needed to complete the super-multiplet. In ref. [4 ], apart from the gauge field, one required additional scalar fields ¢, 2 of U= + 2 in the adjoint representation of the gauge group. In gauge theory, spin zero fields in the adjoint representation can be regarded as generators of the gauge group. In general relativity, the generators of the gauge group correspond to vector fields, so we are led to guess that the necessary bosonic fields, beyond the tetrad, will be vector fields C ~ and BA~ of U= + 2. C has conformal dimension - 1; the choice of conformal dimensions of B and 2 leads to certain problems that will appear later, so in table 1 the conformal dimension of those fields is simply indicated as unknown k. Next we try to work out the fermionic symmetry. With the indicated conformal dimensions and ghost numbers, the only possible transformation laws that one can try for e ~ and ~UABAa(modulo constants

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Table I The fields entering in the supermultiplet. D denotes dimension, U denotes "ghost number", and the statistics are+and-for commuting or anticommuting fields. Field

D

U

Statistics

e,~A,i

-- 1

0

+

C~,~

-- 1

2

+

BA~ ,t~•~ ~.4~,,.ih XA~CD

k k 0 2

-2 - 1 1 -- 1

+ ---

that could be a b s o r b e d in the n o r m a l i z a t i o n o f ~ a n d C) are ~eAA= l ~ e aBB ~AB,AB , •

2 June 1988

(8)

S = e ~ x ~ D a C x~ .

In ( 6 ) , the first term is an infinitesimal general coo r d i n a t e t r a n s f o r m a t i o n generated by the vector f i e l d - 2 i q e C ~ (with C ' ~ = e , ~ C A A ) , the next two terms are local Lorentz transformations with generator r/e/l, a n d the last term is a local conformal transf o r m a t i o n (rescaling o f e) with p a r a m e t e r ½ir/ES. In particular, we see (as expected) that the fermionic s y m m e t r y we are looking for requires local conformal invariance, a n d that the conserved charge Q will obey Q 2 = 0 , acting on objects invariant u n d e r the above described symmetries. It is now tedious but straightforward to see that if we postulate

"

~ C A~ = i ~ I AB,Ai~C Bh ,

~.IAB,A ~ -__ ~e( I Or. . + e BAD~CA~ Or. . e AADaCBB

+ e ~ B D , C~A + e ~ h D , C A A ) .

(3)

Here e is a constant a n t i c o m m u t i n g p a r a m e t e r o f U = - 1. These equations i m p l y that the inverse tetrad must transform as ~e aAA = -- i e e ~ ~[AB,.4B ,

(4)

and the spin connection (which is d e t e r m i n e d in terms o f the tetrad by ( 2 ) ) must transform as ~O)aA B = ½ie ( D ~ C D ' C D ) e o : D b (

eACe#B~ + e B C e ~ d , ) ,

5o9,~a = ½ie ( D # ~ Co,ab) eo~ob ( e~ee~ci~ + ei~ae~c~ ) •

(5) The next step is to find the algebra o b e y e d by the fermionic t r a n s f o r m a t i o n s (3). If 6, denotes a fermionic t r a n s f o r m a t i o n with infinitesimal p a r a m e t e r e, then one computes ( t~n t~, - t~, t~,~) eo~AA = -- 2iqeD~ CaA +r/e(e.abAAb+eo/~AAaD)+½iqee~S,

(6)

(9)

then the algebra closes on ~ and C. "Closing the algebra" on a field q~A,...A,,,~...A,,,means showing that (6~6, - 6~6,) ( ~ ) = - 2ir/eC~D~ qb+ qeA ( ~ )

- ½ir/~D~.~.

(10)

Here D~ is the d i m e n s i o n o f • ( D = 0 for ~u, and D = - 1 for CA~), a n d A(q~) schematically denotes a local Lorentz t r a n s f o r m a t i o n generated by A. The three terms in (10) are diffeomorphisms, local Lorentz transformations and local conformal transf o r m a t i o n s with the same p a r a m e t e r s as in (6). At this point, we have given t r a n s f o r m a t i o n laws for eoA~, q/A~.~B, and CA;~ which close on themselves u n d e r the fermionic symmetry. However, it is not possible to write a sensible lagrangian for those fields only; we will have to introduce a d d i t i o n a l fields. It is convenient to first work out the t r a n s f o r m a t i o n law o f the Weyl tensor. The R i e m a n n tensor (or at least the part o f it self-dual in the last two indices) is defined by Ra#AB=OaOJ#AB--OflogaAB

+ [ O.)a,O)fl]AB .

( 11)

where

One has

AAD = ~AX, Bk ~tDX'B~ + ½i ( e'~kD,~ C o k + e~xDo, C ~ ) ,

e ~ x e ~ ' R o ~ A B = ~Oc~'Rxy,~B + ~xyRAB, R~" •

A~D = ~/,x,B~ ~/BX, D~

Here RAB,ki', o f spin ( 1, 1 ) is the traceless part o f the Ricci tensor. Rxr,AB is not completely s y m m e t r i c in all indices. But the s y m m e t r i z e d version

+ ~1"l ( e x /Ot~ D , ~ C X b + e%z~D,~CX~),

and

(7)

1

W xrAB = ~ ( R x r ~ 8 + R x B, rA + R xA,B r )

(12)

( 13 )

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has spin (0, 2) and is the self-dual part o f the Weyl tensor, which is conformally invariant. (Apart from the Weyl tensor, Rxr,AB contains the Ricci scalar, of spin (0, 0); this is eliminated by the sum over cyclic permutations in ( 13 ) which thus is enough to make Wcompletely symmetric in its indices. ) It is straightforward, but, again, tedious, to show from (3) and (5) that WXYA B =~le[(~/Xy l" YZRaB,~,2--eAAeBi~DaDz~lxy, o~ ff Ah )

+ (5 permutations of X, Y, A, B) ] .

(14)

Since the fermionic transformation that we are studying closes on e,~A;~, it also close on any functional of e~A~, such as WABCD.Therefore ( (~qfe --(~,(~q) WABCD = - 2ir/eC"D~

WASCD

( 15 )

O f course, this can also be checked directly. Now, let us give a name to the right-hand side of (14), say ~,2 '" AABCD = -~i[ (~Uxr, RAB,~z -e~AePsi~D, Dpg/xr, AB) + (5 permutations of X, Y, A, B) ] .

+ ½ie(AAFBFA -bAApBAp) "JrI keBA~ S .

( 19 )

Here k is the conformal dimension of B and 2, which we leave open for the moment. It is then not obvious that the algebra closes on 2a~, but this may be verified ~l. Finally, we consider ZABCO. The only possible transformation law of Z (compatible with diffeomorphisms, local conformal transformations, and ghost number), is

8)(,ABCD= eWAsco •

(20)

(The normalization o f the right-hand side of (20) can be absorbed in the definition ofz. ) This gives (21

)

with AASCO defined earlier. The desired formula is instead

+ cyclic permutations of (ABCD) ]

- iqe WABCD(e%2D,~ C x2) .

82 AA = eCaDa2AA

((Jrt • -- 4 Ort)ZABCD = -- 2rI~4ABCD ,

"q-qe [ W F B c D A A F

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(16)

(15) can be restated as a transformation law for A:

( ~,1t~, --~,~,)ZaBCO = -- 2tN{iCe~Dc~XABCD -- ½[ A a v g V c o + cyclic permutation o f (ABCD) ] + ½iZanco 'S}.

(22)

Therefore, the algebra does not close on ZAnCD without use of equations of motion. The only hope is that the lagrangian should be such that the equation o f motion is

0 = 8~/8X ABcD

8AABcD = i e C " D , WABCD

~AABcD--{iC~D~ZABCO

-- ½E[ WFcDAAF

- ½[AaFZFBcD+ cyclic permutation o f (ABCD) ]

+ cyclic permutations of (ABCD) ] 1"

~.

+~leWABcD(exxD,~C

XX

).

(17)

+ ½iZABcoS}.

(23)

This formula will be convenient later. N o w we need transformation laws for the other fields. First we consider BA~ and 2AA. We may as well postulate

Now, let us try to find an invariant lagrangian. We certainly want to include the Weyl invariant gravitational action

8BAA=ie2a~ ,

~=

(18)

since additional terms added to the fight-hand side of (18) can be absorbed in a redefinition Of 2A,~. The transformation law o f 2A~ is then uniquely determined by asking that the algebra should close on BA~:

604

I d4xdet e ½WA~cDW ABcD.

(24)

~t In (18) and (19) the following generalization is possible. Instead of B and 2 having spin ( 1/2, 1/2) under Lorentz transformations, they could have arbitrary Lorentz quantum numbers. The algebra still closes, with the obvious modification of the A.B term in ( 19 ).

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In addition, we know that the terms containing Z must be such as to give the equation of motion (23). This gives Lf~ = | d4x det e ,)

x[½izA~CD(C,*D,)ZABCD - - i~,ABCD,,, .. A WAB, ~bO ~t'CD,CD ~l_ i)~ABCDe¢~ 1e~BD~ Da~/c~ i~

r.v~',- - ie,~xD.CFk) ]"

, , ,A V , O(/~ [1~ ' --I~,~ABCD,ttF ,~ B C D \[W

2 June 1988

integrand in (26) has definite conformal weight if and only if B and 2 have conformal weight k = 1 (which indeed is the natural value of the conformal weight of those fields in the context of the instanton moduli problem). In that case, though, (26) is not conformally invariant (but has w e i g h t - 2 , taking account of the det e factor). To salvage the situation, we must postulate the existence of a suitable field q) of spin zero and conformal weight two, and let

Z=-

f

d4xdeteq~(e c~AADuB Bb q/AB,Ai~)."

(27)

(25) It is possible to verify that Lf' = LP~+ ~2 is invariant. The main steps are as follows. The terms in 8L¢' independent ofzvanish using (14) and (20), the terms linear in Z vanish using ( 17 ) and (20), and the terms quadratic in Z vanish by identities similar to the ones used in closing the algebra. L~' is thus an invariant lagrangian, but not a sarisfactory one, since the kinetic energy is absent for C and for some components of q/. To redress the situation, we must find a new invariant functional. The basic strategy for doing this is to use the fact that on the fields e,A~, BAn, and q/AB.~B,the algebra closes up to a diffeomorphism, local Lorentz transformation, and conformal transformation. Suppose that we find a functional Z of e,A;~, BAj and ~UAB.~Bwhich is is invariant under diffeomorphisms, local Lorentz transformations, and conformal transformations. Then if we let &a3= {Q, Z}, then L~3 will automatically be invariant. Unfortunately, there is no completely sarisfactory choice for Z without introducing additional fields. One might first try Z o = -- f d 4 x

det e(e°~a;~D,~BBh ~Ua~.AB) •

(26)

Clearly this is invariant under diffeomorphisms and local Lorentz transformations, so the only issue concerns conformal transformations. At this point, we must make a choice regarding the conformal weight of B and 2. It is clear that (26) is globally scale invariant if and only ifB and 2 are assigned the conformal weight k = 3. On the other hand, with this choice, the integrand in (26) does not transform with definite weight under local conformal transformations ~. The ~2 I am indebted to S. Axelrod for correcting an erroneous claim on this point in a previous version of this manuscript.

Then L~3= (Q, Z} will automatically be Q invariant. Explicitly, one finds a rather complicated formula, L~3={Q, z} = f d4x det e q) • BB ( e X [i2

a A) A D a q / A B , A h - -

" aB ~ (eaAAD

B B' )

× (ePAADaCBB+e~i~DpCB~ +e~DaCAh +e~BDpCA~) •~ B i ~ AC.A~ a r~ -- 1u ~l/ e c e ,-., o, ~ A B , A B + ' U B h , , a . . . . UV, Of~r',~ . . . . . L~, C.B B T ~ , L~otWUV, UV .~_1"1"~ l ~ B k ( ot E .. AX,A. ~lx.~ot~, ~ e B ~IAX.AB~/ E + e ~ E h ~ J A B , A k ~ C A E f l k ) + ... ] ,

(28)

where "..." denotes terms proportional to D,q) and to {Q, qb}, which we will not specify because we will not make a definite model for q). The lagrangian L f = ~ + ~ +L¢3

(29)

then contains kinetic energies for all fields of the gravitational multiplet, arranged to give the local fermionic symmetry which (in the BRST sense) eliminates all local degrees of freedom, leaving a unitary theory of purely topological character ~3. To complete the story, however, we must describe how q) is constructed from suitable elementary degrees of freedom and write an action for these degrees of freedom. There are many possible ways to doing ~3 It m a y appear that we could introduce an arbitrary free parameter in (29) by replacing L~ in (29) by an arbitrary multiple of itself. This is not really so; the parameter in question can be absorbed in a rescaling of B and 2.

605

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this. One possibility is to couple to the gauge multiplet o f ref. [ 4 ], which contains scalar fields from which a gauge invariant composite q) o f conformal spin two could be constructed. There are m a n y other possibilities, however. As noted in footnote 1, m a n y possible multiplets can be constructed and coupled to the gravitational multiplet, a n d clearly there m a n y possible choices will lead to the existence o f a suitable (elementary or c o m p o s i t e ) scalar q~ o f conformal spin two. We will not try to analyze here the plethora o f possibilities. These models, however, will all have one thing in common. It is necessary to assume that • has a vacuum expectation value. Otherwise, ( 2 8 ) does not contribute to the kinetic energy o f the gravitational multiplet, and does not cure the deficiencies o f Lf' = L~ + LP2. But since q~ must have a nonzero conformal weight, its expectation value will spontaneously break the local conformal invariance. Thus, we reach the i m p o r t a n t conclusion that local conformal invariance is spontaneously b r o k e n in all models o f this sort. One m a y w o n d e r whether there is a suitable analogue o f D o n a l d s o n polynomials. In ref. [ 4 ], the key to the D o n a l d s o n p o l y n o m i a l s was the existence o f a BRST invariant scalar field q$. In the present setting, the closest analogue o f ~ is the vector field C a = e~AACAA, which is easily seen to be BRST invariant. Perhaps suitable correlation functions o f C a m a y serve as analogues o f the D o n a l d s o n polynomials, but we leave this for the future. One also wonders whether the present considerations could have anything to do with physics. This p r e s u m a b l y depends on finding a version in which local d i f f e o m o r p h i s m invariance is spontaneously b r o k e n (down to the global Poincar6 s y m m e t r y ) a n d local physics re-emerges. We will have to leave this

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too for future investigation. But it is interesting to note that an a n t i c o m m u t i n g spin zero field a with t r a n s f o r m a t i o n law

~a = ~ ( 1 + iC'~aD,~ a)

(30)

furnishes a non-linear realization o f the symmetry. Finally, we should note that the fermionic symmetry u n d e r discussion here is begging to be interpreted via BRST gauge fixing o f some system with higher symmetry. But it is clear that no presently known system will fill the bill, unless "topological gravity" can emerge as the low energy limit o f string theory is some presently unknown phase. I a m greatly indebted to M.F. Atiyah for m a n y observations about Floer and D o n a l d s o n theory and for interesting me in this whole subject. In addition, I thank S. Axelrod for a careful reading o f the m a n u script and for his critical comments.

References [1] S. Donaldson, J. Diff. Geom. 18 (1983) 269. [ 2] A. Floer, An instanton invariant for three manifolds, Courant Institute preprint ( 1988); Bull. AMS 16 (1987) 279; A relative Morse index for the symplectie action; Morse theory for lagrangian intersections, Courant Institute preprints (1987). [ 3 ] M.F. Atiyah, New invariants of three and four dimensional manifolds, in: Proc. Syrup. on the Mathematical heritage of Hermann Weyl (Durham, NC, May 1987 ), to be published. [4] E. Winen, Topological quantum field theory, IAS preprint (1988). [ 5 ] D.Z. Freedman, P. van Nieuwenhuizen, and S. Ferrara, Phys. Rev. D 13 (1976) 3214. [ 6 ] S. Deser and B. Zumino, Phys. Lett. B 62 ( 1976 ) 335. [ 7 ] P. van Nieuwenhuizen, M. Kaku and P. Townsend, Phys. Rev. D 17 (1978) 3179. [8] S. Adler, Rev. Mod. Phys. 54 (1982) 729. [9] M.F. Atiyah, N. Hitchin and I. Singer, Proc. R. Acad. Sci. London A 362 (1978) 425.