Bosonic physical states in N = 1 supergravity

Physics Letters B 321 (1994) 368-371 North-Holland

PHYSICS LETTERS B

Bosonic physical states in N = 1 supergravity P.D. D'Eath

Department ofAppliedMathematicsand TheoreticalPhysics, Universityof Cambridge, SilverStreet, CambridgeCB3 9EW, UK Received 9 November 1993; revised manuscript received 22 November 1993 Editor: P.V. Landshoff

Two bosonic physical wave functions in N= 1 supergravity are found by solvingthe supersymmetry constraints, in the case that the spatial topology is S3. These both have the form const..exp(-I/h), where I is an action functional of the three-metric, and correspond respectively to the Hartle-Hawking state and wormhole ground state of the theory. In the case of spatial topology R3, there is a solution of the form const..exp(-I/h), corresponding to the ground state of the theory. For each bosonic solution of the form const..exp(-I/h), there is a solution in which all fermion levels are completely filled, proportional to exp (l/h). One expects that (e.g.) the semi-classical expansion for the wormhole ground state (assuming it exists) should also be given by pathintegral perturbation theory about a curved background (the classical infilling four-geometry which is asymptotically Euclidean and subject to the boundary data on the sa). This gives an indication that Feynman-diagram perturbation theory for N= 1 supergravity might be finite at all orders beyond one loop.

Usually, q u a n t u m supergravity is treated in terms o f scattering theory using a path-integral approach. However, the full q u a n t u m theory can instead be treated non-perturbatively by studying the q u a n t u m constraints acting on the wave functional [ 1 ], i.e. by taking an approach based on functional differential equations. This approach is used here to obtain bosonic physical states o f the theory. One finds that (e.g. for three-geometrics defined on S 3) there are at least two purely bosonic wave-functions in N = 1 supergravity, which have the simple form const..exp ( - I / h), where I is an action functional o f the three-metric, corresponding to the bosonic part o f the HartleHawking [ 2 ] and wormhole ground state [ 3 ]. For each bosonic solution, there is a corresponding solution at the level filled with fermions, proportional to e x p ( H h ) . The case o f general fermionic states is left for further investigation. A wave function can be taken to be o f the form ~ ( e ~ ' i ( x ) , ~A,.(x)). Here, using two-component notation [ 1 ], eaA'i(x ) is the spatial tetrad, which gives the three-metric as h0-= - e ' ~ ' i e ~ v, and (C/A, ~ ' ~ ) is the spatial gravitino field, taken to be an odd Grassmann quantity. The wave function can equivalently be described by ~ ( e ~ ' i ( x ) , ~/A'i(x) ), which is related to ~ b y a fermionic Fourier transform [ 1 ]. 368

A physical wave function must obey the Lorentz and supersymmetry constraints JaS~=0, Sa~U=0,

JA"'~=0, SA'~U=0.

(1, 2) (3, 4)

The remaining Hamiltonian constraints ~ , 7t=0 will be implied by the above constraints, if we define ~AL4' to be given by the anti-commutator o f SA and Sa, [ 1,4]. The Lorentz constraints imply that ~ is invariant under local Lorentz transformations applied to the arguments ( e ' ~ ' ; ( x ) , ~/Ai(x)). Thus all Lorentz indices must be contracted together in ~. All commutators with j a n or fA,B, as one argument close on constraints. The factor ordering o f ~a, is given naturally by eq. (7) below, which describes the transformation property of the wave function under a local supersymmetry transformation parametrized by ~m(x); there is no factor-ordering ambiguity [ 1 ]. Similarly, S A is given naturally by eq. (14) below. These generators obey the relations [ 1 ]

[SAx),S,(y)]+ =0,

(5)

[~a,(x), S,,(y) ] + = 0 .

(6)

The commutation relations involving -,~,o, and Ss, ~a, or ~ s , have not been evaluated. However, our

0370-2693/94/$ 07.00 © 1994 Elsevier ScienceB.V. All fights reserved. SSDI 0370-2693 (93)E1483-E

V o l u m e 321, n u m b e r 4

PHYSICS LETTERS B

aim is to find solutions of eqs. ( 1 ) - ( 4 ) and ~ , ~ = 0, and it is clear that all (anti-) commutators such as [ oafaa,, Sn] - will close weakly by annihilating any state ~Fwhich obeys eqs. ( 1 ) - ( 4 ) . Hence we restrict attention to solving eqs. ( 1 ) - (4). The constraint ,~A,~ = 0 reads

~ijke4A,i(aSDjqlAk) ~.t ½~lX2q/Ai ~,t

=0.

(7)

Here 3~Djis the three-dimensional covariant derivative on spinors without torsion [ 1 ], and xz = 8n. The constraint (7) is to be interpreted as

3 F e b r u a r y 1994 /.

¢~I---- 2i [ d3x EiJk~A'eaA,i(3SDj~Ak) . ?¢ d

(10)

Then the path integral between the initial and final data, formed by summing terms of the form exp ( - / / h), gives back the transformation law (9) under supersymmetry. This again confirms that ~ transformed under supersymmetry has the form ~1 ( e ~ ' i ,

~uA,, ~A').

The constraint SA ~ = 0 reads

(11)

~ d 3x e~JkgA'e~,~( 3q~j~'4k) ~ where

~ =0, -½hx2~d3xE -A'~,Ai8--~-7i

(8)

DBA'ji ---- - - 2ih - l/2eBB,iecB,jn cA,.

(12)

for all odd ~A'(x). Equivalently, this gives the change under an infinitesimal supersymmetry transformation Je~'~ = - i/¢~"l'q/"li, dh//A i = 0 ,

Here h = det (h o) and n AA'is the spinor version of the unit future-pointing normal n u to a surface x°=constant. This is defined as a function of the eAA'i by

-2i O~= ~ ~ dax

nAA'eAA,i=O, n~'nAA,= 1 .

EijkgA'e.4A,i(asDj~/Ak)~.t.

(9)

It might, at first sight, seem odd that j~u should not depend on the variables ga, and q/A through their product ga,q/A, as they appear in the argument of ~F(e~, _ixgA'q/~, ~A ) [5 ]. But note that one can take the odd quantities gA' and ~ ; to depend on linearly independent Grassmann elements. Then, from knowledge of the quantity g~'q/A and of the argument q/A, one can deduce gA'. Hence the infinitesimally varied quantity ~ ( e ~ ' i - i x g ~' ¥~, ~ ) can equivalently be written in the form ~ (eAa'i, ~/~'~, ga,), and the expression (9) for J ~ m a k e s sense. This shows that there is no contradiction involved in Page's example [ 5 ] gA"--~e~gA', ~bcAi'*e--~qlAi . One can further see how the transformation (9) or differential equation (7) arise, by considering the transformation law for the action I of fields filling in between data given on two boundary surfaces; see section 5 of [ 1 ]. Suppose that (eaa';, ~ a ) are given on the final boundary surface of interest. One applies an infinitesimal supersymmetry transformation at the final surface with parameter gA'. Then [ 1 ] the change in I (whether or not I corresponds to a classical solution) under supersymmetry JeAA'; = --ixgA' ~Uai, J ~ = 0 at the final surface, is

(13)

The Sa constraint is more easily understood in the representation ~(eAA'~, ~A,), where it reads

~ijkeAA'i(3SDj~IA'k) ~'J¢"½hlC2~lA'i 5 e~y.t ~, ' =0.

(14)

Note that the supersymmetry constraints do not mix fermion number. Hence one can study the states containing different numbers of fermions (finite or infinite) separately, de Wit et al. [6 ] have studied physical states in d = 3, N = 2 supergravity, which has some similarities to the present model (and some differences from it). They found that physical states for d = 3, N = 2 must contain an infinite number of fermions. There is a somewhat similar situation in free-field spin -3 theory in four dimensions [ 1 ]; one must again have an infinite product of fermions there. However, as remarked in section 4 of [ 1 ], the structure of the constraints in the full theory should allow very different solutions from the free theory. As we shall see below, there are indeed purely bosonic physical states in the d = 4, N = 1 theory, very different from the solutions of the free theory. The bosonic physical quantum states (for d = 4 , N = 1 ) are described differently, depending on the topology, such as Sa or R3, of the spatial three-surface. 369

Volume 321, number 4

PHYSICS LETTERS B

Take first the case of S3. First consider the classical bosonic solution which fills in, according to the vacuum Einstein equations, between the specified e~'~(x) on the given surface and a large round Sa at infinity. The action I[e'~'i(x) ] gives the leading approximation to the ground wormhole quantum state [ 3 ] through In ~,,, ( - I / h ) . In supergravity, we specify ( e ~ ' i ( x ) , ~ua~(x) ) on the boundary. Let us choose #A, = 0, ~a'o = 0 for the infilling fields. Then there is no fermionic contribution to the supergravity action, which remains I [ e ~ ' , ya~] =I[eaa'i], provided that we take the four-dimensional geometry to be the infilling Einstein solution. Note that the infilling field, however y a is chosen, will in general not be a solution of the classical supergravity field equations, since it is in general impossible to solve for ya~ the constraint

g,t,----~iJkeAA,i(3SDj~lAk)-l-½ix2~"lip.4A,i=o

(15)

given eAA,~ and p~,~; this does not matter. As described earlier, the action I transforms under supergravity at the boundary surface according to eq. (10). Hence ~ = exp ( - i / h ) obeys the transformation law (9), or equivalently the constraint ~A,~=0 [eq. (7)]. The constraint SA~ = 0 [eq. ( 1 1 ) ] is automatic for ~u= exp ( - i / h ) , with I(e, ¥) = I [ e ] , since it involves the derivative 8~/8¥A~=0. Note that, in order to show that ~v= exp ( - i / h ) defines a physical state, one only has to evaluate the constraint generators acting on it, in the case that the argument eaa';(x) is complex hermitian, and ~A~(x) is odd Grassmann. (E.g. it is expected that the inner product will be defined using wave functions with these arguments. Similarly, the path integral is normally defined with such boundary data.) There is no need to consider here the wave functions qJ(e'~'j--il¢,~A'~A i, ~/A i), acted on by constraints. Hence ~ V = e x p ( - I / h ) obeys all the constraints, and so gives a physical state. One might ask more precisely for the relation between the bosonic physical state ~ = c e x p ( - I / h ) and the ground wormhole quantum state [ 3 ]. Assuming that the path integral for the ground wormhole state ~wh exists and obeys the constraints, one can study the asymptotic expansion at the bosonic level:

( ~wh)bo.oni~ ~ (Ao +hAL +h2A2 +...) exp( 370

-I/h).

(16)

3 February 1994

Now a given action I leads to a unique prefactor Ao, and iteratively to a unique An, for all n (e.g. by consideration of the path integral). Hence Ao=c, A~ =A2 . . . . . 0, and c e x p ( - I / h ) is the bosonic part of the wave function for the wormhole state, up to possible exponentially small corrections. Similarly, in the case of S3, one can study the action I given by filling smoothly inside the S3. This leads to another bosonic state ~ = c o n s t . . e x p ( - I / h ) , corresponding to the bosonic part of the HartleHawking state. On R3, the ground state of supergravity is defined (assuming the path integral makes sense) by integrating over all fields which tend to flatness, with spin3 field-.0, for large negative imaginary time [2]. There is a corresponding bosonic action/, which leads to a bosonic physical state const..exp(-I/h) of quantum supergravity. Assuming always that the path integral is well defined, the bosonic part of the ground state should agree with c exp ( - I / h ) up to possible exponentially small corrections. The preceding discussion refers to purely bosonic states 7'(e'~',-(x) ) containing no fermions. One can also examine a "filled" state in which all fermions are present. In the representation ~( e~'i(x), ~/A'i(x) ), this corresponds to the #o part of the wave function. The form of the SA~=0 constraint [eq. (14) ] leads to the solution ~=const..exp (Uh),

(17)

where I is the wormhole or Hartle-Hawking action for S3, or the ground state action for ~3. Exponential solutions of the type const..exp( +I/ h) have previously been found in mini-superspace examples, where supergravity is quantized subject to the Friedmann k = + 1, Bianchi I or Bianchi IX Ansatz [7-11 ]. Note also that a semi-classical Chern-Simons state has been found by Sano and Shiraishi [ 12 ] for supergravity with a cosmological constant, using the Ashtekar representation. It is unfortunately hard to compare with the present approach, based on the variables (e~'~(x), ¢Ai(x)), since a functional integral is needed in transforming from Ashtekar variables to these variables. However, it seems that one should be careful with the interpretation of the Chern-Simons state, since there are no solutions of the constraints

Volume 321, number 4

PHYSICS LETTERS B

for the supersymmetric Bianchi I mini-superspace model with A-term [ 13 ], and since the functional integral above becomes an ordinary integral there. One might expect there to be a correspondence between the results o f this paper and the results o f perturbation theory about curved background four-dimensional spaces. For example, one can look for the one-loop correction to the wormhole ground state, where one sets the boundary data to ( e ~ ' ; ( x ) , 0) and integrates over the linear fluctuations o f the gravitational and gravitino fields about the classical infilling solution between the surface and infinity. Using zetafunction regularization [ 14 ], one finds the prefactor Ao in eq. (16) to depend on the Euler n u m b e r Z [ 15 ] through an unknown mass parameter/z, and also on the boundary data (or equivalently the classical fourgeometry) through terms o f the form ( ' ( 0 ) for the various differential operators involved. The first contribution is simply to an overall constant rescaling o f Ao; the second introduces some dependence on boundary data into Ao. If the wormhole state is to be c exp ( - 1 / h ) , then such boundary data-dependent corrections must be absent. One m a y conjecture that the structure is similar at higher loop order, so that (e.g.) the three-loop divergence [ 15 ] would be absent as would all higher-order divergences, while no finite correction to the semi-classical amplitude should appear. A calculation o f the full finite non-loop correction in supergravity is practically impossible-it would in principle involve very complicated non-local effects. Feynman-diagram calculations can be done practicably at one loop, to investigate whether there is an finite correction, by considering one-loop diagrams with only external graviton legs. An infinite n u m b e r o f diagrams, with different numbers o f external legs, would need to be considered in order to find the full correction. As yet, the finite part o f a one-loop diagram in N = 1 supergravity has not been c o m p u t e d [ 16 ]. With regard to F e y n m a n diagrams in supergravity, one should bear in mind that the (in and out) scattering data correspond only to asymptotic "states", obeying the q u a n t u m constraints o f the linearised theory. These asymptotic "states" are not true states as they do not obey the full q u a n t u m constraints o f supergravity, and hence do not have (e.g.)

3 February 1994

the forms found in this paper. [A similar feature is present in gluon scattering in Yang-Mills theory, since the div E constraint is non-linear. This problem is not restricted to gravity. ] Nevertheless, the results here do give an indication that Feynman-diagram perturbation theory for N = 1 supergravity might be finite at all orders beyond one loop. The special form e x p ( + / / h ) o f the solutions to the quantum constraints found here may be a feature only o f N = 1 supergravity, or perhaps also ofhigher-N supergravity without couplings. When one turns to supergravity coupled to supermatter, it is immediate that m a n y more solutions to the constraints are possible, essentially because the wave function ~ depends on more fields, but does not have to obey any more constraints than in pure supergravity. This is already evident from a study o f the Friedmann sector [ 11 ]. The special form o f the solutions found here may be a property of pure supergravity only. It will also be interesting to examine fermionic states in the N = 1 theory.

References

[ 1] P.D. D'Eath, Phys. Rev. D 29 (1984) 2199. [2] J.B. Hattie and S.W. Hawking, Phys. Rev. D 28 (1983) 2960. [ 3 ] S.W. Hawkingand D.N. Page, Phys. Rev. D 42 (1990) 2655. [4] C. Teitelboim, Phys. Rev. Lett. 38 (1977) 1106. [ 5 ] D.N. Page, University of Alberta preprint. [6] B. de Wit, H.-J. MatschuUand H. Nicolai, Phys. Lett. B, to appear. [ 7 ] P.D. D'Eath and D.I. Hughes, Phys. Lett. B 214 (1988) 498. [8] R. Graham, Phys. Rev. Lett. 67 ( 1991 ) 1381. [9] P.D. D'Eath, S.W. Hawking and O. Obreg6n, Phys. Lett. B 300 (1993) 44. [ 10] P.D. D'Eath, Phys. Rev. D 48 (1993) 713. [ 11 ] P.D. D'Eath and D.I. Hughes, Nucl. Phys. B 378 (1992) 381. [ 12] T. Sano and J. Shiralshi, University of Tokyo preprint UT622. [ 13] P.D. D'Eath, Phys. Lett. B 320 (1994) 12. [ 14] S.W. Hawking, in: General relativity, an Einstein centenary survey, eds. S.W. Hawking and W. Israel (Cambridge U.P., Cambridge, 1979). [15] M.J. Duff, in: Supergravity '81, eds. S. Ferrara and J.G. Taylor ( Cambridge U.P., Cambridge, 1982). [ 16 ] P. van Nieuwenhuizen, personal communication.

371