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UNIFICATION AND EXTENDED SUPERGRAVITY
B. de Wit, H. Nicolai
To cite this version:
B. de Wit, H. Nicolai. UNIFICATION AND EXTENDED SUPERGRAVITY. Journal de Physique
Colloques, 1982, 43 (C3), pp.C3-310-C3-316. �10.1051/jphyscol:1982361�. �jpa-00221916�
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CoZZoque C3, suppZZment au n o 12, Tome 43, dZcembre 1982 page C3-310
U N I F I C A T I O N A N D E X T E N D E D S U P E R G R A V I T Y
B. de W i t and H. ~ i c o l a i *
NIKHEF-If, S e c t i o n H, P.O. Box 41882, 1009 Amsterdam, The NetherZands
*TH ~ ~ i i s i o ; , CEEZ, CH-1211, Geneva 23, SwitzerZand
It is well-known that supersymmetry has important implications for a possible unification of elementary particles and their interactions. Supersymmetry describes fermions and bosons in a unified way as partners of a supermultiplet. Such multiplets necessarily have a decomposition in terms of boson and fermion states of different spins. If one assumes N independent supersymmetries then a supermultiplet covers a range of spins of a t least fiN/4. Therefore, when there are more than two supersymmetries it is no longer possible to restrict oneself to field theorles of orily spin-h and spin-0, but one must include spin-1 fields which are usually described by gauge fields. In this way gauge and matter fields are forced to occur as partners of a common supermultiplet, and are no longer treated separately. In addition the invariance under supersymmetry implies that the theory must be invariant under translations, because the anticommutator of the supersymmetry charges is proportional to the momentum operators which are the generators of space-time translations. Hence if supersymmetry transformations are local then the theory must be invariant under local trandations, which describe arbitrary reparameDizations of space-time w i t h gravity as a corresponding gauge theory. Therefore, local supersymmetry implies gravity, so that it is no longer possible to ignore gravitational interactions a t w i l l , since those have become an intrinsic part of the theory in question. An even larger degree of u d c a t i o n is achieved if we have more than four supersymmetries. In that case supermultiplets can no longer be restricted to states of spin s
2
1, but one must a t least allow spin-3/2. Such fields are usually described by Rarita-Schwinger fields which act as gauge fields for local supersymmetry. Hence adding extra multiplets to the theory w i l l not correspond to coupling matter multiplets to supergravity, but instead leads again to a pure supergravity theory but now based on more local supersymmetries. In the context of N 4 extended supergravity it i s thus not possible to have separate matter multiplets.There are good reasons to restrict oneself to supergravity with states of spin not higher than two. The most important one is that it seems impossible to have a consistent description of massless higher-spin particles coupled to gravity. According to the previous arguments for supermultiplets the largest possible supergravity theory i s then invariant under N=8 independent supersymmetries. There i s an alternative way of expressing this result. Namely the number of supersymmetry generators is also increased if one considers s u p e r g r a m in a higher space-time dimension, because spinors in higher dimensions have more components. In this way four- dimensional N=8 supergravity corresponds to eleven-dimensional N=l supergravity, where the 8 four-component supersymmetry generators are combined into a single 32-dimensional spinor.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982361
B. de W i t e t a l .
In these talks we concentrate on the largest supergravity theory mainly in the context of four dimensions. Let us first sketch the main features of N=8 supergravity. The states that are described by the classical theory are given in the table below and constitute a supermultiplet w i t h spin extending from 0 to 2. Besides the spin-0 and -$ fields we distinguish 28 gauge fields of the group S 0 ( 8 ) , 8 gravitini fields which are the gauge fields of 8 local supersymmeties, and the vierbein field which may be regarded as the gauge field of the general coordinate transformations. According to the table these fields can be assigned to the representations of two internal gauge groups, SU(8) and SO(8). The 70 scalar fields live on the coset space E7/SU(8) and are described by a 56x56 matrix
v
which is an element of the noncompact E7-groupwhere the indices of the 28x28 submatrices u and v are antisymmetric index pairs [ij], [IJ], etc. Under SU(8)xSO (8) the matrix (1) transforms as follows
V (x) + I, (x) V (x) o - ~ ( x )
.
U(x) E SU(8),
O(x) E SO(8),
(2)where the matices U(x) and O(x) are in the appropriate representations. Since the matrix represents 133 degrees of freedom and SU(8) has 63 generators one may use the local SU(8) invariance to fix a gauge and therefore recover the 133
-
63 = 70 scalar degrees of freedom.Let us now discuss the various symmetries under which the N=8 supergraviv action is invariant.
First of a l l we have 8 local supersymmetries and general coordinate transformations w i t h New- ton's constant as the relevant coupling parameter. Then there is a local SU(8) symmetry, which has no independent coupling constant because the SU(8) gauge fields have no kinetic t e r n s in the
Table: States of N=8 supergravity with the corresponding fields assigned to representations
-
of SU(8) and SO(8).
s t a t e s f i e l d s S U ( 8 ) S O ( 8 )
1 graviton e a 1 1
I'
8 gravitini
q
8 1JOURNAL DE P H Y S I Q U E
action. If kinetic terms are generated by quantum effects then those determine the S U ( 8 ) coupling constant dynamically. Finally the theory is invariant under local S O ( 8 ) transformations governed by an independent gauge coupling constant g. If g is equal to zero then the nonabelian S O (8) reduces to 2 8 independent U (1) groups, in which case the field equations are invariant under a noncompact E7 group of duality transformations. These transformations rotate the electric and magnetic components of the field strengths F
P" IJ = 8 I P ~ V ]IJ in a nonabelian fashion. In addition they act on the scalar fietds.
If gfo, then the lagrangian and transformation rules acquire additional terms which are parametrized i n terms of a remarkable S U ( 8 ) tensor
which has a number of surprising properties. For example, T can be decomposed as follows
Here, the first term on the right-hand side is antisymmetric in the upper indices [jkR] and traceless, and therefore corresponds to the representation of SU ( 8 ) ; the second term, Tmimj, is symmetric under interchange of the indices i and j and therefore belongs to the of S U ( 8 ) . To prove this and other properties of the T-tensor, one uses the fact that the matrices u and v constitute an element of E 7 according to (1).
Both N = 8 supergravity and its eleven-dimensional counterpart are completely known; the invariant actions and the full nonlinear transformation rules have been constructed by iterative methods, and by arguments based on dimensional reduction from 11 to 4 dimensions. Alternatively, an intrinsically four-dimensional derivation of the complete theory is possible; the key ingredient in that construction is the E 7 / S U ( 8 ) structure of the theory which allows one to disentangle the otherwise unmanageable norilinearities of the theory. There are also versions of the theory that exhibit spontaneous breakdown of supersymmetry; these are conveniently obtained via dimensional reduction.
There is hope that supergravity is a consistent quantum theory of gravity. A s is well-known gravitational field theories are not of the renormalizable type, and therefore their quantum- mechanical consistency requires the complete absence of ultraviolet divergence. The balanced decomposition in fermions and bosons provides the crucial ingredient for taming the divergences.
It is known that the 1- and 2-loop divergences in the pure supergravity S-matrix vanish. For N = 8 it has been shown that the Green's functions are finite up to 7 loops in the context of an extended-superspace formulation. I n any case there exist invariant counterterms in N = 8 supergravity, so that one faces the difficult task of proving that all the corresponding mergences cancel. The situation is not hopeless in view of the many so-called nomenormalization
B. d e W i t e t a l .
results that have already been found in supersymmetry and supergravity. One of them whirh is relevant within the context of these talks, is the fact that the SO(8) gauge coupling constant g does not require an infinite renormalization. This result has been established in 1 loop and there are arguments that it should hold to all orders.
Hence supergravity theory brings us back to the old trail of finite field theory. Therefore, if supergravity is the underlying theory of Nature, it provides the ultraviolet cut-off for particle physics which is therefore no longer an arbitrary parameter a s in the context of renormaJizable field theories. If broken supergravity leads to a hierarchy of interactions both the hierarchy and the effect of radiative corrections will then be uniquely given, and unless the hierarchy is explained as an accidental result that emerges entirely from radiative effects, one is forced to assume that these effects depend only weakly on the cut-off. Therefore the so-called fine-tuning problem becomes more than a matter of aesthetics in this context. However, it is then conceivable that residual supersymmetry effects are sufficient to ensure the weak dependence of the cut-off. A t that point t h i s scenario w i l l make contact with present attempts to construct realistic grand-unified theories based on rigid supersymmetry, although there remain some rather crucial differences. One of them concerns the breaking of supersymmetry which is not as strongly restricted as in rigid supersymmetry. A second aspect is that if the mass scale of supersymmetry breaking is large (> 10 10 GeV) then supergravity couplings, which are inversely proportional to the Planck mass, can no longer be ignored. However, according to the arguments outlined above, the induced effects must remain small. In other words the breaking must be soft in the technical sense such as not to affect the nature of the cut-off dependence.
The fact that supersymmemy must be broken introduces a new mass scale in the problem which may also explain the existence of hierarchies. Assuming that supersymmetry is broken a t a mass scale m , which is independent of the Planck mass mp, hierarchies could emerge through radiative corrections induced by supergravity couplings. The hierarchy structure results from a different cut-off dependence of various sectors of the theory expressed in powers of m / m In that
P'
context it is important to realize that this field theory i s not of the renormalizable type, and that the cut-off is set by the supersymmetry breaking mass rather than by the Planck mass.
Some of these scenarios have been considered recently i n the context of N = l supergravity coupled t o matter, but we remind the reader that only pure supergravity theories have the possibility of being finite. Therefore those attempts should be interpreted as possible truncations of an extended supergravity theory. However, one of the most outstanding problems is to make contact between extended supergravity and the known "low-energy" theories of quarks and leptons.
The most direct attempt to exhibit a relation between N=8 supergravity and elementary particle phenomenology is to identify the supergravity fields with those of the known constituents of matter. This approach is doomed to fail, however, since the spin-$ representation when decomposed according to the phenomenologically relevant gauge group SU ( 3 ) XSU ( 2 ) XU ( 1 ) does not
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lead t o any realistic assignments. The most obvious problem is that the colour assignments a r e not vectorlike, so that the resulting theory of strong interactions would be parity violating. An alternative approach is t o envisage N=8 supergravity a s a compactification from eleven-dimensional supergravity w i t h the seven extra compactified dimension parametrizing a certain manifold. Such a compactification w j l l i n general lead t o a breakdown of supersymmetry a t low energies, and one may regard this a s a welcome effect. However, the f a c t that Nature prefers one groundstate over another must be explained, and it is not clear a t the present time according t o which principles a groundstate should be selected. One p o s s i w t y is t h a t the groundstate should obey the eleven-dimensional field equations ( w i t h the fermions being put equal t o zero), i n which case there are not many solutions available. For the most interesting solution the internal manifold is the seven-sphere S 7
,
and one expects that this solution will eventually be shown t o be equivalent t o gauged N=8 supergravity in four dimensions; the peculiar geometric properties of S 7 (parallelizability) will then certainly play an important role in further developments of the theory. At any r a t e , the viability of the scheme w i l l crucially depend on the low-energy fermion spectrum. Also in this case there appears t o be a serious problem: how can one avoid a vectorlike theory in such a reduction?It seems therefore that the only remaining viable scenario should be based on composite states. A standard approach is t o embed the SU(3)xSU(2)xU(l) group of strong and electroweak forces in the chiral SU(8) group, and t o conjecture that bound states form supersymmetrically. Hence one considers one or several supermultiplets of bound states, which f a l l i n specific representations of SU(8). Somewhat inconsistently one assumes that most of the original states of supergravity (preons) a r e confined, with the exception of the graviton. The a i m is then t o find a "realistic"
subtheory of the renormakable type, i n such a way that the remaining bound states a r e r e a l under a t l e a s t SU(3)xU(1) so that they can acquire large masses and become irrelevant for present phenomenology. However, it seems impossible t o find any separation into a massless and a massive subtheory which is sufficiently realistic on the basis of any finite number of supermultiplets. Hence, one may be forced t o assume that the remaining states were never bound i n the f i r s t place, which casts doubt on the central assumption underlying this approach.
There is still the alternative possibility that one has to use an infinite set of supermultiplets, which may not be unnatural when one contemplates possible implicalions of the noncompact E7 invariance. Classically this symmetry is realized norilinearly on the physical states, but when bound states are generated those may transform linearly under the E7 group. Indeed one could argue on general grounds that the E7 invariance of the underlying dynamics should become manifest for the spectrum a t a certain energy scale. However, in that case one must have an infinite set of states because dl nontrivial unitary representations of noncompact groups a r e infinite dimensional. The f a t e of SU(8) is not immediately obvious in such a scenario. Classically it becomes entangled with the nonlinearly realized E7 transformations, but whether t h a t remains so f o r the full theory is a dynamical issue. It is conceivable therefore that SU (8) remains independently relevant for the particle spectrum. This aspect is crucial if one wishes t o attach a
B. de W i t e t a l .
special meaning to SU(8) a s the local gauge group that contains the gauge group of strong and electroweak interactions. It also deserves attention when E7 duality invariance is not exact, for instance because of anomalies. The infinite degeneracy of states will then disappear, and one
may wonder whether all bound states remain present.
The most obvious way to break the E7 invariance is by switching on the SO(8) gauge coupling constant. The phenomenological relevance of SO(8) i s , however, not clear. We have suggested in the past that the SO(8) gauge interactions provide the force for Ending the preons. This ldea has a number of interesthg consequences. It leads to a well-defined preconfinement criterion which has been lacking so far. Since only part of the preons is confined, this leads to supersymmetry breaking a t a mass scale which i s a priori independent of the Planck mass. In this scenario E7 is no longer relevant, because SO(8) confinement implies that physical states should also be neutral with respect to E7. We have already mentioned that the SO(8) @-function may vanish to a l l orders, so that the confinement must be of a nonperturbative nature. The zero @- function could also be viewed as an indication that confinement could persist a t a l l energy scales and become permanent. Therefore it appears that the theory w i t h local SO(8) invariance differs rather drastically from the theory w i t h g=O.
If g#O, the theory contains a scalar field potential which is given by
This potential is not bounded from below, and this annoying feature represents a serious problem since it allows arbiaarily large quantum mechanical fluctuations to occur. In the context of constructing a well-defined functional integral this problem has been encountered before: the Einstein action itself i s not bounded from below. A surprising new result is that an anti-de Sitter background can be stable in the presence of an unbounded scalar potential under certain conditions and within natural boundary conditions. In this anti-de Sitter background the scalar fields are evduated around a constant value where the potential has a local exmemum or saddle point. Such stationary points of the potential (5) are governed by a complicated equation; namely the fully antisymmetric part of
must be an anti-selfdual tensor
where =
+
1 is a duality phase present i n N=8 supergravity.JOURNAL DE PHYSIQUE
WMe these r e d s may be sUll insufficient to completely resolve the problem, one may speculate that the potential becomes stabillzed through quantum effects just a s in the case of the hydrogen atom. Assuming that these difficulties will be overcome i n some way or another, one recognizes that this potential provides a beautiful and powerful method t o break both the supersymmetry and the local SO(8) and/or SU(8) invariances. This is so because the T-tensor may acquire a nonvanishhg vacuum expectation value
in wbich case a super-Higgs effect may take place (it is remarkable that the m a x i m a l residual symmetry group of (8) contains a chiral SU(5)). Observe that the potential is uniquely fixed by the requirement of local supersymmetry, so that neither the choice of Higgs representations nor the Higgs couplings are any longer arbitrary a s in conventional grand unjfied theories. Moreover, the b t e n t i a l reveals an unsuspected and very rich structure as one goes to higher extended s u p e r g r d t i e s . While, for N=4, there is only a trivial stationary point, already the N=5 theory exhibits nontrivial s i a H o ~ a q c & t s a t which supersymmetry is broken. Curiously, the N=5 theory also has the feature that the "bad" configurations constirute a set of measure zero. For the N=8 theory, we expect further improvement and the existence of nontrivial stationary points. It would be interesting t o see whether these completely break N=8 supersymmetry or whether one or more supersymmetries survive. Only i n the l a t t e r case one could be able t o make contact w i t h current attempts to construct grand unjfied theories w i t h rigid or local N=l supersymmetry.
We conclude that supergravity and i n partkular the N=8 theory has a rich structure, which may give rise t o an attractive scheme for unification of particles and their interactions w i t h unique features. Although there a r e many ideas on how to make use of these theories, there is a rather serious lack of dynamicat input. It may be that supergravity is the only viable framework for extending unification to the Planck scale, either as an effective field theory based on what happens beyond that scale, or a s the ultimate physical theory. Therefore it is important t o d v e it.
References
N=8 supergravity is discussed in B. de W i t and D.Z. Freedman, Nucl. Phys. (1977) 105;
E.
Cremmer and B. Jdia, Phys. Lett.9
(1978) 48, Nucl. Phys. (1979) 141; B. de W i t , Nucl. Phys.B158
(1979) 189; B. de W i t and H. Nicolai, Phys. Lett. (1981) 285, Nucl. Phys. B.,
to be published. The last work contains many relevant references.For supergravity i n 11 dimensions, see E. Cremmer
,
B. Julia and J. Scherk, Phys. Lett.2
(1978) 409.
Further references may also be found in B. de W i t , "N=8 Supergravlty", i n proc. of the John Hopkins Workshop on Current Problems i n Parti.de Theory 6 , Florence 1982, and i n the contributions of S. Ferrara, M
