Counterterms in extended supergravities

Volume 99B, number 2

PHYSICS LETTERS

12 February 1981

COUNTERTERMS IN EXTENDED SUPERGRAVITIES ILE. KALLOSH

Lebedev Physical Institute, Moscow, USSR Received 3 November 1980

To the memory o f Felix Berezin Geometrical invariants respecting all necessary symmetries of the theory are shown to exist, starting from the 8th (4th) loop approximation in N = 8 (N = 4) on-shen supergravity. 3-loop counterterms are presented on a linearized level for N = 4 and N = 8 theories.The corresponding 3-1oop non-linear invariants are discussed.

There is a considerable reduction o f ultraviolet divergences in supersymmetric theories. In supergravity this reduction is connected, firstly, with the vanishing o f some superinvariants on shell, as it occurs in the one-loop approximation in supergravities with N = 1, 2 [1]. Secondly, it is connected with the fact that not all gravitational invariants have supersymmetric partners, as e.g. in the two-loop approximation in the N = 1 theory [2], where the gravitational invariant looks like k2Vrg-(Ruvxo) 3, and k is the gravitational coupling constant. However, in the threeloop approximation in supergravities with N = 1, 2 supersymmetric invariants were pointed out, which generalize the term k 4 x / ~ ( R u v x o ) 4, and do not vanish on shell [3]. These invariants also have been expressed in terms of linearized superfields [4,5], full non-linear superfields [6], and also have been expressed in terms o f geometrical objects of the theory (like torsion and curvature tensors) with the help of integrals over the whole supermanifold [ 7 ] Let us call the following the standard geometrical invariant in extended supergravity:

=

P 1 f dV22 (RMNPQ , T~IN) ,

(1)

where 22 is some scalar, constructed from curvature and torsion tensors in the tangent space. The integration in (1) goes over the whole supermanifold with invariant volume d V = d4x d4NO Ber E, where the berezinian E is a superdeterminant o f the vielbein, and I is the number o f loops. Dimensional considerations show that dim Z? = 4 - 2N + 2 ( / -

1).

(2)

Taking into account that 22 contains only torsion and curvature tensors one can easily verify that dim22~> 2, i.e. l/> N .

(3)

It follows from (3) that the standard geometrical for the N = 3 supergravity, from a 4th loop for N In an important paper of Brink and Howe [8] shell N = 8 theory were expressed, b y solving the the components o f the torsion * 1,

invariants like (1) give contributions beginning from a 3rd loop = 4 and from an 8th loop for N = 8. all components of the superspace curvature and torsion in the onBianchi identities, through one superfield leg.k, which is one of

,1 We use throughout the paper the notations of ref. [8], where in the tangent space all indices: MNPQ, spinor and internal: ABCD, spinor: abcd ... db~d, internal SU (N): ijk...t, vector: uo .... The corresponding indices in curved space are denoted by Q,~c~ ; ~; ~...; ~t3...; ~...; ~,v.... 122

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Volume 99B, number 2

PHYSICS LETTERS

12 February 1981

T b~ .a ijk = e ~ Wi/k" a

(4)

Thus is follows that in the N = 8 theory the standard geometrical invariants like (1) do not exist up to the 8-loop level but appear in all loops beginning from the 8-loop level, e.g. for 1 = 8

s~28=k14f

d4x d320 BerE(Wi~k~Ok) 2 = k 14 f

d4X(RabcdDu . . . D u s R d ~ d ) 2 + ....

(5)

The first geometrical invariant in the N = 4 theory is the 4-loop counterterm S/--=44 = k 6 f d4x d160 Bet E ( W / I ~ / ) 2 ,

(5')

where

~dl = ~ ,pi/k d ebe Clijk~ bc "

(4')

Superinvariants like (1), and in particular (5) and (5') respect all on-shell symmetries of the theory, i.e. general covariance in superspace and tangent space symmetries, internal [e.g., local SU(8) in (5)] and Lorentz symmetry. In extended supergravities some geometrical invariants different from (1) may also exist since in superspace we have a "constrained" geometry, when some components of the torsion are equal to zero or to their flat space value. This is a consequence of the existence of some prepotentials from which all geometrical objects are constructed, as is known in detail in the N = 1 theory [9]. The simplest example of such invariants are the integrals over the left or right superspace from chiral fields like d4x d20 or d4x d20" [9]. In the Yang-Mills supersymmetric theories the most interesting example is the SU(4) theory [ 10] where

s= f

d4x Di75Dj Tr(X/75Xt),

i.e. instead of integration over d160 we have an integration only over d20. In what follows we try to use the linearized approximation to find the candidates for the three-loop counterterms. Let us note that extended supergravities N ~< 4 [11 ] on shell can be described by means of the following linearized chiral superfields ,2 In the theory with N = 1 there is a tree-component spinor superfield [5] (two-component notations are used)

Wabc : ~abc + OdRabed ,

(6)

where ~babc (Rabcd) is the spin 3/2 (2) field strength. For the N = 2 theory we have

w t,

=

+ Oic ;q)abc + (1/2!)oidofeqRabcd ,

(7)

- .

Fab being the spin 1 field strength. For N = 3 the lowest component of the spinor $uperfield is the spin 1/2 field Xa: = + i . . . . Wa Xa OibFab + (1/2!)OibOjcetlk~kabc + (1/3!)OibOjcOkdetlkRabcd •

(8)

Finally for N = 4 the lowest component of the scalar superficial is the scalar field qs:

W = dp + Oia Xi + (1 ]2 !)OiaOjbFffb + (1/3 !)OiaOjbOkc effkl~ l abe + (1/4!)OiaOjbOkeOldeffklRabed .

(9)

In the theories with N ~< 4 at the linearized level on shell we have the following three-loop counterterm: l=3

=f d4x

2 = f d4x (RabccZff'h]~bcl)2

+

....

(10)

,2 All dependence on k is absorbed by an obvious rescaling of spin 0, 1/2, 1, 3/2 fields. 123

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12 February 1981

In the theories with N = 1, 2, 3 expression (10) has a natural geometrical (i.e. fully non-linear) generalization. For example, in the N = 3 theory we have

S~3=3= f d4xdl20 BerE(Wal~d) 2 ,

(11)

where

1~d = ei/keabTdab ilk ,

(12)

i.e. the linearized volume is replaced by the full one, and the linearized superfield I~d is replaced by the full one which is expressed through the torsion tensor in (12). In the theory withN-- 4 it is non-trivial to pass from (10) to geometrical objects, as was done in (11) and (12), since the scalar superfield W of zero dimension is not connected with the torsion or curvature which have dim ~> 1/2 (except for the zero dimensional TUg = %u~) as follows from their definition by means of the commutator of two covariant derivatives. Therefore we shall try to transform the expression S~__34 = f d 4x d160 (WI~) 2

(13)

to a form fike

f d4x(dl20)~l (DiW) (DiaW) (Dkd W)(Dld W),

(14)

since at a linearized level

n w=w ,

(15)

W/ being defined through the torsion in (4'). We have found that, in fact, S~= /=34 can be represented in the following form: S~34 = f d4xQ)ffl pmm.l, " nnllh iil " Wan Wna l Wmd Wmld '

(16)

where Wan is defined in (4'), and

~ #ux' - 0~/,~ 0/' hbC QiabcQi 1~b~ ,

(I7)

where

Oiabc-ei/gldOia dObk dOct"

(18)

The projector P is defined as follows (brackets denote symmetrization):

p(mm 1) (iil) -- ' -~' (in Uxnill ) ~ / m ~ ] ? (nnl)O'/1)

1)

-

~li~?

[~i'l)Brnl" l)~i ~il -11 - n l ) + - 1~ x" n( m x m U n 1 ~' ( i " i l )

(19) '

and possesses the property S i p , ( m m 1 )(ii~ ) = = 0 i tnnl)(/jl) ....

(20)

The projector P appears because the action of four spinor derivatives D an D anl DmdDm d on W2 I~2 produces not only superfields Wan but also non-geometrical terms like 6nm(aad W) (D nl W) (Odm 1 WI if/. To exclude them we have introduced the projector P with the property (20) and also have used the following property of the invariant (linearized) volume:

f d4x d160 ~ f d4x c~iil]JlP(nn,)(//l )(mm,)D-mdSm,dDanDn, (i,) 124

.

(21)

Volume 99B, number 2

PHYSICS LETTERS

12 February 1981

Further expression (16) can be written down in terms of geometrical objects, by means of differential forms [12, 7] in superspace,

E m = dzc'g Ecff{ = dx" E y + dOME~a~,

(22)

where dx u (d0 sff) are anticommuting (commuting) variables. These differential forms (22) are insufficient for constructing superinvariants. One should also use "dual" differential forms, introduced by Berezin [13],

EM = E ~ l d ~ + E~dOM '

(23)

where cL~ (d0"s~) are commuting (anticommuting) variables. In this case we come from a linearized volume to the full one as follows [14] : 4

d4x d4uO

=, d4x

4N

d4N0 Ber/:~= I-I E u 1-1 EA • u=l

(24)

A=I

Now we are in a position to write the linearized subvolume in (16) in terms of differential forms: 4

sl=3 =f N=4

H EucDJJl iil p(mml)(iil) --(nnl)(jjl) Wan Wan 1 ~]md~mld , u=l

(25)

where ~ and P are defined in (17) and (19), respectively, and the new definition of Qiabc is as follows: _

~'j ~k~l

Qiabc = eijklE aE bE c ,

(26)

Eu, EJa being defined in (22) and (23). Now eq. (25) contains only geometrical objects, on the linear level it coincides with (13) and therefore in this approximation it is superinvariant. However, to be sure that the N = 4 theory has, in fact, a full nonlinear 3-loop counterterm (25) one should verify whether the differential form in (25) is integrable, i.e. the integral does not change with a change of integration variables. Let us note that we deal with the integration over a subsupermanifold, since we have d120 in (25) and not d160. Now let us discuss the linearized approximation in the theory with N = 8 [15]. On shell at this level we have a superfield Wijkl , the lowest component of which is a scalar f i e l d d~ijkl = 1 ~Cijklnpq.e ~mnpq . This superfield is self-dual (in internal space):

Wili2i3i4 = ~1 eili2i3i4JlJ2J3]4 ~]JlJ2J3f4

(27)

Let the indices ik take only values from some set of four numbers, e.g. 1, 2, 3, 4, and let the indicesJk take only values from the complementary set of the other four numbers (from 1, 2 ..... 8), in our example they are 5, 6, 7, 8. Then it is useful to introduce the notion of "proper" basis for the superfield Wili2i3i4 t3,

Wili2i3i4 = Wili2i3i4 (Xad + i ~. Oik Oad'Oik -- i S. . OjkOad0-]k , Oik, "OJk) . tk lk

(28)

In this basis it depends only on "proper" Oik and on "strange" ~Jk (in our example i k = 1, 2, 3, 4;/k = 5, 6, 7, 8). On the contrary, I~/lj2/3j4 depends only on "strange" Oik a n d on "proper" ~Jk in accordance with the self-duality condition (27), i.e. - d Wili2i3i4 = DJak Dik

~/JlJ2J3J4 =

0.

(29)

In the "proper" basis the on-shell superfield has the form (square brackets denote antisymmetrization):

#3 This superfield can be called half-left (in

Oik space)

and half-right (in 0/ k space).

125

Volume 99B, number 2

PHYSICS LETTERS

12 February 1981

Wil i2i3i4 = dPil i2i3i4 + 0 [il Xi2i3i41 + ~ eil i2i3i4h/2/3/4 g]l X/2J3]4 + Oi I aa d "~/t qbj1 i2 i3i4 + gl O[ilaOi2bF~3i4 l a b + ... + ~4 (O , a'ob'oofdi4Rabcd+ 12 4fe2il3i4i1/]213 3/J4l

+

...

d ~ O? d

ob OC od - / 1 - / 2 - 1 3 - J 4

(30)

+ (1/8!)°}~ ;~ i~ ,. 0a 0/, 0e od o~ ~,~o,.e ~ad *hh/3/.

By the use of eq. (30) we present the linearized 3-loop invariant in the "proper" basis as follows:

s 728 = f d4x Qil

(31)

...ia Ojl ...]8 wia ... i4 Wis ...i8 ~Jl ...J4 ff]Js ... J8 ,

where e.g. i k = 1, ..., 4 ; j k = 5 .... ,8, or any other subdivision of the numbers 1, ..., 8 into two groups ik,Jk can be made,

• " =1.1 -- n ial un bi2 L" hi3c ur~di4 Dis D~6 DtcTD~8 Qtl...t8

(32)

Expression (31) has the following properties:

D i k S = D~.k S = O

(33)

by construction, and

D/aks = DdkS = 0

(34)

due to eqs. (27), (29). Thus in eq. (31) we have d160 instead of d320, i.e. the integration over a subsupermanifold, and nevertheless the action (31) is superinvariant (linearized). It contains in particular the term (RabcdRd'bdc~)2 which is scalar in SU(8). This means that the other terms in (31), which are superpartners of this scalar, are also scalars in SU(S), though in eq. (31) we have some (arbitrary) subdivision of eight numbers into two groups, i.e. the action (31) is not a manifest SU(8) scalar. It is even more difficult than in the N = 4 theory to represent the linearized superinvariant (31) in geometrical terms. The scalar superfield 14/ijkl is not a geometrical object. At the linearized level D ia 14/ifkl = W~kl ' where the spinor superfield WTk l is expressed through the torsion in eq. (4). Besides, at the linearized level

Oad Wi]kl = Pad ifkl ,

(35)

where there is an exact relation [8]

=1 ~ .mnpq Pad i]kl = DdiW]kl a 24Ci]klmnpq ~t aa •

(36)

We have not succeeded in operating with a few spinor derivatives (from 16) on eq. (32) to come to geometrical superfields like I~/~k and its derivatives. But ~we have verified that when all the 16 spinor derivatives in eq. (31 ) act on W4, the answer has the form * 4 :

S1~¢38 = f

d 4 x e [ O (ma O bnO cp Wmnpd ) "Oml ( d D- n ~ D p- l d

-mlnlpl]2 W~)

+ ... + [ O a d P b ~ m n p q O a l d l P b l ~ s

a

mnpq] 2 , (37)

and all the terms in eq. (37) are expressed through Wmn p and its derivatives, the indices m, n, p ... = 1, 2 ..... 8. Thus expression (37) contains only superfields which are tensors in the local SU(8) group [ 16] and are global E 7 invariants as distinct from the original scalar superfield Wi]kl, which corresponds to the fixed "symmetrical" gauge in SU(8) [15] and breaks E 7 symmetry. The nonlinear SU (8) covariant conditions generalizing eq. (29) are ,4 We have also introduced e, the determinant of the vierbein, to be closer to the exact nonlinear form.

126

Volume 99B, number 2 O-F W~kl = 5 6 ifklmnpqL,r~mfflnpq a vva , Dma Wnp b q = (N-

PHYSICS LETTERS

12 February 1981

- a W;kl) a =0 , D(i

ab _ ~mMab [~mMab ) 2) -1 (6 nm M~)q - p "-nq + vq ""np.",

b

Wnp q + nonlinear t e r m s .

(38) (39) (40)

These equations could help to find whether the nonlinear 3-loop counterterm (37) is a superinvariant (probably with some nonlinear additional terms). In conclusion we would like to emphasize that in the theory with N = 8 on shell there definitely exist standard geometrical invariants o f the type (1), (5), starting from the 8th loop. Therefore the hope for a fundamental improvement o f the ultraviolet behaviour o f the N = 8 supergravity should not be connected simply with the absence of on-shell superinvariants. Of course, one may still hope that in supergravity some "miraculous" cancellation of divergences takes place similar to the cancellation which occurs in the first 3 loops (and higher?) o f supersymm etric SU(4) Yang-Mills theories [ 17], where the corresponding superinvariant does exist on shell, but appears with a vanishing coefficient. I am very grateful to V. Ogievetsky and S. Ferrara for their interest in this work and for many stimulating and fruitful discussions. N o t e added. After this paper had been written, we have seen the preprint "Counterterms for extended supergravity" by Howe and Lindstr6m, where a 7-1Gop counterterm in the N = 8 theory (3-loop counterterm in the N = 4 theory) has been suggested at the linearized level. However, a nonlinear extension o f the counterterm suggested in their paper violates local SU(8) and global E 7 symmetries for t h e N = 8 theory (the corresponding symmetries for the N = 4 theory are also violated) since it depends on a scalar superfield in the fixed SU(8) "symmetric" gauge. References

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