Volume 129, number 1,2 PHYSICS LETTERS 15 September 1983

C O M P A C T I F I C A T I O N OF d = 11 S U P E R G R A V I T Y ON K3 X T 3

M.J. D U F F 1, B.E.W. NILSSON and C.N. POPE I

Theory Group and Center ]or Theoretical Physics, The University of Texas at Austin, Austin, TX 78712, USA

Received 14 April 1983 Revised manuscript received 14 June 1983

When d = 11 supergravity spontaneously compactifies to d = 4, the number of unbroken sypersymmetries, 0 N N ~< 8, is determined by the holonomy group, ,7C, of the d = 7 ground-state connection. Here we present a new solution: Minkowski spacetime × K3 X T 3, for which .7( = SU(2) and N - 4. The massless sector in d = 4 is given by N = 4 supergravity coupled to 22 N = 4 vector multiplets. Aside from its intrinsic interest, this example throws new light on Kaluza-Klein supergravity. In particular, we note that the 192 + 192 massless degrees of freedom obtained from K3 × T 3 exceed the 128 + 128 of the N = 8 theory obtained from T 7 or S 7.

The field equa t ions o f N = I supergravi ty in d = 11

d imens ions admi t o f c and ida t e g round-s ta te so lu t ions

in which seven d imens ions are compac t i f i ed . Se t t ing

~M = 0 ( M , N = 1 ..... 1 1), these equa t ions are

1 1 R M N -- ~ gMN.R = "~ (FMPQRFN PQR

1 -- g gMNb),(_)RsFPQRS ) (1)

1 ...Mr PQR " VM FMPQR = -- g-7g e M1 FM l ...M4t'Ms ...3/I8 •

(2)

The F r e u n d - R u b i n [ 1 ] cho ice for which FMNPQ van-

ished excep t for

F.~,po = 3m e.,,o~, (3)

yields the p r o d u c t o f a four -d imens iona l Einstein space

t ime

Rla~, = 1 2m 2 gtav (4)

wi th Minkowski s ignature and a seven-d imens iona l

Einste in space wi th eucl idean s ignature

R m n = 6m 2 g,,m , (5)

where g, u-- 1 . . . . . 4 and m , n -- 5,..., 1 1. Eq. (5) im-

plies c o m p a c t i f i c a t i o n w h e n m 4 = 0 and is cons i s t en t

wi th , bu t does no t i m p l y , c o m p a c t i f i c a t i o n w h e n m = 0.

1 On leave from The Blackett Laboratory, hnperial College, London, UK.

As discussed in refs. [2 4 ] , the n u m b e r o f u n b r o k e n

supe r symmet r i e s , N, in the resul t ing four -d imens iona l

t heo ry , is d e t e r m i n e d by the n u m b e r of Kill ing spinors

on the d = 7 man i fo ld i.e. the n u m b e r o f spinors satis-

fying

b, , ,~-=(a ,n -{- co ,# h Cab - ~ m e m a r a ) ~ = 0 , (6)

where I a are the d = 7 Dirac matr ices ,

{r~, r ~ = 2aab. (7)

l ab = F [ a P b l , and oomab and ema are the spin connec-

t ion and s iebenbe in of the g round state so lu t ion to

eq. (5). Such Killing spinors satisfy the in tegrabi l i ty

c o n d i t i o n

1 [D m , D,,] ,~ = - ~ C,,,,ae ruby = 0 , (S)

where Cmn ab is the Weyl tensor . The subgroup o f Spin

(7) genera ted by these l inear c o m b i n a t i o n s of the Spin

(7) genera tors Pub cor responds to the h o l o n o m y group

o f the c o n n e c t i o n o f eq. (6). Thus the m a x i m u m

n u m b e r o f u n b r o k e n supersyn lmet r ies , N, is equal to

tile n u m b e r o f spinors left invar ian t by ,7{7.

it is the excep t ion , r a the r t han the rule, t ha t the

g round state admi t s Killing spinors and, to date , on ly

three examples have been discussed in the l i te ra ture ,

two wi th N = 8 and Cnm ab = 0 and one wi th N = 1

and Cmn ab @ O. The case m = 0 and N = 8 singles ou t

the seven torus o f C r e m m e r and Jul ia [5] for which

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Volume 129, number 1,2 PftYSICS LETTERS 15 September 1983

J£ = 1 ; the case m 4 : 0 and N = 8 singles out the round

seven-sphere o f Duf f and Pope [2,3] for which JC = 1

and then there is the squashed seven-sphere o f Awada

et al. [4] for which m 4: 0, N = 1 and = G 2. (In fact the

full d = 4 theory obta ined f rom the squashed S 7, in-

cluding the massive states, corresponds to a sponta-

neously broken phase o f the one obtained from the round S 7 [6].) Al though T 7 and the round S 7 exhaust

all possible N = 8 solutions (since they are the only

Einstein spaces to be conformal ly flat ~1 : Cmnab = O)

it is o f interest to ask whether there are any others

wi th 0 < N < 8, and this leads naturally to the study

of possible h o l o n o m y groups J£ for the connec t ion o f

eq. (6). For Jf = 1 we have N = 8 and for 3£ = G 2 we

have N = 1 because these are the groups which, in seven dimensions, leave invariant 8 and 1 spinors re-

spectively. In this paper we examine another solut ion o f eq.

(5) wi th m = 0 for which Jf = SU(2). Since this SU(2) leaves invariant 4 spinors we find an effective d = 4

theory with N = 4 supersymmetry . The solut ion is

given by the Ricci flat metr ic on K3 X T 3 where K3

is Kummer ' s quartic surface in CP 3 . A review of the

K3 li terature may be found in ref. [7]. The Ricci flat

metr ic on K3 is no t known explici t ly but there is an

existence proof . Moreover it is known to have 58 pa-

rameters, to have a self-dual Riemann tensor, and no

symmetr ies . Topological ly , K3 has Euler number X =

24 Hirzebruch signature ~- = 16 and Betty numbers

b 0 = 1 , b I = 0 , b 2 = 2 2 , b 3 = 0 , b 4 = 1 . (9)

This in format ion will be sufficient for us to determine

the K a l u z a - K l e i n ans~itze necessary to isolate the

Strictly speaking, the Weyl tensor characterizes the re- stricted holonomy group of Din; i.e., it describes the rota- tion of a spinor parallel transported around a closed loop which is homotopic to zero. If the space is not simply con- nected there may be global obstructions to the existence of covariantly constant spinors, in addition to any local obstruction implied by the Weyl tensor. In addition to the ground state solutions T 7 and S 7, there will also exist solu- tions of the form TT/P (generalizations of Klein bottles) and ST/P (generaUzations of lens spaces), where P is a dis- crete group. These spaces, like their T 7 o r S 7 covering spaces, have Cabcd = 0, but these global considerations imply that they admit fewer than 8 covariantly constant spinors, and hence provide another means of obtaining 0 < N < 8 supersymmetry.

We thank Don N. Page for discussions on these points.

massless particle con ten t o f the resulting N = 4 super-

gravity theory . The number o f massless particles o f

each spin is given by the number o f zero-eigenvalue

modes o f the corresponding mass matrices. These are

given by differential operators on the seven-dimensional

ground state manifold (second order for bosons and

first order for fermions) and are discussed in detail in

ref. [3] for the case rn 4 = 0, where they were applied

to S 7. To apply them to the K3 X T 3 solut ion o f this

paper, we need only set m = 0. The results are given

in table 1, where we compare with the reduct ion on T 7 .

The single guy comes f rom the single zero mode of

the scalar laplacian, the three B u f rom the three Killing

vectors on T 3 (K3 has no Killing vectors) , and the 64

scalarsS from the zero modes o f the Lichnerowicz op-

erator acting on symmetr ic rank-two tensors: 58 from

K3 (the 58 parameters) and 6 from T 3 (the 6 param-

eters o f the metr ic on S 1 X S 1 X S1) .We note that

these 6 are Killing tensors but that the 58 are not.

As far as the fermions are concerned, we first note

that since K3 is half-flat the h o l o n o m y group is SU(2)

rather than the SU(2) X SU(2) o f a generic four mani-

fold and hence it admits two covariant ly constant

spinors (i.e. Killing spinors) which are left or right

handed according as K3 is self-dual or anti-self-dual

[8]. Tile four ~u come from the four Killing spinors on K3 X T 3 (2 on K3 X 2 on T3). To obtain the 92

• 1 spm-~ fields X we note that there are 40 zero-modes

of the Ra r i t a -Schwinge r opera tor on K 3 : 3 8 of

which are r - t race-free and 2 o f which are not but are

covariantly constant , while on T 3 there are 6 such zero-modes which are covariant ly constant but not I'-

trace-free. We note that these 6 are Killing vector-spinors

but that the 40 are not. With these condi t ions the

Table 1

d = 11 d = 4 spin T 7 K3 X T 3

gMN g~v 2 1 1 B~ 1 7 3 S 0 28 64

q;M q;U 3/4 8 4 x 1/2 56 92

.4 MN P A lavp 1 1 AUv 0 7 3 A u 1 21 25 A 0 35 67

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Volume 129, number 1,2 PHYSICS LETTERS 15 September 1983

92 modes, given by 40 on K3 X 2 Killing spinors on T 3 plus 6 on T 3 X 2 Killing spinors on K3, will be

• 1 zero-modes of the spin- 5 mass matrix [3]. The numbers ofAuvp, A~v , A~ and A fields are

given by the zero-modes of the Hodge-de Rham oper- ator acting on 0, 1 ,2 and 3 forms respectively; i.e. by

the Betti numbers b0, b l , b 2 and b 3 of K3 X T 3 . But for a product manifold M = M' X M",

P

bp = ~0_ brb;r ' (10)

where bp, bp, b; are the p ' th Betti numbers of M, M' and M" respectively. Hence from eq. (9), and the Betti n u m b e r s bp = (3) for T 3, we obtain the numbers given

in table 1. A detailed discussion of boson and fermion zero-modes on K3 (and their relation to axial and con- formal anomalies) may be found in refs. [8,9].

To summarize, the spin content is given by 1 spin • 3 1 2, 4 spin g, 28 spin 1,92 spin ~, 67 scalars and 67

pseudoscalars. This corresponds to an N = 4 supergrav- ity multiplet (1 ,4 , 6, 4, 1 + 1) coupled to 22 N = 4 spin-one matter multiplets (1 ,4 , 3 + 3).

Several comments are now in order especially since K3 × T 3 provides a counterexample to many claims to be found in the Kaluza-Klein literature.

(1) The number of massless degrees of freedom (per d = 4 spacetime point) of th i sN = 4 theory ob- tained from K3 × T 3 , namely 192 + 192, exceeds the 128 + 128 of the N = 8 theory obtained from T 7 (or $7). Thus one's naive expectation that the N = 8 theo- ry maximizes the number of zero-modes is seen not to be fulfilled. Note that per d = 4 spacetime point, the d = 11 theory has (128 + 128) X 007 degrees of free-

dora and so there is no contradiction in obtaining more than (128 + 128) when one isolates the massless states from the infinite tower of massive states. We do not know whether 192 + 192 is the maximum.

(2) Note that K3 X T 3 is neither a group manifold nor a coset space, lndeed K3 has no symmetries at all, yet this does not prevent a sensible Kaluza-Klein the- ory with a large number of massless particles. Of course, the 28 massless spin l are only abelian gauge fields, the gauge group being [U(1)] 3 × [GL(1, I:i)] 25. Note also that K3 X T 3 provides the first example of a supersymmetric Kaluza-Klein theory for which the extra dimensional ground-state manifold is not paral- lelizable.

(3) The ansatz for the massless scalars coming from gMN is not in general given by products of Killing vectors. When m = 0, the criterion for masslessness corresponds to zero-modes of the Lichnerowicz oper- ator 2x L. These are in one-to-one correspondence with the number of parameters of the ground state metric

gmn because A L describes the first variation of the Einstein tensor and so its zero-modes preserve eq. (5) when m = 0. Thus T 7 yields 28 and K3 × T 3 yields 64.

This ceases to be true when m ~ 0, however, because the ground state solution ofeq. (4) is now anti de Sitter space. Massless scalars must now obey the conformal wave equation and hence, on the round S 7 for exam- ple, the mass matrix of ref. [3] is (A g -- 16 m 2) which has 35 zero-modes rather than (A L -- 12 m 2) which follows from the first variation of eq. (5) and which has no zero-modes. Hence it was found that 35 mass- less scalars come from gMN even though the S 7 solu- tion of eq. (5) has no parameters.

(4) How do the many parameters of K3 × T 3 show up in the effective four-dimensional theory? The answer is in the expectation values of the scalar fields. Compactification on Ricci flat manifolds yields no ef- fective potential for the scalars and their expectation values are arbitrary. This contrasts with compactifica- tion on Einstein manifolds with m 4= 0. {Note inciden- tally that in this respect ungauged N = 8 supergravity [5] obtained from T 7 has many more parameters than gauged N = 8 supergravity [10] obtained from S 7, con- trary to the claim that gauging increases the param- eters from one (Newton's constant) to two (Newton's constant plus gauge coupling constant).)

Although we have focussed our attention on N = 1 supergravity in d = 11, solutions of the kind discussed here also exist f o rN = 1 in d = 10, 9 and 8 for which spacetime is Minkowski space and for which the extra dimensions are K3 X T 2, K3 X S 1 and K3 respective- ly. Owing to the Weyl condition in d = 10, we have N = 2 rather than N = 4. Omitting the details, we quote the results. Starting from the 64 + 64 components o f N = 1 in d = 10, we obtain the 96 + 96 components in d = 4 o f N = 2 supergravity coupled to 3 N = 2 vec- tor multiplets and 20 N = 2 scalar multiplets. Starting from the 56 + 56 components of N = 1 in d = 9, we obtain the 92 + 92 components in d = 4 o f N = 2 supergravity coupled to 2 N = 2 vector multiplets and 20 N = 2 scalar multiplets. Starting from the 48 + 48 components o f N = 1 in d = 8, obtain the 88 + 88

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Volume 129, number 1,2 PHYSICS LETTERS 15 September 1983

componen t s o f N = 2 supergravity coupled to 1 N = 2

vector mul t ip le t and 20 N = 2 scalar mult iplets . Note

that each drop in dimension from 10 to 8 corresponds

to one less vector mul t ip le t in d = 4.

Returning to the case o f d = 1 1, we recall that so-

lut ions o f eqs. (1) and (2) may be found for which

the Frnnp q componen t s Of FMNPQ are also non-zero. However , the solution of ref. [11 ] is known to break

all 8 supersymmetr ies [12,13] and this is in fact an

inevitable feature [14] ofFmnpq 4 = 0 solutions. In order to find out whether o ther supersymmetry-pre-

serving solutions exist, one can look to the h o l o n o m y

group. For example the SU(3) subgroup of G 2 leaves

invariant 2 spinors but we do not know of any solu-

tions of eq. (5) with ~f = SU(3). Thus the outs tanding

problem is to classify all d = 7 Einstein metr ics o f non-

negative curvature and their ho lonomy groups.

We are grateful to S. Weinberg for s t imulat ing dis-

cussions on K a l u z a - K l e i n theories and to J .A. Wheeler

and S. Weinberg for their hospital i ty at the Center for

Theoret ical Physics and at the Theory Group. This

publ icat ion was assisted by NSF Grant PHY 820571 7

and by organized research funds of The University o f

Texas at Austin. Suppor ted in part by the Rober t A. Welch Foundat ion .

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[41 M.A. Awada,M.J. Duff and C.N. Popc, Phys. Rcv. Lctt. 50 (1983) 294.

[51 E. Cremmer and B. Julia, Nucl. Phys. B159 (1979) 141. [6] M.J. Duff, B.E.W. Nilsson and C.N. Pope, Univcrsity of

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