V01ume 1408, num6er 5,6 PHY51C5 L E 7 7 E R 5 14 June 1984

A NEW 50LU710N 0 F d = 11 5UPER6RAV17Y

W17H 1N7ERNAL 150ME7RY 6 R 0 U P 5U(3) X 5U(2) X U(1)

8r1an D0LAN Department 0 f Natura1 Ph110 50phy, Un1ver51ty 0 f 61a590w, 61a590w 612 8QQ, UK

Rece1ved 16 Fe6ruary 1984 Rev15ed manu5cr1pt rece1ved 21 March 1984

A new 501ut10n 0f the 6050n1c 5ect0r 0 f D = 11 5uper9rav1ty 15 pre5ented 1n wh1ch the 1nterna1 5pace ha5 the t0p0109y CP 2 × 52 × 51 and the 1nterna1 150metry 9r0up 15 5U(3) × 5U(2) X U(1).

7here ha5 6een much 1ntere5t recent1y 1n n0n- a6e11an Ka1u2a-K1e1n the0r1e5 1n wh1ch a 9au9e 9r0up, 6 , act5 0n a c0mpact n-d1men510na1 man1f01d, wh1ch 15 a 5u6-man1f01d 0f a 1ar9er (4 + n)-d1men510na1 5pace-t1me [ 1 - 4 ] . 1n part1cu1ar, e1even-d1men510na1 5uper9rav1ty ha5 rece1ved much attent10n [5]. 7he 6050n1c 5ect0r 0f the act10n f0r th15 the0ry c0nta1n5 n0t 0n1y the e1even-d1men510na1 curvature 5ca1ar, 6ut a150 a f0ur-f0rm (f0ur-1ndex ant15ymmetr1c ten50r) f1e1d. Many 501ut10n5 t0 the 6050n1c 5ect0r are a1- ready kn0wn, w1th var10u5 9au9e 9r0up5 [ 6 - 8 ] . 7he5e 501ut10n5 a11 re1y 0n a mechan15m f1r5t 5u99e5ted 6y Freund and Ru61n [9] where6y the f0ur-f0rm f1e1d (0r at 1ea5t part 0f 1t) 15 a c0n5tant mu1t1p1e 0f the v01ume f0ur-f0rm 0f f0ur-d1men510na1 5pace-t1me. W1th th15 ch01ce 0f f0ur-f0rm f1e1d, the 5even 1nterna1 d1men510n5 can 6e any E1n5te1n 5pace w1th c0n5tant, p051t1ve, 5ca1ar curvature and f0ur-d1men510na1 5pace- t1me 15 ant1-de 51tter 5pace-t1me.

1n th15 1etter a new 501ut10n 15 pre5ented, w1th 1n- terna1 150metry 9r0up 5U(3) X 5U(2) X U(1), wh1ch re11e5 0n a d1fferent ch01ce 0f f0ur-f0rm f1e1d. 7he 1nterna1 5pace 15 CP 2 × 5 2 × 51 , w1th 51 appear1n9 1n a 5119ht1y d1fferent r01e fr0m CP 2 × 5 2. CP 2 × 5 2 × 51 wa5 f1r5t c0n51dered 1n e1even-d1men510na1 the0- r1e5 6y W1tten [4] and CP 2 × 5 2 ha5 6een c0n51dered 1n ten-d1men510na1 the0r1e5 6y Watamura [ 10,1 1 ] ,1

,1 5tr1ct1y 5peak1n9, 1t 15 5U(3) /2 a X 5U(2) /2 2 wh1ch act5 tran51t1ve1y and effect1ve1y 0n CP 2 X 52 .

304

CP 2 × 5 2 15 a 51x-d1men510na1, c0mpact, K/1h1er man1f01d and adm1t5 an E1n5te1n metr1c. 7he f0ur- f0rm f1e1d 15 c0n5tructed fr0m the K/1h1er tw0-f0rm f0r CP 2 × 5 2 and the e1even d1men510n5 6reak d0wn t0 5 + 6 .7he f1e1d e4uat10n5 f0r the f1ve-d1men510na1 part are f1ve-d1men510na1 E1n5te1n, w1th a ne9at1ve c05- m01091ca1 c0n5tant. 7he5e can 6e 501ved u51n9 a J0rdan-7h1ry an5at2 [12,13], t0 y1e1d a t1me depen- dent J0rdan-7h1ry f1e1d and a f0ur-d1men510na1 R06ert50n-Wa1ker 5pace-t1me w1th f1at three-5pace.

7he 6050n1c 5ect0r 0f e1even-d1men510na1 5uper- 9rav1ty ha5 the f0110w1n9 act10n den51ty e1even-f0rm

5~ = R A 8 A #eA8 - F • # F + ~ F A F A A , (1)

where RA8 are the e1even-d1men510na1 curvature tw0- f0rm5 (A, 8 = 0 ..... 10). F 15 a f0ur-f0rm and A a three- f0rm w1th F = dA. # 15 the e1even-d1men510na1 H0d9e dua1 and eA are an 0rth0n0rma1 6a515 0f 0ne-f0rm5, e A8 = e A Ae 9. 7he f1e1d e4uat10n5 der1va61e fr0m (1) are

RA8 • #eA8 c = rc, d(#F) = F•F, (2,3)

t09ether w1th the 1dent1ty dF = 0 .7he r A are the ener9y--m0mentum ten-f0rm5

~A = FA 1A ( # F ) -- 1A (F)A #F, (4)

where 1A 15 the 1nter10r der1vat1ve, tak1n9 (p + 1)-f0rm5 t0 p-f0rm5 and 5at15fy1n9 1A(e8 ) = 6A8, 1A(#e8 ) = #(e8 •eA) and 1A(e8C ) = 6A8 eC - e86A C.

1n a11 that f0110w5, unpr1med 1nd1ce5 (1 ..... 6) 1a6e1

0.370-2693/84/$ 03.00 • E15ev1er 5c1ence Pu6115her5 8.V. (N0rth-H011and Phy51c5 Pu6115h1n9 D1v1510n)

V01ume 1408, num6er 5,6 PHY51C5 LE77ER5 14 June 1984

CP 2 X 5 2 and pr1med 1nd1ce5 (0•, .... 4•) 1a6e1 the 0ther f1ve d1men510n5, a, 6, c, ..., run fr0m 1 t0 6 and #, v, ..., run fr0m 0• t0 4•. ~ repre5ent5 the H0d9e dua1 0n CP 2 X 5 2, and ~ the H0d9e dua1 0n the rema1n1n9 f1ve-d1men510na1 man1f01d. 7he 0r1entat10n5 are 91ven 6y

61* = •e123456, 5 1* = •e0•1•2•3•4•,

# 1 = * * 51•61 ,

and the metr1c 519nature 15 (--++++++++++). U51n9 the 5tandard Fu61n1--5tudy metr1c5 0n CP 2

and 52 = CP 1 , ref. [14] , an E1n5te1n metr1c can 6e c0n5tructed 0n CP 2 × 5 2 a5 f0110w5

6 = ~ ea• e a

9(6) a= 1

where ea, a = 1 .... ,6 are a 6a515 0f 0rth0n0rma1 0ne- f0rm5, 91ven 6y

e 1 = m - 1 r 0 1 / ( 1 + r 2 ) 1/2, e 2 = m - 1 r a 2 / ( 1 +r2) 1/2,

e 3 = m - 1 r 0 3 / ( 1 +r2) , e 4 = m - 1 d r / ( 1 +r2) ,

e 5 = (~)1/2m-1 dp/(1 +p2) ,

e 6 = (~)1 /2m-10d0~/( 1 + 02), (5)

1 . . . . ,4 1a6e1 CP 2 and 5 ,6 1a6e1 5 2. m 15 a rea1 c0n- 5tant 5ca1e fact0r, a 1 (1 = 1,2, 3) are 1eft 1nvar1ant 0ne- f0rm5 0n 5 3 = 5U(2) and 06ey d01 = e1Jk0]A0 k. Ex- p11c1t1y

01 = ~(51n ~ d0 - 51n 0 c05 ~ d~6),

02 = -•(c05 ~ d0 + 51n 0 51n $ d0),

0 3 = {(c050 d~6 + d ~ ) ,

where 0 ~<0 < 7r, 0 < ~ 6 < 27r, 0 ~< ~ < 4 n , 0 ~<r<00 0n CP 2 and 0 ~< a < 21r, 0 </9 < 00 0n 52.

1n 9enera1, 0ne c0u1d make the curvature5 0f CP 2 and 5 2 1ndependent 6y u51n9 a d1fferent 5ca1e fact0r 0n each, and the re5u1tant metr1c w0u1d n0t 6e E1n- 5te1n. 7he f0110w1n9 ana1y515 5t111 h01d5 1n th15 ca5e, 51nce the ene r9y -m0men tum ten-f0rm5 can 6e ad- ju5ted t0 make 5ure that the E1n5te1n e4uat10n5 (2) are 5at15f1ed.

7he curvature tw0-f0rm5 der1va61e fr0m (5) v1a

the Lev1-C1v1ta c0nnect10n 5at15fy

* -a6 = 24m2 ~ec (6) Ra6 A 6 e c

- * a6 = 36m2~1 (7) ~1~a6A 6e

7he K~1h1er tw0-f0rm 0n CP 2 X 5 2 can 6e taken t0 6e

K = e 43 + e 12 + e 56.

N0w we make the an5at2 f0r the f0ur-f0rm f1e1d.

F = X K = ~ X K A K ,

where X 15 a c0n5tant. 0 f c0ur5e there 15 n0 9106a1 three-f0rm p0tent1a1 0n CP 2 X 5 2 f0r 5uch a f1e1d, 51nce K 15 n0t exact, 6ut 0ne can c0ver CP 2 X 5 2 w1th c00rd1nate patche5, 1n each 0f wh1ch a p0tent1a1 15 def1ned, and then re1ate the p0tent1a15 1n the 1nter- 5ect10n5 0f the c00rd1nate patche5 6y n0n-tr1v1a1 9au9e tran5f0rmat10n5. We a55ume that the metr1c 0n the e1even-d1men510na1 man1f01d 5p11t5 1nt0 the d1rect pr0duct 0f the a60ve metr1c 0n CP 2 X 5 2 and 50me 0ther metr1c 0n the f1ve-d1men510na1 man1f01d. 7hen, 51nce # F = * * (51)• 6 F and d(~ 1) = 0, we have

d F = d # F = 0,

51nce the K~1h1er tw0-f0rm 15 c105ed. N0w F • F = 0, hence the e4uat10n5 0f m0t10n (3) f0r F are 5at15f1ed.

7he ener9y--m0mentum ten-f0rm, (4), 5p11t5 1nt0 tw0 p1ece5,

2 * * 7a = X2(~11~ ~(ea), ~ = 3 X (5eu1•(61) .

Under 0ur a60ve a55umpt10n5 Rua = 0, hence, the E1n5te1n e4uat10n5 (2) reduce t0

R 6c • # e 6c a = r a - Ruv A # e uva,

RUvA #euvp = ~p - Ra6 A #ea6p .

U51n9 (6) and (7) the5e 6ec0me

* uv •(X2 +24m2)~1 , R u v A 5 e =

*eUv = 3(X 2 - 12m2)~ep. Ruv A 5 p

U51n9 the 1dent1ty eu • ~(~6~eu) = (5 --p)~,6 f0r any p-f0rm ~6, we 9et

X 2 = 6m 2.

7he pr061em ha5 n0w 6een reduced t0 that 0f 501v1n9 the f1ve-d1men510na1 E1n5te1n e4uat10n5 w1th a c05- m01091ca1 c0n5tant A = --9m 2,

305

V01ume 1408, num6er 5,6 PHY51C5 LE77ER5 14 June 1984

R v~e"vp = - 1 8 m 2 ~ e , . (8)

7h15 pr061em ha5 6een c0n51dered 1n ref5. [ 15] and [ 16]. 0ne 06v10u5 501ut10n 15 f1ve-d1men510na1 ant1- de 51tter 5pace, 1n wh1ch f0ur-d1men510na1 5pace-t1me w0u1d 6e ant1-de 51tter 5pace. H0wever the m0de1 ha5 m0re freed0m than that. We exh161t the 501ut10n 0f ref. [16], m0d1f1ed 51nce A 15 ne9at1ve. A f1ve-d1men- 510na1 metr1c 15 50u9ht 0f the f0rm

9(5) = - d t • dt + R2(t)9~3) + ~62(t)dx 4~ • dx 4•,

where 9~(3) 15 a 5tandard 150tr0p1c and h0m09ene0u5 metr1c 0n three-5pace w1th k = -+ 1,0 def1n1n9 the three-5pace curvature and ~(t) 15 a J0rdan-7h1ry f1e1d. U51n9 a f1ve-d1men510na1 6a515 0f 0rth0n0rma1 0ne-

f0rm5 a5 f0110w5:

e0•=dt, e1•=R(t)dX, e2•=R(t)h(X)d[3,

e 3• = R(t)h(X)51n{3d7, e 4• = ~6(t)dx 4~,

where h(X ) = (51nh X, 51nX, X 2) f0r k = ( - 1 , + 1 , 0 ) and

~3, 3• are an9u1ar c00rd1nate5 1n three-5pace, e4. (8) re- duce5 t0 (w1thR = dR/dt, etc.,)

R/R + (/~/R) 2 - k/R 2 = - 3 m 2 ,

(/~/R) 2 + (R1R)~1~ --k1R 2 = --3m 2,

2R/R + (/~/R) 2 - k/R 2 + (;/(9 + 2(R/R)~/d~ = --9m 2.

F0r the ca5e k = 0, the5e e4uat10n5 have the 501ut10n

R2(t) = A 2 51n c0t,

~2(t) = 8 2 [(c052 c00/51n c0t],

where c0 = X/~rn, and A and 8 are ar61trary 1nte9ra- t10n c0n5tant5.7he f1fth d1men510n can 6e taken t0 6e 51 6y 1dent1fy1n9 the p01nt5 x 4• = x 4• + 27r.

0 f c0ur5e, 1t 15 we11 kn0wn that CP 2 d0e5 n0t ad- m1t a 5p1n0r 5tructure, and 50 0ne can ant1c1pate d1f- f1cu1t1e5 1n try1n9 t0 m0d1fy th15 501ut10n t0 a110w f0r 5p1n0r5. H0wever, 6y add1n9 an add1t10na1 U(1) 5truc- ture 0nt0 CP 2, 0ne can f1t 5p1n0r5 0n 1n a we11-def1ned fa5h10n [ 17,18]. 70 d0 50 1t m19ht n0t 6e nece55ary

t0 1ntr0duce an add1t10na1 U(1) f1e1d, 1t may pr0ve p055161e t0 u5e the 1nterna1 U(1) 9au9e 5ymmetry a1- ready pre5ent 1n the metr1c.

7he a60ve 5cheme can 6e u5ed t0 c0n5truct a w1der c1a55 0f 501ut10n5. A11 0ne need5 15 a 51x-d1men510na1 K~1h1er man1f01d wh1ch adm1t5 an E1n5te1n metr1c and the K/1h1er tw0-f0rm can 6e u5ed t0 c0n5truct a f0ur-f0rm wh1ch 5p11t5 the e1even d1men510n5 1nt0 5 + 6, e.9. CP 3 c0u1d 6e u5ed wh1ch w0u1d re5u1t 1n an 1nterna1 150metry 9r0up 5U(4) × U(1), 0r 52 × 52 × 52, wh1ch w0u1d re5u1t 1n 5U(2) X 5U(2) × 5U(2)

x u(1).

1 w0u1d 11ke t0 thank R.6. M00rh0u5e, D. 5uther- 1and and 1.M. 8enn f0r many he1pfu1 d15cu5510n5.

Reference5

[1] 7. Ka1u2a, 51t2un956er. Preu55. Akad. W155. (1921) 966. [2] 0. K1e1n, 2. Phy5.37 (1925) 895; 46 (1928) 188. [3] A. E1n5te1n and P. 8er9mann, Ann. Math. (NY) 39

(1938) 683. [4] E. W1tten, Nuc1. Phy5.8186 (1981) 412. [5] E. Cremmer, 8. Ju11a and J. 5cherk, Phy5. Lett. 768

(1978) 409. [6] M.J. Duff and D.J. 70m5,1n: Un1f1cat10n 0f the funda-

menta1 f0rce5 11, ed5. J. EU15 and 5. Ferrara (P1enum, New Y0rk, 1982).

[7 ] L. Ca5te11an1, R. D•Aur1a and P. Fr6, Un1ver51ty 0f 7ur1n prepr1nt 1F77 427 (1983).

[8] F. En91ert, Phy5. Lett. 1198 (1982) 339. [91 P.6.0. Freund and M.A. Ru61n, Phy5. Lett. 978 (1980)

233. [10] 5. Watamura, Phy5. Lett. 1298 (1983) 188. [11 ] 5. Watamura, 5p0ntane0u5 c0mpact1f1cat10n and CP N,

Un1ver51ty 0f 70ky0 prepr1nt U7-K0ma6a 83-15. [12] P. J0rdan, A5tr0n0m. Nachr. 276 (1948) 193. [13] Y.R. 7h1ry, C.R. Acad. 5c1. 226 (1948) 216. [14] 7. E9uch1, P.8. 611key and A.J. Han50n, Phy5. Rep. 66

(1980) 213. [15] 7. Dere11 and R.W. 7ucker, Phy5. Lett. 1258 (1983)

133. [16] 7. Appe14u15t and A. Ch0d05, 7he 4uantum dynam1c5

0f Ka1u2a-K1e1n the0r1e5, Ya1e Un1ver51ty prepr1nt Y7P 83-05.

[17] A. 7rautman, 1ntern. J. 7he0r. Phy5.16 (1977) 561. [18] 5.W. Hawk1n9 and C.N. P0pe, Phy5. Lett. 738 (1978)

42.

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