Volume 133B, number 1,2 PHYSICS LETTERS 8 December 1983

SCALAR AND DIRAC EIGENFUNCTIONS ON THE SQUASHED SEVEN-SPHERE

B.E.W. NILSSON a, 1 and C.N. POPE b a Center for Theoretical Physics, University o f TexasatAustin, Austin, TX 78712, USA b Blackett Laboratory, Imperial College o f Science and Technology, Prince Consort Road, London SW7 2BZ, UK

Received 23 August 1983 Revised manuscript received 15 September 1983

The spectrum of massless and massive states in a Kaluza-Klein theory is determined by the eigenvalues of certain dif- ferential operators in the internal space. We calculate the eigenvalues of the scalar laplacian and Dirac operator for the squashed seven-sphere, which has recently been proposed as a ground state solution of eleven-dimensional supergravity. The techniques used are applicable also to the other operators which appear in the formulae for the mass matrices.

A spontaneous compactification o f N = 1 supergrav- ity in eleven dimensions has recently been found in which the ground state solution of the d = 11 field equations is the product of anti-de Sitter spacetime with the homogeneously squashed seven-sphere, equipped with its Einstein metric [ 1 ]. This solution gives rise to two distinct four-dimensional theories * 1 ; one with N = 1 supersymmetry and the other with N = 0 [2]. The two theories are related by a relative re- versal of the orientation of the squashed sphere. The isometry group of the squashed sphere is Sp(2) × Sp(1) [~SO(5) × SU(2)], and so both of the theories have a local Sp(2) × Sp(1) gauge invariance. Thus the massless sector of the N = 1 phase comprises 1 graviton, 1 gravitino, 13 spin-1 fields together with their 13

• ~ possibly further spin-0,-~ multi- spln-~ partners, and plets. For the N = 0 phase the massless sector com- prises 1 graviton, 13 spin-1 fields, and possibly addi- tional spin-0 and/or " 1 spln-~ fields. In general the mass spectra for the fields of each spin are given by the cor- responding mass matrices, which are differential opera- tors on the compact internal space [5]. These are re- lated to the standard Hodge-de Rham operator on p- forms, the Lichnerowicz operator on symmetric ten- sors and the Dirac and Rarita-Schwinger opera-

1 Address from September 1983: Imperial College, London, UK.

,1 There is also a third squashed sphere solution, with non- zero bosonic "torsion" [ 2 -4 ] .

0.031-9163/83/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

tors. The calculation of the eigenmodes and eigenvalues

on round n-spheres is straightforward (for instance by projecting harmonic fields in euclidean R n+l onto the unit sphere), but similar calculations for squashed spheres are much more difficult. Hitchin [6] has cal- culated the spectrum for the Dirac operator on a squashed three-sphere, but using a method not easily applicable to higher dimensional cases. In this letter we calculate the spectra of the scalar laplacian and the Dirac operator on the squashed seven-sphere. The technique used can be carried over to the other opera- tors mentioned above, and will be discussed in greater detail in a forthcoming publication [7].

The metric on the squashed seven-sphere is [1]

ds 2- - -ea®e a ( a - -0 ..... 6) (1)

where

i _ i e 0 = d / J , e - ~ sin/~6o i ,

e i _ 1 )~(u i + cos/2 COg) (2) --5

Here we are splitting the index a as (0, i, [ ), where i = I, 2, 3 and [ = ], 2, 3, = 4, 5, 6 and defining

vi = ° i + Z i , w i = ° i - ~-'i , (3)

where o i and Ei are one-form satisfying

1 1 do i = - ~ e i j k oj A o k , d~ i= - ~ e i j k ~j A Z k . (4)

67

Volume 133B, number 1,2

X is a constant describing the degree of squashing of the sphere. The connection one-form COab, defined by de a = --Wab ^ e b, CO(ab) = 0, is given by

^

COoi = - c o t /t e i + ~ Xe i ,

WO{ = ½ Xe i ,

w 6 = e i / k [ c o t / t e x + (~ X -- 1/X)e/c ] ,

w f f = - - ( 1 / 2 X ) eiik eic ,

1 1 C O i l = --2 X6i] e ° -- ~ Xeiik e k ' (5)

The curvature two-form is given in ref. [1], from which one obtains the Ricci tensor Rab = diag(a, a, a, a,/3,/3,/3), with a = 3 - -~ X 2, 13 = X 2 + 1/2X 2. The metric is Einstein when a =/3; )t 2 = 1 is the standard

i round seven-sphere solution, and X 2 = g is the squash- ed sphere solution.

Defining the inverse siebenbein E a by (e a, E b ) = 6ab , the laplacian acting on a scalar field ¢ is

A¢ - - r - D

= - E a E a ( O ) - 3 cot U E0(~) - (6)

E a may be expressed in terms of O/~/t and the vector fields k i and l i dual to o i and 2; i respectively, where

(d/t , k]) = O, (o i, k]) = ~i] , (Ni, k]) = O ,

(d/t, t j) = O, (oi, l/) = 0 , (~i, ll) = 6i f" (7)

From (4) it follows that k i and l i satisfy

[k i, k / l = e i / k k k , [l i, l]] = e i /k l k , [ki, ll] = 0 , (8)

Eq. (8) may therefore be written as

A~b = A0q~ -- [(1 - X2)/X 21 s?(~b), (9)

where

A 0 = --32/3/t 2 -- 3 co t / t O/a/t

21 2 1/t12 (10) - s e c ~ / t k i - ¢ o s e c 2 ~ ,

and

si = k i + li . ( 11 )

A0, which is independent of X, is the scalar laplacian

PHYSICS LETTERS 8 December 1983

on the X 2 = 1 round sphere. The vector fields s i can be shown to be Killing vectors, and from (8) they satisfy the same SU(2) algebra as do k i and l i. In fact they generate the Sp (1) of the Sp (2) X Sp (1) isometry group of the squashed sphere.

By projecting harmonic functions on I:18 onto the round sphere, one can show that the eigenfunctions of A 0 fall into the (n, 0, 0, 0) representations of SO(8) (n/> 0), with eigenvalues ¼ n ( n + 6). The (n, 0, 0, 0) representation breaks under Sp (2) X Sp (1) to give * 2

In/21 (n, 0 , 0 , 0 ) - + ~ ( n - 2 r , r ; n - 2 r ) , (12)

r=0

where [n/2] denotes the integer part of n /2 , and (p, q, t) denotes a representation of Sp(2) × Sp(1) with Sp(2) Dynkin label (p, q) and Sp(1) Dynkin label (t). It is therefore clear from (9) that the scalar eigenfunction carrying the representation (n - 2r, r; n - 2r) on the squashed sphere the eigenvalue 0 2 given by

02 = 1 n ( n + 6) + [(1 - ~k2)/~. 2 ] j ( j + 1), (13)

where j = n /2 - r. Thus the eigenvalues and degener- acies of the scalar laplacian on the squashed sphere are

o 2 = ¼n(n + 6)

+ [(1 - xE) /4X 2] (n - 2 r ) (n - 2 r + 2) , (14)

1 d = g ( n + 3 ) ( n - r + 2 ) ( n - 2 r + 1 ) 2 ( r + l ) , (15)

where (15) is the dimension of the (n - 2r, r; n - 2r) representation, n = 0, 1 ... . and r = 0, 1 .... [n/2].

Turning now to the Dirac opera tor we recall that - - 1 ~k2

the spinor ,1 which satisfies Dar / - ] m P a ~ on the 1 = g squashed sphere is an eigenfunction of the Dirac

operator for all values of X, and satisfies [2]

= - 3 [(1 + 2X2)/4X] 77 =- K~r/. (16)

[In the metric (1), the constant m, related to the Ricci scalar by R = 42m 2, is given by rn 2 = (28X2) - 1 X (1 + 8X 2 - 2X4).] One can also construct an Sp(1) triplet, ~i, of eigenfunctions satisfying [9]

~ i = [(5 + 2X2)/4X] ~i __-K~i, (17)

where

,2 This, and other relevant properties of group representa- tions are discussed in ref. [8].

68

Volume 133B, number 1,2 PHYSICS LETTERS 8 December 1983

~i = KiaPar/ , (18)

where Kia are the components of the Killing vectors s i (Kio = O, K'I = O, K i/ = 6~). r~ and ~i can be shown to satisfy the equations

Dar/ - 7~ afa r /+ [(I - 5a2)/4X] Kia ~i (19)

1 a[,a~i Da~ i = -7~

+ [(1 - X2)/4X] (3eijk~JKk a -- r /Kia) , (20)

where r/and ~! are Majorana spinors normalized such that f/r/= 1, {z ~j = 6il.

The general idea now is to use 77 and Gias a spinor basis, and to expand any eigenmode ~ of the Dirac operator as products of r/or ~i with scalar eigenfunc- tions. It follows from (12) that this will result in spinor fields carrying the representations (i) (n - 2r, r; n - 2r), (ii) (n - 2r, r; n - 2r + 2) and (iii) (n - 2r, r; n - 2r - 2), since r/and ~i are singlets under Sp(2) and respectively a singlet and triplet under Sp(1). For reasons of clarity we will treat the first of these cases separately from the latter two. Specifically for case (i) we write

= (.4 + gaBPa)r /+ @i(a + V ~ b r a ) ~ i , (21)

where cO i are the anti-hermitian generators of Sp(1) in the representation with Dynkin label (n - 2r), and A, B, a and b are scalar eigenfunctions in the (n - 2r, r; n - 2r) representation of Sp(2) X Sp(1). (The presence of the terms involving Pa is necessary in order to span the full eight-dimensional spinor space with only 4 spinors, r/and ~i). For cases (ii) and (iii) we write

~_+ = P± [(a + VabPa) ~] , (22)

where P_+ denotes the projection onto the representa- tions of cases (ii) and (iii) respectively. Note that there are no terms involving the singlet 7?, since they would be projected out by P+ and P_ .

We now substitute (21) and (22) into the Dirac equation

J0~ = K ~ , (23)

and equate terms proportional to each independent spinor component. After some algebra, we find for the modes of (21)

A = ( K - 3 / 4 a ) B , a = ( K + S / 4 a ) b ,

-r- lB = (K - Kn) (K - 3/4a)B

+ [(1 - a2)/2a 2] j ( j + l ) b ,

-Ulb = (K - K~) (K + 5/4a)b

+ [(1 - 5x2)/2a 2] B + [3(1 - xz) /2a 2] b , (24)

where K n and K~ are defined in ( t6) and (17), and / = n/2 - r. Thus for each scalar representation (n - 2r, r; n - 2r) we obtain 4 spinor eigenfunctions in the same representation, with eigenvalues given by the roots of the secular equation

[(K - Kn) (K - 3/4a) - o 2]

x [ ( K - K ~ ) ( K + 5 / 4 X ) + 3 ( 1 - a 2 ) / 2 a 2 - o 2]

- [(1 - a2) (1 -5a2) /4a2] / , ( / ,+ 1 ) = 0 , (25)

where o 2 is the scalar eigenvalue given by (14). Note that in the case of the two Einstein metrics (a 2 = 1 or a 2 _ 1 - ?), (25) factorizes into two quadratic equations. For the modes of (22), we find

a = (K + 5 /4a )b ,

-I--]b = (K - K~) (K + 5/4a)b

- [3(1 - a2)/4a 2] [ ( / , ' - / ) ( 1 ' +/,+ 1 ) - 2] b ,

(26)

whe re /= n / 2 - r a n d ] ' =/, + 1 o r / ' = / , - 1 for the cases (ii) or (iii) respectively. Thus for each scalar re- resentation (n - 2r, r; n - 2r) we obtain 2 spinor e i g e n f u n c t i o n s in the (n - 2r, r; n - 2r + 2) representa-

t ion w i t h / . ' =/' + 1) and 2 in the (n - 2r, r; n - 2r - 2 )

representa t ion ( w i t h / , ' =/" - 1 ), w i t h e igenvalues given

by the roots of the secular equation

(K - K~) (K + 5/4a)

- [ 3 ( 1 - x 2 ) / 4 x 2] [ ( / , ' - / , ) ( / , '+ / ,+ 1 ) - 2 ] o 2

= 0 . ( 2 7 )

It is of course crucial to check that we obtain all the Dirac eigenmodes by this procedure. To do this, we begin by analyzing the splitting of the SO(8) repre-

69

Volume 133B, number 1,2 PHYSICS LETTERS 8 December 1983

sentations on the round sphere into Sp(2) X Sp(1) representations on the squashed sphere. We recall from ref. [2] that the positive Dirac modes occur in the (n, 0, 1,0) representations, with eigenvalues (n + ~)m, and the negative modes occur in the (n, 0, 0, 1) rep-

7 resentations, with eigenvalues - ( n + ] )m, where m (related to the Ricci scalar by R = 42m 2) is the inverse radius of the round sphere. By l considering the integral of lI~ a ~012, where Da ~ = D a + ]mI'a, one can show that on any seven-dimensional space with R = 42m 2, m constant, the eigenvalues of the Dirac operator must satisfy ~ 7 K t> ]m or K ~< - ~m, with equality if and only if Da ff = 0 respectively, and so when the sphere is squashed the eigenvalues cannot stray inside the "for- bidden region" hKL < ~ m. In particular this means that the positive and negative spectra must remain dis- joint as X varies. The splittings under Sp (2) X Sp (1) of the (n, 0, 0, 1) and (n, 0, 1, 0) SO(8) representa- tions are as follows [8]:

[n/21

(n, 0 , 0 , 1 ) - + ~ ( n + l - 2 r , r ; n + l - 2 r ) r=0

I n ] 2 ]

+ ~ ( n + l - 2 r , r ; n - l - 2 r ) r=O

In/2]

+ ~ ( n - l - 2 r , r + l ; n + l - 2 r ) r=O

[n/Z]

+ ~ ( n - - l - - 2 r , r , n - - l - - Z r ) . (28) r=O

and

[n/21+l

(n, 0 , 1 , 0 ) - + ~ ( n - 2 r , r + l ; n - 2 r ) r=0

l (n - 1)/2]

+ ~ (n - 2r, r ; n - 2r) r=O

In/2]

+ ~ ( n - 2 r , r ; n + 2 - 2 r ) r=0

[ n / 2 ] - 1

+ ~ (n - 2r, r ; n - 2 - 2r) . r=O

(29)

Comparison now reveals that each representation in (28) and (29) occurs exactly once in the set of Dirac

modes constructed in (21) and (22). However, (21) appears to contain an additional "maverick" set of representations (n, 0, n) with positive eigenvalues, while (22) appears to contain a maverick set of repre- sentations (n, 0; n + 2) with negative eigenvalues. A detailed analysis shows that the eigenfunctions corre- sponding to these representations are in fact identical- ly zero. Furthermore, one can show that all the other eigenfunctions are non-vanishing. Thus we have estab- lished that our construction generates the complete set of Dirac eigenmodes, with representations as given in (28) and (29). A more detailed discussion of this construction will be given in ref. [7].

In conclusion, we shall make a few remarks about the physical significance of these results. If one plots the eigenvalues as a function of the squashing param- eter ?`2, one obtains a complicated diagram showing how some of the degeneracy on the round sphere is lifted, and an abundance of "level crossings" of the kind alluded to in ref. [10]. As one would expect from the fact that the squashed sphere does not enjoy the orientation-reversing isometry of the round sphere (which is, for instance, generated by 0 (8 ) transforma- tions disconnected from the identity), the spectra of positive and negative Dirac eigenvalues are different.

1 In particular we note that when ?`2 = ] there is a sing- 7 let mode with eigenvalue - 5 m, corresponding to the

N = 1 supersymmetry of the left-squashed sphere, 7 but there are no modes with eigenvalue + ] m, which

reflects the fact that the right-squashed sphere gives rise to a four-dimensional theory with no supersymme- try [2].

If one looks at the fermion mass matrix, one can see that some at least of the massless spin-½ fields in the four dimensional theory are obtained from the

t ~ ~ t 1 ansatz ~a = ~2aX (x)r~a(Y), ~I'a = q~a - ~FaPb XPb, (see ref. [5]), and that if//" a = Par/a 4 :0 then ~ a = -~mT~ a Furthermore, 9r/a ~ can be written as Da~a + ~mFa~2 a where C a are ~ rn Dirac modes, labelled by a. From our results in this paper, we find that there are indeed Dirac eigenmodes of this type for the ?,2 - - ~ left- squashed sphere, with representations (2, 0; 0) and (0, 0; 2). There are the supersymmetric partners of the gauge bosons associated with the Sp(2) X Sp(1) iso- merry, and were obtained previously by different methods in ref. [3]. However there are no eigenmodes with eigenvalue _9 m, indicating the absence of mass- less spin-~ fields of this type in the case of compactifi-

70

Volume 133B, number 1,2 PHYSICS LETTERS 8 December 1983

cation on the right-squashed sphere. A complete anal- ysis of the massless and massive sectors of the squashed sphere compactifications will require a detailed study of the full mass matrices for all spins. This will be con- sidered further using techniques similar to those of

this paper, elsewhere.

We are very grateful to M. Awada, P. Candelas, M.J. Duff and D.N. Page for valuable discussions, and to S. Weinberg and J.A. Wheeler for their hospitality at the Theory Group and Center for Theoretical Physics at Austin, Texas. This publication was assist- ed in part by organized research funds of the Univer- sity of Texas at Austin, and the Robert A. Welch

foundation.

References

[1] M.A. Awada, M.J. Duff and C.N. Pope, Phys. Rex ~. Lett. 50 (1983) 294.

[2] M.J. Duff, B.E.W. Nilsson and C.N. Pope, Phys. Rev. Lett. 50 (1983) 2043; 51 (1983) 846 (E).

[3] F.A. Bais, H. Nicolai and P. van Nieuwenhuizen, Geom- etry of coset spaces and massless modes of the squashed seven-sphere in supergravity. CERN preprint TH-3577 (1983).

[4] F. Englert, M. Rooman and P. Spindel, Phys. Lett. 127B (1983) 47.

[5] M.J. Duff and C.N. Pope, in: Supersymmetry and super- gravity 82, eds. S. Ferrara, J.G. Taylor and P. van Nieuwenhuizen (World Scientific, Singapore, 1983).

[6] N.J. Hitchin, Adv. Math. 14 (1974) 1. [7] B.E.W. Nilsson and C.N. Pope, in preparation. [8] R. Slansky, Phys. Rep. 79C (1981) 1. [9] M.A. Awada, private communication.

[10] B. Biran, F. Englert, B. de Wit and H. Nicolai, Phys. Lett. 124B (1983) 45.

71