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Space-time geometry and symmetry breaking
Nuclear Physics B135 (1978) 111-130 © North-Holland Publishing Company
SPACE-TIME GEOMETRY AND SYMMETRY BREAKING J.F. LUCIANI Laboratoire de Physique Thborique de l'Ecole Normale Supbrieure, Paris, France *
Received 8 September 1977
We look for solutions of the Einstein-Yang-Millsequations in a 4 + D dimensional space-time. We find solutions where the first 4 dimensions are a flat Minkowskian spacetime, while the D others are a compact, space-like manifold of small size. Such solutions can be obtained for an arbitrary compact gauge group K and are invariant under a subgroup G of K related to the space-time geometry. This shows that 4 + D dimensional gravity can give a mechanism for the super-strong symmetry breaking needed in grand unified field theories without introducing Higgs scalars.
1. Introduction The idea of using an internal space to generate symmetries dates at least from 1921, when Kaluza [1 ] gave a geometrical treatment of the Maxwell-Einstein theory, in the framework of a five-dimensional space. More recently, it has been possible to generalize the idea to an arbitrary gauge group [2]. But this requires the introduction of many extra dimensions, and of a specific structure for spacetime, namely a fibre bundle. Thus, the extra dimensions have lost their physical sense as real space-time dimensions. In our work, we investigate a quite different problem: we start with a theory in which the equations of motion in the 4 + D dimensions are symmetric, namely, an Einstein-Yang-Mills theory, and we search for solutions where one has a Ddimensional internal space, compact, and whose dimensions are small. We do this for two purposes: first, to give a physical meaning to theories containing gravitation and gauge fields in a 4 + D dimensional space, like the spinor dual model [3] for which D = 6, or for a supergravity theory in 10 dimensions, whose existence is indicated by the spinor dual model [4]. Secondly, unified gauge theories [5] of quarks and leptons need a spontaneous symmetry breaking giving rise to huge masses for leptoquarks-type bosons. From * Laboratoire propre du CNRS, Associ~ h l'Ecole Normale Sup6rieure et ~ l'Universit~ de ParisSud. Postal address: 24, rue Lhomond, 75231 Paris C~dex 05, France. 111
1 12
J.F. Luciani / Space-time geometry and symmetry breaking
the lifetime of the proton, and from asymptotic freedom considerations, one can evaluate that the typical order of magnitude needed in a realistic model may be as large as Planck's mass (1019 GeV). Such a huge breaking is very hard to achieve by the means of Higgs scalars, as shown recently by Gildener [6]. So, it was speculated by Georgi, Quinn and Weinberg [7] that gravity may have something to do with this huge symmetry breaking. In a recent paper it was shown [8] that in the framework of a unified gauge theory in a 6-dimensional space-time, this symmetry breaking could indeed arise out of the mechanism of spontaneous compactification. We generalize this idea to the general 4 + D dimensional case. The first solutions of the 4 + D dimensional Yang-Mills-Einstein equations where spontaneous compactification occurred were obtained by Cremmer and Scherk [9], in the case of an SO(D + 1) gauge group, the internal space being an SD sphere. We generalize their solutions for a large class of internal spaces, realized as homogeneous spaces on a group G, such that any simple compact Lie group admits at least one such space. The "size" of the internal space is given by one parameter, of the order of Planck's length ( 1 0 - 3 3 cm). Thus, by expanding any field on the internal space in series of harmonics, we shall get a four-dimensional spectrum of masses classified by G, with only zero and superheavy masses. Only the zero-mass states are observable at usual energies. We shall show further that the group G not only classifies the states, but is also a local gauge group, i.e. that when one expands the fields around our classical solutions, taken as a possible "vacuum" of the system, 4-dimensional massless gauge bosons associated with the group G appear in the spectrum. Sect. 3 of our work can be considered as a mathematical extension of the work done on the fibre bundle approach [2] to the unification of gravity and gauge fields, where instead of working with the group space, we work with homogeneous spaces. In our case, the fibre bundle structure of space-time is not imposed but arises as a classical solution of the field equations. Our models may look less geometrical than the fibre bundle approach since the starting point (Einstein-Yang-Mills theory) is not a unified geometrical theory. On the other hand the situation has changed recently with the advent of extended supergravity theories [10] where the Einstein-Yang-Mills theory appears as the bosonic sector of a unified field theory involving fermions (spin ~ and 1). Also in the context of the spinor dual model [4], the Einstein-Yang-Mills fields appear as a subsector of a unified theory. Finally, the non-renormalizability of 4 + D dimensional theories is not a real drawback in our case, since we merely want to obtain an understanding of how the extra dimensions in a dual spinor model can be made unobservable and used to generate a huge symmetry breaking. In such a realistic model, the Einstein-YangMills theory is automatically short-distance modified and renormalizability is achieved.
J.F. Luciani / Space-time geometry and symmetry breaking
113
2. Compactification of an homogeneous space
2.1. Generalities We start with a theory of gravitation, in interaction with gauge fields (gauge group K), with a cosmological term, in the (4 + D)-dimensional space E:
~G ~,/--~~ " 16__ R
p=
~1 4e x / ~ GM, a GM,a _ V ~
Vo
The/~ indices will always refer to the whole space E (whose metric has a signature - 1 , +1 .... , +1); the structure constants are normalized to: =
They are antisymmetric on the three indices. (The group K is simple and compact). We assume that, in a large enough domain, E can be split as: E=MXS,
M: Minkowski space, S: compact, space-like space.
Relatively to this decomposition, we shall take coordinates (x u, Oi), la indices for M, i for S. The equations of motion are:
Ti~ =1 G~ ^ e 2 M~G~° - g i ~ ( G ~ , G ~ +
Vo)
Di~ Gi~ = 0 , where/}/~ is the Yang-Mills covariant derivative. We search for a solution where:
g~b =
I?
and Ruv = O. This includes the possibility for the M space to have a metric corresponding to freely travelling gravitational waves. It is natural to make the ansatz: B~u=O, =
i(o6 •
Thus the equations become:
1 i Gc~Gc~ij -~Riguv -_ 8riG guy (~e2 -i] + Vo ) ,
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J.F. Luciani / Space-time geometry and symmetry breaking
~1 R kkg q = - 8 r i G I~-~ Gik c~ G~k - gi/ (~e2 Gkl a GC~kt + V o )]
Ri/-
Oi G°~i/= 0 . So we are left with: 87rG
Ri:=---~ GTk G'ff , 1
Vo
4 X 87rG Ri'i
(1) (2) (3)
biGC~i/=o.
We shall now forget for a moment the M space and seach for solutions of (1) and (3) on the S space. Note that, for a solution of this type, the action vanishes, since: 1
i
1
- ] R i = 8zrG 4e 2 GT.I. G °~ij + V 0
So, these solutions can be called "instantons" in a generalized way. Since the metric of S is assumed to be space-like (for the positivity of the spectrum of masses), R~ is constant and negative, and Fo is positive as in the super-symmetric theories where a "mass" term for the spin-~ fields is introduced [10]. For such theories, Vo is constrained, and has a form similar to the one we shall find. 2.2. F o r m o f the solution
In order to find a solution for eqs. (1) and (3), subject to the constraint (2), we shall suppose that S is a homogeneous space on a compact simple group G, that is, we can define an action of G on S such that any two points of S can be connected by an element of G. Furthermore, we shall assume in sects. 2 and 3 that the gauge group K is G, and return to the general case in sect. 4. Later, we shall investigate the mathematical structure of these spaces. Now, we only need a few properties stated as follows: (a) The action of G can be represented by differential operators L a, with: [L ~, L t3] = f ~ V L V (again, f ~ are normalized). Writing L a = A i c ~ i , we get: AiaOiAJ3
Ai3OiAJ c~ = f ~ 3 V A / 7 .
(b) On S, we can build an invariant metric (with upper indices):
g~ (O)= Ai~AJC~
.
(4)
J.F. Luciani / Space-time geometry and symmetry brealcmg
115
The metric is G-invariant, that is, if 0 i ~ 0 'i = R(O j) is a general coordinate transformation induced by an element of G: g~'(O'k) - aO 'i ~O'J kl.0n~
~0 k ~OTgol.
)
(5)
(to prove it, we need only (4)). (c) Then the A ia are Killing vectors:
V;A 7 + VjA = 0.
(6)
(d) We introduce hag = A i a A ~i .
Then, ha~hG'Y = ha'r
and A i a ~ i hG'r =faGSh67 +f~V6hG6
(results from (4) and (6)).
Note that, in the particular case S = G, h aG = 6 aG. Now, we make the ansatz: B ia
= ~ A ia ,
l i" = g i'J =__gg
7
1 Bia B/C~
"
/a is related to the parameter R o having the dimension of a length introduced in ref. [9], and characterizes the size of the internal space by:/~ = R2o/2(D - 1). It will be determined by the equations of motion. X is essentially ~z-1 up to a dimensionless constant to be fixed. Inspecting eq. (1), we see that the dimensionless quantity 87rG/e 2 appears everywhere, so that/J will always be given by ta ~ 87rG/e 2 (up to a numerical constant). We begin with the calculation of Vi/~/• From Bia~T iBJG _ BiG\T tBJ a = Xf~GT BJ~ ,
we get (from eq. (6)) ViB~(8 u ' + h ae) = ~~k/2 1 fGa'~BTB.~ J ! t
"
The inverse of flag + hag is 8 ~G - 1hag (because haGh G'y = ha3'). Then V i l ~j = 1XII fGeq, BGB.r(fa i j e -- l h a e ) "
J.F. Luciani / Space-time geometry and symmetry breaking
116
Now, we can find the Ricci tensor, by
v ' v , ~ -- -v'v,.~,. ---v~v'~ + R y e , viViB~j = -R;t~g
from (6).
A direct calculation, using the preceding formulae, gives:
Ri/= -
gi]
2X2/~2
The tensor K a/3 will have a great importance. It is given by K a/3 =faY6'h66'. Now, we can easily compute:
G~ = 2~7i1~i +fa~'rB~iB 7
and
Di oiIe' ,
by computing the four terms appearing in it. For example:
Bi[JViB/T~(JT = l~k f~flTf~76BJ6 = I x BJ °t " In the same way, in the calculation of G~kGIk, we use:
( v e ~ ) ( v j ~ k) = _x2~ R;i (which also results from the preceding formulae). Thus, we get:
DiGiJ°~ = ( 1 - Xla)(1+ 34a-Ka'B]' - ~ B'°~) ik ~j
= gij( 3~k2 -- ~3~2)
2
We can now solve eqs. (1) and (3): (a) If K ~ B ~ :/: C B~, the Yang-Mills equation admits the solution 1 - X/a = 0 only. But we cannot solve eq. (1). Our ansatz cannot be a solution. (b) Now the interesting case is when
/ ~ B ~ = C8~ (C is a constant). This is constraint on the geometrical structure of the space S, and this will be solved in subsect. 2.3. Assuming it to be realized, we find that the Yang-Mills equation admits two solutions: (~) ~ = 1. Then,
GO, ~ak ik~j
= ~t2gi/(1 _ C) 1
3 _ )-C)
(7)
J.F. Luciani / Space-time geometry and symmetry breaking
117
and 87rG 1 - C P
_
w
e2
Vo
e2 ¼-½C'
D(43 - ½C)2
4×(8~G) 2
1
c
This solution is possible if C # = 1 (we always have C~< 1). Note that 1Io is constrained. 3 (/3) ?q~ = ~ - ~ - 1. Then, At
Gi•k cWa
k
= X2gii( 3 - ~ C ) ( C 1
11 2
3
- R i ] = pgi]($ - ~C) , 87rG P =~-2(c-~)
1( 3 ~--1
)z
_ ,
Vo
ez DC 4X(STrG) 2 4 ( C - 1 ~ X3~ C - 1) •
Thus, we n e e d / l > 0, so C > 1, otherwise the S space would be time-like, and negative (mass) 2 would appear in the spectrum of states. We shall see in sect. 3 that, though G and e are not the constants observed in the 4 dimensions, the ratio 87rG/e 2 is of the same order as 87rG(4)/e (4)2 . Thus, we get R o ~ 10 - 3 a cm (for e (4) ~ 1).
2.3. Homogeneous spaces In order to see if the crucial condition (7) can be fulfilled, we must investigate the mathematical structure of homogeneous spaces [ 11 ]. Let us denote by g" x the action o f an element g o f G on an element x of S. For a given point x o in S, let us call H the subgroup of G such that: H "Xo =Xo • Then S can be realized as the cosets gH = (gh} for h E H (i.e. by identifying all the elements of G of the form g . h, for a given g and arbitrary h E H). Furthermore, we have an application 7r of G to S, such that: 7r(g) = g " H. Thus, 7r(e) : Xo
,
7r(gh) = 7r(g)
for h E H .
We recall that if 7r is an application of E to F, its differential dlr is defined by:
d~rik - a(Tr(x)) i ~X k
'
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J.F. Luciani / Space-time geometry and symmetry breaking
and that the Lie algebra of G can be viewed as the set of right-invariant vector fields on G. Let us denote by Ca a basis of the Lie algebra ~ of G, constituted by such fields. Since the fields C '~ ire right-invariant, we can use dn to transport them on S. Then we have:
~i~ : (d~)~Ck~. The A i~ are just the fields introduced above (this is sufficient to verify eq. (4)). Let us give an explicit example: G is SU(2), S is a 2-dimensional sphere S 2. Thus H = U(1). (7OL ~) being a matrix of SU(2), let us set:
E;II=I jI101• Then (Z 1, Z2) verifies: IZ 112 + IZ212 -- 1. Furthermore, the application (7 ~) -* (Z~v~S one to one. Thus, SU(2) is the sphere $3. can parametrize $3 by: Z 1 = e i~1 c o s l o ' , Z 2 = e i~p2 s i n 1 0 r .
Then, the fields C i'~ generate the rotations on S 3 (the generators are R a = ci°toi). One finds, in the basis of coordinates (0', ¢1, ~P2):
Cz = C+ -
C_
--
1(0, 1 , - 1 ) , i ei(¢ 1 _tp2)(1, --i tg 1 0 ' , - i cotg ½0'), 2x/2 i
e_i(~01_~o2)(1 ' i tg ½0', i cotg ½0')
242 But SU(2) can also be parametrized by three Euler angles (~o, 0, if):
g(~, o, ~ ) = R2(~) Rx(O) R2( ~ ) . So, the cosets g • H, where H is the (0, 0, ~k) rotations, parametrize the sphere $2: (¢, 0) are the polar angles, and the projection is just: 7r(~, 0, ~ ) : (~, 0 ) .
The correspondence between the two sets of coordinates on SU(2) is: 0'=0, ~1 = -~(~0 + ~ ) , ~2 = _ 1 ( ~ _ ~) _ ½~.
J.F. Luciani/ Space-timegeometry and symmetry breaking
119
So, we can find C a in the (0, ~p, ~) base:
Cz =1~(0, 1,0), C+ = 2! V ~2 ei~°(1 , i tg 0,//sin 0) 1 1 C_ = ]M'~-~ e - - i~(-1, i tg 0,//sin 0 ) ,
(Note that C a, being right-invariant, do not depend on if). We obtain (on the sphere in the (0, ~) basis):
Ax = ½(-sin ~0, cos ~ptg 0 ) , Ay = l(cos ¢, -sin ¢ tg 0 ) , Az = ½(o, - 1 ) . This is the ansatz of refs [9]. (It is exactly the angular dependence of the 't Hooft monopole [12]). Now, we can investigate the structure o f K ~ . (Let us denote by ~ the Lie algebra of H, and write: ~ = 9g * Q/. Q/is the space orthogonal to ~ for the Killing metric, that we shall denote by (cd/~). We shall also denote c~E ~ or c~Ec2/, for a direction C a of the Lie algebra in 9 ( o r q / ) . Then, since H • x o = Xo, we have:
Aia(xo)= dTr~Cka(e) = 0
for c~E ~ ,
Aia(xo)'A~i(Xo)=Sa~
for a, 13E Q/.
Then, at the point Xo: Ka~ = j ~ , ~ f~,yChe~ = ~a~ _ ~
f~,y~ y~,~5 .
6C~
Any element X in ~ acts on ~ , by the relation: adX" Y = [X, Y] (in particular, for the subspace ~ of ~ we get the adjoint representation of ~ ) . If we call A the Casimir operator: 2;6 ~ ~ (ad x 6)2 we have: K at3 : 6 a~ - (c~lAl~) • So, at Xo, the condition (7) can be written (because of (8))
(~lAIt3) = ( 1 - C)~ ~ (c~[A I/3) = 0 (c~l A I~3)arbitrary in 9g.
for ~,/3 in q/ ,
for a in q l ,
/3 in ~ ,
(8)
J.F. Luciani / Space-time geometry and symmetry breaking
120
The second relation is trivially verified, because:
[~,ql] c~, [ ~ , ~ ] c go. Furthermore, we can make the same analysis at any point x of S, by a rotation of the Lie algebra ~ , such that: A i~ = 0 , .
a @ c~t',
AiC'A3i = 8 a~,
a, {3 E cg ,
where ~ ' is the Lie algebra of the group H' corresponding to x. (We recall that H' : gHg -1 for s o m e g in G). So a sufficient condition in order to fulfill (7) is: ( a l A l f l ) = (1 - C ) 8 ~
for a, fl in c E .
We shall give two general types of spaces where it works: (i) If c~ = 0, S is a (simple compact) group. Then A = 0, and C = 1. In this case, we must choose the solution 8zrG /~ - 4e 2 '
X/a = ½,
V°
D e2 4 (TrrG) "!-"
(ii) If the Lie algebra ~ has an involutive automorphism a:
o% y] = [o(x), o(Y)], o2=1
.
Then we get a subalgebra ~ such that o = 1 on c~g, and its orthogonal qg, on which o = -1. Furthermore:
[~,g~] c f~,
[~,9/]
c9/,
[~,gZ] cge.
We can construct the homogeneous space G/H (H has the algebra g ( ) . Then, for this space:
(alalt3) = a, 3Cc/Z
because [q/, ~ ] Cqg 6~
,), E ~¢ 6~qt
= ~
8Eclt
fa'r6ft3~.6
= 8 ~ - (~lzXlD.
because [ 9 / , q / ]
C g(
J.F. Luciani / Space-time geometry and symmetry breaking
121
So
(al~l/3) = ~ a ~
for c~,/3Eq/
.
Then the condition (7) is fulfilled, with C = ½. For these spaces we must take the solution X/l = 1 and we get: 8rig / a - e2 ,
D e2 V o - 8 ( 8 r r G ) 2"
A classification of these automorphisms of simple Lie algebras has been made. (All the simple Lie algebras admit such automorphisms.) F o r the usual groups, we get: G = SU(P + Q),
H = SU(P) X SU(Q) x U ( 1 ) ,
dim S = 2 P Q ,
G = SU(N),
H = SO(N),
dim S = ~(N - 1)(N + 2 )
G = SO(2N),
H = U(N),
dim S = N ( N - 1),
G = SO(P + Q ) ,
H = SO(P) × SO(Q),
dim S = P Q .
The solution previously found by Cremmer and Scherk [8] corresponds precisely to G = SO(N + 1), H = SO(N). Note the interesting feature that the two values C = 1 and C = 1 are just the critical values of the Einstein equations. We do not know if the condition (7) can be fulfilled by another type of space. Very likely, it seems that it is not the case. It is easy to find examples in which it is not satisfied. One such example is G = SU(3), H = SU(2). Note that the groups G, H and spaces S can be replaced by locally identical groups and spaces. This can change the spectrum of masses. In the same way, we can replace G by a product of simple groups. For S, we can thus obtain a product of spaces, or something more complicated (with, again, a different mass spectrum). If condition (7) is not fulfilled, a solution might be obtained by writing: B ia =Aa[ ~ A i ~ , gJJ = Ma#AiaA]# , where A a# and M at3 are constant in each irreducible representation o f ~ in ~ .
3. Local gauge invariance
The solution we have written can be considered as the classical vacuum state around which the theory will be quantized. It was shown in ref. [13] that one can first reformulate the theory by expanding all fields around our classical solu-
J.F. Luciani / Space-time geometry and symmetry breaking
122
tion, and integrating the action over the coordinates of the S space. Then one obtains a 4-dimensional field theory, with an intinite set of fields, o f spins 2, 1 and 0 in this case. The squared masses of these fields are quantized and are given by Nile, N being dimensionless quantities related to the eigenvalues of the Casimir operator of G. As ~ turns out to be very small, we are mostly concerned with exhibiting the field content of the theory for N = 0, the fields of (mass) 2 N/I J, with N 4= 0 (for instance ultragravitons) being unobservable at present energy. As shown in ref. [9] one o f the fields with N = 0 is the ordinary gravitational field. As we have a vacuum expectation value to the gauge fields, one may wonder whether or not some gauge fields with zero mass still subsist. We shall prove that, though we have broken the symmetry by choosing a solution of the equations of motion, there remain a local invariance and a set of gauge fields of the group G. The following is only a generalization for homogeneous spaces of the formalism of fibre bundles used to unify gravitation and gauge theories. For the sake of simplicity, we shall suppose that S is a space of type If, and we let 8rrG/e 2 = 1, so that X ; / 1 = I. (We shall restore the units at the end.). The main fact is that the solution found above is invariant under a rotation of G, followed by the same rotation in the Lie algebra of K (here, K ; G) for the gauge fields B~i. Indeed: gij is invariant (eq. (5)). If we make the coordinate transformation 0 'i = 0 i + A~A ic' we get:
30J ~ k A;fl (o'k) =~dTf Af(O ), 5A~i(o,k) = _AC~(aiAj~) A~ -- A"AI"aIA~ i = A s (Ai~VIA ~. - Ai~ViA~i) = _ A a f ~ V A ~ . Thus, we can write: (R + r ) A ~ = O , where R is the above transformation, and
(TAi) ~ = AO~j~VA~[, In fact, the transformation is defined only for the D-dimensional space. In the whole space, a global rotation is defined only by a choice of coordinate (x ", Oi). If we change (x ", 0 i) into (x ~, R(oi)), where R is a local rotation, we get another decomposition o f E, and we can find another solution relative to this new choice. Indeed, it will have the same form, but it will not be the same. This indicates that we have a gauge theory associated with these local rotations. In order to find the gauge bosons o f zero mass, consider the metric:
I
g~u(x x) +g~igvjg q
gf~b =
t.giu(x, O)
gM(x, O) l gij(O)
J.F. Luciani / Space-time geometry and symmetry breaking
123
gi] is the same as before gi, = W;(X) AT(O ). If we make a change o f coordinates: (x 'u, 0 'i) = (x", 0 i + Aa(x) A ia) we get: 3x ~' Oxb g'iu(x', o') - oo, ~ Ox'" gi~i'(x' o) 30 / -- o0'i
30 j
gu/(x, O) + ~
gi/(O) ,
6giu(x', 0') = -A"t(OiAI°~)At~ + AiaO/A~i] W~ u
3uA~AT ,
so we can write
6gi,,(x, o) = AT(O) ~ W~(x) , with
A~. 5 W~(x) = - Ae'f~:~'rA'~ W~(x) -- 3u AS(x) A T , 5W~(x) = ATfWt3Wfi(x) - OuAe(x). Thus I4:~ transforms as gauge bosons (with "coupling constant" - 1 ) . In the same way, we get the right transformation for guy and g6" We shall write, for the gauge bosons: t ~ = (h a~ -
5a~)Wfi .
This ansatz is dictated by the same feature ~ = A7 as above. Now, let us investigate the case D = 1. S is a circle. We need no matter fields. Thus, consider the metric:
g=
gor +A~Av
~°2Av1
tp2A la
~02
I
J
Nothing depends on the fifth coordinate 0 (but qo depends on xU). For the Ricci tensor Ru~, we get: Rs5-
Vkp ~o
ltp2 FuvFU v _ AuRtas ,
RUs = ~Dv~O3FVU , !
Rouv + ~ DUDv~° + ½~o2FUXGx +A~RUs, R uv =*~a R
= R o + 1,~e~uv~-~
--
. t~v+--
212]~0 ~o
124
J.F. Luciani / Space-time geometry
and symmetry breaking
with Fp v = d,A, - avA,, R”l is the curvature calculated withgP,. (This is the result of Kaluza-Klein, for cp= constant.) Thus, if we set cp= constant, the Einstein equations lead to F,,,FpV = 0. This indicates that, in the general case, there are scalars, coupled with the gauge fields. Thus, the ansatz given above cannot be a solution of the equations of motion (without quadratic constraints), and is only useful to get the expansion of the Lagrangian in a series of “harmonics”. We shall give the main results of the computation. First, one checks that the mass of WE vanishes. After that, we compute the Ricci tensor. The easiest way is to use the “vierbein” formalism, writing:
where ni6 is the flat metric (-A, + r ,”r) we must calculate the connection that:
w’;b such
^ ^ ^ ^ ~'~~e~;-w~~;,e~pta~e~;- a,eac=O. Here, eh,, = 4-dimensional
vierbein
gfiV = VA,, e',epv
eii = D-dimensional
vierbein
gij = 6kl eki ezi
we define: F”PV = a,w;-
a,wff-f”PYwPwY M V’ cc
then Wijk
= a?.
(D-dimensional
vk
connection)
Wipj
=
0
Oijp
=
--teik ej’(akA~ - alA”,) WF ,
(4vcc
=
~e,“ei”A~F&
wpvi
=-2
opvp
= qlvp
,
,
I,
p
he PA”FQ
0
(recall haa ’ = A’“A;‘.)
v
-
i
i
e.k I is the inverse of ei k,
, hp’
Wp”e,” eVaI& h”“’
,
0
oMvp =
4-dimensional
connection.
J.F. Luciani / Space-time geometry and symmetry breaking
125
Then the curvature R~bh9 is given by:
RabM, = 3htAab~
- ~coa ~ ~ + ~oa~¢oc ~
-- c o a ~ C o c fi~ .
Furthermore Rb~ = e~Rab~ ~ and R~b = ehbRb~. For the Ricci tensor with ~ indices, we get:
RZj = ROij
--
AiaW~RI~j
Iaaat3r~vc~Fl3
~'i~'j--
" ~v
R ' i = ~(DvF v'c~) A~. , R ~v = R°Uv + ~ha #Fu X~Fv#x + A "- ia 141a - v ' ' iR' # R
= R ° + R S + 1-ha#l~a F"v# 4"
--~v--
'
R ° is the 4-dimensional curvature and R s = curvature of S. Furthermore T~ = 0. (This indicates that the Yang-Mills equation holds when scalars are not taken into account.) We can also calculate the action for this ansatz. We take the contribution given by - ( 1 / 4 e 2) G~,G ai~. It is
1 G~GaiJ
1
4e 2 --ij--
- - ~ e 2 (6
o~#
- h~#)F~v F # ' v "
Remarking that the Lagrangian must be a scalar for the group G (because G acts as a general coordinate transformation, and a gauge transformation) the second term is easy to obtain. Furthermore
gO = det
Iguvl,
gS = d e t l g q l . At zero order in W,, the action vanishes, as mentioned above. Then 1 16~rG
S= - - - f
_
d4xd°O~V~(R
1 fd,~doO~x/~(5~ 2e 2
1-,, hc'#,.~uve".~#
° + 4,-". . . .
~ _ hC4S)F,~vFttV#
(l t = 87rG/e2) . We can do the integration over the internal coordinates:
f dD 6 x/~he~# = e6,~ ,
,~,
J./~: Luciani / Space-time geometry and symmetry breaking
126
(because the results nmst be invariant under G). Because h a s (V s = volume of S, N = dim G.). Then
D we get e = Vs
DIN
S =
167r;~4~:d4xR ° V / ~
4(e:4,)2fd4xx/~OF~v Fuva ,
with G(4) = G
Vs' e2 (e(4)) 2 . . . . .
Vsr? ' 3D 7=2 .... 2N Remark that
G(4)/(e(4)) 2 ~ G/e 2.
The value of/1 can be computed in terms of
G (4) and e (4) (only G (4), e (4) are experimentally observable). So, if we take Newton's constant for G (4) and a coupling constant of order a for e (4)2, we get the result that the size of the internal is always of the order of Planck's length, i.e. 10 - 3 3 cm. So,
the 4-dimensional Lagrangian one obtains by expanding around our classical solution, is just that of gravitation and of a gauge theory. What happens essentially is that our solution breaks gauge invariance in (4 + D) dimensions, but 4-dimensional gauge invariance remains unbroken. But, as mentioned above, the ansatz is not a solution of the equations of motion: so the expansion of the Lagrangian is incomplete. In order to find a better expansion, we shall forget the high-mass states of spin 1 and 2, and keep only scalars. This is not crucial for the following. So, we set X) g;,~ = :iguv(x, +g~igv/ gj/ "t.gi, (x, o)
gM (x' O)1 , g~/(x, O) .J
gq(x, O) is arbitrary,
g.i(x, o)= gqfx, o)A:'~(O)W~(x). As before, one can check that Wu transforms again as a gauge field, for the same action of G as above, gi/(X, O) will represent an infinite set of scalars. For example, one can write
gq (x, o) = Ai'~(O) A ~~(0) M~:(x, 0). Furthermore, we set for the gauge fields
= (A'~B~8 °~) ~ ( x ) ,
J.I,: Luciani / Space-time geometry and symmetry breaking
127
arbitrary, e.g.
B~i
:
L~A~ .
We call the scalars collectively ~ . Because of the gauge invariance the expansion of the Lagrangian will take the form ./2(4) _
~ ROa(gp ) 167rG(4)
X/~
4(e(4)) 2
fl(~b) F~ F uv~ - 1T(~)(Duqb D~qb) +
....
where k~v is the same as before, and a, fi, 3' are scalar functions such that a(0) =/3(0) = 1 . Note that we can eliminate a((I)) (variation of the constant of gravitation) by a conformal transformation on guy" We don't know whether there are massless scalars. If not, we can forget them, and the first expansion is sufficient at usual energies. If there are massless scalars, and with the likely hypothesis that there are no other massless spin-1 states (except the massless spin-1 states we shall find later), the observation at four dimensions is: gravitation + gauge fields + massless scalars coupled in a covariant way. We have an indication that there are no massless scalars: for D = 1, the radius of the circle is a massless scalar, because Einstein's equation does not fix its value. For D > 1, the curvature is fixed by Vo. So, it seems to us that the apparition of such scalars would be accidental.
4. Symmetry breaking Until now, we have assumed that K is G. More generally, we shall consider the case when G 4: K. So, we choose a subgroup G of K, and normalize the structure constants of K such that: for a,/3, 7, 6 corresponding to G,
j ~ ' ~ 2sp'~ = a ~ . Thus, we can find a solution o f the equations described in sect. 2 by setting:
B i~
= ~ A ia
B ia = 0
in the direction of G , otherwise .
(9)
(Then the value ~t = 87rG/e 2 changes a little.) After that, as in sect. 3 we get a gauge theory of G. But if G' is a maximal subgroup of K commuting with G, it will also be left unbroken. Thus, we get a gauge theory: G × G'. We shall now inspect the spectrum of spin-1 fields (besides those already obtained from G) obtained by expanding the B~ fields g~ is not changed; B~ is not changed (same as (9));
128
J.F. Luciani / Space-time geometry and symmetry breaking
is changed (it must decouple from the linearized Einstein equations, for consistency). The linearized Yang-Mills equations, for B~, are: DlaG(1)*tv°e + Di G(l)iv°e + f ~ T B ~ i G ( l ) i v 7 = 0 , DuG (1)ui~ = 0 , G(1)u ~v = ~ . B ~v
OvB~,
giv(,)o~ _=~iB~v + j ~ , B ~ i B ~ .
So, fluctuations such that: G~v = 0 will be massless, and will, at first order, decouple from the Einstein equations (because T(ul) -- 0). A trivial solution is obtained if c~ corresponds to G' and we get massless vector bosons associated with the G' subgroup of K. However, non-trivial solutions may exist: let us set
e.(x, o)= U.(x)~(o). o/
So we must search for scalar functions ~p~(0) on S, such that 3iS0c~ + Xf~ ' y A / ~ y = O.
(10)
The Lie algebra of G acts on that of K, by the relation ad X- Y= [X, Y ] . We can split the Lie algebra of K into irreducible representations R a. If we denote by T ~ the generators for the action of G in R a, (10) takes the form Oi~~ + XA~(Tt~)c~ ¢'Y = 0 . Furthermore: L ~ = AiC~oi (differential operators on S), and Oi
=
1 - - A ~" L~ la
.
Then, in each representation R a (recall X# = 1 for the type-II spaces that we consider) eq. (10) becomes symbolically .4~(L" + r ") ~ = 0 .
The scalar ffmction on S gives representations S b of G. Eq. (1 l) means that we must find a representation S b = R a. If this is the case, one obtains one vector/.~ (x) for the representation R a. In particular, the Lie algebra of G' gives several trivial representations, and the corresponding fields obviously form a gauge field of G'. But other massless bosons can appear. For them, the topological properties of S are important, as they can suppress the representations of S b. Note that the invariance of G' is more than a four-dimensional gauge invariance:
J.F. Luciani / Space-time geometry and symmetry breaking
129
this leads, for example, to massless scalar bosons, in the components B~., for ~ in the "direction" of G'. This generalizes to 4 + D dimensions the results obtained in [8] where the full mass spectrum of the broken generators was explicitly computed. Let us give now an example. The dual spinor model predicts D = 6. We can realize this space by:
S=S 1XS2
Sl-
su(3) SU(2) X U(1) '
su(2)
$2 - U(1) '
dim4,
dim 2 (sphere).
S1 and $2 are spaces of type II. We can find a solution with gauge group K = SU(5), G = SU(3) X SU(2). Then G' = U(1). The massless bosons are obtained from gauge fields by expanding the Lie algebra of SU(5) with respect to SU(3) X SU(2): 24=8®lel®3el®l®3X2e3X2; 1 ® 1 give the gauge boson o f G ' = U(1); 3 ® 2 and 3 ® 2 are not obtained on the space 61 X 62; 8 ® 1 and 1 ® 3 give two massless vectors, and an U(1) 2 invariance. So we get a gauge group: SU(3) X SU(2) X U(1) 3. The other bosons acquire superheavy masses. So, for instance, starting from the Georgi-Glashow [14] model with K = SU(5), we obtain an unbroken gauge invariance which is SU(3) X SU(2) X U(1) 3. Leptoquark bosons naturally acquire a huge mass. On the other hand many questions remain to be solved before such a model can be considered to be realistic. (i) We must look at the fermion spectrum (this has so far been done only for D = 2 (see ref. [8]) in order to see if the flavour could appear as a degeneracy of massless multiplets. (ii) The only mass scale in this model is Planck's mass. Can one introduce other, nmch smaller mass scales, like the W boson mass, by considering solutions which are not exactly, but "almost" invariant under G? More generally, can one adjust exactly the space-time geometry of the classical solution to the pattern of broken symmetries observed in Nature, thus replacing the Higgs mechanism by a geometric symmetry breaking? I am grateful to E. Cremmer and J. Scherk for their constant help during this work.
130
J.F. Luciani / Space-time geometry and symmetry breaking
N o t e added A f t e r c o m p l e t i o n o f o u r w o r k we l e a r n e d t h a t gauge fields o n coset spaces have b e e n c o n s i d e r e d also b y G. D o m o k o s a n d S. K 6 v e s i - D o m o k o s [15] w i t h t h e purpose o f r e d u c i n g the n u m b e r o f i n d e p e n d e n t gauge fields.
References [ 1 ] Th. Kaluza, Sitzungber. Preuss. Akad. Wiss. Berlin, Math. Phys. K1,966 (1921); O. Klein, Z. Phys. 37 (1926) 895; J. Rayski, Acta Phys. Pol. 27 (1965) 89. [2] R. Kerner, Ann. Inst. H. Poincar~ 9 (1968) 143; N.P. Konoplyova and V.N. Popov, Tile gauge fields (Atomizdat, Moscow, 1972) (in Russian); Y.M. Cho and P.G.O. Freund, Phys. Rev. D12 (1975) 1711; Y.M. Cho and Pong Soo Jang, Phys. Rev. D12 (1975) 3789; J. Rayski, Dublin preprint DIAS-TP-76-31, to be published. [3] A. Neveu and J.H. Schwarz, Nucl. Phys. B31 (1971) 86; Phys. Rev. D4 (1971) 1109; P. Ramond, Phys, Rev. D3 (1971) 2415. [4] F. Gliozzi, J. Scherk and D. Olive, Phys. Lett. 65B (1976) 282; Nucl. Phys. B122 (1977) 253. [5] H. Fritzsch and P. Minkowsky, Ann. of Phys. 93 (1975) 193. [6] E. Gildener, Phys. Rev. D14 (1976) 1667. [7] H. Georgi, H.R. Quinn and S. Weinberg, Phys. Rev. Lett. 33 (1974) 451. [8] Z. Horvath, L. Palla, E. Cremmer and J. Scherk, Nucl. Phys. B127 (1977) 57. [ 9 ] E. Cremmer and J. Scherk, Nucl. Phys. B 108 (1976) 409; B 118 (1977) 61. [ 10 ] D.Z. Freedman, P. Van Nieuwenhuizen and S. Ferrara, Phys. Rev. D 13 ( 1976) 3214; S. Deser and B. Zumino, Phys. Lett. B62 (1976) 335; S. Ferrara and P. Van Nieuwenhuizen, Phys. Rev. Lett. 37 (1976); S. Ferrara, J. Scherk and B. Zumino, Phys. Lett. 66B (1977) 35; Nucl. Phys. B121 (1977) 393; D.Z. Freedman, Phys. Rev. Lett. 38 (1976) 105; D.Z. Freedman and A. Das, Nucl. Phys. B120 (1977) 221; A. Das, ITP-SB-77-4 preprint, to be published; E. Cremmer, J. Scherk and S. Ferrara, LPTENS 77/7, to be published. [11 ] S. Helgason, Differential geometry and symmetric spaces (Academic Press, New York and London, 1962). [12] G. 't Hooft, Nucl. Phys. B79 (1974) 276. [13] E. Cremmer and J. Scherk, Nucl. Phys. B103 (1976) 399. [14] M. Georgi and S.L. Glashow, Phys. Rev. Lett. 32 (1974) 438. [15] G. Domokos and S. K~Svesi-Domokos, Desy preprint 77/08, to be published; Preprint JHU HET 774 (April 1977) revised version.
SPACE-TIME GEOMETRY AND SYMMETRY BREAKING J.F. LUCIANI Laboratoire de Physique Thborique de l'Ecole Normale Supbrieure, Paris, France *
Received 8 September 1977
We look for solutions of the Einstein-Yang-Millsequations in a 4 + D dimensional space-time. We find solutions where the first 4 dimensions are a flat Minkowskian spacetime, while the D others are a compact, space-like manifold of small size. Such solutions can be obtained for an arbitrary compact gauge group K and are invariant under a subgroup G of K related to the space-time geometry. This shows that 4 + D dimensional gravity can give a mechanism for the super-strong symmetry breaking needed in grand unified field theories without introducing Higgs scalars.
1. Introduction The idea of using an internal space to generate symmetries dates at least from 1921, when Kaluza [1 ] gave a geometrical treatment of the Maxwell-Einstein theory, in the framework of a five-dimensional space. More recently, it has been possible to generalize the idea to an arbitrary gauge group [2]. But this requires the introduction of many extra dimensions, and of a specific structure for spacetime, namely a fibre bundle. Thus, the extra dimensions have lost their physical sense as real space-time dimensions. In our work, we investigate a quite different problem: we start with a theory in which the equations of motion in the 4 + D dimensions are symmetric, namely, an Einstein-Yang-Mills theory, and we search for solutions where one has a Ddimensional internal space, compact, and whose dimensions are small. We do this for two purposes: first, to give a physical meaning to theories containing gravitation and gauge fields in a 4 + D dimensional space, like the spinor dual model [3] for which D = 6, or for a supergravity theory in 10 dimensions, whose existence is indicated by the spinor dual model [4]. Secondly, unified gauge theories [5] of quarks and leptons need a spontaneous symmetry breaking giving rise to huge masses for leptoquarks-type bosons. From * Laboratoire propre du CNRS, Associ~ h l'Ecole Normale Sup6rieure et ~ l'Universit~ de ParisSud. Postal address: 24, rue Lhomond, 75231 Paris C~dex 05, France. 111
1 12
J.F. Luciani / Space-time geometry and symmetry breaking
the lifetime of the proton, and from asymptotic freedom considerations, one can evaluate that the typical order of magnitude needed in a realistic model may be as large as Planck's mass (1019 GeV). Such a huge breaking is very hard to achieve by the means of Higgs scalars, as shown recently by Gildener [6]. So, it was speculated by Georgi, Quinn and Weinberg [7] that gravity may have something to do with this huge symmetry breaking. In a recent paper it was shown [8] that in the framework of a unified gauge theory in a 6-dimensional space-time, this symmetry breaking could indeed arise out of the mechanism of spontaneous compactification. We generalize this idea to the general 4 + D dimensional case. The first solutions of the 4 + D dimensional Yang-Mills-Einstein equations where spontaneous compactification occurred were obtained by Cremmer and Scherk [9], in the case of an SO(D + 1) gauge group, the internal space being an SD sphere. We generalize their solutions for a large class of internal spaces, realized as homogeneous spaces on a group G, such that any simple compact Lie group admits at least one such space. The "size" of the internal space is given by one parameter, of the order of Planck's length ( 1 0 - 3 3 cm). Thus, by expanding any field on the internal space in series of harmonics, we shall get a four-dimensional spectrum of masses classified by G, with only zero and superheavy masses. Only the zero-mass states are observable at usual energies. We shall show further that the group G not only classifies the states, but is also a local gauge group, i.e. that when one expands the fields around our classical solutions, taken as a possible "vacuum" of the system, 4-dimensional massless gauge bosons associated with the group G appear in the spectrum. Sect. 3 of our work can be considered as a mathematical extension of the work done on the fibre bundle approach [2] to the unification of gravity and gauge fields, where instead of working with the group space, we work with homogeneous spaces. In our case, the fibre bundle structure of space-time is not imposed but arises as a classical solution of the field equations. Our models may look less geometrical than the fibre bundle approach since the starting point (Einstein-Yang-Mills theory) is not a unified geometrical theory. On the other hand the situation has changed recently with the advent of extended supergravity theories [10] where the Einstein-Yang-Mills theory appears as the bosonic sector of a unified field theory involving fermions (spin ~ and 1). Also in the context of the spinor dual model [4], the Einstein-Yang-Mills fields appear as a subsector of a unified theory. Finally, the non-renormalizability of 4 + D dimensional theories is not a real drawback in our case, since we merely want to obtain an understanding of how the extra dimensions in a dual spinor model can be made unobservable and used to generate a huge symmetry breaking. In such a realistic model, the Einstein-YangMills theory is automatically short-distance modified and renormalizability is achieved.
J.F. Luciani / Space-time geometry and symmetry breaking
113
2. Compactification of an homogeneous space
2.1. Generalities We start with a theory of gravitation, in interaction with gauge fields (gauge group K), with a cosmological term, in the (4 + D)-dimensional space E:
~G ~,/--~~ " 16__ R
p=
~1 4e x / ~ GM, a GM,a _ V ~
Vo
The/~ indices will always refer to the whole space E (whose metric has a signature - 1 , +1 .... , +1); the structure constants are normalized to: =
They are antisymmetric on the three indices. (The group K is simple and compact). We assume that, in a large enough domain, E can be split as: E=MXS,
M: Minkowski space, S: compact, space-like space.
Relatively to this decomposition, we shall take coordinates (x u, Oi), la indices for M, i for S. The equations of motion are:
Ti~ =1 G~ ^ e 2 M~G~° - g i ~ ( G ~ , G ~ +
Vo)
Di~ Gi~ = 0 , where/}/~ is the Yang-Mills covariant derivative. We search for a solution where:
g~b =
I?
and Ruv = O. This includes the possibility for the M space to have a metric corresponding to freely travelling gravitational waves. It is natural to make the ansatz: B~u=O, =
i(o6 •
Thus the equations become:
1 i Gc~Gc~ij -~Riguv -_ 8riG guy (~e2 -i] + Vo ) ,
114
J.F. Luciani / Space-time geometry and symmetry breaking
~1 R kkg q = - 8 r i G I~-~ Gik c~ G~k - gi/ (~e2 Gkl a GC~kt + V o )]
Ri/-
Oi G°~i/= 0 . So we are left with: 87rG
Ri:=---~ GTk G'ff , 1
Vo
4 X 87rG Ri'i
(1) (2) (3)
biGC~i/=o.
We shall now forget for a moment the M space and seach for solutions of (1) and (3) on the S space. Note that, for a solution of this type, the action vanishes, since: 1
i
1
- ] R i = 8zrG 4e 2 GT.I. G °~ij + V 0
So, these solutions can be called "instantons" in a generalized way. Since the metric of S is assumed to be space-like (for the positivity of the spectrum of masses), R~ is constant and negative, and Fo is positive as in the super-symmetric theories where a "mass" term for the spin-~ fields is introduced [10]. For such theories, Vo is constrained, and has a form similar to the one we shall find. 2.2. F o r m o f the solution
In order to find a solution for eqs. (1) and (3), subject to the constraint (2), we shall suppose that S is a homogeneous space on a compact simple group G, that is, we can define an action of G on S such that any two points of S can be connected by an element of G. Furthermore, we shall assume in sects. 2 and 3 that the gauge group K is G, and return to the general case in sect. 4. Later, we shall investigate the mathematical structure of these spaces. Now, we only need a few properties stated as follows: (a) The action of G can be represented by differential operators L a, with: [L ~, L t3] = f ~ V L V (again, f ~ are normalized). Writing L a = A i c ~ i , we get: AiaOiAJ3
Ai3OiAJ c~ = f ~ 3 V A / 7 .
(b) On S, we can build an invariant metric (with upper indices):
g~ (O)= Ai~AJC~
.
(4)
J.F. Luciani / Space-time geometry and symmetry brealcmg
115
The metric is G-invariant, that is, if 0 i ~ 0 'i = R(O j) is a general coordinate transformation induced by an element of G: g~'(O'k) - aO 'i ~O'J kl.0n~
~0 k ~OTgol.
)
(5)
(to prove it, we need only (4)). (c) Then the A ia are Killing vectors:
V;A 7 + VjA = 0.
(6)
(d) We introduce hag = A i a A ~i .
Then, ha~hG'Y = ha'r
and A i a ~ i hG'r =faGSh67 +f~V6hG6
(results from (4) and (6)).
Note that, in the particular case S = G, h aG = 6 aG. Now, we make the ansatz: B ia
= ~ A ia ,
l i" = g i'J =__gg
7
1 Bia B/C~
"
/a is related to the parameter R o having the dimension of a length introduced in ref. [9], and characterizes the size of the internal space by:/~ = R2o/2(D - 1). It will be determined by the equations of motion. X is essentially ~z-1 up to a dimensionless constant to be fixed. Inspecting eq. (1), we see that the dimensionless quantity 87rG/e 2 appears everywhere, so that/J will always be given by ta ~ 87rG/e 2 (up to a numerical constant). We begin with the calculation of Vi/~/• From Bia~T iBJG _ BiG\T tBJ a = Xf~GT BJ~ ,
we get (from eq. (6)) ViB~(8 u ' + h ae) = ~~k/2 1 fGa'~BTB.~ J ! t
"
The inverse of flag + hag is 8 ~G - 1hag (because haGh G'y = ha3'). Then V i l ~j = 1XII fGeq, BGB.r(fa i j e -- l h a e ) "
J.F. Luciani / Space-time geometry and symmetry breaking
116
Now, we can find the Ricci tensor, by
v ' v , ~ -- -v'v,.~,. ---v~v'~ + R y e , viViB~j = -R;t~g
from (6).
A direct calculation, using the preceding formulae, gives:
Ri/= -
gi]
2X2/~2
The tensor K a/3 will have a great importance. It is given by K a/3 =faY6'h66'. Now, we can easily compute:
G~ = 2~7i1~i +fa~'rB~iB 7
and
Di oiIe' ,
by computing the four terms appearing in it. For example:
Bi[JViB/T~(JT = l~k f~flTf~76BJ6 = I x BJ °t " In the same way, in the calculation of G~kGIk, we use:
( v e ~ ) ( v j ~ k) = _x2~ R;i (which also results from the preceding formulae). Thus, we get:
DiGiJ°~ = ( 1 - Xla)(1+ 34a-Ka'B]' - ~ B'°~) ik ~j
= gij( 3~k2 -- ~3~2)
2
We can now solve eqs. (1) and (3): (a) If K ~ B ~ :/: C B~, the Yang-Mills equation admits the solution 1 - X/a = 0 only. But we cannot solve eq. (1). Our ansatz cannot be a solution. (b) Now the interesting case is when
/ ~ B ~ = C8~ (C is a constant). This is constraint on the geometrical structure of the space S, and this will be solved in subsect. 2.3. Assuming it to be realized, we find that the Yang-Mills equation admits two solutions: (~) ~ = 1. Then,
GO, ~ak ik~j
= ~t2gi/(1 _ C) 1
3 _ )-C)
(7)
J.F. Luciani / Space-time geometry and symmetry breaking
117
and 87rG 1 - C P
_
w
e2
Vo
e2 ¼-½C'
D(43 - ½C)2
4×(8~G) 2
1
c
This solution is possible if C # = 1 (we always have C~< 1). Note that 1Io is constrained. 3 (/3) ?q~ = ~ - ~ - 1. Then, At
Gi•k cWa
k
= X2gii( 3 - ~ C ) ( C 1
11 2
3
- R i ] = pgi]($ - ~C) , 87rG P =~-2(c-~)
1( 3 ~--1
)z
_ ,
Vo
ez DC 4X(STrG) 2 4 ( C - 1 ~ X3~ C - 1) •
Thus, we n e e d / l > 0, so C > 1, otherwise the S space would be time-like, and negative (mass) 2 would appear in the spectrum of states. We shall see in sect. 3 that, though G and e are not the constants observed in the 4 dimensions, the ratio 87rG/e 2 is of the same order as 87rG(4)/e (4)2 . Thus, we get R o ~ 10 - 3 a cm (for e (4) ~ 1).
2.3. Homogeneous spaces In order to see if the crucial condition (7) can be fulfilled, we must investigate the mathematical structure of homogeneous spaces [ 11 ]. Let us denote by g" x the action o f an element g o f G on an element x of S. For a given point x o in S, let us call H the subgroup of G such that: H "Xo =Xo • Then S can be realized as the cosets gH = (gh} for h E H (i.e. by identifying all the elements of G of the form g . h, for a given g and arbitrary h E H). Furthermore, we have an application 7r of G to S, such that: 7r(g) = g " H. Thus, 7r(e) : Xo
,
7r(gh) = 7r(g)
for h E H .
We recall that if 7r is an application of E to F, its differential dlr is defined by:
d~rik - a(Tr(x)) i ~X k
'
118
J.F. Luciani / Space-time geometry and symmetry breaking
and that the Lie algebra of G can be viewed as the set of right-invariant vector fields on G. Let us denote by Ca a basis of the Lie algebra ~ of G, constituted by such fields. Since the fields C '~ ire right-invariant, we can use dn to transport them on S. Then we have:
~i~ : (d~)~Ck~. The A i~ are just the fields introduced above (this is sufficient to verify eq. (4)). Let us give an explicit example: G is SU(2), S is a 2-dimensional sphere S 2. Thus H = U(1). (7OL ~) being a matrix of SU(2), let us set:
E;II=I jI101• Then (Z 1, Z2) verifies: IZ 112 + IZ212 -- 1. Furthermore, the application (7 ~) -* (Z~v~S one to one. Thus, SU(2) is the sphere $3. can parametrize $3 by: Z 1 = e i~1 c o s l o ' , Z 2 = e i~p2 s i n 1 0 r .
Then, the fields C i'~ generate the rotations on S 3 (the generators are R a = ci°toi). One finds, in the basis of coordinates (0', ¢1, ~P2):
Cz = C+ -
C_
--
1(0, 1 , - 1 ) , i ei(¢ 1 _tp2)(1, --i tg 1 0 ' , - i cotg ½0'), 2x/2 i
e_i(~01_~o2)(1 ' i tg ½0', i cotg ½0')
242 But SU(2) can also be parametrized by three Euler angles (~o, 0, if):
g(~, o, ~ ) = R2(~) Rx(O) R2( ~ ) . So, the cosets g • H, where H is the (0, 0, ~k) rotations, parametrize the sphere $2: (¢, 0) are the polar angles, and the projection is just: 7r(~, 0, ~ ) : (~, 0 ) .
The correspondence between the two sets of coordinates on SU(2) is: 0'=0, ~1 = -~(~0 + ~ ) , ~2 = _ 1 ( ~ _ ~) _ ½~.
J.F. Luciani/ Space-timegeometry and symmetry breaking
119
So, we can find C a in the (0, ~p, ~) base:
Cz =1~(0, 1,0), C+ = 2! V ~2 ei~°(1 , i tg 0,//sin 0) 1 1 C_ = ]M'~-~ e - - i~(-1, i tg 0,//sin 0 ) ,
(Note that C a, being right-invariant, do not depend on if). We obtain (on the sphere in the (0, ~) basis):
Ax = ½(-sin ~0, cos ~ptg 0 ) , Ay = l(cos ¢, -sin ¢ tg 0 ) , Az = ½(o, - 1 ) . This is the ansatz of refs [9]. (It is exactly the angular dependence of the 't Hooft monopole [12]). Now, we can investigate the structure o f K ~ . (Let us denote by ~ the Lie algebra of H, and write: ~ = 9g * Q/. Q/is the space orthogonal to ~ for the Killing metric, that we shall denote by (cd/~). We shall also denote c~E ~ or c~Ec2/, for a direction C a of the Lie algebra in 9 ( o r q / ) . Then, since H • x o = Xo, we have:
Aia(xo)= dTr~Cka(e) = 0
for c~E ~ ,
Aia(xo)'A~i(Xo)=Sa~
for a, 13E Q/.
Then, at the point Xo: Ka~ = j ~ , ~ f~,yChe~ = ~a~ _ ~
f~,y~ y~,~5 .
6C~
Any element X in ~ acts on ~ , by the relation: adX" Y = [X, Y] (in particular, for the subspace ~ of ~ we get the adjoint representation of ~ ) . If we call A the Casimir operator: 2;6 ~ ~ (ad x 6)2 we have: K at3 : 6 a~ - (c~lAl~) • So, at Xo, the condition (7) can be written (because of (8))
(~lAIt3) = ( 1 - C)~ ~ (c~[A I/3) = 0 (c~l A I~3)arbitrary in 9g.
for ~,/3 in q/ ,
for a in q l ,
/3 in ~ ,
(8)
J.F. Luciani / Space-time geometry and symmetry breaking
120
The second relation is trivially verified, because:
[~,ql] c~, [ ~ , ~ ] c go. Furthermore, we can make the same analysis at any point x of S, by a rotation of the Lie algebra ~ , such that: A i~ = 0 , .
a @ c~t',
AiC'A3i = 8 a~,
a, {3 E cg ,
where ~ ' is the Lie algebra of the group H' corresponding to x. (We recall that H' : gHg -1 for s o m e g in G). So a sufficient condition in order to fulfill (7) is: ( a l A l f l ) = (1 - C ) 8 ~
for a, fl in c E .
We shall give two general types of spaces where it works: (i) If c~ = 0, S is a (simple compact) group. Then A = 0, and C = 1. In this case, we must choose the solution 8zrG /~ - 4e 2 '
X/a = ½,
V°
D e2 4 (TrrG) "!-"
(ii) If the Lie algebra ~ has an involutive automorphism a:
o% y] = [o(x), o(Y)], o2=1
.
Then we get a subalgebra ~ such that o = 1 on c~g, and its orthogonal qg, on which o = -1. Furthermore:
[~,g~] c f~,
[~,9/]
c9/,
[~,gZ] cge.
We can construct the homogeneous space G/H (H has the algebra g ( ) . Then, for this space:
(alalt3) = a, 3Cc/Z
because [q/, ~ ] Cqg 6~
,), E ~¢ 6~qt
= ~
8Eclt
fa'r6ft3~.6
= 8 ~ - (~lzXlD.
because [ 9 / , q / ]
C g(
J.F. Luciani / Space-time geometry and symmetry breaking
121
So
(al~l/3) = ~ a ~
for c~,/3Eq/
.
Then the condition (7) is fulfilled, with C = ½. For these spaces we must take the solution X/l = 1 and we get: 8rig / a - e2 ,
D e2 V o - 8 ( 8 r r G ) 2"
A classification of these automorphisms of simple Lie algebras has been made. (All the simple Lie algebras admit such automorphisms.) F o r the usual groups, we get: G = SU(P + Q),
H = SU(P) X SU(Q) x U ( 1 ) ,
dim S = 2 P Q ,
G = SU(N),
H = SO(N),
dim S = ~(N - 1)(N + 2 )
G = SO(2N),
H = U(N),
dim S = N ( N - 1),
G = SO(P + Q ) ,
H = SO(P) × SO(Q),
dim S = P Q .
The solution previously found by Cremmer and Scherk [8] corresponds precisely to G = SO(N + 1), H = SO(N). Note the interesting feature that the two values C = 1 and C = 1 are just the critical values of the Einstein equations. We do not know if the condition (7) can be fulfilled by another type of space. Very likely, it seems that it is not the case. It is easy to find examples in which it is not satisfied. One such example is G = SU(3), H = SU(2). Note that the groups G, H and spaces S can be replaced by locally identical groups and spaces. This can change the spectrum of masses. In the same way, we can replace G by a product of simple groups. For S, we can thus obtain a product of spaces, or something more complicated (with, again, a different mass spectrum). If condition (7) is not fulfilled, a solution might be obtained by writing: B ia =Aa[ ~ A i ~ , gJJ = Ma#AiaA]# , where A a# and M at3 are constant in each irreducible representation o f ~ in ~ .
3. Local gauge invariance
The solution we have written can be considered as the classical vacuum state around which the theory will be quantized. It was shown in ref. [13] that one can first reformulate the theory by expanding all fields around our classical solu-
J.F. Luciani / Space-time geometry and symmetry breaking
122
tion, and integrating the action over the coordinates of the S space. Then one obtains a 4-dimensional field theory, with an intinite set of fields, o f spins 2, 1 and 0 in this case. The squared masses of these fields are quantized and are given by Nile, N being dimensionless quantities related to the eigenvalues of the Casimir operator of G. As ~ turns out to be very small, we are mostly concerned with exhibiting the field content of the theory for N = 0, the fields of (mass) 2 N/I J, with N 4= 0 (for instance ultragravitons) being unobservable at present energy. As shown in ref. [9] one o f the fields with N = 0 is the ordinary gravitational field. As we have a vacuum expectation value to the gauge fields, one may wonder whether or not some gauge fields with zero mass still subsist. We shall prove that, though we have broken the symmetry by choosing a solution of the equations of motion, there remain a local invariance and a set of gauge fields of the group G. The following is only a generalization for homogeneous spaces of the formalism of fibre bundles used to unify gravitation and gauge theories. For the sake of simplicity, we shall suppose that S is a space of type If, and we let 8rrG/e 2 = 1, so that X ; / 1 = I. (We shall restore the units at the end.). The main fact is that the solution found above is invariant under a rotation of G, followed by the same rotation in the Lie algebra of K (here, K ; G) for the gauge fields B~i. Indeed: gij is invariant (eq. (5)). If we make the coordinate transformation 0 'i = 0 i + A~A ic' we get:
30J ~ k A;fl (o'k) =~dTf Af(O ), 5A~i(o,k) = _AC~(aiAj~) A~ -- A"AI"aIA~ i = A s (Ai~VIA ~. - Ai~ViA~i) = _ A a f ~ V A ~ . Thus, we can write: (R + r ) A ~ = O , where R is the above transformation, and
(TAi) ~ = AO~j~VA~[, In fact, the transformation is defined only for the D-dimensional space. In the whole space, a global rotation is defined only by a choice of coordinate (x ", Oi). If we change (x ", 0 i) into (x ~, R(oi)), where R is a local rotation, we get another decomposition o f E, and we can find another solution relative to this new choice. Indeed, it will have the same form, but it will not be the same. This indicates that we have a gauge theory associated with these local rotations. In order to find the gauge bosons o f zero mass, consider the metric:
I
g~u(x x) +g~igvjg q
gf~b =
t.giu(x, O)
gM(x, O) l gij(O)
J.F. Luciani / Space-time geometry and symmetry breaking
123
gi] is the same as before gi, = W;(X) AT(O ). If we make a change o f coordinates: (x 'u, 0 'i) = (x", 0 i + Aa(x) A ia) we get: 3x ~' Oxb g'iu(x', o') - oo, ~ Ox'" gi~i'(x' o) 30 / -- o0'i
30 j
gu/(x, O) + ~
gi/(O) ,
6giu(x', 0') = -A"t(OiAI°~)At~ + AiaO/A~i] W~ u
3uA~AT ,
so we can write
6gi,,(x, o) = AT(O) ~ W~(x) , with
A~. 5 W~(x) = - Ae'f~:~'rA'~ W~(x) -- 3u AS(x) A T , 5W~(x) = ATfWt3Wfi(x) - OuAe(x). Thus I4:~ transforms as gauge bosons (with "coupling constant" - 1 ) . In the same way, we get the right transformation for guy and g6" We shall write, for the gauge bosons: t ~ = (h a~ -
5a~)Wfi .
This ansatz is dictated by the same feature ~ = A7 as above. Now, let us investigate the case D = 1. S is a circle. We need no matter fields. Thus, consider the metric:
g=
gor +A~Av
~°2Av1
tp2A la
~02
I
J
Nothing depends on the fifth coordinate 0 (but qo depends on xU). For the Ricci tensor Ru~, we get: Rs5-
Vkp ~o
ltp2 FuvFU v _ AuRtas ,
RUs = ~Dv~O3FVU , !
Rouv + ~ DUDv~° + ½~o2FUXGx +A~RUs, R uv =*~a R
= R o + 1,~e~uv~-~
--
. t~v+--
212]~0 ~o
124
J.F. Luciani / Space-time geometry
and symmetry breaking
with Fp v = d,A, - avA,, R”l is the curvature calculated withgP,. (This is the result of Kaluza-Klein, for cp= constant.) Thus, if we set cp= constant, the Einstein equations lead to F,,,FpV = 0. This indicates that, in the general case, there are scalars, coupled with the gauge fields. Thus, the ansatz given above cannot be a solution of the equations of motion (without quadratic constraints), and is only useful to get the expansion of the Lagrangian in a series of “harmonics”. We shall give the main results of the computation. First, one checks that the mass of WE vanishes. After that, we compute the Ricci tensor. The easiest way is to use the “vierbein” formalism, writing:
where ni6 is the flat metric (-A, + r ,”r) we must calculate the connection that:
w’;b such
^ ^ ^ ^ ~'~~e~;-w~~;,e~pta~e~;- a,eac=O. Here, eh,, = 4-dimensional
vierbein
gfiV = VA,, e',epv
eii = D-dimensional
vierbein
gij = 6kl eki ezi
we define: F”PV = a,w;-
a,wff-f”PYwPwY M V’ cc
then Wijk
= a?.
(D-dimensional
vk
connection)
Wipj
=
0
Oijp
=
--teik ej’(akA~ - alA”,) WF ,
(4vcc
=
~e,“ei”A~F&
wpvi
=-2
opvp
= qlvp
,
,
I,
p
he PA”FQ
0
(recall haa ’ = A’“A;‘.)
v
-
i
i
e.k I is the inverse of ei k,
, hp’
Wp”e,” eVaI& h”“’
,
0
oMvp =
4-dimensional
connection.
J.F. Luciani / Space-time geometry and symmetry breaking
125
Then the curvature R~bh9 is given by:
RabM, = 3htAab~
- ~coa ~ ~ + ~oa~¢oc ~
-- c o a ~ C o c fi~ .
Furthermore Rb~ = e~Rab~ ~ and R~b = ehbRb~. For the Ricci tensor with ~ indices, we get:
RZj = ROij
--
AiaW~RI~j
Iaaat3r~vc~Fl3
~'i~'j--
" ~v
R ' i = ~(DvF v'c~) A~. , R ~v = R°Uv + ~ha #Fu X~Fv#x + A "- ia 141a - v ' ' iR' # R
= R ° + R S + 1-ha#l~a F"v# 4"
--~v--
'
R ° is the 4-dimensional curvature and R s = curvature of S. Furthermore T~ = 0. (This indicates that the Yang-Mills equation holds when scalars are not taken into account.) We can also calculate the action for this ansatz. We take the contribution given by - ( 1 / 4 e 2) G~,G ai~. It is
1 G~GaiJ
1
4e 2 --ij--
- - ~ e 2 (6
o~#
- h~#)F~v F # ' v "
Remarking that the Lagrangian must be a scalar for the group G (because G acts as a general coordinate transformation, and a gauge transformation) the second term is easy to obtain. Furthermore
gO = det
Iguvl,
gS = d e t l g q l . At zero order in W,, the action vanishes, as mentioned above. Then 1 16~rG
S= - - - f
_
d4xd°O~V~(R
1 fd,~doO~x/~(5~ 2e 2
1-,, hc'#,.~uve".~#
° + 4,-". . . .
~ _ hC4S)F,~vFttV#
(l t = 87rG/e2) . We can do the integration over the internal coordinates:
f dD 6 x/~he~# = e6,~ ,
,~,
J./~: Luciani / Space-time geometry and symmetry breaking
126
(because the results nmst be invariant under G). Because h a s (V s = volume of S, N = dim G.). Then
D we get e = Vs
DIN
S =
167r;~4~:d4xR ° V / ~
4(e:4,)2fd4xx/~OF~v Fuva ,
with G(4) = G
Vs' e2 (e(4)) 2 . . . . .
Vsr? ' 3D 7=2 .... 2N Remark that
G(4)/(e(4)) 2 ~ G/e 2.
The value of/1 can be computed in terms of
G (4) and e (4) (only G (4), e (4) are experimentally observable). So, if we take Newton's constant for G (4) and a coupling constant of order a for e (4)2, we get the result that the size of the internal is always of the order of Planck's length, i.e. 10 - 3 3 cm. So,
the 4-dimensional Lagrangian one obtains by expanding around our classical solution, is just that of gravitation and of a gauge theory. What happens essentially is that our solution breaks gauge invariance in (4 + D) dimensions, but 4-dimensional gauge invariance remains unbroken. But, as mentioned above, the ansatz is not a solution of the equations of motion: so the expansion of the Lagrangian is incomplete. In order to find a better expansion, we shall forget the high-mass states of spin 1 and 2, and keep only scalars. This is not crucial for the following. So, we set X) g;,~ = :iguv(x, +g~igv/ gj/ "t.gi, (x, o)
gM (x' O)1 , g~/(x, O) .J
gq(x, O) is arbitrary,
g.i(x, o)= gqfx, o)A:'~(O)W~(x). As before, one can check that Wu transforms again as a gauge field, for the same action of G as above, gi/(X, O) will represent an infinite set of scalars. For example, one can write
gq (x, o) = Ai'~(O) A ~~(0) M~:(x, 0). Furthermore, we set for the gauge fields
= (A'~B~8 °~) ~ ( x ) ,
J.I,: Luciani / Space-time geometry and symmetry breaking
127
arbitrary, e.g.
B~i
:
L~A~ .
We call the scalars collectively ~ . Because of the gauge invariance the expansion of the Lagrangian will take the form ./2(4) _
~ ROa(gp ) 167rG(4)
X/~
4(e(4)) 2
fl(~b) F~ F uv~ - 1T(~)(Duqb D~qb) +
....
where k~v is the same as before, and a, fi, 3' are scalar functions such that a(0) =/3(0) = 1 . Note that we can eliminate a((I)) (variation of the constant of gravitation) by a conformal transformation on guy" We don't know whether there are massless scalars. If not, we can forget them, and the first expansion is sufficient at usual energies. If there are massless scalars, and with the likely hypothesis that there are no other massless spin-1 states (except the massless spin-1 states we shall find later), the observation at four dimensions is: gravitation + gauge fields + massless scalars coupled in a covariant way. We have an indication that there are no massless scalars: for D = 1, the radius of the circle is a massless scalar, because Einstein's equation does not fix its value. For D > 1, the curvature is fixed by Vo. So, it seems to us that the apparition of such scalars would be accidental.
4. Symmetry breaking Until now, we have assumed that K is G. More generally, we shall consider the case when G 4: K. So, we choose a subgroup G of K, and normalize the structure constants of K such that: for a,/3, 7, 6 corresponding to G,
j ~ ' ~ 2sp'~ = a ~ . Thus, we can find a solution o f the equations described in sect. 2 by setting:
B i~
= ~ A ia
B ia = 0
in the direction of G , otherwise .
(9)
(Then the value ~t = 87rG/e 2 changes a little.) After that, as in sect. 3 we get a gauge theory of G. But if G' is a maximal subgroup of K commuting with G, it will also be left unbroken. Thus, we get a gauge theory: G × G'. We shall now inspect the spectrum of spin-1 fields (besides those already obtained from G) obtained by expanding the B~ fields g~ is not changed; B~ is not changed (same as (9));
128
J.F. Luciani / Space-time geometry and symmetry breaking
is changed (it must decouple from the linearized Einstein equations, for consistency). The linearized Yang-Mills equations, for B~, are: DlaG(1)*tv°e + Di G(l)iv°e + f ~ T B ~ i G ( l ) i v 7 = 0 , DuG (1)ui~ = 0 , G(1)u ~v = ~ . B ~v
OvB~,
giv(,)o~ _=~iB~v + j ~ , B ~ i B ~ .
So, fluctuations such that: G~v = 0 will be massless, and will, at first order, decouple from the Einstein equations (because T(ul) -- 0). A trivial solution is obtained if c~ corresponds to G' and we get massless vector bosons associated with the G' subgroup of K. However, non-trivial solutions may exist: let us set
e.(x, o)= U.(x)~(o). o/
So we must search for scalar functions ~p~(0) on S, such that 3iS0c~ + Xf~ ' y A / ~ y = O.
(10)
The Lie algebra of G acts on that of K, by the relation ad X- Y= [X, Y ] . We can split the Lie algebra of K into irreducible representations R a. If we denote by T ~ the generators for the action of G in R a, (10) takes the form Oi~~ + XA~(Tt~)c~ ¢'Y = 0 . Furthermore: L ~ = AiC~oi (differential operators on S), and Oi
=
1 - - A ~" L~ la
.
Then, in each representation R a (recall X# = 1 for the type-II spaces that we consider) eq. (10) becomes symbolically .4~(L" + r ") ~ = 0 .
The scalar ffmction on S gives representations S b of G. Eq. (1 l) means that we must find a representation S b = R a. If this is the case, one obtains one vector/.~ (x) for the representation R a. In particular, the Lie algebra of G' gives several trivial representations, and the corresponding fields obviously form a gauge field of G'. But other massless bosons can appear. For them, the topological properties of S are important, as they can suppress the representations of S b. Note that the invariance of G' is more than a four-dimensional gauge invariance:
J.F. Luciani / Space-time geometry and symmetry breaking
129
this leads, for example, to massless scalar bosons, in the components B~., for ~ in the "direction" of G'. This generalizes to 4 + D dimensions the results obtained in [8] where the full mass spectrum of the broken generators was explicitly computed. Let us give now an example. The dual spinor model predicts D = 6. We can realize this space by:
S=S 1XS2
Sl-
su(3) SU(2) X U(1) '
su(2)
$2 - U(1) '
dim4,
dim 2 (sphere).
S1 and $2 are spaces of type II. We can find a solution with gauge group K = SU(5), G = SU(3) X SU(2). Then G' = U(1). The massless bosons are obtained from gauge fields by expanding the Lie algebra of SU(5) with respect to SU(3) X SU(2): 24=8®lel®3el®l®3X2e3X2; 1 ® 1 give the gauge boson o f G ' = U(1); 3 ® 2 and 3 ® 2 are not obtained on the space 61 X 62; 8 ® 1 and 1 ® 3 give two massless vectors, and an U(1) 2 invariance. So we get a gauge group: SU(3) X SU(2) X U(1) 3. The other bosons acquire superheavy masses. So, for instance, starting from the Georgi-Glashow [14] model with K = SU(5), we obtain an unbroken gauge invariance which is SU(3) X SU(2) X U(1) 3. Leptoquark bosons naturally acquire a huge mass. On the other hand many questions remain to be solved before such a model can be considered to be realistic. (i) We must look at the fermion spectrum (this has so far been done only for D = 2 (see ref. [8]) in order to see if the flavour could appear as a degeneracy of massless multiplets. (ii) The only mass scale in this model is Planck's mass. Can one introduce other, nmch smaller mass scales, like the W boson mass, by considering solutions which are not exactly, but "almost" invariant under G? More generally, can one adjust exactly the space-time geometry of the classical solution to the pattern of broken symmetries observed in Nature, thus replacing the Higgs mechanism by a geometric symmetry breaking? I am grateful to E. Cremmer and J. Scherk for their constant help during this work.
130
J.F. Luciani / Space-time geometry and symmetry breaking
N o t e added A f t e r c o m p l e t i o n o f o u r w o r k we l e a r n e d t h a t gauge fields o n coset spaces have b e e n c o n s i d e r e d also b y G. D o m o k o s a n d S. K 6 v e s i - D o m o k o s [15] w i t h t h e purpose o f r e d u c i n g the n u m b e r o f i n d e p e n d e n t gauge fields.
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