/

SYMMETRIES OF EINSTEIN-YANC-MILLS FIELDS

AND DIMENSIONAL REDUCTION

R. COQUEREAUX

A. JADCZÏK*

Centre de Physique Théorique** CNRS - Luainy - Caie 907

F-13288 MARSEILLE CEDEX 9 (Fmnee )

o

Let E be a manifold on Hhich a coupact t i e group S a c t s simply (a l l orbits of the sane type) ; E can be written locally as H x S/I , M being the manifold of orbits (space-tine) and I a typi­cal isotropy group far the S action. We study the geometrical struc­ture given by an S-invariant metric and an S-invariant Yang Hills field on E with gauge group R • We show that there i s a one to one corres­pondence between such structures and quadruplets ( ) / j / A^ &*~ / (Î ) of fields defined solely on H ; vVj i s a metric on M , ft are sca­lar f ields characterising the geometry of the orbits (internal spaces),

tft* are other scalar fields (Higgs f ie lds) characterising the S invariance of the Lie(R)-valued Yang Hil ls f ie ld and A , i s a Yang Mills f ie ld for the gauge group N(I) | I x Z(A (I))» N(I) being the normalizer of I in S , X i s a honcrrrphism of I into R associa­ted to the S action, and *5( X(D) i s the centraliser of X ( D in R . We express the Einstein-Yang-Mills Lagrangian of E in terms of the component fields on H . Examples and model building recipes are given.

APRIL 1984

x Institute of Theoretical Physics, University of Wroclaw (Poland)

xx Laboratoire Propre, Centre riational de la Recherche Scientifique

C*JÇ>$ »» CPT-84/P.1611

1

I - INTRODUCTION

1.1 - Several descriptions for the same geometrical structure

Symmetry properties of gravity (metric structure) and Yang-Mills fields (connections) have been often studied separately, both by physicists and mathematicians. These two kinds of geometrical structures are however deeply inter-related and several techniques of "dimensional reduction" al­low us to cast a new light on the subject. Let us suppose that we live in an extended universe U endowed with a metric g(U) invariant under a group G (description 1), then, in many cases, we can also describe the same situation by saying that we live in an universe E (dim E < dim U) endowed with a metric g(E) and a Yang-Mills field A(E) ,both invariant under a subgroup of G (description 2). We can finally describe the same situation by saying that we live in a universe H (space-time, dim M < dim E < dim U) endowed with a metric g(M) , a new Yang Mills field A(M), a fes scalar fields and no symmetries left (description 3). The study of the link between the descriptions 2 and 3 is the aim of this paper. The method that we shall use is the following : a general result |_ 1 J ,which we recall in Section 2, allows us to obtain the link between descriptions 1 and 2 as well as the link between descriptions I and 3, we will there­fore obtain the desired results by comparing the above two relations. Particular examples of the general-situation have-been studied in [2 J , [3]

providing interesting pHenoraenolbgicai models r''the interpretation

of Higgs f ields as Yang-Mills f ields has been emphasized in [ 4 - 7]

where some properties of symmetric Yang-Mills f ields are also studied.

The construction given in the present paper i s a natural application of

the methods developped in |_ I j and may be thought of as an alternative

to that of [ 8 J , where a direct analysis of the link between the des­

criptions 2 and 3 i s made (see also [ 9 J ) . 1.2 - The mathematical framework

Symmetries can be studied globally (group actions) or locally

(vector fields) ; here we study Symmetric configurations of coupled Yang-

CPT-84/P.1611

2

Mills and Einstein fields in the most general case and we perform this ana­lysis both from the local and global point of view. The natural mathemati­cal framework used to study global aspects is provided by the theory of fiber bundles and we will use freely the corresponding terminology. In plain terms, let us only say that a fiber bundle is a geometrical object which can be thought of as a collection of "fibers" glued together and pa­rametrised by a manifold called the "basis" ; besides, often there is a well defined action of a group G on the fibers (the "structural group")-one can think of the base as being space-time and of the typical fiber as being the internal space (there is an internal space above every spa­ce-time point)• Properties of connections are discussed in the mathemati­cal literature in terms of the connection form (which is a Lie algebra valued one-form on the'bundle space) but physicists prefer to use Yang-Mills fields (which are Lie algebra valued one-forms on the basis) ; Yang-Mills fields can only be defined (locally) via the choice of a (lo­cal) gauge ("section" of the bundle). In the following we will express the results in these two languages.

1.3 - Structure of the paper

In Section 2, we show how to construct invariant metrics on fi­ber bundles and recall the Reduction Theorem [1 j . In Section 3, we de­fine and study symmetries of bundles and connections. In Section 4, we analyze the geometrical structure for a space E endowed with an S-inva-riant metric and a S-invariant, Lie(R)-valued Yang-Mills field, we obtain à'generalized redaction theorem and a "dimensionally reduced" Einstein-Yang-Mills action. In Section 5, we discuss examples, model building re-ceipes and give comments and a summary of our results. The reader who only wants to get the main ide&r of the present paper may read only the summary section (ft-1) as well as lables 1,2 p.22,.23.«henever we give a phy­sical interpretation to our results, the signature of space-time is +++-, see also Sect. 5-4.

CPT-84/P.1611

3

2 - INVARIANT METRICS ON FIBER BUNDLES

2.I - Invariant metrics on principal bundles

Let us first recall a result which is well known both in physi­cal and in the mathematical folklore (see e.g. pO , II , 12] ; consider a G-invariant metric g(P) on a principal bundle P of base H and struc­ture group 6 , then this metric can be reinterpreted in terms of objects defined on H : a metric g(M) » ( </ . ) on M , a gauge field A(M) -(A^ (x)), *< - I, 2, ..., n - dim £ , valued in "J - Lie(G) , and «(»»*•) - component scalar field h » (h„. 00) which, for a fixed x, determines a right-invariant metric in the copy of G above x . Reci­procally, the data consisting of these three objects allow us to recons­truct back a unique G-invariant metric on the bundle. For example, if we believe that Che "real world" can be described by a space which is a "lo­cal product" of space-time M and the color group SU(3) - the internal space - then an S0(3)-invariant metric on this 4+8 • 12-dimensional spa­ce F splits into a gravity field on M , a color field, and scalar fields hj. (x) characterizing the "shape" of the internal SU(3) space above x e,M (notice that there is a 8»!} = 36 -parameter family of right invariant metrics on SU(3) ).

The scalar curvature <J associated to a G-invariant metric on P is constant along the fibers and can be written entirely in terms of the fields on M ] >

where C w , being the structure constants for G .

(2.1.1)

CPT-84/P.161I

(J'(M) « c h e scalar curvature of M , and ^(G) i s the scalar curva­

ture of the copy of G over X , which can be interpreted as (minus) the 2)

potential term for the scalar fields

w..Cltc^a\^^c!f $) ., 1.2)

If one assumes that G(SU(3) in our example) takes, at each point x £ H ,

its most symmetric standard shape (the G x G-invariaat Killing metric),

then 3> n,,*'0 and we recover the usual Einstein-Yang-Mills ac-

tion ; in that case ~S is just, a (cosmologlcal) constant.

Let us end this subparagraph with a mathematical application :

construction of all possible SV(2)-invariant metrics on the seven sphere

S . First we realize that S can be written as an SU(2) bundle over

S (the Ropf fibration), therefore a direct use of the above theorem

tells us that in order to construct a general SV(2)-invariant metric on 7 4 S one must choose I) an arbitrary metric on S ,2) an arbitrary Yang-

i

Mills field defined on S with values in Lie (SU(2) ) , and 3) an arbi­

trary St)(2)-invariant metric on each copy of SU(2) above the points of

S (there is a «"-parameter family of such metrics).

5Êmark_2iiil

The last term of the Eq.OM.I) gives no contribution to the field equation if

one takes for the Lagrangian L B S P V l 9 ( n ) | » however, it has to be

taken into account if the volume density is taken to be y ! g ( p M

1) The factor in the kinetic energy term of scalars is - i. and not - 4 as

erroneously printed in [ ll .

2) The last term in (2.1.2) vanishes for all unimodular groups and will be

omitted in the rest of this paper.

CPT-84/P.16U

5

2.2 - Invariant metrics on bundles with homogeneous fibers

What we will need in the following is actually a gep~ralization of the previous theorem to the case when the internal space is not a group

3) G but, more generally, a homogeneous space G/H such a generalization was obtained in [ 1 J , where the following result (which we call the Re­duction Theorem) was proved.

Reduction Theorem : Consider a right action of G on a local product bundle E sf M x(G/H) with base M , whose fibers are homogeneous spaces isomorphic to G/H • Then there is a one-to-one correspondence between G-invariant me­trics on E and triples^( \(L* » A"L « h«*p ) , where V^ is a me­tric on the base M , A is a Yang-Mills potential with gauge group t?(H)|H , N(H) being the normalizer of H in G , and the scalar fields "„,•, (x) describe a G-invariant metric in the copy of G/Il above x .

Let us recall that the normalizer N(H) of H in G is the biggest subgroup of G in which H is invariant ; we have

-I aeG : aH = Ha The tangent space at the origin of G/H (indices •( , % , . . . ) i s isomorphic to *& / ZX. and is ususally not a Lie algebra, but i t contains as a subspa-ce the Lie algebra of N(H)/H (we use £ , £ , • • • • for the indices of this subspace). The scalar curvature of E endowed with such a G-invariant me­tr ic , can be, here again, expressed entirely in terms of M-based f ie lds , and we get [ 1 j an expression similar to Eq. (2.2.1)

* ( e ) _ ^ ( M ) + -srx ( 6 / H ) - 'Ar^ r W

where

3) Throughout this paper G/H denotes the space of right cosets 2Ha:a£G f , This i s opposite to the usual convention. '

CPT-84/P.1611

6

j>„ K, =""} ^ - 4 A* *^ - 4 A / ^ r and

The index jf' in (2.2.2) runs through the Lie subalgebra

the stability group : # C =Lie(H).

Example I : If we take as a model a space E which is a local product of

space-time M and the 16-dimansional complex Stiefel manifold SU(5)/SU(3),

then an SU(5)-invariant metric on this 20-dimensional space E can be cons­

tructed out of a gravity field on M , a Yang-Mills field with gauge group

SU(2)x 0 ( 0 , and a scalar field h w ^ (x) characterizing the shape of the

internal SU(5)/SU(3) homogeneous space sitting above x & M . Notice that

the Lie algebra of the normalizer of SU(3) in SU(5) is indeed the Lie

algebra of SU(3) x SU(2) x U(l) .

ç.7 S 7

Example 2 : We take as a model the 1 l-dimensional space £ = J * J

which is constructed as follows : by using the right action 7 7 7

p e S , g 6 SU2 f g g S of SU(2) on S , we define the following 7 7 7 7

diagonal right action of SU(2) on S x S : (p,p')<= S xS , gfeSU(2j

(pg»p'g)€. S x S and define E as the coset space obtained via the dia­

gonal action ; of course we use the fact that S is indeed a SU(2) bun-4

die over S (Hopf fibration). Let us now show that E can also be written 7 L

as a non trivial S bundle over S ; indeed if we define a left action

of SU(2) on S by gp « pg we can construct an associated bundle

|_class(p,p')/p,p'£ S 7 J where class(p,p*) = [(pg,g"'p')/g6 SU2 j =

{ (Pg»P'g)/g & SU(2)j . This associated bundle therefore coincides with

E but it has now base S and typical fiber S . Now the group G=Sp(2)

aces transitively on S (S = G/H » Sp(2)/Sp(l) ), the normalizer of Sp(l)

in Sp(2) being Sp(l)xSU(2) ; then G - Sy(2) acts also on E , there is only

one stratum , all the orbits are of the same type (S 7) and the manifold of

orbits is S . The most general Sp(2) invariant metric on E can be construc-CPT-8<f/P.1611

7

ted out of a gravity f ield (metric) on S , a Yang Mills f ie ld A de-fined on S , valued in SU(2) and a scalar f ield h- . . (x) characteri-

7 " 4 zing the Sp(2)-invariant metric of the copy of S above x £ S (there i s a 7-dimensional manifold of such metrics j j ] ) • This example could be generalized to the study of G-invariant metrics on spaces of the type E «• ^ H * g / n considered as bundles over G/N with typical fiber G/H.

ilffvlH Other examples are given in [I J . Notice that when G/H is an

isotropy irreducible space (i.e. when Ad(H) acts on *}" /^C real irredu-cibly), then N(H)/H is discrete. In such a case the Yang-Hills potential is trivial. More material on the subject can be found in [_13J , [_14J •

3 - SYMMETRIES OF A PRINCIPAL BUNDLE

In this section we will introduce the principal concepts and no­tation used throughout the rest of the paper.

3.I - Symmetries of Yang-Mills fields

Let (U, 71", E, R) be a principal bundle with base E, projection 77" , and Lie structure group R acting on l) from the right. Let S be a Lie group acting on U from the left by bundle automorphisms, i.e.

s(«.r) • (s «Or , V s e S r e R ,

-see Fig.I. We shall assume that the action of S on U is effective i.e. su » u r u £ D implies s»e . The action of S on U induces an action of S on E :

s 7T(u) S 7T(su)

This induced action on E needs not to be effective - thus we allow for pure gauge transformations also called vertical automophisms (coapare [ 1 5 - 1 7 ] ) .

CPT-84/P.16I1

8

We will also use the right action of S on 0 and on E defined by

us = 8~'U, u £ u , s 6 S ,

ys S s ' y y e E , s * S

When a possibility of confusion can arise, we shall write L and R

for the left and right action respectively.

R s I-r , '3

*(wt)=<f<0'1

u

usir(«) *•(««.) = s^

Denote by the tie algebra of R, and let, for each v é 6 ( , Z denote

the fundamental vector field on U generated by v :

V u ) * l t [ " u e x p ( t v ) ] t-o

Recall that a I-form c<i , defined on II and with values in 6{_ , is a 1-

form of a principal connection if

CO (Z„) - v ,

and

R*0>= Ad(r)~'ft) , r t B ,

The group S, introduced above, i s said to be a symmetry group of &1 if ,

CPI-84/P.I6U

9

for all s e. S ,

HT to U (3.1.1)

The content of the above equation was discussed in many papers and we re­fer the reader to the existing literature [6 J , [7 J , [l8 J - L 2 1 ] • The brief discussion we give below has the purpose of introducing notation and concepts we will use later on.

Global description of a gauge field involves a principal connec­tion I-form 03 . Locally a gauge field is described by an -valued 1-form on E rather than on U. Let C" : E — > 0 be a local cross-section (gau-ge), then A - the Yang-Mills potential in the gauge «- - is defined by

Let S be a symmetry group of co , and let us see what can be ' said about the local representative A . Of course the cross-section" <*~ will not be, in general, invariant under S. Its noninvariance is described by a compensating function r B. r (s. y) taking values in the gauge group K, defined by (see Fig.2) :

«"(ys) • (_<T(y)s ~\ r (3.1.2)

•*v-

r

Because of noninvariance of the gauge <r- , also the Yang-Mills potential A wil l not be invariant j indeed, using (3.1.1) we find

CPT-84/P.I61!

10

.TO. -'»*" -I J^ (3.1.3) = A = n. A a + n, «n. •

Therefore R A differs from A by a gauge transformation. Let ^ be the Lie algebra of the symmetry group S. With J 6 A

let 11-» exp(tj) be the 1-parameter subgroup generated by 3 • D en°te by Z the fundamental vector field generated by T :

25 w ib-^L. With s = s(t) differentiate (3.1.3) with respect to t at t = 0. From the very definitions of Lie and covariant derivatives, we obtain

L^A- . >A"(S) ,

where

ror a f ixed ytf E, A ^. • i j ) i s a l i n e a r map from •A to 6? . I t w i l l play an important r o l e l a t e r on. Let us analyze i t a l i t t l e b i t c l o s e r . Choose yg, E and denote by I the s t a b i l i t y group o f y

I = J s e S : sy « y / .

Choose u & it (y) » and l e t , for every s-6.1, /Au(s) be the unique element of R s a t i s f y i n g

su • u Au(s) ( 3 . 1 . 4 )

( see F i g . 3 ) . Then Au: I —*R i s a homomorphism of Lie groups. Compa­

ring ( 3 . 1 . 2 ) and ( 3 . 1 . 4 ) we f ind

CPT-84/P.1611

Il

(s.y) <r(y)

s<sl

and thus s t. . > Jt (s ; y) , when restricted to se 1 , is a group homos»r-

phism. Let ï sLie(I) be the Lie algebra of I. It follows then that A

restricted to J £, "3 is a Lie algebra homorphism, which coincides

with the derivative *<r/, of restricted to p(«) "* ^ r t o ' W e t h u s s e e t h a e

J depends only on the action of S on U, and not on the connection. What does depend on W i s the restriction of A to a complement, say

, of 1 in >^ . Write / i = with Add)!? C G* (reducti­ve decomposition), and define Q (. . u ) to be the restriction of

A to ( ? . The f ie ld 0 defined in this way wi l l be later on interpreted as the Higgs f ie ld resulting from dimensional reduction of the Yang-Mills f i e ld .

H

"V-

»U. 3 ««-X \>)

u

J - 1 ^

3.2 - Several bundle structures of the principle bundle U

As before, consider a principal bundle (U, E, TT, R) with base E

and structure Lie group R, and l e t S be a Lie group acting on II by bundle

automorphisms. We want to discuss now in more details the structure arising

from such an action. We wil l assume that both R and S are compact although,

CPT-84/P.16I1

12

with proper care, our discussion could be carried through for non-compact groups admitting biinvariant, nondegenerate metric, with essentially the same results.

We recall from the previous subparagraph that there is an induced action of S on the base E. For every y £. E denote by Sy the stability group o f y ! S y - J s € S : s y - y | . There are then two natural equivalence rela­tions in E. One is : y ~ y ' , if Sy and Sy' are conjugate ; the other, stronger one, is : y se y' iff Sy " Sy'. The equivalence classes \y\ • <y'£ E : y'"»y } are called strata. In general E will be an union of seve­ral strata and, since S is compact, one of them will be open and dense (see {22J , J23] ) . Let us restrict our further discussion to one of these strata. Or, better, let us assume that E consists of a single stra­tum only . Thus Sy and Sy' are conjugate for all y.y'fc E . Consider now the second equivalence relation. The equivalence class {Cyj "J y'£ E: y'sfe yl is conveniently called the substratum of y (.24 J • He choose, once for all, one of these substrata, call it P, and denote by V the sta­bility group common to all the points of P :

P - J y £ E s Sy - I ]• ,'

P is a submanifold of E, and E can be thought at as a collection of orbits of type S/I, the collection being parametrized by a manifold H - the mani­fold of orbits. In other words E is a fiber bundle of base M and fiber S/I ~ see Fig. 4. It is important to realize that P is a subbundle of E, it is a principal bundle with the same base H and structure group K s N(I)/I,N(I) being the normalizer of I in S (see [22] , [23j , also Tl J ) .

For our further discussion it is important to notice that the di­rect product group G*=SxR acts on V via the following right action

G a SxR 3 (s,r) : u _». s ur .

With the terminology introduced above we will assume that this action is

4) E is then called simple S-space and the action of S on E is called simple

CPT-84/P.I6I1

13

^ ^ , - r

•t

• M

Fi& _4 : Notice that E - [ y ] and P » f[y]j . P i s a subbundle of E. The Killing vectors Z> , J e A > introduced in Section 3.I are vertical in E, those corresponding to 5 £ Lie(N(I)) are, at the points of P, tangent to the submanifold ? .

CPT-84/P.16J]

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simple i . e . that U consists of a single stratum. lo f ix a substratum Q of

U le t us choose u £ U such that T (u) ç P (see (2.3.1)) and le t H = Gu be

Hie s tabi l i ty group of this u. We take for the substratum QCD the sub­

stratum characterized by H :

| u e U : i f f (s ,r) é H ) •

Since Q - fru^j and TT(u)C V, it follows that TT (Q)C P. As before U

is a fiber bundle with (the same as E) base H and fiber G/H, and (Q,M,N(H)/H)

is a principal bundle. The space II can therefore be fibrated in several

ways and Fig. S below summarizes the results.

u

1 '.-'I u u

-*-*:

7(H) IH

M

M * K

V j

I) Other fibrations are also possible but we wil l not need thefti here, see

[22] •

CPT-84/P.161I

15

In Sect. (3.1) we already introduced the group homomorphisms ' S ^ ^ — » "•

Let us observe that A remains constant while u varies in Q. Indeed, by

the very definitions we have G u - j(s, > u (s)):sfeS „./„)$ • "hen u " H » 3

through Q then & u S H and S fr-ru\S I are both constant and thus \ ï \

is constant too. Thus we have

H - j(i. A U ) ) s i «.I j " diag (I, A ( I ) ) c G

The stability group H is isomorphic to I, but it is not equal to

Ix^e[ . Because of this fact the normalizer of H in G - SxR is not iso­

morphic to N(I)xR. Let us see what is the relation between N(H) and N(I).

By using the definitions we find

,aX(i)«C* a X(sU-) V'x Ê X .

Consider the centraliser Z of the image X(I) of I in R :

By the embedding Z—» Z x ^ ej , Z can be considered as a subgroup of

N(H)/H, and one easily gets the'following important result (. 8 J .

EïSE2§i£i2a_3i2iI-: z i- 8 a n invariant subgroup of K(H)| H, and N(H)J H is

locally isomorphic to (N(I)| I)xZ, N(I) being the normalizer of I in S.

3.3 - Lie algebra decomposition and the vielbein

We introduce the Lie algebra ^ , "3 , JPcD.^of S, I, H(I),

K-N(I)/I and decompose the Lie algebra y^ of S as follows (compare [iJ.M)

CPT-84/P.16I1

<? 16

and let us introduce the Lie algebra bases T„j , T j , T^ . I*, X^ in

3 , TC. @, 2 , and Û^. respectively - see Fig. 6

jrç)<

A

n T,

X Ta >X

x -

>X 9

Fig. 6 : Decompositions of the Lie algebras A and

*L denotes the Lie algebra of the centralizer Z of A(I)-

The honumorphism )i:I ^R introduced in 3.2 induces the homonor-

phism X : Ï , 0 , of Lie algebras, and we define the matrix elements X*"

CPT-84/P.1611

17

of X »>y

* ' ( x i ^ - X i ^ The Lie algebra <fl of the direct product group G * SxK can be decomposed in

two ways s \ ->

= 3 + 36+ + ft = #+*;+*-*- JL

where

is the Lie algebra of Che stability group H. It is Che second decomposition which is used in the reduction theorem for G/H. To apply this theorem we have to introduce a new basis in tt , with tilda, adapted to the second, reductive, decomposition:

X - X - * t "0 w ,.1UT; (f/*0 X * X ( * ) .

ca) (3.3.1)

It is important to notice that the Lie algebra of N(H)| H is composed of 36 and 2(see Proposition 3.2.1 , also [ 8 J )

Lie (N(H)j.H) - 3C + Z, (3.3.2)

We shall use the index A - ( 4 » i) to label generators of this algebra.Thus

CPT-84/P.161I

The structure constants of the adapted (non-product) basis in '6 =j{ +• d , which differ from those of the product (non-tilda) basis, are the follow­ing ones

%$ = ck ** V - C*P- x £ ft.

(3.3.3)

Observe that

C^ - Cl

4 - o <"•*>

Let now gjj be a G-invariant metric on U. Being, in particular, S-invariant, g 0 induces a metric g_ on E (Sect. 2.2), and g„ is S-invariant. Thus g_ induces a metric g„ on H. We recall that H - E/S * U/G , therefore g„ can be also induced directly from g„. For our purpose it will be enough to assume that g„, restricted to the fibers of the principal bundle U — > E, induces a fixed biinvariant metric k on R. He will call it the Killing me­tric.

Let u be a point in Q, y - its projection in F, and let x be the projection of y on M (see Fig. 5). We introduce the following vector fields:

0^ - a holonomic moving frame (vierbein) around xgM e - the horizontal, i.e. orthogonal to the fibers of E-»M,

lift of " ^ e^ - fundamental fields e x s z_ in E e M " ^ef* ,e** ~ t' i e vi-el°e'-n around y E K - fundamental fields E K s Z_ in U E. - fundamental fields E.e- Z_ in U E. » (E^ , E.)-vertical vielbein around u in V E* - (E£ , E£)-vertical vielbein around u in Q E, - the horinzontal, i.e. orthogonal to the fibers of U—» E,

lift of e .

CPT-84/P.161I

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The components of the fields appearing in the following discussion will refer to the vielbeins introduced above. In particular we will use the following notation (some of the formulas will be explained later) :

8AB s *u' EA , EB* " *8*|* '***• * 8ij*' 8 = ( t h e inverse of g A B)

bMN W V = ( V J *h*<P ) ' h M N* ( t h e " ^ " e of h^) (3.3.5) A A u*IL h«*p*«E (V' e

f> = B U(E^ ,E p ) , h =r(the inverse of h^)

/ • • *

^ = 8M(f/.»£^) * 8 E ( > 'e«> > * V » ( t h e i O T" S e ° E P Let td be the principal connection form on II induced by g (considered as R-invariant, see Sect. 2.1).

We define the fields <p by

«* ( c - ( -> ) = - <£c~) T. , <3-3-6> Then one easily finds the following relations

i *f "? g » h + 0 ^ J k - . . g * b ,

ii . *i u*1!4 M i » * j - - # « k i j • 8 " h P * «

(3.3.7)

where k.. are the components of the Killing metric of R. The G • SxR-inva-riance of the metric gy implies that <p » ( P*), considered as a linear map from ^p to 5? » satisfies the constraint of Ad(H)-invariance

j6o Ad(i) •= Ad(XCO) o j6 , i £ I (3.3.8)

or, infinitesimaly,

CPT-8A/P.I61!

20

In particular ^ - 0 i f i ^ i and 0 ^ - 0 i f ^ ^ £ , which means

that <p<X) C x anà that 0 (X) /J £ * 1° \ •

We also have

C S p ")f£ •+• ^"iS" " P î f = ° (3.3.9)

which expresses the I-invariance of h^. .

4 - REDUCTION OF THE EINSTEIN YANG-MILLS ACTION

4.1 - Outline of the method

As it was already explained in the Introduction, our aim is to inves­

tigate dimensional reduction of S-invariant Einstein-Yang-Mills fields in a

multidimensional universe. Thus we start with an S-invariant metric g £ on E,

and an S-invariant principal connection CO on U. We also fix a biinvariant

metric k. . on the initial gauge group R. The logic followed in this section

is summarized by the Tables I and II.

Table I stresses the equivalence between three possible descrip­

tions of the same geometrical structure (we called them Descriptions 1, 2

and 3 in our Introduction). The links between any two boxes of this chart

are provided by the use of the Reduction Theorem (recalled in Sect. 2). The

main goal of our work is to connect the Descriptions 2 and 3 (Link N*l),

and this will be done via the Description I.

Table 2 summarizes the formulae associated to the links LI, L2,

L3, L4, L5 of Table 1 via the use of the Reduction Theorem ; all these re­

lations are of course obtained and analyzed in the subsequent paragraphs.

We introduce the following notations : YM (base, group) denotes the lagran-

gian for a Yang-Mills field defined on the space "base", valued in the Lie-

CPT-84/P.1611

21

algebra of "group" ; KE (base, fiber) denotes the kinetic energy term contain­ing "base" derivatives" (eqs. 3.1, 3.4) of the metric on the space "fiber" ; finally, the scalar curvature of a space F is denoted by ~& p.

4.2 - The Link H°2

We start with the Description 2 : a multidimensional Universe E furnished with an S-invariant metric g(E) * (lu.,) and an S-invariant Yang-Mills field Aj. with values in the Lie algebra Jt of R ; the S-invariance was discussed in Sect. 3.1. The Einstein-Yang-Hills Lagrangian is given by the expression

EYM(E) - "5(E) - \ ^. h 1 4 0 h 1* FJfljFJp , (4-2.1)

where 7(E) is the scalar curvature of E for the metric K-, and F„„ is the Yang-Hills field strength associated to A?; . Provided that we add to this expression a constant ~GW with value equal to the scalar curvatu­re of the group R (endowed with the Killing metric) we can use the Reduc­tion Theorem of Sect. 2.1 to construct, out of these three pieces, an R-invariant metric g(U) on U considered as an R-principal bundle over E.This metric g(D) will be actually SxR-invariant because we started with the in­gredients which were themselves S-invariant (here one also exploits biinva-riance of the Killing metric k). The scalar curvature of U, associated to this metric g(U) is

T5(U)U - S m y + 'S'(R) + YM(E,R)y (4.2.2)

where

VM(E,R) - { k l j . » l . ' ' 4 ( 7 ) 4 W l

u being any point in the R-fiber of D over y. This geometrical structure, described in terms of the space U, constitutes what we called the Descrip­tion I in our Introduction.

CPT-84/P.1611

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Table 1

A. G » SxR invariant metric on U 1 !

la

ï L2 rr__:

. S invariant metric h ™ on E.

_ .S invariant Yang Mills field Ai on E " valued in Lie (R).

.Killing metric k>- on 1.

.S invariant metric h on S/I parametrized by x e M. A r

. Yang Mills field A* on M valued in Lie N(I)/I.

. Metric V on M. th.*

L3

Metric if', on M.

Yang Mills field (A,. ,b\.) iHH)/H S« NC

G invariant metric g.- o n G/H parametrized by xeH

Yang Mills field (Au ,k*i) on M valued in the Lie algebra of H(B)/HSi N(I)/I x Z(A(«)

1 R invariant metric k.. on R

S invariant metric h^. on S/I (parametrized by xcH).

S invariant Yang Mills field 0^ on S/I, valued in Lie(R) and parametrized by x«M ; the Higgs field.

L5

where

U

and

Vr

_ _ _ _ _ _ . - _ - -a

I • typical stabilizer of the S.'action on E

H(I) • normalizer of I in S

^(1) - homomorphic image of I in R characteri­zing the S action on U.

Z(A(I))- centralizer of >,(!) in G = SxR.

M

23

Table 2

" ! *(n> " r ( E ) + zm *™(E'R)

L3 : r ( E ) - Ï M + 2 - ( s / I ) + Vli(M,N(I)['l) • KE(M,S/I)

L4 : ^(U) " (M) * ^CG/u) + ™<M»N<H>|H> + KE(M,G/H)

" ! ^G/H) " r ( S / I ) + ^R> * Y M < S / I ' R >

Also

ÏM(M,N<H)/H) - YM(M,N(I)JI) + YM(M,Z(^(I)) + ù. cf. Eqs.4.4.2 to 4.4.5

YM(S/I,R> - - V ( A , $ ) cf. Eqs. 4-5.3, 1.5.4

KE(M,G/H) - KE(M,S/I) + KE($) cf. Eqs. 4 .3 .5 , 4.4.6

CPT-84/P.16I1

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4.3 - The Link N*3

This link i s a standard application of the Reduction Theorem of Sect. 2.2 to a S-invarisnt metric g(E) on E. For the scalar curvature of E, endowed with this metric, we get (see (2.2.1) and (2.2.2)) :

S (E) - "SOI) + -5 (S / I ) + YM(M,N(I)l I) + KE(M,S/I), (4.3.1)

where, with the notation given in Sect. 3 .3 ,

R(M) - R( ) £ j ) (4.3.2)

•**' 1 * <* 1 ft' * < ' t I1 . R(S/I) - -h (•£ CM|. C , V + r h r h C„. Cy.'+C* C . v

/•/>' > " » « _ $ (4.3.4) YM(M,N(I)J I) - - { h ~ , y ^ F ^ 7 . . , ,

KE(M,S/I) - - •£ fc?P h f (D h.,f l T h - j + D ^ h D"h s )

^ D V (4.3.5)

CPT-84/P.I6I1

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4.4 - The Link M*4

As explained in Sect. 3.2, U is also a G/H bundle over H (recall that, since V a ExR, the manifold M of G=SxR-orbits in U is the same as the manifold of S-orbits in E) ; here again we can use the Reduction Theo­rem. The G-invariant metric g(U) on U can be expressed as being built out of the following three pieces : a metric g(M) = ( Vlj) on M (usually in­terpreted as space-time metric), a Yang-Mills field A^ valued in Lie (N(H)/H) « X + * (see (3.3.1)), and a scalar field h A B(x) which can be interpreted as a G-invariant metric in the internal space U ~ G/H above

x xtM. The scalar curvature "J» (U) of U can now be written entirely in terms of M-based quantities :

C u(U) » ^(M) + "BX(G/H) + YM(M,N(H)| H) + KE(M,G/H) (4.4.1)

where

Z f r / m , « ( , I ? C ?» I BB" -C AC' ? C ? B 5(G/H) » -g ( ? C A B C A , C + j g g c c , C A B C A < B , + C-B C A , £ ) ,

YM(M,N(H)| H) -~\Y Y gjg £ F ^ ' ,

i . i AM rn *** ^ # ^/ KE(M,G/H) - - { ^ • " « " «V 8AC ^ gBD + V 6AB D>> W •

and

V 8AB " \ 8 A B + AC A £ ha * *lt A £ «, 'AD

These formulae may be understood as referring te a certain local cross-sec­tion <S~ : H » * - * r ( i ) e Q c O . The ti ldas refer to the reductive basis in

<£. (3 .3 .1 ) . To reduce further the above expressions we apply the relations (3.3.3) and (3.3.7) with the result

YH(H,H(H)| H) - YM(M,K) + YM(M,Z) + & (4.4.2)

CPT-84/P.16JI

26

where

™(M,K)=-{y ^ ^ V V ^ ( 4 * 4 , 3 )

YM(M,

and

, / t ' '>' à - - i V f kîj V V ^ Î V "*/-J ) ( 4 ' 4 , 5 )

He also get

KE(M,G/H) « KE(M,S/I) + KE(0 ) ,

where

with

« < • > - - £ h - P y ^ k y D^ <£„ i D,, ^>", (4.4.6)

V*-1 » V*1 + c*r A £ p* + 4* * £ ' - (4>4-7>

Notice that when K » N(I)/I (resp. Z • Z(X(I)) is discrete, then the 2nd

(resp. 3rd) term of (4.4.7) vanishes as well as (4.4.5). We omit the result

one gets for 3 (G/H) since it will be derived in a different way in the

discussion of the Link n*5 below.

CPT-84/P.161J

27

4.5 - The Link H*5

The tern S"(G/H) in (4.4.1) is the scalar curvature of the fiber U of the bundle (U,H, G/H). It is easy to see that the projection

TT : U — > E makes U S£ G/H into a principal bundle with base E — S/I and structure group S (Fig. 7) :

^ V I

Figure 7

The metric g . ( see (3.3.4) i s SxR-invariant on U and therefore, a fort ior i , It-invariant. He can apply therefore the Reduction Theorem in i t s principal bundle version (Sect. 2.1) : g .- can be expressed entirely in terms of an (S-invariant) metric h j * on E , an (S-invariant) Yang-Mills f ie ld d> p x f*< on E , valued in Ji , and the metric k.. on R (the Killing metric). The x i j expl ic i t expression of g . , in terms of i t s building blocks has been already given in (3 .3 .S) . Applying the Reduction Theorem of Sect. 2.1 ue also get

Z (G/H) = - S , (S/I) + "S"(R) + YM(S/I,R) (4.5.1)

where ~C (R) and 7(S/I) already appeared in (4.2.2) and (4.3.1) respecti­vely. Let us discuss now the Yang-Mills term. Since S acts transitively on the base E —S/I of our bundle, we are in the situation of the Hang Theo­rem (see e.g. £ 2 S J ) , and we could simply refer to the literature. How­ever, it is both instructive and convenient to obtain the desired expres­sion directly. Denoting by tij - (<»£) and F » ( F * ) the connection and

CPT-84/P.1611

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curvature form of the induced Yang-Mills field, from the very definition of the curvature we get

F * p - D r t ( E - • E P > - E « " < V " E n f t , ( E J - 4 j ( I E K • SV+

+ [«<£„), A)(Ep )] ,

while the S-invariance of CO implies

0 - ( L ^ c d ) ( E | 1 ) - E^ CO ( E p ) - *> ( [ E v , E^-J ) ,

combining the two formulae we get

F * -«"'Cj E^ , Ep] ) + p ( E ^ ) , «J(E,i ) ] (4 .5 .2 . )

Now

v IT o' V

and we can use the formula (3.3.6) as well as the fact that Ey- + X^E. vanishes on Q (see (3 .3 .1 ) ) , to get ~ ~

and consequently

The Yang-Mills term of (4.5.1) gives therefore the potential energy for the Higgs f ield ft :

-V(0> =• YMCS/I, R) - - ^ h""* h " k £ j F^p 1 F j (4.5.4)

with F l given by (4 .5 .3 ) .

CPT-84/F.1611

29

4.6 - The Link »°1

By simple collecting of the results obtained so far we find the

following result .

Einstein-Yang-Mills Reduction Theorem. There i s one-to-one corres­pondence between pairs (g(E), A(E)) of S-symmetric Einstein-Yang-Mills sys­tems on E and the quadruples (g(M)> A(M), jt ,h) of f ields on M. The Einstein-Yang-Mills Lagrangian EYM(E) of E, when expressed in terms of the component f ields on M reads

EYM(E) = « ( y / ) + YM(A^,A^ ) + KEfl»^ ) + KEC^1)

- V(h) - V ( £ ) (4.6.1)

where

"S5(YJt is the scalar curvature of the metric g(K) = ( "jj j) YM(A* ,A* ) = YM(M,N(H)/H) i s given by (4.4.2 - 4.4.5) KEOIAP ) - KE(M, S/I) i s given by (4.3.5) KE(^ X) - KE(<M is given by (4.4.6 - 4.4.7) V(h) = Z(S/I) i s given by (4.3.3) V(< ) i s given by (4.5.4)

As a by-product we also get the- reduction formulae for the Yang-Mills term alone :

YM(E,R) = YM(M,Z(A.(D) + KE( ) ) * A - V(<j>) , (4.6.2)

where, however, the terms KE(4) and A depend on the connection A,_ in the (P,M,K) bundle too.

One has to remember that the scalar fields <j>, V and h . satisfy the algebraic constraints (3.3.8) and (3 .3 .9 ) .

Remark 4.6.I : The expressions (4.6.1) and (4.6.2) can be s t i l l multiplied by |det h | ' , compare Remark 2.1.1

CPT-84/P.1611

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5.1 - Summary of the results

Space-time i s , in this paper, identified with the manifold H of orbits of a compact group S (global symmetry group) acting on a manifold E (extended space-time, multidimensional Universe). Thus each space-time point x&M has internal structure of a homogeneous space S/I. I i s the i so -tropy group characterizing the orbit of S over x.

Let g(E), A(E) be an Einstein-Yang-MilJa system in E, consisting of a fcseudo) Riemannian metric g(E) and a Yang-Mills f ie ld A(E) on E with gauge group R. We show how such a system can be interpreted in terns of f ields on M, when a constraint of S-symmatry is imposed. It i s proved that there is a one-to-one correspondence between S-invariant Einstein-Yang-Mills systems (g(E), A(E)) and quadruples (g(M), A(M),£,h), where g(M) is a. metric on M, A(M) i s a Yang-Mills f ield on M with the effective gauge group N(H)/H described below, while <p and h are scalar f i e lds . g(M) and h originate from g(E), $> originates from A(E), while A(M) takes i t s ori­gin from both g(E) and A(E).

The effective gauge group i s the quotient N(H)/H, where N(H) i s the normaliser of H in G = S x R, and H C G i s defined as H = d i a g ( I x > ( l ) ) ,

X : I —» R being the group homomorphism determined by the action of sym­metry group S on the gauge f ield (see below). Locally N(H)/H i s isomorphic to the product (H(I)/I)xZ, where N(I) i s the normaliser of 1 in S, and Z i s the centraliser of A (J) in R.

The homomorphism A ! l —»R, where I i s the s tabi l i ty subgroup of S at ye E, i s defined by the action of S on the principal R-bundle U over E on which the i n i t i a l gauge f ield A(E) l i ves . Let u £ U be such that n"(u)«y. Then for each s 6. I

su = u A ( s ) .

The derivative A : 0 —> K of X i s a homomorphism of Lie alge­bras. A and p are parts of the map J\\ .A—»S%_ defined as follows : every J f i - i i s an infinitesimal symmetry of Ag. Thus the Lie derivative

CPT-84/P.161I

31

Z»A_ i s an infinitesimal gauge transformation - there exists a func­

tion A s y - » ^ y < " 5 ) e */£ such that L * A(E) = D A ( Ç ) . For J e I , A ( f ) - ) / ( $ ) . On the other hand write A- ï t * P with |J> such that

A d d ^ C <? . Then <}> ( J ;x) - A y ( $ ) for x - 7T(y). Thus 0 (x) i s

a linear map 0 (x)s <?—"><X . I t i s constrained to satisfy

<)> o Ad(s) « Ad( A(s)) o £ , s £ I .

The f ield h, originating from g„, describes an S-invariant metric on the

homogeneous space S/ T at x ; algebraically h "(h, , . ) i s an Ad(I)-invariant

scalar product on ? •

The Einstein-Yang-Kills lagrangian for (g E , A_) on E when repre­

sented in terms of (g(M), A(M), (p ,h) i s given in equation 4 .6 .1 . .

The f ield <j> sp l i t s into two parts ^ta a n c ' $\sr ' w n e r e

i s the space of 1-singlets in !> and ^C i s a complement of X in <4 . The

f ields h and fix interact with A(M) by the minimal interaction while 0

interacts also directly with the Yang-Mills strength. The potential energy

term for f> and p\* i s quartic.

The results improve those of Ref. L " " " J " the n e w ingre­dient i s the part N(I)/I of the gauge group (with Lie algebra isomorphic to X ) which may be present i f S/I i s not isotropy irreducible.

5.2 - The apace of Higgs fields 0

JR Let us f i r s t remember that we have a linear map A from A i ï l - X t - ï C into <5f » Lie(R) and that A has to coincide on the subal-gebra TJ of A with the algebra homomorphism ,X which caracterizes the S action. The Higgs f ield <p was defined as the restriction of A to the complement 5* of "3 in A ; i t i s also convenient to sp l i t fi in

^ = ^ 3 C + ^C where 4>j^'(^J') naps X into a C <H a n d P^ "^P8

% into a complement W of z in 0{_ (see also sect . 3 .3) . Schematically :

CPT-84/P.161]

^ =

/-

where JT (>'(!)) i s d » normalizer of /(I) in j{ , C(\'(I)) the center

of A' (I), and W a supplementary subspace.

It is easy to show that, under the action of the group N(H), the

map A transforms as follows : let ( <T,P ) £ N(H)C SxR and « 6 / i then

A(s)s7l > f ' A ^ s r ' J ^ fc. *R. (6.2.1)

Moreover (see (3.3.8)), A satisfies the constraint

A (isi" 1 ) - > ( i )A(s ) A(i)" 1 for i f i l (6.2.2)

From this we can deduce the transformation properties of jD— and &.

under the gauge group N(H)/H whose Lie algebra is z+fc ; in particular :

.) K acts on ^ y via the coadjoint representation/and Z via the

adjoint - this can also be seen from the fact that <f> =(& )

has lower indices inX and upper indices in SL.. There are no

constraints on p « coming from 6.2.2.

.) K and Z act as above on the space of jliy fields however, ,

now, 4L, has to satisfy the constraint pi (isi~ )=X(i) 4> (s)^ (i)

for iej; s é A .

The representation of the gauge group on the space of <p fields is usually

reducible ; in order to find which irreducible representations appear, it is

convenient to decompose X into irreducible representations of the product

ÏK by looking at the branching rule for ad S into I.K and to decompose W

into irreducible representations of \(I). Z by looking at the branching ru­

le for adK into ,\(I)Z.

The potential V(£) for the Higgs field can be written as the

norm of the Yang Hills strengh F,. for a connection & l on G/H considered

CPT-84/P.1611

33

as a R-principal bundle over S/I (see Sect. 4.5 ) ; V(^) i s therefore au­tomatically invariant under R and in particular under the subgroup Z.

Therefore the zeros of V ( + ) , which are at the same time absolute minima, are those maps <f> which extend " to Lie algebra homomorphisms ; this observation was already made in J_ 19j . Then i t i s not too diff icult to prove (using (6.2.1)) that the "unbroken" gauge group (the stabi l i ty group of ^ ) has Lie algebra isomorphic to a + % where z C 9 i s the commutant (centralizer) of A (k ) in land Tfe • Lie(N(I)/I) .

b) The potential V(h) = - o"(S/I) for the scalar fields h . , (x) "P

i s more di f f icul t to analyze. First , the scalar curvature 'S (S/I) for the metric h = (h^,) i s not necessarily of a fixed sign (for example, i t i s not necessarily positive even i f S/I i s compact and h 4 g i s positive def inite) . Next, i t i s well known that the saddle points of the functional h —% / Z(S/I ;h)d vol(h) when h varies in the space of a l l metrics with

**• i" t fixed volume element, coincide with Einstein metrics on S/I [_ 29 J .In ma­ny cases, however, saddle points of S 1 C / T ^ when h varies in the space of S-invariant metrics (with fixed volume) coincide also with S-invariant Einstein metrics on S/I. Notice that these saddle points are usually nei­ther minima nor maxima. The potential V(h) is clearly invariant under the whole group of diffeomorphisms of the differentiable structure of S/I, it is in particular invariant under the group N(I)/IxS. All the metrics we are considering (all the fields h^,, ) are, by assumption, S-invariant ; their full isometry group can of course be bigger ; if h° is a saddle point of V(h) and if the isometry group of h* is included in N(I)/IxS (it will then be of the kind FxS, with FcN(I)/I) we will say that the N(I)/I piece of the gauge group is "broken" to F.

5.4 - Normalization and units

a) Sien conventions

If H i s interpreted as a four dimensional space-time with signa­ture -+++, then, the Einstein lagrangian with cosmological term i s —rr—=r U?(H)-4AJ.

GFT-84/P.I6II

34

For positivity reasons, the signature of the "internal" metric on S/I has

to be spacelike, i.e., of positive sign with the above convention. Notice

that the cosmological "constant" is in our case a function of xe.M, how-o

ever, if we expand the internal metric h.,» around some background h — , ,

we obtain indeed a constant (coming from the "5 (S/I) term) to be identi-

fied with the cosmological term, however, if "Ç (S/I) = ^(S/Ijh) is a

positive scalar curvature, then A will be negative and vice versa.

If we choose the signature + on M, the "physical" conclusions

are of course the same but one has to remember that, in order to be space­

like, the signature of the "internal" metric h .- has to be taken nega-1 2

tive : the scalar curvature of a standard 2-sphere S , for example, would

therefore also be negative. In what follows, we assume that our choice for

M is -+++, therefore, h.- (and K..) are positive definite. i|* IJ

b) Dimensions

When computing the Einstein-Yang-Mills action on Eef Mxl/S for

a symmetric configuration of the metric and of the Yang-Mills f ie ld , we

have to integrate over the "internal" space S/I. The integration i t s e l f

i s quite tr iv ia l (because of the symmetry under the S group) and we obtain,

at each point xe M, the volume V(x) of the "internal" space. This quantity

V(x) needs not be a constant unless we impose this new constraint. Let us

however assume that, in the following, V = V(x) i s constant ; i f this is

not the case we wi l l just have to multiply the final four-dimensional la -

grangian by a real valued function on H (compare Remarks £-.1.1) and (4.6.1))

He start with the following dimensionless Einstein-Yang-Hills action :

8

— 2 . . . where K and g are a priori independent quantities. Let us then make the

following changes (in that order) : 2 ~

1) Set h*» « R h„. , R being some (length) scale.

2) Set V - R nV , V being the volume of S/I for the metric "h

3) Define jgVjf - RnV/l6 irK .

4) Define 1/g* - R n + 2 V/16 nK and 1/g* - RnV/g 2

5) Rescale the fields Y 3 I A ^ A ^ , A * J * J<!

«k CPT-84/P.161I

35

We find

J 1

• * n.i * * ,' >»>'£ ^ ?•< PI* T ' " ft ^

-1 H ) «.' t s ÇÎ. H- - Î " f*1*vV *v«

where y = J ^ t X H ) I **"* " * * / ! = "*$* ( R ^

and where . * . ; ,

The pcevious action is still dimensionless but now, all quantities have stan­

dard dimensions : let FLJ be some length, then

i -i- f *• Ï ~V„£ , $;^\y~L" , tf-K-L*

Notice that a l l coupling constants can be expressed in terntof the indepen-••66 2 dent quant i t ies K, R and g R ; K is the Newton constant K « 16.10 cm , R

i s a fundamental length (in cm) and g„ is a dimensionless constant ; the r e -

CPT-84/P.16U

36

lation g 2 - ' 6 ^ K indicates that if g „ ^ 1 then R » I 0 - cm. If we expand s n ->» -*a

the "internal" • metric h,,. around some fixed background h > we ob­tain a constant s "s/I t o *'e i a e n ti fi- e < 1 with the cosma logical term ^ ^ , l 6 l r * R that is A - -^'s/J2** • H e r e > w e d o n o t 8 e t

any contribution from the Biggs mechanism for $ since the potential is zero at the minimum (see sect. 6.3).

c ) Normalization of generators

Notice that in a conventional theory, one introduces usually dif­ferent coupling constants for the simple components of the gauge group, also, one makes use of the Killing metric associated to each simple compo­nent, finally covariant derivative, acting for example on the Higgs field which belong to some representation P of the gauge group, are written by using a representation of the generators with some standard normalisation. In our case, however, the simple components of N(I)/I (or of Z(A(I)) are coupled to the Higgs fields via the same g ( or g R' > also, the covariant derivative acting on the Higgs field is written by using the structure constants of S - or G - (see eq. 5.4.1 ) . i n order to make contact with a conventional theory, one has therefore to specify the representation content for the Higgs fields and to rescale our generators (or the fields) according to some conventional normalisation.

One obtains different coupling constants for the simple compo­nents of Z( \(I)) (or of N(I)/I) but their ratios (mixing angles) become entirely computable quantities.

5.5 - Model building

One can distinguish two classes of models : those where the group H(I)/I is discrete and those where this is not the case ; all models studied so far in the literature belong to the first category |_ 2 j , i_ 3 j [5 J , 130J . One has just to choose an extended space-time E which can be written locally as MxS/I ; when the pair (S,I) (symmetric or not) is an isotropy irreducible space, the group N(I)/I is indeed discrete and we are in the first situation ; notice that tables 1 and 2 of f 1 ] can be help-full to provide examples where N(I)/I is not discrete. In any case, this

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choice being made, one has to choose a group R containing a subgroup isomor­

phic with I (or I divided by a normal -.ubgroup of I), this allows one to de­

fine the group homomorphisra X : I > X (*>c R- *« already stressed,

this map X does not characterize the "geometry" but rather the action of

S on the local product ExR. If we now choose a Lie(R) valued, S invariant

Ïang-Hills field on E and a S invariant metric on E, the dimensionally redu­

ced Einstein-Yang-Mills field lagrangian will in particular contain a Lie

(2( X(I)) + Lie(N(I)/I) valued Yang-Mills field on M. In general one chooses

the group R big enough, in such a way that the centralizer Z(X(D) of X (I)

is not discrete, but one could of course find an extreme situation where

2.1 M D ) is discrete and where N(I)/I is the only piece left ! (see example

below). The Higgs field will now belong to some representation of the final

gauge group Z( A (D)xN(l)/I, and this representation can be found through

the technique explained in Sect. 5.2. If one is now interested in possible

"symmetry breaking", one should use the comments of Sect. 5.3 ; in particu­

lar, if one wants to be in a situation such that the potential V(f>)=0,

then, rather than choosing \ , one can directly construct a global homo-

morphism A from S into R by choosing a group R containing a subgroup iso­

morphic with S (or divided by a normal subgroup). The final work is of

course to restaure the dimensionful constants (using sect. 6.4) and to ana­

lyze the physical spectrum of the model. Let us now analyze several exam­

ples :

1) S - S03 (resp. SU2), I » S0(2) « U(l), R - SU(3). These mo­

dels have already been studied in (_ 3 J . S/I is the two sphere S . The

homomorphism X maps I - SO(2) - U(l) onto a U(l) subgroup of SU(3).

From the one hand, the centralizer of the image is Z = SU(2)xUl/Z_, from

the other hand, the normalizer N(I) of S0(2) in S0(3) (resp SU(2)) is

SO(2)xZ2 (resp S0(2)), therefore N(I)/I = Z 2 (resp. [ej) is discrete.

The Lie algebra of the final gauge group emerging from the reduction of

the Einstein-Yang-Mills system is Lie(SU2xUl) ; in this case the Einstein

part of the Lagrangian on MxS does not bring anything new

(but for a factor Z 2) ; also the h^_ field sopearing in eq. 5.27 is

quite trivial since S admits only one~ up :o scale- S0(3) (resp SU(2>) in­

variant metric : h 4» (x) is therefore a real valued fu ion h(x) and

is even a constant if we keep fixed the volume of S/I ; cne-.u KE(h) - 0

and V(h) is just a (cosmological) constant. The branching ruies for the

CPT-84/P.I6U

adjoint representation of S0(3) or S02 into U(l) and of the adjoint of SU(3) into SU(2) are 3—> T + [ V + l] and 8 -» 3 + f + [2 + 2] where the upper subscript refers to the U(]) eigenvalues ; these eigenvalues can be obtained by writing 3x3 - 8 + 1 and by specifying the eigenvalues of the U(l) genera­tor (hypercharge) in the fundamental representation (3) of SU(3), a conven­tional choice is diag (2/3; - 1/3, - 1/3). The most general A 1 field would map the [ 1 + 1 J subspace of Lie(SU2) into the [2 + 2 J subspace of Lie (SU(3)) ; however we can further specify the model by considering only those <p fields which map 1 into 2 and whose restriction to 1 is just zero.

This last choice allows us to make contact with the phenomenology of the Weinberg Salam model where p is a doublet of S0(2) with hypercharge +1 . Notice that here there is no direct coupling of ^ to the field strengh (eq. (4.4.5)) <f£ is zero since N(I)/I is discrete. Absolute minima V(*) » 0 for the Higgs potential are associated with the existence of alge­bra homomorphism A from Lie(S) into Lie(R) - see sect. 5.3 -. The group SU(3) has two maximal subgroups (defined up to conjugacy) = SU(2>>rtKn and S0(3) ; correspondingly, the Lie algebra of SU(3) has two maximal simple subalgebraj that we call Lie(SU(2)) and Lie(S0(3)) although they are isomor­phic. The homomorphism X from I « U(1)C S - SU(2) into Lie(SU(3)) can be extended in two possible ways : either we set A (Lie(SU2)) » Lie(SU(3)) or we set A (l>ie S) • Lie(S0(3)). Only in the first case the stabilizer of the minimum (the "unbroken" gauge group)- i.e. the centralizer of

A(Lie(SU(2))) in Lie(SU(3))- is not zero, we get Lie(U(l)) ; indeed the centralizer of SU(2) in SU(3) is U(l) whereas the centraliser of S0(3) in SU(3) is discrete ((SU(3)/SO(3) is an irreducible symmetric space). Notice that there is a difference between the two cases because an algebra homomor­phism cannot necessary be lifted to a group homomorphism : there are homo-morphisms S0(3) —> S0(3), SU(2) —>SU(2) and SU(2)—>S0(3) but no homomor­phism S0(3)—» SU(2). One can now compute mixing angles ! 3 . .

2) S » U(2,H) = Sp(2) - 2 by 2 unitary matrices over the quater­nions H. I - tl(l,H) - Sp(l) - SU(2) ; R = SU(5). The "internal" space S/l is the seven sphere S . The maximal subgroup of SU(5) are SU(4)xU(l), SD(3)xSU(2)xU(l) andSp(2)(in this section we no longer mention the discrete Zp f a c">rs>- We choose X as a homomorphism from SU(2) onto an SU(2) sub­group of SU(5). From the one hand, the centralizer of the image is SU(3)xU(l), from the other hand, the normalizer N(I) of I = Sp(l) in Sp(2) is Sp(|)xSp(l) - SU(2)xSU(2), as it is clear by representing Sp(2) by 2x2 matrices over the quaternions ; therefore we get in this case a non discrete Lie group N(I)/I - SU(2). The full gauge group emerging from the reduction of the Eins-CPT-84/P.16U

39

tein-Yang-Mills system is therefore SU(3)xSU(2)xU(l). Besides the SU(2), the Einstein part of the lagrangian on MxS brings us also a non trivial h •_ field ; indeed, the space of Sp(2) invariant metrics on S has di-

r . . 7 mension d » 7 (one decomposes the tangent space at the origin of S into Ad fSp(l)\ real-irreducible representations : 7 - (t+ 1 + I) + 4 and cons­truct an Ad(Sp(l)) invariant bilinear symmetric form, therefore d= —=-+1=7, see also [lj, Sect. 4.1 ) .

Let us now consider the Higgs field <t> ; the branching rule for the adjoint representation of Sp(2) into Sp(l) - SU(2) and of SU(5) into SU(3)xSU(2) are £lOl -~» U~\ + f 1 + 1 + ll + f 2 + "5l and 2 4 — * (8,1) + (1,3) + (JT2) + (3,2) + (1,1)* where the upper subscript re­fers to the eigenvalue of the 11(1) generator Y, these eigenvalues are ob­tained by specifying (arbitrarily) Y = (2/3, 2/3, 2/3, -I, -1) in the fun­damental representation S of SU(5) - as in the conventional SU(5) model of Georgi Glashow - and writing 5x5 - 2 4 + 1 . The most general Higgs field {&* would map the j_ 1 + 1 + I ] + [2 + ? } subspace of Lie(Sp(2)) into the (3,*2) + (T,2) + (1,1) subspace of Lie(SU(5)) J however we can further specify the model by imposing that the restriction of 0 to 1 + I + 1 + 2 vanishes ; in such a way we obtain an SU(3) triplet of Higgs with hyper-charge 5/3. In this theory we obtain a Weinberg angle equal to the one found in the conventional SU(5) model but the analogy stops there : we can find a homomorphism A of S « Sp(2) onto A (S) « Sp(2)c".Su*(4)C SU(5), the centralizer of A (S) in SU(5) is then U(l), the SU(3) group is "bro-en" and the N(I)/I - SU(2) piece of the gauge group stays unbroken ; this example is therefore for illustration only but cannot be used in phenome­nology. Notice finally that, with the above assignent for the Higg„ field, there is no direct coupling between the Higgs field and F*.-» or "£,, (eq. 5.17) , this would not be true if we suppose that ( is not zero.

3) Let us conclude this section by just giving an example where NU)/I is not discrete but where Z( M I ) ) is discrete ; S - S0(5), I = S0(3) and R- S0(4). S/I « V. 2 is the Stiefel manifold of 2-planes in R 5 and M i l » S0(2) » U(l) but if \(I) = S0(TlcS0(4) then Z(A(D) is discrete. Notice that V & 2 » | ° | jfe s 7 - IE! although S0(5) -resp S03 - is isomorphic with S|i - resp 5|i . P

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40

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