Eic/83/32

INTERNAL REPORT(Limited distribution)

International Atomic Energy Agency

and

United Nations Educational Scientific and Cultural Organization

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

THE KALUZA-KLEIK IDEA: STATUS AND PROSPECTS

W, Mecklenburg

International Centre for Theoretical Physics, Trieste, Italy.

MIRAMARE - TRIESTE

January 1983

* Submitted for publication.

TABLE OF CONTENTS

ABSTRACT

The present status of the Kaluza-Klein idea is reviewed,

including a discussion of the fermionic sector». Formalism and

geometrical methods necessary for the formulation of high-dimen-

sional field theories are described in some detail. The mechanism

of spontaneous compactification is explained and a description

of presently available models is given. A chapter on coset space

reduction of high dimensional Yang-Mills theories is included.

The review concludes with an outlook on supersymmetric models.

NOTE

Authors who feel that relevant work of .theirs has not

been included into the bibliography, may send references (includingi

titles) to the following address* Dr. W. Mecklenburg; 11, Contastr.j

20; Msst Germany.

I. INTRODUCTION 1

II. FORMAL CONSIDERATIONS 3

A. The setting of high-dimensional field theories 3

B. Harmonic expansions 6

1. Introductory remarks 6

2. Harmonic expansions: General formalism 9

J. Harmonic expansions: Examples 11

III. SPONTANEOUS COMPACTIFICATIOM 13

A. Models of spontaneous compactification 13

1. General remarks 13

The CS model of spontaneous compactification 14

The minimal gauge group in the CS model 16

Further models of spontaneous compactification 19

The CD model 21

A lattice calculation 22

B. Stability considerations 23

1, General remarks 23

2. Positive energy theorems and instability of the

Kaluza-Klein vacuum 26

IV. THE CLASSICAL KALUZA-KLEIN THEORY 29

A. Introductory remarks 29

B. Th« Kaluza-Klein ansatz 30

1. The five-dimensional case 30

2. The general case 32

3. JBD scalars 36

4. Kaluza-Klein ansatz for the CS model 39

". delation to fibre bundles 'iO

C. Spinors k2

1. Dimensional reduction of the Oirac Lagrangian 42

2. The zero made problem and parity violation 46

3. Rarita-Schwinger fields 50

•V..' HIGH-DIMENSIONAL YANG-MILtS THEORIES 52

A. Introductory remarks 52

B. Dimensional reduction of Yang-Mills theories 5^

,1. Symmetric gauge fields 5*1

2. The bosonic sector 57

3- Symmetric spinor fields 59

C. Modified hi^-dimensional Yang-Mills theories 62

1. Higgs fields and geometry 62

2. Ghosts and geometry 64

VI. SUPERSYMMETRY 66

A. Supersyrometric Yang-Mills theories 66

3. Supergravity 69

ACKNOWLEDGEMENTS 73

APPENDICES

A. The Lagrangian of eleven-dimensional supergravity 7k

B. Generalised Dirac matrices 75

C. Coset spaces, I 76

Dv Coset spaces, II 78

E. Differential forms 8l

BIBLIOGRAPHY 83

mmmtmsMBBium ii: Ivii m m». . 1

ISTHODUCTIOM

In 1921, Kaluza [ii^Jsuggested to unify the gravitational

and electromagnetic interaction within the framework of a

five dimensional gravity theory. The fifth coordinate was made

invisible through a "cylinder condition": it was assumed that in

the fifth direction, the world curled up into a cylinder of

very small radius (10 JJcm, Planck'a length).

During the classical period of "unified" theories,

Kaluza's ansatz was studied from many different angles. An

account of this may be found in the reviews by Ludwig j_/l(j]and

icnrautzer f~ f-TJ ,

In retrospect, Kaluza's Idea that the unifying stage of the

world be a high dimensional space, seems to have appeared

precociously since neither strong nor weak interactions were

known at the time (at least in the theoretical sense). Still,

already thmugb the classical period, the idea has fascinated

many people.

During the last few years the idea has gained in popularity,

event though1 it has not yet provided an answer to any real physical

problem.

But the question of the "true" dimensionality of the world

is surely qrie of the deepest one can possibly ask within the

framework of physics. It is of course far from obvious that

the answer can actually be found there. One would keep in mind

however that relativity theory ^ells us that space-time is not

just the stage whereon the play Called "physics" takes place.

Consider for example the question of critical dimensions.

Their occurence is common in systems of statistical mechanics,

also one encounters them in the lattice versions of field theories.

In field theory, dual string models are a classical example

for theories with ciritcal dimensions: they exist only in a

finite number of dimensions. Actually it has been argued recently

C^zT} that this is probably generally so for field theories of

extended objects. Finally, not unrelated to dual theories, and of

particular importance, w« have superBvmmetric theories which do

not seem to exist in physically sensible verions beyond elevett-

dimensionaJ space-time.

The case for higher dimensional theories is presently

strong, even though the reasons for this are theoretical rather

than phenomenological, This situation is different from that

of ten or fifteen years ago when only occasional references

on the aubjpct could be found.

The objective of this review is to document developments

that have taken place as from approximately the mid-seventies

on. At around this time, Cremmer and Scherk [_ 4-0, Iflj (f-2.

provided the remarkable idea of spontaneous compactification

which opened a new avenue to the understanding of higher dimensions.

It is from tin.'- modern point of view that the Kaluza-Klein

theory will be presented. To put it shortly, we will study

gravity and the coupling of matter fields to gravity in Ds't+K

dimensions. Space-time £ will be of the form £ sM^xB^., where

B is a compact space, mostly chosen to be a coset space. Four

dimensional fields will appear a* coeffi cents in a Fourier

series over Bv and we will be particularly concerned with ths

zero mode sector.

Some formal preparations are made in chapter II. In chapter

III, a detailed discussion of spontaneous compactification will4 be

given. Chapter IV is devoted to the classical Kaluza-Klein theory

including the incorporation of spinors. In Chapter V, the per-

spective changes somewhat »s we turn to a closer look at higher

dimensional Yang-,Mills theories.

Chapter VI on supersymmetric theories i« comparatively

shore as on)' very recently the first results specifically related

to the methods presented in this review have become available.

It is precisely from this point that future developments will depart.

Referencing throughout the review is sometimes sparse.

However, a referencing section can be found as well as an extensive

bibliography which contains even some papers whose contents could

not be included into thi» raviaw.

- 2 -

M I .

A.

FORMAL CONSIDERATIONS

The setting of high dimensional field theories

We parametrize D==4+K dimensional space-time by local

coordinates z = (x , yf). At each point we define a local frame

of reference with the help of a vielbein with components E .

Frame (tangent space) indices and world indices are taken from the

early ajvd middle part of the alphabet respectively. The splitting

into Minkowskian and internal indices la denoted by M = (m,u ) and

A = (a,«,.)..

We have two basic symmetries. Under general coordinate trans-M M

formations a ~"> z1 the vielbeins transform according to

We will also have invariance under local (as - dependent) framerotations,

the frame group being the pseudo - orthogonal group 50(1,3+K).

Let /)7A be an S0(l,3+K) tensor which by a convenient choice of

basis can be given the form ty = diag(+, ; - . . . - ) . Vo can

then form the frame - independent quantities

A r gz (3-3

AZ )which may serve an a metric tensor. Indices are raised and loweredin the usual way using Gvejj and ^respectively. E and G artdefined as t i i f Edefined as matrix inverses of E and respectively.

As a typical example for the coupling of D - dimensional

•gravity" to matter consider a fundamental SO(l,3+K) spinor field.

Such a spinor has 2 -'components, with [PJ the integer part of

P. <Cp. App. B for details.) Under general coordinate transfor-

mations J£(V)is a scalar, U'(»)J> i£(z') = (£• (z), whereas under

local frame rotations we have f (z) ><// (z) = S(a ) t/» (z), where

S(a) denotes the spin representation of a € SO(l,3+K). The partial

derivative V„ |f(z) is n world vector, but transforms inhomo-

geneously under local frame rotations. A covariant derivative

is specified by the requirement that it transform homogeneously

under local frame rotations. This fixes the transformation beha-

viour of the spin connection B under frame rotations to be

LB takes its values in the Lie algebra of SO(l,3+K-)> ThusAB ^ tT i the generators of7 B w h e r eB H = 7 B M TAB"!

S0C1.3+K) in the spin representation.

A spinor Lajrangian density which is invariant under local

frame rotations and a scalar density under coordinfte transforma-

tions is given by

(B-,det(EN°). The curvature scalar density provides awith detE

Lagrangian for the gravitational field itself

with the components of the curvature tensor being

Besides the latter, there is a second tensor with the character of

a field strength, the torsion tensor with components

A riemannian geometry is one for which the torsion constraints

T ^ * = 0 hold. These are algebraic constraints for BJJ rABl a n d

allow the solution

with

As a specimen for a theory involving vector fields consider

a nonabelian gauge theory. An invariant Lagrangion density is now-

provided by

- 4 -

with PAB = E A ,EB" F M

We can rewrite F aaAD

with cUf^'df!' T h e

invariant are Vi

and noticing that the vector representation

transformations that leave ("3 -IZ)

i » + A ' K where it is crucial to notice

that pi ,Og J --_£L. ,_,CQ.• Contact with our general prescription

can be made by writing for the generic vector Vn

given by

The terms linear in V in eqn. (i.l'S) can then be replaced bydAvB - djjV . More generally one considers the totally antisymmetric

tensor A* » a n d chooses th« totally antisymmetric field

strength to be

F". L ^ ( A n <^A A -i • / n \r\

As a final example, consider the Rarita - Schwinger field*

The invariant Lagrangia« density is given by

where -r The co variant derivative ±* given as

where ?f>\ and J^-^denote the spin and vector representation for

the Lie algebra of 30(1,3+K) reopectavely. Because'of the anti-

of I ' ~ the setae simplification as above occours. Thus

.-,• ,>fhen dealittg- w4,t*l• spi'nQrs,. it Is <?nnyeni«nt to Itnpv . th.a:t . ;

deriv.ative of generalized Dirac matrices vanishaa,.

A useful relation for the discussion of invariance properties

is (detE)dAVA = d M^(detE) E^" V

A J , where VA are the frame

components of some generic vector. For the proof of this formula

one requires the identity "b (detE) = (detEltrE^O £ =

(detE)E^ {3 „ K, • A» *» example consider the scalar curvatureAB

density, (detE)R = <detE) H ^ ^ with R C D £ka~i =

dCBD[AB^- BcfA^ ^ ^ " < D^> C>- " follow, from the above

that the terms linear in the connection coefficients are a

total divergence. In the action integral J (detE)R this would

contribute just a surface term. If we are willing to discard

such a term, we can replace the action integral by ^ (detE)R

with R = (Bn.E BA,,B - B * B n

E B ) . The variation of such an action

integral would produce a surface integral rather than vanish.

B. Harmonic Expanaiona

1. Introductory remarks

In this section ve will introduce the method of harmonic

expansion as a meana to obtain effective four dimensional field

theories trom high dimensional ones through the simple example of

a scalar field theory. We will see in particular that the mass

spectrum o the four dimensional theory reflects the choice of

the internal space.

The simplest case is £, a M^ x S^, with S^ the circle with

circumference 1. In order to hay« a non tachyonic mass spectrum

the internal metric will have to spacelike; on M^xS^ we can choose

it to be of the blockdiagonal form G^j - bdi«g(Gmn(x),-1). The

five dimensional acticn for- the real w,»l«r. fi«ld__'_5(.».> we. .... •

t a k e t o b e .. .. • .,., V . • •'-.•.' "'"•••' • ••-"-. '•••

• • (Ml).- ;•

where surface terms have beeji naglect-ed. As before, z=<x,y) and

Qs is the five dimensional d'Alemberti«n. S^ is compact; therefore

I> (i) i> periodic in y, and we may write down a harmonic expansion

-' 5-'-- 6 -

1ft —— o

where O f y<£ L = (2T/M). The reality condition^ (z) =3>*(z)

implies 4>*n(x) = Jf^x). Noting that the five dimensional

d'Alembertian ia ^ = CL-^/^?". w e obtain from (2.21) after

performing the y -integration,

s- i IJ1"* „,„,This is the dimensionally reduced action. It features an

infinite number of four dimensional scalar fields^ (x) with

masses nM, the complex massive modes, and a massless field,

A»o(x), a real massless mode. In the limit L-'JO, the massive modes

become very heavy. As a first approximation they may be

discarded at sufficiently low energies. Keeping only the aero

mode a amounts to using y - independent five dimensional fields

only. This procedure is often called "ordinary dimensional

reduction" in contrast to "generalized dimensional reduction",

where some y - dependence ia kept.

la generalization of the above consider now the real scalar

field on the product manifold £. = M^ x Bj, with metric

= bdi«5<Gmn(x) , G (y)). A suitable action is now

D

where

! d e t Su*! dDz = jdet Gmnl d^x

we have for the d'Alembertian

the curvature scalar RD = R^ +

T h e volume element splits,

l d^x-ldet Grv| dKdKy = Also,

o ^ Q + •„ and for

D = R^ + RK. Now let {% n(y)] be a

complete orthonormal set of eijpenfunctions of f — Di, f 'Vi»i+-F D "\

with corresponding eigenvalues . ^ , that is *•

X d VK

(no sum)

with normalization X d VK / ^ y ) % m^

y^ = &nm' I n fsi"nl9 o f these

eigenfunctions, the generalized harmonic expansion for tj (x,y)

is then simply (x, y) a/^n^ x^ "^n^y^' I n s e r t l nS this expansion^ /n n

into (2.24) and using the reality condition Z<t>**(x) !£* (y) =

2" n(x) % n(y) we obtain the dimensionally reduced action

We have again an infinite number of fields with masses (A• We see•

immediately how the mass spectrum reflects the choice for

the eigenvalue equation (1

through

A typical example is Br =

0. Then X-* X, -K

S2 (the two - sphere), with

i1 faO,!r where is a mass unit

giving the inverse radius of S2" The four dimensional fields(§, lm(x) provide (21 + 1) dimensional representations of S0(3). This

follows from the transformation properties of "X , (y) and the

requirement that ^ (z) be a scalar field.

Compact spaces, like B = G/H, with G and H compact groups,

allow the introduction of nontrivial topological structures. As a

typical example consider the concept of a "twisted11 field on B^

Above, we had the periodicity condition i(x,y) = ct(x,y+L).

By contrast, for twisted fields we would require ij(x,y) =

•J, (x,y+L). Note that this eliminates in particular the zero

mode in the harmonic expansion.

An alternative description is the following. Consider a

circle with circumference 2L, and identify antipodal points on

this circle. Twisted fields are those whose values differ in sign

at antipodal points. For twisted fields, only odd n occur in

the harmonic expansion. If in a similar way we identify

antipodal points on $„, we obtain the real projective space RP2*

For twisted fields, we keep in the harmonic expansion the modes

with odd 1 only and for untwisted fields those with even 1.

In this section we have studied only free field theories.

There is no coupling between different modes. Conversely, had

we started with an action containing a self interaction of the

scalar field, then there would -occur couplings between different

V

- 8 -

- 7 -

2. Harmonic expansion: General formalism

We generalize here the observations of the previous section

to general coset spaces B^ = G/H. In this section, material from

appendix 0 will be used.

As. a first example take H -= 1 so that B_ = G. Consider a

scalar field J (x,y). Under coordinate transformations y_j,y',

^ (x,y)->j (x,y') = Cp (x,y). Consider now the ansatz

5 (x,y) = D(L~ (y)) <t> (x). Here L(y) is a group value function

which is generically of the form L(y) = expdCO (y) Qfi ) a n d D

denotes some representation of G. The fundamental property of L(y)

is that under left translations g: L(y)—^ L(y') = gL(y),

g €. G. Now under coordinate transformations y—=>y' we have

r(-*1{y))<J>{x) -T- D(L"1(y') c|>' (x) = D(L'1(y)) D"1 ( g) 4>' (?) ,

so that for scalar fields $ <x,y) we have c$>{x) => D(g) Cp (x).

The anaatz given here is nothing but a symbolic form for the

harmonic expansion as written down in the previous section. We

find indeed that the four dimensional fields carry representations

of G.

In a less formal way we use representation matrices Dn

D(L~ (y) ) <^> (x) is replaced byinstead of a simple D. Then J(x,y)

Here dd is the dimension of the representation labelled by n;

p and q specify labels within this representation. If G=SU(2),

the Dn are the Wigner rotation functions.pq

On coset spaces we have to proceed in a slightly different

way. The fundamental transformation property for L(y) is now

(compare appendix D) g: L(y)—5 ^ ( y 1 ) ^ gL(y)h~ (y^g). Therefore

ui.'ar coordinate transformations y—1 y',

H.We have a factor D(h) on the right hand side of (J.2* ), which

can not be compensated by requiring a suitable transformation

behaviour for <p (x)• We would not want to do this in any case,

as we want cf> (x) to carry a representation of G. Thus we have

- 9 -

to require that under a coordinate transformation the world

scalar J(x,y> transforms according to

where T) denote some particular representation of H. Writing

out some indices, this becomes J^Cx^' ) =|) (h) <£. (x, y) . Only

if J) happens to be the trivial representation, we have the old

definition of a scalar field. The indices i and j are by definition

indices associated with H. Since<^>(x) carries a representation

of G, the index associated with H must be carried by the factor

D(L (y)') in the symbolic formula, for the harmonic expansion.

Therefore, on G/H, the correct formula for the harmonic expan-

sion is

Here d_ is the dimension of the representation /Q of H. The sura

over n is a sum over all representations that contain the repre-

sentation D of H. If in a given representation! D appears

several times (Example: the doublet representation of isospin

SO(S) appears twice in the adjoint representation of SUOi), then

5 labels these different representations. The representation J)

will be specified below. We note now that it is related to the

group of frame rotations. In particular, for a frame scalar,

J) is the trivial representation.

In order to obtain the inversion of (2 - o) we have to

evaluate for summed! tlte- integral- •••"'... . - ..; ;

with G - invariant neasuro. d.u (y) =. (I jr. iit («„•') . Because «•* .

G- invarian.ce-, we may replace LfyJ by1 L(y'), with g>y-Vy'j

Ho« L(y) - gLlyJh"1, so that

The dependence on h cancels (because the O

tation matrices for H) and we obtainare represen-

T Jr'1

- io -

which marka I . as the cjD"mporient« of an invariant tensor.: P ? iP? .' ' : • •

Accordingly) by S-chur"'.'• lemma,.

The factor \£ \ depends only on representations and"can be found

by Setting n=n' , p=.p' ( aftd summing over p.. Then

We thus obtain the desired inversion formula,

lAs before, the dimensional reduction procedure consists

effectively in replacing the fields of the higher dimensional

theory by expressions like CilQ) and integrating out the internal

variables. In general, algebraic complications will occur,

mainly due to the appearance of Clebach-Gordan factors which

not necessarily recombine in a simple way.

3. Harmonic expansion: Examples

For tb.e discussion of the Kaluga - Klein theory below it

will be convenient to have certain specific examples of the

above formalism available.

Consider for example the one form e"(y) = eJC^d/ on coset

spaces, where e „ f* (y) are the vielbein coefficients. In appendix

D we have given the transformation rule (cp. eqn. (D.12))

Since e' 'j/is a world scalar, this is obviously nothing but a

special case for ( W 1 ). From O.Si,) we infer that frame rotations

and group transformations h € H are related. In fact_, ID is

that representation of H that is obtained by the natural imbedding

of H into SO(K), viz.

where Q denote the generators of H and C

Gowider • also. a ; high dimensiflnal'.spinor . It may b6

looked, upon ffs a rtHiltlpl.e-t. of 'Dtir*.c s^nsrs. The internal label

arising in this way la t:o ij±e:ntd,f ied with/the label i.i*- in the

formula fox the ha-rmoni-c expaDSion-. The high di-me-nsional spinor

thu-B constitutes a direct sum of.H t multi.pl*-ts of Dir&c spiJtiors..

On a group space> we hav« two poasibilities. Either (i/difl, H=l.

In this case, the internal degree of freedom and the group G

remain quite unrelated. On the other hand we may have GxG/G = G.

In this case G=H and the spxnor actually decomposes into & -

multiplets.

As an example for the harmonic expansion itself, let iis .

consider the vielbein, and in particular the components E * (x,y.) •

Since these transform like the components of a K—vector under

SO(K), even the lowest term in the harmonic expansion must display

some y - dependence. In fact, application of the general formula

and retaining the first term Only gives

In a similar way, higher rank tensors and teasor/upinori under

the frame group may be treated.

References for chapter II

The material in this chapter is to great extent

collated from references (~l69j , T^^J «""< j 55 ~^\ •

are those of SO(K).

- 11 - - 12 -

rr i «§:!»* ft

III. SPONTANEOUS COMPACTIFICATION 2. The CS model of spontaneous coropactification

A. Models of apontaneuua _ -immt U.1 lcntlon

1. General remark*

The concept of spontaneous compactification has been intro-

duced in the mid - seventies by Cremmer and Scherk T'-'O, lf V?. 1

in order to provide on answer to the question of what might happen

to the extra dimensions in a high dimensional field theory.

The question was relevant at the time since dual models had turned

out to be consistent in 10 and 26 dimensions only. It may be of

even more importance today because of the advent of supersymmetric

theories which often are at least particularly simple if for-

mulated in higher dimensional space-time.

We have seen above that in a field theoretic context small

(internal)radius corresponds to large masses for the massve modes

of a harmonic expansion. Thus it may be argued that for sufficiently

small internal radius even though the extra dimensions would mani-

fest themselves through a mass spectrum of the four dimensional

theory, they would nevertheless be unobservable at least at

presently available energies. *

The suggestion that the extra dimensions are not being seen

because they curl up to extremely tiny spaces, is already inherent

in Kaluza's original work. The new suggestion due to Cremmer and

Scherk is that the presumed fact that space-time is of the formMi,xHg is not to be an input of the theory. Rather, such spaces

should correspond to ground state solutions. Physical fields are

then introduced as fluctuations around these ground state solutions.

The formalism of the proceeding chapter comes in as the fluctuations

are parametrized through harmonic expansions.

This chapter deals with a description of the~presently

available ground state solution:and contains also a discussion

of the equally relevant question of thoir stability.

In the CS icifmej Stherk) - model \_^ 0 " 4-! J gravity

is coupled to Yang-Mills fields in D=*t+K dimensions; there is

also a cosmological constant. The Lsgrangian of the model is

The constants H and e are related to but not identical to the

Newton constant and the four dimensional Yang-Mills self coupling.

In addition to the conventions of appendix A we have that

late Latin indices with caret refer to the gauge group R of the

Yang-Mills Lagrangian. In (1- I ), the Yang-Mills tensor is

The field equations are

- i SB. R.

c - 0where the erfrgy momentum tensor is given by

For a spontaneously compactifying solution the D-dimensional linoS 2 = CLdzMd»N = G dxmdxn + G_ dy1* dy11 tCL,wdz

Md»N = Gpus m

with M, four dimen-

element takes the form <JS2

giving a D-dimensional space-time

sional Minkbwski space (or, more generally, a solution of the

four dimensional Einstein equations) and BK = G/Ht a coset space.Note that there is as yet no stipulation that G and R be identical.

Choosing for M, Minkowsltiv space means G (x) = *» =H \ > mn , I mn

dii-g(f, ). vhereas a f oruiinvariant metric (cp. App. C) on G/H

is given by G^" (yj = 6/jO Kfil* (y) K** <y) . Here fl/j1) is a constant

that is essentially the radius of G/H. The K P (y) are the Killing

field components for G on G/H. The metric on the coset space is

forminvariant under left G - translations, much as y m n is

forminvariant under Poincare transformations. Rather than D -

dimensional Poincare invariance as for D-ditnensional Minkowski

space, the ground state we are seeking has four dimensional

Poincare and (local) G - invariance.- 13 -

The CS ansatz (in the general form as given by Luciani (JOfJ)

is now completed by specifying the Yang-Mills fields to be

EL * (*,y) =• 0 and Bu (x,y) = B Hi(y) = K 5^ (y). It is to be noted

that in the CS ansatz one specifies R = G, in the abstract group

sense. Later we will discuss how this restriction can be circum-

vented. Using the CS ansatz, tha field equations turn out to be

equivalent to

CM-)

)

From ( 1-d ) we infer in particular that we can not find a

compactifying solution for vanishing Yang-Mills fields. This is

so since we have already specified that the four dimensional

cosmological constant should not vanish.

. K^ty) K1 ~ (y) and

£< h one can work out by using general properties

of Killing fields,

Introducing the notations h

«'* - fSvw ?

whereas ( "5 - 8 ) becomes

''* 3(3.1/ )

One reads off that the field equationsican be satisfied if the

-kMat.-i.ili HS E . " - C S-J is satisfied for some constant C.V t

Solving this constraint constitutes the final step in the

CS construction of a compactifying solution. Since^J.t is an

equation among invariant tensors, it is sufficient if one finds

a solution at one point in G/H. This then is an algebraic problem.

For this, two essentially different solutions have been de-

scribed by Luciani. The first corresponds to H=l. In this case

the coset space is n group space. One has C=l and the remaining

constants a n fixed as Au- 7a , U — (?TT*</(te.i ) /\;lf<

- 15 -

The other is for groups Q = H that allow a Cartan decomposition

with respect to H. Denote the generators of H by h, those of G

by g and the set of generators associated with G/H by _v. The

Cartan decomposition of generators is such that the commutation

relations fh, Jj CJSj i [l!i IJ C v ] £v, y j c tj hold. Such a

coset space is called a symmetric space. In this case, C =1/2,

and the 6ther constants are >|*« \ _J (*. -- ?FJt A,1 ; A ~ k e V O CViT 2 .Examples for such solutions are G=SU{P+Q), H=SU(P)xSU(Q)xU(l),

dimBv=2PQ; G=SU(N), H=SO{N), dimB^tN-l)(N+2)/2; G=SO(2N),

H=U(N), dim BK=N(N-1); G=SO(P,Q), H=SO(P)xSO(ft), dimBK=PQ.

A further specification of constants arises if one studies

the zero mode fluctuations around the ground state solutions.

(See below.) Here we just note that i-f/e2^ 4*. / 0^< Q

Therefore u is of the order of Planck's mass (10 GeV) and

the internal radius isvthe order of Planck's length, 10 J J cm.

3• The minimal gauge group in the CS model

The CS construction was for the case H=6, i.e. the

internal space-time group is identified (in the abstract group

sense) with the gauge group of the external Yang-Mills fields.

The construction immediately provide* solutions R 3 G if we

specify the gauge fields B^ to be nonvanishing in the direction

of the Lie algebra of G only. Still it should be noted that

the structure of high dimensional Yang-Mills theories with

B 3 G is interesting in its own rifffe. It will be discussed below.

Here we turn to the case RC G. In particular, one may search

for the minimal gauge group R such that compa.ctification of

M^ „ into M.xB occurs. Clearly^one feels that, the smaller R,

the better. It has been shown that this minimal group ia not

bigger than H. Indeed, R may be smaller in rank than G, and need

not act transitively on G/H.

The available compactifying solutions which use groups R

smaller than G are topologically nontrivial, unlike the CS

solution discussed above. This is to be understood in the

following way. For the gauge fields BM(z) = (Btn(x, y) ,Bj,,(x,y) ) , the

compactifying solution takes the form BM(a) = (0,B^ (y)). Now B^(y)

are gauge fields on the internal space Bg. The latter may be used

- 16 -

as'a base' space : for a nontriy.ia.1 fibre bundle, and BL (y ) are the

.connections for "this bundld.'Ths nantrivial topology is then

netlecrted in the existence- &f".nx>nvanishing topo'T&gieal quantum

-ntimbers like the instanton.charge in four dimensions (on S^)

"and its generalizations (the characteristic classes)*

A typical example, of such a solution is the f ollowingLH£, 84 j.

We take B =S2- On S , we introduce polar coordinates ©((p. Then

the potentials B o , B, leading to the compactifying solution are

given by

Here the N.P solution is valid in the region of S that contains

t.1% nor a pole (0--Q) but not the south pole, the other for a

region overlapping with the first that covers the south pole.

'X is a generator of the gauge group R; in general it would be

of the form J- «,, \^

The N.P. and the S.P. potentials are to be connected by a

gauge transformation exp(i2vp£). This is just the condition

that ( 3- (1 ) describes a potential albeit a topologically non-

trivial one. In fact, all the information concerning the non trivial

structure of the solution is coded into this gauge transformation,

the so-called transition function. The transition function has

to be single-valued, thus exp(4tfi r ) = 1 or <r • n, with 2ne"IN.

Therefore, instead of (1.12),

v •*Note that in this case, r generates just a U(l) - subgroup of R.

Ii1- this sense, spontaneous compactification M , —"> M.xS occurs

with R=U(1). Note that S =G/H=SU(2)/U(1), so that this is an

example where R=H. For n f- 0, the solution may be visualized as a

magnetic monopole on S in the Wu-Yang formalism.

A local description of the general solution for the case

R=H is easily given using the formalism of appendix D. Recall

that e<y)=L"1(y)dL(y) = (eKok (y)Q, + e,." (y) ft- )dyf" . We then

r o( r •have for the external gauge fieldsBm

S(x,y)3O; B. *(x,y)» e '(y) and for the metric

G j ; J :-, i „-e *(y,)' e f tyj . ,, whereas G is a solution of the

four dimensional Einstein equations. Solutions.exist in .this, case

if either the four dimensional or the D dimettsioiia;l cosmological

constant vanishes; but both can not vanish sihiultaneously•

The nontrivial topology reflects itself in the fact that '

only a local but not a global definition for the function L(y)

exists. A global definition of L(y) exists if <J is a trivial

bundle over Q/H with structure group H ["UPT J • Above we had

H — ^ R=G so that the bundle indeed would be trivial.

Some remarks are in order.1 £J |

The solutions given in this section L/Sr u. which use H=H

are valid only if G/H is symmetric (like in the previous section).

So definite proof exists as yet that spontaneously corapacti-

fying solutions with R C G must.be topologically nontrivial. All

the known solutions have this property however. It'also an

open question whether solutions with R C H exist..

It has been noted that the nontrivial topology o:f

B R may be employed in the following way QSS" J. Even though the

CS construction given above is topologically trivial (i.e., the

appropriate characteristic classes vanish), they still may be

split into topologically nontrivial solutions in a manner analogous

to the construction where a q=0, 30(4) instanton solution .is

written as thp sum of a q=n and a q=«-n instanton lying in the

two S0(3) factors of S0(4) respectively.

An important feature of topologically nontrivial

corapactifying solutions to which we will refer below is that they

may serve to trigger left-right asymmetry in the fermionic

sector. In fact, the characteristic classes give the difference

of the number of left and right ferroionic Dirac zero modes

(Atiyah-Singer index theorem).

- 17 --• 1 8 -

k .,. Further ;models of apontjuiequs1. compact! f.ication -..

Besides the GS model, a number Of .models featuring•'

spontaneous compactification axe availble and will now be described.

In the first by Omero and Percacci Q^fJ, a sigma model coupled to

gravity. Here, no cosmological constant is required. Physical

space-time, M, , can be any solution of the Einstein equations,

R = 0 . The solutions are further described by saying thatmn

the scalar fields of the nonlinear sigma model have to comprise

the same space as the internal coordinates. (One mauthus get the

idea of "coordinates as dynamical variables".) More formally,

the scaJai- fields may be visualized as identity map from BK ast-.Terns! space to Dv as field space.A.Incidentally, the construction

is possible for general Ricmannian spaces H . Classical stabili-

ty is analyzed in particular for the so-called Grassmann spaces

and stability is found for B^ (Fn)=CF(n)/tTF(n-k)UF(k), with

UjjCnisSOtn) and U,(n) = SU(n). Stability is also found for B ^ S j , :

K > 2, but not for Graasmann spaces G.{ |H ), where |H is the space

of quaternions.

A further model is by Freund and Rubin(j^j. It is interesting

in particular as it contains as a special casj an exact solution

of eleven dimensional supergravlty. Also it displays the intriguing

feature that a preferential number of dimensions compactifiea.

The model consists of D-dimensional gravity coupled to antisymmetric

tensor fields of rank s-1 with field strength tensor.F

A spontaneously compactifying solution for which G^^bdiag(G (x),

G f., (y)) is found with the help of the ansatz

j. = £ j. „ f } |G_| , X= c o n s^ a n t• T n e solution is valid

if B-ddlfttriSionai ipace-time splits into an S- and a (D-S\-

dimensional factor, respectively. In this case (and only in this case)

the Einstein equations of the D-dimensional theory simplify to

Rn _ =- (S-D(D-S) >/(D-2) and R_ = -S(DrS-l) > /(D^2), > =8lT4ff2.

For D S+l, the signs of R „ and R^, the curvature scalars of the

two subspaces, are different, so that either the S-dimensional or

the (D-S)-dimensional factor are compact, but not both. In this

sense, preferential compactification into spaces of dimension S

or D-S occurs. It is to be noted however that both spaces feature

a cosmological constant of approximately the same absolute size.

- 19 -

'*<.

Ai ail example, let,us consider the case D'lt,- which is relevant

for super^ravity. (The field equations considered by Freund und

Rubin are the same as for the bosonic sector of l.i dimensional

sup*rvgravity..) We have now preferential Compactif ication of eitlier

four or Seven dimensionsi Pbysd&al space -tdmw may or may not be

the four dimensional factor. In the latter case, when physical :

space-time is contained in the seven dimensional factor, further

(hierarchial p2.£j) spontaneous compactif ication has to occur.

Aspects of this possibility have been discussed recently by

Salam and Sezgin

All the above spontaneously compactifying solutions where

solutions of high dimensional gravity coupled to matter. The

occurrence of matter is probably enforced by the stipulation that

xne four dimensional etymological constant be small. In fact,

pure gravity in D dimensions does have, spontaneously compactifying

solution?but they all correspond to larga values of the four

dimensional cosmological constants.

Perhaps an ansatz with nonvanishing offdiagonal elements for

the metric, 5^ , could improve this situation. In the spirit of

the Kaluza-Klein ansatz such an ansatz would involve constant

Yang Mills potentials. For the pure Yang-Mills theory such solutions

ejfist] \ t*f land therefore the attempt may not be hopeless.

Quite a different possibility has been investigated recently

by Wetterich[Zoo jwho starts off with an action containing

terms quadratic in the curvature tenaor. This provides for extra

terms that can take the place previously taken by the matter terms.

From a physical point of view this possibility, is also interesting,

as one1 expects such terms to arise as counterterms in a renor-

malized theory. Prom this point of view, one may even see this

as a starting point to answer the question whether spontaneous

corapactification is more than a classical phenomenon and survives

in the full quantum theory.

- 20 -

1

5. The CD model

The idea that the small size of the internal space may

be related to the large age of the universe has been introduced

by Chodos and Detweiler ^Isrj. They present time-dependent

spontaneously compactifying solutions for the vacuum Einstein

equations Rj_.= O, where R ™ is the D=4+K - dimensional Hicci

tensor.Their solutions are generalized Kasner solutions, which

are of the form .

with £; p = l_ p. a 1. Because of the latter condition at least

one of p. has to be negative. One therefore infers that some

space-time dimensions will be expanding, others contracting in

time. While this ia a poor description of the four dimensional

universe, for a high dimensional theory it may Just be what the

doctor ordered.

Chodos and Detweiler describe in detail a five dimensional

case with P1 = P2 = P-i = "P4=1/2 • This choice ensures isotropy in the

first three spatial dimensions. Further, all spatial dimensions

are being taken periodic with L. For this case, the line element

becomes '

,-it^ [ft (it?t=t , th,e universe ia flat and extends for a distance L in

For t= "£ , where £ is the present age of the

At

every direction

universe, the first three dimensions extend very far ( tV) 0

the effective period being ( T I k ) L whereas the fifth curies

up very tightly with an effective period (4-0/t ) t- Comparing

this vlth the adjustment of constants in the Kaluza-Klein

picture (see below), one infers that (t / T ) L has to be of the

order of Planck's length.

The Chodos-£etweiler (CD) solution has recently been much

extended by Freund (__ 7-/ J* " e argues that scenarios of this type

would be quite important for coamological considerations, as the

"effective" dimension of space-time might depend on time and at

an early time the uniierse could indeed have been high-dimensional

with all dimensions of conparable •iza.

He also discusses the possibility that in eleven dimen-

sional supergravity, preferential expansion of three space-

dimensions occurs, much in continuation of the ideas put forward

in connection with the Freund-Hubin model (see above). Ia addition,

he discusses scenarios for the whole procedure to take place in

hierarchiaI steps.

6. A lattice calculation

It is difficult to foretell whether spontaneous compacti-

fication persists as a quantum phenomenon beyond semiclassical

approximations. It has recently been suggested by Skagerstatn rf?TJ

to approach this problem by studying lattice versions of

high dimensional field theories.

He assembles Monte Carlo data for the Wilson loop average

for the following two models: (1) a five dimensional Yang-Mills

system; (2) a four dimensional Yang-Mills-Higgs system with

Higgs field in the adjoint representation that corresponds to

five dimensional Yang-Mills system with periodic fifth coordinate.

He compares the two sets of data and finds very good

agreement. Correspondingly, as far as Wilson loop averages are

concerned, the two systems behave in the same way. This indicate*

that spontaneous compactification of the fifth coordinate may

indeed occui* in the five dimensional Yang-Mills system.,

- 21 -

B. Stability considerations

1. General remarks

Spontaneously compactifying solutions have been sought

as minimal energy solutions (as much as energy is defined in

gravity theories) of classical field equations * In order to

ascertain further the nature of these solutions, stability

considerations hove to be taken into account.

In the context of the Kaluza-Klein theory, this analysis

ha* only been carried out for the most simple cases. As we will

see, stability considerations are of non neglegible impact.

Energy-like (conserved) quantities play a prominent role

in stability considerations. A typical example io Dirichlet's

theorem which asserts that a solution of the equations of motion

is stable if it provides a strong minimum of some conserved

quantity. Note that stability is not guaranteed if the minimum is

not a strong one. Also note that froms; the point of stability,

energy may be replaced by another conserved quantity.

In a quantum theory, barrier penetration from one local

minimum to another may also occur. For stability we would have

to require that a minimum be a strong absolute one• If two

minima have appare -vt,tly the same energy, we have to demonstrate

that no semiclasslcal decay from one minimum to the other occurs.

Let us elaborate some of these statements for an example

Consider a single scalar field (ic,- in 1 + 1 dimensions, with

Lagrangian V -_ I ~ i**^Vfa).The corresponding field equations are'

Suppose, these have a classical solution <£> = U) , Coc4.

Perturbations around this solutions are described by the ansatz

4>(*it^ = fcL -H <f(*)«*Y>(fwfcV I n s e r t i n S this lnto t h« fieldequation and retaining terms of first order only one has

(•2.19.)

For stability, all (xy- should be non-negative, otherwisewe would have exponentially growing modes. This requirement can

actually be translated back into a criterion that the energy have

a strong local minimum for for static solutions« for static solutions lp

In fact, for such solutions the energy functional becomes

(x) .

<p + Vclassical solutions so that

the requirement C^vO

It is minimized by the

=0 and further

translated into•4*1

0.The above considerations are complicated by subtleties

concerning degenerate ground states in general and zero mode

oscillations in particular. If we have separated strong local

minima of the energy then they will classically all correspond

to stable equilibrium points. To proceed further we will have to

resort to semiclassical procedures (see below.) It can also happen

that a minimal energy solution can be deformed continuously into

another one without change of energy. Such degeneracies occur

typically for Goldstone-type potentials. In the formalism above

they will manifest themselves as zero frequencies. Physically,

the corresponding zero mode fluctuations are associated with mass-

less particles (Goldstone bosons), provided, the degeneracy can be

associated with an internal symmetry of the Lagrangian. If no

symmetry is associated with the zero mode there may be instabilities,

in accordance with Dirichlet's theorem.

Consider now the field equation for static solutions and

differentiate once more with respect to x. One obtains

and we se that j = As 1 is a zero mode solution of ( "£ • '"V) .

This zero mode solution, which'always, exists, can be related to

the iranalational invariance of the system. In fact, consider small

translations, X-->X+X Q. Then ^(/.O- ^cl (< + Vo") - 4c« 60+ 4^*") ' *o

whence by comparison with ("3-n- ) we get Ca — 0 and

For stability, the translational mode has to be the lowest

mode. In more then one dimension, there is a translational zero

mode in every spatial direction. Because of rotational invariance,

the eigenvalue &0^0 i* degenerate, and can therefore not be the

.'•1 mum !„•

lowest. In particular, for a spherically symmetric ||tl, the

translation modes form a p state and there is an s-state mode

with negative <A .

Semiclassical instability can occur if two local minima

are separated by a finite potential barrier. Intuitively, the

system may tunnel from one configuration into the othrr. Quantum

mechanically it will do so in such a way as to minimize the energy

•functional. Field theoretically, one considers minima (extrema)

of the Euclidean action functional. The tunneling is then associated

with a "bounce" solution that interpolates between the two

minima. The bounce solution is a solution of the Euclidean field

equations with specific boundary conditions. The typical features

can already be seen ir. the one dimensional case; we then have

t as (1 (t). The boundary conditions for this case are such thatb i b

for t=0 we are at one (local)minimum of the Euclidean action, and

for t= ± CD at the other. Thus, beginning at t=0, the bounce

solution will just reach the other minimum at t= tcO . Now if

this solution is stable, barrier penetration will not occur,

as the bounce solution ia constructed to be the borderline case

for this not to happen.. Conversely, if the bounce solution is

unstable, decay of the "false" vacuum occursL 2 3^ ??/,

Thus, for the analysis of semiclassical .stability in

field theory, the following recipe emerges, (l) Find a ground

state solution of the Euclidean field equations, whose stability

is to be analyzed. (3) Find a bounce solution, i.e. a solution

of the Euclidean field equations that coincides asymptotically

with the ground state solution. (3) Analyze the stability of the

bounce solution, i.e. look for jnodes with negative txf~ in the analog

of (1 1>). In fact, the differential equation to bfe studied is

(1.1V) with ? c l-*f b.

Note that for step (3) it is often not necessary to actually

solve the analog of (1. PH. In fact, as we have seen, sometimes

general arguments indicate the existence of modes with negative CO •

2. Positive.energy theorems and instability of the

Kaluza-KIein vacuum

In gravity theories, energy can be defined only with respect

to a given background. Typically, one can not compare the energies

of, say, M_ and MiixS1 since even though both spaces are solutions

of the vacuum Einstein equations (M and K, denoting five and four

dimensional Minkowski space, respectively) and thus naively might

be associated with zero energy, they do not coincide asymptotically,

In the spirit of Dirichlet's theorem, stability of a certain

configuration can be ascertained, if this configuration minimizes

(in the strong sense) a conserved (energy-like) quantity. For

gravity theories, such a quantity (with respect to a given back-

ground) can be introduced in the following wayL * J •

Consider the Einstein equations with cosmological term,

Km - (1/2) am R + A G™ =0. Now write G™ = 5™ + H™, where5 is any solution of the Einstein equations, and it represents

derivations that vanish at infinity. Inserting into the field

equations, the terms of zeroth order in H drop out by construction,

and if we define all terms of second or higher order in H to make

up the gravitational energy momentum tensor, then we can rewrite

the Einstein equations as

R } y^n^l"* A 11 \ s? l~ L —I— M tJ_ r w + / \ p | - 1 ( 1 I

where T will act as energy momentum density tensor of the

gravitational field, the subscript L refers to linear in H.

Let us now see why T can act as an energy-momentum tensor.

By virtue of the field equations we find d T =0, where d is the

S.;ni;n; - jovtriant derivative with respect to the background.

Now, if the background metric has a Killing vector ~JM associated

with some symmetry, then it follows from d § + d ? M = 0

and the symmetry of T M N that dJrMN f „} =cL(T f „ ^ = 0.

This is the desired conservation law. For instance, in four

dimensions, with every Killing vector ZM there is associated

a conserved quantity E(^ ) = (l/8Tdf)^d3x T°Ng . If f is

timelike, then this quantity is referred to as Killing energy.

It can be rewritten as a two-dimensional surface intergral

- 26 -

An >>:;nsple for the- typical statement of a positive energy

theorem is the following. For Minkowskian background, A- Oi

in four dimensions, with ^ = (l;0,0,0) E vanishes for Minkowaki

space only and is nonzero for all other spaces that are

asymptotically Minkowskian. (The proof of the theorem involves

a number of technical assumptions*)

Let us now turn to the discussion of stability properties

of Kaluaa-Klein vacua E 2- °t, 2.0fl

As has been remarked above, it makes no sense to compare

M_ and M.xS on energetic grounds, We therefore have to consider

H,x3 by itself. ClassicaUy, M,xS is stable. The small oscil-

lations around M; xS are (compare also f:he following chapter)

some massless states: ^riiviton, photon, and a Jordan-Drans-Dicke

scalar and an infinite number ofmassive states. There are no

exponentially growing modasi The iiiassLeas statfia reflect symmetries

of the Lagrniigian (four dimensiona 1 Poincare, and gauge invariflnce,

also an invariants under a roscaling of tho radius of S ).

Let us now investigate the question of semiclassical stability.

A bounce solution is specified through the line element

where d X2_ is the three dimensional solid angle, c( is a constant.

Indeed, for large values of r, r = (x,l4-»i +-*£),l the solution ( 3-ZO )

approaches the Kaluza-Klein vacuum, ds = dr + T ailt + d ^

In the latter solution, <fy is periodic with periodicity 2TTR,

H being the radius of S , the internal sphere. Further, in order

that ( 1 2 0 ) be nonsingular, A has to be

periodic with periodicity 2 T / ( ^ . Thus we have tp identify

V T" = 8 *, It is important to note that in (I.ZQ), r is restricted

to run from R to infinity. Because of this latter property, the

Kaluza-Klein vacuum solution and the bounce solution describe

spaces with different topology.

The bounce solution displays spherical symmetry. The existence

of an unstable mode (Coi< 0) follows therefore from general arguments

in the spirit of those described in the previous section

from the vacuu-i topology is possible. Note that It is not otvioua

that such a process is possible. This is unlike the situation in

Yang-Mills theories where say, antimonopole-monopole pairs can

be created smoothly out of the vacuum and thus topologically

nontrivial configurations must be considered. Here an analogous

argument can not be applied as there is no such thing as an

antispace.

Intuitively, the semiclassical decay associated with the

bounce solution is the formation Of holes. It is likely that

seitiiclassical instability of the Kaiuza-Klein vacuum occurs only

through such a me chanisrr;. In fact, it has been argued by Wilten

^O^^that all excitations of M^x^ with the same topology have

positive energy,. Technically this is shown by considering solutions

of the Oirac equation on initial value surfaces. Also if elemen-

tary spd.nors are present in the theory they may stabilise the

ground state. The reason for this is basically that the definition

of spinors is not independent of the topology of the underlying

cianifold. In particular, Kaluza-Klein spinors could be introduced

in such a way as to be incompatible with the topology of the

bounce solution.

For the six dimensional case, £=. M^xS2, a first stabili-

ty analysis is now also available. It concerns the minimal

solution in the CS model 1^UZJ]J where classical stability is

ascertained. One recalls that this solution displays topologically

nontrivial features corresponding to &n internal "magnetic charge".

It is hoped that the existence of such a charga provides for

stability also in the gemral case.

NOTE

0n« concludes that the Xaluaa-Klein vacuum is semiclassically

unstable if spontaneous creation of manifolds witr. top^lcgy <3i:f.fereat

A modification of the Freund-Rubin solution, which provides

for 11 dimensional supergravity the parallelized seven sphere as

internal space has been given recently by Englert (i'J .

•- 2 8 -

i'UE CLASSICAL KALUZA-KLEIN THEORY B. The Kaluza-Klein ansatz

Introductory remarks 1. The f i ve-dimensiona1 case

In 1921, Kaluza O s u S S e s t e d t h e unification of gravity

and eleetromagnetism into a five dimensional gravity theory.

Some special conditions were to be met by the latter: the five

dimensional space was to be M, xS with S having a very small_ T *l

circumference (^j 10 cm, Planck's length) indeed. The small-

nesa of this internal space was thought to render it unobservable.

In a field theoretic context a four dimensional theory may be

obtained by harmonic expansion over 5 ; the massive particles

chen have messes of at least 10 GeV (Planck's mass). Kaluza's

idea then means in this language that at presently available

energies only the aero maan sector of the theory would be relevant.

It ia mostly this sector that will be of concern to us in this

chapter.

Even though the zero mode approximation will be good for

small energies, on principle grounds the massive modes can not

be neglected entirely |j',SSSy/£<f J. One reason is that the massive modes

are likely to give effects comparable to those of quantum gra-

vity. The other is that the k+K dimensional field equations

are simply not compatible with the truncation implied by the

Kaluza-Klein ansatz (itr ZogJ- Actually, It is interesting to note

that if on the other hand all the massive modes are kept, one

expects to obtain a consistent theory. For five and six dimensions,

the corresponding investigations have been carried out f Hi f£>2~l

with positive answer. These are. the only known examples for a

ciMisistsn-t interacting theory of massive spin two fields.

In the following, the classical Kaluza-Klein theory will be

described from the point of view of a zero mode approximation

around a spontaneously contpactifying solution. Only briefly will

I comment on the relation to the picture of fibre bundles.

Finally, an account of the incorporation of spinor» in the Kaluza-

Klein framework is given.

In this case no extra matter fields are needed for spon-

taneous compactification since both M as well as M^xS are so-

lutions of the five dimensional vacuum Einstein equations.

There is no reason in this case why one should prefer one of the

solutions as a ground state. As we have seen, even stability

considerations do not help. In fact, M^xS is less likely to be

stable than M . For the time being we therefore just assume that

M, xS is the true ground state of the theory.

The metric of the ground state is G,™- ( + . ;-) where we

anticipate that internal dimensions have to be spacelike. The

metric is forminvariant under P^xU(l), where F^ is the four

dimensional Poincare group. Among the zero modes we therefore

expect the gravitational field (spin 2) and one gauge field

(spin 1). There will also be a massless scalar, whose existence

probably reflects the invariance of the theory under rescaling

of the internal radius. These are the zero modes whose massless-

ness is enforced by a symmetry and which therefore are expected

to be present in a quantum theory.

As we have anticipated, the massless modes just give the

zero modes in a harmonic expansion of the metric tensor. In fact,

2t

exp(iny) 6 and in the present case the zoro

mode approxiamtion consists simply in dropping ail y-dependence.

The following form of the ansatz exhibiting the zero modes is

convenient

g - A Amn rn n

-*lA_

It turns out that a rather G =xTim mn mn

f-A A plays the role ofm nphysial gravity (see also below). Now a straightforward calcu-

lation using the formulae provided in chapter II yields the

result for the five dimensional curvature scalar,

" HEinstein (

- 29 -

Here R n. ^ . is toe curvature s^aidr ? L T X , -'o:~ >~ .-. d V ^ ^ L ^£inst?in ?nn

just get the Einstein-Maxwell Lagrangian with unit gravitational

constant. In this case the Einstein field equation R =0 readsF Fml1 = 0. Therefore the ansatz (4.1) is not compatible with themnfield equations.

Five dimensional gravity is invariant under general coor-

dinate transformations, whereas the zero mode approximation

reflects the symmetry of the ground state. To see this, remember

first that the symmetry relating to the masslessness of If) is

not a symmetry of the groundstate. We therefore set for the

time being j * | and consider the ansatz

smn " Am An/ V " AmAn I "Am

V -A^ i -1

It is now easy to see how U(l) gauge transformations emerge as

special cases of cooordinate transformations of the five dimensional

theory. In fact, consider

Um, y) » (x'ra, y') = (xm, y + £ U)) . < <t.<+ >Using (<f, i|, ) and applying the general formula

we find in particular

of ( <*,"* ) is A (x)—fA + £->)'(x) - £, (x) which in view

Thus, the coordinate trans-

One obtains this if Q is indeed the matrix inverse of { f i'

-fA"

-fA"

where we have rescaled the gauge field, A 4fA , and if— i m to

( <f

we identify <p(xty) • <p(x)exp(i U,y) . We have to set

l«.f - « . ( <f,? )f2

Since the ansats ( ) leads to R,. = R E i n s t e i n - T~ ^ F .

we identify f as gravitational constant, f 10 /GeV.

Ve therefore find that for e-j 1 as befits a reasonable gauge1*}coupling that we must have u = e/f -~* 10 QeV.

It is in this way that the Planck mass enters into the

Kaluza-Klein theory. In fact, the bare mass of the dimensionally

reduced theory is Mo f*is the lower bound for the four dimensional mass.

ra , so that the Planck mass really

formation ( "• </ ) can be compensated by the gauge transformation

The relation |i f = e is determined by the coupling of

A^ to a matter field. In the nonabelian case (see below) it is

already provided by the selfcoupling of the gauge field. The

ansatz for the scalar field implies that u is the inverse radius

of S . It i's fixed after we have set the values for f and e, that

is o*vLy after matter fields have been introduced. In the nonabelian

case matter fields are not necessary for this.

Another way to see this is to study the line-element

_2 „ J_MJ^N g d xm

d xn ( ^ y + A d x

m ) 2ds" «mnd* dx " <r +\,^ > Under (<>. ( ) we have

so that in order to ensure invariance of the

line element we have to require again A »A - £, . Thism ' m m

exhibits clearly the necessity of the term quadratic in A in { V.3 ) .

Another way to motivate the Kaluza-Klein ansatz is the

following. Consider the coupling of five dimensional gravity

to a scalar field £ ^if) • A "free" Lagrangian in five dimensions

would be G*1" <J ,,£„ - m 2 J* X . Here G is defined as matrix inverse— W W o L ^

of <\rfi' 0 n e n o w requires that the free five dimensional Lagrangian

reproduce the four dimensional Lagrangian for a scalar field

- 31 -

2. The general case

In the previous section we\have seen that in order to

identify gauge transformations as coordinate transformations

It was important that the five dimensional line element could

- (dy+A dx m)'m

be given the form ds =

From this, we read off suitable vielbein components. In fact,

let E = fcL. dz replace dz • The vielbein components are deter-

mined by the requirement that the line element take the form

"2 ^AB £* £ B ' With tABds" diag(+, ; - ) . We then have

M)

- 32 -

Note that the appearance of the complete square (dy + A dx™1)

in the line element is reflected in the appearance of the 0 in

Ci.3), The components E are to be identified with the vierbeinm

components of four dimensional gravity. Recall that this form

of the ansatz clearly displays the gauge symmetry as a special

case of invariance under general coordinate transformations.For the D - dimensional case we now make the ansatz

In (4.12) the abbreviations g

E/CO- '/Vis

0If we write down a harmonic expansion for the vielbein ( k- 1 ),

then the first term for E ° will simply be E a(x) by the general

arguments of chapter TI, since t. • >e behave like components of

i. frai.ie s:aiar under internal rota.jOas, As we have noted above.

the arguments of chapter II provide for E the ansatz

aa the first term of a Fourier series. The A p1 (x) will turnm

out to be four dimensional gauge fields associated with the gauge

group G (recall that internal space is presumed to be of the

form G/H)* The E will again he the vierbein components for

four dimensional gr.ivity. These are the zero modes we expect for

symmetry reasons. The ansatz for the vielbein is completed by setting

for E *• simply the vielbein for B = G/H, that is EL* = e * (y)

(cp. appendix D). The ground state corresponds to the block

diagonal vielbein E ^ = bdiag (§ a , e * ( y ) ) . As e * (y) is

forminvariant under group-coordinate transformations, viz.eii ' v' = e 'y. (y) t a n d similarly r) a is unaffected by four di-

mensional Poincare transformations, the ground state is invariant

r.der ?,xG. The Kaluza-Klein ansatz preserves this symmetry for

the zero mass fluctuations. Fully, it reads,

r A/

The metric equivalent is

( kit )

E dEm r

and

. .,.-„. Cy)g.^ have been used; the

coset space metric is (cp. Appendix D) g (y) =e "My) e^f" (y) -^

The components of the Killing fields had been introduced as

IU r" (y) = Dj Is (L(y) )e^f- (y) and K/j (y) = g Kj v (y) . The Killing

fields have the fundamental property that they serve to

express infinitesimal group transformations g=l+ £* Q^ as

coordinate transformations, viz. y^ ? yl* + £""K/jV{y). One thus

expects that the coordinate transformations

can be re-expressed as loc^l infinitesimal gauge transformations

for the would-be gauge fields A M x ) . That this is indeed the

rn.=-e . s •shown in appendix C in the metric formalism for infini-

tesimal transformations. In the vielbein formalism for finite

transformations the same result has been proven in ref. L l i J•

The calculation in appendix C is carried out for G , the matrix

inverse of G^. The inverse vielbein i«

(0

where E and e_T are the matrix inverses of E

respectively, and A *• (x)MN M

tensor G then is

and e. *

m f-E n A '* (x). The inversea n

<The change of the metric tensor under coordinate transformations

at a given point is O G (z!

nata transformation ( , '3 ) we find in particular

GiMN(z) - G^Cz). For the coordi-

with

Again we find that coordinate transformations can be reexpressed

as gauge transformations.

Tlje curvature scalar corresponding; to the Kaluza-Klein ansatz

is evaluated using the zero spin connections which are given by

(cp. the general formulae given in chapter II)

J d y(flete ^ )DA' (L(y) )Da^(L(y) ) =k V £ j « , with

k -(dim G/H)/dimS. The left hand side here is a group in-

variant Symmetric tensor of rank two. Only one such tensor is

available., the Killing-Cartan metric tensor, which in our case

is up to a factor 0 » A .

Therefor* the four dimensional action corresponding to the

Kaluza-Klein ansatz comprises gravity coupled to a Yang-Mills field,

the gauge group being G. There is also a cosmological constant

because of the presence of IL. in (4.23). This constant is veryT ft 2

large ( ssj 10 GeV ) . It can not be compensated by a D-dimensional

cosmological constant, since from the point of view of the

D-dimensional field equations, a cosmological constant and Rg

ars really quite different objects.

o

far Vwhere = E mE. n

a b - E. mEba and the Yang-Mills

field strength tensor is defined as usual, F *n r * Jk " ' _ mn

The structure constants are defined through

with Q the generators of G. Using ( <f. \3 - ."it.

the 4+K dimensional curvature scalar

Here

), one finds for

Einstein " ¥ abREinstein i S t h e four dimensional curvature scalar

expressed in terms of the vierbein E a(x), and R_. is the constantva it _^

curvature scalar of S/H. "To proceed further, one notices first that because of the

triangular form for EJJA we have det(E A) = detCEna)det(Ev*)

det(En )det(e*^ ) . The x- and y- integrations therefore separate

Moreover, d« (y) dette^'ld y is a G-invariantcompletely. Moreover, d«measure on G/H. We therefore have in particular

3. JBD scalars

In the previous section we have chosen for the internal

metric the metric of the coset space. Thus, unlike for the five

dimensional case, we have excluded £ordan-Brans-£icke scalars m<|| \t\'i. \

{JBD scalars) from the discussion. This section now will contain a

short discussion of JBD scalara.

Unlike for the five dimensional case-there is no reason

that in the general case JBD scalars should be tnassless. In

five dimensions, there was an accidental symmetry under, rescaling

of the internal radius that could be related to the masslessness

of the scalar.

Historically, the JBD theory has motivated the idea of an

^'•ffctive'y apace-time dependent gravitational "constant"

this actually being the main novel feature of the JBD theory

as compared to the Einstein theory. Another not unrelated idea

is that the gravitational constant acquires its vaXvie by a

spontaneous symmetry breaking mechanism I [**>() \- In fact, one

may ask the more general question of what symmetry breaking patterns

JBD scalars may provide. This question is certainly of relevance

for some supergravity theories containing JBD scalars / ft^ J.

JBD scalars are being introduced by allowing the internal

- 35 -- 36 -

i.mm-m m -* m

metric tensor g in the Kaluza-Klein ansatz to become x-dependent.

Such a procedure does not affect coordinate transformations which

are associated with transformations corresponding to elements of

the internal group G. In particular we expect that g v (x,y)

transforms undMG - transformations as g o (y) before, that is

like a second rank tensor whith is forminvariant under internal

coordinate transformations. Above, we wrote for the internal metric

tensor g,.<)(y) = e^Cy) euMyJ'o n, = diag(-...-). Here we

now replace the constant tensor <* by &S*} ' L i k e %a •> t n e fields.

^p (x) have to G-invariant, ie. they transform like group scalars.

In the Kaluza-Klein ansatz (V, 12) we replace g (y) by

5 *(y) e,"(y) y (x); for the off-diagonal elements^ P- * *• -7 /

of the metric tensor we replace A <y(x)Kjt(y) by A <f Kj. (y)g. ,(xil«m / r m f hv ' iff 1

It is then principally strsightforward to evaluate the D-dimensional

where

g|u(x,y) 3

curvature scalar. I merely quote the result from the original

paper i_ _j . Some notations will be needed. We have y (x)K0(x) tj*^. (x) with L?( x )3

where the latter defines k

(x)), and g (x,y) -

We denote( * , y ) .

V 5*It then turns out that

•V « • * »

One notices that the potential for the JBD seniors is not

unlike in structure to those for nonlinear sigma models.

The symmetry breaking patterns for a potential similar to

that of equation Ci.26) have been investigated in some detail

by Scherk and Schwarz [ /"KJJ. However, their treatment of the

invariance properties of w a s different from the one

sketched here. Actually, the same remark applies also to an

earlier paper by Cho and Freund L ^ J- However, as has been argued

by Randjbar-Daeml and Percacci f(S^J, it ia the present treatment

that seems to be the consistent one.

A discussion of the symmetry breaking properties of

the scalar sector of Ct.26) is still lacking.

It ia not easy to read off from ('1.26) whether the scalar*

are massive or not. What is needed is an investigation of the

spectrum of the theory. For the six dimensional case the results

of such an investigation have recently been presented ( /fc , J.

In this cas-e, the scalars turn out to be massive.

Thus, the role of the JBD scalars in the Kaluza-Klein

framework remains somewhat ambiguous. Experimentally they

seem not to be really needed (eft. ref.(.H8 1) except perhaps

as a possible explanacion for Dirac's large number hypothesis

However, supergravity may enforce their incorporation .(yj,}/!,

even though they do not seem to appear in the zero mass sector

of the theory.

- 37 -- 38 -

Kaluza-Klein ansatz for the CS model Relation to fibre bundles

Above, we have described in some detail the Kaluza-Klein

ansatz (zero mode approximation) for the high dimensional metric

tensor. However, since no spontaneously compactifying solutions

with nonabelian internal symmetry and vanishing four dimensional

cosmological constant have been found, this discussion lacks some-

what in consistency. It would be desirable to discuss the zero

mode approximation for a model that actually does display the

phenomenon of spontaneous compactification, like the CS model

{the gravity-Yang-Mills-systera) which we have described above.

In fact, an early formal discussion of the zero mode

approximation has already been given by Lucianl [_ 10^ 1.

A general observation is that the metric gouge field and the

external gauge fields get mixed. However, the gauge coupling

constant for the external gauge fields tenda to infinity just

•a the masses for the massive modes; in the four dimensional

theory, the two types of gauge fields therefore behave quite

differently.

A detailed analysis of the Einstein-Maxwell system in six

dimensions has recently been presented by Randjbar-Daemi,

Salam and Strathdee f It 2-7. In particular, these authors dicuss

the classical stability of the ground state solution (with

positive answer). They also analyze the spectrum of the theory

after harmonic expansion. The massless states found are just

the graviton, the SU(2> (internal) and U(l) (external) gauge fields;

there are no massless scalars. In addition, spinor zero modes are

being constructed. Their existence follows from the general arguments

we have reported above; the construction given here continues

similar work done by Horvath and Valla £* ?£" laome time ago.

To some extent, Kaluza-Klein theories may be viewed as the

physicists' way to understand fibre bundles. This connection

was probably first obseved by deVfitt (2oy[and later much elaborated

on in particular by Trautmannfi J, Kjjand also by many others (/*, 2Y, 3»

'O^M, «"7j . Here, I will only very briefly describe in what sense

the Kaluza-Klein ansatz /V '/ ^reflects the structure of the fibre

bundle. For more detailed expositions I refer to the literature.'

Rougly speaking, a bundle is a space £ which locally

looks like the product of two spaces, M, the base space,

and F, the fibre,Z£_ M x P . In our formalism above, the local

product structure is reflected in the possibility to choose an

orthonormal set of frame vectors. There will be a projection

TT,-?-*M with the properties thatT(S) = H and 1*'(x) " F for all

i( M. The existence of a set of homeomorphisma £ •*£ which among

some other requirements, have to form a group G, makes the bundle

a fibre bundle. If the fibre space is the same as the group space,

we have a "principal fibre bundle", otherwise we speak of an

"associated fibre bundle".

It is now a nontriVial result that a fibre bundle, once made

into a Riemarinian space (i.e. endowed with a metric), has a metric

of the form ( <M2. ) .

Consider firstly the case where the fibre is identical to

the structural group. Such • principal fibre bundle underlies

a gauge theory with G as a gauge group. Local triviality of the

bundle means that G ^ can locally be given block diagonal form,

^ bdiag(^ mn, g (y)), which incidentally may be recognized

as the statement that locally gauge fields can be transformed away.

The metric we identify as the metric part of a spontaneously

compactifying solution, with g ^ l y ) the metric of G. Note now

that the zero mode fluctuations contain gauge fiel<i« correspon-

ding to the gauge group GxG, since G admits left as well as right

transflations - in the framework of spontaneous compactification,

automatically takes the role of a coset space for GxG: we thus

really do not have a principal fibre bundle. .

m

V<? thus realize that within the picture of spontaneous

compacttfication, a principal fibre bundle can not be expressed

as a Kaluza-Klein ansatz [ZOgl. Bather, a .Vors bundle treatment

of the Kaluza-Klein picture ought to employ associated bundles

from the very beginning.

Whereas a considerable body of work is available on metri-

sized principal fibre bundles, the couching of theKaluza-Klein

picture with coset spaces as fibres ("homogeneous fibres") into

the language of fibre bundles has received much lees attention.

Early attempts are [ 3 £, 2T- jand I 4s? ^ , more recent formulations

ore given by (f^^and [3,2. '.

Let us put soiie of the remarKs made into a slightly mor.>

formal language. Introduce the vector fields 01 = E M "i .We noted

in chapter IIA that by means of the relation[2 , 3 1 = -Xh~Bl°cL

we can obtain the connection coefficients. Far simplicity choose

Minkowskian space as base space (i.e. disregard physical gravity),Ea = 0 a

m* I n this case we have ic^, 13,, \ = (x)K/, (y) £) =

a n d C9A J'CL l---t.|>«-j "^ i where the latter simply provides us with

the connection coefficients on the coset space. This construction

clarifies how tht particular form of the D-dimensional vielbein

relates to the appearance of the Yang-Mills density in the decom-

position of the D-dimensional turveture scalar (use the definition

of the modified curvature scalar R at the <sna of section IIA).

The vector fields 3 and <jt are referred to as "horizontal"a *.and "vertical" vector fields respectively. It is characteristic

that vertical vectors commute among themselves, horizontal vectors

commute TO give a vertical vector and that the commutator of a

horizontal with a vertical vector vanishes. The construction of

horizontal vector fields amounts to the same as the construction of

a "connection on a fibre bundle". In fact, the gauge fields are

to be identified as connections on the bundle. This construction reali-

zes explicitly the local triviality of the bundle. The associated

bundle corresponding to the Kaluza-Klein bundle may be viewed as a

subapace of the "frame bundle" provided by the tangent vectors of £. .

The specific structure expressed by the properties of horizontal and

vertical vectors singles out the associated fibrebundle with

structural group G.

C. Spinors

Fb ^ D**(L(y) ) c . Quite generally, we also have ft) , 9 7= 0

For spinors, the process of dimensional reduction is rather

involved. Only quite recently, general formulae have become

available C

In the next section, I will report the result for the dimen-

sional reduction of the Dirac Lagrangiau. After that, I will discuaa

the issue of zero modes for the internal Dirac operator and the

related question of parity violation in the dimensionally reduced

theory.

Explicit formulae for the dimensional reduction of the

Rarita-Schwlnger Lagrangian (relevant for supergravity) are not

./°; available. Therefore, only some general properties of the

Rarita-Schwinger equation in k+K dimensions will be discussed here.

1. Dimensional reduction of the Dirac Lagrangian

The general form of-the Dirac Lagrangian in D dimensions

has been given in chapter II to be = i-dtt E k-c. .

Here U T <Mx,y) is an SO(1,3+K) - spinor with 2 -* compo-

netns. It carries a label associated with H; the generators Q—

of H are given by the imbedding into SO(K), Qj » - I s Z?f .

If the label on <£ is exposed, j£ -? <+-,- then the H generators

wil act on <i; as ID. ( 2 1*) U' .. For the spinor we use the ansatz

|L(j,y) = D(L (y)) (x) which is symbolic for the Fourier expansion

(explained in chapter II) v

\Let us consider the covariant derivative on G/H. It is to be

defined as (note that the imbedding H —">SO(K) involves a minus sign)

Since the y-dependence of the spinor is given through the Fourier

expansion, we can eliminate the derivative 2l • I n fact, using

the definition of e(y), we obtain

- kZ -

After integration over y one gets

with ¥ -1= G^fjJj,11 • Using the imbedding of H into the framegroup

as well as a suitable connection on G/H, one obtains a mutual

cancellation of terms containing T\,* . In fact, in ref. L'^T 1

the connection coefficients B*£»-•=-4,/ i. * •?- - IF 3 -f

are being used. In general, they correspond to nonvanishing

torsion coefficients, T^* =- ~ £ ** . Note that these vanish on

symmetric spacesj however. Now the imbedding of H into SO(K) is

provided by -\ ^ ^ (Z**) - D ^ (Q.$ ") . Inserting this, one

obtains for the covariant derivative of Dn(L~\y)):

We see that the operation of covariant differentiation on coset

spaces is turned into an algebraic operation.

In order to apply the Kaluza-Klein ansatz to the Oirac

Lagrangian, it is convenient to denote 2^ - !5f£r ^ etc,

by the generic symbol Q^ (similarly for P the totally

antisymmetric product of P- matrices) and to drop labels on

The Dirac lagrangian then becomes

. c.

Here we now have to insert the various expressions for B r -i ,A1 B CJas given in the Kaluza-Klein picture (see above). One eventually

obtains, using the connection on G/H as given in this section

with

and

Even though the actual derivation of ( 0, ?3) is cumbersome,

the emergence of a four dimensional Lagrangian with invariance

under four dimensional frame rotations, general/coordinate trans-

formations ,and gauge transformations does not come as a surprise.

In fact, recall that these are just the symmetries left from

the D-dimensional ones after imposing the Kaluza-Klein ansatz.

The results as presented here are taken fromf/6? ~J. More

detailed derivations for certain special cases can be found in

(_2il'f%tll.~l. A l s o , in a somewhat different setting, a treatment is

given in [H, to3ij.

In equation (U.Yi) , the first three terms in the curly

bracket provide a Riemann- and gauge- covariant derivative. The

terms linear in the gauge field can be traced back to originate in

EJ"tL and 8af(1 ] P " • In addition to these, we have-a- Fierz-Pauli

type term and a mass term. The first indicates that we have a

non-renormalizable dimenaionally reduced theory, a not too sur-

prising observation since we started off with a non-renormalizable

theory (D-dimensional gravity) in the first place. The coupling

of the Fierz-Pauli term is small, and the term may or may not play

a role physically. Incidentally, the term actually disappears

under certain specific circumstances n l j . J.

The mass term, on the other hand, is causing troubles,

which we will describe in more detail below. In fact, if it

does not vanish, spinors will have a mass not smaller than Planck's

mass.

Some final remarks are in order.

For the imbedding problem, let us consider an example f"/£<J 1-

We choose G/H =. SU(3)/U(2) (= CP 2). The adjoint representation

§_ at SU(3) decomposes under U(2) = SU(2)xU(l) according to

. 8 = 1 ^ + 2 ^ + 2 ^ + 2_t, where the indices refer to the U(l) factor '

(hypercharje). We identify the genrators of SU(2) and U(l) with

the 2^ a n a t n e J jt respectively. Accordingly, the H - content of

SU(3)/U(2) is 2• 1 *

As 5Ui'3)/U(2) is four dimensional, the frame group will beSOCl) . The label i on <x,y) will therefore take on four values.

Now, two four dimensional representations of S0(4) are avai-

lable. SOCl) is locally isomorphic to SU<2)xSU(2). The SU(2)xSU(2) -

contents of the two four dimensional representations is (2,2)

(vector representation) and (2,l)+(l,2) (bispinor representation),

respectively. The vector representation may be rewritten as

(2,2) = (2,l)x(l,2). The H=SU{2)xU(l) - contents of (2,1) is

2^, and the H-content of (1,2) is 1 + 1 . The vector repre-

sentation is then _2 x(.ly * A Y ) = Su-f Y + —v ¥ ' which

must be _2 + ,2_,- Thus the vector representation is obtained

with Y2 =-T. => 1. Correspondingly, the S0(4) four vector transforms

und H as ,2. '+ £_1 whereas the bispinor transforms is 2 . H . t i-i*

Whereas the SU(2)xU(l) representation 2_ + _2_ is obviously

contained in some SU<3) multiplets (it is containedin the adjoint,

for example), the same does notNapplyto the SU(2)xU(l) repre-

>9.taii ">?• J^ •<- .j i_i' T h i a h a S the important consequence that

no Fourier expansion for the bispinor of SOCt) exists on

G/H = SU(3)/U(2) = <CP2.

In this sense, apinors do not exi3t (can not be defined)

on CP . The transformation properties of an S0(4) spinor are not

compatible with the symmetry properties of <CP . This incompatibility

must have its roots in the global properties of CP , as locally the2

frame group of CP is of course just S0(4).

A precise criterion as to which spaces allow a definition

of spinors will therefore be a topological one. In fact,(compare

t 51 J) a manifold admits a spin structure if and only if its

second Stiefel-tfhitney class vanishes. Unfortunately, the definition

of the Stiefel-Whitney classes is rather abstract; in particular,

no simpl* integral representation like for the characterisitic

classes has been given for them.

Finally, a remark on the transformation properties of the

spinor (J, (x) a ( (l» (x) ) as it appears in the dimensionally reduced

Dirac Lagrangian. We notice that only the index p is associated with

a gauge field. This is to be expected, as the index on <£ .(x,y) is

a H - index rather than a G - index, and the gauge invariance is

with respect to the group G. This means that the internal degree of

freedom of ti/(x) originating in the use of higher dimensions is

not being gauged. (Under certain specific circumstances there

are exceptions, cp. ref.[ 2-2-J. A typical example requires

the coset space to be of the form HxH/H.) Now, in grand unified

theories also only some portion of the available symmetry appears

to be gauged. Whether the structure of KaLuza-Klein theories is

relevant for such situations is not yet clear.

2. The zero mode problem and parity violation

We have seen in the previous section that the dimensionally

reduced Dirac Lagrangian contains a mass term. In fact, if we

decompose the D-dimensional Dirac operator, F ol = P <%. + ' *d, \

then the internal piece will act as a four dimensional mass operator,

providing the Dirac field with'a mass of the order of Planck's

mass. Therefore oniy the zero mode sector, that is the solutions of

would be of interest to us.

The solutions of equation ( &%,) form a representation of G.

This is so since the operator " d is G-invariant by construction.

The zero mode sector gives a parity invariant four dimensional

Dirac Lagrangian, unlike the massive sector. Zn fact, let us

choose a representation for the generalized Dirac matrices such that

- 46 -

I a % ®'I (CP- also appendix B), where ^ are standard four

dimensional Dirac matrices. From the definition of cLj one then

reads off that = 0 for all a, ao that the eigenvalue

equation for the internal Dirac operator can be given the form

P*^* 4l = TS/Vn, (f with Vs *• yS © A , The mass term therefore

has the form \Z V ft* u, • Since no other sources of P-violation

are present we indeed conclude that the massive sector of the

dimensionally reduced Dirac Lagrangian is P-violating and the

raassless sector is not. *

This argument is not quite complete as it stands, since we

have left out the possibility that the eigenvectors of equation

C(|,?fc) come in left-right asymmetric pairs. This subtlety can

presently be neglected as it turns out that P** d^ has no zero

modes at all. This ia a consequence of the identity

which can be derived by using the definition of the curvature

tensor, [[Jutl J4; = ^n^rrffil^* t a n d i t s sy m m e* ry properties, the

vanishing of the covariant derivative of the P -matrices and the

anticommutation relations for the P - matrices. Now, on coset

spaces of compact groups", -d d and R have the same sign, moreover

(compare appendix E), R is a nonvanishing constant. Therefore the

right hand side of ('f.Tf-) is never zero, thus (P*a(ljran<i conse-

quently P dj have no zero modes.

Now, from the physical point of view, we would rather have

zero modes _and_ parity violation. We want the latter, as the funda-

mental fermions in unified theories t eno* 16 come in left-right

symmetric pairs. Therefore the question arises how we could get

out of the impass outlined above1* '

Por scalar fields, the following method has been suggested[fl?].

Imagine we introduce a negative (mass) in the D-dimensional

theory which would just compensate the mass of th«"^ightest massive

mode. We can then prevent the appearance of tachyonic modes by

using twisted fields. In fact, recall that for twisted fields we

can arrange matters such that the lowest mode does not appear*

In this way one would arrive a__t a masaless four dimensional theory

if the higher modes are neglected.

Such a procedure does not work in the present case. In fact,

we could have to introduce imaginary masses which would spoil

hertneticity requirements. What we could still do however is to

impose a chirality condition; the troublesome mass term may then

disappear identically.

According to our general philosophy, the chirality condition

must be imposed on the high dimensional spinors Cp(3c,y) and it

must be compatible with the requirements of local Lorentz invariance

and general coordinate invariance in D dimensions. „

Denoting P D + 1 a P •... *"D> s u c h a constraint is given by

the Weyl condition f £ = t £ . Weyl spinors exist in even

dimensions only (for odd dimensions, i is a multiple of the

unit matrix). Note that for spinors subject to the D-dimensional

Weyl constraint, four dimensional chirality (eigenvalues of

P j-... -P.) and internal chirality (eigenvalues of P.: . . PQ)

are related.

Another constraint by which we may obtain masslessness for

spinors in four dimensions, is the Majorana condition. The Majo-

rana condition, imposed in D dimensions, may actually also enforce

chirality in four dimensions. Majorana spinors exist in

2,3,4,8 and 9 mod 8 dimensions.

A detailed analysis of the relation between high dimensional

Weyl and Majorana conditions and four dimensional chirality condi-

tions has recently been carried out [ZdM, compare also ("*/ *0 J.

The nfet result is that Weyl spinors in Da6,8 mod 8 ,

Majorana spinors in D=9 nod 8 and Weyl-Majorana spinors in

D = 2 mod 8 dimensions are promising candidates for the provision

of the desired objective of chirality in four dimensions.\ i

Qjitf s. different possibility to obtain masslessness for

spinors and at the same time left-right asymmetry is indicated

by the Atiyah-Singer index theorem. (cp.^S^j). This theorem

relates the socalled index of the Dirac operator, ind (f*D), to

be defined as the difference in the number of left-handed and

right-handed zero modes to topological quantum numbers referring

to the underlying manifold. In a typical example, the index

may be expressed in a linear way through the characteristic classes.

The theorem applies to compact manifolds only.

This however 1-< perf>ctty sufficient for oar present needs, as

we are interested in the zero modes of the intenial Dirac

operator on i y,

A typical scenario where left-right asymmetric zero modes

have actually been realized is the following j?£~ /(-j. 1. Recall

that in the CS modet (the Einsteln-Yanf-Mills system) an external

Yang-Mills field appears. Further we have seen that spontaneously

compactifying soJutions exist with nontrivial topological properties

coded into the would-be Higgs part B^(y) of the external Yang-Mills

field. In this case, the internal Dirac operator is modified,

I d*""* ' (<** * CU^H,)) - This modified Dirac operator now

does have zero modes, on account of the Atiyah index theorem,

since thf> characteristic classes belonging to B^ (y) do not vanish.

Also, tlic zero modes do not come in left-right symmetric pairs.

In this context it is to be noted that actually the addition

of Bo^(u) spoils in any case our previous argument on the posi-

tivity of the internal Uirac operator.

It is to be stressed that implicitly underlying the present

discussion of the zero mode and parity violation problem for Dirac

spinors is the assumption that fundamental fermions are (nearly)

massless and come in left-right asymmetric combinations. This need

not be so and actually grand unified theories with left-right

symmetric fermion multiplets have been discussed tftC25!.

As to the masses it has been suggested ( fS" J that funda-

mental fermions retain their high masses, but neutral (colour

neutral) bound states may turn out to be nearly masslessj this

would be a confinement mechanism.

Further, in a supersymmetric context, the relevant high

dir.iwnsi.o, ! fieia may be a Ilanta-Schwinger field. This also

alters the picture, as the properties of the Dirac operator are

quite different from that of the Rarita-Schwinger operator. Some

properties of the latter will be discussed in the following section.

3. Rarita-S-:hwinger fields

"The" supergravity theory, the N=8 theory, contains in itseleven dimensional formulation just one fermionic field, a Rarita-Schwinger field 4- (x,y).

"AAs a spinor in eleven dimensions this has 32 components.

These are subject to a Majorana condition in order to provide the

correct counting of states for a supersymtnetric multiplet. From

the four dimensional point of view, U< decomposes into eigth

Rarita-Schwinger fields (£• (gravitinos) and 56 Dirac fields <y

The dimensional reduction of the Rarita-Schwinger Lagrangian

for "£. =M,xG/H has not yet been carried out. This section will there-

fore contain only a few general remarks.

The candidates for the elementary fermions of some grand

unified theory may be taken to be the Jf , for these we face

again the zero mode as well as the left-right asymmetry problem.

Let us consider the Rarita-Schwinger equation in the form

P A dgf'c <x>y) = °t with dQ = 5 B + Bg. The connection B0 carries

a vector and a spinor representation. By definition, I =

(ViV P - P P P + cycl.), which can be rewritten as

Contracting the Rarita-Schwinger equation with f* and d respectively,

one finds two constraints. Employing these one finds that the

Rarita-Schwinger equation may be replaced by

Unfortunately, the Rarita-Schwinger operator M^ lacks a simple

positivity property as possessed by the Dirac operator. On the

other hand, the Rarita-Sehwing«r equation simplifies considerably

if the constraint ? 5f• 0 is imposed. (Such a constraint may

actually be necessary in order to obtain the correct number of

states.) Instead of ( f, 3g ) we then have fS^g (£*= 0 . This looks

very nearly like a set of Dirac equations, exc-ept that dD carrieso

also a vector representation. On the other hand, we may equally

well study the Rarita-Schwinger equation in the holonomic basis,

>

Now, because of the

- 50 -

Therefore, in this form, no connection coefficients will be

attached to the vector index. Therefore ( V. 'H ) is just a set

of decoupled Dirac equations, and we have a zero mode problem as

before for the Dirac operator.

Also it is likely that problems with left-right asymmetry

will have to be overcome. In fact recall that for the particularly

interesting case o^ eleven dimensions, Weyl spi'nors do not exist,

which deprives us at least of this way to obtain left-right

asymmetry in four dimensions.

V. HIGH-DIMENSIONAL YANG-MILLS THEORIES

A. Introductory remarks

Yang-Mills theories are, like gravity theories, of

intrinsically geometric structure and are thus naturally formulated

for space-time! of arbitrary dimensions.

We have seen that high dimensional gravity theories provide

a possible framework for the unification of gravitational and

gauge interactions. We will see now that high dimensional Yang-Mille

theories provide a possible framework for a geometrical unification

of gauge and Higgs fields. In fact, many people consider the Higgs

sector in unified theories as unnatural, artificial and arbitrary,

and many attempts have been made to improve on such features.

The basic idea of the present attempt is to introduce gauge and

Higgs fields as different components of a high dimensional Yang-

Mills field. This clearly restricts the number of possible Higgs

structures.

Recall also that the CS model of spontaneous compactification

contains high dimensional Yang-Mills fields. In addition, in a

supersymmetric context, Yang-Mills theories often occur naturally

in high dimensional space-time. All this provides ample motivation

to study high-dimensional Yang-Mills theories.

We will see below that one can use a high-dimenaional Yang-

Mills theory to unify gauge and Higgs fields in the Weinberg-

Salam model. It then turns out that the characteristic mass scale

determining the internal radius is not Planck's mass mass like

in the Kaluza-Klein picture above, but rather of the order of the

mass fct the lightest Higgs boson, tha't is, probably around 100 GeV.

This observation suggests the fascinating speculation that already

at such low energies we may be able to actually see the internal

dimensions directly since at this energy high dimeiLaiona1 space-

time rotations would transform four dimensional and internal

coordinates into each other.

It is easy to see that a high dimensional Yang-Mills theory

contains a four dimensional Yang-Mills-Higgs theory. Let the gauge

field in D=4+K dimensions be VM<x,y) = (Am(x,y),A, (x,y)) and let

^ (the field strength tensor be - ~^NVM + (-VH»

VN

Now choose all fields to be independent of the internal coordinates,

OfVM — 0 • The action density tr(W N« ) then contains the usual

kinetic term for the four dimensional Yang-Mills fields, tr(W W ™ ) ,

and the A^ serve as Higgs fields in the adjoint representation

of the gauge group, with a minimal coupling term tr(W W r ),

1H u its r I Ttl f*v

e2tr(CAp A,]1).

Note that as a result of the D-dimensional rotational invariance

of the theory, the self couplings of the Higgs field and the gauge

field appears to be the same.

The Higgs potential obtained in this way provides minimalenergy solutions of the field equations AK = const, CA

,.~ 0.

i-ci. soittions are degenerate; A may be replaced by A AM i A =const.

As a result of this degeneracy, the solution is instable in the

classical sense O^Q- Quantum corrections tend to lift the degeneracy

entirely. The effective potential (including first quantum corrections)

in general does not have nonvanishing minimal energy solutions at

all. Therefore the above nontrivial solution is not expected to

trigger a Goldstone-Higgs-Kibble mechanism [ I2C J.

Certain care has to be taken for supersymmetric theories,

where the following theorem holds : If a degeneracy like the

one above occurs in jeroth order perturbation theory, and if super-

symmetry is not spontaneously broken, then the degeneracy persists

to all order's in perturbation theory. In such a case one may still

entertain hope that the exact dynamics of the system in such

that a Goldstone-Higgs-Kibble phenomenon persists.

Early attempts to construct a unified picture for gauge

and Higes fields using a simple pVocediire as Outlined above are

given in references [^i^f^O^2, til - ISf J. Eventually it turned out

thnt in order to obtain a suitable Higgs sector, a number of ad hoc

assumptions had to be made. There was some hope that-these could be

justified by the use of graded gauge groups. This however has

caused new problems (£>p)fj. Because of the anticommutation relations

in the graded gauge algebra, the theory will contain wrong statistica

fields. Also, gauge field kinetic terms appear with wrong signs.

Other attempts have been made to cure these problems by identifying

the wrong statistica fields with ghosts (see below).

- 53 -

In the following X will in some detail describe the most

attractive scheme which has been advocated mostly by Manton

and collaborators g.1|6?-. ll?,l|6 k 'So ~\ • There a method of

dimensional reduction is employed where the condition 'cL U =0 is

required to hold only up to a gauge transformation. In this way, some

promising results can be obtained, t£p.«i*e C^SY, h-(, J J

After that, some remarks on modified Yang-Mills theories

and a possible geometrical interpretation of ghosts follow.

B. Dimensional reduction of Yang-Mills theories

The requirement that Vu^L - 0 is to hold up to a gauge transfor-

mation provides an interrelation between y-dependence and gauge

symmetry. In this way, the constraint is weak enough to let us

probably escape the stability problems mentioned above and strong

enough to restrict severly admissable Higgs sectors. In fact, it

is not yet clear whether the restrictions are not too strong to

allow for the description of physically sensible situations.

In the following section, G-nymmetric gauge fields will be

introduced as solutions, to the constraint 9^1/^ ~ 0 (UP to a gauge

transformation). The general form of the constraint for the four

dimensional: fields will be given in the following section, and

its solutions are described in the subsequent section. A further

section is devoted to a discussion of spinors.

1. Symmetric gauge fields

We otasider Yang Mills theories with gauge group R over

space-time of the form O = M. xB with B = G/H, where G and H

are compact groups. Note that not necessarily G=R (in the abstract

group sense). £, is endowed with a metric ^(ilsbdiagfa^lx) ,G ,(y))-

The gauge fields are V^z) = (Am<x,y), A, (x, y)) and the field strength

tensor has components W ^ = 3 ^ - V^ + e f~VM i V^ J . The

D-dimensional action we choose to be

(SM )

with ^ - Ja %. dd-(£^)1' i M4 w e assume to be Minkowski space.

^ -Q I

The action of g £ G on C is g,: (x,y) —>(x, g y) . We will now

call a gauge field V^(z) to G-aymmetric, if for any g^ G we can

find a gauge transformation r£R compensating the action of g.

Note that the coordinates y do not depend on x, and the action of

G on £, should not introduce such a dependence. Therefore, the

compensating gauge transformation r should be x-independent.

Let us denote the compensating gauge transformation by r(y,g).

Then G-symmetric gauge field are specified by the requirements

(S-2)

that is

Then (CM) and ($S) become, with r( | ,h) •= (h),

Here7Xj (y,g) is the Jacobian for y1

Ju (y,s) = ^ L W l TNow it is important to observe that for G-symmetric gauge

fields, the action density tr(W ) becomes y-independent. To see

this one has to remember that on coset spaces like Bv = G/H

there exists a fixed pornt |e B with the property hj = f for

all h£ H and the further one that any y e Bv can be written in

the form y = g£ with some jgQ. Inserting this form for y into

(SI ) and ( 5,1) respectively, we obtain

V*»> --" (From (£.<f) we now easily infer W. (x,y) = r(y,g)W (x, £ )r~ (y,g).

Accordingly, we have indeed tr(W (x,y)Wmn(x,y)) =

tr(¥mn(x,J' ) K™(x,|)) for any y 8R. Similarly one proceeds with

the remaining terms in the action (recall that raised indices are

transformed with inverse Jacobian).

The relations i( 5-f ) and ( ) are constraints to be ful-

filled by the would-be gauge field A and the would be Higgs fields

A^ , respectively. To see how this works in more detail let us

consider a special example where we set g=h£ H so that y=gf-^e -T.

- 55 -

which are algebraic relations for the fields at the point (x,£ ).

The mapping u. :H-)R constitutes a group homorph'ism. Actually, it i*

to be required that, in the abstract group sense, G is a subgroup

of H.

From the arguments above we can identify Ara(xi t ' a s tlie

gauge field of the four dimensional theory. Let the effective

gauge group of the four dimensional theory be T, with elements t.

Then the constraint (S"6) implies ["u W.), i. ~] - 0 ' Therefore we

identify T as the maximal subgroup of H that commutes with H. (T ia

called the "cantraliaer of H in H". We shall write R D T X H . )

As an example consider the case H=SU(3) and BK=G/H=SU(2)/U(1)=Sg.

Then T=SU(2)xU(l).

Principally in the same way we proceed for general gS <S.

Also, ( S 5 ) will put constraints on the Higgs sector of the

four dimensional theory.

Further constraints emerge if we set y • ^ + 5 .

in (9-2) and (?•"*) and compare terms of first order in (Ty.

(Equations (C-t) and (S'T-) correspond to terms of zeroth order

in &j, in this expansion.) One get* (j> (, ~\

(?•?)

)

Here Q j are the generators of the subgroup H. From ( '_. g) we infer

that vector bosons get their masses through a Higgs-Kibble mechanisd

only; there is no specific mass generation by dimensional reduction

for the vector bosons* Equation ( 5'f > tell us that unlike for the

situation with y-independent fields we now have quadratic terms

in the Higgs potential of the four dimensional theory.

For the calculation of the effective four dimensional action

- 56 -

we need V (x,y) only at the point y= | . Therefore all infor-

mation we need is provided by (£•?) and (S.*)) and the solutions

of (54 ) and (.$•%). Symmetry breaking patterns are determined

inasmuch as they follow from the form of a Higgs potential and

it* parameters.

The solution of the constraints is a difficult group

theoretical exercise. Nevertheless, quite general results have

become available and will be described in the next section.

These results refer to the symmetry structure of the four dimen-

sional theory rather than to the actual form of the four dimensional

action. The latter is not easily obtained for unconstrained

fields; for some special cases however, compare £/K, /(fc ,3pj.

In terms of constrained fields, the action density is simply

irt'aL-jC , % /"* " (x, ? )) where we may insert ( S'S? ) and ( £•*} ) for

simplification.

2. The bosonic sector

As a first example for a solution of the constraints in the

bosonic sector we noted above that the effective gauge group T

of the four dimensional theory is the centralizer of H in H,

R O T.xH. After spontaneous symmetry breakdown, the exact gauge

group T is given as the centralizer of G in R, RDTxG, ¥ C T .

In the example from the previous section, we had R=SU(3) and

G/H=SU(2)/U<1)=S2 whence T = SU(2)xU(l) and f = U(l). One notices

that this example fits the requirements of the Weinberg-Salam model.

The detailed structure of the Higgs sector is as follows.

The adjoint representation of G, adG, decomposeses into irreducible

representations of H according tb the branching rule

adG » (adH) + ^ n, (5.1O)

where each n. denotes an irreducible representation of H.

Further, adH decomposes into irreducible representations of HxT

according to the branching rule

adH (5.11)T n\ x mi

Now the rule is that for each time when a representation 11. appears

- 57 -

in (5.10) as well as in (5.11), the corresponding m. in (J.ll)J

denotes the irreducible representation of a Higga field of thefour dimensional theory.

For clarification, consider again the above example,

G=SU(2), R=;SlI(3), H=U(.l)v Since all irreducible representations

of H are one dimensional, we have adG = {!) + i + i- F o r the

decomposition of adH into irreducible representations of HxT we

have adR = JLx2 + JLx£ + Ix2_ + !*•}_• We notice in particular that in

this case the Higgs sector is not unique. We may however choose

the first two terms in the decomposition of adS and thus obtain

two Higgs doublets, corresponding to one complex Higgs doublet.

In this way we are able to reproduce the entire bosonic sector of

the Veinberg-Salam model.

We have gained something as compared to the usual situation.

The bosonic sector of the Weinberg-Salam model emerges from a nice

geometrical construction even though there is some arbitrariness left.

The Higgs self coupling and the gauge field selfcoupling are iden-

tified as a residuum of the D-dimensional rotational invariance

of the original theory. Also, the generators of T are related to

each other like generators of R. We thus obtain a means to fix

the Weinberg angle (after identifying the electromagnetic direction).

The result has been worked out for the rank two groups [//S J:

one obtains for SU(3), 0(5) and G(2) the values * w * 60°. (t5°t3O°

respectively. Note that the last result is seemingly the best

from the experimental point of view.This is of course not the last

word as presently the entire discussion is really a (semi-)classical

one. Also, the imbedding into a grand unified theory is still

missing and spinors have not yet been included.

As a final example for the general rule given above let

us consider one relating to the structure of grand unified theories.

Following [21 j, we choose R=E(8), G=S0<10) and HsSUfJJj, where the

label on the latter distinguishes it from another~SU(5) group to

appear shortly. H is chosen to be some fixed subgroup of G.

One then finds TsSUtS)^ as some subgroup of H which is not identical

with H, and T = SU(4). We notice in passing that up to now no model

with T=SU(5) and T*=SU(3 )xU(i) has been found. However, the Higgs

sector in the present scenario turns out to be the one of the

- 58 -

of ..ne Oitorgi-Glishow model. In fact, the branching rules are

adG = h%. = C2^) + 12. + 12.' + i t

adE(8) = 248 = <2*,!£) + <2'i£*' + <12>1> + (1°.* .5.*

whence we conclude that the four dimensional theory contains

a 5_, a 2.* a n d a ih.i o r equivalently a complex %_ a n d a Si-

lt con also be shown that a hierarchy with large spacing can be

obtained at the semiclaasical level.

3. Symmetric spinor fields

We have discussed above in the context of the Kaluza-Klein

theory that left-right asymmetry for the four dimensional theory

has to be coded into the D-dimensional theory. In particular,

for even D, D-dimensional Weyl spinors may be used to obtain

four dimensional Weyl spinors. Now, in addition, we have the

choice to put the spinors into self-conjugate or non self-conjugate

representations £ of the D-dimensional gauge group B. In four

dimensions we would want the spinors to be in non self-conjugate

representations.

Both choices for the D-dimensional spinor representations

have been "discussed. Manton [l[fe Juses the Weyl constraint only

and non self-conjugate representations £, whereas Chapline and

S Ian sky I *3>0 Juse both Weyl and Majorana constraints and employ

self-conjugate representations f_.

The internal Dirac operator is1selfadjoint and correspondingly

Its zero modes form self-conjugate representations. It is comfor-

ting to know that we may choose such representations for _f and still

obtain left-right asymmetry in the four dimensional theory.

Let the fermion field transform according to some represen-

tation £_ H, with r £ _f. Then, for G-symmetric fermion field,

we have the constraint

in analogy with the definition of a G- symir.e t ric gauge field. The

action of g on y is compensated by the compensating gauge trans-

formation rf(y,g). D has been defined in chapter II;-it takes

the role of the Jacobian in (S'3)•

The constraint (5.11) is employed principally in the same

way as the constraint for the bosonic sector. In presenting the

available results, the approach of Chapline and Slansky (lO ( i a

being followed. For a different approach consider the paper by

Manton Tilt J; he also discusses the relation to the zero mode

problem.

The K - vector representation of SO(K) decomposes as

K = 2. a. , where adG= (adH) + 2T TI. (cp. the previous section).

The spinor representation of SO{K) on the other hand decomposes

into irreducible representations of H according to the branching

rule 5 = 5. <±i . Let i , = "'...• ff> a The positive chirality

spinor (b ^ [2 «• in D dimensions will contain a left-handed

four dimensional piece which will be in some representation f,.

This representation will be non self-conjugate if ^-!rt is

non self-conjugate. Necessary conditions for this to be possible

are^3oj(l) The spinor^oT^§0(K) must be non self-conjugate.

This happens for D=4n+2, nf IN, which are juat dimensions where

Majorana-Weyl spinors exist. (2) H must have non self-conjugate

representations. Note that these conditions are not sufficient

as can be shown by constructing appropriate examples. A conjecture

for a necessary and sufficient condition for ?. to be non

self-conjugate for self-conjugate _£ is that rankll = rankG.

The conjecture is motivated by examples.

ATo obtain the rule that actually describes _f recall theiS rule for adH into irreducible representations of HxT,

with T the effective gauge group of the four dimensional theoryf

viz. adH = £ li.xm. . Now, for each time a representation of HV1 J J •*_ —.

appears here as well as in the branching rule }&-~2rS'^ . , a/v.

representation ni is contributed to £ . Denote this subset of

as I li /, then _f = 2. it\ •In a similar way, the right handed four dimensional spinora

coming from po£sitive chirality D-dimensional spi-itors comprisef . J h(H) _ ~ Z(D

i

- 59 -- 60 -

These are the basic results and it now remains to work

out examples which are sufficiently convincing phenomenologically.

One notices first that the six dimensional theory with

G=SU(3), G(2) or 0(5) (which gave the bosonic sector of the

Weinberg-Solam model) does not allow the incorporation of fermiona

in a satisfactory way. The reason for this is that H=U(1), via

the imbedding into the gauge group (see above); acts also as weak

hypercharge group. This in turn fixea the charge assignments

for the complex Higgs doublet as well as for the leptons, and

these two assignements can not be made right at the same time.

However, examples exist of ten dimensional theories that

reproduce the entire Weinberg-Salam theory in four dimensions.

lajse for instance G/H = G(2)/SU(3), and R=SU(5) or Sp(8). This

provides the right Higgs sector. For R = SU(5), one can choose

the self-conjugate fermionic representation £ = J^ and eventually

one obtains one lepton family with correct quantum numbers.

Note that the lepton sector comes out wrong for f = 24, the

adjoint representation of SU(5). This spoils a possible super-

symmetric ahsatz. Quite a similar situation occurs for R = Sp(8),

for which not the adjoint, _f =» 2i< °u* L = 1Z provides a

satisfactory assignement.

For a discussion of unified theories and further details

omitted here I refer to the original papers C^f.^J

I also nott that a paper exists that is concerned particularly

concerned rfith aupersyrametric LagrangianaO('3j .

C. Modified high-dimensional Yang-Mills theories

In addition to the attempts to exploit the structure of high

dimensional Yang-Mills theories in order to construct a unified

picture for gauge and Higgs fields, there have been related

ones usiTig modified version of high dimensional Yang-Mills

theories.

A particularly drastic change that will however not be

reported hero would be the introduction of Yang-Mill3 theories

over spaces with an11.commuting internal coordinates (cp. \"2- j

for references).

More conservatively, the first attempt to be described

oeiow ,-nay be looked upon as an attempt to extend the structure of

a high dimensional Yang-Mills theory to that of an interacting

antisymmetric tensor theory. Part of the motivation here is to

minimize the number of internal coordinates (see below).

In a following section I will describe a simple formal

argument that has raised the hope for a possible geometrical

interpretation of ghosts.

1. Higgs fields and geometry

The gauge fields of a high dimensional Yang-Mills theory

have components A, = (A , A ). Here the label s,t,u, ....A a ^referes to the gauge group R, and o( to some external space which

up to now we have chosen to be G/H. For the rest of this section

we set H=l so that in accordance with our conventions the gauge

fields are labelled Oy (A , A» }.

Geometrically, we are dealing with a theory over M.xGxR.

One may now ask whether a construction is possible where we identify

G and R in such a way that we are dealing with a theory over

My xG only. More precisely, we are looking for a theory for fieldsH b ft

with components (A , A F , A^ * ) , which after reduction to four

dimensions provides us with a Yang-Mills-Higgs theory. I will

now proceed to describe a provisional construction which seems to

display some promising features.

- 6i -

- 62 -

The basic field will be introduced via a two form, A =

AAg E /*, E rather than an algebra-vaL ued one form. In this

way all indices are treated on an equal footing. The coefficients

A jj are components of an antisymmetric tensor. Since we want

gauge and Higgs fields only, we set A . = 0 for the time being.

and A- I* will be treated as invariant tensors of first and

second rank under G. This implies conditions of the form

3*i>:/.j h , and 7). A * * - -k - ' A* * ^ Jt . $ d- *.

aABC

respectively, with 9*- p ri-.* rWe now wish to find an action density of the form F

where the tensors F are of the form F,n_ = L.A,,,, - L-,A.ABC AI3C A BC B A1

<j,a,_ is to be determined by invariance requirements, and L.ABC i Ais a suitably defined differentiation. It turns' out that one has

to set L = 9 and L* - Njr where Nj- * is a G-invariant

second rank tensor, independent of x. Now, if U=(U^r) specifies

an element of the gauge group, then the gauge tranufirmations are

A*

It 13/ important to realize that the would-be Higgs fields

Ajft transform inhomogeneously as befits physical Higgs fields;

this is in accordance with the fact that the vector bosons are

massive from the very beginning^. Closer inspection shows that

N ttkes the role of the vacuum expectation value of the

Higgs field. Indeed, the construction turns out to be identical

to the conventional hidden symmetry picture.

F _ F turns out to be the action densityTbr a Yang-Mills

Higgs system with gauge group G. However, the Nj r have to satisfy

a constraint. This is easily and satisfactorily solved for GaSU(2)

only. For higher groups, only special solutions with diminuished

physical relevance are known. Still it remains that, on account of

(f'1"! ) and ( 'Y) , a CIOJS e resemblence between gauge and physical

Higgs fields emerges in this picture. For more details,

For a related construction, see£V'/~l

2. Ghosts and geometry

The purpose of this section is to describe a simple argu-

ment (cp. the paper by Thierry-Mieg, {jj_ J and references therein)

that indicates a possible way to a geometrical description of ghosts.

Starting point is again the gauge field AS(x)

whtch is now extended to a high dimensional object,

^ {x)dxra.

It will be attempted to identify the coordinates y with the

vertical coordinates of a fibre bundle (for a short description

of fibre bundles, see sec. (W.fS.'f.) ) .

Let us define an exterior differentiation <T = dx™ 3 mAs a field strength tensor we set

The requirement that (x,y) are the coordinates on a fibre bundle

is now implemented by imposing a "fibration" through the

"Maurer-Cartan condition"

F

1* -1 FX m (x,y )dxra A dx11. In order that1 FX mn

F be of tbis form we have to demand that the coefficients

of dy'' "• dxm and dy'n dy in the expansion of the right hand side

of (?-lt> vanish. This gives

(*?. n-)These relations are reminiscent of the BRS transformations of

s s i<the quantized Yang-Mills theories, with C = C dyr the ghost3 u

fields. Note that the C anticommute because the dyl do so.

Antighosts may be incorporated in a similar way. In this

case we enelarge the space further, (x,y) —> (x,y,y) , where the

Now

and 8 = dyp 2^ , and

A" = A "dx" + C^ "dyl + C - "dj;" • F is defined as in (£. It).

Now the Maurer-Cartan condition gives

y comprise another copy of the vertical space.S = dx;n? + s + i, with s sdy" o

9dyr+ c^ adX«-

^ o- 63 - - 6k -

Or:p now sets sC - b by definition which we have the liberty

to do even though it is not quite clear why one should do so.

Then (5i>) and (5.1? ) actually reproduce the structure of the

well-known BRS and anit-BRS t.ransfo.rmation.-:,, (k,p. reference £?2.~]

and references therein.)

The formal argument displayed here seems to be the basis

for quite a few papers which appeared during the last few years

and where the problem of a possible geometrical interpretation

of ghosts is attacked.

The reason that not all is obvious and done with is that

the above considerations are formal and closer inspection shows

that a proper mathematical setting is difficult to obtain.

In fact, a bvsic difficulty is hidden at the following

point [toTJ. The would-be ghosts C, d^are, as one forms, elements

of a finite dimensional Grassmann algebra, unlike the ghosts fields

of the quantum Yang-Mills theory. It has therefore been argued

that a true geometrical theory of ghosts ought to be a theory

involving an infinite dimensional Internal space. Otherwise,

Green's functions involving a sufficiently high but still finite

number of ghosts would Tanish.

HoweVer, one may still hope that this impass is temporaryand will be overcome in due time.

Indeed, even greater hopes than those outlined here are

connected with these theories. In fact, the theories as given

here may also be formulated for graded gauge groups. The ghosts

asso ciated with odd generators are then bosons with the right

spin-statistics connection. It is then hoped that these

"exorcised gho3ts;l could play tAe role of Higgs bosons.

No definite picture has as yet emerged. (Cp. f f-jjfor further

references.)

VI. SUPERSYMKETRY

This chapter will contain some comments on sapersymetric

Yang-Mails1 theories and supergravity. The reason for brevity is

not only lack of space but also that up to now not many results

concerning supersymmetric theories which employ the geometrical

structures described in this review are as yet available.

On the other hand, even though only the method of "ordinary

dimensional reduction" has mostly been used up to now, super-

symmetric theories often find 9 natural that is particularly simple

formulation in higher dimensional space-time. This can be seen

in the examples given below.

Supersymmetric Yang-Milla theories

Consider the action

A.

(S.I)

where the Yang-Mills tensor F „ is defined as before, and

D ) S = d ) - fS*<- A \V. The indices s, t, v refer to the

gauge group R. Note that the Dirac spinor is in the adjoint re-

presentation of R. The simple theory ( fi, | ) turns out to be

supersymmetric on the mass shell without addition of extra fields

in D = >t, 6, and 10 dimensions, if the spinor is subject to a

Majorana or Weyl condition (0=4), a Weyl condition (D=6) or the

Majorana and Weyl condition (D=10). Supersyminetry transformations are

where the infinitesimal parameter £ is a spinor subject to th«

stune constraints as A .

The action ( (,) ) is invariant under (&Z ), ( £, 3) only if

surface terms are discarded and probably only for flat spaces.

In fact, for flat space the spinor parameter £~ has to be constant,

d £= w E. = 0. Repeating the proof of invariante for curved

A- 6=5 - 66 -

spac n one finds after discarding surface terms that the correspon-

condltion is now

= o . (4. <nIt is unlikely that (£ <f } has solutions on such curved spaces

that are of interest to us. In fact, from ( vM ) we obtain upon

contraction with ™i the Dirac equation (T-cljE=0. As we

have seen above, the latter has no solutions for the particularly

interesting case of "£ = M; xG/H, where G is a nonabelian compact

group. This becomes clearer if one considers the more restrictive

equation d E = 0 . This has the integrability condition

R^gfP ,f I = 0, which also is unlikely to have solutions for interesting

situations.

We therefore conclude that supersymmetric Yang-Mills theories

over curved (internal) spaces are probably consistent only

in connection with supergravity. However, even in such a framework,

one may study supersymmetric Tang-Mills theories in their own rightoAn intriguing feature of the high dimensional approach is

that it provides a possible geometrical interpretation of central

charges. This comes about as follows. Consider the algebra of

extended sujpersytnmetry with central charges,

where i..,N. Here and are spinor indices,

=-U , V^ =-V .

symmetry algebra without central charges. In this case the spinors

are eight dimensional objects and we may label them by (c J, ) ,

^ -1...4, L=l,2. The supersymmetry algebra is now,

It is suggestive to associate the central charges U and V,

(V±-= A-Ui Vij = i V ^ o f t h e f o u r dimensional N= 2 Theory with ttvp

fifth and sixth component of momentum of the six dimensional

N=l theory. One notices actually that after dimensional reduction,

the N=l six dimensional theory yields a four dimensional N=2 theory.

Closer inspection U.o j now shows that V and V are to be identified

with the electric and magnetic charge operator of the theory after

spontaneous symmetry breakdown. The latter is provided by the

- 67 -

would-be Higg? fi«!ld, the fifth and sixth component of the Tang-

Hills field. The etctric and magnetic charge operators are expressed

as surface integrals; the magnetic charge is that of the t'Hooft-

Polyakov monopole.

Recoil that the dimensional reduction is here implemented

through 95 9 -- 0 • This leads to « potential of the form

trfA A,T~ - We have noted above that the question whether

spontaneous symmetry breakdown occurs for such a potential is

a subtle one.

A typical example for the remarks made so far is provided

by the N=2 supersymmetric Yang-Mills theory which may be visualised

as obtained by dimensional reduction from six dimensions. In four

dimensions, it contains a Yang-Mills field, a doublet of Majorana

spmoro i£.. , i = l,2, a scalar A' and a pseudoscalar B . The

action is

The supersymmetry current for this model is

U % + €'i^ffV^t;

which gives rise to the commutation relations ( fc-5) for the

integrated charges, where now U and V are specified, to denote

the surface integrals

("1 ) l £ V t C * % 1-

Clearly wo obtain contributions to U and V only if A and B do

- 68 -

not vanish asymptotically. If indeed they do not vanish, we obtain

t'Hooft's definition of electric and magnetic charge, respectively.

These observations have many exciting consequences beyond

a concretization of the idea that electric and magnetic charge might

be manifestations of the fifth and sixth component of momentum

respectively. In particular it follows from { £,$") that for any

particle state in the theory, M ^ U2 + V2 (M being the mass),

which is nothing but a quantum version of the Bogomolny bound (_2 0^ I,

For a further discussion of these topics I refer to the

original papers [N>, m(7.O<? 2.|1, I If J.

B. Supergravity

In this section, a short account will be given of some

recent work concerning a synthesis of supergravity with the

Kaluza-Klein idea. As a typical example for a supergravity Lagran-

gian, Appendix A contains an explicit description of the eleven

dimensional theory.

For supergravity, eleven dimensions is somewhat of a magical

number L °'-l • In fact, for once supergravity in more than eleven

dimensions would after dimensional reduction provide a fouT dimen-

sional theory containing particles of spin higher than two (.'f£j.

It is not explected that such theories would turn out to be con-

sis tent (."•*_!: • Conversely, Witten (2o#J observed that eleven is the ,

minimal number for which coset spaces (as seven dimensional internal

spaces) exist that admit the symmetry SU(3)xSU(2)xU(1) which is

considered to be the physically relevant gauge group below

the Planck (grand unification) s^cale.>Actually, the close vicinity

Of grand unification and Planck scale has triggered some

interesting speculations (_'"rc>j •

Let us now have a look at the Lagrangian of-e4even dimensional

supergravity (see appendix A for details). We have a curvature

scalar, a Rarita-Schwinger Lagrangian, and Lagrangian for an

antisymmetric tensor field A^pj. which takes the form

If now as a first attempt, the internal space is assumed

to be the seven - torus(S ) 7 , then the zero mass sector is

obtained by ordinary dimensional reduction; no y-dependence is

retained. This procedure was employed for the first construction

of N=8 supergravity in four dimensions (_3(J " N o t e that the

counting of states in the four dimensional theory would change in

general once we are dealing with curved internal spaces, in particu-

lar, the number of bosonic and fermionic states might not match

any longer. In such a situation one must face the perspective

that the eleven dimensional theory is the true one rather than

the four dimensional one, and the true dimensionality of space-time

would be 11.

Let us return for the moment to the case where the internal

space is the seven-torus. One may modify the procedure of dimen-

sional reduction by retaining some y-dependence, on the fifth

coordinate,say. Note that one has now available a global

rotational symmetry as residuum from the symmetry undef~-frame

rotations. Denote a generator of this symmetry by M. Wo may

then give certain fields a dependence on the fifth coordinate

which will characteristically be of the form exp(iy M ) . This

breaks supersymmetry, but only spontaneously* some fields will

get masses and others not. This mechanism has been suggested by

Scherk and S.-.hwarz I'M j <f 1 J ,

The torus (S )7 has the symmetry (U(l))7. If we want a nonabeliar

internal symmetry we have to look for non-flat solutions of the

field equations of eleven dimensional supergravity.

We have remarked above that the ,Freund-Hubin solution

j.:- t\e fir:t fwr.pit n " such a solution. The ground state would

be anti-deSitter (ADS) spatexS. or the seven dimensional anaLogue

of anti-deSitter space times 5.. The latter version would require

further (hierarchical \JiB J) spontaneous compactification in order

to produce a four dimensional theory. Sezgin and Salam ( ??• J have

recently suggested to pursue this way.

The other possibility, £• =ADSxS has been developed in

a number of recent papers \J,( 5 1 J

- 69 -- 70 -

>t£hematically, the seven-sphere is a very interesting space.

In particular, it is the only parallelizable coset space (S =

S0{8)/S0(7)} which is not a group space. Here, "parallelizability"

means that inclusion of a torsion term to the connection coefficients

{the torsion coefficients being antisymmetric even in a holonmic

basis) the geometry of S? can be modified in such a way that the

curvature tensor of S vanishes. Now torsion occurs naturally in

supergravity theories, and that torsion as it dccurs in supergravity

can be used for the purpose one has in mind here has been argued

by Destri, Orzalesi and Ros*i(_Hj. As first application recall that

the very large cosmological constant in Kaluza-Klein theories has

its root in the curvature scalar of the internal space. If the latter

vanishes one may hope that one can solve the riddle of the cosmo-

logical constant f^V, 'V?~7 .

A solution of the field equations of eleven dimensional

gravity which is essentially a modified version of the Freund-

Rubin solution and which corresponds to the parallelized seven-

sphere as internal space has been given recently by Englert • (_ C I J *

For this solution, the components

tensor F., of the field "strength

MNPK are nonvanishing. From the four dimensional point

of view, the F^v>p a r e just scalars; the Englert solution therefore

involves scalars with nonvanishing vacuum expectation value. One

therefore expects a connection between the property of paralleli-

J.izability and spontaneous symmetry breakdown. For more general

arguments on this point compare £%/ J •

One can'squash' the seven-sphere (ohange the radius in one

or more directions). The resulting space is still a solution of the

field equations. Closer inspection £ f 4J shows that the latter

property is an outcome of parali^elizability again.

r\s a simplified example consider S which has the symmetry

SU(2)xoU(2). If we squash in one direction, the resulting symmetry

is SU(2)xU(l), and the Kaluza-Klein ansatz will produce, according

to our general arguments above, a correspondingly reduced number

of massless vector bosons- This can only be due to a spontaneous

symmetry breaking mechanism; we are discussing merely different

ground states of the same theory. For more details see

Also, on the squashed spere the Dirac operator is no longer

positive definite; correspondingly, on such spaces, zero modes

- 71 -

for the internal. Dirac operator may actually be obtained.

The Kaluza-Klein picture has also some bearing on spontaneous

breaking of supersymnstry ( |2°?J. To see this recall that super-

symmetry is spontaneously broken, if the vacuum is not left

invariant by some generators of supersymmetry, Q /o'p* - Q

Now under supersymmetry transformations, the Rarita-Schwinger field

^ Nvaries into i o+ ^ w £. • Unbroken supersymraetry requires

<«l!lutlo"> — O, or, in the form of a classical field equation,

C^E=Oi which has integrability condition [D^Of-l^ — 0 •

On coset spaces with nonabelian compact symmetry group, one does

not expect solutions of this condition {j£°3J.

For the seven sphere (which has symmetry S0(8) and correspon-

dingly should give S0(8) supergravity after dimensional reduction)

the situation is as follows ["?1j : 1) no spontaneous breakdown

of supersymmetry for the round seven-sphere without torsion;

2) breakdown of supersymmetry for the round and squashed seven-

sphere with torsion.

This concludes a description of presently available results on the

connection between Kaluza-Klein and supergravity ideas. Much more

work will certainly be produced in the near future.

Similarly, quantum properties of high dimensional theories will

(have to) be developed. This will happen on the basis of the

requirement of "finiteness" rather than "renormalizability".

First indications how this might work have recently been given

by Duff [^1^2/SfJ, cp. also [~U'l/Z90'3 •

- 72 -

APPENDIX A LAGRAJJGIAN OF ELEVEN DIMENSIONAL SUPERGRAVITY

ACKNOWLEDGEM£NTSThe Lagrangian of eleven dimensional supergravity is

given by (cp. ref.^5\| for further details on the notation)

Thanks are due to Prof. Abdus Salam, the International

Atomic Energy Agency and UNESCO for hospitality at and support i =. C«LJt t ^ )J ~ F ** f * D **V "\ i_ iT n^'JK ft , A 71

from the International Centre for Theoretical Physics, Trieste. L ' li " hW L —VI • tjLl ^ '••'O JJ ^

Thanks are also due to Prof. P. Budinich and Prof". JL. Fonda for

support from the International School for Advanced Studies, Trieste,

with the help of which some of this work was done. ™ TTJ, '--

Special thanks are for Prof. J. Strathdee; I am particularly

indebted to him in connection with chapters II and IV.C.

Dr. E. Sezgin shared with me his knowledge of supersymmetric 4. 2. U I (i ' 4* 4- 12. P^O, I / C~ +. p \ f (A l)

theories. Dr. T.N. Sherry hepled me understand the basics of 1

Kaluza-Klein theories some time ago. Some elucidating discussions Here i-ti

with Prof. E. Witten helped considerably with the understanding hw| .and f1 *'" are antisymmetrized products of P - matrices. Further

of the matter. »

Somewhat dubious thanks are for Dr. Leah Mizrachi; without f — P* (tz\ , V ,. ,,

her encouragement this work might not have been undertaken.

Finally, •nd above all, without the support from Dr. Annette

Holtkamp it would not h«v« been finished. • ^Mi.0 ~ r (~ XL +-.? - -CL^ . ). /!.. A = 1 P A

j <A.3)

The constants are c=2/(12)\ b=l/(12)2.

(A.5)

(A.6)

y ~- R " i<~^ r , . ut^ . < A - 7 >

- 73 -

The Lagrangian (A.I) is invariant under

1. General coordinate transformations in 11 dimensions;

2. Local SO(l,10) Lorentz transformations -

3. N=l supersymmetry transformations,

APPENDIX C - COSET SPACES I

4. Abelian gauge transformations,

t/4.io )

(A-ll )5. Time reversal together with "*yyp"

APPENDIX B GENERALIZED DIRAC MATRICES

The generalized Clifford algebra, 7P^ I

with D elements has one finite dimensional irreducible repre-

sentation only with dimension 2*- , where [_D/3lis the biggest

integer smaller than or equal to to D/2. This is also the dimen-

sion, of the spin representation of SO(l.D-l). The generators of

SO(l,D-l) in the spin representation are given by X ij ~ [ "A , ' IJ 1 .

If D is odd we can define a generalized ft5 - matrix, P^ -

r,'..." f1^ and Weyl spinora, \ (|i - (f * p \«, ., Similarly,

charge conjugation can be defined and a Fieri rearrangement formu-

la can be provenL^°j.

Generalized Dirac matrices are constructed t+eratively.

Clearly, once a Clifford algebra with 2n, n€ IN, elements ia found,

we also have one with 2n+l elements, since we sim ly can add |

to the set. A Clifford algebra with 2n+2 elements is given by

where -j are the Pauli matrices. a c

For more material on generalized Dirac matrices compare

for example \_1o

MConsider a Riemannian space with coordinates z and metric

tensor G (z)t Under infinitesimal coordinate transformations.M M 'S , the metric tensor transforms according to

^

Neglecting terms of second or higher order in 3> i one obtains

S'*M to - %"" (l) 1- frg^f*) (C.2)with

If for a generic vector with components v we define the covariant

derivative

(C.4)

with the help of the Christoffel symbols

r MP(C"5)

- 75 -

one can rewrite (C.3) a*

0 U UJ : i 4 i t (C.6)For the covariant components G.M(i) one obtains the inverse

formula

Consider now coset spaces BK=G/H. We then have characteristi-

cally: {l) Ciroup transforr.iations associated with elements of G

can be expressed as coordinate transformations within G/H;

(2) Any two points in B can be connected by a group transformation

("transitivity"); (3) Group transformations leave T:n~e metric tensor

invariant, on account of (l) and (C.6) this implies the

"Killing equations"

where the choice of indices reflects that in the present context,

coset spaces will be internal spaces.

- 76 -

As the are associated with group transformations,

we may write (cp. also appendix D)

which, using (C.10), can be expressed as

.?-,<yo-

(en)

CC.9)

Here N is the order of the group G and the u>*specify the group

element. The If* ** are referred to as the components of the

Killing fields. They satisfy the differential equation

which states that the differential operators •! ~ J T<

can serve as generators of G,

with ~i\h^ t n e structure constants of G. The Killing fields

also provide an invariant metric through the prescription

<r0 l^. / — j w <$ <j; ^ J i.c.lti;

Invariance of the metric under coordinate transformations CC.9)

is readily checked using(C3). In appendix D, a formal construction

of the Killing fields as well as proofs for some of the statements

given here will be provided.

The material presented no far is sufficient to demonstrate

the equivalence of gauge and coordinate transformations in the

Kaluza-Klein formalism (see section ji.J.J, ). The Kaluza-

Klein ansattf for the off-diagonal elements of the metric tensor is

(C.13)

Conaider now the coordinate transformation•

7^ "" 0 •*> v ^ x - , ^ (

Clearly this corresponds to local (i.e. x-dependent) group

transformations on BK= G/H. Applying the general Cttrmula (C.3)

with ^^=.(<\S^ one finds

(C.15)

-T

- 77 -

may therefore write

with

(c.i6)

(c.17)

(C.18)

Equation (C.17) states that the coordinate transformation (C.l'j)

can be re-expr-eaBed as a gauge transformation, (C.18).

APPENDIX. D - COSET SPACES II

the group G with generators Q AA generic group element Le U may be parametrized locally by

a set of coordinates yT , L=L(y)= exp(iW* (y)Qjl, Let G act on

L(y) by left translation, g:L(y) H> gL(y) = L(y').

Now consider a subgroup H C G, with associated coordinates

y* . A left coset is the set of all elements of the form hg,

g^G fixed, h« H. Equivalently, a coset is the set of L(y',y^ )

that hove the same y* ; y may take on any value. A coset is an

equivalence-class under H - translations. For many purposes we

may confine our attention to representatives (from each class),

L(y) =. L(y* , 0). These functions transform under left translations

according to g:L(y)-~i gL(y) = L^y|A,y'*) - L(y')h(y,g) whence

L(y') = g L(y) (D.I)

Assuming the local existence of the group valued function L(y)

and specification of its transformation behaviour Ctf.1) suffice

to discuss many relevant properties of coaet spaces.

Infinitesimally, with y' = y + » , (D.l) reads

C\ 4- ^

(D.2)

•There Q^ denote the generators associated with H. The parameters

0* specify h(y,g). Multiplication of (0.2) by L (y) and comparing

terms up to first order gives

. (DO)

This is a relation among Lie-algebra valued quantities. We may thuswrite in particular

and introduce the adjoint representation U»'ifL)of G by

The e *1 (y) form a rectangular matrix at any point y. The e^^(y)

give a nonsingular part (with e4v(y) as matrix inverse); they may

sei-ve as vialbaina on £!„.

We introduce Killing field components through "*§ - to Kj 14 J_

<cp. also appendix C). From (D.3), using (D.4) and (D.5),

we obtain

This provides a definition of the Killing fields in terms of

the basic groupY—function L(y), The commutation relation

[^i, ~ia~\- *fjji V^ f o r t h e differential operators ^ t k j ^ ^

follows from iterating (D.l). It may also be obtained by explicit

though lengthy and not quite straigtforward computation.

From (D.5) one obtains by differentiating with £^ the

differential equation

(D.7)which has as integrability condition

Above, we have introduced an invariant metric on coset

spaces, g^ (y) = K<;r(y)K* (y). Since for the matrices D.r(L)

we have DDtr 1, we have also

(D-9)

On coset spaces, the invariance of the metric (D.9) is to

be understood as left invariance (i.e. invariance under left

translations). In general, a right invariant metric would be

different (albeit constructed in a similar way). An exception

is the case where G/H is again a group space.

For the discussion of transformation properties it is

convenient to introduce the algebra valued differential formA

•1%)= a - * ^ J *<j w;~ - (D.IO)

Under a coordinate transformation y —1 y'(x,y) we have

e(y)--)e(y'} = L~1(y')'3 L(y'). Using (D.I) one obtains

e(y') = h 3 h"1 + h e(y) h'1 + h L'1(y)(g'1<9g)L(y) h"1 (D.ll)

By writing this formula in components one extracts the transfor-

xstion rules for the vielbein coefficients and correspondingly

for the Killing field components. The corresponding calculations

can be found in ref ,\\(fl I. One can then in particular reproduce

the result from appendix C that coordinate transformations

y-^y'(x,y) associated with group translations are re-expressible

as gauge transformations in the Kaluza-Klein theory.

From (D.ll) it follows in particular

On the other hand, from the general arguments of section (II.l)

we have

(D.13)

Comparing (D.ll) and (D.12), we obtain e*-(y') = e'y,*iy'), which

expresses the forminvariance of the vielbein on coset spaces

under coordinate transformations-. This is equivalent to the

etate-nent that the metric tensor (D.9) is forminvariant under

coordinate transformations.

All considerations above referred to left G-tranaUtions

on G/H. Similar formulae as those above hold for right translations<

By their respective definitions, [X* 'XjJJ * ^ > where ift are the

analogs of afj for right translations. For the particular case

H = 1, the Z£ turn out to be given as £j - - •** 3^ .

- 80 -

- 79 -

APPENDIX E DIFFERENTIAL FORMS

For certain calculations, Cartan's language of differential

forms is a particularly suitable one. Here the basic objects are

the basis 1-forms, like E = E M dz ! We define a skew-symmetric

product dzMA dzN= -dzN/\dzM which induces E*AE , and the operation

'S of exterior differentiation, 3 E = 9JJE*. dZl *-dz i having the

property "& •*• 0 . The notation is clearly basis-independent.

For instance, the components of the torsion tensor are collected

within the torsion 2-form T = T.-J dz A dz which is now

defined through

TA = (E.I)

with Bc = BKCAdzM the "connection 1-form". (For the definition

of and BH .compare chapter II.) In a similar way

n ft Vfwe introduce the curvature 2-form B = 'SiNA d z A d z whichsatisfies

RAB = 5 B B +B A

CAB CB. (E.2)

Equations (E.I) and (E.2) are referred to as Cartan's first and

second identity respectively. They are particularly useful on

Rlemannian spaces, i.e., when the torsion vanishes. In this case,

(E.I) determines the connection 1-forms by simple differentiation

of the basis 1-forms, whereas (E.2) provides the elements of the

curvature tensor.

As an application let us consider the case of coset spaces.

Here the basis 1-forms may be introduced through (recall App. D)

e(y) = L"1(y)dL(y> = e*(y)QA = dyf e * Qj , where #;• with [&- 3sl *

lie. * Q? a r e the generators of G. F,rom the definition of e(y)r o

we obtain defy) = e(y) Ae(y), the "Maurer-Cartan formula's Itmay also be written in the form

Je' = (1/2) e^/ve^f * (E.3)

For vanishing torsion Cartan's first Identity is 0»d«"* + eC», Bj .

We thus obtain for the connection 1-forms

with '(<i)- ' W ^ / W • I n order to derive (E.4),

one uses the subgroup property of H, "fajj - 0 , and the

identity (f- p ""iL Jr t. C^e^^C^ f . If one further assumes

£ - 1-0 (for non-compaot' groups, this is not al#ways true)

one obtains for the curVature 2-form

v AFrom (E.5) one reads off that the components oiT the curvature

tensor and consequently of the Hicci tensor are

constants. Also in particular, coset spaces are spaces of constant

curvature•

For symmetric spaces, where lxn ^ • we have for the

Hicci tensor ft^^ •=, - tj, J3, % , For maximally symmetric spaces

wnich are defined to ae symmetric spaces of dimension K such that

they admit the maximum number of Killing fields, namely K(K+l)/2,

one can show that the curvature tensor takes the simple form

7i • (E.6)

Above were given the connection and curvature assignement3

for vanishing torsion. One may also choose B such that the torsion

2-form becomes T

and

Then

If G/H=G, H=l, then this assignement gives canishing curvature.

(Group spaces are "parallelisable, cp. chapter VI.) For symmetric

spaces the two assignements are identical (cp. Eqn. (£.4)).

REFERENCES

Appendices C and D are based on material that can be found

in ref. |jt69 j . Appendix E is collated from results that can be

found in references |l69jand [

- 81 - - 82 -

BIBLIOGRAPHY

This bibliography contains papers relating to the Kaiuza-Kleintheory and/or paper's that have been made use of within this review.

1.

2.

3.

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6.

7.

8.

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11.

12.

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14.

15.

E.S. Abers and B.W. Lee, "Gauge theories",Phys. Rep. £ 1 (1973)

L.F. Abbot and S. Qeser, "Stability of gravity with acoamological constant", Nucl. Phys. B195 76 (1982)

J.L. ALonso, V. Azcoiti and A. Cruz, "Origin of inertia atrest and the number of generations", Phys. Rev. D26 691 (I982)

R. d'Auria and P.Fre, "Spontaneous generation of OSpCl/8)symmetry in the spontaneous corapactification of d=llsupergravity", University of Geneva preprintUGVA-DPT 1982/09-365

R.d'Auria, P. Fre and P. van Nieuwenhuizen, "Symmetrybreaking in d=ll supergravity on the parallelizedseven-sphere", preprint CERN-TH 3*153

R. d'Auria, P. Fre and T, Regge, "Group manifold approachto gravity and supergravity theories", Trieste preprintI/8/4

M.A. Awada,"Kaluza-Klein theory over coset spaces",Imperial College preprint IC/82-83/5

M.A. Awada, M.J. Duff and C.N. Pope, "N=8 supergravitybreaks down to N=l", Imperial College preprint ICTP/82-83A

N.S. Baaklini, "Eigtheen-dimensional unification",Phys. Rev. D25 ^78 (1982)

A.P. Balachandran and A. Stern, "Nonlocal conservation lavsfor strings", Phys. Rev. D26 l't36 (1982)

T, Banks, Y. Dothan and D. Horn, "Geometric fermions",Phys. Lett. 117B 413 (1982^

L. Baulieu amd J. Thierry-Mieg, "The principle of BRSsymmetry: An alternative approach to Yang-Mills theories",Nucl. Phys. B197 <*77 (1902)

I.M. Benn and R.W. Tucker, Phys. Lett. 112B (1982)

I.M. Benn and R.W. Tucker, "A Riemannian description ofnon-abelian gauge interactions", Lancaster preprint, March 1982

S. Bergia and G. Venturi, "Confinement and Kaluza-Kleintheory", Bologna preprint IFUB 18-82

P.G. Bergmann, "Unitary field theory, geometrization of physiss,or physicalization of geometry?", talk, Einstein Symposium,Berlin 1979, P«84

17. P.G. Bcrjmnnn, "Unitary field theories", Phys. Tod. _3_2 **'* (1979*1

18. T, Uradfield and R. Kantowski, "Jordan-Kaluza-Klein typeunified theories of gauge and gravity fields", ,Journ.Kath.Phys. 2J_ 128 (1982)

19. C. Brans and R.H. Dicke, Phys. Rev. 12k 925 (1961)

20. L. Brink, J.H, Schwarz, and J. Scherk, "SupersymmetricYang-Mills theories", Nucl. Phys. B121 77 (1977)

21. P.F. Browne, "On five-dimensional unified field theory andprojective relativity", Gen. Rel. Grav. _1_2 503 (l98l)

22. J. Buralaff and U. Haison, "On stationary, axiallysymmetric solutions of Jordan's five-dimensional, unifiedtheory", J. Math. Phys. j2£ 1352 (1979)

23. C.G. Callan and S. Coleman, "Fate of the fals vacuum, II:?ir;t MiiiirtiiTi corrections", Phys. Rev. Dl6 17^2 (1977)

2k. R. Casalbuoni, G. Uomokoa and S. Kbvesi-Domokos',"Internal symmetries and ghost symmetries: Models onhermitian spheres", Phys. Rev. D17 20^8

25. W.E. Casweil, J. Milutinovic and G. Senjanovic, "Predictionsof left-right symmetric grond unified theories'!Phys. Rev. U26 l6l (1932)

26. L.N. Chang and F. Mansouri, "Nonlinear gauge fields and thestructure of gravity and supergravtty theories"Phys. Rev. D17 ")l6S (1978)

27. L.N. Chang, K.I. Macrae and F. Mansd-uri, "Geometricalapproach to local gauge and supergauge invariance: Localgauge theories and supersymmetric strings",Phys. Rev. U13 235 (1976)

28. G.F. Chapline, "Higgs fields and the origin of gravity",Phys. Lett. ?4B 39^ (i960)

29. G.F. Chapline nnd N.S. Manton, "The geometrical significanceof certain Higgs potentials: An approach to grand unification"Nucl. Phys. Blft't 391 (198>)

30. ii. OhapJine anu R. Slansky, "Dimensional reduction andflavor chirality", Los Alamos preprint LA-UR-82-IO76

31. Y.M. Cho, "Higher dimensional unifications-of gravitationand gauge theories", J. Math. Phys. jj6 2029 (1975)

32. Y.M. Cho, "Gauge theories on homogeneous fibre bundles"preprint CEBN - TH. 3^1^

33. Y.M. Cho and P.G.O. Freund, "Non-abelian gauge fields asNambu-Goldstone fields", Phys. Rev. D13 1711 (1975)

34. Y.M. Cho and P.S. Yang, Phys. Rev. D13 3789 (1975)

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• • • • * • - — <••-•*••»•

35. A. Chodos and S. Detweiler, "Where has the fifth dimensiongone?", Fhys. Rev. 021 2167 (198O)

36. E.A.B. Cole, "Particle decay in six-dimensional' relativity",J.Phys. AIJ. 109 (1980)

37. E.A.B. Cole, "Centre-of-mass frames in six-dimensionalspecial relativity", Lett. Nuov. Cim. 2&_ 171 (1980)

38. E. Cremtner, "Dimensional reduction in field theory andhidden symmetries in extended supergravity",Paris preprint LPTENS 8l/l8

39. E. Cremtner and B. Julia, "The 50(8) supergravity",Nuc-1. Fhys. B159 141 (1979)

40. E. Cremmer and J. Seherk, "Dual models in four dimensionswith internal symmetries", Nucl. Phys. B103 399 (1976)

41. E. Cremraer and J. Scherk, "Spontaneous compactification ofSpace in an Einstein-Yang-Hills model",Nucl. Phys. B108 409 (1976)

W2.. E. Cremraer and J. Scherk, "Spontaneous compactif ication ofextra space dimensions", Nucl. Phys. Bll8 6l (1977)

43. E. Cremmer, J. Scherk and J.H. Schwarz, "Spontaneouslybroken N=8 supergravity", Fhys. Lett. B84 83 (1979)

44. T. Dereli and S.W. Tucker, "A spin(l,'i) gauge theory modelwith Kaluza-Kleiir symmetry", Nucl. Phys. B2O9 217 (19°"2)

45. C. Dsstri, C.A. Orzalesi and P. Rossi, "Origin and fate ofthe Kaluza-Klein mechanism", New York Univ. prepr. NYU/TR4/82

46. C. Destri, C.A. Orzalesi and P. Rossi, "Matter fields and metricdeformations in multidimensional unified theories"New York Univ. prepr. NYU/TR8/82

47. P. Oobiasch and D. Maison, Gen. Hel. Grav . lk_ 231 (1982)

48. G. Domokos and S. Kiivesi-Dotnokos, "Internal symmetries andghost symmetries", Phys.NRev. Dl6 3060 (1977)

49- G. Domokos and S. Kdveai-Domokos, Nuov. Cim. 44A 318 (1978)

50. P.H. Dondi and P.D. Jarvis, "A aupersymmetric Weinberg-Salam model", Phys. Lett. 64B 75 (1979)

51. M.J. Duff, "Ultraviolett divergences in extended super-gravity", preprint CERN - TH 3232

52. . M.J. Duff, "Kaluza-Klein Kounterterots", prep-. CERN-TH 3259

53. M.J. Duff, "Supergravity, the seven-sphere, and spontaneoussymmetry breaking", CEHN-TH 3451

54. M.J. Duff and C.A. Orzalesi, "The cosmological constant andspontaneously compactifled d=ll supergravity", CERN-TH 3473

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55. M.J. Duff and D.J. Toms, "Divergences and Anomalies inKaluza-Klein theories", prepr. CERN - TH 3248

56. M.J. Duff, P.K. Townsend, and P. van Nieuwenhuizen,"Spontaneous compactification of supergravity on the3-sphere", pepr. CERN - TH 3465

57. R. Ecclestone, "A critique of supersymmetric Weinberg-Salam models", J. Phys. Al_3_ 1395 U98o)

58. R. Ecclestone, "The Weinberg angle in the supersymmetricWeinberg-Salam model", Phys. Lett. I16B 21 (1982)

59. T. Eguchi, P.B. Gilkey and A.J. Hanson, "Gravitation, gaugetheories and differential geometry", Phys. Rep. (&_ 213 (1980)

60. A. Einstein and P. Bergmann, Ann. Math. 22. 683 <1938)

61. F. EngJert, "Spontaneous compactification of 11-dimensionalsupergravity", preprint CERN-TH 3394

62. H. Essen, "Particle kinematics in a unified geometricdescription of all fields", Nuov. Cim. 66B 1 (1981)

63- D.B. Fairlie, "Two consistent calculations of the Weinbergangle", J. Phys. G^ L55 (1979)

6k. D.B. Fairlie, "Higgs fields and the determination of th»Weinberg angle", Phys. Lett. 82B 97 (1979)

65* B. Felsager and J.M. Leinaas, "Geometric interpretation ofmagnetic fields and the motion of charged particles",Nuci:. Phys. Bl66 162 (1980)

66. G. Forgacs and Z. Horvath, "On a static solution of theEinstein-Yang-Mills system in six dimensions", Gen.Rel.Grav. _10_ 931 (1979)

67. P. Forgacs and N.S. Manton, "Space-time symmetries in gaugetheories", Comm. Math. Phys, J2^ 15 (198O)

68. P.G.O. Freund, "Superunification and electric chargenon-conservation", Chicago Univ. prepr. EFI 80-37

69. P.G.O. Freund, "Dynamical spin", Chicago Univ. prepr. EPI 8l-O7

70. P.G.O. Freund, "Grand unification near the Kaluza-Kleinscale", Chicago Univ. prepr. EFI 82-23

71. P.G.O. Freund, "Kaluza-Klein cosmologies", ChicagoUniv. prepr. EFI 82-24

72. P.G.O. Freund and F. Mansouri, "Critical dimensions forstrings, bags, ...", Chicago Univ. prepr. EFI 82-03

73. P.G.O. Freund and R.J. Nepomechie, "Unified geometry of anti-symmetric tensor gauge fields and gravity", Nucl. Phys.B199 482 (1982)

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74. P.6.0. Freund and M.A. Rubin, "Dynamics of dimensionalreduction", Phys. Lett. 97B 233 (1980)

75. If. Fujii, "On the charge-mass relation in the extendedKaluza-Kiein theories", Phys. Lett. 98B 179 (1981)

76. T. Fukuyama, "Gauge theory of gravitation- 4+N dimensionstheory", Osaka Univ. (Toyonaka) prepr. Oct. 198l

77. P. Furlan and H. Raczka, "A new approach to unified fieldtheories", Nuov. Cim. 65A 509 (1981)

78. P. Furlan and R. Raczka, "On the origin and forms of internalsymmetry groups in models of unified theories",Phys. Scr. 2k_ 902 (1981)

79- G.W. Gibbons and B.F. Whiting, "Constraints on unificationtheories imposed by measurements of Newtonian gravity",Cambridge Univ. prepr., 1981

80. V.D. Gladuah, "Five dimensional theory of interacting scalar,electromagnetic, and gravitational fields",Sov. Phys. J. 22_ II72 (1979)

81. J. Goldstone and R. Jackiw, "Quantization of nonlinearwaves", Phys. Rev. Dll i486 (1975)

82. L. Halpern, "On group covariance and spin in gravitationaltheory", Florida State Univ. prepr. (Tallahassee) 1981

83. J. Harnard, S. Shnider and L. Vinet, Journ. Math. PhyB.21 3719 (1980)

84. J. Harnard, S. Shnider and J. Tafel, Lett. Math. Phys-ic 1Q7 (.1990)

85. Z. Horvath, and L. Palla, "Spontaneous compactification and'monopoles' in higher dimensions", Nucl. Phys. BtkZ 327 (1978)

86. Z. Horvath, L. Palla, E. Creamer and J. Scherk, "Grand unifiedschemes and spontaneous compactification" Nucl. Phys.B127 57 (1977)

pi7. A. Hosoya, Y. Ohkuwa and K. Omote, "Kaluza-Klein theory andBBS symmetry", Phys. Lett. 107B 412 (I98D

88. H. Ishikawa, "Effects of nonstatistical gravitation on thedimensional reduction", Lett. Nuov. Cim. .3_2__.Jl86 (198l)

89. R. Jackiw, "Quantum meaning on classical field theory",Rev. Mod. Phys. _4j9_ 63l (1977)

90. R. jackiw and N.S. Manton, "Symmetries and conservation lawsin gauge theories", Ann.Phys. (N.J.) 127 257 (1980)

91- P. Jordan, "Schwerkraft und Weltall", Vieweg, Braunschweig,1955

92. M.W. Kalinowski, "Gauge fields with torsion",Warsaw preprint IST-2-80, part I, part II.

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93-

94.

95-

96.

97-

98.

99.

100.

101.

102.

103.

104.

105.

106.

107-

108.

109.

110.

112.

M.W. Kalinowski, "CP-nonconservation and a dipole electricmoment of fermion in the Kaluza-Klein theory"Acta Phys. Austr. 22 229 (I98l)

M.W. Kalinowski, "Rarita-Schwinger fields in nanabelianKaluza-Klein theories", Warsaw preprint IFT-14-1981

M.W. Kalinowski, "On vanishing of the cosmological constantin nonabelian Kaluza-Klein theories", Warsaw prepr. IFT-8-1981

M. tf. Kalinowski, "Spinor fields in nonabelian Klein-Kaluza theories", Warsaw preprint IFT-lO-1981

M.W. Kalinowski, "The nonaymmetric Kaluzn-Klein theory"Toronto Usiiv. prepr., 19&2

M.W. Kalinowski, "The nonsymmetric nonabelian Kaluza-Kleintheory", Toronto Univ. prepr., 1982

Th. Kaluga, "Zutn Unitatsproblem der Physik", Sitzungsber.preufi. Akad. Wiss., 966 (1921); Englisch transl.: cf. ( |fc? )

A. Karlhede and T. Tomaras, "Spontaneous Symmetry breaking inKaluza-Klein theories", CALTECH prepr. CALT-68-968

T. Kawai, "A five dimensional unification of the vierbeinand electromagnetic field, I" Osaka prepr. OCU-82

T. Kawai, "A five dimensional unification of the vierbein andelectromagnetic fields, II: Geometric structure with regardto affine connection of the bundle space", Osaka prepr. OCU-87

R. Kerner, Ann. Inst. Poinc. (1968)

R. Kerner, ."Geometrical background for the unified fieldtheories: thei Einstein-Cartan theory over a principle bundle"Ann. Inst. Poinc. 4_ 437 (198l)

0. Klein, "Quantentheorie und fiinfdimensional Relativi-tatstheorie", Z.Phys. _37_895 (1926), Engl. transl.: cf. (US )

A. Lapedes and E. Mottala, "Another gravitational instabilityof flat space-time", Phys. Lett. HOB 114 (1982)

J.M. Leinaas and K. OLaussen, "Ghosts and geometry",Phys. Lett. 10 In 1S9 (19^2)

C.Y. Lo and G.R. Goldstein, "Radiation reaction force andunfication of electromagnetic and gravitational fields",Tufts Univ., Medford, prepr. TH-Sl-101

J.F. Luciani, "Space-time geometry and symmetry breaking",Nucl. Phys. B135 111 (1978)

G. Ludwig, "Fortschritte der projektiven Relativitatstheorie",Vieweg, Stuttgart 1951

K.I. Macrae, "Origin of Higgs fields", Phys. Rev. 1)22 ,1996(1980

M.D. Maia, "The special Relativity of Kaluza-Klein"Brasilia Univ. prepr. (1982)

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1'•*,,#' »*!'

1'.'- M.D. Maia and V.L.H. Silvirs, "Gravitational induction ofHiggs fields", Brasilia Univ. prepr. (I9fi2)

114. F. Mansouri and L.N. Csang, "Gravitation as a gauge theory",Phys. Rev. £±2 3192 (1975)

115. N.S. Manton, "A new six dimensional approach to the Weijiberg-Salam model", Nucl. Phys. BI58 I'll (1979)

116. N.S. Manton, "Fermions and parity violation in dimensionalreduction schemes", Nucl. Phys . B193 502 (19fll)

117. T. Marinucci and P.A. Zlzzi, "The soliton supermultipietin 4 + 1 dimensions", Nucl. Phys. B195 466 (1982)

118. M.E. Mayer, "Symmetry breaking in gauge theories (geometricaspects)',' Irvine preprint 1979

119. B. McClain, A. Niemi, C. Taylor and L.C.R. Wijewardhana,"Superapace, dimensional reduction, and stochastic quanti-zation", MIT prepr. CTP 996, April 1982

120. W. Mecklenburg, "Mass splitting in high dimensional extensionsof general relativity", Acts Phys. Pol. B£ 793 (1978)

121. W. Mecklenburg and D.P. O'Brien, "Gauge field masses anddegenerate potentials", Lett. Nuov. Citn. £2 566 (I978)

122. W. Mecklenburg, "Aspects of ssven-dimensional re-±*tivity",Phys, Sev. D21 2l49 (I98O)

123. W. Mecklenburg, "Geometrical Unification of gauge and Higgsfields", J. Phya. £6 1049 (1980)

124. W. Mecklenburg and P.P. Martin, "Constant Yang-Mills fields",Phya. Rev. D22 3088 (1980)

125- W. Mecklenburg, "Comment on stability properties ofdegenerate systems", Phys. Rev. D24 2766 (I98l)

126. W. Mecklenburg, "Towards a unified picture for gauge andHiggs fields", Phys. Rev. D24 3194 (l98l)

127. W. Mecklenburg, "Attempts at a geometrical unificationof gauge and Higgs fields*?1, Ac^a Phys. Austr. Suppl .' _g_26=rj (1981)

128. W. Mecklenburg, "Hierarchical spontaneous coropactification",Trieste prepr. IC/82/32

129- W. Mecklenburg, "Zero modes of the internal Dirac operatorin Kaluza-Klein theories", Phys. Rev. D26 1327 (1982)

130. P. Minkovski, "On the spontaneous origin of Newton'sconstant", Phys. Lett. 71B 419 (19 77)

131. J.W. Moffat, "Extended Kaluza-Klein unified gauge theories",Acta Phys. Austr. Suppl. 23_ 665 (1981)

132. J.W. Moffat, "Unified gauge theory including gravitationin higher dimensions", Toronto Univ. pepr. (1981)

13J.

134.

135.

136.

137.

138,

100.

140.

l4l.

142.

143.

144.

145.

146.

147.

148.

l49«

150.

A.E. Moose and H.F. Abner, ''Inequivalence of five-dimensionaLrelativita and Brans-Diclte theory", Phys. Rev. D2 6 1275 (1982)

T. Moriya, "Induced gravity and spontaneous compactification",Tohoku Univ. preprint TU/82/248

T. Moriya, "Induced gravity from heat kernel expansion andspontaneous compactification", Tohoku Univ. prepr. TU/82/251

W. Nahm, "Suporsymmetry and their representations",Nucl. Phys. B135 I'i9 (1970)

S. Naka and S. Kojima, "Mass structure of the string inhigh-dimensional unified theory", Tokyo (Nikon) prepr.NUP-A-32-04

3, Naka and S. Kojima, "High-dimensional formulation of aspinning particle interacting with external gauge fields",Progr. Theor. Phya. _6jj_ 1732 (198I)

N.K. Nielsen, Nucl. Phys. BI67 249 (1980)

P. von Nieuwenhuizen, "Supergravity", Phys. Rep. 68_ 189 (1981)

Y. Ohkuwa, "Kaluza-Klein theory and extended BRS - symmetry"Phys. Lett. ll4B 315 (1982)

0. Olive, "The electric and magnetic charges Is^xtra com-ponents of four-momentum", Nucl. Phys. B153 1 (1979)

D. Olive and P. West, "The N=4 supersymmetric E(8) gaugetheory and coset space dimensional reduction".Imperial College prepr. ICTP 81/82-25

C. Omero and R. Percacci, "Generalized non-linear ^ -modelsin curved space and spontaneous compactification",Nucl. Phys. Bl65 351 (I960)

C.A., Orzalesi, "Multidimensional unified theories",Fortschr. Phys. J2£ 4i3 (l98l)

C.A. Orzalesi, "Gauge field theories and the equivalenceprinciple", New York Univ. prepr. NYU-TR6-81

C.A. Orzalesi and M. Pauri, "Geodesic motion in multi-di:iicn3ional unified gauge theories", New York Univ.prepr. NYU-TR7-81

C.A. Orzalesi and M. Pauri, "Spontaneous compactification,gauge symmetry and the vanishing of the - cos"niological constant",Phys. Lett. 107B 186 (I98D

H. Osborn, "Topologicol charges for N=4 supersymmetric gaugetheories and monopoles of spin 1", Phys. Lett. 83S 321 (1979)

W.W. Osterhage, "Geometric unification of classical gravitatio-nal and electromagnetic interaction in five dimensions: Amodified approach", Z. Naturforsch. 35A 302 (1980)

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151. P.T. Piippas, "The 'three dimensional time' - equation",Lett. iNuov- Cim. 2J_ ;t29 (1979)

152. P.T. Pappas, "On velocity and the transverse Doppler effectin the six-dimensional space. A hidden variable model?"Lett. Nuov. Cim. _2ji 209 (1930)

153. W. Pauli, "Uber die Formulierung der Naturgesetze mit funfhomogBnen Koordinaten", Ann. d. Phys. lS 337 (1933)

15^. M. Pavsic, "The prospects of higher dimensional unifiedtheories", Ljubljana prepr. 19B0, Obr 6-7/80

155. M. Pavsic, "Unified kinematics of baryons and tachyons insix-dimensional space-time", J. Phys. Al*t 3217 (1981)

156" M. Pavsic, "Solution of the Planck-mass problem inKaluza-Klein theories", J. Phys. G3 L 8 ? (1982)

137. R. Percacci and S. Randjbar-Daemi, "Kaluza-Klein theories onbundles with homogeneous fibres", Trieste preprint IC/82/18

158. P- Pigeaud, "Unified fields in pentadimensional theory",Rep. Math. Phys. Jjj_ 26l (1978)

159. D. Pollard, "Antlgravity and classical solutions—of five-dimensional Kaluza-Klein theory", Imperial College pepr.ICTP 8l/82-l6

160. G. Preparata, "Beyond four-dimensional space-time; a possibleanswer to the riddle of color confinement", Phys. Lett.102B 327 (1981)

161. S. Randjbar-Daemi and R. Percacci, "Spontaneous compactifi-eation of a{ 'i + d) -dimensional Kaluza-Klein theory intoMU)xG/H for symraetr c G/H" , Phys. Lett. 11?B kl (1982)

162. 5. Randjbar-Daemi, Abdus Salam and J. Strathdee, "Sponta-neous compactification in six-dimensional Einstein-Maxwelltheory", Trieste prepr. IC/82/208

163. J. Rayski, Acta Phys. PoK 2J. 9^7 (19&5)

164. J. Rayski, Acta Phys. Pol. £8_ 8? (1965)

165. J- Rayski, "Multi-dimensional unified theory",Acta Phys. Austr. Suppl. 0 869 (1978)

166. J. Rayski and J.M. Rayski Jr., "On a fusion of supersymmetrieswith gauge theories", Trieste prepr. IC/83/76

l67' P- Roman and J. Haavisto, "Higgs fields, curved space andhadron structure", Int. J. theor. Phys. JUS, 915 (1977)

168. V. deSabato (Ed.), "Unified field theories of more than fourdimensions including exact solutions", Froc, 6th. Int. courseErice, Trapani, Sicily, 20 May- 1 June 1982, World Scient.Publ., Singapore, scheduled to appear Dec. 1982

I69.

170.

171.

172.

173-

17**.

1~"> -

176.

177-

178.

179-

ISO.

l8l.

l82.

18".

134.

185.

186.

A. SnLa::i and J. Strnthdee, "On K.a 1 uza-K I. ei n theory",Ann. Phy-. ( N . .J . ) I'll il6 (lQl'3)

A. Schafer, J. Rafelski, and W. Greiner, "A five-dimensionalDirac theory and its relation to nonlocal theories in fourdimensions" Frankfurt Univ. prepr. UFTP-8it/1982

J. Scherk, "Extended supersymraetry and extended supergravitytheories", prepr. LPTENS 78/21 (Cargese lectures 1978)

I. Scherk and J.H. Schwarz, "Spontaneous breaking of super-symmetry through dimensional reduction", Phys. Lett.82B 60 (1979)

J. Scherk and J.H. Schwarz, "How to get masses from extradimensions", Sucl. Phys. BI53 6l (1979)

E. Schmutzer, "Relati.vistische Physik", Teubner, Leips5ig( 19-68)

A.S. Schwarz, "Are the field and space variables on an'. ;;ua 1 rnoti.ig1.1" . Nuc ! . Phys. 13171 l^k (190O)

A.S. Schwarz and Yu. S. Tyupkin, "Dimensional reduction of,the jauge field theory", Nuc1. Phys. Bl8? 321 (I981)

E. Sezgin and A. Salara, "Maximal extended supergravity inseven dimensions", Trieste- prepr. IC/82/94

T.N. Sherry and D.H. Tchrakian, "Dimensional reduction of a6-dimensional self-dual gauge field theory", Dublin Inst.Adv. Stud, prepr. DIAS-STP-82-O7

Bo-Sture K. Skagerstam, "On spontaneous compactificationof space-time in quantum field theory", Goteborg prepr. 1982

M.F. Sohnius, K.S. Stella and P.C. West, "Dimensional reductionby Legendre transforraetion generates oof-shell SupersymmetricYangV*iills theories"

J.M. Souriau, Nuov. Cim. 22. (1963)

E.J. Squires, "On a derivation of the -Salam-Weinberg model",Phys. Lett. 02B 395 (1979)

T, Tnjina, "Solutions with both magnetic and electric chargesi.(i a \ i fN)-dimensional geometric field theory",Phys. Rev. IKH ?20i (1981)

S. Tanaka, "Spontaneous dimensional reducti.an in generalizedKaluza's high-dimensional theory", Progr. Theor. Phya.66 l'*77 (1931)

P. Tataru-Mikai, "Grand unification via Kaluza-Klein approach",Lett. Nuov. Cim. 21 551 (1982)

J.G. Taylor, "Electroweak theory in SO(2/1)1Phys. Lett. j JL 331 (1979)

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187- J.G. Taylor, "Gauging SU(n/m)", Phys. Lett. 84B 79 (1979)

188. J.Q.Taylor, "Superunification in SU(5/l)",Phys. Rev, Lett. 3 83 <i (1979)

lfig, J.G. Taylor, "Do electroweak interactions imply sixextra time dimensions?'1, J.Phys. A13 l86l (198O)

190. W. Thirring, "Five dimensional theories and CP-violation" ,Acta Phys. Austr. Suppl. £ 256 (1972)

191. Y. Thiry, C.R. Acad. Sci. Ber. B226 2l6'(19<*8)

192. M. Toller, "Geometric field theories with a given set ofconstant solutions", Nuov. Cim. 58B l8l (1980),Err.: Nuov. Cim. 62B 423 (198l)

193. A. Trautmann, Rep. Ma tti . Phys. jU> 2026 (1975)

19^. A. Trautmann, "The geometry of gauge fields", Csech.J.Phya. 022 107 (1979)

195. S.D. Unwin, "Nontrivial field configurations in fivedimensional relativity", Phys. Lett. 103B 18 (1981)

196. S.D. Unwin, "Gravitational aspects of a conformally coupled,five space-time dimensional field theory", Newcastle Univ.prepr., to appear in Proc. Roy. Soc• A)

197. F.T. Vertosick, "Remarks on stratifying interactions in fivedimensions", Found. Phys. JJ 93 (1978)

198. S. Weinberg, "Gravitation and Cosmology", John Wiley & Sons,Inc.;, New York, London, Sydney, Toronto, 1972

199. C. Wetterich, "SO(lO) unification from higher dimensions",Phys. Lett. HOB 379 (198a)

200. C. W«tterich, "Spontaneous corapactification in higher dimen-sional gravity", Phys. Lett. 113B 377 (1982)

201. C. Wetterich, "Massleas spinors in more than four dimensions",prepr. CEHN-TH. 3 310

202. C, tfetterich, "Unified gaige theories from higher dimensions",prepr. CEHN-TH. 3420

203- B.deWit and P.van Nieuwenhuizen, "Quantization of 11 dimen-sional supergravity", Phys. Lett. 104B 27 £1981)

20*1. B.S. deWitt, "Relativity, groups and topology", Lea Houcheslectures 1963

205. E. tfitten, "A new proof of the positive energy theorem",Comm. Math. Phys. QQ 381 (1981)

206. E. Witten, "Kaluza-Klein theory and the positive energytheorem", Lect. at the BANFF Inst. (C.A.P.) on Particles andFields, August 1981.

E. Witten, "Instability of the Kaluza-Klein vacuum",Nucl. Phys. B19^ <tfll (1982)

207

208. E. Witten, "Search for a realistic Kaluza-Klein.theory",Nucl. Phys. B136 h\2 (I98D

209. E. Witten and D. Olive, "Supersymmetric algebras that includetopological charges", Phys. Lett. 78B 97 (1978)

210. Li Xinzhou, Wang Kelin and Zhang Jianau, "J(ackiw)-H(ebbi)solitons in B(rink)-S(cherk)-S(chwarz) supersymmetrictheory", Trieste prepr. IC/82/193

211. Y. Yamaguchi, "On the unification of all interactions",Tokyo Univ. prepr. UT-389

212. A. Zee, "Grand unification and gravity-selected topics",Proc. '(th Kyoto Summer Inst. on Grand Unified Thuriesand related topics, M. Konuma and T. Maskawa, eds.

213. P.A. Zizzi, "The USp(2) supersymmetric SU(2) Yang-Millstheory in ('l + l) dimensions and the central charge",Nucl. Phys. Bl39 317 (198l)

21k. P.A. Zizzi, "Topological content of dimensional reduction",preprint CEHN - TH. 3237

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CURRENT ICTP PliBLTC/G .

I .M. REDA and H. BAI-ICliRT - VJI-^S r c - i n , ; aLil±-;y and k i n e t i c s offormat ion of metEl^Sisc::..

I.M. REDA and A. WAGKNEKIdisordered so l id s .

M.Y. EL-AGHRT - Cn the ki>..r. •r e l a t i v i o t i c plasma,

LI XIHZHOU, WANG KEL1N a.:-: •'.liAin t h e preoii model .

S.M. PI2TRUSZK0 - Aaovphc",*., .;m i x t u r e .

J . ZIELINSKI - Seri.L-phet:,..:iu :iv ;d i s c o n t i n u o u s change of •/••ile:i :

- .^••si:! .: '::'i. :.noi': J.n h i g h l y

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IC/82/156 H.B. GHASSIB and S. CUA'LV. J.:

He-3He th in filiriii,

G,B. GELMINI, 3 , KUKC^Jvj/ ,uio.Goldstone bosons?

AjG.F. OBADA and M.il. MAHi\, .. -

IC/82/157

IC/82/158IBT.BUt'.*

IC/82/159INI.HEP.*

IC/62/l60

IC/82/161

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IC/82/16UIHT.REF.*

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IC/83/169

IC/82/170

'!'r. 1 •.••£!£ i ",y L lac1; ^:it iony In d i l u t e

1, ..I.-. iPA - Jueci aa ture l i k e iiairib1;-

t;Lt.vi-..1 a:.-i :-iiAgnetic a ipo le cent r ibu t i cnft o a t h e o r y of r a d i a t i o n r e a c t i o n 1 L!.:O.L: •.-.-, :ir,nm Eelf-er .or j r / ; anoperator reaction field.

R. KUKARAVADIVEL- Ther:u;;ly!;:mic :JI\ ;I .:.•!, U?3 of l l q . a id :-.et;;l.J.

K.H. WOJCIECHOWSKI, P . PI2aAi;::i.- ^:'1 •;. .-.^.LIXKI - An in i i tQ 'o i l i t y i na h a r d - d i s o :;ysteTti iri a n:-u'rov bo, .

J . CHELA FL0RE6 - Abrupt oaseL oi ,.^^1'.;^ V I - j J a t i o n .

J . CHELA FLOEES and V. VARLLA - VA .ow,- - r ^ v i i . y : ;m approiLjh t o i t ssource.

S.A.H. ABOU STEIT - Additional evidur.ee for the existence of the Cnucleus as three a -par t i c les and not as Be_ , + a.

M- EOVERE, M.P. TOSI and H.H. !t\HCH - Freezing theory of RbCl.

C.A. MAJID - Radial distr ibutiosi analysis d ' :^norphoua carbon.

P. BALLOHE, Q. SEJiATOKE iinrl M.P. 'I'C'oI - i:u face doniiiLy i-^oi'Lle andsurface tension of the orie-coirjponenl c lass ica l pla^jnu.

G, KAMIENIARZ - Zero-teniperatur" :'cnorj.alisation of the 2D wamrverieIsing model.

G. KAMIEIIIARZ, L, KOWALEWSKI am: «. FiECHQCKI - The spin 3 quantumIsing model al 1 il.

J . GOEECKI and u. POP (ULAWSKI - On the appLloability -if r.-..;irly free-electron model Cor r " s i s t t v i t y onl'.:ui a~ tons Ln liquid metals.K.F. WOJCIECHOWaKt - Trial charge dr-nsity p-O.files at tin: metall icsurface and work function ca lcula t ions .

THESE P R E P R I N T S AflE AVAILABLE FROM THE P U B L I C A T I O N S O F F I C E , I C T P , P . O . B o x 5 B 6 ,

I-3U100 TRIESTE, ITALY,

* (Limited distribution) .

IC/82/171IMT.REP.*

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Ii-J'l1. ri^'r , *

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H.B. CKALiSLj a:-, 5 A..H. ICKL'.,r:in _ ?:wirds a - . I ^ ^ i C i i v e L;ii-i?ry: o r lie I I : I - i, •:, r c - t t:: 'ip^;;a^f: [.ylirid -.pcrofali.

J . CHELA FLOEKo -.id !i .B. .JHASStB - iovrar-Ic >, 'xi^v •hei.a iv« theoryfor He 11 : - A -emptr^t1,:>•-*-dec-en:- _-nt f i - i id -1-heore t ic appi-unc-h.

C.A. ' « J I D - i'"&crt.ricaJ.,of •lino rr.houa ^oli.:.,, - A:-,

il ana riu".-i t i - n a forma: : T beh .?,vi our:ucti; ju ; J r;ir.d.<;.T .ntulr ; .

K. PA.C, : . PAliAGiOIAKOro1 Uf; mid Q. OUAFT - ;.''«--:et L: no- poivand haryo. -; if-;.!;-;/,

M . : Z - : R - ? : j r n i i j ; i " : : : : : s _ • ) . - i o . ; - n d L - . , • - • , . . :-,;• • • • • : : . . ; •-. , : ' i - ' . l o r .

T . G I K A ) ' . : . . [ , : , U - : • ; ; . . 0:1 : j \ , o . : i : . : . ; . . : ; . : - n ; - ! - : - . . - ; e i ;S . CLCOITI

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rL. SAEED - .[dont L ;'J .•r?.1'J. i i o r . - . - o g a t " •••••:. s

:' ... .••:as(, •/ : n l -.u:'fi •: H-.KL .:• .r.dit ir i t i o n a . ,.rit:-'-.>:i.

K. SAEED - He ii1 ;.•.•-•! ._.:X].L,S ui.--r;n^Liari-jion o:"1 polynooi: • •• tna.tr:.

A . O . i . A^iMAL7.1 - ":U-U-K .,pnrn..-.••:-. - 0 - . • i c i l l i ' .• - o t i j :-•:-. ;>'^ o f ^ a d r f i . i as t r u c t u r e ~ i i .

M. niJEGU1:1 ' ,nd U . K . 'AK - . :rv,fy;n a n d b a r y o n i i u n i- i i':1"!:,.

[ j .w . : ; ; iA ' iA - •.-•le r u l . - - u r a r t i c l u p h y s i c s i n uc-:.:r. a , : ;y a n d g a l u c t i ca.j Li^j-jomj",

l i . F . I . WIV..!J - On -.i.f r'jt,i.-j .::' l.i. .•••LiEaa for t h e D and D'' meson.

J . VJKIl-T'LVrCT - '.'cr.j; • ..-i--:- ;r-i-;i!y -:;d composi te 3i!ji:.ri:-r-1v-i.-1.y,

J . Jjl-urr.L _ r;. :-s .-'Marks :.:; t h e 13,; o r the e t f e c l i v " tr.i.'diuiapprox-. n a t i o n i'or fun r u s i i t i.vity ^-a leu la t iona i n j - iq : ; i J :c:,le n.i%?.lj,

R .P . HAiCJitj; - A Th-ory f-jr Lh - o r i e n t a t i o n n i o rdc r in i ; Ln neiaaticliqi.iid:; :i_d for r.nt:- • i.xse di^^rajn of t h e r v i - ^ t i c - i ^ o t o p i .: t r a n s i t i o n .

R. '£: QUi:iiO and J . i 1 . :.'ARCGT"'" - Tli•" thoriaodyn.'imics of ::up. '-- .-onli-ti j ialanthanun.

k.-'.W. OBADA and M.::. '-L'lir'.AiJ - l^-^pcns" and normal n.odt-.: of ?. -yst-arc

in the electric '-n;i n -' nij1 ic-dip.;"' c approximation .

M.K. KOLEVA and I . ' i . K' JTAiJiliOV - Absence if [[a.11 t'nv.'CC due t ophonon a s s i a t e d hopping \;i .iijot\ier=':a syKtrass.

L. JURCZYi:ZYH and M. :iT.-:,V.l,V:;K.1 - Sur f ic ; . J ta l . es Ln t h a p r e u w a d ofabsorbed riLom£ - 1'1 ; 'ipjji^ i::Lat:ion of T-.ne smal l L-adii.i:; j-.c^.^rir.iais.

M.S. AJLAGLOU^LI, Y,.\\. -..^Wlk. V.a . GARaaVi'iiilSHVTLI, A.M. ;::U.;AD::E,G.O. KUMTAG1IVIL1 and •'•.:.I'. L'CL'iJRIA - Perhaps a new m.if :•.; s c a l i n gv a r i a b l e i"or d e ^ e r i b i n g th..1 low- !i:vL ^-i^h-a.,, oroceo^cLi'.'

A.f>. 0K3-i:L-[iAE •-.! 1 M.:-;. :•: -;-;HAZ:,T - L'h.i i n t e g r a l r-ipreL,. n - i t i o n oft h e expoiicji-itiall.1 mv^x ; .i.icf io:.;; <•]' i i i f i n i t e l y many v a r i a b l e s onH i l h e r t upacea .

LI XINZHOU, MAMG KELiH and V.'.iAnG •! 1 .Ml'.'.'J - J-K T, -J .!. L t.oni in UHSnupercyctnetry i . Ivory.

W. KECKLENEliH'.] and L. MIzr.Ai.'iir - ::ur:'.-. :s terms and dual fo rmula t ionsof gauge t h e o r i e o ,

I . ADAWI - lnt,-..-raet:.o:i ••>.' .1 i.os',,• ii: and magnet ic cna r^es - [ i .

I . AC.Vn1: ti::d V. :• . GOD'.-dli - .:.. .-:ni71-M:r of Lhe e l e c t r o n a e n s i t y near aniinpuriLy jit!"; exohan<^_- anil c o r r e ! t t i o n .

IC/82/197

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1C/82/20U

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TC/B2/2O6INT.REP.*

[C/82/207ERRATA

IC/82/20a

IC/8P/209

IC/82/210

IC/82/211IUT.REP.*

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:C/82/2lU

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IC/S2/220INT.REP.*

PREM P. SRIVASTAVA - Gauj;u and non-gauge curvature tensor copies.

R.P. HAZOUME - Orientaticnal ordering in choiesterlcs and smectics,

S. RANDJBAR-DAEMI - On the uniqueness of the 3U(-2) instanton.

M. TQMAK - On the photo-ionization of impurity centres in semi-conductors.

0. BALCIQIiLU - Interaction between lonization and gravity waves in theupper atmosphere.

S.A. BAHAM - Perturbation of an exact strong gravity solution,

H.N. BHATTAfiAI - Join decomposition of certain spaces.

M. CARMELI - Exter.t:ion of the principle of miniraal coupling to particleswith magnetic moments,

J.A, de AZCARRAGA and J. LUKIERSKI - Supersymmetric particles in H = 2superspuce: phase space variables and hamilton dynamics.

K.G, AKDENIZ, A, IIAC TNL1Y A..-J and J. KALAYCI - Stability of merons ingravitational models.

H.S, CRAIGIE - Spin physics and inclusive processes at short distances.

S. RASDJBAE-DAEMI, AEDU3 SALAM and J. STRATHDEE - Spontaneouscompactification in six-dimensional Einstein-Maxwell theory.

J. FISCHER, P. JAKES and M. HOVAK - High-energy pp and pp scattering 'and the model of geometric scaling.

A.M. HAEUH ar RA3HID - On the electromagnetic polarizabilities of thenucleon.

CHR.V. CHRISTOV, l.J. PE'i'HOV and l.i. DELCHSV - A quasimoleculartreatment of light-particle emission in incomplete-fusion reactions.

K.G. AKDEHIZ and A. SMAILAGIC - Merons in a generally covariant modelwith Gursey term.

A.R. PEASAKKA - Equations or motion for 3. radiating charged particlein electromagnetic fieldr, on curved 3pacc-time.

A. CHATTERJEH and S.K. GUPTA - A::g,ulur dLstributions in pre-equilibriumreactions.

ABDUS SALAM - Physics with 1C0-L0Q0 TeV accelerators..

B. FATAH, C. BE3HLIU and G.S. CRISIA - The longitudinal phase space(LPS) analysis of the interaction np •*• ppfr at Pn - 3-5 GeV/c.

BO-YU HQU and GUI-liliASG Til - A formula relating infinitesimal tacklundtransformations to hierarchy gt-neralin : operators.

BO-YU iiOU - The gyro-electric racLo of Gupersynunetrical monopoledetermined by its structure.

V.C.K. KAKAJJF, - Ionospheric at: intillation observations.

R.D. BAETA - Steady state creep in •juartz.

G.C. GILIliABDI, A. RIMINI and 7. WiiBER - Valve preserving quantummeasurements: impossibility theorems and lower hounds for the distortion.

IC/lZ/222 BRIAH W. LIN1I - Order ally corrections to the parity-viol it ingelectron-quark potential in the Weinterg-Salam theory: c;\•*i+.'violatiori in one-electrori atoifls,

IC/82/223 G.A. CHRISTOS - Note on the n ,.._,.- dependence of < 5q > iTor. o.draiINT REP * Q U

perturbation theory.IC/82/Z2U A.M. SKULIMOWSKI - On the optimal exploitation of csiLr. L---V:IOUIMT.REP,* atmospheres.

IC/S2/225 A. KOWALEWSKI - The necessary and sufficient condition; o.;' * -:eIHT.REP.* optimality for hyperbolic systems with non-differsntiahie per•'onranc

functional.

IC/82/226 S. GUNAY - Transformation methods for constrained jpt iml :,at i on.INT.REP.*

IC/82/227 S. TAMGMAHEE - Inherent errors in the methods of appro;: L':,at ...::• ;IHT.REP.* a case of two point singular perturbation problem.

IC/82/229 J. ETRATHDEE - Symmetry in Kaluza-bilein theory.IMT.REP.*

IC/82/22O. S.B. KHADKIKAR and S.K. GUPTA - Magnetic tnomentc •-:•- .. U:,n". a i-y.. ••.in harmonic model.

IC/82/230 M.A. NAMAZIE, ABDUS SALAM and J. STRATHDEE - Finitcnor;.. :f lrak?nN = k super Yang-Mills theory,

IC/82/231 H. MAJLIS, S. SELZER, DIEP-THE-HUHG and H. PUSZ'rSJiiKj - •irfi.--parameter characterization of surface vibrations in lir> -•••v ••-•jl^j,.

IC/82/232 A.G.A.G, BABIKER - The distribution of sample e ij-eour.t ..uid ir,IMT.REP.* effect on the sensitivity of schistqsomiasis test..

IC/82/233 A. SMAILAGIC - Superconformal current multiplet.INT.REP.*

IC/82/234 E.E. RADESCU - On Van der Waals-like forces in spontaneously broker,supersymmetries.

IC/82/235 G, KAMIEKIARZ - Random transverse Ising model in two dimvii:;ior..;.