ic/83/32 INTERNAL REPORT E (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 I. INTRODUCTION 1 II. FORMAL CONSIDERATIONS 3 A. The setting of high-dimensional field theories 3 The present status of the Kaluza-Klein idea is reviewed, B. Harmonic expansions 6 including a discussion of the fermionic sector». Formalism and 1. Introductory remarks 6 geometrical methods necessary for the formulation of high-dimen- 2. Harmonic expansions: General formalism 9 sional field theories are described in some detail. The mechanism J. Harmonic expansions: Examples 11 of spontaneous compactification is explained and a description III. SPONTANEOUS COMPACTIFICATIOM 13 of presently available models is given. A chapter on coset space A. Models of spontaneous compactification 13 reduction of high dimensional Yang-Mills theories is included. 1. General remarks 13 The review concludes with an outlook on supersymmetric models. 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 NOTE B. Th« Kaluza-Klein ansatz 30 1. The five-dimensional case 30 2. The general case 32 Authors who feel that relevant work of .theirs has not 3. JBD scalars 36 been included into the bibliography, may send references (including i 4. Kaluza-Klein ansatz for the CS model 39 titles) to the following address* Dr. W. Mecklenburg; 11, Contastr.j ". delation to fibre bundles 'iO 20; Msst Germany. 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: Ivi1 i m m» ISTHODUCTIOM not seem to exist in physically sensible verions beyond elevett- dimensionaJ space-time.
The case for higher dimensional theories is presently In 1921, Kaluza [ii^Jsuggested to unify the gravitational strong, even though the reasons for this are theoretical rather and electromagnetic interaction within the framework of a than phenomenological, This situation is different from that five dimensional gravity theory. The fifth coordinate was made of ten or fifteen years ago when only occasional references invisible through a "cylinder condition": it was assumed that in on the aubjpct could be found. the fifth direction, the world curled up into a cylinder of The objective of this review is to document developments very small radius (10 JJcm, Planck'a length). that have taken place as from approximately the mid-seventies During the classical period of "unified" theories, on. At around this time, Cremmer and Scherk [_ 4-0, Iflj (f-2. ^ Kaluza's ansatz was studied from many different angles. An provided the remarkable idea of spontaneous compactification account of this may be found in the reviews by Ludwig j_/l(j]and which opened a new avenue to the understanding of higher dimensions. icnrautzer f~ f^-^TJ , It is from tin.'- modern point of view that the Kaluza-Klein In retrospect, Kaluza's Idea that the unifying stage of the theory will be presented. To put it shortly, we will study world be a high dimensional space, seems to have appeared gravity and the coupling of matter fields to gravity in Ds't+K precociously since neither strong nor weak interactions were dimensions. Space-time £ will be of the form £ sM^xB^., where known at the time (at least in the theoretical sense). Still, B is a compact space, mostly chosen to be a coset space. Four already thmugb the classical period, the idea has fascinated dimensional fields will appear a* coeffi cents in a Fourier many people. series over Bv and we will be particularly concerned with ths zero mode sector. During the last few years the idea has gained in popularity, event though1 it has not yet provided an answer to any real physical Some formal preparations are made in chapter II. In chapter problem. III, a detailed discussion of spontaneous compactification will4 be But the question of the "true" dimensionality of the world given. Chapter IV is devoted to the classical Kaluza-Klein theory is surely qrie of the deepest one can possibly ask within the including the incorporation of spinors. In Chapter V, the per- framework of physics. It is of course far from obvious that spective changes somewhat »s we turn to a closer look at higher the answer can actually be found there. One would keep in mind dimensional Yang-,Mills theories. however that relativity theory ^ells us that space-time is not Chapter VI on supersymmetric theories i« comparatively just the stage whereon the play Called "physics" takes place. shore as on)' very recently the first results specifically related
Consider for example the question of critical dimensions. to the methods presented in this review have become available. Their occurence is common in systems of statistical mechanics, It is precisely from this point that future developments will depart. also one encounters them in the lattice versions of field theories. Referencing throughout the review is sometimes sparse. In field theory, dual string models are a classical example However, a referencing section can be found as well as an extensive for theories with ciritcal dimensions: they exist only in a bibliography which contains even some papers whose contents could finite number of dimensions. Actually it has been argued recently not be included into thi» raviaw. 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 - 2 - MI. FORMAL CONSIDERATIONS
A. The setting of high dimensional field theories is specified by the requirement that it transform homogeneously under local frame rotations. This fixes the transformation beha- We parametrize D==4+K dimensional space-time by local viour of the spin connection B under frame rotations to be 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 LB takes its values in the Lie algebra of SO(l,3+K-)> Thus early ajvd middle part of the alphabet respectively. The splitting AB the B = BB where ^ tT ^i generators of into Minkowskian and internal indices la denoted by M = (m,u ) and H 77 M TAB"! S0C1.3+K) in the spin representation. A = (a,«,.).. A spinor Lajrangian density which is invariant under local We have two basic symmetries. Under general coordinate trans- frame rotations and a scalar density under coordinfte transforma- M 1 M formations a ~"> z the vielbeins transform according to tions is given by
( We will also have invariance under local (as - dependent) frame B-, with detE det(EN°). The curvature scalar density provides a rotations, Lagrangian for the gravitational field itself
the frame group being the pseudo - orthogonal group 50(1,3+K). with the components of the curvature tensor being / 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 Besides the latter, there is a second tensor with the character of A r g a field strength, the torsion tensor with components z AZ ) (3-3 which may serve an a metric tensor. Indices are raised and lowered A riemannian geometry is one for which the torsion constraints definein thed usuaas lmatri way x usininverseg Gvejsj oanfd ^respectively. E and G art E and respectively. T^* = 0 hold. These are algebraic constraints for BJJ r l and defined as ti i f E AB As a typical example for the coupling of D - dimensional allow the solution •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.
The terms linear in V in eqn. (i.l'S) can then be replaced by d v A B - djjV^. More generally one considers the totally antisymmetric tensor A* » and chooses th« totally antisymmetric field B. Harmonic Expanaiona strength to be
1. Introductory remarks F". L ^ (An <^A A -i • / n \r\
As a final example, consider the Rarita - Schwinger field* In this section ve will introduce the method of harmonic The invariant Lagrangia« density is given by 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 where -r The co variant derivative ±* given as 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 where ?f>\ and J^-^denote the spin and vector representation for circumference 1. In order to hay« a non tachyonic mass spectrum the Lie algebra of 30(1,3+K) reopectavely. Because'of the anti- the internal metric will have to spacelike; on M^xS^ we can choose of I ' ~ the setae simplification as above occours. Thus 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. .... •
take to be .. .. • .,., V. • •'-.•.' "'"•••' • ••-"-. '••• .-,• ,>fhen dealittg- w4,t*l• spi'nQrs,. it Is
deriv.ative of generalized Dirac matrices vanishaa,. • • (Ml).- ;• where surface terms have beeji naglect-ed. As before, z=
-' 5-'- - 6 - We have again an infinite number of fields with masses (A• We see • 1ft —— o immediately how the mass spectrum reflects the choice for through where O^f y<£ L = (2T/M). The reality condition^ (z) =3>*(z) the eigenvalue equation (1 implies 4>*n(x) = Jf^x). Noting that the five dimensional A typical example is B = S (the two - sphere), with Kr 2 d'Alembertian ia ^ = CL-^/^?". we obtain from (2.21) after 0. Then 1 where is a mass unit X-* X, - i faO,!r performing the y -integration, giving the inverse radius of S2" The four dimensional fields 1 (§, lm(x) provide (21 + 1) dimensional representations of S0(3). This s- i IJ "* „,„, follows from the transformation properties of "X , (y) and the This is the dimensionally reduced action. It features an requirement that ^ (z) be a scalar field. infinite number of four dimensional scalar fields^ (x) with Compact spaces, like B = G/H, with G and H compact groups, masses nM, the complex massive modes, and a massless field, allow the introduction of nontrivial topological structures. As a A»o(x), a real massless mode. In the limit L-'JO, the massive modes typical example consider the concept of a "twisted11 field on B^ become very heavy. As a first approximation they may be Above, we had the periodicity condition i(x,y) = ct(x,y+L). V discarded at sufficiently low energies. Keeping only the aero By contrast, for twisted fields we would require ij(x,y) = mode a amounts to using y - independent five dimensional fields •J, (x,y+L). Note that this eliminates in particular the zero only. This procedure is often called "ordinary dimensional mode in the harmonic expansion. reduction" in contrast to "generalized dimensional reduction", where some y - dependence ia kept. An alternative description is the following. Consider a circle with circumference 2L, and identify antipodal points on la generalization of the above consider now the real scalar this circle. Twisted fields are those whose values differ in sign field on the product manifold £. = M^ x Bj, with metric at antipodal points. For twisted fields, only odd n occur in = bdi«5 antipodal points on $„, we obtain the real projective space RP2* For twisted fields, we keep in the harmonic expansion the modes The where volume element splits, with odd 1 only and for untwisted fields those with even 1. det DD KK ! Su*! d z = jdet Gmnl d^x-lded^x t Grv| dd y = Also, In this section we have studied only free field theories. we have for the d'Alembertian o ^ Q + •„ and for There is no coupling between different modes. Conversely, had the curvature scalar RD = R^ ++ RK. Now let {% n(y)] be a we started with an action containing a self interaction of the complete orthonormal set of eijpenfunctions of f — Di, f 'Vi»i+-F D "\ scalar field, then there would -occur couplings between different with corresponding eigenvalues .^ , that is *• (no sum) dV y = In fsi nl9 of with normalization X K /^y) % m^ ^ &nm' " these eigenfunctions, the generalized harmonic expansion for tj (x,y) is then simply ^ (x, y) a/^n^/n x^ "^n^n y^' InsertlnS this expansion into (2.24) and using the reality condition Z 2" ^ n(x) % n(y) we obtain the dimensionally reduced action - 8 - - 7 - to require that under a coordinate transformation the world 2. Harmonic expansion: General formalism scalar J(x,y> transforms according to We generalize here the observations of the previous section to general coset spaces B^ = G/H. In this section, material from where T) denote some particular representation of H. Writing appendix 0 will be used. out some indices, this becomes J^Cx^' ) =|) (h) <£. (x, y) . Only As. a first example take H -= 1 so that B_ = G. Consider a if J) happens to be the trivial representation, we have the old scalar field J (x,y). Under coordinate transformations y_j,y', definition of a scalar field. The indices i and j are by definition ^ (x,y)->j ^ (x,y') = Cp (x,y). Consider now the ansatz indices associated with H. Since<^>(x) carries a representation 5 (x,y) = D(L~ (y)) H. The dependence on h cancels (because the O are represen- tation matrices for H) and we obtain We have a factor D(h) on the right hand side of (J.2* ), which can not be compensated by requiring a suitable transformation T J behaviour for (x) to carry a representation of G. Thus we have - io - - 9 - which marka I . as the cjD"mporient« of an invariant tensor. : P ? iP? .' ' : • • Gowider • also. a ; high dimensiflnal'.spinor . It may b6 Accordingly) by S-chur"'.'• lemma,. looked, upon ffs artHiltlpl.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 The factor \£ \ depends only on representations and"can be found thu-B constitutes a direct sum of.H t multi.pl*-ts of Dir&c spiJtiors.. by Setting n=n' , p=.p' ( aftd summing over p.. Then 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. We thus obtain the desired inversion formula, In this case G=H and the spxnor actually decomposes into & - multiplets. l As an example for the harmonic expansion itself, let iis . As before, the dimensional reduction procedure consists consider the vielbein, and in particular the components E *^ (x,y.) • effectively in replacing the fields of the higher dimensional Since these transform like the components of a K—vector under theory by expressions like CilQ) and integrating out the internal SO(K), even the lowest term in the harmonic expansion must display variables. In general, algebraic complications will occur, some y - dependence. In fact, application of the general formula mainly due to the appearance of Clebach-Gordan factors which and retaining the first term Only gives not necessarily recombine in a simple way. In a similar way, higher rank tensors and teasor/upinori under 3. Harmonic expansion: Examples the frame group may be treated. For tb.e discussion of the Kaluga - Klein theory below it will be convenient to have certain specific examples of the above formalism available. References for chapter II Consider for example the one form e"(y) = eJC^d/ on coset The material in this chapter is to great extent spaces, where e „ f* (y) are the vielbein coefficients. In appendix collated from references (~l69j , T^^J «""< j 55 ~^\ • 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 (W1 ). 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 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 In the CS icifmej Stherk) - model \_^ 0 " 4-! J gravity is coupled to Yang-Mills fields in D=*t+K dimensions; there is 1. General remark* also a cosmological constant. The Lsgrangian of the model is The concept of spontaneous compactification has been intro- duced in the mid - seventies by Cremmer and Scherk T'-'O,^ lf V?. 1 The constants H and e are related to but not identical to the in order to provide on answer to the question of what might happen Newton constant and the four dimensional Yang-Mills self coupling. to the extra dimensions in a high dimensional field theory. In addition to the conventions of appendix A we have that The question was relevant at the time since dual models had turned late Latin indices with caret refer to the gauge group R of the out to be consistent in 10 and 26 dimensions only. It may be of Yang-Mills Lagrangian. In (1- I ), the Yang-Mills tensor is 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. The field equations are We have seen above that in a field theoretic context small - i SB. R. (internal)radius corresponds to large masses for the massve modes of a harmonic expansion. Thus it may be argued that for sufficiently - 0 small internal radius even though the extra dimensions would mani- c where the erfrgy momentum tensor is given by fest themselves through a mass spectrum of the four dimensional theory, they would nevertheless be unobservable at least at presently available energies. * For a spontaneously compactifying solution the D-dimensional lino The suggestion that the extra dimensions are not being seen 2 MM N m n 1 11 element takes the form 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. CM-) 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 From ( 1-d ) we infer in particular that we can not find a Therefore u is of the order of Planck's mass (10 GeV) and v JJ compactifying solution for vanishing Yang-Mills fields. This is the internal radius is the order of Planck's length, 10 cm. so since we have already specified that the four dimensional 3• The minimal gauge group in the CS model cosmological constant should not vanish. Introducing the notations h . K^ty) K1 ~ (y) and The CS construction was for the case H=6, i.e. the «'* - fSvw ? £< h one can work out by using general properties internal space-time group is identified (in the abstract group of Killing fields, sense) with the gauge group of the external Yang-Mills fields. The construction immediately provide* solutions R3G 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 whereas ( "5 - 8 ) becomes B3G is interesting in its own rifffe. It will be discussed below. ''* 3 Here we turn to the case RC G. In particular, one may search (3.1/ ) for the minimal gauge group R such that compa.ctification of One reads off that the field equationsican be satisfied if the M^ „ into M.xB occurs. Clearly^one feels that, the smaller R, -kMat.-i.ili HS E . " - C S-J is satisfied for some constant C. V t the better. It has been shown that this minimal group ia not Solving this constraint constitutes the final step in the bigger than H. Indeed, R may be smaller in rank than G, and need CS construction of a compactifying solution. Since^J.t is an not act transitively on G/H. equation among invariant tensors, it is sufficient if one finds The available compactifying solutions which use groups R a solution at one point in G/H. This then is an algebraic problem. smaller than G are topologically nontrivial, unlike the CS For this, two essentially different solutions have been de- solution discussed above. This is to be understood in the scribed by Luciani. The first corresponds to H=l. In this case following way. For the gauge fields BM(z) = (Btn(x, y) ,Bj,,(x,y) ) , the the coset space is n group space. One has C=l and the remaining compactifying solution takes the form BM(a) = (0,B^ (y)). Now B^(y) ( i constants an fixed as Au- 7a , U — (?TT* - 15 - - 16 - as'a base' space : for a nontriy.ia.1 fibre bundle, and BL (y^) are the G j ; J :-, i „-e *(y,)' e f tyj . ,, whereas G is a solution of the .connections for "this bundld.'Ths nantrivial topology is then four dimensional Einstein equations. Solutions.exist in .this, case netlecrted in the existence- &f".nx>nvanishing topo'T&gieal quantum if either the four dimensional or the D dimettsioiia;l cosmological -ntimbers like the instanton.charge in four dimensions (on S^) constant vanishes; but both can not vanish sihiultaneously• "and its generalizations (the characteristic classes)* The nontrivial topology reflects itself in the fact that ' A typical example, of such a solution is the f ollowingLH£, 84 j. only a local but not a global definition for the function L(y) We take B =S2- On S , we introduce polar coordinates ©((p. Then exists. A global definition of L(y) exists if the potentials Bo , B, leading to the compactifying solution are bundle over Q/H with structure group H ["UPT J • Above we had given by 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 Here the N.P solution is valid in the region of S that contains are valid only if G/H is symmetric (like in the previous section). t.1% nor a pole (0--Q) but not the south pole, the other for a So definite proof exists as yet that spontaneously corapacti- region overlapping with the first that covers the south pole. fying solutions with RC G must.be topologically nontrivial. All 'X is a generator of the gauge group R; in general it would be the known solutions have this property however. It'also an of the form J- ^«,, \^ open question whether solutions with RC H exist.. The N.P. and the S.P. potentials are to be connected by a It has been noted that the nontrivial topology o:f gauge transformation exp(i2vp£). This is just the condition BR may be employed in the following way QSS" J. Even though the that ( 3- (1 ) describes a potential albeit a topologically non- CS construction given above is topologically trivial (i.e., the trivial one. In fact, all the information concerning the non trivial appropriate characteristic classes vanish), they still may be structure of the solution is coded into this gauge transformation, split into topologically nontrivial solutions in a manner analogous the so-called transition function. The transition function has to the construction where a q=0, 30(4) instanton solution .is to be single-valued, thus exp(4tfi r ) = 1 or - 17 - -• 18 - k .,. Further ;models of apontjuiequs1. compact! f.ication -.. 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 Besides the GS model, a number Of .models featuring•' sup*rvgravity..) We have now preferential Compactif ication of eitlier spontaneous compactification axe availble and will now be described. four or Seven dimensionsi Pbysd&al space -tdmw may or may not be In the first by Omero and Percacci Q^fJ, a sigma model coupled to the four dimensional factor. In the latter case, when physical : gravity. Here, no cosmological constant is required. Physical space-time is contained in the seven dimensional factor, further space-time, M, , can be any solution of the Einstein equations, (hierarchial p2.£j) spontaneous compactif ication has to occur. R =0. The solutions are further described by saying that Aspects of this possibility have been discussed recently by mn Salam and Sezgin the scalar fields of the nonlinear sigma model have to comprise the same space as the internal coordinates. (One mauthus get the All the above spontaneously compactifying solutions where idea of "coordinates as dynamical variables".) More formally, solutions of high dimensional gravity coupled to matter. The the scaJai- fields may be visualized as identity map from BK as occurrence of matter is probably enforced by the stipulation that t-.Terns! space to Dv as field space. Incidentally, the construction A. xne four dimensional etymological constant be small. In fact, is possible for general Ricmannian spaces H . Classical stabili- pure gravity in D dimensions does have, spontaneously compactifying ty is analyzed in particular for the so-called Grassmann spaces n solution?but they all correspond to larga values of the four and stability is found for B^ (F )=CF(n)/tTF(n-k)UF(k), with dimensional cosmological constants. UjjCnisSOtn) and U,(n) = SU(n). Stability is also found for B^Sj, : K > 2, but not for Graasmann spaces G.{ |H ), where |H is the space Perhaps an ansatz with nonvanishing offdiagonal elements for of quaternions. the metric, 5^ , could improve this situation. In the spirit of the Kaluza-Klein ansatz such an ansatz would involve constant A further model is by Freund and Rubin(j^j. It is interesting Yang Mills potentials. For the pure Yang-Mills theory such solutions in particular as it contains as a special casj an exact solution ejfist] \ t*f land therefore the attempt may not be hopeless. of eleven dimensional supergravlty. Also it displays the intriguing feature that a preferential number of dimensions compactifiea. Quite a different possibility has been investigated recently The model consists of D-dimensional gravity coupled to antisymmetric by Wetterich[Zoo jwho starts off with an action containing tensor fields of rank s-1 with field strength tensor.F terms quadratic in the curvature tenaor. This provides for extra A spontaneously compactifying solution for which G^^bdiag(G (x), terms that can take the place previously taken by the matter terms. G f., (y)) is found with the help of the ansatz From a physical point of view this possibility, is also interesting, 1 j. = £ j. „ f } |G_| , X=cons^ant• Tne solution is valid as one expects such terms to arise as counterterms in a renor- '*<. if B-ddlfttriSionai ipace-time splits into an S- and a (D-S\- malized theory. Prom this point of view, one may even see this dimensional factor, respectively. In this case (and only in this case) as a starting point to answer the question whether spontaneous the Einstein equations of the D-dimensional theory simplify to corapactification is more than a classical phenomenon and survives 2 in the full quantum theory. Rn _ =- (S-D(D-S) >/(D-2) and R_ = -S(DrS-l) > /(D^2), > =8lT4ff . 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. - 20 - - 19 - 1 5. The CD model He also discusses the possibility that in eleven dimen- sional supergravity, preferential expansion of three space- The idea that the small size of the internal space may dimensions occurs, much in continuation of the ideas put forward be related to the large age of the universe has been introduced in connection with the Freund-Hubin model (see above). Ia addition, by Chodos and Detweiler ^Isrj. They present time-dependent he discusses scenarios for the whole procedure to take place in spontaneously compactifying solutions for the vacuum Einstein hierarchiaI steps. equations Rj_.= O, where R™ is the D=4+K - dimensional Hicci tensor.Their solutions are generalized Kasner solutions, which are of the form . 6. A lattice calculation It is difficult to foretell whether spontaneous compacti- with £; p = l_ p. a 1. Because of the latter condition at least fication persists as a quantum phenomenon beyond semiclassical one of p. has to be negative. One therefore infers that some approximations. It has recently been suggested by Skagerstatn rf?TJ space-time dimensions will be expanding, others contracting in to approach this problem by studying lattice versions of time. While this ia a poor description of the four dimensional high dimensional field theories. universe, for a high dimensional theory it may Just be what the He assembles Monte Carlo data for the Wilson loop average doctor ordered. for the following two models: (1) a five dimensional Yang-Mills Chodos and Detweiler describe in detail a five dimensional system; (2) a four dimensional Yang-Mills-Higgs system with 1 2 case with P1 = P2 = P-i = "P4= / • This choice ensures isotropy in the Higgs field in the adjoint representation that corresponds to first three spatial dimensions. Further, all spatial dimensions five dimensional Yang-Mills system with periodic fifth coordinate. are being taken periodic with L. For this case, the line element He compares the two sets of data and finds very good becomes ' agreement. Correspondingly, as far as Wilson loop averages are ,-it^ [ft (it? concerned, the two systems behave in the same way. This indicate* that spontaneous compactification of the fifth coordinate may At t=t , th,e universe ia flat and extends for a distance L in indeed occui* in the five dimensional Yang-Mills system., every direction For t= "£ , where £ is the present age of the 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. - 21 - For stability, all (xy- should be non-negative, otherwise B. Stability considerations we would have exponentially growing modes. This requirement can actually be translated back into a criterion that the energy have 1. General remarks a strong local minimum for « for static solutions lp (x) . In fact, for such solutions the energy functional becomes Spontaneously compactifying solutions have been sought It is minimized by the as minimal energy solutions (as much as energy is defined in nature of these solutions, stability •4*1 considerations hove to be taken into account. The above considerations are complicated by subtleties concerning degenerate ground states in general and zero mode In the context of the Kaluza-Klein theory, this analysis oscillations in particular. If we have separated strong local ha* only been carried out for the most simple cases. As we will minima of the energy then they will classically all correspond see, stability considerations are of non neglegible impact. to stable equilibrium points. To proceed further we will have to Energy-like (conserved) quantities play a prominent role resort to semiclassical procedures (see below.) It can also happen in stability considerations. A typical example io Dirichlet's that a minimal energy solution can be deformed continuously into theorem which asserts that a solution of the equations of motion another one without change of energy. Such degeneracies occur is stable if it provides a strong minimum of some conserved typically for Goldstone-type potentials. In the formalism above quantity. Note that stability is not guaranteed if the minimum is they will manifest themselves as zero frequencies. Physically, not a strong one. Also note that froms; the point of stability, the corresponding zero mode fluctuations are associated with mass- energy may be replaced by another conserved quantity. less particles (Goldstone bosons), provided, the degeneracy can be In a quantum theory, barrier penetration from one local associated with an internal symmetry of the Lagrangian. If no minimum to another may also occur. For stability we would have symmetry is associated with the zero mode there may be instabilities, to require that a minimum be a strong absolute one• If two in accordance with Dirichlet's theorem. minima have appare -vt,tly the same energy, we have to demonstrate Consider now the field equation for static solutions and that no semiclasslcal decay from one minimum to the other occurs. differentiate once more with respect to x. One obtains 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). 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 corresponding field equations are' the iranalational invariance of the system. In fact, consider small translations, X-->X+XQ. Then ^(/.O- ^cl (< + Vo") - 4c« 60+ 4^*") ' *o whence by comparison with ("3-n- ) we get Ca — 0 and Suppose, these have a classical solution <£> = U) , Coc4. Perturbations around this solutions are described by the ansatz 4>(*it^ = fcL -H .'•1 mum !„• 2. Positive.energy theorems and instability of the lowest. In particular, for a spherically symmetric ||tl, the translation modes form a p state and there is an s-state mode Kaluza-KIein vacuum with negative ). In fact, the differential equation to bfe studied is S.;ni;n; - jovtriant derivative with respect to the background. (1.1V) with ?cl-*fb. Now, if the background metric has a Killing vector ~JM associated Note that for step (3) it is often not necessary to actually with some symmetry, then it follows from d § + d ?M = 0 MN MN solve the analog of (1. PH. In fact, as we have seen, sometimes and the symmetry of T that dJr f „} =cL(T f „ ^ = 0. general arguments indicate the existence of modes with negative CO • 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 from the vacuu-i topology is possible. Note that It is not otvioua theorem is the following. For Minkowskian background, A- Oi that such a process is possible. This is unlike the situation in in four dimensions, with ^ = (l;0,0,0) E vanishes for Minkowaki Yang-Mills theories where say, antimonopole-monopole pairs can space only and is nonzero for all other spaces that are be created smoothly out of the vacuum and thus topologically asymptotically Minkowskian. (The proof of the theorem involves nontrivial configurations must be considered. Here an analogous a number of technical assumptions*) argument can not be applied as there is no such thing as an Let us now turn to the discussion of stability properties antispace. of Kaluaa-Klein vacua E 2- °t, 2.0fl Intuitively, the semiclassical decay associated with the As has been remarked above, it makes no sense to compare bounce solution is the formation Of holes. It is likely that M_ and M.xS on energetic grounds, We therefore have to consider seitiiclassical instability of the Kaiuza-Klein vacuum occurs only H,x3 by itself. ClassicaUy, M,xS is stable. The small oscil- 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 lations around M; xS are (compare also f:he following chapter) some massless states: ^riiviton, photon, and a Jordan-Drans-Dicke positive energy,. Technically this is shown by considering solutions scalar and an infinite number ofmassive states. There are no of the Oirac equation on initial value surfaces. Also if elemen- exponentially growing modasi The iiiassLeas statfia reflect symmetries tary spd.nors are present in the theory they may stabilise the of the Lagrniigian (four dimensiona 1 Poincare, and gauge invariflnce, ground state. The reason for this is basically that the definition also an invariants under a roscaling of tho radius of S ). of spinors is not independent of the topology of the underlying cianifold. In particular, Kaluza-Klein spinors could be introduced Let us now investigate the question of semiclassical stability. in such a way as to be incompatible with the topology of the A bounce solution is specified through the line element bounce solution. For the six dimensional case, £=. M^xS2, a first stabili- ty analysis is now also available. It concerns the minimal where d X2_ is the three dimensional solid angle, c( is a constant. solution in the CS model 1^UZJ]J where classical stability is Indeed, for large values of r, r = (x,l4-»i +-*£),l the solution ( 3-ZO ) ascertained. One recalls that this solution displays topologically approaches the Kaluza-Klein vacuum, ds = dr + T ailt + d^ nontrivial features corresponding to &n internal "magnetic charge". In the latter solution, Introductory remarks 1. The f i ve-dimensiona1 case In 1921, Kaluza ^OsuSSested the unification of gravity In this case no extra matter fields are needed for spon- and eleetromagnetism into a five dimensional gravity theory. taneous compactification since both M as well as M^xS are so- Some special conditions were to be met by the latter: the five lutions of the five dimensional vacuum Einstein equations. dimensional space was to be M, xS with S having a very small There is no reason in this case why one should prefer one of the _ T *l solutions as a ground state. As we have seen, even stability circumference (^j 10 cm, Planck's length) indeed. The small- considerations do not help. In fact, M^xS is less likely to be nesa of this internal space was thought to render it unobservable. stable than M . For the time being we therefore just assume that In a field theoretic context a four dimensional theory may be M, xS is the true ground state of the theory. obtained by harmonic expansion over 5 ; the massive particles The metric of the ground state is G,™- ( + . ;-) where we chen have messes of at least 10 GeV (Planck's mass). Kaluza's idea then means in this language that at presently available anticipate that internal dimensions have to be spacelike. The energies only the aero maan sector of the theory would be relevant. metric is forminvariant under P^xU(l), where F^ is the four It ia mostly this sector that will be of concern to us in this dimensional Poincare group. Among the zero modes we therefore chapter. expect the gravitational field (spin 2) and one gauge field (spin 1). There will also be a massless scalar, whose existence Even though the zero mode approximation will be good for probably reflects the invariance of the theory under rescaling small energies, on principle grounds the massive modes can not of the internal radius. These are the zero modes whose massless- be neglected entirely |j',S^SSy/£ " HEinstein ( - 29 - Here R . ^ . is toe curvature s^aidr ?LT X , -'o:~ >~ .-. d V^^L^ n£inst?in ?nn just get the Einstein-Maxwell Lagrangian with unit gravitational One obtains this if Q is indeed the matrix inverse of { fi ' constant. In this case the Einstein field equation R =0 reads F Fml1 = 0. Therefore the ansatz (4.1) is not compatible with the mn -fA" field equations. ( The relation |i f = e is determined by the coupling of Um, y) » (x'ra, y') = (xm, y + £ U)) . < Another way to see this is to study the line-element 2. The general case m n m 2 ds_"2 „ J_MJ^N gdx«mnd*dx dx " - 31 - - 32 - d In (4.12) the abbreviations g E E and Note that the appearance of the complete square (dy + A dx™1) m r . .,.-„. Cy)g.^ have been used; the in the line element is reflected in the appearance of the 0 in coset space metric is (cp. Appendix D) g (y) =e "My) e^f" (y) -^ Ci.3), The components E are to be identified with the vierbein m The components of the Killing fields had been introduced as components of four dimensional gravity. Recall that this form IU r" (y) = Dj Is (L(y) )e^f- (y) and K/j (y) = g Kjv (y) . The Killing of the ansatz clearly displays the gauge symmetry as a special fields have the fundamental property that they serve to case of invariance under general coordinate transformations. express infinitesimal group transformations g=l+ £* Q^ as For the D - dimensional case we now make the ansatz coordinate transformations, viz. y^ ? yl* + £""K/jV{y). One thus expects that the coordinate transformations E/CO- '/Vis 0 If we write down a harmonic expansion for the vielbein ( k- 1 ), can be re-expressed as loc^l infinitesimal gauge transformations a then the first term for E ° will simply be E (x) by the general for the would-be gauge fields A Mx). That this is indeed the arguments of chapter TI, since t. • >e behave like components of rn.=-e . s •shown in appendix C in the metric formalism for infini- i. frai.ie s:aiar under internal rota.jOas, As we have noted above. tesimal transformations. In the vielbein formalism for finite the arguments of chapter II provide for E the ansatz 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« aa the first term of a Fourier series. The A p1 (x) will turn m out to be four dimensional gauge fields associated with the gauge ( group G (recall that internal space is presumed to be of the 0 and e_T are the matrix inverses of E and e. * form G/H)* The E will again he the vierbein components for where E respectively, and A *• (x) m f- four dimensional gr.ivity. These are the zero modes we expect for n MN M E A '* (x). The inverse tensor G then is a n 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. The change of the metric tensor under coordinate transformations iMN eii 'v' = e 'y. (y) t and similarly r) a is unaffected by four di- at a given point is OG (z! G (z) - G^Cz). For the coordi- nata transformation ( ^, '3 ) we find in particular 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, with r A/ ( kit ) The metric equivalent is Again we find that coordinate transformations can be reexpressed as gauge transformations. Tlje curvature scalar corresponding; to the Kaluza-Klein ansatz J d y(flete ^^ )DA' (L(y) )Da^(L(y) ) =k V £j« , with is evaluated using the zero spin connections which are given by k -(dim G/H)/dimS. The left hand side here is a group in- (cp. the general formulae given in chapter II) 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 very T 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 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. \ far V {JBD scalars) from the discussion. This section now will contain a short discussion of JBD scalara. where = E mE.n - E. mE and the Yang-Mills a b ba Unlike for the five dimensional case-there is no reason field strength tensor is defined as usual, F * n r * Jk " ' _ mn that in the general case JBD scalars should be tnassless. In five dimensions, there was an accidental symmetry under, rescaling The structure constants are defined through of the internal radius that could be related to the masslessness with Q ^ the generators of G. Using ( - 35 - - 36 - i.mm-m m -* m metric tensor g in the Kaluza-Klein ansatz to become x-dependent. where 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 gv (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 •V «•*» tensor g,.<)(y) = e^Cy) euMyJ'o n, = diag(-...-). Here we Like tne now replace the constant tensor <* by &S*} ' %a •> fields. One notices that the potential for the JBD seniors is not ^p (x) have to G-invariant, ie. they transform like group scalars. unlike in structure to those for nonlinear sigma models. In the Kaluza-Klein ansatz (V, 12) we replace g (y) by The symmetry breaking patterns for a potential similar to g|u(x,y) 3 5 *(y) e,"(y) y (x); for the off-diagonal elements ^ P- * *• -7 / that of equation Ci.26) have been investigated in some detail of the metric tensor we replace A - 38 - - 37 - Kaluza-Klein ansatz for the CS model Relation to fibre bundles Above, we have described in some detail the Kaluza-Klein To some extent, Kaluza-Klein theories may be viewed as the ansatz (zero mode approximation) for the high dimensional metric physicists' way to understand fibre bundles. This connection tensor. However, since no spontaneously compactifying solutions was probably first obseved by deVfitt (2oy[and later much elaborated with nonabelian internal symmetry and vanishing four dimensional on in particular by Trautmannfi J, Kjjand also by many others (/*, 2Y, 3»^ cosmological constant have been found, this discussion lacks some- 'O^M, «"7j . Here, I will only very briefly describe in what sense what in consistency. It would be desirable to discuss the zero the Kaluza-Klein ansatz /V '/ ^reflects the structure of the fibre mode approximation for a model that actually does display the bundle. For more detailed expositions I refer to the literature.' phenomenon of spontaneous compactification, like the CS model Rougly speaking, a bundle is a space £ which locally {the gravity-Yang-Mills-systera) which we have described above. looks like the product of two spaces, M, the base space, In fact, an early formal discussion of the zero mode and F, the fibre,Z£_ M x P . In our formalism above, the local approximation has already been given by Lucianl [_ 10^ 1. product structure is reflected in the possibility to choose an A general observation is that the metric gouge field and the orthonormal set of frame vectors. There will be a projection external gauge fields get mixed. However, the gauge coupling TT,-?-*M with the properties thatT(S) = H and 1*'(x) " F for all constant for the external gauge fields tenda to infinity just i( M. The existence of a set of homeomorphisma £ •*£ which among •a the masses for the massive modes; in the four dimensional some other requirements, have to form a group G, makes the bundle theory, the two types of gauge fields therefore behave quite a fibre bundle. If the fibre space is the same as the group space, differently. we have a "principal fibre bundle", otherwise we speak of an "associated fibre bundle". A detailed analysis of the Einstein-Maxwell system in six dimensions has recently been presented by Randjbar-Daemi, It is now a nontriVial result that a fibre bundle, once made Salam and Strathdee f It 2-7. In particular, these authors dicuss into a Riemarinian space (i.e. endowed with a metric), has a metric the classical stability of the ground state solution (with of the form ( m V ./°; available. Therefore, only some general properties of the 1 M formal language. Introduce the vector fields 0 = E "i .We noted Rarita-Schwinger equation in k+K dimensions will be discussed here. 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), E m In a = 0a * this case we have ic^, 13,, \ = ^ ^ (x)K/,^ (y) £) = 1. Dimensional reduction of the Dirac Lagrangian F b ^ D**(L(y) ) c ^ . Quite generally, we also have ft) , 9 7= 0 and The general form of-the Dirac Lagrangian in D dimensions C9A J'CL l---t.|>«-j "^ i where the latter simply provides us with has been given in chapter II to be = i-dtt E k-c. . the connection coefficients on the coset space. This construction clarifies how tht particular form of the D-dimensional vielbein Here U T and 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 Even though the actual derivation of ( 0, ?3) is cumbersome, antisymmetric product of P- matrices) and to drop labels on the emergence of a four dimensional Lagrangian with invariance The Dirac lagrangian then becomes 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. Also, in a somewhat different setting, a treatment is given in [H, to3ij. . c. In equation (U.Yi) , the first three terms in the curly Here we now have to insert the various expressions for B r -i , A1BCJ bracket provide a Riemann- and gauge- covariant derivative. The as given in the Kaluza-Klein picture (see above). One eventually terms linear in the gauge field can be traced back to originate in obtains, using the connection on G/H as given in this section 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 nlj. J. The mass term, on the other hand, is causing troubles, A precise criterion as to which spaces allow a definition which we will describe in more detail below. In fact, if it of spinors will therefore be a topological one. In fact,(compare does not vanish, spinors will have a mass not smaller than Planck's t 51 J) a manifold admits a spin structure if and only if its mass. second Stiefel-tfhitney class vanishes. Unfortunately, the definition Some final remarks are in order. of the Stiefel-Whitney classes is rather abstract; in particular, For the imbedding problem, let us consider an example f"/£ - 46 - Such a procedure does not work in the present case. In fact, I a % ®'I (CP- also appendix B), where ^ are standard four we could have to introduce imaginary masses which would spoil dimensional Dirac matrices. From the definition of cLj one then hertneticity requirements. What we could still do however is to reads off that = 0 for all a, ao that the eigenvalue impose a chirality condition; the troublesome mass term may then equation for the internal Dirac operator can be given the form disappear identically. P*^* 4l = TS/Vn, (f with Vs *• yS © A , The mass term therefore has the form \Z Vft* u , • Since no other sources of P-violation According to our general philosophy, the chirality condition are present we indeed conclude that the massive sector of the must be imposed on the high dimensional spinors Cp(3c,y) and it dimensionally reduced Dirac Lagrangian is P-violating and the must be compatible with the requirements of local Lorentz invariance raassless sector is not. * and general coordinate invariance in D dimensions. „ such a This argument is not quite complete as it stands, since we Denoting PD+1 a P •... *"D> constraint is given by have left out the possibility that the eigenvectors of equation the Weyl condition f £ = t £ . Weyl spinors exist in even C(|,?fc) come in left-right asymmetric pairs. This subtlety can dimensions only (for odd dimensions, i is a multiple of the presently be neglected as it turns out that P** d^ has no zero unit matrix). Note that for spinors subject to the D-dimensional modes at all. This ia a consequence of the identity Weyl constraint, four dimensional chirality (eigenvalues of P j-... -P.) and internal chirality (eigenvalues of P.: . . PQ) are related. which can be derived by using the definition of the curvature Another constraint by which we may obtain masslessness for tensor, [[Jutl^ J4; = ^n^rrffil^* t and its symme*ry properties, the spinors in four dimensions, is the Majorana condition. The Majo- vanishing of the covariant derivative of the P -matrices and the rana condition, imposed in D dimensions, may actually also enforce anticommutation relations for the P - matrices. Now, on coset chirality in four dimensions. Majorana spinors exist in spaces of compact groups", -d d and R have the same sign, moreover 2,3,4,8 and 9 mod 8 dimensions. (compare appendix E), R is a nonvanishing constant. Therefore the A detailed analysis of the relation between high dimensional right hand side of ('f.Tf-) is never zero, thus (P*a jranfermions in unified theories t eno* 16 come in left-right D = 2 mod 8 dimensions are promising candidates for the provision symmetric pairs. Therefore the question arises how we could get 1 of the desired objective of chirality in four dimensions. out of the impass outlined above * ' \ i Qjitf s. different possibility to obtain masslessness for Por scalar fields, the following method has been suggested[fl?]. spinors and at the same time left-right asymmetry is indicated Imagine we introduce a negative (mass) in the D-dimensional by the Atiyah-Singer index theorem. (cp.^S^j). This theorem theory which would just compensate the mass of th«"^ightest massive relates the socalled index of the Dirac operator, ind (f*D), to mode. We can then prevent the appearance of tachyonic modes by be defined as the difference in the number of left-handed and using twisted fields. In fact, recall that for twisted fields we right-handed zero modes to topological quantum numbers referring can arrange matters such that the lowest mode does not appear* to the underlying manifold. In a typical example, the index In this way one would arrive a__t a masaless four dimensional theory may be expressed in a linear way through the characteristic classes. if the higher modes are neglected. The theorem applies to compact manifolds only. 3. Rarita-S-:hwinger fields This however 1-< perf>ctty sufficient for oar present needs, as we are interested in the zero modes of the intenial Dirac "The" supergravity theory, the N=8 theory, contains in its operator on i y, eleven dimensional formulation just one fermionic field, a Rarita- Schwinger field 4- (x,y). A typical scenario where left-right asymmetric zero modes "A have actually been realized is the following j?£~ /(-j. 1. Recall As a spinor in eleven dimensions this has 32 components. that in the CS modet (the Einsteln-Yanf-Mills system) an external These are subject to a Majorana condition in order to provide the Yang-Mills field appears. Further we have seen that spontaneously correct counting of states for a supersymtnetric multiplet. From compactifying soJutions exist with nontrivial topological properties the four dimensional point of view, U< decomposes into eigth coded into the would-be Higgs part B^(y) of the external Yang-Mills Rarita-Schwinger fields (£• (gravitinos) and 56 Dirac fields - 50 - V. HIGH-DIMENSIONAL YANG-MILLS THEORIES Therefore, in this form, no connection coefficients will be attached to the vector index. Therefore ( V. 'H ) is just a set A. Introductory remarks of decoupled Dirac equations, and we have a zero mode problem as before for the Dirac operator. Yang-Mills theories are, like gravity theories, of Also it is likely that problems with left-right asymmetry intrinsically geometric structure and are thus naturally formulated will have to be overcome. In fact recall that for the particularly for space-time! of arbitrary dimensions. interesting case o^ eleven dimensions, Weyl spi'nors do not exist, We have seen that high dimensional gravity theories provide which deprives us at least of this way to obtain left-right a possible framework for the unification of gravitational and asymmetry in four dimensions. 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 1H u its r I Ttl f*v promising results can be obtained, t£p.«i*e C^SY, h-(, J J 2 1 e tr(CAp A,] ). After that, some remarks on modified Yang-Mills theories Note that as a result of the D-dimensional rotational invariance and a possible geometrical interpretation of ghosts follow. 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 minimal B. Dimensional reduction of Yang-Mills theories energy solutions of the field equations AK = const, CA ~ 0. ,. i-ci. soittions are degenerate; A may be replaced by A AM i A =const. The requirement that Vu^L - 0 is to hold up to a gauge transfor- As a result of this degeneracy, the solution is instable in the mation provides an interrelation between y-dependence and gauge classical sense O^Q- Quantum corrections tend to lift the degeneracy symmetry. In this way, the constraint is weak enough to let us entirely. The effective potential (including first quantum corrections) probably escape the stability problems mentioned above and strong in general does not have nonvanishing minimal energy solutions at enough to restrict severly admissable Higgs sectors. In fact, it all. Therefore the above nontrivial solution is not expected to is not yet clear whether the restrictions are not too strong to trigger a Goldstone-Higgs-Kibble mechanism [ I2C J. allow for the description of physically sensible situations. Certain care has to be taken for supersymmetric theories, In the following section, G-nymmetric gauge fields will be where the following theorem holds : If a degeneracy like the introduced as solutions, to the constraint 9^1/^ ~ 0 (UP to a gauge one above occurs in jeroth order perturbation theory, and if super- transformation). The general form of the constraint for the four symmetry is not spontaneously broken, then the degeneracy persists dimensional: fields will be given in the following section, and to all order's in perturbation theory. In such a case one may still its solutions are described in the subsequent section. A further entertain hope that the exact dynamics of the system in such section is devoted to a discussion of spinors. that a Goldstone-Higgs-Kibble phenomenon persists. 1. Symmetric gauge fields Early attempts to construct a unified picture for gauge and Higes fields using a simple pVocediire as Outlined above are We otasider Yang Mills theories with gauge group R over given in references [^i^f^O^2, til - ISf J. Eventually it turned out space-time of the form O = M. xB with B = G/H, where G and H thnt in order to obtain a suitable Higgs sector, a number of ad hoc are compact groups. Note that not necessarily G=R (in the abstract assumptions had to be made. There was some hope that-these could be group sense). £, is endowed with a metric ^(ilsbdiagfa^lx) ,G ,(y))- justified by the use of graded gauge groups. This however has The gauge fields are V^z) = (A - 53 - Then (CM) and ($S) become, with r( | ,h) •= (h), with ^ - Ja %. dd-(£^)1' i M4 we 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 which are algebraic relations for the fields at the point (x,£ ). compensating gauge transformation r should be x-independent. The mapping u. :H-)R constitutes a group homorph'ism. Actually, it i* Let us denote the compensating gauge transformation by r(y,g). to be required that, in the abstract group sense, G is a subgroup Then G-symmetric gauge field are specified by the requirements of H. A x as tlie (S-2) From the arguments above we can identify ra( i t ' 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 1 identify T as the maximal subgroup of H that commutes with H. (T ia Here7Xj (y,g) is the Jacobian for y that is called the "cantraliaer of H in H". We shall write RDTXH.) Ju (y,s) = ^LWlT As an example consider the case H=SU(3) and BK=G/H=SU(2)/U(1)=Sg. Now it is important to observe that for G-symmetric gauge Then T=SU(2)xU(l). fields, the action density tr(W ) becomes y-independent. To see Principally in the same way we proceed for general gS - 55 - - 56 - we need V (x,y) only at the point y= | . Therefore all infor- in (5.10) as well as in (5.11), the corresponding m. in (J.ll) mation we need is provided by (£•?) and (S.*)) and the solutions J of (54 ) and (.$•%). Symmetry breaking patterns are determined denotes the irreducible representation of a Higga field of the four dimensional theory. inasmuch as they follow from the form of a Higgs potential and it* parameters. For clarification, consider again the above example, G=SU(2), R=;SlI(3), H=U(.l)v Since all irreducible representations The solution of the constraints is a difficult group of H are one dimensional, we have adG = {!) + i + i- For the theoretical exercise. Nevertheless, quite general results have decomposition of adH into irreducible representations of HxT we become available and will be described in the next section. have adR = JLx2 + JLx£ + Ix2_ + !*•}_• We notice in particular that in These results refer to the symmetry structure of the four dimen- this case the Higgs sector is not unique. We may however choose sional theory rather than to the actual form of the four dimensional the first two terms in the decomposition of adS and thus obtain action. The latter is not easily obtained for unconstrained two Higgs doublets, corresponding to one complex Higgs doublet. fields; for some special cases however, compare £/K, /(fc^,3pj. In this way we are able to reproduce the entire bosonic sector of In terms of constrained fields, the action density is simply the Veinberg-Salam model. irt'aL-jC^, % /"* " (x, ? )) where we may insert ( S'S? ) and ( £•*} ) for simplification. 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. 2. The bosonic sector The Higgs self coupling and the gauge field selfcoupling are iden- tified as a residuum of the D-dimensional rotational invariance As a first example for a solution of the constraints in the of the original theory. Also, the generators of T are related to bosonic sector we noted above that the effective gauge group T each other like generators of R. We thus obtain a means to fix of the four dimensional theory is the centralizer of H in H, the Weinberg angle (after identifying the electromagnetic direction). R O T.xH. After spontaneous symmetry breakdown, the exact gauge The result has been worked out for the rank two groups [//S^J: ( group T is given as the centralizer of G in R, RDTxG, ¥CT. one obtains for SU(3), 0(5) and G(2) the values * w * 60°. t5°t3O° In the example from the previous section, we had R=SU(3) and respectively. Note that the last result is seemingly the best from the experimental point of view.This is of course not the last 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. word as presently the entire discussion is really a (semi-)classical one. Also, the imbedding into a grand unified theory is still The detailed structure of the Higgs sector is as follows. missing and spinors have not yet been included. The adjoint representation of G, adG, decomposeses into irreducible representations of H according tb the branching rule As a final example for the general rule given above let us consider one relating to the structure of grand unified theories. adG » (adH) + ^ n, (5.1O) 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 where each n. denotes an irreducible representation of H. appear shortly. H is chosen to be some fixed subgroup of G. Further, adH decomposes into irreducible representations of HxT One then finds TsSUtS)^ as some subgroup of H which is not identical according to the branching rule 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 adH T n\ x m (5.11) i sector in the present scenario turns out to be the one of the Now the rule is that for each time when a representation 11. appears - 58 - - 57 - of ..ne Oitorgi-Glishow model. In fact, the branching rules are 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- adG = h%. = C2^) + 12. + 12.' + i t formation rf(y,g). D has been defined in chapter II;-it takes adE(8) = 248 = <2*,!£) + <2'i£*' + <12>1> + (1°.* .5.* 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 whence we conclude that the four dimensional theory contains available results, the approach of Chapline and Slansky (lO ( ia a and a o r and a a 5_, 2.* ih.i equivalently a complex %_ Si- being followed. For a different approach consider the paper by lt con also be shown that a hierarchy with large spacing can be Manton Tilt J; he also discusses the relation to the zero mode obtained at the semiclaasical level. problem. The K - vector representation of SO(K) decomposes as K = 2. a. , where adG= (adH) + 2T TI. (cp. the previous section). 3. Symmetric spinor fields The spinor representation of SO{K) on the other hand decomposes into irreducible representations of H according to the branching We have discussed above in the context of the Kaluza-Klein rule 5 5. <±i . Let i^, = "'...• ff> a The positive chirality theory that left-right asymmetry for the four dimensional theory = spinor (b ^ [2 «• in D dimensions will contain a left-handed has to be coded into the D-dimensional theory. In particular, four dimensional piece which will be in some representation f,. for even D, D-dimensional Weyl spinors may be used to obtain This representation will be non self-conjugate if ^-!r is four dimensional Weyl spinors. Now, in addition, we have the t non self-conjugate. Necessary conditions for this to be possible choice to put the spinors into self-conjugate or non self-conjugate are^3oj(l) The spinor^oT^§0(K) must be non self-conjugate. representations £ of the D-dimensional gauge group B. In four This happens for D=4n+2, nf IN, which are juat dimensions where dimensions we would want the spinors to be in non self-conjugate Majorana-Weyl spinors exist. (2) H must have non self-conjugate representations. representations. Note that these conditions are not sufficient Both choices for the D-dimensional spinor representations as can be shown by constructing appropriate examples. A conjecture have been "discussed. Manton [l[fe Juses the Weyl constraint only for a necessary and sufficient condition for ?. to be non and non self-conjugate representations £, whereas Chapline and self-conjugate for self-conjugate _£ is that rankll = rankG. S Ian sky I *3>0 Juse both Weyl and Majorana constraints and employ The conjecture is motivated by examples. self-conjugate representations f_. A To obtain the rule that actually describes _f^ recall the The internal Dirac operator is1selfadjoint and correspondingly iS rule for adH into irreducible representations of HxT, Its zero modes form self-conjugate representations. It is comfor- with T the effective gauge group of the four dimensional theoryf ting to know that we may choose such representations for _f and still viz. adH = £ li.xm. . Now, for each time a representation of H obtain left-right asymmetry in the four dimensional theory. V1 J J •*_ —. appears here as well as in the branching rule }&-~2rS'^ . , a /v. Let the fermion field transform according to some represen- representation ni is contributed to £ . Denote this subset of tation £_ H, with r £ _f. Then, for G-symmetric fermion field, as I li /, then _f = 2. it\ • we have the constraint In a similar way, the right handed four dimensional spinora coming from po£sitive chirality D-dimensional spi-itors comprise f . J h(H) _ ~ Z(D i - 60 - - 59 - These are the basic results and it now remains to work C. Modified high-dimensional Yang-Mills theories out examples which are sufficiently convincing phenomenologically. One notices first that the six dimensional theory with In addition to the attempts to exploit the structure of high G=SU(3), G(2) or 0(5) (which gave the bosonic sector of the dimensional Yang-Mills theories in order to construct a unified Weinberg-Solam model) does not allow the incorporation of fermiona picture for gauge and Higgs fields, there have been related in a satisfactory way. The reason for this is that H=U(1), via ones usiTig modified version of high dimensional Yang-Mills the imbedding into the gauge group (see above); acts also as weak theories. hypercharge group. This in turn fixea the charge assignments A particularly drastic change that will however not be for the complex Higgs doublet as well as for the leptons, and reported hero would be the introduction of Yang-Mill3 theories these two assignements can not be made right at the same time. over spaces with an11.commuting internal coordinates (cp. ^ \"2- j However, examples exist of ten dimensional theories that for references). reproduce the entire Weinberg-Salam theory in four dimensions. More conservatively, the first attempt to be described lajse for instance G/H = G(2)/SU(3), and R=SU(5) or Sp(8). This oeiow ,-nay be looked upon as an attempt to extend the structure of provides the right Higgs sector. For R = SU(5), one can choose a high dimensional Yang-Mills theory to that of an interacting the self-conjugate fermionic representation £ = J^ and eventually antisymmetric tensor theory. Part of the motivation here is to one obtains one lepton family with correct quantum numbers. minimize the number of internal coordinates (see below). Note that the lepton sector comes out wrong for f = 24, the In a following section I will describe a simple formal adjoint representation of SU(5). This spoils a possible super- argument that has raised the hope for a possible geometrical symmetric ahsatz. Quite a similar situation occurs for R = Sp(8), interpretation of ghosts. for which not the adjoint, _f =» 2i< °u* L = 1Z provides a satisfactory assignement. For a discussion of unified theories and further details 1. Higgs fields and geometry omitted here I refer to the original papers C^f.^J I also nott that a paper exists that is concerned particularly The gauge fields of a high dimensional Yang-Mills theory ( concerned rfith aupersyrametric LagrangianaO '3j . 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 fields with components (A H , A bF , A^ft * ) , 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 - 2. Ghosts and geometry The basic field will be introduced via a two form, A = A g E /*, E rather than an algebra-vaL ued one form. In this A The purpose of this section is to describe a simple argu- way all indices are treated on an equal footing. The coefficients ment (cp. the paper by Thierry-Mieg, {jj_ J and references therein) A^jj are components of an antisymmetric tensor. Since we want that indicates a possible way to a geometrical description of ghosts. gauge and Higgs fields only, we set A . = 0 for the time being. S ra and A- I* will be treated as invariant tensors of first and Starting point is again the gauge field A (x) ^ {x)dx . second rank under G. This implies conditions of the form whtch is now extended to a high dimensional object, 3*i>:/.j h , and 7). A * * - -k - ' A* * ^ Jt . $ d- *. respectively, with 9*- p ri-. It will be attempted to identify the coordinates y with the * r aABC We now wish to find an action density of the form F vertical coordinates of a fibre bundle (for a short description where the tensors FAB C are of the form F,AI3n_C = L.A,,,A BC, - L-,AB A.1 of fibre bundles, see sec. (W.fS.'f.) ) . (*?. n-) It 13/ important to realize that the would-be Higgs fields These relations are reminiscent of the BRS transformations of Ajft transform inhomogeneously as befits physical Higgs fields; the quantized Yang-Mills theories, with C s = C sdy i - 63 - - 6k - Or:p now sets sC - b by definition which we have the liberty VI. SUPERSYMKETRY 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 This chapter will contain some comments on sapersymetric well-known BRS and anit-BRS t.ransfo.rmation.-:,, (k,p. reference £?2.~] Yang-Mails1 theories and supergravity. The reason for brevity is and references therein.) not only lack of space but also that up to now not many results The formal argument displayed here seems to be the basis concerning supersymmetric theories which employ the geometrical for quite a few papers which appeared during the last few years structures described in this review are as yet available. and where the problem of a possible geometrical interpretation On the other hand, even though only the method of "ordinary of ghosts is attacked. dimensional reduction" has mostly been used up to now, super- The reason that not all is obvious and done with is that symmetric theories often find 9 natural that is particularly simple the above considerations are formal and closer inspection shows formulation in higher dimensional space-time. This can be seen that a proper mathematical setting is difficult to obtain. in the examples given below. In fact, a bvsic difficulty is hidden at the following point [toTJ. The would-be ghosts C, d^are, as one forms, elements A. Supersymmetric Yang-Milla theories of a finite dimensional Grassmann algebra, unlike the ghosts fields of the quantum Yang-Mills theory. It has therefore been argued Consider the action that a true geometrical theory of ghosts ought to be a theory involving an infinite dimensional Internal space. Otherwise, (S.I) Green's functions involving a sufficiently high but still finite where the Yang-Mills tensor F „ is defined as before, and number of ghosts would Tanish. D )S = d ) - fS*<- A \V. The indices s, t, v refer to the HoweVer, one may still hope that this impass is temporary gauge group R. Note that the Dirac spinor is in the adjoint re- and will be overcome in due time. presentation of R. The simple theory (fi, | ) turns out to be Indeed, even greater hopes than those outlined here are supersymmetric on the mass shell without addition of extra fields connected with these theories. In fact, the theories as given in D = >t, 6, and 10 dimensions, if the spinor is subject to a here may also be formulated for graded gauge groups. The ghosts Majorana or Weyl condition (0=4), a Weyl condition (D=6) or the asso ciated with odd generators are then bosons with the right Majorana and Weyl condition (D=10). Supersyminetry transformations are 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.) 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 invariantAe for curved - 6=5 - 66 - spac n one finds after discarding surface terms that the correspon- would-be Higg? fi«!ld, the fifth and sixth component of the Tang- condltion is now Hills field. The etctric and magnetic charge operators are expressed as surface integrals; the magnetic charge is that of the t'Hooft- = o . (4. The supersymmetry current for this model is where i..,N. Here and are spinor indices, € =-U , V^ =-V . U % + 'i^ffV^t; 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, which gives rise to the commutation relations (fc-5) fo r the It is suggestive to associate the central charges U and V, integrated charges, where now U and V are specified, to denote U V = V of the four (V±-= A- i ij i ^ dimensional N= 2 Theory with ttvp the surface integrals 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. ("1 )l£ V t C * % 1- 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 Clearly wo obtain contributions to U and V only if A and B do spontaneous symmetry breakdown. The latter is provided by the - 68 - - 67 - not vanish asymptotically. If indeed they do not vanish, we obtain t'Hooft's definition of electric and magnetic charge, respectively. If now as a first attempt, the internal space is assumed These observations have many exciting consequences beyond to be the seven - torus(S )7, then the zero mass sector is a concretization of the idea that electric and magnetic charge might obtained by ordinary dimensional reduction; no y-dependence is be manifestations of the fifth and sixth component of momentum retained. This procedure was employed for the first construction respectively. In particular it follows from { £,$") that for any of N=8 supergravity in four dimensions (_3(J " Note that the particle state in the theory, M ^ U2 + V2 (M being the mass), counting of states in the four dimensional theory would change in which is nothing but a quantum version of the Bogomolny bound (_2 0^ I, general once we are dealing with curved internal spaces, in particu- lar, the number of bosonic and fermionic states might not match For a further discussion of these topics I refer to the any longer. In such a situation one must face the perspective original papers [N>, m(7.O For supergravity, eleven dimensions is somewhat of a magical then give certain fields a dependence on the fifth coordinate number L^°'-l • In fact, for once supergravity in more than eleven which will characteristically be of the form exp(iy M). This dimensions would after dimensional reduction provide a fouT dimen- breaks supersymmetry, but only spontaneously* some fields will sional theory containing particles of spin higher than two (.'f£j. get masses and others not. This mechanism has been suggested by It is not explected that such theories would turn out to be con- Scherk and S.-.hwarz I'M j - 69 - - 70 - >t£hematically, the seven-sphere is a very interesting space. for the internal. Dirac operator may actually be obtained. In particular, it is the only parallelizable coset space (S = The Kaluza-Klein picture has also some bearing on spontaneous S0{8)/S0(7)} which is not a group space. Here, "parallelizability" breaking of supersymnstry (^^|2°?J. To see this recall that super- means that inclusion of a torsion term to the connection coefficients symmetry is spontaneously broken, if the vacuum is not left {the torsion coefficients being antisymmetric even in a holonmic invariant by some generators of supersymmetry, Q^/o'p*^- Q basis) the geometry of S can be modified in such a way that the ? Now under supersymmetry transformations, the Rarita-Schwinger field curvature tensor of S vanishes. Now torsion occurs naturally in ^ varies into i + ^ £. • Unbroken supersymraetry requires supergravity theories, and that torsion as it dccurs in supergravity N o w <«l!l tlo"> — O, or, in the form of a classical field equation, can be used for the purpose one has in mind here has been argued u C^E=Oi which has integrability condition [D^Of-l^ — 0 • by Destri, Orzalesi and Ros*i(_Hj. As first application recall that On coset spaces with nonabelian compact symmetry group, one does the very large cosmological constant in Kaluza-Klein theories has not expect solutions of this condition {j£°3J. 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- For the seven sphere (which has symmetry S0(8) and correspon- logical constant f^V, 'V?~7 . dingly should give S0(8) supergravity after dimensional reduction) the situation is as follows ["?1j : 1) no spontaneous breakdown A solution of the field equations of eleven dimensional of supersymmetry for the round seven-sphere without torsion; gravity which is essentially a modified version of the Freund- 2) breakdown of supersymmetry for the round and squashed seven- Rubin solution and which corresponds to the parallelized seven- sphere with torsion. sphere as internal space has been given recently by Englert • (_ C I J * This concludes a description of presently available results on the For this solution, the components , of the field "strength connection between Kaluza-Klein and supergravity ideas. Much more tensor F.MNPK are nonvanishing. From the four dimensional point F are work will certainly be produced in the near future. of view, the ^v>p just scalars; the Englert solution therefore involves scalars with nonvanishing vacuum expectation value. One Similarly, quantum properties of high dimensional theories will therefore expects a connection between the property of paralleli- (have to) be developed. This will happen on the basis of the J.izability and spontaneous symmetry breakdown. For more general requirement of "finiteness" rather than "renormalizability". arguments on this point compare £%/ ^ J • First indications how this might work have recently been given One can'squash' the seven-sphere (ohange the radius in one by Duff [^1^2/SfJ, cp. also [~U'l/Z90'3 • 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 - - 72 - APPENDIX A LAGRAJJGIAN OF ELEVEN DIMENSIONAL SUPERGRAVITY ACKNOWLEDGEM£NTS The 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.5) (A.6) y ~- R " ^ i<~^ r,. ut^ . The constants are c=2/(12)\ b=l/(12)2. - 73 - The Lagrangian (A.I) is invariant under APPENDIX C - COSET SPACES I 1. General coordinate transformations in 11 dimensions; M 2. Local SO(l,10) Lorentz transformations - Consider a Riemannian space with coordinates z and metric 3. N=l supersymmetry transformations, 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 t/4.io ) S'*M to - %"" (l) 1- frg^f*) (C.2) 4. Abelian gauge transformations, with (A-ll ) 5. Time reversal together with "*yyp" If for a generic vector with components v we define the covariant derivative (C.4) with the help of the Christoffel symbols APPENDIX B GENERALIZED DIRAC MATRICES MP r (C"5) one can rewrite (C.3) a* The generalized Clifford algebra, 7P^ I with D elements has one finite dimensional irreducible repre- 0 U UJ : i 4 i t (C.6) sentation only with dimension 2*- , where [_D/3lis the biggest For the covariant components G.M(i) one obtains the inverse integer smaller than or equal to to D/2. This is also the dimen- formula 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^ - Consider now coset spaces ^BK=G/H. We then have characteristi- r,'..." f1^ and Weyl spinora, \ (|i - (f * p \«, ., Similarly, cally: {l) Ciroup transforr.iations associated with elements of G charge conjugation can be defined and a Fieri rearrangement formu- can be expressed as coordinate transformations within G/H; la can be provenL^°j. (2) Any two points in B can be connected by a group transformation Generalized Dirac matrices are constructed t+eratively. ("transitivity"); (3) Group transformations leave T:n~e metric tensor Clearly, once a Clifford algebra with 2n, n€ IN, elements ia found, invariant, on account of (l) and (C.6) this implies the we also have one with 2n+l elements, since we sim ly can add | "Killing equations" to the set. A Clifford algebra with 2n+2 elements is given by where -j are the Pauli matrices. a c where the choice of indices reflects that in the present context, For more material on generalized Dirac matrices compare coset spaces will be internal spaces. for example \_1o - 76 - - 75 - which, using (C.10), can be expressed as As the ^ are associated with group transformations, we may write (cp. also appendix D) (c.i6) CC.9) .?-, (C.18) which states that the differential operators •! ~ ^J T< can serve as generators of G, Equation (C.17) states that the coordinate transformation (C.l'j) can be re-expr-eaBed as a gauge transformation, (C.18). (en) with ~i\h^ tne structure constants of G. The Killing fields also provide an invariant metric through the prescription APPENDIX. D - COSET SPACES II l 7^ "" 0 •*> v^x-,^ ( L(y') = g L(y) (D.I) Clearly this corresponds to local (i.e. x-dependent) group Assuming the local existence of the group valued function L(y) transformations on BK= G/H. Applying the general Cttrmula (C.3) with ^^=.(<\S^ one finds 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.15) -T (D.2) C\ 4- ^ - 77 - •There Q^ denote the generators associated with H. The parameters On coset spaces, the invariance of the metric (D.9) is to 0* specify h(y,g). Multiplication of (0.2) by L (y) and comparing be understood as left invariance (i.e. invariance under left terms up to first order gives translations). In general, a right invariant metric would be different (albeit constructed in a similar way). An exception . (DO) is the case where G/H is again a group space. This is a relation among Lie-algebra valued quantities. We may thus For the discussion of transformation properties it is write in particular convenient to introduce the algebra valued differential form A and introduce the adjoint representation U»'ifL)of G by •1%)= a -*^J * This provides a definition of the Killing fields in terms of From (D.ll) it follows in particular the basic groupY—function L(y), The commutation relation [^i, ~ia~\- *fjji ^V^ for the differential operators ^ t kj^^ follows from iterating (D.l). It may also be obtained by explicit On the other hand, from the general arguments of section (II.l) though lengthy and not quite straigtforward computation. we have From (D.5) one obtains by differentiating with £^ the (D.13) differential equation Comparing (D.ll) and (D.12), we obtain e*-(y') = e'y,*iy'), which expresses the forminvariance of the vielbein on coset spaces (D.7) under coordinate transformations-. This is equivalent to the which has as integrability condition etate-nent that the metric tensor (D.9) is forminvariant under coordinate transformations. All considerations above referred to left G-tranaUtions Above, we have introduced an invariant metric on coset on G/H. Similar formulae as those above hold for right translations< By their respective definitions, [X* 'XjJJ * ^ > where ift are the spaces, g^ (y) = K<;r(y)K* (y). Since for the matrices D.r(L) we have DDtr 1, we have also analogs of afj for right translations. For the particular case H = 1, the Z£ turn out to be given as £j - - •** 3^ . (D-9) - 80 - - 79 - with '( B A M wnich are defined to ae symmetric spaces of dimension K such that with Bc = KC dz the "connection 1-form". (For the definition they admit the maximum number of Killing fields, namely K(K+l)/2, compare chapter II.) In a similar way of and BH. n ft Vf one can show that the curvature tensor takes the simple form we introduce the curvature 2-form B = 'SiNA dz A dz which satisfies 7 B B C B (E.6) RA = 5B +BA ABC . (E.2) i • Equations (E.I) and (E.2) are referred to as Cartan's first and Above were given the connection and curvature assignement3 second identity respectively. They are particularly useful on for vanishing torsion. One may also choose B such that the torsion Rlemannian spaces, i.e., when the torsion vanishes. In this case, 2-form becomes T Then (E.I) determines the connection 1-forms by simple differentiation and of the basis 1-forms, whereas (E.2) provides the elements of the curvature tensor. If G/H=G, H=l, then this assignement gives canishing curvature. (Group spaces are "parallelisable, cp. chapter VI.) For symmetric As an application let us consider the case of coset spaces. spaces the two assignements are identical (cp. Eqn. (£.4)). 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Tataru-Mikai, "Grand unification via Kaluza-Klein approach", l67' P- Roman and J. Haavisto, "Higgs fields, curved space and Lett. Nuov. Cim. 21 551 (1982) hadron structure", Int. J. theor. Phys. JUS, 915 (1977) 186. J.G. Taylor, "Electroweak theory in SO(2/1)1 168. V. deSabato (Ed.), "Unified field theories of more than four Phys. Lett. j^JL 331 (1979) dimensions including exact solutions", Froc, 6th. Int. course Erice, Trapani, Sicily, 20 May- 1 June 1982, World Scient. - 92 - Publ., Singapore, scheduled to appear Dec. 1982 187- J.G. Taylor, "Gauging SU(n/m)", Phys. Lett. 84B 79 (1979) 207 E. Witten, "Instability of the Kaluza-Klein vacuum", 188. J.Q.Taylor, "Superunification in SU(5/l)", Nucl. Phys. B19^ - 93 - CURRENT ICTP PliBLTC/G . IC/82/171 H.B. CKALiSLj a:-, 5 A..H. ICKL'.,r:in _ ?:wirds a -.I^^iCiive L;ii-i?ry IMT.REP.* :or lie II: I - i, •:, rc-t t::'ip^;;a^f: [.ylirid -.pcrofali. IC/82/150 I.M. REDA and H. BAI-ICliRT - VJI-^S rc-in,; aLil±-;y and kinetics of IC/82/172 J. CHELA FLOEKo -.id !i.B. .JHASStB - iovrar-Ic >, 'xi^v •hei.a iv« theory INT.REP.* formation of metEl^Sisc::.. INT.REP.* for He 11: - A -emptr^t1,:>•-*-dec-en:- _-nt fi-iid-1-heoretic appi-unc-h. IC/82/151 I.M. REDA and A. WAGKNEKI - .^••si:! .: '::'i. :.noi': J.n highly IC/82/173 C.A. '«JID - i'"&crt.ricaJ., il ana riu".-i ti-na forma: : T beh .?,vi our INT.REP.* disordered solids. IMT.REP.* of •lino rr.houa ^oli.:.,, - A:-, :ucti;ju ;J r;ir.d.<;.T .ntulr;. IC/82/152 M.Y. EL-AGHRT - Cn the ki>..r. •• ".: .•.:.•!•;•• •..•: • •.ir-irKjtric re%n:\m\ee ih IC/82/1TH K. PA.C, :. PAliAGiOIAKOro1 Uf; mid Q. OUAFT - ;.''«--:et L: no- poiv relativiotic plasma, and haryo. -; if-;.!;-;/, IC/SS/153 LI XIHZHOU, WANG KEL1N a.:-: •'.liA IC/82/17^ M. :Z-:R - ?:jrniij;i ":::::s _•).-io.; -nd L -.,•-•,.. :-,;• ••••::..; •-. ,:'i -'.lor. IHT.REP.* in the preoii model. IC/82/176 S. CLCOITI T. GIKA)'.: ..[,:,U - :•;;.. 0:1 :j\,o.:i:.: .;..: ;.:-n;-!-:-..-;ei; ic/82/15"* S.M. PI2TRUSZK0 - Aaovphc",*., .; lattice l INT.REP.* mixture. IC/82/177 S. AAJPOCL - !.,-,: F.-±.; ;-••;•.:_• :;;...;-ity [•• -.;V;>-M i xpand:^ gauge IC/82/155 J. ZIELINSKI - Seri.L-phet:,..:iu :iv ; IMT.REF.* discontinuous change of •/••ile:i : IC/82/178 rL. SAEED - .[dont L ;'J .• :' ... .••:as(, •/ nl -.u:'fi •: H-.KL .:• .r.dit i 1 : IC/82/156 H.B. GHASSIB and S. CUA'LV.^J.: '!'r. 1 •.••£!£ i ",y L lac ; ^:it iony In dilute IMT.RE?,* r?.1'J. iior.-.-ogat" •••••:. s ritiona. ,.rit:-'-.>:i. He-3He thin filiriii, IC/82/179 K. SAEED - He ii1 ;.•.•-•! ._.:X].L,S ui.--r;n^Liari-jion o:"1 polynooi: • •• tna.tr:. 1 1 IMT.REP.* IC/82/157 G,B. GELMINI, 3, KUKC^Jvj/ ,uio. , ..I.-. iPA - Jueci aature like iiairib ;- Goldstone bosons? IC/82/lSQ A.O.i. A^iMAL7.1 - ":U-U-K .,pnrn..-.••:-. -0 -.•icilli ' .• -otij :-•:-. ;>'^ of ^adrfi.ia 1 structure ~ ii. IC/82/158 AjG.F. OBADA and M.il. MAHi\, .. - t;Lt.vi-.. a:.-i :-iiAgnetic aipole cent ributicnf IBT.BUt'.* to a theory of radiation reaction 1 L!.:O.L: •.-.-, :ir,nm Eelf-er.orjr/; an IC/82/18] M. niJEGU1:1 ',nd U.K. 'AK - .:rv,fy;n and baryoniiun i-i i':1"!:,. operator reaction field. IC/32/182 [j.w. :;;iA'iA - •.-•le rul.- -u rarticlu physics in uc-:.:r. a,: ;y and galuctic IC/82/159 R. KUKARAVADIVEL- Ther:u;;ly!;:mic :JI\;I .:.•!, U?3 of llq.aid :-.et;;l.J. a.j Li^j-jomj", INI.HEP.* IC/82/16J li.F.I. WIV..!J - On -.i.f r'jt,i.-j .::' l.i. .•••LiEaa for the D and D'' meson. IC/62/l60 K.H. WOJCIECHOWSKI, P. PI2aAi;::i.- ^:'1 •;. .-.^.LIXKI - An iniitQ'oility in IC/82/181* J. VJKIl-T'LVrCT - '.'cr.j; • ..-i--:- ;r-i-;i!y -:;d composite 3i!ji:.r :-r- v-i.- .y, a hard-diso :;ysteTti iri a n:-u'rov bo, . i 1 1 IC/02/185 J. Jjl-urr.L _ r;. :-s .-'Marks :.:; the 13,; or the etfecliv" tr.i.'diui IC/82/161 J. CHELA FL0RE6 - Abrupt oaseL oi ,.^^1'.;^ VI-jJation. IHT.REP,* approx-. nation i'or fun rusiit i.vity ^-aleulationa in j-iq:;iJ :c:,le n.i%?.lj, IC/82/162 J. CHELA FLOEES and V. VARLLA - VA .ow,- -r^vii.y: ;m approiLjh to its IC/82/186 source. R.P. HAiCJitj; - A Th-ory f-jr Lh^- orientationni ordcrini; Ln neiaatic INT.REP.* liqi.iid:; :i_d for r.nt:- • i.xse di^^rajn of the rvi-^tic-i^otopi .: transition. IC/82/163 S.A.H. ABOU STEIT - Additional evidur.ee for the existence of the C 1 IC/82/187 R. '£: QUi:iiO and J.i . :.'ARCGT"'" - Tli•" thoriaodyn.'imics of ::up.'--.-onli-tijia IHT.EEP.* nucleus as three a-particles and not as Be_ , + a. iilT.REP.* lanthanun. IC/82/16U LC/82/1S8 k.-'.W. OBADA and M.::. '-L'lir'.AiJ - l^-^pcns" and normal n.odt-.: of ?. -yst-arc IHT.REF.* M- EOVERE, M.P. TOSI and H.H. !t\HCH - Freezing theory of RbCl. Ii-J'l1. ri^'r , * in the electric '-n;i n^-'^nij1 ic-dip.;"' c approximation . IC/82/165 C.A. MAJID - Radial distributiosi analysis d' :^norphoua carbon. IC/82/189 M.K. KOLEVA and I.'i. K' JTAiJiliOV - Absence if [[a.11 t'nv.'CC due to INT.REP.* INT.REP.* phonon assiated hopping \;i .iijot\ier=':a syKtrass. P. BALLOHE, Q. SEJiATOKE iinrl M.P. 'I'C'oI - i:u face doniiiLy i-^oi'Lle and L. JURCZYi:ZYH and M. :iT.-:,V.l,V:;K.1 - Surfic;. Jtal.es Ln tha preuwad of IC/82/166 surface tension of the orie-coirjponenl classical pla^jnu. IC/82/190 INT.REP.* absorbed riLom£ - 1'1 ; 'ipjji^ i::Lat:ion of T-.ne small L-adii.i:; j-.c^.^rir.iais. G, KAMIENIARZ - Zero-teniperatur" :'cnorj.alisation of the 2D wamrverie IC/82/191 M.S. AJLAGLOU^LI, Y,.\\. -..^Wlk. V.a. GARaaVi'iiilSHVTLI, A.M. ;::U.;AD::E, IC/82/167 Ising model. G.O. KUMTAG1IVIL1 and •'•.:.I'. L'CL'iJRIA - Perhaps a new m.if :•.; scaling IC/8E/168 G. KAMIEIIIARZ, L, KOWALEWSKI am: «. FiECHQCKI - The spin 3 quantum variable i"or de^eribing th..1 low- !i:vL ^-i^h-a.,, oroceo^cLi'.' INT.SEP.* Ising model al 1 ^ il. IC/82/192 A.f>. 0K3-i:L-[iAE •-.! 1 M.:-;. :•: -;-;HAZ:,T - L'h.i integral r-ipreL,. n-ition of 1 IC/83/169 J. GOEECKI and u. POP (ULAWSKI - On the appLloability -if r.-..;irly free- IMT.REP.* the expoiicji-itiall. mv^x ; .i.icf io:.;; <•]' iiifinitely many variables on electron model Cor r"sisttvity onl'.:ui a~ tons Ln liquid metals. Hilhert upacea. IC/82/170 K.F. WOJCIECHOWaKt - Trial charge dr-nsity p-O.files at tin: metallic IC/82/193 LI XINZHOU, MAMG KELiH and V.'.iAnG •! 1 .Ml'.'.'J - J-K T, -J .!. L t.oni in UHS surface and work function calculations. nupercyctnetry i.Ivory. IC/82/19U W. KECKLENEliH'.] and L. MIzr.Ai.'iir - ::ur:'.-. :s terms and dual formulations of gauge theorieo, THESE PREPRINTS AflE AVAILABLE FROM THE PUBLICATIONS OFFICE, ICTP, P.O. Box 5B6, IC/82/195 I. ADAWI - lnt,-..-raet:.o:i ••>.' .1 i.os',,• ii: and magnetic cnar^es - [i. I-3U100 TRIESTE, ITALY, IC/82/196 I. AC.Vn1: ti::d V. :• . GOD'.-dli - .:.. .-:ni71-M:r of Lhe electron aensity near an * (Limited distribution). iinpuriLy jit!"; exohan<^_- anil corre! ttion. IC/82/197 PREM P. SRIVASTAVA - Gauj;u and non-gauge curvature tensor copies. IC/lZ/222 BRIAH W. LIN1I - Order ally corrections to the parity-viol it ing IC/82/198 R.P. HAZOUME - Orientaticnal ordering in choiesterlcs and smectics, + INT.REP.* electron-quark potential in the Weinterg-Salam theory: c;\•*i .' violatiori in one-electrori atoifls, IC/82/199 S. RANDJBAR-DAEMI - On the uniqueness of the 3U(-2) instanton. IC/82/223 G.A. CHRISTOS - Note on the n ,.._,.- dependence of < 5q > iTor. o.drai IC/82/200 M. TQMAK - On the photo-ionization of impurity centres in semi- INT REP * QU conductors. perturbation theory. IC/82/201 0. BALCIQIiLU - Interaction between lonization and gravity waves in the IC/82/Z2U A.M. SKULIMOWSKI - On the optimal exploitation of csiLr. L---V:IOU^ IHT.RE?.* upper atmosphere. IMT.REP,* atmospheres. IC/S2/225 A. KOWALEWSKI - The necessary and sufficient condition; o.;' * -:e IC/82/202 S.A. BAHAM - Perturbation of an exact strong gravity solution, IHT.REP.* optimality for hyperbolic systems with non-differsntiahie per•'onranc functional. IC/8S/?O3 H.N. BHATTAfiAI - Join decomposition of certain spaces. IC/82/226 S. GUNAY - Transformation methods for constrained jpt iml :,at i on. INT.REP.* M. CARMELI - Exter.t:ion of the principle of miniraal coupling to particles 1C/82/20U with magnetic moments, IC/82/227 S. TAMGMAHEE - Inherent errors in the methods of appro;: L':,at ...::• ; IHT.REP.* a case of two point singular perturbation problem. J.A, de AZCARRAGA and J. LUKIERSKI - Supersymmetric particles in H = 2 IC/82/205 superspuce: phase space variables and hamilton dynamics. IC/82/229 J. ETRATHDEE - Symmetry in Kaluza-bilein theory. IMT.REP.* TC/B2/2O6 K.G, AKDENIZ, A, IIAC TNL1Y A..-J and J. KALAYCI - Stability of merons in INT.REP.* gravitational models. IC/82/22O. S.B. KHADKIKAR and S.K. GUPTA - Magnetic tnomentc •-:•- .. U:,n". a^i-y.. ••. in harmonic model. [C/82/207 H.S, CRAIGIE - Spin physics and inclusive processes at short distances. ERRATA IC/82/230 M.A. NAMAZIE, ABDUS SALAM and J. STRATHDEE - Finitcnor;.. :f lrak?n N = k super Yang-Mills theory, S. RASDJBAE-DAEMI, AEDU3 SALAM and J. STRATHDEE - Spontaneous IC/82/20a compactification in six-dimensional Einstein-Maxwell 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,. J. FISCHER, P. JAKES and M. HOVAK - High-energy pp and pp scattering ' IC/8P/209 and the model of geometric scaling. 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.. A.M. HAEUH ar RA3HID - On the electromagnetic polarizabilities of the IC/82/210 nucleon. IC/82/233 A. SMAILAGIC - Superconformal current multiplet. INT.REP.* IC/82/211 CHR.V. CHRISTOV, l.J. PE'i'HOV and l.i. DELCHSV - A quasimolecular IUT.REP.* treatment of light-particle emission in incomplete-fusion reactions. IC/82/234 E.E. RADESCU - On Van der Waals-like forces in spontaneously broker, supersymmetries. tC/82/212 K.G. AKDEHIZ and A. SMAILAGIC - Merons in a generally covariant model INT.REP.* with Gursey term. IC/82/235 G, KAMIEKIARZ - Random transverse Ising model in two dimvii:;ior..;. IC/82/213 A.R. PEASAKKA - Equations or motion for 3. radiating charged particle INT.REP,* in electromagnetic fieldr, on curved 3pacc-time. :C/82/2lU A. CHATTERJEH and S.K. GUPTA - A::g,ulur dLstributions in pre-equilibrium reactions. iC/82/215 ABDUS SALAM - Physics with 1C0-L0Q0 TeV accelerators.. IC/82/216 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 tacklund transformations to hierarchy gt-neralin^: operators. 1C/82/21U BO-YU iiOU - The gyro-electric racLo of Gupersynunetrical monopole determined by its structure. ic/82/219 INT.REP.* V.C.K. KAKAJJF, - Ionospheric at: intillation observations. IC/S2/220 INT.REP.* R.D. BAETA - Steady state creep in •juartz. G.C. GILIliABDI, A. RIMINI and 7. WiiBER - Valve preserving quantum measurements: impossibility theorems and lower hounds for the distortion.