REFERENCEIC/81/95

INTERNATIONAL CENTRE FORTHEORETICAL PHYSICS

INTERNATIONALATOMIC ENERGY

AGENCY

UNITED NATIONSEDUCATIONAL,

SCIENTIFICAND CULTURALORGANIZATION

C O N F E R E N C E ON

DIFFERENTIAL GEOMETRIC METHODS IN THEORETICAL PHYSICS

(30 June - 3 J u l y 1981)

1981 MIRAMARE-TRIESTE

IC/81/95

International Atomic Energy Agency

and

United Hations Educat ional S c i e n t i f i c and Cu l tu ra l Organizat ion

IMTERKATIONAL CENTRE FOE THEORETICAL PHYSICS

C O N F E R E N C E OH

DIFFERENTIAL GEOMETRIC METHODS HJ THEORETICAL PHYSICS

(30 June - 3 Ju ly 198l)

E X T E N D E D A B S T R A C T S

(Edited Tiy G. Denardo and H.D. Doebner)

MIRAMARE - TRIESTE

July

The Conference on Differential Geometric Methods in Theoretical

Physics took place from 30 June to 3 July 1981 at the International Centre

for Theoretical Physics, Trieste.

Besides the International Centre for Theoretical Physics, the

International School for Advanced Studies of the University of Trieste

and the Technical University of Clausthal (Federal Republic of Germany)

have given their contribution to the success of the conference. The

conference consisted of lectures ontl seminars and these vill be published

in the proceedings. Herewith ve want to give an outline of the subjects

treated. This collection of informal summaries of the lectures and seminars

is published just with the aim of giving an immediate account of the

meeting and this is realized thanks to the kindness of the speakers vho

all agreed to deliver their manuscript before the end of the conference.

G. Denardo and H.D. Doebner

i is r a u > A i* t it x A i. A

UNITEU NA I UIN» LiUIUAHUB*!. 1 i » U CUMUKA1. <JUli AMI/.ATIOf

INTEBNATIONAL SCHOOL FOB ACOU4CBD STUDIES OV TBIESTE

nnXBNATKMALCSarraBFQRTHEXlRETEAL PHTOK8O* TRIESTE

TECHNICAL UNIVERSITY OF CLAU8THAL

CONFERaiCB OH OIFFIKBJTIAL GHDMETRIG METHODS IV THHDHiTIGAI, H1YSIC3

30 June - 3 July 1981

P fl O G R A M M E

l^SSday. ^0 June 19dl

Otairman 1 K. Blauler

9.13 *-"' B.O. Doebner I S T R O D U C T I O H

9.30 a.m. T. Begge Action Principle for Unified Theoriesin the Group Manifold Approach

Stability of Gravity Theories

Supergravity and Unitary FieldTheories

Relativity of Hotating Systems

Unified Gauge Theories IncludingGravitation

Gauge Theories and Principle ofEquivalence

T. Begge

(University of Turin, Italy)

10.30 ».D. C O F F E E B B E i l

11.00 a.m. 3 . Deoer(CERir, Siiitierland)

12.00 noon S. Perrara

(CBOr, Switzerland)

Oiairman t H.H. Fatry

2.30 p^n. K. Canwli(Ben Curion University, I srae l )

3*30 f d . J . K>ff»t(Univ. of Toronto, Canada)

4.30 p^a. C. Or»ale«i(UniT»r«ity of Pama, I t a l y )

5.30 p .« . t I C I F I I O l

Hadnasilav. 1 Jnl-v 1<J81

Chairman j H.D. Jtoebner

95 I.B. Segal

( M . I . T . , Carabridge, U . S . A . )

1 0 . 3 0 a . n . C O F F E E B I I 1 I

11.00 a.a. s . Stemberg

(Harvard University, U.S.A.)

12.00 neon B. Kbstant

(M.I.T., Canbridge, U.S.A.)

Chairman t 3 . Sternberg2.30 p^a. C.T.J. Sodson

(University of Lancaster, U.K.

3.30 priB. S.I. Andersson(Claaathal, F.R.O.)

4.30 p.m. L. 0'aaifeartaigh

(ajblin loot. Advanced Studies)

3*30 P^». 3 seminars

Particle Theory in Alternative Cosmos

Supergravityt idiys and horn

Coadjoint orbits and a new synbolcalculus for line bundles

Second Order Tangent Structures forSpacetime

KMhler Structures for Solution Varie-ties of PBeudodifferential Operators

Twisted Axial Symaetry of SeparatedMonopoles

84100 TRIESTE <ITALY) - p. o. B. ua - M:OABLEs CBMTIUTOU • TILIX 480393

-ii-

i D * COBTIEBA 11 -TELEPHONES: 3343* l / l ««s sLIX 480393 1

Thursday. 2 July 1981

Chairman ; I.E. Segal

9.15 a.m. P. Sudinich On Conformal Splnor Geometry(3.I.S.S.A., Trieste, Italy)

10.30 a.m. C O P P E E BREAK11.00 a.m. a. Parian Gravity at Short Distance

(University of 0!ries1;e, Italy)12.00 noon. M. Duff Quantum Differential Forms

(Imperial College, U.K.)Chairman : C.T.J. Dodson2.30 p.m. P. Hajicek

(University of Bern, Switzerland)3.30 p.m. C. Ishan

(imperial College, U.K.)4.30 p.m. 3 seminars3.00 p.m. C O N C E R T

Friday. 3 July 1981

Chairman t 0. Denardo

9.15 a.m. A. Trautman(University of Warsaw, Poland)

Quantum Theory of Black-Holea

Quantum Field Theory and SpatialTopology

10.15 a.m. C 0 F F E BREAK10.30 a.m. Y. He'eman

(Tel' Aviv University, Israel)

Hon-ahelian charges and timedependent fields

Progress in Supergravity

The Geometrical Meaning of the12.00 noon W. Thirring(University of Vienna, Austria) Nailer and Scattering Transformations

Chairmen 1 E. Binz (A.)X. Canneli (B)

2.30 p.m. 2 parallel seminar sessions8.00 p.m. F A R E W E L L P A R T Y

SEMINAR SPEAKERS

Wednesday. 1 July 1981

A.O. Barut(university of Colorado, U.S.A.)

P. Fasemaan(University of Clausthal, F.R.G.)

U.K. Sen(University of Toronto, Canada)

Group Action on Homogeneous Spacesand Solution of Nonlinear DynamicalSystemsGeometry of Gauge Theories, Secondquantization of Hon-Ahelian GaugeTheoriesTang-Hills gauge and the gravitationalgauge

Thursday. 2 July 1981Q> Knost attar(Imperial College, U.K.)X. Kartelliai(University of Milan, Italy)3« Unwin(Univ. of Newcastle upon Tyne, U.K.)

Friday. 3 July 1981(A) Chairman : E. Binz

2.30 p.m. H. Urbantke(University of Vienna, Austria)

3.00 p.m. I.S. Anandan(University of California, U.S.A.)

3.30 p.m. F.J. Bloore(University of Liverpool, U.K.)

4.00 p.m. 3.R. Komy(Riyadh University, Saudi Arabia)

4.30 p.m. L. Halpern(Florida State University, U.S.A.)

(jB) Chairman t X. Carmeli

2.30 p.m. S. Paneita(University of Clausthal, F.R.G.)

3.00 p.m. J.F. Fommaret(Ecole Hationale des Fonts et

Chaussees, France)

3.30 p.m. T. Palev(Inst. for Unclear Research,

Bulgaria)4.00 p.m* P. Zizsi

(S.I.S.S.A., Trieste, Italy)4.30 p.m. J.K. Lasry

(University of Paris-IX-Sauphine,France)

Canonical Vacuum Structure of Tang-Kills Theories on Arbitrary 3-Spacea

quantum Gravity on Kegge Skeleton

Applications of Twisted Fields

A Quasi Metric and a DifferentialClassification Associated with SU(2)Tang-Kills FieldsHolonomy Groups in Gravity and GaugeFieldsSU(n) Bundles over Configuration Spaceof 3 Identical ParticlesStudies of First Cohomology Groups forLie Groups and Classification ofSoli-ton Sectors in 2 and 3 DimensionalField TheoriesOn Group Covariance and Spin inGravitational Theory

Invariant (Jpantisation of Time-Dependent Have EtruationsNon-Linear Differential Sequencesin Gauge Theory

I&namical Quantization

Topological Charges in SupersymmetricTang-Kills Theories

Periodic Solutions for HaailtonianSystem

- " • - • - • . ->J* i r ..„» li.

C O N T E N T S

J . AHANDAH

S . I . ANDEKSSON

A.O. BAHUT

F . J . BUX)RE

P . KJDIKICHP . FURLAN

M. CAKMELI

S . EESER

C.T .J . DODSON

H.D. n)EBTEHF . B . P1SQUHK

M.J. IHJFP

S. FERRAHA

G. fUKLAHS . OTHHIV. d e ALFAfO

P . HAJI&K

L. HALFHUt

C . J . ISBAM

S.R. mm

B. KDSTAHT

G. KUH3TATTER

J.K. USBT

H. MABTHJLINI

J.M. HOPFAT

I . ITE'EHAM

L* O'HAIFEABEAIGH

Holonoay groups in gravity and gauge fields 4

Kahler structures on "the solution varieties ofpseudodifferential operators 5

Solution of nonlinear dynamical systena by groupaction on homogeneous spaces 6

SU(n) bundles over the configuration space of 3identical particles i i S ' 7

Kg spinor geometry 3

Relativity of r o t t i n g systems 10

Stabil ity of gravity theories 11

Second order tangent structures for spacetime 13

Algebraic gauge quantum f ie ld theory on generalized

Kaiuza-ELein spaces 14

Quantum differential forms 15

Ultraviolet properties of extended supergravities 16

Some remarks about quantum theory of gravity 17

Quantum theory of black holes 18

On group covariance and spin motion in gravitationaltheory 19

Quantum f ie ld theory and spatial theory 20

Studies of f i rs t cohonology groups for Lie groupsand classif ication of soliton sectors in 2 and 3dimensional f ield theories 21

Coadjoint orbits and a new symbol calculuB for l ine

bundles 22

How spatial topology affects the Yang-Hills vacuua 25

Periodic solutions for Hamiltonian system 26

Quantum gravity on a Regge skeleton 26

Unified gauge theories including gravitation 30

Supergravityt whys and hows 32

Twisted axial symmetry of separated monopoles 34

C.A.. Q8ZALE5I Gauge f ie ld theories and the equivalence

principle

T.D. PALEV Dynamical quantization

S.M. PAHEIT2 Invariant quantization of time-dependent wave

equations

J .F . PDMMARET Non-linear d i f ferent ia l sequences in gauge theory

T. RB3GE Action principle for unified theories in the groupmanifold approach

I . E . SEGAL Partic le theory on alternative cosmos2.K. SOI Yang-Kills gauge and the gravitational gauge

3 . STERHBHE Symplectic analogies

H. 1HISHIK0 The geometrical meaning of the KBller- andscattering-transformations

A. TRAUTKAH Non-abelian charges and time-dependent gaugefields

S.D. ONKDI Applications of twisted f ields

H . URBAHTKE A quasi-metric and a differential classificationassociated with SU(2) Yang-Mills fieldB

P. ZIZZI Topological charges in supersymmetric Tang-Hillstheories

35

36

37

33

39

40

41

42

43

44

47

48

49

HOLOSOKY GBDOT3 IN GRA.Vm.AHD GAUGE Y1SJB

Jeeva Ananrtan

Department of Mathematics, University of California, U.S.A.

A uaif ied approach t o gravity and gauge f i e ld* , i n which the

feolonomy group plays a fundamental mathematical ro le and quantum i n t e r -

ference provided a physical motivation ia suggested. I t i s shown that the

holonomy map from the loop space, at any given reference point, t o the

entire symmetry group determines the gravitational and gauge f i e l d

connections up to gauge transformations [ l ] . The metric, at each point,

i a determined by the flaffinrir operator of the loca l Poincar* group £ 2 ] .

These ideas naturally lead t o a recent ly proposed geometry on the

group manifold [ 3 , 2 ] . A unified c la s s i f i ca t i on of gravity and gauge

f i e l d s in terms of the holonony group, which i s a subgroup of the direct

product of the fbincare1 group and the gauge group, i s proposed, l

d i s t inc t ion i s made between the "c lass ica l" and "quantum1* holonomy

groups. I t ia shown, using these ideast that Newtonian gravity nay be

regarded as a gauge f i e l d corresponding t o the group of Gal i l e i boosts .

[1] 3 . KobayaBhi, C.fi. 23_8, 318 (1954); J. Anandan, Int. J. Bu Phys.12, 537 (1980).

[2] J. Anandan, Found, of Physics. 10, 601 (I960).

[3] T. He'eoan and 1. Regjje, Fhys. Lett. ?[£, 54 )

KAHLIS STSUCTDHSS OH THE SOLUTIJH VARIETIES

OF PSffittODIPFfflHrriiL OPERATORS

S . I . AndersBon

University of CLauethal, Federal Bepublic of Germany

KBhler structures are constructed for a c la s s of non-linear,

l inear isat ion stable (paaudo-) d i f ferent ia l operators. Locally ( in the

tangent bundle) we attach connections to the l inearized operators with

• o s t of the standard properties of connections. In particular, these

connections are d i f ferent ia l operators of order ^(order of original

operator) - 1 . from properties of these connections one can ( l o c a l l y in

the tangent bundle) construct almost KMhler structures. Integrabi l i ty

depends on non-linearity-type and so lvab i l i t y properties . In th i s way

information of a priori type for the solut ion variety of the non-linear

operator i s obtained. Compared with our ear l i er construction using

Fourier integral operators, t h i s construction has the advantage of being

exact (not only "exact modulo (?*•) and t o extend t o other categories

than the 0°° one.

souraros OF NQNLIBE&R matxica, SYSTEMS BY QBOOFACTIOS on BOUDGHTTOUS SPACES

A.O. BarutUniversity of Colorado, Boulder, Colorado 80309, USA.

nonlinear dynamical systems («•£• Gause-Lotke-Volterra eqs.)exhibit as a function of parameters widely different qualitative behavioursuch as periodic limit cyolea, non-periodic oscillations, chaos, strangeattractors. He show that such systems are exactly soluble for specialsyametries of the coefficients by nan-linear realizations of groups onhomogeneous spaces* These special values fora boundaries between regularand irregular regions. Contents:

I* Introduction.U i linear action of a, group C and linear dynamical system.XII. Konliaear action of C and nonlinear dynanical systems,

Protective Case,

Conforaal Case >0(11 + 1) oa S1 .

IT* Honlinew Superposition Principle.

7 . Applications and Examples.

3 U ( n ) BDKDLES OVER THE OOMPIGUfiiTIOH SPACE

OF 3 IDHITIC1L PABTICKBS B) ffi3

F.J. BLoorethiiversity of Liverpool, U.K.

1 spectral sequence calculation gives the following oohonolgy forthe configuration space C^R3) s {R3 x it3 x B^MVs, of 3 identical

•l J J

particles on tri

2 0 1 2

C,, Z) Z 0 Z_

So there are just three principal SU(n) bundles over i - 0, 1,Modulo 3* It i s convenient to regard elements of C, as triangles,(possibly colinearl) C~ nay be retracted to the stthcomplex WuD, whereB ia the 3-gpace of equilateral triangles of side 1, centroid 0 B},and V i s the 4-epace of isosceles triangles of bose 1, centroid 0 andheight h with 0 ^ h <Jl/2. Then 9W - 3B; U generates H,(C,, Z) - Z

J 7 J A JThe three bundles

given maps f±

F. are pull backs of the universal bundle, (C , , Z)^ 7 A by

Kg SPIHOfi GB3HEFKI

P. BodiBich and. P. Furlan

International School for Advanced Studies

Trieste, Italy

In attempt of "understanding" internal symmetry in geometrical terms

i t presented,!)? using the Cartan definition of pure spinora as equivalent to

the sum of two polarised, isotropic seai hyper planes in cooplei Swlidean

spaces. These seai-plaaes and their interaaotions build up "The spinor

geometry of f lat space" • By this we man the geometrical frame which

•ay be deftpad in the space starting from only one pure spinor of the space.

•r* spinor geoajatry consists of two iootropic seai-planes Jl^ t £ ^

and their intersection t an isotropio four vector \L . The isotropy

eqaation «ay be expressed as

E.S-0

Pron conforms! spinor geoaetry we also obtain extended Poincare

snpei'Bjiaietry or confonnal supersjnsBttry<

Spinor geoaetry say provide an alternative method to dimensional

reduction to descend to Kinkowski space.

The equation defining the apinor may be identified as the spinor

Weyl equation* fa • 0 pa • 0 from which Maxwell's equation for f^t and

tuv Hay formally be obtained with the condition that f^y (photon) i s not

interpreted aa a superposition of apinora (neutrinos).

Superayflmetry Ibinoare equation may also be formally obtained frost

" spinor geooetry but not its physical interpretation since this geoaetry

contains only that one of the light cone (v^ v/*1 - 0 ) .

To go out of it one needs U. - or coofaraal spinor geometry. It

is shown that nassleaa Diroo equations nay not be "the confoimal apinor

equation" for which the 0(4,2) covariant

is proposed. It has solutions on the cone2 0 of the form

where ^ ( x ) and l^-(x) are canonical Birac spinors (both of dimension

' ). Furthermore a parameter m having the dimensions of a mass appears

which spontaneously breaks confornal (dilatation) covariance. The resulting-

uasaive DLrac equations are Poiaoare covsriant and present an internal 30(2)

sjroetry represented by the two inversions and their product (coomuting with

the Poincare algebra).

••••*+

RELATIVITY OF HOTATIBtt SYSTEMS

H. Carneli

Center for Theoretical Pbysica, Ben Our ion University of the Bogev,Beer Shera 84105, Israel

The special theory of relativity i s based on the assumption thatthe speed of light c i s constant which i s independent of the state of(linear) notion of the source of light or that of the measuring apparatus.Lavariance of the expression o t - x under inertial frame (in which aparticle will have a constant velocity Khan the total forces acting on i tvanish) transformations then leads to the Lorentz transformation and to theLorentz group. The concept of the (constant) MgnTaT* aoMentum (aPin) ofthe photon appears in the representations theory of the Loreivtz group.' . In this talk we explore the constancy of the angular momentum (spin)of the free photon (and of other "fundamental" particles) and assume thati t i s a constant 'fi which i s independent of the state of rotational motionof the source of light or that of the measuring apparatus. Spinning framesare subsequently defined as those frames in which a particle will have aconstant angular momentum when the total torques acting on i t vanish.Invariance under spinning frame transformations then leads to a tranaforaatioo6f the LorentE form but where Planck's constant -* replaces the speed oflight c and angular momentum replaces the velocity* Fbr example i f onechooses as "coordinates" the energy £ and the "angular velocity* o>,

2 2 2invariance of the expression E --fte* under spinning frame transform-ations -then leads to the transformation

where T - l / ( l - j 2 /n 2 ) a n d i i s * "specific angnlar momentum"defined by j «••&&)/£ t with j<-n, in analogy to the Lorent* transform-

ation

where now V i s given by Y - l / ( l - T^/O2)

v - c p/fe , with p as the l inear momentum and v < c .

and the velocity

STABILITY OF GBAVITI THEORIES

S . Deser

CEBS - Geneva, Switzerland

We discuss stability properties of gravitation ia presence of aoosmological constant -A. of either sign, corresponding to inaTi msi symmetrysolutions of the aati-De Sitter 0(3,2), or of De Sitter 0(4.1) typ«.f irs t , we must define energy of perturbations or solutions with respect tothese backgrounds (and which tend asymptotically to them). The results aredifferent in the two cases. Par the 0(3,2} model, one may derive directly,from the anti-De Sitter version of supergravity [ l ] , that the translationgenerator along the time-like kill ing direction defined by the backgroundis always positive. This result follows as in -A- - 0 supergravity £2],froa the fact that the global generator* obey £§, Q3 " ff

Ag'X'rf*wj/'*'+&JL. fV . Consequently, P ^ trQ2 > 0; a generalisation of Kitten'srecent related proof [3] for A - 0 should also be possible. One mayfurther check explicitly that small oscillations about 0(3,2) have aconserved Haadltonian which i s of the free linearized field type for trans-varse-traceleea excitations, but for an extra mass term which i s reallythere to ensure light cone propagation for the invariants (as is generallytrue for spin > 0 fields in anti-De Sitter spaces); there are s t i l l only 2degrees of freedom.

In the 0(4,1) case, things are more involved for several reasons.First, there i s no longer a supergravity embedding (the "mass" of the fermionwould be iiMginttry —/-IA1 , for example), and there ia no non-perturbativeproof that E ? 0. Indeed, there had better be none, because there is anevent horison in Be Sitter space (even apart from global embedding), andbecause the energy i s in any case not the correct (conserved) generator whosedefinition i s with respect to the killing direction. In the simplestrepresentation of (on half of) De Sitter via da2 - f2(t)(dx2 + dy2 + d*2) - dt2 ,with f { t ) - e x p f ^ t ) , the killing vector i s 0/t - (-1, t). It i s tiae-likeonly when G2 - 1 -'i2f2;?.O. The energy, J T V X , »f saall oscillation*i s positive but non-conserved [4 ] , while the correct generator OS yT£ f "e^x ,i s conserved. However i t i s not always positive, as can be seen in terms ofappropriate variables, G .SOfob<tfT&t)'ZTTV$ • At th . horieon, |xf | - 1, the-triangle inequality P2 + Q2 T 2PQ breaks down and we believe this i s theEamiltenian signal for the onset of Hawking radiation connected with the eventhorisoa (in this example i t would be graviton emission). Jhe above work vasin collaboration with L. Abbott, and will appear as a CE8M preprint.

1 1

Finally, we surveyed an old unsolved problem related to stability in

the asns« that the energy should have only on« sxtremun as a functional of

field configuration*, naisely at vacuum. This ia equivalent

to proving there are no static solutions, since the field equations are f

for the iinBtsin-Iang-Kills system, this ia at i l l an open problem, unlikefor tbe par* Einstein, Einstein-Maxwell [5] or flat-space Tans-Kills [6]oases. P">of of this conjecture for E-T.K. i s left as a challenge to

mathematicians.

[1] P. Tomaend, Phys. Bar. £ £ 2908 (1977).

(2] 3. DeBer sad C. Teitelboist Ays. aev. U t t . ^ 249 (1977)*

£3] C Kitten, Coma. Math. Phys. (in press).

[4] H. Wariai and T. Xuanra, bog . Tneor. Phys. 28 52$ (1962).

[5] A. Liahnerowici, Lea Theories Belativistes, Maason, Paris (1955).

[6] 3. Ooleman 1975 &iee Lectures} S. Baser, Phya. Lett. 64J 463 (1976).

SECOND OBDEB TAHGIHT SIKUCWEES FOfl SPACBTIXE

C.T.J. Dodson

Department of Mathematics, Oniversity of Lancaster, U.K.

It ia fundamental to the concept of a smooth manifold If that one has

the canonical tangent vector bundle IN, aluo a smooth aanifold, and hence MM

over Hi and so on. Geometrically we view TM as the apace of a l l possible

v e l o c i t i e s , or tangent vectors , for curves in It but analyt ica l ly i t i s more

convenient t o handle as the apace of a l l f i r s t order derivations on functions

defined on M. ftoraally we interpret T as a functor from tbe category of

smooth Manifolds t o the category of vector bundlest

H mT 1 Han —» Ybtfci 1 4 f 1—» I Df

H* W

and Df (or f,) is the derivative of f, which is at each point a linear

approximation to f.

There are then two ways to generalise T, geometrically to give the space

Tt of all possible accelerations for curves in II and analytically to give

JT( the apace of all second order derivations. It turns out that & is alwayB

a functor from Nan to Tbqn [1] but IT is only aush a functor in the presence

of a linear connection, on M [2]. Moreover, given a connection, TOt can be

decomposed [3] as a vector bundle over E, it is always a vector bundle over

M of course.

Vbr a spacetiae (K,g) the Lorenta structure g determines the unique

Lett Civite. connection V e in the frame bundle LH, and in ita reduced sub-

bondlss. Hence all of the above second order tangent structures are available

in this oase. Is is to be expected, the extra structure of a connection on

spaoetis* K makes T 2* more simple (with fibre I?8) than J2* (with fibre

R ) . Biese vector bundles are associated respectively to principal bundles

LTI [2] and PT1 [4]. In this lecture we compare the constructions of the

various bandies of second order. They have significance for spacetime 'because

of the impertanoe attached in physics to differential equations of second order.

Geometrically, a differential equation of first order on H is a section of

Bl and, a differential equation of seoond order on H is a section of TIM

(over <m). Ibe presence of a connection simplifies the geometry by reducing

the number of essential dimensions.

[1} MbrOM, tf.t Palais, K.3. and Singer, I.K. "Sprays- Inais da Acadeawia•• Brasileira le Citnoias 32, 2 (i960) 1«3-1?8.

[23

[3]

[43

Bodaon, C.T.J. and BadiToiovici, K.S. "Tangent and frame bundles oforder two", inal. Stiintifioe Univ. Iasi Bomania (in press). Cf. also"Seoond order tangent structures9 Int. J. Theor. Pays, (in press).

Dombrowki, P. mOa. the geometry of the tangent bundls" J. Bains and ing.Math. 210 (19^2) 73-88.

flarayaahi,lin 1972

In differential Spxinger-Verleg

i

ALGEBRAIC GAUGE QUAHTCK FIELD THB3ET Off

KAUKA-KLEW SPACES

H.D. Soebner odd P.B. Fasemana

University of Claus-thal, Federal Republic of Gernany

Starting from the differential geometric formulation of c lass ica l

gauge theories and using the Kaluza-Klein approach to these theories , an

algebraic quantum f i e l d theory for gauge f ie lds i s constructed. This

construction 1B carried out following Borchers* approach to an axiomatic

QPT for the scalar f i e l d .

The geometry and topology of a c lass ica l gauge theory i s transported

to the algebraic leve l by defining appropriate Barchers algebras for the

potential description and the f i e l d strength description, respectively, of

the associated algebraic gauge QST. Subalgebras reflecting the information

contained in the c lass ica l equations are constructed.

The result ing theory recovers the standard results for the U(l) -

theory on Kinkowaki apace. Par the nos-ebelian cases l inearised versions

of gauge QFT's are obtained.

1 4

QU1HTOJC DiynfliEtfTlAli FORKS

M.J. Doff

Imperial College, London, U.K*

He discuss -the quantum properties of antisynnetric tensor f ie lds

and their application t o the questions of anomalies and the oosmological

constant in quantum gravity and supergravity.

Mxrt we show that , contrary to naive expectations, the gauge

theory of a ranlc-two aatisjnmetrio tensor potential A o> i s not

equivalent to the theory of a single scalar f i e l d A even though each

describes one degree of freedom* In particular, the gravitational conforms!

anomalies are differentt

where T i s the anomalous contribution t o the trace of the regularised

energy aomentum tensor. Similarly the gauge theory of a ranlc-ihree ant i -

symmetric tensor potential Ip-af , which has no degrees of freedom at a l l ,

s t i l l has a non-^ranishing trace nnnaaly given by

This quantua inequivalence of different f i e l d retrosontations may be

derived in a topological fashion. Idea* from differential geoxetry ( e . g .

the Hodge decomposition of differential forms, the asymptotic expansion of

the heat kernel, Betti numbers, the Biler-Poincare" chwraoteristie, the

Gauss-Bonnet theorem, the HiriebSuch signature theorem) then oaks their

appearance in the functional integral quantisation of the antisymmetric

tensor f i e l d s .

A second feature of the i-pvf f i e ld i s that , idran ooupled t o

gravity, i t forces the appearance of a oosmological constant in £Lnstein*s

equations.

B * most interest ing application of these resul t s l i e s in extended

super gravity theor ies . In particular the version of H - 8 supergravity

obtained by dimensional reduction contains 63 A f i e l d s , 7 J^m f ie lds and

one Aytvp f ie ld* This magic combination i s just such as to y ie ld a

vanishing trace anomaly when combined with the remaining contributions

froa f i e lds of spin l / 2 , 1 , i/z and 2 . moreover, the single A vj> f i e ld

provides a eosmological constant and gives r i s e t o sons interesting symmetry

breaking e f f ec t s .

ULTRAVIOLET PROPERTIES OF fflCTHTDED SUFKfiGRAvTrlES

S. ferrara

CIBB - Geneva, Swat«arland

I t has been recent ly argued that bidden symmetries oould prevent the

presence of u l t r a - v i o l e t quantum divergences in It-extended superaymmetric

f i e l d t h e o r i e s , beyond naive Bjnmetry arguments. Ecamples of hidden sym-

metries are already present i n the maximally extended S - 4 supersymmetric

Tang-Mills theory where e x p l i c i t ca lcu la t ions have proven that the P

fnnotion, re la ted t o charge renormalixation, i s zero at the f i r s t three

loops of perturbation theory. Conjectures have Been Bade i n order t o

further support the vanishing of the B function t o a l l orders of perturb-

at ion theory [ l ] . The relevance of hidden symmetries i s even more dramatic

i n extended supergravity where e x p l i c i t poss ib le an-shel l count ert eras have

been constructed at the three- loop l e v e l in l inear ized form as wel l as f u l l

non-l inear couaterterms at the 8-loop order in H - 8 supergravity.

laaarkably i t has been shown [ 2 ] that i n X > 4 extended supergravity with

an 30 ( l ) lang-Mil ls group, charge renornal izat ion ia absent a t the one-

loop l e v e l , s ince i n these theor ies the Tang-Kills charge i s r e l a t e d ,

through auperBymmetry, t o the cosnological constant , A . - 6 * e 2 / ^ • *hl"

r e s u l t a l s o implies that the coBaologioal constant i s one-loop f i n i t e and

oaloolable i n these models.

We derive [ 3 ] general h e l i o i t y and spin BUN ru les far massless and

massive supermultiplets us ing properties of the spinor representations of

the 30(2x) (3O(4K)) group o f authokorphlsms of the K-extended eupersjamotry

algebra at f ixed l i g h t - l i k e ( t i n e - l i k e ) Momenta. Then* include observed [ 4 ]

sum rules whioh are responsible for the softening of quantum divergences of

model f i e l d theor ies with K-extanded supersjssaetrjr, the connection of mass

formulae obtained in ^-extended spontaneously broken supergravity models [ 5 ]

with these spin sum ra l e s i s pointed out .

SOME SHARKS ABOUT QDASTUM THHOBT OF

T. de Alfaro*, 3 . fub ia l* and G. Parian**

* CHffl - Oeneva, dept . o f Ibeoret ioal rnys i c s , Universi ty o f Torino, I t a l y

IHSSi Sezione d i Torino, I t a l y .

** Sept. o f Theoretical Physios , Universi ty of Tr ies te , I t a l y

IOTJI» Sesione of Tr ies te , I t a l y .

We discuss the r o l e of the Kewton constant i n gravity theory. A

natural consequence of the general covariance of thft theory i s t o not i ce

that the act ion i s independent of any dimensional constant . The Howtou

constant i s introduced In gravi ty through the c l a s s i c a l f l a t so lut ion

g (x) a j " ) ^ which displays the dilatation breaking of this roincare

invariant solution.

nothing changes for the usual results of classical gravity, since

the tfewton constant i s responsible for the low energy limit, but there are

other solutions for gravity coupled to matter uhere the Hewton constant

does not appear* These non Newtonian solutions may be relevant to the

hadron struoture*

Arguments are also given to show that the small distance behaviour

of the theory can be greatly improved, leading to a situation very similar

to usual ramormalixable theories.

[ l ] S. Ferrara and B. Zuaino, unpublished.

I.F. Sohnius and P.C. West, rnys. Lett. B100 (1981) 245-

[2] S.K. ChriBtensen, ».J. Doff, G.W. Gibbons and X. Booek, Fhyo. Rev.

Lett. 45 (1980) 161.

[3] 3 . Perrara, C.A. Savoy and, L. Qirardello, CUM preprint TH-3094 (»V 1981).

C4] T.L. Curtright, (nys. Lett. 102B (1981V U .

n>* *» t^.

THEDST OP JUCX HOLES

P. Hajicek

Inst i tute for Theoretical Fhysics, University of Berne,

Berne, Switzerland

We generalize the well-known quantum theory of so l i tons so that

i t becomes applicable to the blade hole solution* of Qeneral Belativity*

The generalization ia not straightforward, as the blade hole solutions

are either incomplete or singular, * minor d i f f i c u l t y i s also the general

covariance, because the translat ion no(!n becomes a pure gauge. He work

only Kith a part of the spaeetime, the so-cal led "domain of outer conoun-

icat ion", which i s globally hyperbolic, asymptotically Xinfcovakian and,

of course, regular* Choosing natural boundary condition at the surface

of the holes leads t o a regular canonical foraalisB and, for extreme ho les ,

to a regular propagator. The boundary conditions a lso guarantee that the

c las s i ca l s o l i t o n solution ia unique and that so xero aodes are present

(on* so l i ton c l a s s i c a l solut ions are four-parametric). He concentrate on

one-sol i ton sector in the Einstein-Maxwell theory as * nodel, but the

method can e a s i l y be transformed t o other models (supergravity for more

f i n i t e theory) and t o isore-soliton sec tors .

OH CROUP COVARIAHCE AHD SPIH MOTION DT GRAVITATIOHAL THEOHT

L . H a l p e r n

Florida State UniversityU.S.A.

Departing froa the hypothesis that the local invariance group G ina generalised relativistic theory i s sinple (or at least semi-simple) i t isargued that space-tine should fora an organic part of the group manifold andnot simply a realization; this can be achieved by identification with theelements of a system of iaprimitivity with respect to a suitable subgroupH of 0.

To arrive at a generalised theory the group manifold with i t s Cartan-Killing metric i s considered as the source free solution of a modified non-Abelian Kaluza-Klein theory. This theory differs from conventional theoriesof this kind by non-vanishing "cosmologies!" Tang-Kills fields which give riseto non-geodesic notion of "charged* particles even in the source-free case.The case of Q f 0(3,2) and H ^ 0(3,1) is considered where the Tang-Hillsfields of the theory can be interpreted to mediate an interaction betweenspinning particles. It i s speculated that the non-geodesic notion yieldsa description of test partioleB with spin i f quantum theory i s properly takeninto account.

QUAHTuH Pli lD THEOHT ADD SPATIAL TOPOLOGY

C.J. Isham

Imperial Gollegs, London, U.K.

The study of the role of spatial and/or spacetine topology in quantum

gravity i s currentlj of Bach i n t e r e s t . A ainple example i s given by "twisted

scalar f i e l d s " — cross sections of nan t r i v i a l l i n e bandies over J-or 4-space ^

Such bundles are c l a s s i f i e d by H (Z 1,Zv)*H<v<(nil2),'2^Bnd the different s e c t o r s /

bundles lead to di f fer ing quantum e f f e c t s .

A deeper example i s afforded by canonically quantized Tang-Mills

theory on as arbitrary compact and orientable j-apace 2. • The &" and n-state

c l a s s i f i c a t i o n nay bs computed in a way wLich i s adaptable to eanonically

quantized gravity. With suitable res tr ic t ions the 3pa.ce of connections -C

i s a principal bundle with, f ibre the function, apace c"&, a) (assuming the

0-bundle o v e r j . i s t r i v i a l ) and base space Q= c/c""(2( <»)• Q i s the

configuration apace i TTO a o a t generalquantisation schemes/on a space of cross-sections of a flat but nontrivialcomplex line bundle over Q and these are classified, by Horn fl fi/), ^ ' jwhose elements label the generalised "©-States". JJ, 0] may be explicitlyexpressed in terms of Z © Bora (TT,(X^1ti(^V

In canonically quantized gravity S, C*"^ Q) and q are replacedby respectively the space of rierananiaa 3-aetres, the diffeomorphiam groupof X and the quotient apace of 3-oetres modulo dlff (Z)s^S (Hheelerts"superapa.ce"). The *n-atates" are now labelled by7?I W) = 7TO fOt-ffZ.)and the "©-Btates" by Horn l'TTotcitff"£-1> l) • Recent pure mathematical workrenders these groups explicitly calculable.

When considering Taag-Hills vacuum states i t needs to be appreciatedthat the existence of holonomy leads to solutions to F. .- 0 which are notpure gauge* The general solution i s of the form A, (1) =• E(y)9.s(y)where S maps £ (the universal covering space of X ) into G and»(y<0 - B(y) h(V) VV&¥.Li.) for some a t f i = HOrrvCff7t^J&)-If the space of al l such continuous functions i s denoted (S then $ can beshown to be a principal fibre bundle over H with fibre C°(£, c) and the"n-classifioation" of the vacuum states is in t«rat of U^(S) . This maybe computed from the homotopy exact s e < i u e n c e ^ ' n l | ( f t V 5 C S ) ^ l T f ^ r t )and the labelling of q*~^ (a component) i s in terms of [ i ,rather than the [£, a] of the general n-state.

OF FIBST ooftnouiaT GHDUPS FOR LIE GBOUPS ABB CLASSIPICATIOB OF

SOLITOB SECTORS IH 2 AND 3 SUCOTSIONAL FISLD THBDHIES

S.B. fomy

Riyadh University, Saudi Arabia

Quantum so l i tons are realized as coherent s t a t e s , and can be

interpreted as vectors ly ing outside the Hilbert space of the theory.

Moreover, so l i ton s tates are in one—to—one correspondence t o f i r s t order

cocycles of the fbincare" group r e l a t i v e t o i t s various actions on the

various c las ses of solutions of f i e l d equations.

In 2-epaceti»e dimensions the Ibincare1 group ia a. aoluable Lie

group and i t s f i r s t cohonology group i s computed. For a connected and

simply connected Lie group, we show that R* (G, 7 ) £ H» ( » j t v ) . Star

a semi-simple Lie algebra where the second order Casimir operator i s

invert ib le , we show that H' ( * j , T) « 0 . Other cases are also mentioned.

eiGOAflJODIT OHBITS AND A HEH SIMBOL CALCULUS IDE LIMB BUBBLES

Bertram Eootaat

M.I.T., Cambridge, U.S.A.

the symbol calculus for d i f ferent ia l operators i s the inverse of

quantization. That i s , l o t

l>e a differential operator of degree k on

U a multi-index, „,. ^ and ( | ^

symbol 0

Here 6=

l o t l l e function on phase space

the

ven by

The purpose of this note i s to announce a new symbol calculus forline bundle* which overcomes the difficulty mentioned above. Let (L, H) bea pair Mhexe M i s any manifold and L i s a Hemitian line bundle over X.Let m a dl>* 1 . He introduce what we refer to as the L-ahifted cotangentbundle X - I (L, H) of M. this was introduced independently and earlierby Meinstein (in generalising some results of Stsraberg) for other purposeshaving to do with connections in L. One has a fibration

where the fibres are the leaves o f -tha polarisation. X i s defined simplyby taking the cotangent bundle of L, and than reducing by the IL* action.

Now on* has a surjective map

One mast consider only the terms where \t) m It in order to makecorrespondence 3-?<?kC*) invariant under canonical transformations andalso to mate . connotation go to Poiseon bracket. However in ignoringterns where jcj<)l there i s clearly a great loss of information. Par scalarvalued functions vf (u.) this loss can be dealt with in many cases (e.g. theel l iptic theory). On the other hand the symbol calculus i s valid for linebundles (or even vector valued)functions f(>n). Here the loss of informationcan be very serious indeed. An important example of this i s the case ofcoadjoint orbits of Lie groups.

In more detail let C be a connected Lie grouptf i t s Lie algebraand aj' the dual space t o t ) . One knows that any orbit Q g<T)' of a (thatiB3 a coadjoint orbit) has the structure of a sympleotic manifold (that i s t

a generalized phase apace) which i s invariant under 0. Assume that 0 i sprequantizable and that there i s a G-invariant polarisation P. Lets(o-j) and U(*)) be respectively the symmetric and enveloping algebra* of <rjThen there is a line bundle L(O/P) and a representation

TT :VCy)

6 D ) differential operators on L(O/F) ( i . e . an induced representation).The operators ~K"(p(fjj) can be regarded as a quantisation of the functions-Sf*)) 1 0 t obtained by restricting 5 £9) to 0. The failure of thesXubol calculus is manifest in that one does not obtain S(»\) I 0 *>y applyingthe symbol calculus toTTfU ci-jY). In fact 0 does not appear, in general, inthe cotangent bundle T*(o/p).

where Biff(X, L) are the differential operators on H operating on sectionsof L and DiffL are ordinary differential operators On L whichconmute with the (L* action. The point, however, is that any 3 in D ff(»,L)only has a symbol °:n T*(M) whereas any <j£ Dtff*L has a symbol on X*The latter i s much sharper. For example i f 9 i s a first ***wr operator thencan be trivialised u i = > q j l + J • The usual symbol ignores b.But for the new symbol =1

one has 6~, CM,*-) 2 => J. 0The reason why this works i s that the multiplication operator b i s actuallya vertical vector field on L« Applied to groups this recovers the coadjointorbit theory.

One has now the following theorem, characterizing the subgroups of aLie group which polarize coadjoint orbits.Theorem. Lgt G be any connected Lie group and let (H,}() be any pair whereH i s a closed subgroup of G and jC H *-» £ i s a unitary character on H.Let L(4/H) be the line bundle over /H defined by putting

= &

Then H is a polarization at some f fe «T' where Sj ia the Lie algebra of

qt and £ - tKC 2TTL'4- 9Ji H i f and only i f there exists an open

orbit of G in the shifted cotangent bundle Y.C&IH , !•(£•/H) •

Furthermore ire can solve a problem on the kernel of certain induced

moduleiof L>(V). Let 6 6 f j ' a n d fys0) b e a P ° l a r i a a t i o t t a* f B0

(£, becomes a AiQ{) module. Let

so that V i s an induced li^-module. Let "X* Oft))be the kernel of the

representation of tffoj^on V. Let fi-I^S^be the associated graded ideal

in SfoV

•be the usual moment map on the cotangent bundle T*Theorem. Assume that C i s algebraic and 0 ( the orbit -through f( i sclosed, or more generally assume that 0 i s a normal variety. Then (jV Xia the ideal in Sla\ which vanishes on the image

Bemarlc. In the semi-simpl» case «-f 0 i s closed the image o f * i s theclosure of the ELchardson orbit defined by the parabolic I) . The theoremabove solves a problem rained, I believe, by H. Borho.

S4

HOW SPATIAL TOFOLOCT A&BCfSS THE IAMG-KHA3 TACDTJN

C. Kuns ta t t er

Imperial C o l l e g e , London, O.K.

It i s shown with a few simple examples that when three space i smultiply-connected, the canonical vacuum structure of Tang-Mills theoriesi s considerably more complex than expected by looking only at the hoaotopyclasses of gauge functions. In particular, there may be non-trivial 0-bundles over three space which admit flat connections • Even i f the 0-bundle i s trivial , connections which are not pure gauge globally nay exist .The examples are chosen to illustrate a method of analysis whioh allowssignificant progress to be made in the general classification of Tang-Hills vacua with arbitrary gauge group on an arbitrary three space.

- . , . ••.» * » . *

asPERIODIC SOLUTIONS fOB KAMILTOHIiH SYSTEM

Jean-niche]. Lasry

Oniversite' da Ru-ia-lX-Dauphine, France

The Maupertuis principle states that the Hamilton Equations

(HE) (-#, 4) - VE (q, p)

are the variational aquations for

Jo

given energy level (again vaiw reasonable hypothesis on H).

Ho appear or recently appearedF. Clarke, Periodic solutions of Haniltonian inclusions, J. Biff. Eq.P. Clarice and I . B»landf Hsniltonian trajectories having prescribed

minimal period, Conn. Bare and 1pp. lath.

I . aaland and J-N Lasry, On the mmber of periodic trajectories for aHamiltonian flow on a oonrex energy surface, Annals of Hath, 112, 1980,

pp. 28>JL9.(and other papers quoted therein).

-Bit the functional 7 appearing in (JC) has both technical and geometrical

properties: for example both subsets ?F < o ] and £p > oj containinfinite dittensional subspaces.

1 new variational principle for (HE) was introduced recently byF. Clarke and I . Bceland in the eaaa where the Haadltonian funotion a iaconvext

where H* i s the Legendre-Fttnohel transfom of the Haailtonian fnnotion Ht

Lrt as see briefly and informally see how this comes about. The BOlerequations of (CB) are

(q, P) -

Hy Fenchel-Legeadre reciprocity fomula (i.e. the functions <7H*

and VE are the inverse of eaoh other) we see that (E) and (HE) are equivalent*

Ihe functional F£ appearing in (CE) has bath good technical and

geometrical properties. For example under reasonable hypothesis on the

function H, the functional F 2 reaches its •<«<—™» on the set °g ot 1-

periodio functions and if (q, p) is suoh that ?2(q, p ) - sdniasm of F.

on *g , then (q, p) is a periodic solution of (HE) of •••'-•<—i period T.

A modification F, of leads to another new variationalprinoiple of the type where the hoaologj of sot o|

functional space leads *• at least n periodio solutions (BE) of

tin

QUANTUM. GEA7ITT OB A HEGGE SKELETON

X. Karte l l in i

I a t i t u t o di F i s i c a , Univeraita d i KLlano, I t a l y ,and

I a t i t u t o Hazionale di F i s i c a Nucleare, Sezione di Pavia, I t a l y .

Quantum gravity is a non-renormalizable theory. This means

that the cancellation of the ultraviolet divergences would

require an infinite number of terms in the Lagrangian, proportional

to arbitrary powers of the curvature tensor and i t s :<»variant

derivatives. Despite the non-renormalizability of the gravitational

interaction, the leading theories of elementary particles may be

expected to be renormalizable. Furthermore, the Einstein theory,

based on the Lagrangian (-g) R/1USG seems in perfect agreement with

the experimental data. We can think to resolve this "puzzle" if

we admit the existence of a fundamental mass M (possibly to be

identified with the Planck mass 10~9GeV) and understand the

quantum gravitational effects as relevant only on such an energy

scale. Prom this point of view the validity of our particle

theories, as well as of the Einstein theory of gravity, ""Ids because

all the possible non-renormalizable interactions are suppressed

«* tha *n«r?les (Aiatancee ) where they are retimBrt. Ona might thus

* * * ! • th« problem of quantum gravity by imposing a

o**«* . 4l*tafttffe t , a oarMiB multiple of l/n in c4fcL units, a n d thus

to con»iA«r quantum fluctuations on a scale larger than I .

Clearly the f i r s t atep becomes that of understanding thewith-Buch a. cutoff

gravi ty theor / . More e x a c t l y , s ince the Einste in theory contains

natural ly the Pla nek length 1/M , the correct s trategy would be

that of emphasising .; the geometrical or topologica l s tructures which

make i t d i r e c t l y . an "already-cutoff-theory". Our suggest ion i s to

look at the "skeleton structure", Regge c a l c u l u s 1 , of space-

time. More exaatly, we consider a modified version of the2

Regge calculus, in order to understand i t as a lattice gauge theory. Inthis sense, our approach i s not canonical: we start directly fron a discretetheory, which we "translate" into the lattice dictionary. Furthermore, herethe development i s opposed t« that based on a lattice structurein which the cutoff i s introduced a priori by imposing a hypercubical latticewith spacing I , How i t i s not at al l clear i f the topological structureof the spacetine usually considered in gravity i s compatible with such anunderlying lattice* Howftrar, Our modal also raises some trouble at theclssBical level . In fact, we derive the new classical skeleton action,making use only of the curvature invariants. An a consequence of this, thecontinuous theory is more like an affin* gravitational theory than a metricone. Only for weak gravitational fields i t i s reasonable to believe that thecontinuous limit gives the Einstein theory. The second step consists, afterquantisation, in ex£loriig the behaviour of the theory. The underlying ideai s that the quanta* gravity at very short distances becoaes a finite theory;only at low-energies i s i t nonrenoraalizable. He find a close formal connectionwith an Ising-lika gauge model.4 On the baais of this result and using therecursion formula developed by Migdal for the lattice gauge theories, whichwe conjecture (anaats^ to be valid again, we show that quantum gravity, withoutcosmological constant, ia an "asymptotically safe" theory with % uniquedimensional free parameter K . Since this merely defines our units of massor length, we shall identify K with the Planck mass. All this means that thereexists only one (two, i f there ia also a cosmological constant) arbitrarycoupling parameter which i s an attractor for the infinite charges -requiredfor the ranormalination of the quantised gravitational interaction.

1 . T. Regge, Huovo Cinento 1 ^ 558 (196I) .

2. K. X&rtell ini , "(taantua Gravity on a Regge Skeleton", BAJfTF (Cambridge, UK)

preprint (September i 960 ) ; I . Rooek et a l . , DJUCTP (Cambridge, UK) prep. (1981).

3 . L. Swolin, Itocl. *hys . B148. 333 ( 1979)»

i . Das, K. Kalcu and PJt. Townnend, Phys. Letters 3TB, 11 (1979);

C U T . Katmion and J.O. Taylor, "General Relat iv i ty on a Plat Lattice*,

King's College (London) preprint (November 1930).

4 . Leo P. Kadanoff, Rev. Mod. Phys. .Jgi 267 (1977).

5 . A.A. Migdal, Zh. Eks. Teor. P i s . 6^, 1457 (1975).

6 . S. Ifeinberg, in Gravitational Theories since Einstein. Eds. 3.H. Hawking

and W. Israe l , Cambridge Hnivarsity PresB (1979).

GAUGE THTOHIES IKCLUDIHG aBAVITATIOS

J.W. Jloffat

Department of R iys i c s , Universi ty of Toronto, Ontario, Canada

i n attempt i s made t o construct r e a l i s t i c uni f i ed gauge theor ie s i n

higher dimensional spaces , based on a b i l i n e a r symmetric m e t r i c A complete

un i f i ca t ion of fundamental in teract ions moat include gravi tat ion and the

uni f i ca t ion energy of a l l the f i e l d s i s expected t o occur at the Planck

energy ( n c ' / o j ' ~ld^ OeV. At t h i s energy, the f i e l d s are transformed

in to one another under the i s o a e t r i c s of a Lie group G of the 4 + H

dimensional manifold. Only one dimensionless parameter, |^p/(f ic ) ' ,

where p i s a momentum and K.™ (4nG) ' , ia expected t o appear i n the

theory which i s broken down by spontaneous compactifieation t o the four

coupling constants of the "low-energy" electroweak, strong and gravi tat ional

forces.

The total action of our theory i s

ithere !*• » {xt* , y°* ) are the coordinates of the 4 + N dimensional

space, It) are fermion f i e l d s , B - DET ( f 2 ) , B i s a scalar curvature i n

4. + B dimensions, and A i s a cosmologies! constant . The minimal unifying

group for a symmetric metric i s 30(13, 1 ) > 30(10) ® S0(3, 1 ) i n 14

dimensions. For a sesqu i - l inear (Hermitian) metric the minimal group i s

SU(8, 1 ) > SU(5) <C U ( l ) <S> SU(3, 1 ) i n 16 (or 18) rea l dimensions.

After dimensional reduction, the act ion 3 contains the 4-dimensional

scalar curvature R. and the correct k i n e t i c energy term 5. C>v> ) ;

a Higgs symsietry breaking operator V ( ^ ) and a Yukawa-type coupling that

breaks the fermion masses. The act ion S becomes t ru ly uni f i ed at the

Flanck energy ~ 1 0 ' CeV, when S i s invariant under the ajmnetris of 0

i n 4 + H dimensions.

A theory of grav i ta t ion suggested by the uni f i ed gauge theory, based

on a sesqui l inear (Hermitian) metric (g/ti> =• g%ti- ) i n 4 -diaensions , i s

considered. In the 4-dimensional real manifold M, the principal bundle

F(Nt Q) has the associated f ibre bundle E(X, OL(4,(^) /u(3, 1 ) , Q L ( 4 , ^ ) , P)

which admits the f ibre metric

1 *- V .

30

Bi« classical theory of grwlty i s found mto bt ia agreenent with all

relativity experimntal data, including the binary pulsar gravitational

radiation data. The theory possesses a Caochy init ial value formulation.

The cosaologicel oonseqoenoes for the sesquilinear gravity theory

art i w i e w d . The special init ial conditions of the Jriedmann-ltobertaon-

Halker universe at the big-bang singularity (zero amisotropy etc . ) are

predicted by the theory.

3-1

SUPBRGRAVITTr IBIS AHB HOTS

Yuval Ifo'eman*

Tel Aviv university, Israel and University of Texas, m a t i n , U.S.A.

B » Whys

Sinoe I960! General Relativity has "re-centred" physics, in the

sense of being influenced by observations - and influencing thorn. The

discovery of the 3°K background radiation destroyed the consensus*

adoption of Steady-State theory. Black hol« physics i s now guiding the

search for (tray) pulsars. Grand Unification Theories (GUIs) are applied.

t o CoMofonjr and for the f i r s t t ine explain how the Universe got r id of

antimatter, and nhy the photon* outnumber baryons by a factor 10 . However,

GOT physics step* art lCr* GOT, just below the Planck nass

and i s completely cured of the Johnson-Sudarshain and Telo-Zvaaziger

paradoxes re lat ing t o spin 3/2 f i e l d s , although i t contains such f i e l d s

( l ike a l l super gravity t h e o r i e s ) .

Ojxrept mrk pursAes the proof of f i a i t e n e s s , and attemptB t o

r*l«te the theory -to the phenomenology ( i . e . identify quarks and leptons

in the extenlM supweynuwtry mul t ip le t s ) . •Bain i s s t i l l only a beginning.

Bik I •> 8 -theory has a lso opened up a new interest the geometry of the

Bteeptional Ofwpe, because i t involves BL _ and a sequence of sub-groups

mud originates in Eg.

At amargies of that order, aravity i e the strongest foroe, and in

t o be treated quantum-necliaaioalljr (and n*\ as an external "background"

f ie ld . ) . A * physics of oowaogoay require » -theory d*»cribing gravity

T YI pos^^najtion uttn tlX o-ther ^rt*Ty^t^ t*m» BMff ifco ELd erpiiftin (beyond

O-aaynsietry and n y/* - 10 ) iho orwjtion of wff *T gravitons, and

a l l in vwuikiMd fora. Such s> desm>ptica of qttfcMai gcvrlty has t o b#

not jusrt renernaliaBbl*, Iteoan** a ha» dipnoaions and w* wo«ld

require an i n f i n i t e sequence «f c«w*»rtems for r » n « w a l i i a t i o n .

Snpergrarity holds s»co hopes bocaose th« sup»ralg»bra of attended

Bup«rsjn«*-*ry i s -fen* only knew MUT in which ejrtfraal and internal degrees

of freodoa are ailowed t o b« constrained tos«*h#r ( i » a aiaple supergroup,

the Bupereonforqol 30(2, i/s) for the aaaslafa • i tuot ionB). Moreover,

a certain o lass of representations of « t e n 4 « 4 sityarsyqpietry, in wnich1 " ^ o a x ' " i l o r * b

max i B 'tae - f ^ " t heli«4*y, are sigenreprassntatiens

of 3aparuoitary Parity SD(l/») j CPZ (a eVfV extension) and nay be se

constrained nathematically as -to cancel ajU c e w t e r t o m o . Vhi.9 has already

been shorn t o happaa in h » 1 t e >-»4fp oplsr* lb s sans might be

hopefully true of 1> » 2 , mtpetgrrnitji **.*ii 1 - 8 therefore, mat

theory has now Veaa written up by Crt9*|^ «•& Ju l ia . He already know

that i t i s at least 2-looo f i n i t e , T<mryi ordinary gravity i s only one-

loop f i n i t e , and far the vacuum oao« orn y* K » 8 supergravity sontains

matter of a l l spias iMtnew 2 and ^ Tk« N > 2 case i s the f u l -

fil lment of SLnvtaia.** search* i t pjf» a completely constrained

unification at grvrtty a ^ e l tetMMpttAiao, a1"0 ^ least 2-loop f i n i t e ,

• Supported in »»rt >y ths QaS««l*s«Mil Knational Science Foundation and bythe U.S. tDK W4sr » < P t tX-T*-»yOV3992.

- 32 -

TWISTED AXIAL STJfflETRr OF SEPARATED JOUOIOLES

Dublin Ins t i tu te for Advanced Studies , Ireland

By monopoles tie mean here solut ions of the s t a t i c Tang-Mills Higgs

system which have f in i te-energy and non-zero topo log ies ! (magnetic) charge.

At the -time of the Clausthal conference i t was known t h a i , in the l imi t of

zero Higgs potent ia l , such solut ions corresponding t o separated monopoles

ex i s t ed . But no such solut ions had been found e x p l i c i t l y , and at that

conference we reported the unexpected resul t that the 2-monopole system

(or, more generally, any col inear E uopole system) could not be a x i a l l y

symmetric.

In the intervening year, separated monopole solut ions have been

found e x p l i c i t l y , mainly due to the work of Hard, and with these solut ions

in hand, i t i a poss ible t o re-examine the symmetry of the 2-oonopole

system* l i t turns out that , as predicted, i t i s not axi-symraetric, but

that the axia l symmetry i s replaced by a one-parameter symmetry group which

acts non-l inearly on Biclidean space. More prec i se ly , i f D(x) i a the

central determinant of the system, then D(X) i s a function only of the2 2 2 2 2 2 2

two quadratic variables x + y + s and z cos* + i Bind , where2 2 2

tan <*- is the separation parameter. The lines of constant x + y + a2 2 2 2and z CO B ^ + x since turn out to be closed non-planar loops which

encircle both the z and x axes, and are such that the properties of

D(I) are completely described by its properties in the nz-plane. The

latter result is used to show that the system describes separated monopoles

(located on the x-axis) for all values of the parameter tan c*. , and that

the solution is regular for finite separations of the order tana^-l

(which probably corresponds to monopoles separated by a distance of

approximately the monopole size),

3 4

GAUGE F l a i l THEORIES AHD THE BQOIYALBTCE PRIHCIHE

Claudio A. Orioles!Istituto di Fisioa dell'University di Parma

andDIRT - Sesions di Nilaao, Italy

We review the nonabelian generalizations of the Xaluza-Kleiamultidimensional unified theories of gauge fields interacting withgravitation. Such theories are deeply interrelated with general relativityon a (4 «- R)-dijBBnsional space Ji. Cf, G) which i s a. principal fibrebundle with base </t (• ordinary spacetime) and structural H-dimensionalgauge group Q: the bundle connection i s defined through a o-invariantsetric ~£ and the Lagrangian ia the curvature scalar for a gauge-covariantconnection on •A. . Through dimensional reduction, achieved by takingaverages over the fibres, one recovers on v/l the theory of gravitationwith coupled Tang-Mills fields and Jordan-fThiry-type scalar fields.

The equivalence principle, generalized to A * , states thatpoint-particles, having mass and Tang-Hills chargea, move along geodesiesin J\7 • Thia principle leads to the correct point-'paxticle equations on

JC, for Taog-Kills-LorentE forces and gravitational focus. A. predictionis that charged particles cannot have zero mass.

Uy imposing gauge invarianc* out the Einstein (4 + ») - fieldequations for the vacuum, the vertical components of the connection arefixed in an essentially unique way. A resulting prediction i s that thecoamological constant for the 4-vacuum must vanish.

as

DTHAJCECAL

TJ). PalevInstitute for Unclear Research, Sofia, Bulgaria

Following the ideas of Wigner, a definition of a quantization is given,

e to

operators p^ end

which allows one to ascribe to the classical canonical variable* p and q.

that do not necessarily satisfy tae canonical commutation relations

CijfPfc] " 5 j j , [<Ltq^3 • [PjtP-] " 0 • (2)

The quantization we consider i s more general on one side and more restricted,on the other, since i t tarns out that the mapping ( l ) depends in general onthe dynamics, on the interaction. The mathematical background is of a Liesuperalgebraic nature. This stems from the observation that the canonicalp and q± generate a representation of the simple orthosymplectic Liesuperalgebra (LS).

Is an example we study a noncanonical quantization of a system of twononrealtivistie point particles, interacting via harmonic potential. Thecentre-of-mass variables are quantized in a canonical way, whereas the internalmomentum and coordinates are assumed to generate the simple LS s i { l , 3 ) . Itturns out that the distance between the particles i s preserved and can havefour different values. The angular momentum of the system i s either sere orone (in units ~&/2).

IHVARIAHT gjAtrrizAxioif OP TMB-EEPBHJBIT HIVE BQUAXIOHS

Stephen H. PaneitiUniversity of dansthal, Federal Republic of Germany

The well-known C*-algebraic quantiiation of a generic class oftime-dependent wave equations, for example,

ia completed by the (rigorous) specification of a unique invariant vacuuastate. Invariance here i s with respect to the scattering automorphism,that i s , the vacuum i s defined, as the unique regular state which isinvariant under the scattering autoaorpbisa of the field, or eunplj thestate of 'no physical particles** The existence and uniqueness of suchstates i s shown equivalent to certain stability properties of the classical(sjnplectic) scattering operators, or, in other terms, related to certainproperties of a generic class of orbits in the infinite-dimensionalsjmplectic group. An explicit sufficient stability criterion in terms ofbounds on the V(tt x) i s given.

Similar results apply to arbitrary •causal' symplectic differentialequations, e .g . , perturbations of positive-energy wave equations of evenspin.

Those developments represent wort done in collaboration withI.E. Segal, and also results of the lecturer's doctoral thesis . They havebeen suwnarised in "Quantisation of wave equations and hemitiaa structuresin partial differential varieties", 3.H. Faneits and I.E. Segal, Proo. Sat,lead. Sei. USA, Vol. 77, Ho. 12, pp. 6943-6947f December I960.

3 8

KMM.IMR4R XaFPEBBSttja. SEQUEHCiB HT GAUGE TKH>RT

J.F . Fommaret

Bepartement de Meoanicpie, Boole Nationals des Fonts et Chansse'es

I t o i s , France

ACTION PEKCIFLE JOR UHIFIED THHDHIES

XX 1HE QHOUP KAHIFOLD APPHDACH

T. Regge

University of Turin, Italy

He show how to extend the differential geometry of classical gouge

theory by introducing the non-linear Spencer sequences. The use of groupoids

or peeudograups instead of groups iB somewhat richer and, at the sane time,

we simplify greatly the original exposition of Spencer. As a striking

result, the passage from the first Spencer sequence to the second is

breaking down the use of connections in this framework. The way to escape

from this dilemma is to introduce a nev non-linear sequence called Janet

sequence bat the physical interpretation will have to be done by future

field theory.

Since the formulation of general relativity, differential geometry

has played, a toy role in physics. PhysicistB however have limited themselves to

using ordinary tensor calculus paying little attention to the exterior

calculus of fonts introduced by Cartan. In this talk we propose to use

the exterior calculus BjfBteoatically. Furthermore we propose a clasB of

unified theories of gravity based on the following principles.

1) She theory is formulated on the group manifold of a suitable Lie

super group 0 including as a Bubgroup either the Baincore or He Sitter

groups* The dimension of the super group is (n, a).

2) The field of the theory is a set of (n, a) 1-forms yu. or a

single fora p- with values in the algebra of 0. He define curvature aa

3) The action is a polynomial in the forms fn, and curvatures of a

degree p equal to the dimensions of a submanifold of 0.

4) The field equations should admit flat space, 8 - 0 as a. solution.

Is this case ji> • ePt the left-invariant Cartan forms on Q.

5) (Hheonomy). The theory should not be trivial (rigid). It should

admit solutions other than flat space.

It is shown that this frame incorporates many of the most interesting

unified gravity theories including super gravity.

4 B

FAEHCLK THEDRT OM ALT^iTIVE COSMOS

I . E . 3«gal

K.I .T . , Cambridge, U.S.A.

(Work in collaboration with S.M. Faneitz and, e a r l i e r , H.P. Jakobsen,

B. /tfrsted, and B. Speh.)

He study a variant of elementary part ic le theory i n which Hinkowaki

space MQ i s replaced by a natural a l ternat ive , the unique 4-dioensional

aanifold X enjoying comparable properties of causal i ty and symmetry* II

i s the basis of the chronometric redshift theory, which f i t s cosaological

observations on large objective samples of galaxies and quasars very we l l ,

without any sustainable counter-indications of a s c i e n t i f i c nature.

free part i c l e s are considered to be associated with posit ive-energy

actions in bundles of prescribed spin over ft of the group of a l l causa l i ty -

preserving transformations on X, and with corresponding wave equations

enjoying causal propagation of Cauchy data re la t ive to the given notion of

causality* He deta i l basic aspects of these bandies, actions and wave

equations, treat ing the bundles of spins 0 and -x as examples. Issues

of covariance, nni tar i ty and of the nature of symmetry breaking and mass

are treated; appropriate quant™ numbers are indicated; and physical

applications are discussed.

XAHQ-XILLS (HOOK AID THE GBAVITATIO1I1L QHJOK

D.K. 3en

Department of Mathematics, University of Toronto, Toronto, Canada M5S 1A1

In c l a s s i c a l gauge f i e l d theor ies , whereas the electromagnetic or

the Tang-Kills f i e l d arises from connections on 0 { l ) or SD(2) -

principal bundles F(X) over the spacetiae manifold X, the gravitational

f i e l d ar i ses as a l inear connection on the 0L(4) - principal frame bundle

L{]|) over X (or in particular on the 30(3,1) - principal bundle of

Lorent* frames on X). According to the General theory of Helat ivi ty

every non-gravitational f i e l d , v i a i t s energy-momentum tensor, should

generate a gravitational f i e l d . Thus from the gauge view point every

SU(2) connection should determine a OL(4) connection. Since there

e x i s t s the well-known 2-1 covering homomorphism of SB(2) into ao(3)

and thus QL(4), i t suggests that we consider homomorphisms of principal

bundles. Then there e x i s t s a. mapping theorem of connections on bundles

whenever there i s a diffeomorphism of the base manifolds. Ihi» paper

considers a simple application of such a theorem and shows how, i n

principle , one can compute the l inear connection components from a Tang-

X i l l s connection given a base diffeomorphiaa.

BEREffiEHCES

( c f . The b ib l iograph ie s of t h e s e papers for many addi t ional re l evant r e f e r e n c e s ) .

1 . J . P . X i c o l l , B. Johnson, I . E . Segal and H. Sega l , S t a t i s t i c a l

i n v a l i d a t i o n of the Bobble Law. Proc . Hat. Acad. S c i . USA 2X» 6275-6279

( 1 9 8 0 ) .

2 . I . E . Segal , t ime, Biergy, R e l a t i v i t y and Cosmology, from •Symmetries

i n Sc ience* , e d . B. Gruber, Plenum, 1980*

3* H.P. Jakobsen, B. fiirsted, I . E . S e g a l , B . Speh and X. Vergoe, Symmetry

and Causa l i ty p r o p e r t i e s o f phys ica l f i e l d s . Proc . Nat . Acad. S c i . USA

22, 1609-11 (1?T8).

4. The sane authors, + S.M. Faneitz, Covariaat chrcnogeometry and extremedistanoes. l i t Elementary particles. Proc. Nat. Acad. Sci . , in press.

5. I.E. Segal, contributions to Proceedings of: (a) 1990 Tutzing Conference;

(b) 19&0 Clausthal Conference.

4 1

STMFLECTIC MALOCIiS

Shlooo SternbergHarvard Obiversity, U.S.A.

The purpose of t h i s ta lk i s t o lay out some schemes in the programme

of geometric <piantization which emphasise the role of algebras with involution.

This point of view was introduced by Keinstein [ l ] (with a throat on the

group theoretic Betting) and emphasised t o us by Sfllch of [ 2 ] in tha se t t ing

of quantization and polarizat ion. He are grateful to both fop many useful

discuss ions .

In both se t t ings one has a symplectic manifold T, a coisotropic

• submauifold Zc)f and a Lagrangian submanifold f t y x y x y * which can be

regarded a* defining & binary composition on Lagrangian sutmanifolds of r ,

sanding -/L, and - A ^ i n t o PapL^Jl^) , The coisotropie 2 s a t i s f i e s

&($.%%) c £ «w> the se t of JLc"Z forma a "aubalgebra" and, in fact a

commutative one, as w i l l be seen. In the polarisation s e t t i n g , T. - x XX»"

where X i s a symplectic manifold, P i s the usual composition and 2 i s

the "leaf re lat ion" of a polarisat ion on X. Tha de ta i l s for t h i s case wi l l

be spel led out in sect ion 3 . In the group case , I - T*Q where 0 i s a

Lie group, P i s the (twisted) normal bundle t o the graph of the Mult ipl i -

cat ion nap and £ i s the inverse image of the origin ng* under the moment

map o f the conjugation ac t ion .

In the group oase the ooisotropio £ ™" f i r s t introduced by

Kostant (unpublished) in conjunction with h i s study of character fonmlas

for semi-simple Lie groups. Many of the ideas in th i s ta lk are the outgrowth

of extended conversations and correspondence with him and we are happy to

express our indebtedness to him.

THE GHMCETRICAI. KEUfHTC 0 ? THE Jt&LLEH- AND

H. Shirring

I n s t i t u t e for Theoretical R iys i c s , University of Vienna, i u s t r i a

For c l a s s i c a l dynamical systems there are canonical transformations

i n phase space. She Wlller-^transforoation transforms straight l i n e s in to

the t r a j e c t o r i e s , g lobal iz ing the straightening out theorem. She s c a t t e r i n g '

transformation transforms straight l i n e s into straight l i n e s , rotat ing them

by the scattering angle and shifting them by Telocity tiaes tiae delay.

The latter i s generated by the quasi-classical phase shift which suggests

looping trajectories as analog to quantum mechanical resonances. From the

voluae preserving nature of canonical transformation one infers the

classical version of Schwinger's relation between phase shift and energy

shift and of Levinson theorem.

[ l ] S. Abraham and J. Harsden "Foundations of mechanics*.

£2] G. Etach "Prequantination etc.

4 3

NOH-ABELIAN CHAHOEE ASD TOiE-DEFEHDQIT GAUGE FIELDS

Aadrzej Trautman

Inatytut Fiiyki Teoretycznej UW, Hoia 69, Warsaw, Poland

There are analogies - and important differences - between c l a s s i c a l

theories of electromagnetim, gravitation end non-Abelian gauge theories of

the Tang-Mills type . All these theories are based on inf ini tes imal connections

on principal bundles over spacetime. Except for the Maxwell theory, their

f i e l d equations exhibit natural non—linearities re f l ec t ing the non-Abelian

character of the structure groups.

In a l l three cases under consideration, the f i e l d equations may be

written in terns of d i f f erent ia l forms as

JU - (1)

where j i s a 3-form describing the sources, i i s a "paeudo" - 3-formcorrsaponding to the charge density of the gauge field i tself , and

f*F in the Tang-Hills or Maxwell case,

(_ 3-form of Von Jteud's superpotential [ l j .

Moreover, F » dA + i [A, A] i s the Lie algebra valued 2-fors of the fieldstrengths corresponding to the gauge potential A. Eq. ( l ) leads to the

law for the total charge,

( 2 )

For a static situation, one presumes the existence of a gauge such that

A =

so that (2) converges. However, a gauge transformation

i • S^AS + S^dS

with 3 - S(0fyf>) preserve* (3) and may drastically change q (Ulscher,

Schliedor). Ihia difficulty arises in neither electrodynamics nor Einstein'sgravitation, where P end a are O(l/r ) for static fields. If,gauge transformations are restricted to

where a e G,and u = t - r, then the asymptotic expansion

Nr sit,

(where K, Q, X and H are functions of u, 6 and H7) leads to

and q is defined up to a global change of gauge, qi-^a^q a [ a ] . Such

is the case of the non-Abelian Lienard-Wiechert solution, where the total

charge q i s subject to the equation [3]

implying q - 0 unless Z i s non-Abelian sod non-compact.In a non-Abelian theory, time dependence i s a subtle notion. For

example, in the 3L(2, B)-theory, there i s the spherically synnetricsolution A - r~ (en + b)dt which, however, is static because i t may begauge transformed to A' - r txit + adu, where a, b are elements ofgl(2, H) such that [a, b] - a.

It i s interesting to note that, from the point of view of symmetries

of plane waves, Einstein's theory resembles more Harwell's than T.ana-Mj.lla*»

a generic non-Abelian plane-fronted wave,

A - (a (u)r + b(u)y + c(u))du, a » t - * ,

<x(u), h(u), c(u) eayj

bas no full plane symmetry unless

[ • t (5)

In Einstein's theory, condition (5) on the polarization vectors a, b i s a

consequence of the vanishing of torsion [4]•A similar comment can be made concerning total charges: they are

clearly well defined in Maxwell's theory, but in Einstein's theory for anisolated system one can also compute from (2) i t s total energy-momentum.

. - - . - .—•- .-'Tjti-.-» ' * . '

4B

BEFEHSICB5

[ 1 ] •*. Thirring, A course o f mathematical phys ics , v o l . 2 , Springer-

Verlag, Hew York-Mien, 1980.

[ 2 ] A. Irautman, Lectures i n SchlBdming (February 1981) , t o be published

i n Acta Fhys. Austriaca (Suppl .)*

[ 3 ] A. Traxttnan, Phya. Bar. L s t t . , April 1981.

[ 4 ] A. Trautnan, J . Fhya. A 1^ ( i 9 6 0 ) U .

4 6

AFFUCATIOH3 OF TWISTED FIHJJS

3.D. Umiin

Univers i ty o f Vewoastle upon TJno, U.K.

Sinoe i t was originally suggested that in spaces with certainunderlying noa-triTial topologies, the generalisation, of the usualdefinition of a tensor field leads to the possibility of the existence ofao-called 'twisted' fields, the direct physical interpretation of thesestructures has not been clear* They Bay satisfy identical local dynamicalconstraints as the corresponding untwisted fields and yet when incorporatedinto familiar field theories, lead to give different physical effects^

Here, two situations are considered in which the incorporationof twisted field structures into the theory i s not only an alternative tothe uraal untwisted field theory, but for technical, and perhaps even•philosophical* reasons, a preferable alternative. Firstly, a model inwhich the oosmological term depends upon the syaswtry broken vacuum stateof a real, scalar field i s investigated* 0 - 0 i s the oaly constanttwisted scalar field and hence in general, the oosaologicsl tern i s space-tiae dependent, suggesting a possible explanation of the •ayetery" of thesmall cosswlogical constant! It i s typically a sdcropnysioal nnsbar butof necessity, we inhabit an atypical region of the universe. Secondly,i t i s desonctrated that an untwisted real scalar field theory in fivedimensions, which, upon dimensional redaction describes scalar electro-dynamics, displays instabilities even at the tree graph level which arenot displayed by the corresponding twisted field theory*

A QUASI-METRIC AHD A ELEFERaiTIAL CLASSIFICATION ASSOCIATED WITH

3U(2) YANG-MILLS FIELDS

E. Urbantke

University of Vienna, Austria

The work of frtiyah and Ward cm finding Tang-Kills f i e l d s with self-dual

curvature i s based on the observation that for these the parallel transport

i s integrable on t o t a l l y null 2-planes of the a n t i - s e l f dual type in

(Complexified) Kinkowski space; the same fact i s also the basis of Yang's

approach which introduces these planes as coordinate planes, thus reducing

the number of non vanishing curvature components and allowing the introduction

of simplifying gauges* i l e o . i n Coleman'B work on plane—fronted T.H. waves,

the integr&bility of parallel transport along the wave hypersurfaces allows

for a gauge that removes a l l non l i n e a r i t i e s . This suggeststhe invest igation

of those Bubspaces of the tangent spaces to spacetime where parallel transport

along infinitesimal closed loops resu l t s into the ident i ty , and further,

whether these subspacee can be f i t t e d together to form ( loca l ) sub-manifolds.

(Nothing ia said about global aspects of t h i s ) . This leads to an algebraicand d i f ferent ia l e lassi£iea±ion of Sll(2) laug-Hil ls curvatures which i s

independent of a given metric on gpacetime. in particular, one can, in

the general case , write down a (conforms!) metric with respect to which a

given SU(2) curvature i s s e l f -dua l . The technical too l s used are taken from

l i n e geometry for the infinitesimal part, and Frobieus1 theorem far the

loca l problen.

as

TOJOLOGICAL CHAEGES IK SUFESSTMtETHIC

TAMO-ELLS THEDRTES

P. ZizsiInternational School for Advanced studies, Trieste, Italy

It is wall known that in supersynoietric theories, the graded Liealgebra can be modified by the presence of central charges. In the case ofBuperuymraetric theories with BOIitems, one of the possible central chargesis of topologic&l origin. For supersymmetrio Tang-Hills theories inparticular, this fact has been show in 3 + 1 dimensions by Mitten andOlive and in 4 + 1 dimensions by the author.

In the 3 + 1-dimansional case, they found an electric charge and

the topologies! charge of magnetic monopoles (the «t Hooft Folyakov

monopoles, pseudoparticlea in 2 + 1 dimensions) both in the phase of the

theory where the gauge group 0(3) i s spontaneously broken down to an

abelian subgroup U(l) . In that case, both the electric charge and the

topologioal charge are the usual central charges defined by Fayet aa the

generators of the unbroken gauge subgroup*In the 4 +l-dioenaional case, instead, i t ia found that the

topologies! charge of the solitons (the BPST instaatons in the 3 + 1dimensional Euclidean •pacetime) modifies the supersyametry algebra as acentral charge in the phase of tbe theory where the gauge group 30(2) i«unbroken. (The eleotric charge obviously appears in the usual phase ofbroken gauge »ymnetry)» So, in this case, the definition of the centralcharge given by Fayet, does not apply to the topological oharge.

i n ' W 1 • I IK.