P R O C . 19th I N T . C O N F . H I G H E N E R G Y P H Y S I C S

T O K Y O , 1978

C 6 Review of Supersymmetry and Supergravity

D . Z . F R E E D M A N

Institute for Theoretical Physics, State University of New York,

Stony Brook, New York 11794

T h e g e n e r a l s t a t u s a n d r ecen t d e v e l o p m e n t s i n t h e subjec ts o f s u p e r s y m m e t r y

a n d s u p e r g r a v i t y a r e r ev iewed for t h e benefi t o f non-spec ia l i s t s . T h e r e p o r t

i s d iv ided i n t o t h r e e p a r t s a s fo l l ows : I . Bas ics a n d G e n e r a l S t a t u s ; I I .

Se lec ted R e c e n t W o r k i n G l o b a l S u p e r s y m m e t r y ; a n d I I I . Selected R e c e n t

W o r k i n S u p e r g r a v i t y .

§1. Basics and General Status

Motivation: If supersymmetry or supergra­vity is the answer, then wha t is the quest ion? There is a serious reply to this existential query, namely : is there a symmetry principle powerful enough to raise hope for complete unification of the e lementary particles and their interac­t ions?

At the global level, supersymmetry is the only invariance compat ib le with q u a n t u m f ie ld theory which unifies particles of different spin and in ternal q u a n t u m numbers . I t does this by relat ing bosons and fermions, the two b road classes of part icles found in na ture .

Local supersymmetry is a gauge principle of a un ique type. It is fermionic—the gauge field is a spin 3/2 Rar i t a -Schwinger field—and it is a significant extension of the general covariance g roup of relativity. Local super-symmetry au tomat ica l ly requires gravity. The corresponding supergravi ty f ie ld theories there­fore l ink the concept of space-time geometry with the q u a n t u m mechanica l no t ions of spin and statistics.

After two years of active research in super-gravity, the following key results have emerged :

1) The gravi t on can be unified with very restrictive combina t ions of lower spin particles in irreducible representa t ions of supersym­metry algebras. T h e highly gauge invariant field theories of these systems raise hope for the unification of gravi ta t ion with the other interact ions of e lementary part icles.

2) An early sign of encouragement was the cancellat ion in these unified theories of some of the divergences t h a t have always plagued previous mat ter -gravi ty systems. Specifically,

the divergences oi b-matr ix elements in one and two- loop order cancel.

3) Previous consistency problems of the spin 3/2 field such as a causal p ropaga t ion and negative probabil i t ies are all overcome in supergravity because the interact ions respect a fermionic gauge principle.

Despite these achievements there are difficul­ties. Non-renormal izabi l i ty looms at the three-loop level a n d beyond. Phenomenologica l appl icat ions of bo th supersymmetry and super-gravity have interesting features bu t presently seem unna tu ra l . This r epor t will describe some of these difficulties a n d some of the dazzling theoret ical features of the formalism which have been the subject of recent work .

Basics: The essential idea of supersym­metry is to cons t ruc t q u a n t u m f ie ld theories with conserved (Majorana) spinor charges Qa. These are elements of graded Lie algebras which also involve the Poincaré g r o u p genera­tors Pfl a n d Mflu and o ther opera tors . The var ious supersymmetry algebras which have h a d appl icat ions in theoret ical physics are listed in Table I.

The fundamenta l Poincaré supersymmetry algebra conta ins TV-spinor charges Qi i=\,

- • % N which satisfy

( i )

(2)

(3)

The presence of the t rans la t ion genera tor ir the basic a n t i c o m m u t a t o r (1) a l ready suggests the connect ion between supersymmetry anc the s tructure of space-time which is full) realized in supergravity.

To see tha t (1-3) imply a symmetry betweer

536 D . Z . F R E E D M A N

T a b l e I . S u p e r s y m m e t r y a l g e b r a s .

I . P o i n c a r é S u p e r s y m m e t r y : G o l f a n d a n d L i k h t m a n 1

Y o l k o v a n d A k u l o v 2

W e s s a n d Z u m i n o 3

S i m p l e : N=l

E x t e n d e d : 7 V = 2 , 3, . . .

n a t u r a l SO(7V) in t e rna l

s y m m e t r y

C e n t r a l C h a r g e s : UiJ, F ° H a a g , L o p u s z a n s k i a n d

S o h n i u s 4

I I . C o n f o r m a i S u p e r - W e s s a n d Z u m i n o 3

s y m m e t r y

I I I . D e S i t t e r S u p e r s y m m e t r y K e c k 5

M a c D o w e l l a n d

M a n s o u r i 6

r e l evan t t o c o s m o l o g i c a l t e r m s i n s u p e r g r a v i t y

particles of different spin is extremely simple. O n e simply applies the spinor charge Q a to a part icle state \p, of definite m o m e n t u m a n d helicity. E q u a t i o n (3) and simple addi­t ion of angular m o m e n t a then imply

(4)

T h e r ight side is a superposi t ion of particles of the same m o m e n t u m a n d energy—ergo the same m a s s — a n d helicities s i 1/2. Supersym­met ry therefore relates bosons and fermions.

The algebra with one conserved spinor charge is often referred to as "s imple super-s y m m e t r y " and algebras with m o r e t h a n one spinor charge as "ex tended supersymmetr ies . " In the lat ter case there is a unification with in ternal symmetry. Extended supersymmetry field theories generally have a na tu ra l global SO(N) in ternal symmetry , under which the QÎ t r ans form as an jV-dimensional vector. A larger U(7V) internal symmetry involving j b ro ta t ions o f the Ql

a can usual ly be defined.

There are n o w three possible viewpoints , each of which leads deeper into the subject.

1) T h e algebraic view which emphasizes the part icle representa t ions of supersymmetry algebras and proper t ies of the cor responding f ie ld theor ies ;

2) The f ield mult iplet view which emphas i ­zes the s t ructure of sets of fields which t rans­form irreducibly under supersymmetry ;

3) The geometr ic view whose major b ran­ches are superspace and a group- theore t ic ap ­p r o a c h via curvatures .

We shall a d o p t 1) in this pa r t of the repor t because i t leads mos t rapidly to an overall p ic ture of the subject. Recent work in 2) will be described in Pa r t I II , while 3) will be discussed by other speakers at this session.

T h e irreducible representa t ions of super-symmetry always consist of " towers of sp ins" conta in ing equal number s of boson and fer-m i o n states. They can be derived quite easily using W i g n e r s m e t h o d of induced represen­ta t ions . One f inds tha t the little algebras of spinor charges are mathemat ica l ly equivalent to complex Clifford algebras of N e lements in the massless case and 2N e lements in the massive case. T h e massless representa t ions (of d imens ion 27V) are therefore simpler t h a n the massive representa t ions (of d imens ion 2 2 V ) . The n u m b e r of dist inct spins in a re­presenta t ion increases wi th N. T h e lowest spin representa t ion for a given TV is a tower of spins s = 0 , 1/2, 1 , s m a x . In the mas ­sless case £ m a x = ( l / 4 ) ( A r + l ) or (1/4)JV, for odd and even N respectively. F o r massive part icles sm&s:=(l/2)N.

Higher spins are a notor ious ly difficult p rob lem in q u a n t u m f ie ld theory , a n d a t present consistent local f ie ld theor ies wi th in teract ions can be const ructed only for spins s<2. Therefore a pract ica l l imita t ion for supersymmetr ic field theories is the restric­t ion t o representa t ions with ^ m a x < 2 . T h e spin conten t of all massless representa t ions with ^ m a x < 2 i s given in Table I I . Part icles always appea r in total ly ant i symmetr ic t ensor representa t ions of the SO(A^) groups , so t ha t the multiplicit ies are b inomia l coefficients. F o r N=l supersymmetry , the representa t ions are double ts (s, s—1/2) of adjacent spin bosons and fermions.

A lmos t all theoret ical w o r k in the subject concerns proper t ies of the f ield theories cor­responding to the representa t ions in Table I I .

a ) ^ m a x = 1/2. F o r N= 1 we have the W e s s -Z u m i n o mode l wi th its str iking decrease in the n u m b e r of independent renormal iza t ion cons tan ts . F o r N=29 t he representa t ion of

Supersymmetry {Including Supergravity) 537

T a b l e II . Mass less r ep resen ta t ions " Po inca ré supe r symmet ry .

the algebra must be doubled to be described by a Lagrangian field theory. In both cases a mass term is allowed. (For N=2 this gives a supersymmetry algebra with central charges for which the restriction sm&x=(l/2)N need not apply. See Pa r t II).

b) ^ m a x ^ l - These theories are called the globally supersymmetric Yang-Mills theories. The arbitary Yang-Mills internal symmetry group appears as a direct product with the supersymmetry algebra. It is completely in­dependent of the unified SO(N) internal sym­metry which is realized globally. The Lag-rangians include minimal coupling kinetic terms plus Yukawa and quart ic interactions and are therefore conventional renormalizable theoreies. They possess both Poincare and conformai supersymmetry at the classical level. Some of the many interesting properties of these theories will be discussed below.

c ) ^ m a x = = 3 / 2 . Only free f i e l d theories exist for these representations because the rudimentary form of spin 3/2 gauge invariance which remains in the absence of gravity is insufficient for consistent interactions.

d) s m a s = 2 . The representations include a massless spin 2 particle whose interactions respect the gauge principle of general covari-ance. The corresponding f ield theories must

possess local rather than only global super-symmetry because the not ion of a constant spinor parameter is no t covariant. The field theories indeed are the locally supersymmetric unified supergravity theories.

There are more general supersymmetric theories which involve the coupling of different representations. For example, one can couple ^ m a x ^ l / 2 and smax=l representations using a supersymmetric generalization of Yang-Mills minimal coupling. Futher globally supersym­metric theories with . v m a x < l can be extended to local invariance by coupling to the fields of the supergravity gauge multiplets with sm.iX

=2.

A Global Supersymmetric Theory. The N=

1 super-Yang-Mills theory 7 corresponds to the (1, 1/2) doublet representation of simple supersymmetry. The Lagrangian is just the minimal coupling of Yang-Mil l s potentials A%pc) and Majorana spinor fields %a(x) in the adjoint representation of the arbitrary gauge group G:

(5)

Before the invention of supersymmetry it

would have been very surprising to discover

538 D . Z . F R E E D M A N

tha t this rather conventional theory had ad­ditional fermionic symmetries. Indeed, for Majorana spinors of the form 3 a=e+ifMX7],

where s and rj are constant , the Lagrangian changes by a total derivative under the varia­tions

(6)

which embody the bose-fermi character of

supersymmetry t ransformations.

Variat ions with parameter e correspond to

Poincaré supersymmetry t ransformations with

conserved Noe ther current

(7)

Variations with parameter TJ are superconformai t ransformations with Noether current

(8)

There are no local invariants in global super-symmetry or supergravity because of the close connection with space-time translations. At best there are local densities such as the Lagrangian above. Care must be taken to include the tota l derivative term in the calcula­t ion of Noe ther currents.

Basic Supergravity Theory. The N=l

supergravity theory is the gauge theory of the algebra spanned by P " , Mflv and Qa. The spin content is given by the (2, 3/2) represen­tat ion. The spin 2 graviton is the gauge quan tum of the Poincare group, while its par tner , the spin 3/2 graviton, is the gauge particle of supersymmetry. The gauge fields are the vierbein and Rarita-Schwinger fields Vafl and <pit.

The locally supersymmetric field theory of this system 8 can again be expressed as a minimal coupling Lagrangian 9 using the Car-tan-Palat ini formalism of relativity. Speci­fically

(9)

with curvature tensor

and local Lorentz covariant derivative

(10)

(11)

where ojuah is the spin connection including

torsion terms

(12)

while F ^ d e t V. The gravitational coupling constant is K^=ATZG.

The Lagrangian changes by a total deriva­tive under the variations

(13)

where e(x) is an arbitrary Majorana spinor function. The variation of <ptl involves the covariant derivative of the gauge parameter as is the case for the gauge potentials in Yang-Mi l l s theory and ordinary gravitation. This shows tha t (pfl is indeed the gauge field of supersymmetry t ransformations.

The t ransformation rules (13) are the simplest realization of the gauge principle of local supersymmetry which has several unusual theoretical features. For example, the com­m u t a t o r 1 0 of two variations with parameters ci and e 2 is (when acting on either VAFT or étt)

(14)

It involves a general coordinate t ransformation with displacement parameter f ^ z e ^ - ^ toge­ther with f ield dependent local Lorentz and supersymmetry t ransformat ions and equat ions of mot ion terms which vanish when zl9tt*y*l[p=§. At first glance this seems considerably more complicated than the super-symmetry ant i -commuta tor (1). However, global algebraic relations such as (1) are to be unders tood as acting on the particle states of the theory in an asymptotically flat back­ground geometry. In this limit (14) does imply (1) because the field dependent t rans­format ion and equat ion of mot ion terms do no t contribute. This can be shown using functional W a r d identities.

The proof of invariance of global and local supersymmetric field theories (such as (5) and (9)) can be obtained by direct calculation of the variat ion hSf using the appropria te field t ransformation laws (such as (6) and (13)). M o r e systematic methods based on f ie ld multiplets or super space are available for N=l

Supersymmetry {Including Supergravity) 539

theories, and some of them will be discussed in Par t I I I . At the nitty-gritty algebraic level all of these methods involve

a) detailed Dirac algebra, such as the relations

and

b) recognit ion of basic identities such as the gauge field Bianchi identity or the gravi ta t ional Ricci identity

c) use of Fierz rear rangement to show that expressions such as fahc(^T!lla)(Xhïi^lc) vanish because of over-ant isymmetrizat ion of spinors.

It is essential, part icularly for c) tha t all spinorial quantit i t ies such as e 9 % o r be t reated as an t i -commut ing Gras smann vari­ables. It is here tha t the connection between spin and statistics enters intimately in super-symmetry and supergravity.

Types of Supergravity Theories: The kinds of theories which have been developed so far include

i) Unified supergravity theories which are field theories of particles in representat ions with ^ m a x = 2 of Table II . These representations can be called the gauge multiplets of super-gravity.

ii) Mat te r coupling theories in which the gauge multiplets are coupled to mat ter mul­tiplets with . y m a x < l . One can hope to obtain such theories for N<4 where the relevant representat ions exist.

iii) Conformai supergravity theories which are gauge theories of the superconformai algebras and which can also be viewed as supersymmetric extensions of Weyl 's conformai invariant theory of gravitat ion.

We will now discuss the general properties and status of these theories wi thout detailed reference to Lagrangians , f ield variat ions, etc.

Unified Extended Supergravity Theories : These theories first exhibited the promise of the gauge principle of local supersymmetry for the unification of particle interactions and for a renormalizable theory of gravity. There are 8 distinct theories which may be classified according to the n u m b e r TV of gauged super-symmetry charges. Each theory has manifest SO(A0 internal symmetry and describes a set

of particles where the spin 2 graviton is unified with N graviton and lower spin part i ­cles in ant isymmetric tensor representat ions of S O ( J V ) (see Table II). Complete theories are known for A ^ 2 , u A ^ 3 , 1 2 N=4,l*>1* and there are par t ia l results for N=%.]6

In each theory all particle states are con­nected by supersymmetry and SO(N) t rans­formations. It is no tewor thy tha t this uni­fication can include spins 1/2 and 0 which are no t associated with gauge f ields bu t are necessary to construct f ield theories which gauge the extended supersymmetry algebras for T V > 3 . An approach to supergravity based on algebraic geometry which clarifies the ap­pearance of lower spin particles has recently been proposed by MacDowe l l . 1 6

Unfor tunate ly there is a very complicated non-polynomial s tructure due to the spinless fields in the extended supergravity theories for N>4. In recent work we now have two equivalent forms of the 7V=4 theory. The first fo rm 1 2 has a non-polynomia l structure in which functions such as [\—/c2(A2j\-B2)]~1 2

appear . In the second fo rm 1 3 there is a manifest SU(4) global internal symmetry and a simpler non-polynomia l structure with ex­ponent ia l functions eKip of a single scalar field. The equivalence of the two forms involves t ransformat ions of the fields which include dual t ransformat ion of vector field strengths.

The SO(8) extended supergravity theory is thought to be the largest of the class because the lowest spin representat ions of extended supersymmetry for N>9 include s p i n s > 5 / 2 and we are presently unable to describe such high spins in q u a n t u m field theory. Theories with 7V=5, 6 are expected to be subcases of the SO(8) theory while the N=l representat ion has the same particle content as iV=8 and is expected to be equivalent. The N=S theory has therefore been approached as the next barrier beyond TV—4. Due to the complicated non-polynomia l structure, there are at present only part ial resul ts 1 5 which give all terms in the Lagrangian th rough order /c2 in the gravi­ta t ional coupling. However a supergravity theory of the simple supersymmetry algebra in 11 space-time dimensions has recently been constructed by Cremmer , Julia and Scherk. 1 7

After compactification of 7 spatial dimensions one expects a 4-dimensional supergravity

540 D . Z . FREEDMAÎ

theory with SO(7) internal symmetry which

should be related to the 80(8) theory.

It may seem that the SO(8) supergravity

representation is large enough to accomodate

all the elementary particles which are known

or suggested by present experiments. How­

ever, the opposite is true; the SO(8) theory

is too small. In a classification of the states

with respect to an assumed exactly conserved

SU(3) color subgroup of SO(8), Gell-Mann 1 8

has found that the spin 1 multiplet breaks up

into SU(3) (electric charge) quantum numbers

as 2 8 = 8 ( 0 ) + 1 ( 0 ) - l ( 0 ) + 3 ( - l / 3 ) + 3 ( - 1 / 3 ) + 3 (2 /3 )+3 ( l / 3 )+3 ( l / 3 )+3 ( -2 /3 ) . These particles can be identified with colored gluons, the photon, and weak neutral Z boson plus fractionally charged superheavies of the same type as occur in other grand unification attempts. The spin 1 /2 states have the follow­ing decomposition as Dirac spinors: 3(2/3)+3( — 1 / 3 ) + 3 ( — 1 /3 )+3(2 /3 )+6( 1 /3) + 8 ( 0 ) + l ( — 1 ) + 1 / X 0 ) + 1 L ( 0 ) and can accomo­date the u, d, s, c quarks, a fifth quark flavor which is a color sextet, a neutral octet, one charged lepton (the electron?) and two neutri­nos. The particles missing from the scheme are charged vector bosons W ± and leptons such as fi and r. The essential reason for this deficiency is that SO(8) is not big enough to contain the product subgroup SU(3)x S U ( 2 ) x U ( l ) of color with weak and electro­magnetic flavor. From a purely phenomeno-logical standpoint the A^=9 and 7 V = 1 0 super-symmetry algebras have identical . y m a x = 5 / 2 representations whose lower spin sector is favorable for a grand unification. However, there is so far no indication that a consistent field theory with a "pentat ino" and 10 ''gravi­tons" can be formulated.

It is hard to see how to overcome the pro­blem of c o n s t r u c t i n g a u n i f i e d r e a l i s t i c super-gravity theory. Possible lines of approach, each with its own difficulties, are a) field theories based on several coupled . y m a x = 2 representations ; b) superconformai theories which have a natural gauged U(AQ internal symmetry but which are plagued with ghosts (see below), and c) interpretation of the present set of elementary particles as effective low energy excitations which do not correspond exactly to the fields of supergravity theories.

The latter might be evident only near the Planck energy of 10 1 9 GeV.

We now turn back to the theoretical features of the unified theories and note that the theories were generally constructed in two stages. In the first stage the only coupling constant i s the gravitational constant t c = ( 4 ; r G ) 1 2 ^ 1 0 - 1 9 G e V - 1 and the Lagrangians and transformation rules were found by an iterative procedure in fc. At this stage the SO(7V) internal symmetry is global and there are (l/2)N(N—l) vector fields in the adjoint representation of SO(A^) (as can be inferred from the particle content given in Table II) which interact non-minimally via field strengths F)i=dtlAV—duAiJ. At the next stage of con­struction SO(7V) is gauged using Yang-Mills minimal coupling with charge e and then determining additional terms necessary to maintain the fermionic gauge invariance. 1 9

In this way a marriage of the gauge principles of local internal symmetry and local super-symmetry is achieved, but it is a troubled marriage as we will discuss shortly.

The renormalizability properties of the ex­tended supergravity theories are studied (for e=0) by expanding the vierbein Vafl about a flat background geometry and considering the properties of radiative corrections in loop graphs involving virtual gravitons, gravitinos, etc. interacting with strength tc. Since tc car­ries negative dimension, high powers of virtual momenta occur in Feynman integrals and one cannot hope for traditional renormalizability where divergences are absorbed in redefinition of a finite number of parameters. Instead one hopes for a finite theory in which diver­gences may be present offshell but cancel in S-matrix elements. Prior to the development of supergravity it was known that such cancella­tion occurs in one-loop order in pure general relativity (self-coupled gravitons only) but fails when conventional lower spin matter couplings are added.

One-loop finiteness of the unified theories was first demonstrated 2 0 by arguments which used global supersymmetry to relate 4-point amplitudes with external lower spin to ampli­tudes with external gravitons for which a generalization of the known proof of finiteness could be applied. These theoretical arguments were confirmed by explicit calculation of the

Supersymmetry {Including Supergravity) 541

divergent pa r t of an ampl i tude in the N=2

theory and , subsequently, by other one-loop ca lcula t ions . 2 1 F u r t h e r a rguments showed tha t the loca l 2 2 and non- loca l 2 3 divergences a t the two- loop level cancel on-shell in simple N= 1 supergravity. It is still u n k n o w n whether this result also holds in pure general relativity.

An app roach to renormalizabi l i ty based on locally supersymmetr ic counter terms was also initiated at an early s tage . 2 4 A locally super-symmetr ic counte r te rm is an integral ex­pression of schematic form

(15)

involving p roduc t s of variously contracted curvature tensors Rftuab and terms involving <p(l. The integral is invariant under local supersymmetry t ransformat ions . W h e n there are L + l factors of R such an integral cor­responds by power count ing to the possible opera tor form of a divergence of the sum of all F e y n m a n graphs with L loops in N=\ supergravity. I t was argued tha t for L = l and 2 the only possible counter terms vanish when equat ions of mot ion are used so tha t all S-matrix elements are finite. However indi­cat ion was found for an L = 3 counter term which does no t vanish on-shell and is a candi­date for a genuine divergence of scattering ampli tudes in three- loop order .

There are two more recent developments in the counter te rm approach . Firs t the L=3

counter term ment ioned above has been gene­ralized to N=2 supergravi ty , 2 5 so tha t unifica­tion with internal symmetry does no t appear to cure the threa tened non-renormalizabi l i ty . Second, general techniques have been develop­ed for the const ruct ion of complete locally supersymmetr ic counter terms for N= 1 super-gravi ty 2 6 (see P a r t III) which show tha t there are invariants which correspond to possible divergent scattering ampl i tudes for any order in per tu rba t ion theory beyond L = 3 . At this point i t c anno t be excluded tha t the coefficients of the candida te divergent counter terms vanish, b u t no theoret ical reason for such a miracle is apparen t .

T h e difficulties of unified extended super-gravity theories with gauged internal sym­metry arise from cosmological terms, such as the t e rm (3/2)Pe 2 /c~ 4 which appears in the

Lagrangian density of the SO(2) and SO(3) invariant theories with value dictated by the requirement of local supersymmetry. Quan­titatively this te rm canno t be related to a macroscopic cosmological cons tant because it is more than 100 orders of magni tude larger t han the as t ronomical upper limit. A field theory with cosmological term canno t be quan­tized in a flat background geometry because Minkowski space is no t a solution of the back­ground f ield equat ions . In the present case the maximal ly symmetr ic background geometry is a de Sitter space of 0 ( 3 , 2) signature. Since this geometry does no t have global Cauchy surfaces, it has been though t tha t a well-posed initial p rob lem and f ie ld quant iza t ion are no t possible. Recent work indicates tha t these difficulties m a y be overcome. 2 7

There are even more puzzling features as­sociated with the gauged SO(vV) internal symmetry for N=4 which are also likely to occur for N>4. Because there are spinless f ields the cosmological te rm is no longer con­stant, b u t is replaced by a scalar field potent ial Vf(A9 B). The two forms of the N=4 theory discussed above which are equivalent for zero gauge charge become inequivalent when the internal symmetry is gauged, and the second form leads to a theory with S U ( 2 ) x S U ( 2 ) gauge g r o u p . 2 8 The prob lem is tha t the potentials f(A, B) in all of these theories are u n b o u n d e d from below, and at the naive level this suggests the absence of a stable vacuum state.

There is now a gl immer of hope tha t the formidable problems associated with the cos­mological t e rm can be overcome. This hope arises in the app roach to q u a n t u m gravity taken by Hawking and several col laborators in which the topological aspects of geometries summed in the pa th integral play a key role. Even for pure general relativity, the Eucl idean action is u n b o u n d e d from below, and it is necessary to m a k e a con tour ro ta t ion in func­tion space to define the pa th in tegral . 2 9 Fur ­ther H a w k i n g 3 0 has been led to in t roduce a cosmological t e rm in the microscopic Lagran­gian as a Lagrange multiplier. A picture of the gravi ta t ional vacuum as a "space- t ime f o a m " of virtual b lack holes emerges in which the geometry is highly curved on the micro­scopic scale but appears nearly f lat on length

542 D . Z . F R E E D M A N

scales larger than the Planck length. A microscopic cosmological constant of the same sign and order of magni tude as in super-gravity is predicted. These ideas must be developed considerably further before the questions associated with the extended super-gravity theories can be settled, bu t it is im­por tan t tha t there is a new avenue of approach involving such challenging and elegant con­cepts.

Matter Coupling Theories involving the (1 , 1/2) and (1/2, 0) multiplets of N=\ super-symmetry were constructed quite early. The successful extension of field theories from global to local supersymmetry by coupling to the (2, 3/2) gauge fields was impor tan t in demonstra t ing tha t local supersymmetry is a t rue gauge principle and in determining the basic terms of supergravity Lagrangians. However, mat ter and gravity are not unified in such theories and it is known by explicit calculation in several cases 2 1 that one-loop scattering ampli tudes are infinite.

Recent progress in the mat ter coupling in­cludes the discovery 3 1 of models with a super-Higgs effect in which the gravitino acquires a mass as a consequence of spontaneous break­down of local supersymmetry. Cosmological terms cancel at the min imum of the scalar field effective potential , so tha t there is no difficulty in the interpretat ion of the theory at least at the classical level. Readers should refer to the contr ibut ion of J. Scherk to these Proceedings which describes a very general fo rmula t ion 3 2 of the super-Higgs effect. There have also been recent constructions of mat te r coupling theories for N=2 supergravity. One theory involves an arbitrary number of massless mul t ip le ts 3 3 with sm^=l. Another describes a massive multiplet with sm^=\j2 which has a central charge in its a lgebra . 3 4

Superconformai Theories: The general con­viction tha t symmetry is powerful is sufficient to motivate the study of field theories invariant under super conformai algebras which are the largest permit ted in the framework studied by H a a g et ai (and which for N= 1 is an algebra with 24 generators). Globa l Poincaré super-symmetric field theories of massless particles are automatical ly superconformai invariant as is indicated by the example of super Yang-Mi l l s theory given earlier. Supercon­

formai invariance fails for the supergravity theories discussed until now because the general relativity Lagrangian is not invariant under local conformai or Weyl t ransforma­tions.

The construction of local superconformai the­ories has been considered by Kaku , Townsend, and van Nieuwenhuizen . 3 5 The iV= 1 theory is known in closed form and there are partial results for N>1. The construction involved the consideration of potentials and curvatures for the full superconformai group. Constra­ints on the curvatures were imposed in order to construct the locally invariant action which in final form involves the spin 2 vierbein and self-conjugate spin 3/2 and spin 1 fields (p,t

and A^. The full Lagrangian is complicated, bu t the bilinear kinetic terms are

;i6)

No te that the gravitat ional terms are those of Weyl 's conformai invariant theory, and the superconformai theory can be regarded as the supersymmetric extension thereof.

The full Lagrangian involves one coupling e which is the U( l ) gauge coupling constant , and one may hope tha t all forces can be derived in terms of this single parameter . Two problems make it difficult to fulfill this hope. First, there is no known way to re­cover the macroscopic predictions of general relativity from the Weyl theory and, second, there are negative metric ghosts due to the higher derivative terms, as has been clarified in a recent study of the linearized theory . 3 6

Progress on these points would open a major new avenue for unification in physics.

§11. Selected Recent Work in Global Super-symmetry

Superconformai Anomalies: The generators S a of superconformai t ransformations satisfy {S9 S}=yK where Kp is the conformai gene­rator . Kp is conserved classically in a massless field theory of spin 0, 1/2, and 1 but anomalies occur at the q u a n t u m level because a length scale is inevitably introduced to define Feyn-m a n amplitudes with loops. One should expect analogous anomalies for Sa9 and such anomalies were independently discovered by

Supersymmetry {Including Supergravity) 543

several g roups , 3 7 a l though with somewhat con­fused initial interpretat ions.

In the supersymmetr ic Yang-Mi l l s theory discussed earlier, there is a complete parallel between the di la ta t ion and superconformai anomalies as indicated in the following table :

Noe the r Divergence Dimensional current regularization

Dilatations

Superconformai

The terms which do not vanish for n^4 indicate schematically how an anomaly can arise in a dimensionally regulated calculation of one-loop ampli tudes . This is merely suggestive because dimensional regularizat ion is invalid in supersymmetr ic theories. However , one may calculate the one- loop ampl i tude for / l l using a regulator- independent m e t h o d which gives

(17)

to one- loop order , where the renormal izat ion group function /3(g)=3g 3CV/167r 2 appears .

The superconformai anomaly m a y be com­pared with the axial current and di latat ion anomalies

(18)

which are k n o w n to all orders in gauge theories of vectors a n d spinors independent of super-symmetry.

In a class of supersymmetr ic theories with classical b r eakdown of conformai invariance via mass te rms, the opera tors above t ransform together under supersymmetry t ransforma­t ions . 3 8 F o r example

(19)

At the one- loop level, the q u a n t u m anomalies writ ten above also obey these t ransformat ion rules . 3 9 The y J? anomaly is no t known beyond one- loop order bu t one can already see a conflict between (17), (18) and (19) be­cause the axial anamoly contains no higher order radiat ive correct ions (the Adle r -Bardeen theorem) bu t /3(g) a n d therefore 0£ do . The

resolut ion of this conflict is an interesting open prob lem which may involve only technical aspects of the renormal iza t ion procedure in supersymmetric gauge theories, bu t may possibly lead to new physics.

Vanishing Two-Loop /3(g): The largest glo­bal supersymmetr ic theory is based on the ^ = 4 representat ion with £ m a x = 1 and describes Yang-Mi l l s potentials A% 4 Majorana spinors XXa, and 3 scalars and pseudoscalars AXa and BXa all in the adjoint representat ion of the gauge group . The Lagrangian was found by dimensional reduc t ion 4 0 of the dual fermion model in 10 dimensions and consists of the minimal coupl ing kinetic terms plus Yukawa and quar t ic coupling terms all involving the single coupl ing cons tant g. Earlier resul ts 4 ' in related theories indicated tha t /3(g) vanished in one- loop order for this set of fields, bu t did not vanish in two- loop order , and the question was reopened in connect ion with the N=4 theory with its high degree of symmetry. A vanishing two- loop con t r ibu t ion 4 2 to /3(g) was found! The JV=4 supersymmetr ic Y a n g -Mills theory is the only known field theory with no coupl ing cons tan t renormal izat ion th rough two- loop order. I f this remarkable p roper ty remains t rue to all orders the theory would be exactly conformally covariant .

Central Charges: The concept of central charges in supersymmetry was first discussed by Haag , Lopuszanski and Sohnius 4 who found a consistent modification of the Poincare super-symmetry algebra in which (1) is replaced by

(20)

Here Uij and Vij are Hermi tean scalar and pseudoscalar central charges which are anti­symmetric in indices //. By definition central charges commute with all elements of the algebra. Since the charges have dimension 1, they can only occur in field theories with a dimensional pa ramete r such as a mass . The N=2 massive mult iplet wi th ^ m a x = l / 2 was the first example of a field theory with central charge . 4 3

Our unders tanding of the physics of central charges has improved greatly recently due largely to w o r k of Wit ten and Ol ive . 4 4 They considered the A r = 2 supersymmetry algebra where b o t h central charges are p ropor t iona l to the a l ternat ing symbol eij and (20) takes

544 D . Z . F R E E D M A N

the form

(21)

One result of Witten and Olive is a lower

bound on the mass M of particles in a re­

presentation of this algebra in terms of central

charge eigenvalues. The bound

(22)

is easily derived in the rest frame using the

positivity properties of the anti-commutator.

Fur ther analysis of (21) shows that when the

bound is not saturated, the irreducible repre­

sentations are 16 dimensional as in the case

of M ^ O representations of N=2 super-

symmetry without central charges. When the

bound is saturated, the 8 x 8 matrix on the

right side of (19) has 4 vanishing eigenvalues,

so that the Clifford algebra is smaller. One

then finds massive irreducible representations

of dimension 4, as in the case of m=0 re­

presentations without central charges. The

phenomenon of reduced size massive represen­

tations had also been noted by Sohnius 4 5 for

N=2 and by Faye t 4 6 for both N=2 and N=4.

Witten and Olive study the supersymmetry

algebra in the N=2 super Yang-Mil ls theory

with gauge group SU(2). The fields are the

gauge potentials A%{x\ two triplets of Majo-

rana spinors ^ ? ( / = l , 2 ) , and scalar and pseu-

doscalar triplets Aa(x) and £a(x). The Lag­

rangian

(23)

consists of covariant kinetic terms and Yukawa and quartic couplings. They calculate the ant icommutator {Q, Q) and find that the central charges appear as surface terms of the form

(24)

Spontaneous breakdown of the SU(2) gauge symmetry can occur in this model preserving a U ( l ) subgroup, which defines an electro-magnetism, and preserving supersymmetry. It is also known 4 7 that the model contains classical soliton solutions which are magnetic monopoles and dyons and zero energy fermion solutions which are their supersymmetric

partners and can be called solitinos. In this situation the central charges become

where {A} is the vacuum expectation value

and E and G are electric and magnetic charges.

Central charges are therefore related to topolo­

gical charges !

The final point of Witten and Olive derives from the observation that for all known Higgs mechanism and soliton states the mass formula

(26)

holds through first order in quantum correc­

tions. This corresponds to saturation of the

lower bound discussed earlier, and the massive

states therefore span reduced size super-

symmetry representations. Higher order quan­

tum corrections are unlikely to introduce new

particle states. The reduced size multiplets

persist and the mass formula must be exact to

all orders, which is a remarkable result in any

quantum field theory !

There are several other subjects of recent

research in global supersymmetry which can­

not be discussed in detail because of the page

limitations on this report. Among them are

interesting work on two dimensional super-

symmetric theories 4 8 especially the construction

of some exact S-matrices 4 9 the derivation of a

supersymmetric non-linear <7-model 5 0 in four

dimensions, and an extensive research pro­

gram 5 1 on realistic models for hadron and

lepton phenomenology. Many problems have

been overcome in constructing phenomeno-

logical models and, al though complicated,

they make characteristic predictions for a

new class of particles which carry an additive

quan tum number R.

§111. Selected Recent Work in Supergravity

Many new results have already been men­tioned in Part I, and we discuss a few additional topics here.

Auxiliary Fields and Multiplet Calculus for

N=l Supergravity: The main results here are systematic procedures to construct locally supersymmetric invariants for simple super-gravity. There have been applications to matter coupling theories and to locally super-symmetric counter terms.

Supersymmetry {Including Supergravity) 545

The recent development was motivated by the calculation of the commuta tor of two supersymmetry variations, see eq. (14), in which equation of mot ion terms appeared. Although supergravity can be perfectly well formulated in terms of the physical fields Vafl(x) and (p/t(x), the equation of motion terms mean that Va[x(x) and <pti{x) are an incomplete multiplet of fields, which do not realize a closed algebra of local supersymmetry, Lorentz and coordinate transformations independent of dynamics. Similar situations occur in global supersymmetry. For example, the commutator algebra of the super-Yang-Mills theory does not close on the physical fields Ap(x) and %{x\ bu t an auxiliary pseudoscalar D(x) can be introduced to form a complete multiplet.

In supergravity three g roups 5 2 have recently found that a complete multiplet of fields can be formed from Vatl9 <pfi and auxiliary axial vector, scalar and pseudoscalar fields called A,l9 S and P. This set of 6 auxiliaries super­cedes a larger set found earlier. 5 3 The local supersymmetry variations of the new multiplet are

(27)

where is the spin 3/2 Euler variation in supergravity. The com­mutator of two of these transformations again takes the form of (14) with no equation of motion terms and a more complicated local Lorentz parameter involving the auxiliary fields. Thus a non-linear but closed local algebra has been obtained. It is the universal algebra of all JV=1 supegravity theories en­compassing pure supergravity and matter coupling models.

The Lagrangian

(28)

is invariant under (27). F r o m the equations of motion one finds that all auxiliaries vanish, and everything reduces to the form of super-gravity discussed in Par t I (with new con­ventions). Matter coupling theories can be reformulated using auxiliary fields with some simplification of s t ructure. 5 4 The auxiliary field equations relate Aft9 S, and P to the matter fields.

Fur ther progress has emerged from the deri­vation of the "tensor calculus" based on scalar multiplets by Ferrara and van Nieuwenhuizen 2 6

and based on vector multiplets by Stelle and West . 5 5 Both are direct generalizations of well-known aspects of global supersymmetry. We can describe only the basic notions of the scalar multiplet calculus. The vector multiplet calculus is entirely parallel, and has advantages in some applications.

A local scalar multiplet is any set of com­ponent fields A, B, x> F> G with the following transformation rules

(29)

where Dfi is a supercovariant derivative 5 6 de­fined to transform without derivatives of e(x).

For example DfiA=d/tA—l/2ic<pflx. Scalar multiplets can be formed from matter fields, or from any combinations of Vafi9 Afl, S,

P 9 for which the transformation rules (29) can be established.

A calculus of scalar multiplets can be set up in which the product of two multiplets, the derivative of a multiplet, and a local scalar density can be defined. The density formula tells us that , given the components of any scalar multiplet, the integral

(30)

is invariant under local supersymmetry trans­formations. Therefore the problem of con­structing supergravity Lagrangians is trans­formed into the problem of finding local scalar multiplets and applying (30). This is not

546 D . Z . F R E E D M A N

intrinsically easier, but it does permit input from other approaches. For example, scalar multiplets involving matter fields have been found using global supersymmetry results, while other multiplets involving various com­binations of the supergravity gauge multiplet Vaft9 <pft, A!l9 S, P have been constructed using results of the superspace approach . 3 6

One such multiplet, called the scalar cur­vature multiplet, has components

(31)

The scalar density formed from this multiplet according to (30) is .? S G of (28). F rom similar multiplets containing the Weyl tensor C,ivpa. one can construct the locally supersymmetric counter terms discussed earlier in connection with the renormalizability question.

The several applications of the multiplet calculus formalism, both in the scalar and vector case, illustrate the power of the for­malism. On the other hand, it is clear from (31) tha t it is not entirely simple. It seems unlikely to me that any simpler procedure will be found because of the close relation of the multiplet calculus to global supersymmetry and superspace. 5 7 The formalism is presently limited to simple supergravity and we must hope that extensions to N>2 can be found.

New Ghost Couplings : Another curious con­sequence of the non-closure of the commu­tator algebra (14) is that the naive generalization of the Faddeev-Popov procedure to super-gravity formulated with physical fields Va/l

and <pft is incorrect. The usual gauge fixing t e rm 5 8 is F((ft)=y!t(f>!L and both fixed flat space and covariant y matrices may be used. Naive­ly, one would expect a ghost Lagrangian of the form c^j!l£2)t c, where c and c* are com­muting Majorana spinor f ields, and the addi­tional te rm (c*y!l(pv) (cp^c), from the vierbein variation of F(x), for covariant fs. It is now known tha t one must add the quartic coupling of the ghost fields {c*rac*)(cYac). This term has now been derived from the viewpoints of

the canonical quantization formalism, 5 9 BRS identities and unitar i ty, 6 0 and from auxiliary fields. 6 1 In fact, the necessity for quartic coupling terms is quite easily seen from the auxiliary field viewpoint. Naive covariant gauge quantizat ion is correct for the Lagrangian (28), if the auxiliary field terms in the trans­formation ô(p!t in (27) are included in the com­putat ion of the Fadeev-Popov determinant. However, it is then clear that quartic ghost couplings will arise when auxiliary fields are eliminated, using the equations of mot ion of the effective Lagrangian.

The Lagrangian of a Rarita-Schwinger field in a fixed external background metric which satisfies the Einstein equations i ? / i v = 0 gives a consistent spin 3/2 field theory. The covari­ant gauge condition F(cp)^j!lc})/£ and weight term à-y^jj-cp are then very natural . Niel­sen 6 2 has shown that a third spin 1 /2 ghost is then necessary for the unitarity of one-loop amplitudes.

Anomalies'. The c o n f o r m a i 6 3 - 6 5 and a x i a l 6 4 - 6 7

anomalies have now been calculated for par­ticles of spin 3/2, and even for arbitrary spin. 6 4

Tabulat ions of the conformai anomaly for various supergravity theories have been made. Curiously the anomaly vanishes in the unified SO(3) theory indicating that this theory will have finite one-loop "S-matr ix" even in a background geometry with non-trivial Euler characteristic.

The form of the gravitational axial anomaly is

and the value / = 1 obtains for a Majorana spin 1/2 particle. Several groups using dif­ferent methods have found the value À=—2\ for a Majorana gravitino in an external field with covariant gauge fixing. It has also been found that this result is correct for super-gravity, 6 8 and tha t the value Z=—4 found earlier is in er ror . 6 9 Finally, there has been a recent s tudy 7 0 of eigenmodes of wave operators up to spin 2 in a self-dual gravitational instan-ton background, which have an interesting relation to supersymmetry.

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C 6

Supersymmetry assigns equal masses to bo ­sons and fermions in the same multiplet. Since such a degeneracy is no t observed in Na tu r e , i t is impor tan t to break supersymmetry either spontaneously or explicitly. We opt for spontaneous symmetry breaking since the in t roduct ion of n o n symmetrical terms would m a k e the theory lose all predictive power . The recent advances in supergravity, namely the discovery of a minimal set of auxiliary fields, and the establ ishment of a tensor cal­culus, allow us to construct the most general coupl ing of supergravity (2, 3/2) to the scalar mult iplet (1/2, 0 + , 0 - ) . 1 We recover a s special cases all the previously derived couplings and show tha t the mode l depends u p o n an arbit­rary function G(A9 B) of the scalar (A) and pseudoscalar (B) fields.

Fu the r we show tha t for a very large class of such functions, spontaneous symmetry breaking of supersymmetry takes place. The spinor x field of the scalar mult iplet plays the role of a Golds tone fermion of super-symmetry (Goldst ino) . I t is then absorbed by the spin 3/2 gauge field of supergravity (gra-vitino), jus t like in the Higgs model , after which banque t the gravit ino becomes massive. In addi t ion, this can occur wi thout develop­ing a cosmological constant , due to a cancella­t ion between terms of opposite signs.

W h e n supergravity 2 was f irst discovered, i t

(1978) 5 7 1 .

65 . M . P e r r y : C a m b r i d g e Un ive r s i ty D . A . M . T . P .

p r ep r in t .

66. P . Cr i t ch l ey : Un ive r s i ty o f M a n c h e s t e r p r e p r i n t .

67 . N . K . Nie l sen , M . G r i s a r u , H . R o m e r a n d P . v a n

N i e u w e n h u i z e n : C E R N p r e p r i n t T H 2 4 8 1 .

68 . N . K . N i e l s e n : N O R D I T A p r e p r i n t .

69. M . G r i s a r u , D . F r e e d m a n a n d P . v a n N i e u w e n ­

h u i z e n : P h y s . Le t t e r s 7 1 B (1977) 377.

70. S . H a w k i n g a n d C . P o p e , C a m b r i d g e U n i v e r s i t y

D . A . M . T . P . p r e p r i n t

was r emarked 3 t ha t the algebra of local super-symmetry t ransformat ions did not close unless one used the equat ions of mo t ion of the spin 3/2 field, a p h e n o m e n o n which occurs also in f lat space supersymmetry when auxiliary (non-propaga t ing) f ields are el iminated by use of their equat ions of mot ion . This si tuation was cured by the discovery 4 of a very simple set of 6 auxiliary fields, consisting of an axial vector A, a scalar S and a pseudoscalar P.

This led in tu rn to the development of a tensor calculus 5 which generalizes to curved space the results originally obta ined in flat space by Wess and Z u m i n o . Tensor calculus applies b o t h to scalar and vector mult iplets . Since here we are interested in the coupling of supergravity to a scalar mult iplet , we give a very short summary of the tensor calculus for scalar mult iplets .

A scalar mult iplet is a set of 5 objects Ï~~(A, B9 x> F\ G') which have well defined propert ies under local supersymmetry t rans­format ion. F o r instance dA=e(x)x9 3B=

-Hx)rsX, e t e The fields A 9 F' {B9 G') are scalars (pseudoscalars) and % is a Ma jo rana spinor.

The tensor calculus 5 consists of two basic operat ions . The first is the mult ipl icat ion, which to two scalar mult iplets I l 9 2 2 associate their p roduc t 2=21®22. In componen t form, one has A=A1A2—B1B2 and so on for the

Super-Higgs Effect in Supergravity

J. SCHERK

Laboratoire de Physique Théorique de VEcole Normale Supérieure, Paris

Supersymmetry {Including Supergravity) 549

other components . This operation is commu­tative and does not involve any derivatives. It is purely algebraic.

The second operation is called derivation and associates to a multiplet I its derivative T(£). The third component of T(I) contains a supercovariant generalization of the Dirac operator applied on x a n < i the fourth and fifth components supercovariant generaliza­tions of the d 'Alembertian operator • applied on A and B.

Finally an invariant action can be obtained from any scalar multiplet S by the formula:

In this formalism, the construction of invariant actions under supersymmetry trans­formations is very straightforward and is ana­logous to the construction of covariant action in curved space.

The most general interaction between super-gravity and a scalar multiplet which involves no more than one derivative on the fermi field and 2 derivatives on the bose field can thus be written as :

anm can be taken real and symmetric as one can show that I(Z®T(A)-A®T(Z))=0 and we wish to have a parity conserving model. The action will thus depend a priori on two functions of the spin 0 fields.

where First one computes explicitly / as a function

of eafi9 <pfi9 Aft9 S9 P9 A, B, z, F\ G'. The fields Ap9 S9 P9 F\ G' are auxiliary fields and appear only quadratically in the result. Thus they can be eliminated by solving a set of linear equations and one obtains a reduced action depending only on the physical fields efla,

<p[l9 x, A, B.

The reduced action needs still to be put in a canonical form, as for instance, at tha t stage, the Einstein scalar curvature R and the Ra r i t a -Schwinger Lagrangian appear multiplied by the function 0(z, z).

Thus one redefines the field by a Weyl

rescaling such that the scalar curvature R

appears in the pure Einstein form with the correct normalization. Similarly one rede­fines the fields <pfi and x s u c h that the kinetic terms of the spin 3/2 and spin 1/2 fields are the canonical correctly normalized terms mini­mally coupled to gravity.

These transformations are compatible with the reality of the vierbein and the Majorana property of the spinor fields under very general conditions, namely:

^ ( z , z ) < 0 and G9Zz<0

where G(z, z ) = 3 In [ - ^ l - l n ^ v L 3 J 4

Remarkably enough, the final action and transformation laws involve only the function <7(z, z) rather than <f> and g separately. Fur­ther the condition G9ZZ-<0 also implies that the A 9 B fields kinetic terms have the right sign, />. , that these fields have positive metric and are not ghost-like.

The final action is given by:

where the first term is the supergravity actioi and

where the bosonic action is given by :

and the potential K b y :

The fermionic action I°F contains the kinetic term of the x field and terms bilinear in the fermi fields <pfl and but does not contain derivatives of the scalar fields:

where by definition if M = R e M+iy6 Im M. Finally 7 I N T contains quart ic terms in the

fermi fields (fift and x and bilinear of the fermi fields multiplied by derivatives of the scalar fields, for instance of the type (G. ^3 ; iz—

This action is the most general coupling

550 J . S C H E R K

of supergravity to the scalar multiplet and depends upon one arbi t rary function G(z9 z) of two real variables. It includes as special cases all previously derived c o u p l i n g s . 6 - 1 1

The t ransformat ion laws of the redefined

f ie lds can also be computed . F o r the discus­

sion of the super-Higgs effect, we shall only

need the t ransformat ion law of y :

where the dots indicate cubic terms in the

fermi fields. Using this very general result we can discuss

the super-Higgs effect 1 2 ' 1 3 in a mode l indepen­

dent way. In flat space a necessary condi t ion

for spontaneous supersymmetry breaking to

occur is t ha t the auxiliary field F' (resp. D in

the vector multiplet) can pick up a non-zero

vacuum expectation value. I t then follows tha t

d% contains a term d%=--(\/a)£+ • • • where a is

a constant .

The supersymmetry charge Q a does no t

annihi late the vacuum, and a zero mass ex­

citation of spin 1/2 (Goldst ino) is seen to be

present in the theory. In addit ion, a cosmo­

logical constant of f ixed sign ( + ) is induced by

the supersymmetry breaking.

In curved space, Deser and Z u m i n o 1 3 used as a model for spontaneous breaking of super-symmetry the coupling to supergravity the non-l inear Vo lkov-Aku lov Lagrang ian 1 4 which contains only one fermi field % transforming as 0%=(l/#)e+/a£?' / ix9/£%. As d% contains a (l/a)e t e rm, % is a candidate to represent a Golds t ino f ield. The presence of this te rm implies a negative cosmological cons tan t ; however if one introduces a mass term for the spin 3/2, ano ther cosmological constant of positive sign arises. Since experimentally the cosmological constant is very small, one im­poses t ha t the net cosmological constant vanishes. One then finds 1 3 t ha t m\=k2j6a2

and is very small. Since 8% contains a con­stant t e rm propor t iona l to e(x) and we have one spinor gauge degree of freedom, x can be gauged away completely and is absorbed by the gravit ino which becomes massive. A massive gravit ino indeed has 4 helicity states ± 3 / 2 , ± 1 / 2 .

In our general model we can see the same

phenomenon occurring for a large class of functions G(z9 z).

The potent ial V(z9 z) reaches its m in imum for a certain value z 0 such t h a t :

If we require tha t the f inal theory does no t

violate par i ty z 0 =<(z ) mus t be real.

Fur the r we can impose tha t the absolute m i n i m u m of V is reached for V(z09 z 0 ) = 0 which implies the absence of a cosmological constant , and tha t V>0 everywhere. It is indeed possible for V to be n o n negative as it contains two terms of opposite sign ( remember tha t G 9 zz<0 is a necessary condi t ion for the A 9 B fields no t to represent ghost particles of negative metric).

Finally the condi t ion for spontaneous break­

ing of supersymmetry is t h a t at the min imum,

ôx contains a cons tan t t e rm times e(x). Look ­

ing at 8% we see t h a t the requi rement is t ha t

l / a = ( e x p - G / 2 ) { G , W2) 1 / a }U=z 0 *0. The condi t ion t h a t and tha t V(z09

z 0 ) = 0 are compat ible a s e v a n i s h e s precisely

if

in which case 1 /a2=6 exp (—G).

Unless G becomes infinite at tha t po in t t o o

1J a is n o n zero.

W h e n these very general condi t ions for supersymmetry breaking to occur are met , one can explicitly show by a further redefini­t ion o f the fermi f i e ld s t h a t x c a n be com­pletely el iminated from the act ion and tha t the spin 3/2 acquires a mass given by m}, = l/6a2 =

e~G (in k=\ units) , recovering the result obta ined by Deser and Z u m i n o . 1 3 I t is easier however to write the full act ion in the % = 0 gauge where the Golds t ino has been "ea ten u p " by the gravi t ino :

In general A a n d B acquire different masses. In the simple case where G9ZZ-= — 1/2 (canonical

Supersymmetry (Including Supergravity) 551

kinetic term for A, B) one can establish the

general mass formula :

which is independent upon the remaining arbitrary function of one variable g(z). Par­ticular examples of functions exhibiting the super-Higgs effect are easily found.

Our investigation shows thus tha t the super Higgs effect can take place in a large class of models, and that realistic models where supersymmetry is spontaneously broken even in curved space can be constructed, without having huge cosmological constants. It would be interesting to extend these result to the 0(N) supergravity theories which once they are gauged, create their own potential for N>4 since they contain spin 0 fields. Unfor­tunately the potentials found in the 0 (4 ) and SU(4) theories 1 5 do not lead to spontaneous symmetry breaking, being unbounded below, which is a disappointing result. However, the discovery of auxiliary fields for 0(N) super-gravity theories may reveal more flexibility in this construction than originally thought, as similarily it was believed that the coupling of supergravity to the scalar multiplet was unique.

References

1. E. C remmer , B. Jul ia , J . Scherk , P . van Nieu-

wenhuizen , S . F e r r a r a a n d L . G i ra rde l lo :

L P T E N S 78/17 p repr in t ( to be publ ished) and

C E R N prepr in t (in p repa ra t i on ) .

2 . D . Z . F r e e d m a n , P . v a n Nieuwenhu izen a n d

S. F e r r a r a : Phys . Rev . D 1 3 (1976) 3214; S. Deser

a n d B. Z u m i n o : Phys . Let ters 62B (1976) 335.

3 . D . Z . F r e e d m a n a n d P . v a n N ieuwenhu izen :

Phys . Rev . D .14 (1976) 912.

4 . S . F e r r a r a a n d P . v a n N i e u w e n h u i z e n : Phys .

Let ters 74B (1977) 3 3 3 ; K. Stelle a n d P . C. W e s t :

Phys . Let ters 74B (1977) 330.

5 . S . F e r r a r a a n d P . v a n N i e u w e n h u i z e n : L P T E N S

prepr in t 78/14, to be publ i shed in Phys . Let ters

B; S . F e r r a r a a n d P . v a n N ieuwenhu izen : Phys .

Let ters 76B (1978) 404 ; P . C. Wes t a n d K. S .

Stel le: I C T P repor t /77-78/15 a n d I C T P / 7 7 -

78/24.

6 . S . F e r r a r a , F . Gliozzi , J . Scherk a n d P. v a n

N i e u w e n h u i z e n : Nuc l . Phys . B117 (1976) 133.

S . F e r r a r a , D . Z . F r e e d m a n , P . v a n Nieuwenhu i -

gen, F . Gliozzi , J . Scherk a n d P. Bre i ten lohner :

Phys . Rev . D 1 5 (1977) 1013.

7 . E . C r e m m e r a n d J . Scherk : Phys . Let ters 69B

(1977) 97.

8 . J . P o l o n y i : Budapes t p repr in t K F K I - 1 9 7 7 - 9 3 .

9 . A . D a s , M . Fischler a n d M . R o c e k : Phys .

Let ters 69B (1977) 186; a n d Phys . Rev . D 1 6

(1977) 3427.

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B130 (1977) 105.

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76B (1078) 5 4 ; A . D a s , M . K a k u a n d P . K .

T o w n s e n d : Phys . Rev . Let ters 40 (1978) 1215.

12. D . V . Vo lkov a n d V . A . S o r o k a : J E T P Let ters

18 (1973) 312.

13. S . D e s e r a n d B . Z u m i n o : Phys . Rev . Let ters

38 (1977) 1433.

14. D . V. Vo lkov a n d V. P . A k u l o v : Phys . Let ters

46B (1973) 109.

15. D . Z . F r e e d m a n a n d J . H . Schwarz : Nuc l . Phys .

B137 (1978) 133.

P R O C . 19th I N T . C O N F . H I G H E N E R G Y P H Y S I C S

T O K Y O , 1978

C6 Ghost-Fields, BRS and Extended Supergravity as Applications of Gauge Geometry

Y . N E ' E M A N

Tel Aviv University, Tel Aviv

We review t h e m e t h o d s o f G a u g e G e o m e t r y , inc lud ing t h e sequence Lie

g r o u p / F i b e r B u n d l e / G a u g e Man i fo ld . W e der ive t h e G h o s t F ie lds a n d B R S

t r a n s f o r m a t i o n s geomet r ica l ly ( a n d classically), s h o w i n g t h a t the G h o s t fields

a re n o t h i n g b u t the ver t ical c o m p o n e n t s o f t h e gauge -po ten t i a l one - fo rms . W e

s ta te t he t h e o r e m of S p o n t a n e o u s F i b r a t i o n of a W e a k l y - R e d u c i b l e a n d

S y m m e t r i c G r o u p M a n i f o l d a n d a p p l y t he m e t h o d t o t h e de r iva t ion o f E x t e n d e d

Supergav i ty (Chi ra l or Par i ty -conserv ing) for a n y TV.

§1. Geometrical Introduction

The geometrical t rea tment of gauge theo­ries 1 has provided numerous insights and precise results at bo th the classical and quan­t u m levels. In this note , we present two new applicat ions, one for any Internal Gauge theory, and the other in the realm of the N o n -internal ones. In the first , we derive the exist­ence of the Ghost-fields, needed for Uni tar i ty and Renormal iza t ion , 2 as a classical geometrical result on a Principal Bundle. The BRS t ransformat ions 3 come cut natural ly, and one may hope tha t this approach will p roduce further insights in the study of the renormal i ­zat ion of Gravi ty and other theories. Our second example consists in a direct derivation of all allowed "geomet r ica l" theories of Extended Supergravity, using a theorem of Spontaneous F ibra t ion , 4 in the me thod of gauging on a G r o u p Manifold which we recently in t roduced. 5 6

It is instructive to compare a Lie g roup G, a Principal Bundle P with fiber F=G9 and the G r o u p Manifold G. A Lie group G is a space in which finite mot ions are specified, up to global topological considerat ions, by its Lie algebra A. The connect ion forms over G

are the Car tan Left-invariant forms w. They provide a rigid t r iangulat ion and a vanishing curvature

0.1) The Left-invariant algebra D y is an Or thonor ­mal basis to the w,

(1.2)

(1.3)

In the Principal Bundle P(P9 £, TC9 G9 -)9

with a horizontal base space B in addi t ion to G, and a projection TZ on to the base space,

(1.4)

the t ransformat ions of G are vertical by (4). The connect ion forms m are only vertical, and m a p tangent vectors on to A. The curvature two-forms Q are purely hor izonta l :

(1.5)

(1.6)

P is specified by the knowledge of the fields over a section I

The G r o u p Manifold G is punctual ly iden­tical to G. However, the connect ion one-forms p are locally independent . 5 In G any direction is a gauge direction (as in G=Fc:P)9

but all directions are also curved (as in J 5 c P ) . The Killing Lie algebra is no t isomor­phic to A 9

(1.7)

(1.8)

The curvature

(1.9)

is unrestricted. However, the Jacobi identity does give rise to Bianchi identities

(1.10)

Gauging corresponds to a local m a p in the group ' s r ight act ion, with

(1.11)

(1.12)

Supersymmetry {Including Supergravity)

§2. Ghosts and BRS

Take f irs t an internal gauge, described by P. We m a y decompose to with respect to a section I 7 in to the sum,

(2.1)

<j>tt is thus the (horizontal) Yang-Mi l l s potential . X is the F e y n m a n - D e W i t t - F a d d e e v - P o p o v ghost. It an t icommutes jus t because it is a one-form. A differential df can be expanded as

(2.2)

Equat ions (5)-(6) can be rewrit ten as

(2.3)

(2 .4 )

(2.5)

with

(2.6)

These are the B R S equat ions . The appli­cation to p a t h integrals and renormal izat ion will be described elsewhere. 7

§3. Gauging a Non-Internal Group G

3.1. Quadratic {simple G) and linear (G con­tracted) Lagrangians

For a simple G, use the quadra t ic Lagrangian in t roduced in ref. 8),

(3.1)

Let G be Weakly-reducible and symmetric decomposi t ion of its algebra A=FQ)H; [F, F] czF. The F-metr ic cab is the mos t , general tensor satisfying, [F, H]aH; [H, H]dF

(3.2)

The equat ions of mot ion are,

(3.3)

(3 .4 )

where = implies appl icat ion of the equat ions of mot ion ("pseudo- invar iance 9 ") .

U n d e r cont rac t ion over " p s e u d o - c u r v a t u r e 5 , 6 " :

(3.5)

553

(3.6)

where the latter condi t ion guarantees tha t the "f lat" Lie g roup G will satisfy the equat ions of mot ion , or in the language of Genera l Relativity, t ha t the vacuum will coincide with the " t angen t mani fo ld ." The equat ions of mot ion will be

(3.7)

Eq. ( 3 . 4 ) plays the role of BRS in establishing the Q u a n t u m version.

3.2. The spontaneous fibration theorem (The proof is given in ref. 4) Given (1) G a semi-simple G r o u p o r Supergroup

(2) G is Weakly reducible and Symmetr ic 6

(WRS) , i.e., there exists a decomposi t ion of

its Lie algebra

(3.8)

(3) The complete F i s generated by [H9 //], and the dimension of F

d(F)>(d(H)-l)/3,d{H) the dimension of H.

( 4 ) The Lagrangian i s F-gauge invar iant but no t v4-gauge invariant , Then , G undergoes Spontaneous Fibra t ion with respect to F, i.e., up to global topological considerat ions, (classically)

(1) the F variables factorize, in the sense of ref. 6.

(2) all torsions (curvatures in the H direc­tion) vanish,

R i H ^ 0 (3.9)

(3) all curvatures become hor izontal , name­ly in the decomposi t ion over a cobasis

(3.10)

(3.11)

where (||) denotes 2?, C e F, ( = ) s tands for B, C e H and (|-) for the mixed case.

§4. Extended Supergravity

4.1. OSp(N/4) Take G - O S p (JV/4), the Supe rg roup 1 0 pre­

serving a metr ic ^ 0 © ( ^ 2 ® ^ o ) , where v0 and a0

stand for the identity in N and 2 dimensions respectively.

We use a, è, • • • for space-time direct ions; a, /3, • • • and d, /3, • • • for spinors, i9j, • • • for Q(A0 TV-vectors.

5 5 4 Y . N E ' E M A N

4.2. Extended left-Chiral supergravity

where the dash s tands for a restr ict ion of (1.9) to F. Both theories violate par i ty even in the gravi ta t ional sector. L L vanishes for the flat G=G a n d LL(À

2) obeys (3.6.). Spon taneous f ib ra t ion guarantees t h a t

(4.3)

To recover supersymmetry we still have to cont rac t inside F. We cont rac t over F2, wi th

t), H2(Sdi, Ta). Tak ing H2-+ juH2 we find (R denotes a restr ict ion of (1.9) to F.nF,)

The cosmological t e rm violates (3.6) and in fact the Lagrang ian is n o t of the Gauge type (3.1) or (3.5). The Eai and equa t ions will have source te rms which do no t vanish for the f lat g roup , i.e., the vacuum does n o t cor respond to the tangent manifold.

There is no i V = 8 limit in this formal ism which deals with the gauge f ields in the adjoint representa t ion (the usual "field c o n t e n t " are ho lonomic composi te constructs of the gauge fields).

N o t e t h a t (4.3) does n o t conta in either R l i n o r p l i n

9 i.e., the in ternal symmetry i s no t locally gauged.

4.3. Parity conserving extended supergravity To recover Par i ty conservat ion, const ruct

f i rs t Right -Chi ra l Extended Supergravity by tak ing a decomposi t ion F2: («/[<*&]? IIS-9 Sai),

H2(Sai, Ta). Cons t ruc t LR and cont rac t H2->

{AH2, then cont rac t again wi th H1-^ÀH1 a n d find LR (/LtU2). A d d to (4.16) and find,

(4.4)

This Lagrangian appears to be of the type (3.5). However , the act ion of the in ternal symmetry in calculat ing for instance makes a "cosmolog ica l " t e rm reappear .

The g roup is no t a W R S (since [H, H](zH)

and the equa t ions of mo t ion are of the type studied in refs. 5, 6 and n o t as in equa t ion (3.7). N o t e t ha t for N=l, the t e rm vanishes identically and the vacuum fits G=G.

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P h y s . 1 7 (1976) 2 2 8 ; V . G . K a t z : F u n c . A n a l y s i s

a n d its A p p . ( U S S R ) 9 (1975) 9 1 .

FROC. 19th I N T . C O N F . H I G H E N E R G Y P H Y S I C S

T O K Y O , 1978

C6 The Superspace Method for Supergravity

B . Z U M I N O

CERN, Geneva

Superspace 1 " 3 was introduced by Salam and Strathdee as a technique for the study of representations of the supersymmetry algebra in terms of supermultiplets of fields. The points of superspace are parametrized by coord­inates zA = (xa, 6a) where xa are the commuting space-time coordinates while 6 a are anticom-muting variables (whcih are Majorana spinors). Supersymmetry transformations can be realized as rigid motions in superspace

( I )

where £ a are the infinitesimal anticommuting spinorial (Majorana) parameters . The com­muta tor of two transformations (1) of param­eters Ci and Co

(2)

is a translation of parameter —li^yj^^ as required by the supersymmetry algebra. A superfield V(x, 6) is a function in superspace. Since the four components 6 a ant icommute, in particular the square of each one is zero, a power series in 0 a is at most a polynomial of fourth degree. Therefore a superfield is equi­valent to a finite collection of ordinary fields (a supermultiplet), carrying various spinorial indices and satisfying corresponding statistics. When the superfield transforms like a scalar

(3)

the coefficients of its expanision in 6 a t rans­

form into each other. The "covariant deri­

vatives"

( 4 1

commute with the infinitesimal operator in (3) and can be used to impose supersymmetric constraints on superfields. They satisfy

(5)

showing tha t superspace has torsion, even though its curvature vanishes. This "flat" superspace has been extensively used in the study of supersymmetric field theories in

Minkowski space. Just as Einstein's theory is obtained when one generalizes from flat Minkowski space to urved space-time, one expects supergravity (the supersymmetric theor} of gravitation) to arise from the differential geometry of curved superspace. Indeed this superfield approach to supergravity preceded t h a t 4 5 based on ordinary f ie lds .

The first work on curved superspace was that of Arnowit t and N a th , 6 who developed the differential geometry of a super-Riemannian superspace. However, the flat superspace of rigid supersymmetry does not fit naturally as a special case of a super-Riemannian geometry. For this reason a different geometry of super-space was introduced by Wess and the a u t h o r 7 - 1 3 and by Akulov, Volkov and Soro-ka . 1 4 In this geometry the tangent space group at each point of superspace is no t taken to be the graded Lorentz group (orthosimplec-tic (4, 4)), but just the ordinary Lorentz group. Fur thermore the superspace is allowed to have torsion. More recently this geometry ha s been adopted by other au thors . 1 5 In this lec­ture we shall follow our own work, as deve­loped in refs. 7"* 1 3. Ours is a second order formulation, in tha t we impose on the super-torsion certain conditions ("constraints") , which are sufficient to express the superconnec-tion in terms of the supervielbein.* These conditions also restrict to a certain extent the supervielbein itself. The dynamics is given in terms of a superspace action which must be varied keeping the constraints on the torsion satisfied. Our formulation is exactly equival­ent to the recently developed formalism for supergravity with a minimal set of auxiliary

* It w o u l d be preferable to have a f irst o r d e r ac t ion

in superspace , f rom which b o t h cons t ra in t s a n d equa ­

t ions of m o t i o n follow. A possible choice for a first

o rder ac t ion was suggested in ref. 8 a n d is a lso dis­

cussed in t h e lec ture by J . H. Schwarz in these Proceed­

ings . 1 9 Howeve r , d imens iona l a r g u m e n t s indica te

tha t 6 in tegra t ion of tha t supe r space ac t ion does n o t

r ep roduce the correct supergravi ty ac t ion in c o o r d i n a t e

space.

556 B . Z U M I N O

fields. 1 6 - 1 8 Indeed this auxiliary field formali­sm can be derived from our geometrical formulat ion. 1 3

A non-geometric description of supergravity in superspace is favored by Ogievestsky and Sokatchev 2 0 and by Warren Siegel. 2 1 The supergravity fields are contained in a superfield Um(x9 6) endowed with a vector index m. The theory, based directly on this superfield, has a certain appeal because it is free of constraints. However, the lack of a complete geometrical structure in such a theory makes the con­struction of invariants a difficult task. On the other hand, from our point of view, the superfield Um emerges in solving the constraints on the torsion. The supervielbein and the superconnection are expressed in terms of it and we can use the powerful techniques of differential geometry for the construction of invariants.

The geometric superspace approach gives a complete and satisfactory method for the study of simple supergravity and its couplings to supersymmetric m a t t e r 1 1 , 1 2 (both for the vector-spinor and for the spinor-scalar-pseu-doscalar multiplets). All possible invariants of given dimension are easily constructed and their knowledge can be used 1 6 to discuss the divergences which arise in perturbation theory. The situation is quige different in the case of extended supergravity. Clearly one must use here a larger superspace in which the fermionic coordinates carry, in addition to the spinorial index, an internal symmetry index, and the tangent space group should be the product of the Lorentz group with the internal symmetry 0(AO. The difficulty consists in finding the correct separation of the equations into kinematical constraints (which are par t of the specification of the geometry) and equations of motion (which should follow from the variation of an action, subject to the con­straints). The results obtained so far 2 2 for A r = 3 extended supergravity in superspace do not make this separation. These difficulties in the superspace method correspond to our present ignorance as to the correct auxiliary fields for extended supergravity.

Affine superspace

The points of superspace are parametrized

by coordinates zM = (xm

9 6ft). Latin letters

denote vectorial (bosonic), Greek letters spi­norial (fermionic) indices. The supervielbein matrix EM

A(z)9 where A=(a9 a ) , and its inverse EA

M(z)9 can be used to transform world tensors into tangent space tensors and vice versa (early alphabet letters denote tangent space indices). Covariant derivatives involve the superconnection 0 y l t A

B

9 as in

(6)

Covariant derivatives with tangent space in­dices, &A=EA

MfSM9 satisfy the graded com­mutat ion relations

(7)

where TAB

C

9 are the supertorsion coefficients and RAB is the Lie algebra valued super-curvature operator. The bracket in eq. (7) is an ant icommutator if both indices A and B are spinorial, a commutator otherwise. Torsion and curvature satisfy the Bianchi identities

(8)

(9)

The cyclic sums are taken with appropriate signs: a permutat ion of two vectorial or one vectorial and one spinorial index gives rise to a change in sign, while for two spinorial indices there is no change in sign.

Specification of the geometry (kinematics)

We now restrict the tangent space group from the general graded linear group to the (x and 0 dependent) Lorentz group. Since the curvature operator belongs to the algebra of the tangent space group, this implies restric­tions on the curvature coefficients

(10)

(similarly for the superconnection). We also impose the constraints on the supertorsion

(11)

but leave Tab,r and Tab,

7 undetermined. Using

the Bianchi identities (8) and (9) one can show

Supersymmetry {Including Supergravity) 557

tha t (10) and (11) imply tha t all components of the supercurvature and supertorsion are expressible in te rms of the three superfields Ga$, WapT, R and their conjugates. Here we have used two-componen t spinor nota t ion, G„p is hermitean, Wafi7 total ly symmetric. Fo r instance

(12)

The remaining content of the Bianchi identities

is expressed by the differential relat ions

(13)

The superfield Gaà has a simple physical mean­ing, which can be exhibi ted 1 6 by considering the coefficients of its expansion in 0. Fo r instance, at the 00 level one finds a tensor which contains the Einstein tensor, at the 000 level a spinor which is the Rar i ta -Schwinger ope­rator . Similarly, R contains the contracted (scalar) curvature tensor, at the 00 level. Wan

contains the Weyl conformai spinor, at the 6

level; for 6=0, it contains the Rari ta-Schwinger f ie ld strength. Observe t ha t some components of R are obta ined from those of Gaà by tracing over certain indices: in superspace this is ex­pressed by the differential identities (13) in 6.

Finally we observe 1 1 t ha t the constraints (11) on the tors ion imply

^dxd0E^AvA=O (14)

for any vector vA, where

E=detEM

A

is the graded de te rminant of the supervielbein as defined in the first of ref. 6. This formula permits integrat ion by par ts with covariant derivatives carrying tangent space indices.

(15)

Conformai transformations The correct conformai t ransformat ions in

superspace mus t preserve the constraints (11). These t ransformat ion involve a covariantly

chiral infinitesimal parameter function 2

(16)

and are (Howe and Tucker 2 3 )

07)

(18)

F o r the de terminant of the supervielbein,

imply (17)

(19)

while curvature and torsion behave as given by

(20)

Since the constraints are invar iant under (17) and (18), we can say tha t the geometry (kine­matics) is invariant under conformai t rans­format ions . The dynamics m a y or may not be, depending on the choice of Lagrangian. F o r instance, the action of supergravity (21), given below, is only invar iant under general coordinate t ransformat ions in superspace and under local Lorentz t ransformat ions , bu t not under conformai t ransformat ion. The action of conformai supergravity (28), instead, is also conformally invariant .

Dynamics

The supergravity equat ions of mot ion are derived from the action

(21)

Varying (21) subject to the constraints gives 1 1 the equat ions

o n

(22)

Because of the identities (13) they give

(23)

and this, in tu rn , implies

(24)

On the supergravity mass shell all componen ts of the supertorsion and of the supercurvature

5 5 8 B . Z U M I N O

a re expressed in te rms of the symmetric , chiral spinor superfield Wa^r which now also satisfies (23), and of its derivatives (and, of course, of the complex conjugate quanti t ies) . I t is there­fore very easy to list all invar iants of given d imens ion and to discuss quest ions of renor­malizabi l i ty . 1 6

Chiral superfields as Lagrangians

In flat superspace any chiral superfield 0

can be wri t ten in the form 0=DàDàU. One

can use 0 as a Lagrangian by integrat ing only over Q a (and x) bu t n o t over dà. Alter­natively, one can integrate U over all four 0

componen t s (and x) wi th the same result. The f i rs t p rocedure has no analogue in curved superspace in a general gauge, the second does. One can show 1 1 t h a t any chiral superfield 0

(25)

can be wri t ten in the form

where the new t e rm comes from the com­muta t i on relat ions a m o n g the covar iant deri­vatives. T h e act ion obta ined from U can be t ransformed as follows

(26)

(27)

where the last equali ty follows by par t ia l in tegrat ion because R is chiral . We see tha t , in order to const ruct a Lagrangian density f rom a chiral superfield, we mus t mult iply it by —E/SR, no t by E. As an example , con­sider the chiral superfield Wa^rW

a^r. The cor responding act ion, p ropor t iona l to

(28)

is invar iant under the conformai transfor­ma t ions (19) and (20). I t is the superspace fo rm of the act ion for conformai supergravity.

Coupling to vector-spinor matter11

In analogy with f la t superspace, this mat te r supermul t ip le t is described by a real scalar superfield V, a n d a supersymmetr ic gauge t r ans fo rmat ion by

(29)

T h e gauge invar iant superfield which conta ins t h e e lectromagnet ic field is defined as

(30)

It satisfies identically

(31)

a n d

(32)

which is the supersymmetr ic form of the

electromagnet ic Bianchi identities. We can

take the act ion, up to a propor t iona l i ty con­

stant .

(33)

(observe tha t we use the density E/R a p p r o ­pr ia te to a chiral Lagrangian) . The cor­responding equat ion of mot ion is

(34)

which, combined with (32), gives

(35)

The action (33) is invar iant unde r the gauge t rans format ion (29). It is also invar iant under conformai t ransformat ion provided one assu­mes the conformai p roper ty

(36)

which implies

To see it, combine (37) with (19) and (20) and integrate by par t s .

Coupling to spinor-scalar-pseudoscalar matter12

Let 0 be a chiral superfield

(38)

One can take the act ion

(39)

where F{0) is a (real) function of 0 (potential) . In order to derive the equa t ions of m o t i o n i t is simplest to satisfy the condi t ions (38) by setting

(40)

The superfields U a n d J7* can be subjected to arb i t rary var ia t ions . One f inds the equa t ion of m o t i o n

(41)

a n d its complex conjugate. It is n o t difficult to verify tha t (38) a n d (41) imply t ha t the super-current

Supersymmetry {Including Supergravity) 559

(42)

satisfies

(43)

where

(44)

The mat ter superfields Jaà and S, S* (which contain the energy m o m e n t u m tensor and the spinor cur ren t 2 4 ) satisfy an identity analo­gous to tha t [see (13)] satisfied by the super-gravity superfields Gaà and R, R* (which con­tain the Einstein tensor and the R a r i t a -Schwinger opera tor) . Indeed the supergravity equat ions state the propor t ional i ty of these two sets of fields

If one at t r ibutes to the superfield 0 the con-

formal proper ty

(45)

(46)

one finds tha t the first (kinetic) te rm in the action (39) is conformally invariant . F o r the potential term, one obtains

(47)

The superfield .S, given by (44), appears as a measure of the violat ion of conformai inva­riance in the mat te r act ion. F o r F(@)oc@3,

one has S=0 and the entire mat te r action is conformally invariant . In this case the super-gravity equat ions of mot ion give R=0. In general, however, for instance if the mat ter supermult iplet has a mass (Foc02)R will no t vanish.

A quest ion which deserves considerat ion in future work is tha t of the quant izat ion of the theory directly in terms of superfields. The na tura l objects here appear to be the uncon­strained superfield U m and its superspace pro­pagator . A per turba t ion theory in super-space will be simpler t han tha t in ordinary space, in tha t the n u m b e r of indices and of d iagrams will be m u c h smaller. In this way one may hope to find a reduct ion in the divergences of the theory beyond tha t establi­shed so far on the basis of invariance argu­ments .

References

1 . D . V . V o l k o v a n d V . P . A k u l o v : Phys . Le t te rs

4 6 B (1973) 109.

2 . A . S a l a m a n d J . S t r a t h d e e : N u c l . Phys . 76B

(1974) 477.

3 . S . F e r r a r a , B. Z u m i n o a n d J . Wess : P h y s . Le t te rs

5 1 B (1974) 239.

4 . D . Z . F r e e d m a n , P . v a n N i e u w e n h u i z e n a n d S .

F e r r a r a : P h y s . Rev . D 1 3 (1976) 3214.

5 . S . Dese r a n d B . Z u m i n o : Phys . Let te rs 62B (1976) 335.

6 . R . A r n o w i t t , P . N a t h a n d B . Z u m i n o : Phys .

Le t t e r s 56B (1975) 8 1 ; R . A r n o w i t t a n d P . N a t h :

P h y s . Le t t e r s 56B (1975) 117; P h y s . Le t te rs 65B (1976) 7 3 . F o r a recen t a c c o u n t on this a p p r o a c h

see t h e lec ture by P . N a t h in these P roceed ings .

7. B. Z u m i n o : Proc. of the Conf. on Gauge Theories

and Modern Field Theory, N o r t h e a s t e r n Univer ­

sity, S e p t e m b e r 1975, eds . R . A r n o w i t t a n d

P . N a t h ( M I T Press) .

8 . J . W e s s a n d B , Z u m i n o : Phys . Le t te rs 66B (1977) 3 6 1 .

9 . R . G r i m m , J . Wess a n d B . Z u m i n o : Phys . Let­

ters 7 3 B (1978) 415 .

10. J. W e s s : Supersymmetry—Supergravity, Lec­

tu res given a t t he V I I I G . I . F . T . S e m i n a r (Sala­

m a n c a , 1977), K a r l s r u h e Univer s i ty p rep r in t , t o

be pub l i shed in Spr inge r T rac t s .

11 . J . W e s s a n d B . Z u m i n o : Phys . Le t te rs 74B (1978) 5 1 .

12. B . Z u m i n o : Ca rgèse S u m m e r Schoo l Lec tures

(July 1978), to be pub l i shed in t h e Proceed ings ,

P l e n u m Press .

13. J . W e s s a n d B . Z u m i n o : C E R N p rep r in t T H .

2553 (1978).

14. V . P . A k u l o v , D . V . V o l k o v a n d V . A . S o r o k a :

J E T P Le t t e r s 22 (1975) 396 (Engl ish , 187).

15. L . B r i n k , M . G e l l - M a n n , P . R a m o n d a n d J . H .

S c h w a r z : P h y s . Le t t e r s 74B (1978) 336.

16. S . F e r r a r a a n d B . Z u m i n o : N u c l . P h y s . B134 (1978) 301 .

17. S . F e r r a r a a n d P . v a n N i e u w e n h u i z e n : P h y s .

Le t t e r s 7 4 B (1978) 3 3 3 ; 7 6 B (1978) 4 0 4 ; Eco le

N o r m a l e p r e p r i n t L P T E N S 78/14.

18. K . S . Stelle a n d P . C . W e s t : P h y s . Le t t e r s 74B (1978) 330 ; I m p e r i a l Col lege p r e p r i n t s I C T P / 7 7 -

78/15 a n d 24.

19. M . G e l l - M a n n , P . R a m o n d a n d J . H . S c h w a r z :

s u b m i t t e d t o t h e X I X I n t e r n a t i o n a l Confe rence o n

H i g h E n e r g y Physics , T o k y o , A u g u s t 1978.

20. V . Ogieve tsky a n d E . S o k a t c h e v : N u c l . P h y s .

B124 (1977) 3 0 9 ; D u b n a p r e p r i n t E 2 - 1 1 7 0 2

(1978).

2 1 . W . Siegel : va r i ous H a r v a r d Unive r s i ty p rep r in t s

(1977-78) .

22. L . B r i n k , M . G e l l - M a n n , P . R a m o n d a n d J . H .

S c h w a r z : P h y s . Le t t e r s 7 6 B (1978) 417 .

2 3 . P . S . H o w e a n d R . W . T u c k e r : C E R N p r e p r i n t

T H 2524, t o b e pub l i shed i n P h y s . Le t t e r s .

24 . S . F e r r a r a a n d B . Z u m i n o : N u c l . P h y s . B87 (1975) 207 .

P R O C . 19th I N T . C O N F . H I G H E N E R G Y P H Y S I C S

T O K Y O , 1978

C6 Lagrangian Formulation of Supergravity in Superspace

Presented by J. H. S C H W A R Z

California Institute of Technology, Pasadena, California 91125

T h e equa t i ons of N=\ supergravi ty in superspace , wi th e-?§ a n d wi th

externa l v e c t o r - s p i n o r ma t t e r , were derived previously , b u t in r e d u n d a n t fo rm

(420 e q u a t i o n s for 112 i ndependen t superfields VA

A a n d hA

rs). H e r e we derive

a min ima l set of 112 equa t i ons f rom a L a g r a n g i a n l inear in t he to rs ions a n d

curva tu res . T h e r ema in ing 308 re la t ions can p r e sumab ly be der ived us ing

Bianchi identi t ies. T h e l inear L a g r a n g i a n is found by a sys temat ic m e t h o d f rom

the s t ruc tu re cons t an t s o f t he re levan t supera lgebra , a n d the s a m e m e t h o d can

be used for 7V=-0 to der ive the Eins te in theo ry of g rav i ta t ion .

The reformulation of supergravity theories in superspace is motivated in par t by the hope that the systematics of those theories will become clearer, tha t a simple formula can be found to describe all of them, and tha t more will be learned about the nature of the ap­parent restriction 7V<8. Fur thermore , the great success of the geometrical interpretation of Einstein's theory of gravitation indicates tha t a full geometrical interpretation of supergravity in superspace will be illuminating. Finally it is possible tha t such further insights might suggest some generalization of supergravity tha t is no t now apparent and tha t might come closer to describing physical reality.

Equat ions of mot ion for the curvatures and torsions in superspace have been given for N=l supergravi ty , 1 , 2 including self-coupling parameter e^O2 and vector-spinor external matter , and also for T V = 3 supergravity. 3 In this communicat ion we restrict ourselves to the case N= 1 and employ the nota t ion of ref. 2, which we briefly review and extend.

Latin indices refer to the tangent space and Greek indices to the base space. Early letters (a, /3,7% • • • and a, b, c, • • •) are spinorial and late letters v, />, • • • and m, n, p, • • •) are spatial. Capital Latin indices up to U re­present all eight coordinates of the tangent space, while V—-Z take 14 values and run over the six Lorentz rotat ions as well. Capital Greek indices (A, FF, I, • • •) represent all eight coordinates of the base space.

The superfields we employ are the vielbein Vj, with inverse VA satisfying the relation

a )

and the connection hrJ, antisymmetric in r

and s, which gauges local Lorentz t rans­formations. We define the tangent space derivative L/ A covariant with respect to these transformations :

(2)

where Xrs are the generators of the Lorentz transformations. In "flat" superspace with no supergravitational disturbances, the four­teen operators £y A and Xrs (collectively called Gy) generate the superalgebra OSp( l ,4 ) :

(3)

where the bracket is a commuta tor unless X

and Y are bo th fermionic, in which case it is an ant icommutator , and the non-zero components of the structure constants C | F are given in Table I.

In the general case of supergravity, we can define superspace-dependent quantities i?f Y(z)

by the analogous formula

(4)

where RAB are the curvatures defined in ref. 2, RC

AB are minus twice the torsions defined in ref. 2, and the remaining i?f Y are just equal t o the corresponding constants C | r .

An intermediate situation is also wor th con­sidering, in which supergravitational d i s t u r ­bances are turned off except for the coupling of supergravity to external vector-spinor mat­ter, described by an axial vector superfield J£(z), satisfying the conservation condit ion

(5)

Supersymmetry (Including Supergravity)

Table I.

The / 5 -dependent structure "cons tan ts" f%Y.

The cases no t listed are either zero or related

to ones in the list b y / # r = - ( - l ) X I 7 / z . (The

ordinary structure constants C | F are obtained

by putting / 5 = 0 . )

We use the metric rjr8=(-\ ) and Dirac

algebra conventions in which ^ s = ^ r s + 2 o - r s ?

supplemented by the rule that Dirac indices

are raised and lowered with the charge con­

jugation matrix. Thus, for example, (jr)ah =

{jr)acCcb is a symmetric matrix. The dimen-

sionless constant e is related to the de Sitter

radius R by e=icjR9 where K=\IATZG is the

gravitational coupling (Planck length), which

we set equal to one.

stronger than ordinary conservation. In this

case we have

(6)

where the f£Y are J 5 -dependent extensions of

the structure constants C?v and are siven in

Table I.

The graded Jacobi identities

(7)

give rise to Bianchi identities for the quantities

RXYI these are obeyed in the general case (4)

and, of course, also in the particular case (3),

where they reduce to the usual algebraic

equations for the structure constants of a Lie

superalgebra, and in the particular case (6),

where they give an example of an interesting

generalization of such a superalgebra, with

5 6 1

explicit superspace-dependent structure con­

stants ." (We employ the usual convention that,

in the exponent of — 1 , a fermionic index is

read as 1 and a bosonic index as zero.)

In our general case (N= 1 supergravity with

e=fO and external vector-spinor matter) , the

non-trivial components RZ

AB of 7?f Y obey the

420 equations of ref. 2, which can be trans­

cribed as follows. The components Rr

ab9 Rc

ab9

Rr

a

8

b9 Rla9 Rb

ra9 and i?* s are just equal to the

corresponding components of fAB9 as given in

Table I. The remaining components obey

weaker equations than those for the cor­

responding Ts , namely

(8)

(9)

(10)

In our "first o rder" formulation of the theory,

we regard the components of the inverse

vierbein Vi and of the Latin connection 1fA

s as

independent field variables, giving 112 real

superfields, for which we expect 112 indepen­

dent equations of motion in superspace. Evi­

dently the 420 equations are highly redundant

and can be replaced, in many different ways,

by a list of 112 equations from which the

remaining 308 can be decuced with the aid of

the Bianchi identities that arise from the Latin

graded Jacobi identities

a i )

Nearly identical minimal lists of 112 equations

have been given by MacDowel l 4 and the

authors , 5 and the full list of 420 equations

derived.

We can go further and find a Lagrangian

for the "first o rder" theory in superspace,

from which a different minimal list is obtained

by varying with respect to the superfields Vj

and hrl independently. Since the equations

of motion are all linear algebraic relations for

the RAB9 we consider the most general Lag­

rangian density linear in these quanti t ies:

(12)

where V is the graded determinant of the vielbein, and obtain the 112 equations of motion

562 J . H . S C H W A R Z

(13)

We can then determine the coefficients A and B by setting B equal to the most general l inear function of Jr

5(z) using covariant constants , and then compar ing the resulting equat ions with the 420 equat ions for R^B and requiring com­patibility. The same result can be obtained by

substituting the special values Rxz^fxz in

Eq. (13) and solving for A and B. The Lag­

rangian can thus be found from the generalized

"structure constants" f To within a constant of proport ional i ty , we find the unique solution

(14)

Substi tuting back into eq. (13), we find the list

of 112 explicit equat ion of mot ion given in

Table II . We have not verified tha t with the

aid of the Bianchi identities these equat ions

imply the full list of 420 equat ions, but all the

checks we have made are consistent with tha t

Table II .

Equat ions of mot ion obtained from the Lag­

rangian of eq. (14).

I t is for tunate tha t vector-spinor mat ter can be treated as external and described by the superfield Jl(z). Of course, when we include the dynamics of the vector-spinor mat ter , we must add terms to bilinear in .the mat te r superfield strengths, bu t those terms give vani­shing contr ibut ions to the gravitat ional equa­t ions of mot ion when the mat ter equat ions of mot ion are used.

The use of J\(z) has permit ted us to avoid certain indeterminacies. As / 5 - > 0 , the ra t io of coefficients of the last two terms in eq. (14) for the Lagrangian becomes undetermined. Fur the rmore , when we search for the par t i ­cular solution of the Bianchi identities and the Lagrangian equat ions of mot ion of Table II tha t is given by the "s t ructure cons tan t s" of Table I we find tha t the solution is unique when / 5 ^ 0 but t ha t a two-fold ambiguity develops as / 5 - > 0 .

A ra ther trivial indeterminacy develops as e->0, namely tha t the scale of Rr

ab becomes a free parameter , and a rescaling of the system of equat ions becomes possible.

Our Lagrangian density for 7V=1, which lacks completely the Einstein term Rr

r

s

s, does no t appear to resemble the corresponding quant i ty for the case N=0 of plain gravita­t ion. It is therefore instructive to apply our me thod to the 7V=0 prob lem and show tha t the usual theory emerges. We take eq. (13) with R=f and take the c o n s t a n t / ' s or c's from Table I with spinor indices suppressed. Solv­ing for A and the B coefficients and sub­stituting back into eq. (12) for the Lagrangian density, we find the unique solution

(15)

tha t gives the correct theory.

It is very impor tan t to discover the syste-matics of the Lagrangian density as TV varies and especially to f ind out what happens a t N=8. While equat ions of mot ion can always be obtained by t runca t ion from those for higher N, tha t is no t t rue of the Lagrangian density, which is integrated over a different number of variables as TV changes. F o r tha t reason the systematics of ^f/V may well be non-trivial . In par t icular , for T V > 1 we expect

to contain quadra t ic terms in the f ield strengths, at least those corresponding to the internal SOx g roup. (In our present work , such terms would have shown up for the vector -sp inor mat te r if we h a d t reated i t com­pletely, including its equat ions of mot ion , ra ther t han as an external source.) F o r 7V=0, quadra t ic terms in the torsions are normal ly no t present. Should we have constructed a Lagrangian for N= 1 containing such quadra t ic terms in the tors ions? We are no t certain, bu t i t appears tha t our linear me thod is

Supersymmetry (Including Supergravity) 563

justified by the results.

It should be rioted tha t a very simple superspace Lagrangian for N=l supergravity has been given by Wess and Z u m i n o 6 for a kind of "second o rde r " formalism in which there are constraints relating the vierbein and the connection. We hope tha t our unconstrained "first o rde r" approach will also lead ultimately to a simple description and will be suitable for generalization to extended supergravity.

We would like very much to be able to relate our work to the impor tan t research of Arnowi t t and N a t h ; 7 in a recent art icle 8 they have thrown new light on the relat ionship between the two approaches and have apparent ly shown that the vierbein formalism can be obtained as a special case of their metr ic formalism.

We are happy to acknowledge the many con­tr ibut ions of Lars Brink to our unders tanding of supergravity in superspace, useful and enlightening conversations and correspondence with Samuel MacDowel l , interesting com­municat ions from Richard Arnowi t t and Pran

Na th , and the agreeable hospitali ty of the Aspen Center for Physics.

References

1 . J . Wess a n d B . Z u m i n o : P h y s . Le t te rs 66B

(1977) 3 6 1 .

2 . L . B r ink , M . G e l l - M a n n , P . R a m o n d , a n d J . H .

S c h w a r z : P h y s . Le t t e r s 7 4 B (1978) 3 3 6 ; e r r a t u m

t o b e p u b l i s h e d .

3 . L . B r i n k , M . G e l l - M a n n , P . R a m o n d , a n d J . H .

S c h w a r z : Phys . L e t t e r s 76B (1978) 417.

4 . S . M a c D o w e l l ; Y a l e p r ep r in t . We w o u l d l ike

t o t h a n k Professor M a c D o w e l l for c o m m u n i c a t ­

ing resul ts p r i o r t o p u b l i c a t i o n .

5 . M . G e l l - M a n n , P . R a m o n d a n d J . H . S c h w a r z :

in Proceedings of the Dirac Symposium, 1978 ( to

be pub l i shed ) .

6 . J . Wess a n d B . Z u m i n o : P h y s . Le t te rs 74B

(1978) 5 1 .

7 . R . A r n o w i t t a n d P . N a t h : P h y s . Le t te rs 56B

(1975) 177 a n d 65B (1976) 73 .

8 . R . A r n o w i t t a n d P . N a t h : N o r t h e a s t e r n U n i v .

p r e p r i n t N U B N o . 2361 . W e w o u l d l ike t o

t h a n k t h e a u t h o r s for c o m m u n i c a t i n g the i r resul ts

p r i o r t o pub l i ca t ion .

P R O C . 19th I N T . C O N F . H I G H E N E R G Y P H Y S I C S

T O K Y O , 1978

C6 Superspace Formulation of Local Supersymmetries

Presented by P. N A T H

Department of Physics, Northeastern University, Boston, Mass. 02115

§1. Introduction

The first realization of a local supersym­metry was achieved in superspace 1 in the form of Gauge supersymmetry. 2 After the develop­ment of the supergravity theory, 3 i t was demons­trated tha t the k-+0 limit of gauge super-symmetry p roduced supergravity in the super-space formalism. 4 Al ternate superspace for­mulat ions of supergravity using supervierbeins and without the limiting procedure have been given by Wess and Z u m i n o 5 and by Brink, Gel l -Mann, R a m o n d and Schwarz. 6 How­ever, an equivalence of these vierbein formula­tions and the metric formulat ion can be demonst ra ted . 7

We shall begin first by a brief discussion of the basic principles of gauge supersymmetry and next discuss the more recent developments in this theory which relate to quant izat ion, vacuum symmetries with q u a n t u m effects in­cluded 8 and the f ield content of the theory. 9

The superspace formulat ion of supergravity and the equivalence of the vierbein and metric formulat ions will then be discussed.

§2. Principles of Gauge Supersymmetry

The theory is based on the fundamental assumption tha t all the gauge fields arise from the tensor superfield gAn(z) which plays the role of the metric tensor in superspace, i.e.,

ds2^dzAgAn(z)dzn where zA=(xft

9 6aa) are the

564 P . N A T H

supersymmetry space coordinates (x!t are the

bose and 6aa a=l, • • -4, a=l9 • • -N, a set of

N Majorana coordinates. We shall usually

suppress the internal symmetry index.). The

theory has as its gauge group the general

coordinate group in superspace, zA=zA' +

çA(z), which gives rise to the gauge change

(we use right derivatives)

(1)

where /1=(0 , 1) in ( — 1) J depending on whether

the subscript A=(/A, a). The remarkable as­

pect of eq. (1) is the "g roup theoretic unifica­

t ion" of many normally unrelated gauge groups

such as Yang-Mil ls , Einstein and supersym­

metry groups. The assumption of an inverse

metric leads to a unique set of second order

differential equations

(2)

where x is a constant and RAn is the contracted

curvature in superspace similar to the curvature

Rltv in Einstein theory. An invariant action

can also be given which generates these field

euqa t ions : 1 0

(3)

Thus the principles of gauge invariance

and the assumption of an inverse metric deter­

mine completely the dynamics except for the

arbitrary constant A. Equat ions (2) and (3)

contain all the sources as well as Higgs inter­

actions responsible for a possible self-con­

tained symmetry breakdown. Symmetry break­

ing then must be an output of the theory

rather than a contrived input through addi­

tional Higgs interactions in the action. This

phenomenon is a unique feature of gauge

supersymmetry not found in either supergravity

or Yang-Mil ls theories. It should be noted

that spontaneous breakdown is crucial for the

extraction of the physical content of gauge

supersymmetry. This is so because initially

the theory has a very large gauge group with

all the gauges of f A(z) being preserved. It is

interesting to ask if spontaneous breakdown

can pluck out of the myriad gauges of ç J (z) ,

a set of residual preserved gauges which are

relevant to physics. We have established a

theorem in this regard with all quan tum

corrections included to the vacuum symmetry

breaking conditions and we shall return to this

later on.

§3. Path Integral Quantization

Gauge supersymmetry is most conveniently

quantized using the path integral formulation

which can also be made manifestly globally

supersymmetric. The procedure is analogous

to that used in quantizing globally supersym­

metric gauge theories . 1 1 One introduces the

vacuum to vacuum transition amplitude

(4)

where yjjb r]A* are the Faddeev-Popov ghost

superfields and 7 = / 0 + / c ; + / G is the total action

with 7 C , the gauge fixing term and TG9 the cor­

responding ghost action. As usual the gauge

fixing term is chosen so that Ic=(~l)A/2\dz

CAg[o)AnCu where gi0>An is the inverse of

Si0)AII which is introduced as a (fixed) back­

ground f ie ld: g ( o ) j / 7 =<0|g-^ / 7 | 0> . Analogous

to the Einstein theory, a convenient choice

of C A is the linearized harmonic gauge. 1 2 The

corresponding Faddeev-Popov ghost ac t ion 1 3

involves the (fermi-bose) ghost superfields

rjA(z) and TJA(Z)* whose integral spin components

are treated as fermions and half-integral spin

components as bosons in the functional in­

tegral. This quantization scheme then incor­

porates global supersymmetry manifestly if

one uses for the background field a globally

supersymmetric g { 0 )

l n .

§4, Symmetry Breaking Equations

Equations that determine spontaneous bre­

aking in gauge supersymmetry may be ob­

tained using the effective potential method . 1 4

The total action involves the metric tensor

gAn, the ghost fields as well as the background

metric g% so tha t one may write I=I[gAn+ h{An'> VA> VA*l SAn\ where we have made the

translation g-AU=g;

J7/+hAn. The symmetry

breaking equations arise from variations of

the effective potential r(gAn; g°An) which deter­

mines gAn—g°An. Thus one has

(5)

In eq. (5) r=I+ W, where / is the total action

and Wis a solution of a functional equa t ion . 8 , 1 5

Defining GAn=ÔI/ôgAn the full content of the

Supersymmetry (Including Supergravity) 565

symmetry breaking, eq. (5), takes the fo rm 1 5

(6)

The first t e rm on the left h a n d side in eq. (6) is the tree approx imat ion and the remainder te rm QAn gives the q u a n t u m loop correction.

In looking for solut ions to eq. (6), we shall require for simplicity tha t global supersymme­try be explicitly mainta ined. (More generally, there may exist addi t ional solutions to eq. (6) tha t violate global supersymmetry which is of course desirable since supersymmetry must eventually be b roken for a physical theory.) The t ransformat ion functions £A(z) t ha t main­tain global supersymmetry have the form

(7)

where X a is a cons tan t spinor. It is then easily established tha t the invariance of the vacuum state requires tha t dgfy vanish and that the most general form of such a g% is 8

(8)

In eq. (8), K and Ffl are matrices in the Dirac and internal symmetry space with the proper ty tha t (rjFft) is symmetr ic and (rjK) is anti-sym­metric where y]=— C " 1 and C is the charge conjugation matr ix . Fur ther , it is always possible to set K=l in eq. (8) by linear t ransformat ions in the 6 space and we shall assume this to be the case. The in eq. (8) are to be determined by the symmetry break­ing equat ions eq. (6) which involve the full evaluat ion of the q u a n t u m correct ion te rm QAn. Since the quant iza t ion of the theory has been carried out in a globally supersymmetric fashion, QAn mus t in general be a global tensor and we may then write QAn in the form

(9)

where (j]Kf) is an ant i -symmetr ic matr ix . The quanti t ies 1' and K' are given by the functional integral of eq. (6) and depend on Ffl and À 9

i.e., X'=V(X9 r / 4 ) and Rf=K'(l\ Ffl).

Use of eqs. (8) a n d (9) in eq. (6), reduce the symmetry breaking equat ions to the set

(10)

are real symmetric and ant i-symmetric matrices

and K0A are real symmetric matrices in the

internal symmetry space.)

§5. Residual Symmetries after Spontaneous Breaking

To determine the residual symmetries pre­sent in the spontaneously b roken theory, we examine the content of the symmetry breaking equat ions eq. (10). One may proceed by expanding F{s,a) in terms of the symmetric and ant i-symmetric matrices of the internal sym­metry space; J T ( , ) = a i 7

T

i

( J , ) and r{a)=firT{

r

a\

The (ai9 /3r) consti tute a set of N 2 vacuum ex­pectat ion values of the Higgs fields for the theory. No te , however, tha t eq. (10) represent a set of N(N+l)+\ equat ions which is an impossible si tuation unless some of the equa­t ions are r edundan t as a consequence of an internal symmetry group . Thus one has the following fundamenta l result :

T H E O R E M ( i) : The globally supersymmetr ic solutions of the symmetry breaking eqs. (10) require a r igorously preserved internal sym­metry subgroup.

Thus the spontaneous breaking of gauge supersymmetry with globally supersymmetric vacuum solutions contains in an au tomat ic fashion a set of preserved gauges and the theory thus contains in it the possibility of a natura l explanat ion of why some gauges such as the Maxwell and Q C D gauges are exactly preserved in na ture . We discuss now the quest ion of precisely wha t gauges are preserv­ed and wha t are b roken in %\z). This p ro ­blem is equivalent to determining which f A(z)

leave gfy invar iant so tha t dg{

AI)=0. Recently, we have obta ined a general solution to this p roblem and the results a r e 8

( H )

are ant i-symmetric in­ternal symmetry matrices which obey the condi t ion

( 1 2 )

Thus the preserved internal symmetry t rans­format ions are subgroups of 0(A0 which com-

566 P . N A T H

mute with Ftl. The above analysis then

leads us to the second fundamenta l result

given below :

T H E O R E M (ii): The vacuum symmetries of

the theory with full q u a n t u m correct ions in­

cluded consist of the Poincaré t ransformat ions

(simultaneously on x[l and 6a), global super-

symmetry t ransformat ions generated by X a

a n d linear or thogonal internal symmetry

t ransformat ions in d a space generated by M

obeying eq. (12).

§6. Field Content

F o r each of the preserved global symmetries

after spontaneous breaking there corresponds

a local symmetry and a preserved gauge field.

T h u s one has as the preserved gauges the

Einstein gauge and the corresponding f ie ld

g ;< v(x), the supergravity gauge and the cor­

responding gravit ino field </)[t, and a set of

Y a n g - M i l l s gauges and associated Y a n g -

Mills fields in the linear 6 pa r t of gfla(z). All

o ther gauges such as those of higher order in

0 a are spontaneously b roken for a globally

supersymmetr ic vacuum. Fur ther , one can

also show tha t for each of the b roken gauge

fields there resides in gAn(z) a fictitious Gold-

stone f ie ld which may be absorbed by the

b roken gauge field to lead to a mass growth.

Such f ic t i t ious Golds tone f ie lds do no t exist

for the preserved gauges so tha t the correspond­

ing fields remain massless after spontaneous

breaking. The analysis has been carried to

all orders in 6 a and arbi t rary N and signals

progress towards determining the full particle

content of the theory. Fur the r the analysis

shows tha t gauge supersymmetry possesses a

n o r m a l Golds tone and Higgs structure.

§7. Superspace Supergravity Formalism

We have remarked earlier tha t the spon­

taneously b roken gauge supersymmetry con­

tains for arbi t rary values of K, a supergravity

gauge which emerges as one of the preserved

gauges in the theory. The existence of such a

gauge in the b roken gauge supersymmetry

theory is quite remarkable since it represents

a t rue ou tpu t of the spontaneous b reakdown.

Suppose one discarded in the metr ic tensor

gAn(z) all fields except the gravi tat ional vier­

bein e%(x) and the spin 3/2 field <f)%x) (which

form an N= 1 global supersymmetry mult iplet)

and constructed a gAn=gAn(e™> <Pf) satisfying

eq. (1) and the supergravity t r ans fo rmat ion 1 6

law for e™ and (pfi. The metr ic gAn(z) would

then be m a d e gauge complete with respect

to the supergravity gauge group generated by

a single spinor Za(x). This procedure should

then lead to a superspace formulat ion of super-

gravity.

To implement the p r o g r a m one expands

gAn(z) and ÇA(z) in powers of 0 a with the lowest

non-vanishing terms given by

(13)

and Equa t ion (1) is

then integrated order by order in 0a. Calcula­

t ions to the order needed to generate super-

gravity field equat ions yield :

(14)

where A£A and AgAn are te rms of order k or

higher. Fur ther , the analysis determines the

vierbein affinity r^ to be correctly tha t of

supergravity; r\t=(il4)orsœ™ where co™ cor­

rectly includes the supergravity tors ion. T h u s

the process of gauge complet ing the metr ic

natural ly brings the tors ion into the metr ic

wi thout having to modify the R iemann ian

na ture of the superspace. Inser t ion of the

gauge completed metr ic in eq. (2) then leads

to the supergravity field equat ions in the k-+0

limit.

Recently, a l ternate superspace formulat ions

of supergravity have been given which use the

super vierbein Vj(z) (the Lat in indices represent

local quantit ies) and do no t require the limit­

ing p r o c e d u r e . 5 , 6 The equivalence of these

vierbein approaches to the previous metr ic

formulat ion of supergravi ty 4 can now be estab­

lished 7 for physical (mass-shell) spaces. The

analysis uses the fact t ha t one may always

Supersymmetry {Including Supergravity) 567

construct a Riemannian metric given a set of supervierbein: 6

(15)

where rjAB is the tangent space metric with rjmn and 7]ah being the bose and fermi matrices. The vierbein approach of ref. (6) uses the technique of auxiliary Breitenlohner fields 1 7

in superspace. If one eliminates these Brei­tenlohner fields and uses the supergravity field equations as required for consistency, one finds that the supervierbein of ref. (6) so reduced when inserted in eq. (15) produces precisely the metric of eq. (14) when mass-shell conditions are also imposed there. This result holds for arbitrary values of k. Further, the transformation functions of ref. (6) are seen to be identical to those of eq. (14) since ât;A all vanish on the mass-shell. Thus the vierbein and the metric approaches are equi­valent for physical mass-shell manifolds.

Further, it is seen possible to gauge com­plete the vierbein even off-shell without the use of Breitenlohner fields by relaxing the tangent space assumptions of refs. (5) and (6) while maintaining correctly the supergravity transformation group. In general the super­vierbein obey the transformation law

(16)

where we have included the tangent space transformations generated by eA

B. Commonly, the tangent space transformations are chosen to be super-Lorentzian, 5 ' 6 i.e., em

a=0=ea

m and em

n(z), ea

h(z) generate Lorentz transformations. However, it is not possible 7 to gauge complete this vierbein off-shell using only the fields em

fi(x) and (p^x). In ref. (6) Breitenlohner fields were introduced to overcome this dif­ficulty. However, this difficulty may also be circumvented by an expansion of the tangent coefficients eA

B while still preserving the super-gravity group. We thus abandon the super-Lorentzian condition and utilize the full eA

B

in the process of guage completing the vierbein with the only restriction that the invariance of S An b e preserved. This restriction requires that

(17)

Using the restrictions of eq. (17) (rather than the super-Lorentzian condition), it has been possible to gauge complete the vierbein to the same order as the metric. 7 Further , using this gauge completed supervierbein one may ge­nerate supergravity field equations. This can be done either by use of the gauge super-symmetry field equations RAB(V

A

A)—lr)AB=0

in the limit k->0 or using the superspace equat ions 6

( 1 8 )

without taking the limit &->0. [Here the Xrs

are Lorentz rotation matrices and Rr8

AS are given in ref. (6).] Thus once again one finds the equivalence of the vierbein and metric for­malisms.

§8. Quantum Loop Corrections

Calculations of quantum loops in gauge supersymmetry are currently underway. We have established the remarkable fact that the one-loop corrections to the effective potential are finite. This arises due to the global supersymmetry which sets to zero the divergent parts of loop integrals. Calculations of pro­pagator corrections are more complicated but preliminary analyses show tha t here again global supersymmetry acts to eliminate most (and perhaps all) divergences. Since quantum calculations are now technically feasible, a more realistic investigation of the scope of gauge supersymmetry theory can now be undertaken.

References and Footnotes

1. A. Sa lam a n d J . S t r a thdee : Nuc l . Phys . B76

(1974) 477.

2 . P . N a t h a n d R . A r n o w i t t : Phys . Let ters 56B (1975) 171 ; R. Arnowi t t a n d P . N a t h : G e n . Re l .

G r a v . 7 (1975) 89.

3 . D . Z . F r e e d m a n , P . van Nieuwenhuizen a n d S .

F e r r a r a : Phys . Rev . D 1 3 (1976) 3214; S . Dese r

a n d B . Z u m i n o : Phys . Let ters 62B (1976) 335.

4 . P . N a t h a n d R . A r n o w i t t : Phys . Let ters 65B (1976) 73 .

5 . J . Wess a n d B. Z u m i n o : Phys . Let ters 66B (1977)

361.

6 . L . Br ink , M. G e l l - M a n n , P . R a m o n d a n d J . Sch­

w a r z : C A L T 68-644, 6 8 - 6 4 9 ; M . G e l l - M a n n

a n d J . Schwarz : Aspen W o r k s h o p in Super-

symmetry , 1977.

7 . R . A r n o w i t t a n d P . N a t h : "Super space F o r m u ­

lat ion o f Supergrav i ty , " N U B N o . 2361 .

8 . P . N a t h a n d R. A r n o w i t t : " P a t h In tegra l Q u a n t i ­

za t ion o f G a u g e Supe r symmet ry , " N U B N o . 2343,

568 P . N A T H

t o be publ i shed in Phys . R e v .

9 . R . A r n o w i t t a n d P . N a t h : "Prese rved G a u g e s ,

B r o k e n G a u g e s a n d F ic t i t ious G o l d s t o n e F ie lds

i n G a u g e S u p e r s y m m e t r y , " N U B N o . 2344, s u b .

t o N u c l . P h y s . B . 10. R . A r n o w i t t , P . N a t h a n d B . Z u m i n o : Phys .

Le t te rs 5 6 B (1975) 8 1 .

1 1 . A . A . S l avnov : N u c l . Phys . B97 (1975) 155 ;

B . de W i t : P h y s . R e v . D 1 2 (1975) 1628; S .

F e r r a r a a n d O . P i q u e t : N u c l . P h y s . B93 (1975)

2 6 1 ; J . H o n e r k a m p , F . K r a u s e , M . Scheuner t

a n d M. Schl indwein : N u c l . Phys . B95 (1975)

397.

12.

wher " I " m e a n s covar ian t differentiation wi th

respect to t h e b a c k g r o u n d me t r i c g°An.

13. / 0 = $ d ^ ^ z ) b ( ^ ; / r , , ^ ( ^ ^ - ( i / 2 ) ( - i ) ^ ( / / ;

I)g(°y>2nu] whe re " ; " m e a n s covar ian t differen­

t i a t ion w i th respect to the full met r i c gAn(z) a n d

V(A;ID=yA;n+(-l)A+n+AIIyn;A.

14. R . J a c k i w : Phys . R e v . D9 (1974) 1701.

15. W{gAn)—— i l n $ d / ^ / 7 d ^ d ^ * e * ? wi th / given by

ï=i(gAn+hAn, VA,_yA*',s^0)An)-KsAn\ va, yA*\

g{0)An)—hAn{dridgAn). Z is t h e s a m e funct ional

integral as Z b u t w i th / r ep laced by / .

16. dettn=i<Ptt(x)rmX(x); ( 1 / 2 ) 5 ^ = ( 3 ^ + ^ = 1 ) ^ .

17. P . Bre i t en lohner : Phys . Le t t e r s 67B (1977) 49.

Session C7: Quantum Electrodynamics

Chairman: Y . O H N U K I

Organizer: T . K I N O S H I T A

Scientific Secretaries: A . U K A W A N . N A K A G A W A

1 . R e c e n t D e v e l o p m e n t o f Q u a n t u m E lec t rodynamics

T . K I N O S H I T A

2 . M u o n i u m Hyperf ine S t ruc tu re V . W . H U G H E S

(F r iday , A u g u s t 25 , 1978; 15 : 4 0 - 1 7 : 00)