LETTERE AL NUOVO CIMENTO VOL. 13, N. 14 2 Agosto 1975

Scalar Superfield D y n a m i c s f r o m the A r n o w i t t - N a t h S u p e r g e o m e t r y (*).

G. Woo (**)

Laboratory /or iVuclear Science and Department o/Physics Massachusetts Institute o] Technology - Cambridge, Mass.

(riccvuto il 19 Maggio 1975)

The use of the group algebra (1) to generate representations of supersymmotry has led naturally to the introduction of four Majorana spinor co-ordinates which, together with the four Bose co-ordinates of space-time, define an eight-dimensional supcrspacc. This may be accepted merely as a technical mathematical procedure, not necessarily rich in physical content, but it would obviously be more sat isfactory if a purely geometrical s tudy of superspacc could reproduce the dynamics of super- symmetr ic theories.

Recently (8), the concept of global supersymmctry was enlarged to tha t of a local gauge symmetry in order to construct a supergravitational theory. This theory is invariant under arbitrary co-ordinate transformations in the eight-dimcnsional super- space. ARNOWITT and NATH have been able to formulate a modified Ricmannian geometry for this space by suitably adapting the definitions of standard terms to allow for ant icommuting quantities.

The existence of this systematic geometry in general curved superspaee prompts the investigation of the structure of geometries which ignore gravitational effects. These geometries would then be nondynamical and flat in space-time but curved in the other sectors. Conventional supersymmetry has such a geometry.

In this paper the important quantities of geometrical interest are calculated to exhibit the features of this geometry, and are used to find the motion of a scalar super- field. I t is found tha t the standard equation of motion is obtained for a limiting metric.

The eight co-ordinates of superspace are z ~ = (xa, 0a}, where 0 a is an anticommuting Majorana spinor.

The supersymmetry transformations of this space are (3)

i

(*) This work is supported in part through funds provided by the ERDA under Contract AT(11-1)-3069. (**) Kennedy Memorial Scholar. (1) A. SALA• and J. STRATHDEE: ICTP preprint IC/74/11 (1974); B. ZUMz~O: Proceedings o! the X V I I International Con/erence on Iligh-Energy Physics (London, 1974). 0) PRA~ NXTIZ and R. AR~OWITT: Northeastern preprint NUB 2246 (1975). (a) A. SALA~ and J. STRATIIDEE: ICTP preprint IC/74/42 (1974); ICTP proprint IG]74/85 (1974).

546

SCALAR S U P E R F I E L D DYNAMICS FROM THE A R N O W I T T - N A T H SUPERG]~OM~.TRY ~ 7

where e a is an an t ieommut ing Majorana spinor, independent of z a. Expressed in terms of differentials, we have

i dx'~ = dx~ + ~ ~yv dO, d0'a = d0 a .

The two l inear forms d0 a and d x ~ - - ( i / 2 ) @ ~ d O arc invar ian t . The line clement of second order in differentials, which is invar ian t under the entire

algebra of Poincar4 and supersymmetry transformations, is wri t ten

ds ---- l dx~ - - ~ OT~ ] -t- K dOdO ,

where K is a constant, and the coefficient of the first term has been normalized to the Minkowski expression for a line element in the absence of 0-spacc.

On expanding, this is

ds ~ ---- dxg ~ dx ' - - ~ dx~(0y~)~ d0 a + ~ dOa(Oy~) a dx~ + -~ d0a(0y~) a (07~)~ d0~ + K d0a#a~ dO~,

where ~ = - - C ~ as in ref. (~) (C is the charge conjugation matrix). The metric superfield g~B(z) is defined by

ds 2 = dz~ g ~ ( z ) dz n ,

so tha t

i .

g~, = - - ~ (or~,):,

i g ~ . = ~ (o~#)~ ,

1 _ g ~ = K~a~ + ~ (O~,,)a (07") ~ .

The symmetry requirements g ~ = gv~, g~a = - - g ~ , , g ~ = - - g ~ are satisfied. The inverse metric g~.B(z) can be constructed from the definition g~g~C = ~ .

The components are

1 g~v= ~v§ ~O7~YvO'

i g~== - - _ _ (y~0) = ,

2K

i g ~ = - - 2--K ( ~ 0 ) = '

~at~

g ~ = -K- '

where ~ a ~ = ~ .

548 o. woo

Again the symmet ry requirements guy= 9vu, gun= gnU, gO~=_g~a arc satisfied. The affine connections can be computed from the formula

F , an= ( - - 1 )be �89 [(__ 1)~'ag,~, n + (-- 1 )a%abg$~,a-- gan,~] g~O ,

where ~?~b= (--1) ~+b+=b and the small l e t te r corresponding to a cap i ta l - le t te r index is zero i f t h a t index is Bose and un i ty if i t is Fermi .

We thus have

/ ' v~ , = 0 , 1

G"~ = ~ (@,r"0)= ,

i r , ~ = ~ {(Or.o)= ( @ ' r ~ o ) ~ - ( @ . % - ( @"7~'o)=} ,

1". =, = o . r, .=a = ~-~ (r,,)=,~,

1 r,,t~ = ~ {-- (r (r')0~ + (@~0)~ (r%=} �9

The geomet ry is fixed and independent of fields and can be computed fur ther . The formula for the Ricci tensor :B~ is (2)

R ~ = - - ( - - 1 ) ~ t l ~ t + b ~ r ~ • ( _ _ l ) b t T , A ~ ( _ _ l ~ e / , o r lX~tF E F o I . ~ ~ l ~ .dB~O I I .~ 0 - - ~. - I A 0 .B .B "

Thus

Buy = - - ~pv i i R~,,, = ~ - ~ (@u)=, R,,~, = - - (Ors)=,

and

R,,~= K 4K ~

Le t us now consider the mot ion of a scalar superfield in the presence of th is par t icu la r geometry . (No renormalizable in terac t ions are known for o ther superfields.) The generic symbol r is used for a scalar superfield of e i ther chira l i ty .

The r igh t eont ravar ian t der iva t ive of qO(z) is defined b y

8RO ~,a = g** ( z )

~z s

where the /Lsuffix denotes r ight der ivat ive. The r igh t eovar iant der ivat ive of a vector V a is defined by (2)

~n V, A A V; .= ~ + r'%r'v~.

SCALAR SUPERFIELD DYNAMICS FRO~ THE A_~NOWITT-NATH SUPERGEOMETRY ~ 9

The scalar contraction is

an lT-a (-- 1)aV#~ = (--i) a - + (-- I)aV'Fd~

~z,~

The kinetic term in the equation of motion for r will be

~R~.~[ (--1)a~b'~= ( - - 1 ) a - - ~ - (--1)"o,~rJ~.

, ~ . a

From the above expressions for the affine connections we have

for all indices G. Hence

Using (a)

we find tha t

( -1) ' feZ= o

[~RO Ba 1

We now rescale O(z) so tha t r Since our geometry is fixed and non- dynamical, this operation has no effect on the geometry.

Then the kinetic term for ~(z) becomes

[---~-(/gy~D)~ + / ) D ] ~(z) .

A mass term for ~(z) and a cubic interaction term can be added without difficulty. The same equation can be arrived at by start ing from the alternative definition

1

~_~a L~ g~(~) ~ (-~)o

and using the definition of ~ given in ref. (~). For our metric

1 ~:'~\1} 1

(4) R . ~kRNOWITT, PR•2q NATIt and B. ZIrMINO: Northeastern prepr int NUB 2247 (1975),

550 G. woo

Now, the t e rm in the different ial o p e r a t o r - (K/4)(D?I,D) 8 leads to quar t ic deriv- a t ives for field components, and hence is not permissible if ghosts are to be avoided. Hence we are obliged to consider the l imi t K - > 0 .

In th is l imi t the s tandard field equation is regained by geometr ical means. The l imi t ing singular metr ic then corresponds to the line e lement dsS= (dxg--( i /2)0?~d0) 8 observed b y SALAM and STRATt~D~. (8) to be invar ian t under the supe r symmet ry algebra.

$ $ *

I wish to thank Lord HARLEC~ and the Kennedy Memorial Trus t for the award of a scholarship.

I am indeb ted to Profs. P. NATH and M. FRIEDMAN Of Nor theas tern Univers i ty for discussions.