ON

THE

DEFiNiTION A.S.

OF

SUPERSPACE

Shvarts

A t t e n t i o n is d r a w n to a m a t h e m a t i c a l f o r m a l i z a t i o n of the c o n c e p t s of s u p e r m a t h e m a t i c s e m p l o y e d in p h y s i c s in a l a n g u a g e t h a t d o e s not u s e s h e a f t h e o r y . F . A. B e r e z i n w a s the f i r s t who u n d e r s t o o d t h a t b e s i d e s o r d i n a r y a n a l y s i s and a l g e b r a t h e r e e x i s t a n a l o g o u s t h e o r i e s in w h i c h a n t i c o m m u t i n g q u a n t i t i e s p l a y the p a r t of r e a l n u m b e r s . He s h o w e d that in the s t u d y of the m e t h o d of s e c o n d q u a n t i z a t i o n it is h e l p f u l to u s e d i f f e r e n t i a t i o n and i n t e g r a t i o n of f u n c t i o n s of a n t i c o m m u t i n g v a r i a b l e s [1]. F u r t h e r , in [2] t h e r e w a s p r o p o s e d the c o n c e p t of g r o u p s with a n t i c o m m u t i a g p a r a m e t e r s , and t h e c o r r e s p o n d i n g c o n c e p t of L i e a l g e b r a s w a s i n t r o d u c e d (at the p r e s e n t t i m e , t h e s e o b j e c t s a r e c a l l e d L i e s u p e r g r o u p s and L i e s u p e r a l g e b r a s ) . T h e s e c o n c e p t s c a m e to be w i d e l y u s e d in p h y s i c s a f t e r the d i s c o v e r y of s y m m e t r i e s that m i x b o s o n s and f e r m i o n s ( s u p e r s y m m e t r i e s ) [3-5]. It w a s shown in [6] that in the s t u d y of s u c h t h e o r i e s it is v e r y c o n v e n i e n t to c o n s i d e r a f i e l d on s p a c e s h a v i n g a s c o o r d i n a t e s both c o m m u t i n g and a n t i c o m m u t i n g q u a n t i t i e s ( s u p e r s p a c e s ) . T h e p h y s i c s s t u d i e s s t i m u l a t e d a m a t h e m a t i c a l i n v e s t i g a t i o n of the c o r r e s p o n d i n g q u e s t i o n s , in p a r t i c u l a r , [71 i n t r o d u c e d the c o n c e p t of a s u p e r m a n i f o l d , t h i s c o n c e p t h a v i n g o c c u r r e d i m p l i c i t l y in p h y s i c s s t u d i e s a s w e l l . T h e d e f i n i t i o n of a s u p e r m a n i f o l d g i v e n in [7] i s b a s e d on s h e a f t h e o r y and is d i f f i c u l t f o r p h y s i c i s t s . T h e a i m of the p r e s e n t p a p e r is to g i v e a d e f i n i t i o n of a s u p e r m a n i f o l d and the c o n c e p t s a s s o c i a t e d with it t h a t , on the one hand, a r e m a t h e m a t i c a l l y r i g o r o u s and, on the o t h e r , a r e a s c l o s e a s p o s s i b l e to the i n t u i t i o n of p h y s i c i s t s . One m a y s a y t h a t the p r o p o s e d d e f i n i t i o n s a r e a m a t h e m a t i c a l f o r m a l i z a t i o n of the c o n c e p t s e m p l o y e d in p h y s i c s . T h e b a s i c i d e a of t h e s e d e f i n i t i o n s is c o n t a i n e d in [8]. F r o m the p o i n t of view of p h y s i c s , the (p, q ) - d i m e n s i o n a l s u p e r s p a e e ~P,q is a s p a c e in w h i c h a point h a s p even c o o r d i n a t e s and q odd c o o r d i n a t e s . T h e e v e n c o o r d i n a t e s a r e r e g a r d e d a s even e l e m e n t s of a G r a s s m a n n a l g e b r a , the odd c o o r d i n a t e s a s odd e l e m e n t s of a G r a s s m a n n a l g e b r a . F o r a m a t h e m a t i c i a n , t h i s d e f i n i t i o n i m m e d i a t e l y p o s e s the f o l l o w i n g q u e s t i o n : W h a t p r e c i s e G r a s s m a n n a l g e b r a do the c o o r d i n a t e s b e l o n g to ? The p h y s i c i s t m u s t a n s w e r t h a t in t h i s d e f i n i t i o n the G r a s s m a n n a l g e b r a is not fixed but a r b i t r a r y . T h e n the m a t h e m a t i c i a n c o n c l u d e s t h a t it is not one (p, q ) - d i m e n s ! o n a l s u p e r s p a c e but an i n f i n i t e s e t of s u p e r s p a e e s N~'q, d e f i n e d by d i f f e r e n t G r a s s m a n n a l g e b r a s A (a p o i n t in ~ ' q is d e t e r m i n e d by a row of p even and q odd e l e m e n t s of the a l g e b r a A). T h e p h y s i c i s t w i l l p r o b a b l y not a g r e e with t h i s , e x p l a i n i n g t h a t a l l the s p a c e s N~' q b e l o n g to one and the s a m e s u p e r s p a c e . It now r e m a i n s to t a k e the final s t e p : The r e s o l u t i o n of the d i s p u t e b e t w e e n the m a t h e m a t i c i a n and the p h y s i c i s t is t h a t the s u p e r s p a c e ~p,q is to be r e g a r d e d a s the s e t of a l l the s p a c e s F.~'q, but t h e s e s p a c e s m u s t be a s s u m e d to be r e l a t e d to e a c h o t h e r . N a m e l y , tt m u s t be noted that f o r e v e r y p a r i t y - c o n s e r v i n g h o m o m o r p h i s m p of the G r a s s r n a n n a l g e b r a A into the G r a s s m a n n a l g e b r a A' one c a n c o n s t r u c t in a n a t u r a l m a n n e r a m a p p i n g g of the s p a c e ~ ' q into N~'.~. G e n e r a l i z i n g t h i s s c h e m e , we i n t r o d u c e a g e n e r a l c o n c e p t of s u p e r s p a c e , on the b a s i s of w h i c h we d e f i n e I a t e r t h e c o n c e p t s of s u p e r m a a i f o l d , L i e s u p e r g r o u p , and so f o r t h . W e note that in [9) a s o m e w h a t d i f f e r e n t a p p r o a c h w a s p r o p o s e d . In it, one c o n s i d e r s s p a c e s N_~'q f o r fixed A, but the c o m p l e t e t h e o r y is c o n s t r u c t e d in s u c h a w a y that nothing d e p e n d s on the c h o i c e of A . We s h a l l s a y t h a t a s u p e r s p a c e ~ f is g i v e n if e a c h G r a s s m a n n a l g e b r a A is a s s o c i a t e d w i t h a s e t F ~ (the s e t of A p o i n t s of the s u p e r s p a c e 8") and w i t h e a c h p a r i t y - c o n s e r v i n g h o m o m o r p h i s m p of the G r a s s m a n n a l g e b r a A into the G r a s s m a n n a l g e b r a A' t h e r e is a s s o c i a t e d a m a p p i n g of the s e t s ~: ~ ' ~ - ~ , , in s u c h a w a y t h a t the p r o d u c t olo 2 of h o m o m o r p h i s m s c o r r e s p o n d s to the p r o d u c t 0 , ~ of the m a p p i n g s (i. e . , P,P'-~='Pt0"~). T h e s p a c e ~'0 c o r r e s p o n d i n g to the G r a s s m a n n a l g e b r a A = N (the G r a s s m a n n a l g e b r a h a v i n g z e r o g e n e r a t o r s ) is c a l l e d t h e u n d e r l y i n g s p a c e of the s u p e r s p a c e ~f. F o r e v e r y G r a s s m a n n a l g e b r a A t h e r e is d e f i n e d a h o m o m o r p h i s m m: A ~ N, w h i c h a s s o c i a t e s an e l e m e n t of the a l g e b r a with i t s n u m e r i c p a r t . The m a p p i n g ff~ c o r r e s p o n d i n g to t h i s h o m o m o r p h i s m a s s o c i a t e s with e a c h A p o i n t x ~ a the p o i n t gt(z)~/$o ( " n u m e r i c p a r t " of the A p o i n t ) . The d e f i n i t i o n of su~perspace j u s t g i v e n is too g e n e r a l f o r the c o n s t r u c t i o n of a t h e o r y w i t h c o n t e n t . To c o n s t r u c t s u c h a t h e o r y , it is n e c e s s a r y to a s s u m e that in the s e t s ~ a t h e r e Moscow Engineering Physics Institute. Translated from Teoretieheskaya i Matematicheskaya F i z i k a , Vol. 60, No. 1, pp. 3 7 - 4 2 , J u l y , 1984. O r i g i n a l a r t i c l e s u b m i t t e d O c t o b e r 5, 1983.

0040-5779/84/6001-0657r

9 1985 P l e n u m Publishing

Corporation

657

is i n t r o d u c e d a c e r t a i n a d d i t i o n a l s t r u c t u r e , and c o r r e s p o n d i n g c o n d i t i o n s a r e i m p o s e d on the m a p p i n g s b'. F o r e x a m p l e , one c a n define the c o n c e p t of a s u p e r g r o u p by s u p p o s i n g that all the s e t s ~ a r e g r o u p s and all the m a p p i n g s g h o m o m o r p h i s m s ; then it is n a t u r a l to c a l l the s u p e r s p a c e 'eT a s u p e r g r o u p . We now give s o m e e x a m p l e s of s u p e r s p a e e s in the s e n s e of the g e n e r a l d e f i n i t i o n g i v e n above. T h e n we s h a l l give d e f i n i t i o n s of a l i n e a r s u p e r s p a e e , a Lie M g e b r a , a s u p e r m a n i f o l d , and a Lie s u p e r g r o u p , and we shall i n d i c a t e in which e x a m p l e s the c o r r e s p o n d i n g s t r u c t u r e s c a n be i n t r o d u c e d . 1. Let M be a Z , - g r a d e d l i n e a r s p a c e , i . e . , a l i n e a r s p a c e r e p r e s e n t e d in the f o r m of the d i r e c t s u m MoSM,, w h e r e M 0 is c a l l e d the e v e n s u b s p a c e and M~ the odd. We define the s e t of A points Jt'~ as the set of f o r m a l l i n e a r c o m b i n a t i o n s Ea,e,+Eb~f~, w h e r e e~Mo, filM,, ar a r e the even and bj the odd e l e m e n t s of the a l g e b r a A (it is a s s u m e d that (a'+a")m=a'm+a'}m, a(m'+m")=am'+am", a, a',a"~A; m, m', m"OM). The m a p p i n g ~ c a r r i e s the point Ea~e,+Eb~f[ to EfJ(aOe~+Ep(b~)]~. The c o l l e c t i o n of s e t s ~ , and m a p p i n g s y d e f i n e s the s u p e r s p a c e J / , which c o r r e s p o n d s to the Z , . - g r a d e d s p a c e M. In the s p e c i a l c a s e when M~ and M~ are respectively, p- and q-dimensional linear spaces, it is natural to identify the superspace ./// with the superspace ~,~.

space

An example of an infinite-dimensional super~pace is the superspace B~,~, the elements of which are expressions of the form

constructed

from

the

Z 2-graded

w h e r e f~..... , a r e i n f i n i t e l y d i f f e r e n t i a b l e f u n c t i o n s of the r e a l v a r i a b l e n x ' , . . . , x ~, and ~ ' , . . . , ~ a r e the g e n e r a t o r s of the G r a s s m a n n a l g e b r a Aq (an e x p r e s s i o n of the f o r m (1) is a s s u m e d to be e v e n if all the t e r m s in (1) c o n t a i n an e v e n n u m b e r of g e n e r a t o r s ~, and odd if the n u m b e r of ~ is odd). F x p r e s s i o n s of the f o r m (1) f o r m an a l g e b r a , w h i c h c a n be r e g a r d e d as an a l g e b r a with p c o m m u t i n g and q a n t i c o m m u t i n g g e n e r a t o r s (it is n a t u r a l to call such a l g e b r a s B e r e z i n a l g e b r a s ) , it is i m p o r t a n t to note that the e x p r e s s i o n (1) is s t i l l m e a n i n g f u l if i n s t e a d of the r e a l n u m b e r s x ~, . . . , x ~ one s u b s t i t u t e s a r b i t r a r y even e l e m e n t s of the G r a s s m a n n a l g e b r a A, and i n s t e a d of ~*,..., ~ a r b i t r a r y odd e l e m e n t s of h (to give a m e a n i n g to ]~..... ~(x' . . . . , x~), if x i a r e even e l e m e n t s of the G r a s s m a n n a l g e b r a , it is n e c e s s a r y to u s e an e x p a n s i o n in a T a y l o r s e r i e s with r e s p e c t to the n i l p o t e n t p a r t s of the e l e m e n t s x ~). O u r r e m a r k m a k e s it p o s s i b l e to r e g a r d an e x p r e s s i o n of the f o r m (1) as a f u n c t i o n on the s u p e r s p a e e ~,~. The l i n e a r c o m b i n a t i o n s of e x p r e s s i o n s of the f o r m (1) with coefficients in the Grassmann algebra h we shall call h functions on [~ ~,~. The set of h points of the superspace corresponding to the 7.~-graded space B~,~can be identified with the set of even A functions on the space ~,~. 2. We consider in ~ ' ~ the set AA, which consists of the points determined by the equations

f, (x', . . . . x,, ~' . . . . . ~ ) = 0 , ]~(z' . . . . . x,, ~' . . . . . ~ ) = 0 , g~(z', . . . . x~, ~ ' , . . . , ~ ) = 0 ,

(2)

g. (z' . . . . . x~, ~' . . . . . ~ ) =0, w h e r e x i and ~J a r e , r e s p e c t i v e l y , even and odd e l e m e n t s of the G r a s s m a n n a l g e b r a , and 1,. . . . , l, and g , , . . . , g, a r e , r e s p e c t i v e l y , even and odd f u n c t i o n s on the s u p e r s p a c e ~.~,* (we r e c a l l that a f u n c t i o n on ~p,q is defined by an e x p r e s s i o n of the f o r m (1)). It is e a s y to s e e that the m a p p i n g ~ of the s e t ~ q into 0 ~ "q c o n s t r u c t e d f r o m the h o m o m o r p h i s m p c a r r i e s the s e t AA into A~,. T h e r e f o r e , the s e t Ah t o g e t h e r with the m a p p i n g s ~ d e f i n e s the s u p e r s p a c e ~ . 3. The construction of the superspace ~ just given is a special case of a more general construction. Namely if for the superspace ~r we separate in each of the sets ~A the subset ~'A', which satisfies the condition ~(~A')~A'~ for each of the homomorphisms p:~A-~A', then the set ~A~ together with the mappings ~^ forms a superspace that can naturally be called a subsuperspace of the superspace ~. In particular, if v-//~'0 is a subset of the underlying space ~0, we consider the sets ~A~, which consist of points X~A such that ~(x)Ea// for the homomorphism m: A-+~. These sets define the superspace ~,~x, which is called the subsuperspaee over ~ . [f ~=~P.q, and ~ is a domain in 0~~,~ then ~'~ is called a superdomain (putting it differently, one can say that the superdomain (0~P.q)~ consists of the points whose numeric parts belong

6~8

to ~ ) . 4.

We c o n s i d e r the s e t a/t'.~'*1~''r

which c o n s i s t s of the block m a t r i c e s

CD

'

w h e r e A and D c o n s i s t o f even e l e m e n t s of the a l g e b r a A and have d i m e n s i o n s p • p

t

r

and q • q , and B

and C c o n s i s t of odd e l e m e n t s of the a l g e b r a A and have d i m e n s i o n s q • p ' and p • q ' . The s e t s ~'~'~'~'"' d e t e r m i n e the s u p e r s p a c e ./Kp,ql~',q'. T h i s s p a c e is i s o m o r p h i c to the s u p e r s p a c e R PP'+qq',qP'+'q' In the e a s e when t

r

p = p , q = q , the m a t r i c e s in .,~,~lp',~' c a n be m u l t i p l i e d in a c c o r d a n c e with the u s u a l r u l e s .

We denote

the s e t of i n v e r t i b l e e l e m e n t s in off~p'qlp'' by GL A(p, q ) . T h i s set f o r m s a g r o u p . It is r e a d i l y v e r i f i e d that GLA(p , q ) c o n s i s t s of m a t r i c e s w h o s e n u m e r i c p a r t s a r e i n v e r t i b l e . T h i s m a k e s it p o s s i b l e to r e g a r d the s u p e r s p a c e G L ( p , q ) as a s u p e r d o m a i n o v e r the d o m a i n GL(p)XGL(q)~./K~ ,qlp,~. Note that the s e t s R~ q c a n be r e g a r d e d as l i n e a r s p a c e s . M o r e o v e r , e l e m e n t s in NP,~ can be m u l t i p l i e d not only by n u m b e r s but a l s o by even e l e m e n t s of the G r a s s m a n n a l g e b r a . In o t h e r w o r d s , N~,e c a n be r e g a r d e d as a A 0 m o d u l e (by A 0 we d e n o t e , as u s u a l , the r i n g of even e l e m e n t s of the a l g e b r a ~). T h e r e f o r e , we s h a l l call the s u p e r s p a c e ~o a I i n e a r s u p e r s p a c e if all s e t s ~ , 9a r e A 0 m o d u l e s and the m a p p i n g s ~ a r e h o m o m o r p h i s m s of A 0 m o d u l e s . T h e c o n d i t i o n s of this d e f i n i t i o n a r e s a t i s f i e d not only by the s u p e r s p a e e s ]~p,q but a l s o by the m o r e g e n e r a l s u p e r s p a c e s /If c o n s t r u c t e d f r o m a Z 2 - g r a d e d s u p e r space. The l i n e a r s n p e r s p a c e ~T is c a l l e d a Lie s u p e r a l g e b r a if each of the s e t s ~'A is a Lie a l g e b r a and the m a p p i n g s ~ a r e h o m o m o r p h i s m s of Lie a l g e b r a s ( m o r e p r e c i s e l y , it is r e q u i r e d that ~A be a A 0 Lie a l g e b r a , i . e . , it is r e q u i r e d that lea, b ] = k [ a , b] hold for any k ~ A0). The s u p e r s p a c e jfp,qlp,q is a Lie s u p e r a l g e b r a with r e s p e c t to the u s u a l m a t r i x c o m m u t a t o r . Suppose that in the Z ~ - g r a d e d s p a c e M t h e r e is i n t r o d u c e d the s t r u c t u r e of a Z ~ - g r a d e d Lie a l g e b r a , i . e . , we have i n t r o d u c e d the o p e r a t i o n [,~, s a t i s f y i n g the m o d i f i e d J a c o b i i d e n t i t y . T h e n the c o r r e s p o n d i n g l i n e a r s u p e r s p a c e ~/t' is t r a n s f o r m e d n a t u r a l l y into a Lie s u p e r a l g e b r a . It can be v e r i f i e d that the c o n v e r s e is a l s o t r u e : E v e r y s t r u c t u r e of a Lie s u p e r a l g e b r a in the l i n e a r s u p e r s p a c e ./K can be g e n e r a t e d by m e a n s of this c o n s t r u c t i o n . If all the s e t s ~TA a r e g r o u p s , the m a p p i n g s T a r e h o m o m o r p h i s m s of g r o u p s , the s t t p e r s p a c e is c a l l e d a s u p e r g r o u p . The s i m p l e s t e x a m p l e of a s u p e r g r o u p is the s u p e r s p a e e G L ( p , q ) d e s c r i b e d above (we r e c a l l that, as we h a v e a l r e a d y s a i d , the s e t s GLA(P, q) a r e equipped with g r o u p s t r u c t u r e ) . B e f o r e we define the c o n c e p t of a s m o o t h s u p e r m a n i f o t d , we r e c a l l that for a l i n e a r s u p e r s p a c e we required that on the sets ~TA there be define the structure of a A 0 module. Therefore, to define a smooth supermanifold it is not sufficient to require ~, to be smooth manifolds. We also require that each of the tangent spaces to ~, be equipped with the structure of the A0 module. Of course, the mappings ~ must be smooth. However, on them it is also necessary to impose the additional requirement that the mappings of the tangent spaces generated by them be homomorphisms of A 0 modules. We introduce the concept of a A 0 manifold as a smooth manifold such that each of the tangent spaces to it is equipped with the structure of a A 0 module. We shall say that a smooth mapping of A 0 manifolds is A ~ smooth if the mappings of the tangent spaces generated by it are homomorphisms of A 0 modules. Using this concept, one can find the concept of a smooth supermanifold as a superspace for which ~TA are A 0 m a n i f o l d s and ~ a r e A 0 - s m o o t h m a p p i n g s of A 0 m a n i f o l d s . As an e x a m p l e of a s m o o t h s u p e r m a n i f o l d we c a n take a l i n e a r s u p e r s p a c e , and a l s o the s u p e r s p a c e defined by the e q u a t i o n s (2) in the e a s e when the n u m e r i c p a r t s of the m a t r i c e s (0Haa ~) and (0g~/0~0 h a v e , r e s p e c t i v e l y , the r a n k s r and s. E v e r y s u p e r m a n i f o l d in the u s u a l s e n s e [10] c a n be r e g a r d e d as a s m o o t h s u p e r m a n i f o l d in the s e n s e of the d e f i n i t i o n g i v e n a b o v e . T h e r e is a n a t u r a l d e f i n i t i o n of the a c t i o n of the s u p e r g r o u p ~ on the s u p e r s p a c e 3~: F o r e v e r y A, the g r o u p ~ m u s t act on ~F~, and t h e s e a c t i o n s for d i f f e r e n t A m u s t be m a d e c o n s i s t e n t by m e a n s of the m a p p i n g s ~ (if q~g is the m a p p i n g of the s p a c e ~ . c o r r e s p o n d i n g to the e l e m e n t g ~ . , then ~p~=~05(~)). The s p a c e of o r b i t s (the f a c t o r space) ~Y/$ of the a c t i o n of the s u p e r g r o u p ~ in the s u p e r s p a c e ~" is d e t e r m i n e d by m e a n s of the s e t s ~ ' . / ~ . (the m a p p i n g ~" is c o n s t r u c t e d in the n a t u r a l m a n n e r ) .

659

We note that the definition of s u p e r s p a c e given above is e s s e n t i a l l y m o r e g e n e r a l than the s t a n d a r d definition. In p a r t i c u l a r , this is r e f l e c t e d in the fact that, using the s t a n d a r d c o n c e p t s , it is not in g e n e r a l p o s s i b l e even in s i m p l e s i t u a t i o n s to define what is the f a c t o r s p a c e ~'/~. We c o n s i d e r a s i m p l e e x a m p l e . Let {~ be the s u p e r g r o u p G L ( 1 , 0) acting on the s u p e r s p a e e g = ~ ~ (the set GLA(1 , 0) c o n s i s t s of the i n v e r t i b l e even e l e m e n t s of the a l g e b r a A; with each such e l e m e n t X t h e r e is a s s o c i a t e d a t r a n s f o r m a t i o n of the s e t ~ q c o n s i s t i n g of m u l t i p l i c a t i o n of all the c o o r d i n a t e s by X). It is r e a d i l y v e r i f i e d that the f a c t o r s p a c e ~',vl GL(i, 0) is not a s u p e r m a n i f o l d in the s e n s e of the p r e s e n t p a p e r and a f o r t i o r i is not a s u p e r manifold in the s t a n d a r d s e n s e . An e x a m p l e of a smooth s u p e r m a n i f o l d that is not a s u p e r m a n i f o l d in the usual s e n s e can be c o n s t r u c t e d by s l i g h t l y modifying the definition of the s u p e r s p a c e ~ P'q. N a m e l y , one m u s t Consider in ~_gv the s u b s e t

~'~

c o n s i s t i n g of the points f o r which all c o o r d i n a t e s a r e nitpotent.

T h e s e s u b s e t s define the s u p e r s p a c e ~ ' q ,

which can be r e g a r d e d as a l i n e a r s u p e r s p a c e (in o t h e r w o r d s ,

it is p o s s i b l e to define IR~"q, as the s u b s u p e r s p a c e of ~ P.q o v e r the o r i g i n in the u n d e r l y i n g manifold: R p). We shall say that a s m o o t h s u p e r m a n i f o l d g is a (p, q ) - d i m e n s i o n a l s u p e r m a n i f o l d if for e v e r y point of the u n d e r l y i n g manifold g0 t h e r e e x i s t s a neighborhood oZ/ such that the s m o o t h manifold ~.vz o v e r o// is equivalent to a s u p e r d o m a i n . In [11], Voronov showed that the concept of a (p, q ) - d i m e n s i o n a l s u p e r m a n i f o l d as just defined is equivalent to the s t a n d a r d definition. LITERATURE 1. 2. 3. 4. 5. 6. 7. 8. 9. I0.

CITED

F. A. B e r e z i n , The Method of Second Quanttzatton, New York (1966). F. A. B e r e z i n a n d G . [. Kats, Mat. S b . , 82, 343 (1970). Yu. A. Gol'fand and E. P. L i k h t m a n , P i s ' m a Zh. E k s p . T e o r . F i z . , 13, 452 (1971). D. V. Volkov and V. P . Akulov, P h y s . L e t t . B, 46, 109 (1973). J. W e s s and B. Zumino, Nucl. P h y s . B, 70, 39 (1974). A. S a l a m a n d J. S t r a t h d e e , P h y s . Rev. D, 11, 1521 (1975). F. A. B e r e z i n a n d D. A. L e i t e s , Dokl. Akad. N a u k S S S R , 224, 505 (1975). A. S. S c h w a r z , Commun. Math. P h y s . , 87, 37 (1982). [. V. Volovich, DokI. Akad. Nauk SSSR, 269, 524 (1983). A. A. Voronov, T e o r . Mat. F i z . , 60, 43 (1985).

MAPPINGS A.A.

OF

SUPERMANIFOLDS

Voronov

The concept of a mapping of s u p e r m a n i f o l d s is a n a l y z e d . In [1], S h v a r t s d e s c r i b e d a m a t h e m a t i c a l f o r m a l i z a t i o n of the definition adopted in p h y s i c s of a s u p e r s p a c e , a s u p e r m a n i f o l d , and the o t h e r c o n c e p t s of s u p e r m a t h e m a t i c s . It is b a s e d on the idea that with each G r a s s m a n n a l g e b r a A t h e r e is a s s o c i a t e d the s e t ~ of h points of the s u p e r s p a c e , t h e s e s e t s being r e l a t e d by mappings y c o r r e s p o n d i n g to the h o m o m o r p h i s m s p of the G r a s s m a n n a l g e b r a s (see a l s o [2]). In the f r a m e w o r k of the definitions of [1], it is r e a s o n a b l e to define a mapping oe of the s u p e r s p a e e $" into the s u p e r s p a c e ~ as a s e t of mutually c o n s i s t e n t m a p p i n g s aA of the s e t s (gA into ~A- F o r smooth s u p e r m a n i f o l d s , the concept of a s m o o t h mapping can be n a t u r a l l y defined in the s e n s e of [1]. The main r e s u l t of the p r e s e n t p a p e r is the p r o o f that the concept of a s m o o t h mapping of (p, q ) - d i m e n s i o n a l s u p e r m a n i f o l d s is i d e n t i c a l to the concept of a m o r p h i s m of s u p e r m a n i f o l d s in the s t a n d a r d a p p r o a c h (based on s h e a f theory) [3]. It follows in p a r t i c u l a r that the definition of a (p, q ) - d i m e n s i o n a l s u p e r m a n i f o l d given in [1] is equivalent to the s t a n d a r d definition. Let (g and ~" be two s u p e r s p a e e s defined, r e s p e c t i v e l y , by m e a n s of the s e t s ~'A and ~ and mappings ~ a s s o c i a t e d with the h o m o m o r p h t s m s o of the G r a s s m a n n a l g e b r a s . We use the s a m e notation for the mapping of ~',. into ~Ta, and the mapping of ~'A into ~ a ' that a r e g e n e r a t e d by the h o m o m o r p h i s m p. We shall say that a mapping o~ of the s u p e r s p a c e g into the s u p e r s p a c e ~" is defined if with each G r a s s m a n n Moscow State U n i v e r s i t y . T r a n s l a t e d f r o m T e o r e t i c h e s k a y a i M a t e m a t i c h e s k a y a F i z i k a , Vol. 60, N o . l , p p . 4 3 - 4 8 , July, 1984. O r i g i n a l a r t i c l e s u b m i t t e d O c t o b e r 5, 1983.

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0040-5779/84/6001-0660.e08.50

9 1985 Plenum

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