superstrings in higher order extensions of finsler superspaces

8/14/2019 Superstrings in Higher Order Extensions of Finsler Superspaces

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ELSEVIER Nuclear Physics B 494 [ PM] (1997) 590-65613

Superstrings in h igher ord er exten sions o fFins ler superspaces

S e r g i u I . V a c a r u 1Institute of Applied Physics, Academy of Sciences, 5 Academy str., Chi~in~tu2028, Moldova

Received 6 November 1996; accepted 21 January 1997

A b s t r a c tT h e w o r k p r o p o s e s a g e n e r a l b a c k g r o u n d o f t h e t h e o r y o f f i e ld i n t er a c ti o n s an d s t ri n g s i n s p a c e s

w i th h i g h e r o r d e r a n i so t ro p y . O u r a p p r o a c h p r o c e e d s b y d e v e l o p i n g th e c o n c e p t o f h i g h e r o r d e ra n i s o t r o p i c s u p e r s p a c e w h i c h u n i f i e s t h e l o g i c a l a n d m a t h e m a t i c a l a s p e c t s o f m o d e m K a l u z a -K l e i n t h e o r i e s a n d g e n e r a l i z e d L a g r a n g e a n d F i n s l e r g e o m e t r y a n d l e a d s 1o m o d e l l i n g o f p h y s i c a lp r o c e s s e s o n h i g h e r o r d e r f ib e r b u n d l e s p r o v i d e d w i t h n o n - l i n e a r a n d d i s t in g u i s h e d c o n n e c t i o n sa n d m e t r i c s t r u c tu r e s. T h e v i e w a d o p t e d h e r e i s t h a t a g e n e r a l f ie l d t h e o r y s h o u l d i n c o r p o r a t ea l l p o s s i b l e a n i s o t r o p i c a n d s t o c h a s t ic m a n i f e st a t io n s o f c l a s s i c a l a n d q u a n t u m i n t e ra c t io n s a n d ,i n c o n s e q u e n c e , a c o r r e s p o n d i n g m o d i f i c a t io n o f b a s i c p r i n c i p l e s a n d m a t h e m a t i c a l m e t h o d s i nf o r m u l a t i o n o f p h y s i c a l t h e o r ie s .

T h e p r e s e n t a t i o n i s d i v i d e d i n t o t w o p a r t s . T h e f i r s t f i v e s e c t i o n s c o v e r t h e h i g h e r o r d e ra n i s o t r o p i c s u p e r s p a c e s . W e f o c u s o n t h e g e o m e t r y d i s t in g u i s h e d b y n o n - l i n e a r c o n n e c t i o n v e c t o rs u p e r b u n d l e s , c o n s i d e r d i f f e r e n t s u p e r s y m m e t r i c e x t e n s i o n s o f F i n s l e r a n d L a g r a n g e s p a c e s a n da n a l y z e t h e s t r u c tu r e o f b a s i c g e o m e t r i c o b j e c t s o n s u c h s u p e r s p a c e s . T h e r e m a i n i n g f iv e s e c ti o n sa r e d e v o t e d t o t h e t h e o r y o f h i g h e r o r d e r a n i s o t r o p i c s u p e r s tr i n g s . In t h e f r a m e w o r k o f s u p e r s y m -m e t r i c n o n - l i n e a r s i g m a m o d e l s i n F i n s l e r e x t e n d e d b a c k g r o u n d s w e p r o v e t h a t t h e l o w - e n e r g yd y n a m i c s o f s u c h s t ri n g s c o n t a i n s e q u a t i o n s o f m o t i o n f o r l o c a l l y a n i s o t ro p i c f ie l d i n t e ra c t io n s ,O u r w o r k i s t o b e c o m p a r e d w i t h i m p o r t a n t p r e v i o u s v a ri a n ts o f e x t e n s io n s o f F i n s l e r g e o m e t r ya n d g r a v i t y . T h e r e a r e s u b s t a n t i a l d i f f e r e n c e s , b e c a u s e w e r e l y o n m o d e l i n g o f h i g h e r o r d e ra n i s o t r o p i c in t e r a c t io n s o n s u p e r b u n d l e s p a c e s a n d d o n o t p r o p o s e s o m e " e x o t i c " F i n s l e r m o d e l sb u t a g e n e r a l a p p r o a c h w h i c h f o r t r i v i al o r c o r r e s p o n d i n g p a r a m e t r iz a t io n o f n o n - h n e a r c o n n e c t i o ns t ru c t u r es r e d u c e s t o K a l u z a - K l e i n a n d o t h e r v a r ia n t s o f c o m p a c t i f i e d h i g h e r - d i m e n s i o n s p a c e -t im e s . T h e g e o m e t r y o f n o n - l in e a r co n n e c t io n s ( n o t b e i n g c o n f u s e d w i t h c o n n e c t i o n s fo r n o n - l i n e a rr e a l i z a ti o n s o f g a u g e s u p e r g r o u p s ) i s f ir s t c o n s i d e r e d f o r s u p e r s p a c e s a n d p o s s i b l e c o n s e q u e n c e so n n o n - l i n e a r c o n n e c t i o n f i e l d s f o r c o m p a t i b l e p r o p a g a t i o n s o f s t r i n g s i n a n i s o t r o p i c b a c k g r o u n d sa r e a n a l y z e d . F i n a l l y , w e n o t e t h a t th e d e v e l o p e d c o m p u t a t io n m e t h o d s a r e g e n e r a l ( i n s o m ea s p e c t s v e r y s i m i l a r t o t h o s e f o r E i n s t e i n - C a r t a n - W e y l s p a c e s w h i c h i s a p r i o r it y c o m p a r i n g w i t ho t h e r c u m b e r s o m e c a l c u l a ti o n s i n F i n s l e r g e o m e t r y ) a n d a d m i t e x t e n s io n s t o v a r i o u s C l if f o r d a n ds p i n o r b u n d l e s . 1 9 97 E l s e v i e r S c i e n c e B . V.0550-3 213/97/$1 7.00 t~) 1997 Elsevier Science B.V. Al l rights reserved.PH S 0 5 5 0 - 3 2 1 3 ( 9 7 ) 0 0 0 8 9 - 8


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S.I. Vacaru/N uclear Physics B 494 [PM] (1997) 590 -656PACS: 02.40.-k; 04.65 .+e; I 1.90.+t; 12.10.-g; 12.60.J; 12.90.+bKeywords: Extended F insler superspaces; Loc allyanisotropicsigm a mod els; Superstrings

591

1 . I n t r o d u c t i o n

T h e d i f f e r e n t i a l s u p e r g e o m e t r y h a s b e e n f o r m u l a t e d w i t h t h e a i m o f g e t t i n g a g e o -m e t r i c f r a m e w o r k f o r t h e s u p e r s y m m e t r i c f i e l d t h e o r i e s ( s e e f o r t h e t h e o r y o f g r a d e dm a n i f o l d s R e f s . [ 1 9 , 6 6 , 6 7 , 6 5 ] , f o r t h e t h e o r y o f s u p e r m a n i f o ld s R e f s . [ 1 2 6 , 8 8 , 1 3 , 5 8 ]a n d f o r d e t a i l e d c o n s i d e r a t i o n s o f g e o m e t r i c a n d t o p o l o g i c a l a s p e c t s o f s u p e r m a n i f o l d sa n d f o r m u l a t i o n o f s u p e r a n a l y s i s , R e f s . [ 3 3 , 2 6 , 7 2 , 5 3 , 1 2 0 , 1 2 2 ] ) . I n th i s w o r k w e a p p l yt h e s u p e r g e o m e t r i c f o r m a l i s m f o r a s t u d y o f a n e w c l a s s o f ( h i g h e r o r d e r a n i s o t r o p i c )super spaces .

T h e c o n c e p t o f l o c a l a n i s o t r o p y i s l a r g e l y u s e d i n s o m e d i v i s i o n s o f t h e o r e t i c a l a n dm a t h e m a t i c a l p h y s i c s [ 1 2 1 , 5 6 ,5 7 , 7 8 ] ( s e e a l s o p o s s i b l e a p p l i c a t io n s i n p h y s i c s a n d b i o l -o g y i n [ 6 , 5 ] ) . T h e f ir s t m o d e l s o f lo c a l l y a n i s o t r o p i c ( l a ) s p a c e s ( l a - s p a c e s ) h a v e b e e np r o p o s e d b y F i n s le r [ 3 8 ] a n d C a r ta n [ 2 9 ] ( e a r l y a p p r o a c h e s a n d m o d e r n t re a tm e n t s o fF i n s l e r g e o m e t r y a n d i t s e x te n s i o n s c a n b e f o u n d , f o r i n s ta n c e , i n [ 9 0 , 7 , 8 , 7 4 ] ) . I n o u rw o r k s [ 1 0 2 - 1 0 4 , 1 0 6 , 1 0 9 , 1 1 3 , 1 1 9 , 1 1 6 ] w e t r y t o f o r m u l a t e t h e g e o m e t r y o f la - s p a ce s i na m a n n e r a s t o i n c l u d e b o t h v a r i a n t s o f F i n s l e r a n d L a g r a n g e , i n g e n e r a l s u p e r s y m m e t r i ce x t e n s i o n s a n d h i g h e r d im e n s i o n a l K a l u z a - K l e i n ( s u p e r ) s p a c e s a s w e ll as to p r o p o s eg e n e r a l p r in c i p l e s a n d m e t h o d s o f c o n s t r u c t i n g m o d e l s o f c la s s ic a l a n d q u a n t u m f ie l di n t e r a c t i o n s a n d s t o c h a s t i c p r o c e s s e s o n s p a c e s w i t h g e n e r i c a n i s o t r o p y .

W e c i t e h e r e t h e w o r k s [ 1 5 , 1 6 ] b y B e j a n c u w h e r e a n e w v i e w p o i n t o n d i f f e r e n -t i a l g e o m e t r y o f s u p e r m a n i f o l d s i s c o n s i d e r e d . T h e a u t h o r i n t r o d u c e d t h e n o n - l i n e a rc o n n e c t io n ( N - c o n n e c t i o n ) s t ru c t u re a n d d e v e l o p e d a c o r re s p o n d i n g d i s ti n g u i s h e d b yN - c o n n e c t i o n s u p e r t e n s o r c o v a r i a n t d i f f e r e n t i a l c a l c u l u s i n t h e f r a m e w o r k o f t h e D eW i t t [ 1 2 6 ] a p p r o a c h t o s u p e r m a n i f o ld s i n t h e f ra m e w o r k o f t h e g e o m e t r y o f s u p e r-b u n d l e s w i t h t y p i c a l fi b re s p a r a m e t r i z e d b y n o n - c o m m u t a t i v e c o o r d i n a t e s . T h i s w a s t h ef i r s t e x a m p l e o f s u p e r s p a c e w i t h l o c a l a n i s o t r o p y . I n o u r t u r n w e h a v e g i v e n a g e n e r a ld e f i n i t io n o f l o c a l l y a n i s o t r o p i c s u p e r s p a c e s ( l a - s u p e r s p a c e s ) [ 1 0 6 ] a n d h i g h e r o r d e ra n i s o t r o p i c su p e r s p a c e s ( h a - s u p e r s p a c e s ) [ 1 0 8 ] . I t s h o u l d b e n o t e d h e r e t h a t i n o u rs u p e r s y m m e t r i c g e n e r a l i z a t i o n s w e w e r e i n s p i r e d b y t h e M i r o n , A n a s t a s i e i a n d A t a n a s i uw o r k s o n t h e g e o m e t r y o f n o n - l i n e a r c o n n e c t io n s i n v e c t o r b u n d l e s a n d h i g h e r o r d e r L a -g r a n g e s p a ce s [ 7 5 - 7 7 ] . I n t h is w o r k w e s h all f o r m u l a t e th e t h e o r y o f h i g h e r o rd e r v e c t o ra n d t a n g e n t s u p e r b u n d l e s p r o v i d e d w i t h n o n - l i n e a r a n d d i s t i n g u i s h e d c o n n e c t i o n s a n dm e t r i c s t r u c tu r e s ( a g e n e r a l i z e d m o d e l o f l a - s u p e r s p a c e s ) . S u c h s u p e r b u n d l e s c o n t a i n a sp a r t i c u l a r c a s e s t h e s u p e r s y m m e t r i c e x t e n s i o n s a n d v a r i o u s h i g h e r o r d e r p r o l o n g a t i o n so f R i e m a n n , F i n s l e r a n d L a g r a n g e s p a c e s .

I E-m ail: [email protected]


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592 S .L Vaca ru /Nu clear Phys ics B 494 [PM ] (1997) 590- 656S u p e r s t r i n g t h e o r y h o l d s t h e g r e a t e s t p r o m i s e a s t h e u n i f i c a t i o n t h e o r y o f a l l f u n d a -

m e n t a l i n t e r a c t i o n s . T h e s u p e r s t r i n g m o d e l s c o n t a i n s a l o t a c h a r a c t e r i s t i c f e a t u r e s o fK a l u z a - K l e i n a p p r o a c h e s , 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 , l o c a l f i e l d t h e o r y a n d d u a lm o d e l s . W e n o t e t h a t i n t h e s t r i n g t h e o r i e s t h e n o n - l o c a l o n e - d i m e n s i o n a l q u a n t u m o b -j e c t s ( s t r i n g s ) m u t u a l l y i n t e ra c t in g b y l i n k i n g a n d s e p a r a t in g t o g e t h e r a r e c o n s i d e r e d asf u n d a m e n t a l v a l u e s . P e r t u r b a t i o n s o f t h e q u a n t i z e d s t r i n g a r e i d e n t i f i e d w i t h q u a n t u mp a r t i c l e s . S y m m e t r y a n d c o n s e r v a t i o n l a w s i n t h e s t r i n g a n d s u p e r s t r i n g t h e o r y c a n b ec o n s i d e r e d a s s w e e p i n g g e n e r a l i z a t i o n s o f g a u g e p r i n c i p l e s w h i c h c o n s i s t s t h e b a s i s o fq u a n t u m f i e l d m o d e l s . T h e n e w p h y s i c a l c o n c e p t s a r e f o r m u l a t e d i n t h e f r a m e w o r k o fa " n e w " f o r p h y s ic i st s m a t h e m a t ic a l f o r m a l i s m o f t h e a lg e b r a ic g e o m e t r y a n d t o p o l o g y[48] .

T h e r e l a t i o n s h i p b e t w e e n t w o - d i m e n s i o n a l o - - m o d e l s a n d s t r i n g s h a s b e e n c o n s i d e r e d[ 7 1 , 3 9 , 2 7 , 9 6 , 4 ] i n o r d e r t o d i s c u s s t h e e f f e c t iv e l o w e n e r g y f i e ld e q u a t io n s f o r t h e m a s s -l e s s m o d e l s o f s t r i n g s . N o n - l i n e a r t r - m o d e l s m a k e s u p a c l a s s o f q u a n t u m f i e l d s y s t e m sf o r w h i c h t h e f i e l d s a r e a l s o t r e a t e d a s c o o r d i n a t e s o f s o m e m a n i f o l d s . I n t e r a c t i o n s a r ei n t r o d u c e d i n a g e o m e t r i c m a n n e r a n d a d m i t a l o t o f a p p l i c a t i o n s a n d g e n e r a l i z a t i o n si n c l as s i ca l a n d q u a n t u m f i el d a n d s t r in g t h e o r ie s . T h e g e o m e t r i c s t r u c tu r e o f n o n - l i n e a rs i g m a m o d e l s m a n i f e s t s t h e e x i s t e n c e o f t o p o l o g i c a l n o n - t r i v i a l c o n f i g u r a t i o n , a d m i t sa g e o m e t r i c i n t e r p r e t a t i o n o f c o u n t e r t e r m s a n d p o i n t s t o a s u b s t a n t i a l i n t e r r e l a t i o n b e -t w e e n e x t e n d e d s u p e r s y m m e t r y a n d d i f f e r e n t i a l s u p e r g e o m e t r y . I n c o n n e c t i o n t o t h i sa n e w a p p r o a c h b a s e d o n n o n - l o c a l , i n g e n e r a l , h i g h e r o r d e r a n i s o t r o p i c c o n s t r u c t i o n ss e e m t o b e e m e r g i n g [ 1 0 1 , 1 0 7 , 1 0 8 ] . W e c o n s i d e r t h e r e a d e r t o b e f a m i l ia r w i t h b a -s ic r e s u l ts f r o m s u p e r g e o m e t r y ( s e e , f o r in s t a n c e , R e fs . [ 3 3 , 6 7 , 1 2 6 , 8 8 ] ) , s u p e r g r a v i t yt h e o r i e s [ 4 1 , 9 2 , 1 2 5 , 1 2 3 , 1 2 4 ] a n d s u p e r s t r i n g s [ 5 4 , 9 5 , 6 3 , 6 4 ] .

I n t h i s w o r k w e s h a ll p r e s e n t a n i n t r o d u c t i o n in t o t h e t h e o r y o f h i g h e r o r d e r a n i s o t ro p i cs u p e r s t r i n g s b e i n g a n a t u r a l g e n e r a l i z a t i o n t o l o c a l l y a n i s o t r o p i c ( l a ) b a c k g r o u n d s ( w es h a ll w r i t e i n b r i e f l a - b a c k g r o u n d s , l a - s p ac e s a n d l a - g e o m e t r y ) o f t h e P o l y a k o v ' s c o -v a r i a n t f u n c t i o n a l - i n t e g r a l a p p r o a c h t o s t r i n g t h e o r y [ 8 5 ] . O u r a i m i s t o s h o w t h a ta c o r r e s p o n d i n g l o w - e n e r g y s t r i n g d y n a m i c s c o n t a i n s t h e m o t i o n e q u a t i o n s f o r f i e l de q u a t i o n s o n h i g h e r o r d e r a n i s o t r o p i c s u p e r s p a c e s a n d t o a n a l y z e t h e g e o m e t r y o f t h ep e r t u r b a t i o n t h e o r y o f t h e l o c a l l y a n i s o t r o p i c s u p e r s y m m e t r i c s i g m a m o d e l s . W e n o t et h a t t h i s w o r k i s d e v o t e d t o s u p e r s y m m e t r i c m o d e l s o f l o c a l l y a n i s o t r o p i c s u p e r s t r i n g s .

T h e w o r k i s o r g a n i z e d a s f o l l o w s : S e c t io n 2 c o n t a i n s a b r i e f r e v i e w o n s u p e r m a n i f o l d sa n d s u p e r b u n d l e s a n d a n i n t r o d u c t i o n i n t o t h e g e o m e t r y o f h i g h e r o r d e r d i s t i n g u i s h e dv e c t o r s u p e r b u n d l e s . S e c t i o n 3 d e a l s w i t h t h e g e o m e t r y o f n o n - l i n e a r a n d l i n e a r d i s -t i n g u i s h e d c o n n e c t i o n s i n v e c t o r s u p e r b u n d l e s a n d d i s t i n g u i s h e d v e c t o r s u p e r b u n d l e s .T h e g e o m e t r y o f t h e t o t a l sp a c e o f d i s t in g u i s h e d v e c t o r s u p e r b u n d l e s is s t u d ie d i n S e c -t i o n 4 ; d i s t i n g u i s h e d c o n n e c t i o n a n d m e t r i c s t r u c t u r e s , t h e i r t o r s i o n s , c u r v a t u r e s a n ds t r u c t u r e e q u a t i o n s a r e c o n s i d e r e d . G e n e r a l i z e d L a g r a n g e a n d F i n s l e r s u p e r s p a c e s a n dt h e i r h i g h e r o r d e r p r o l o n g a t i o n s a r e d e f i n e d i n S e c t i o n 5 . S e c t i o n 6 c o n t a i n s a n i n t r o -d u c t i o n i n t o t h e g e o m e t r y o f t w o - d i m e n s i o n a l h i g h e r o r d e r a n i s o t r o p i c s i g m a m o d e l sa n d a l o c a l l y a n i s o t r o p i c a p p r o a c h t o h e t e r o t i c s tr in g s . I n S e c t i o n 7 t h e b a c k g r o u n d f i e ldm e t h o d f o r o - - m o d e l s i s g e n e r a l i z e d f o r a d i s t i n g u i s h e d c a l c u l u s l o c a l l y a d a p t e d t o t h e


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S.l. Vacaru/Nuclear Physics B 494 [PM ] (1997) 590-656 59 3N - c o n n e c t i o n s t r u c t u r e i n h i g h e r o r d e r a n i s o t r o p i c su p e r s p a c e s . S e c t i o n 8 i s d e v o t e d t oa s t u d y o f G r e e n - S c h w a r z a c t i o n in d i s t in g u i s h e d v e c t o r s u p e r b u n d l e s . F e r m i s tr i n g s i nh i g h e r o r d e r a n i s o tr o p i c s p a c es a r e c o n s i d e r e d i n S e c t io n 9 . A n e x a m p l e o f o n e - l o o p a n dt w o - l o o p c a lc u l u s f o r a n o m a l i e s o f lo c a l l y a n i s o t r o p i c s tr i n g s i s p r e s e n t e d i n S e c t i o n 1 0.A d i s c u s s i o n a n d c o n c l u s i o n s a r e d r a w n i n S e c t i o n I 1 .

2. Dist inguished superbun dlesI n t h is s e c ti o n w e e s t a b li s h th e n e c e s s a r y t e r m i n o l o g y o n s u p e r m a n i f o l d s ( s - m a n i f o l d s )

[ 1 2 6 , 8 8 , 8 9 , 5 8 , 1 2 0 , 5 3 , 7 2 , 1 3 , 2 6 ,3 3 ] a n d p r e s e n t a n i n t r o d u c t i o n i n t o t h e g e o m e t r y o f d i s -t in g u i s h e d v e c t o r s u p e rb u n d l e s ( d v s - b u n d l e s ) [ 1 1 0 ] . H e r e w e n o t e th a t a n u m b e r o fd i f f e r e n t a p p r o a c h e s t o s u p e r m a n i f o l d s a r e b r o a d l y e q u i v a l e n t f o r l o c a l c o n s i d e r a t i o n s .F o r s i m p l i c i t y , w e s h a l l r e s t r i c t o u r s t u d y o n l y t o g e o m e t r i c c o n s t r u c t i o n s o n l o c a l l yt r i v i a l s u p e r s p a c e s .2 .1 . S u p e r m a n i f o l d s a n d s u p e r b u n d l e s

T o b u i l d u p s - m a n i f o l d s [ 8 8 , 5 8 , 1 2 0 ] o n e u s e s a s b a s ic s tr u c tu r e s G r a s s m a n n a l g e b r aa n d B a n a c h s p a c e . A G r a s s m a n n a l g e b r a i s i n t r o d u c e d a s a r e a l a s s o c i a t i v e a l g e b r aA ( w i t h u n i t y ) p o s s e s s i n g a fi n it e ( c a n o n i c a l ) s e t o f a n t i c o m m u t a t i v e g e n e r a t o r s /3A,[ f l A , f l h ] + = f l A f l ~ + f l d f l ; t = 0 , w h e r e A , / ~ . . . . . 1 , 2 . . . . . L . In th i s c a s e it i s a l s od e f i n e d a Z 2 - g r a d e d c o m m u t a t i v e a l g e b r a A 0 -4- A 1 , w h o s e e v e n p a r t A 0 ( o d d p a r t A 1 )is a 2 L - l - d i m e n s i o n a l r ea l v e c t o r s p a c e o f ev e n ( o d d ) p r o d u c t s o f g e n e r a to r s f lA . A f te rse t t i ng A 0 = 7E -4- A 0 , w h e re 7~ i s t he r ea l n um be r f i e l d an d A 0 i s t he s ub spa ce o f Ac o n s i s t i n g o f n i l p o t e n t e l e m e n t s , t h e p r o j e c t i o n s ~ : A ---+ 7 ~ a n d s : A ~ A o ' a r e c a l l e d ,r e s p e c t i v e l y , t h e b o d y a n d s o u l m a p s .

A G r a s s m a n n a l g e b r a c a n b e p r o v i d e d w i t h b o t h st ru c tu r e s o f a B a n a c h a l g e b r a a n dE u c l i d e a n t o p o l o g i c a l s p a c e b y t h e n o r m [ 8 8 ]

LX - " , . , A , . . . L ~ .I I~II = ~ I a ;~ ' "' A ~ I .Z g = Z ._ .,~ , - % . . 3 ~ .Ai r--O

A s u p e r s p a c e i s i n t r o d u c e d a s a p r o d u c tA n'k = A 0 A o A , ... AI

Y

w h i c h i s t h e A - e n v e l o p e o f a Z 2 - g r a d e d v e c t o r s p a c e V " 'k = VoVt = ~ n G 7 ~k iso b t a i n e d b y m u l t ip l ic a t io n o f e ve n ( o d d ) v e c to r s o f V o n e v e n ( o d d ) e l em e n t s o f A .T h e s u p e r s p a c e ( a s t h e A - e n v e l o p e ) p o s s e ss ( n + k ) b a s i s v e c t o r s { /3 i, i = 0 , 1 . . . . n - 1 } ,a n d { f li ,~ = 1 , 2 . . . . k } . C o o r d i n a t e s o f e v e n ( o d d ) e l e m e n t s o f V n'k are e v e n ( o d d )e l e m e n t s o f A . W e c a n c o n s i d e r e q u i v a l e n t l y a s u p e r s p a c e V "'k a s a ( 2 L - 1 ) ( n + k ) -d i m ens i ona l r ea l vec t o r spaces w i t h a ba s i s { / ~ i (A ) , / ~ / (A )} "


5/67

594 S .I . V a c ar u / N uc l e a r P hy s i c s B 494 [ P M ] ( 1997 ) 590 - 656F u n c t i o n s o f s u p e r s p a c e s , d i f f e r e n t i a t i o n w i t h r e s p e c t t o G r a s s m a n n c o o r d i n a t e s , s u -

p e r s m o o t h ( s u p e r a n a l y t ic ) f u n c ti o n s a n d m a p p i n g s a r e i n t ro d u c e d b y a n a l o g y w i t h t h eo r d i n a r y c a s e , b u t w i t h a g l a n c e t o ce r ta i n s p e c i fi c it y c a u s e d b y c h a n g i n g o f r ea l ( o rc o m p l e x ) n u m b e r f ie ld i n to G r a s s m a n n a l g e b r a A. H e r e w e r e m a r k t h a t f u n c ti o n s on as u p e r s p a c e A ~ ' t w h i c h t a k e s v a l u e s i n G r a s s m a n n a l g e b r a c a n b e c o n s i d e r e d a s m a p p i n g so f t h e s p a c e T {2~L-'~ {n+k) i n t o t h e s p a c e 7"~2L . F u n c t i o n s d i f f e r e n t ia b l e o n G r a s s m a n nc o o r d i n a t e s c a n b e r e w r i t te n v i a d e r iv a t iv e s o n r e a l c o o r d in a t e s , w h i c h o b e y a g e n e r a l iz e df o r m o f C a u c h y - R i e m a n n c o n d it io n s .

A ( n , k ) - d i m e n s i o n a l s - m a n i f o l d M c a n b e d e f in e d a s a B a n a c h m a n i f o l d ( s e e , f o re x a m p l e , R e f . [ 6 8 ] ) m o d e l l e d o n A n'k e n d o w e d w i t h a n a t l a s ~b = {U { i ) , { i ) : U{ i )A n'k, (i) E J } w h o s e t r a n s i t i o n f u n c t i o n s ~ ( i ) a r e s u p e r s m o o t h [ 8 8 , 5 8 ] . I n s t e a d o fs u p e r s m o o t h f u n c t i o n s w e c a n u s e G o o - f u n c t i o n s [ 8 8 ] a n d i n t ro d u c e G - s u p e r m a n i f o l d s( G o o d e n o t e s t h e c l a s s o f s u p e r d i f f e r e n t ia b l e f u n c t i o n s ) . T h e l o c al s t r u c tu r e o f a G o o -s u p e r m a n i f o l d i s b u i l t v e r y m u c h a s o n a C ~ - m a n i f o l d . J u s t a s a v e c t o r f i e l d o n an - d i m e n s i o n a l C o o - m a n i f o l d w r i t t e n l o c a l l y a s

n - 1Eii(x ) e--tgX 'i=0w h e r e f i are C - f u n c t i o n s , a v e c t o r f ie l d o n a n ( n , k ) - d i m e n s i o n a l G o o - s u p e r m a n i f o l dA T / c a n b e e x p r e s s e d l o c a l l y o n a n o p e n r e g i o n U C h T / a s

n - l + k n - I k# a o3) a ~ ', ( x ' ) =1=0 i=O i=1

w h e r e x = ( ~ , 0 ) = { x I = ( . ~ i , 0 ~ ) } a r e l o c a l ( e v e n , o d d ) c o o r d i n a t e s . W e s h a l l u s ei n d i c e s I = ( i , ~ ) , J = ( j , ) ) , K = ( k , ~:) . . . . f o r g e o m e t r i c o b j e c t s o n /17/. A v e c t o r f ie l do n U i s a n e l e m e n t X C E n d [ G o o ( U ) ] ( w e c a n a l s o c o n s i d e r s u p e r s m o o t h f u n c t i o n si n s t e a d o f G o o - f u n c t i o n s ) s u c h t h a t

X ( f g ) = ( X f ) g + ( - )l fl lX l f X g ,f o r a ll f , g i n G o o ( U ) , a n d

X ( a f ) = ( - )I X l la l a x f ,w h e r e IX [ a n d l a l d e n o t e c o r r e s p o n d i n g l y th e p a r i t y ( = 0 , 1) o f v a l u e s X a n d a a n d f o rs i m p l i c i t y in t h i s w o r k w e s h a ll w r i t e ( - ) I f l l x l i n s te a d o f ( - 1 )IfllX l.

A s u p e r L i e g r o u p ( s l - g r o u p ) [ 8 9 ] i s b o t h a n a b s t ra c t g r o u p a n d a s - m a n i f o ld ,p r o v i d e d t h a t t h e g r o u p c o m p o s i t i o n l a w f u lf il ls a s u i t ab l e s m o o t h n e s s c o n d i t io n ( i. e . t ob e s u p e r a n a l y t i c , f o r s h o r t , sa [ 5 8 ] ) .

I n o u r f u r t h e r c o n s i d e r a t i o n s w e s h a ll u s e th e g r o u p o f a u t o m o r p h i s m s o f A ( n ' k ) ,d e n o t e d a s G L ( n , k , A ) , w h i c h c a n b e p a r a m e t r i z e d a s t h e s u p e r L i e g r o u p o f i n v e r t i b l em a t r i c e s


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S.L Vacaru/Nuclear Physics B 494 [PM ] (1997) 590-656 595

D 'w h e r e A a n d D a r e r e s p e c t i v e l y ( n n ) a n d ( k k ) m a t r ic e s c o n s i s ti n g o f e v e n G r as s -m a n n e l e m e n t s a n d B a n d C a r e r e c ta n g u l a r m a t r ic e s c o n s is t in g o f o d d G r a s sm a n ne l ement s . A mat r ix Q i s i nve r t i b l e a s soon as maps o -A and o -D a re i nve r t i b l e ma-t r i c e s . A n s l - g r o u p r e p r e s e n t s a n o r d i n a r y L i e g r o u p i n c l u d e d i n t h e g r o u p o f l i n e a rt r a n s f o r m s G L ( 2 L -1 ( n + k ) , R ) . F o r m a t r ic e s o f ty p e Q o n e d e f in e s [ 1 9 , 6 6 ,6 7 ] t h es u p e r d e t e r m i n a n t , s d e t Q , s u p e r t ra c e , s t r Q , a n d s u p e r r a n k , s r a n k Q .

A L i e s u p e r a l g e b r a ( s l - a l g e b r a ) i s a Z 2 - g r a d e d a l g e b r a A = A 0 @ A j e n d o w e d w i t hp r o d u c t [ , } s a t i s f y i n g t h e f o l l o w i n g p r o p e r ti e s :

[ l , I ' } = - ( _ ) I tl ll 'l [ 1 , , I } ,[ 1 , [ i ' , i " } } = [ [ 1 , t ' } , i " } + ( - ) I ~ l l ~ ' l [ i ' [ 1 , i " ) ) ,

I E A [ / I , l I c A i t , i , wh ere I l l , I '1 = 0 , 1 enum era t es , r e spec t ive ly , t he p oss ib l e p a r i t y o fe l e m e n t s I , F . T h e e v e n p a r t A 0 o f a n s l -a l g e b r a i s a u s u a l L i e a l g e b r a a n d t h e o d dp a r t A I i s a re p r e s e n t a t i o n o f t h i s L i e a l g e b r a . T h i s e n a b l e s u s t o c l a s s i fy s l - a l g e b r a sf o l l o w i n g t h e L i e a l g e b r a c l a s s i f i c a t i o n [ 5 9 ] . W e a l s o p o i n t o u t t h a t i r r e d u c i b l e l i n e a rr e p r e s e n t a t i o n s o f L i e s u p e r a l g e b r a A a r e r e a li z e d in Z 2 - g r a d e d v e c t o r s p a c e s b y m a t r i c e s( A D 0 ) f r e v en e le m e n ts a n d ( 0 B ) r d d e l e m e n t s a n d t h a t ' r u g h l y s p e a k i n g ' C 0

A i s a s u p e r a l g e b r a o f g e n e r a t o r s o f a n s l - g r o u p .A n s l - m o d u l e W ( g r a d e d L i e m o d u l e ) [ 8 8 ] i s i n t ro d u c e d a s a Z z - g r a d e d le f t A -

m o d u l e e n d o w e d w i t h a p r o d u c t [ , } w h i c h s a ti sf ie s t h e g r a d e d J a c o b i i d e n t i ty a n dm a k e s W i n t o a g r a d e d - a n t ic o m m u t a t iv e B a n a c h a l g e b r a o v e r A . O n e c a l ls t h e L i em o d u l e G t h e s e t o f t h e l e f t- i n v a r i a n t d e r i v a ti v e s o f a n s l - g r o u p G .

Th e t an g en t supe rbun d le ( t s -bu n d le ) T / f / ove r an s -m an i fo ld .~7/, 7 r : T / f / ~ A7/ isc o n s t r u c t e d i n a u s u a l m a n n e r ( s e e , f o r in s t a n c e , [ 6 8 ] ) b y t a k i n g a s t h e t y p i c a l f ib r e th es u p e r s p a c e A n '~ a n d a s t h e s t r u c t u r e g r o u p t h e g r o u p o f a u t o m o r p h i s m s , i. e. t h e s l - g r o u pG L ( n , k , A ) .

L e t u s d e n o t e b y .T" a v e c t o r s u p e r s p a c e ( v s - s p a c e ) o f d i m e n s i o n ( m , l) ( w i t h r e s p e c tt o a c h o s e n b a s e w e p a r a m e t r i z e a n e l e m e n t y E C a s y = ( 9 , ( ) = { ya = ( fa , (~ ) } ,w h e r e a = 1 , 2 . . . . . m a n d h = 1 , 2 . . . . . l) . W e s ha ll u se i n d ic e s A = ( a , & ) , B =( b , b ) . . . . f o r o b j e c t s o n v s - sp a c e s . A v e c t o r s u p e r b u n d l e ( v s - b u n d l e ) ~ o v e r b as e /Qw i t h t o t a l s u p e r s p a c e / ~ , s t a n d a r d f i b re f " a n d s u r je c t iv e p r o j e c t i o n 7 rE :/ ~ - -* M i s d e f in e d( s e e d e t a i l s a n d v a r i a n t s in [ 2 6 , 1 2 2 ] ) a s in t h e c a s e o f o r d i n a r y m a n i f o l d s ( s e e , f o ri ns t ance , Ref s . [ 68 ,7 5 ,76 ] ) . A sec t i on o f ,~ i s a supe r sm ooth m ap s : U - -- ,/~ such t ha t7re . s = idu .

A s u b b u n d l e o f ~ i s a t ri p le ( /3 , f , f / ) , w h e r e / 3 i s a v s - b u n d l e o n / f / , m a p s f : / 3 - - -~an d f : AT/---~Ma r e s u p e r s m o o t h , a n d ( i ) 7 r e o f = f%TrB; ( i i ) f : 7 r~ l ( x ) _ o T r - ~ l o f ( x )i s a v s - s p a c e h o m o m o r p h i s m . W e d e n o t e b y

u ( x , y ) (Y c, O , ~ 9 , ( ) { u a ( x l , y a ) (~ ci, o ~, ^a h x ' i ^a. . . . y , ( ) = ( $ i , , y , y ~ ) }


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5 9 6 S.I. Vacaru/Nuclear Physics B 494 [PM] (1997) 590-6 56t h e l o c a l c o o r d i n a t e s i n 8 a n d w r i t e t h e i r t r a n s f o r m a t i o n s a s

x t' = x ~ ' ( x l ) , s r a n k \ - ~ - x l ] = ( n , k ) , ( 1 )

y A' = y 2 ' ( x ) y a , w h e r e Y 2 ' ( x ) C G ( m , l, A ) .F o r l o c a l c o o r d i n a t e s a n d g e o m e t r i c o b j e c t s o n t h e t s - b u n d l e T i f / w e s h a l l n o t d i s -

t i n g u i s h b e t w e e n i n d i c e s o f c o o r d i n a t e s o n t h e b a s e a n d i n th e f i b r e a n d w r i t e , f o ri n s t a n c e ,

^ i i ^ i iu = ( x , y ) = (Yc, O , ~ , ( ) = { u a = ( x l , y 1 ) = ( X , 0 , y , ( ) = ( 2 i, x i , ~ i , y i ) } .W e s h a l l u s e G r e e k i n d i c e s f o r m a r k i n g l o ca l c o o r d i n a t e s o n b o t h s - v e c t o r a n d u s u a lv e c t o r b u n d l e s .2 .2 . D is t ingu i s hed v ec to r supe rbund le s

S o m e r e c e n t c o n s i d e r a t i o n s i n m a t h e m a t i c a l p h y s i c s a r e b a s e d o n t h e s o - c a l l e d k - j e ts p a c e s ( s e e , f o r i n s t a n c e , R e f s . [ 9 4 , 9 3 , 9 ] ) . I n o r d e r t o f o r m u l a t e a s y s t e m a t i c th e o r yo f c o n n e c t i o n s a n d o f g e o m e t r i c s t r u c t u re s o n k - je t b u n d l e s, i n a m a n n e r f o l l o w i n g t h ea p p r o a c h e s o f R e f s . [ 1 2 8 , 7 5 , 7 6 ] , M i r o n a n d A t a n a s iu [ 7 7 ] i n t r o d u c e d t he c o n c e p t o f ak - o s c u l a t o r b u n d l e f o r w h i c h a f i b e r o f k - j e t s i s c h a n g e d i n t o a k - o s c u l a t o r f i b e r r e p r e -s e n t i n g a n e l e m e n t o f a k - o r d e r c u r v e . S u c h c o n s i d e r a t io n s a r e c o n n e c t e d w i t h g e o m e t r i cc o n s t r u c t i o n s o n t a n g e n t b u n d l e s o f h ig h e r o r de r . O n t h e o th e r h an d , f o r d e v e l o p m e n t si n m o d e r n s u p e r s y m m e t r i c K a l u z a - K l e i n t h e o ri e s ( s e e , f o r i n s ta n c e , R e f. [ 9 2 ] ) a su b -s t a n t ia l i n t e r e s t w o u l d p r e s e n t a v a r i a n t o f " o s c u l a t o r " s p a c e f o r w h i c h t h e h i g h e r o r d e rt a n g e n t s - s p a c e d i s t r ib u t i o n s a r e o f d i f f e re n t d i m e n s i o n s . T h e s e c o n d p a r t o f t hi s s e c t io ni s d e v o t e d t o t he d e f i n it io n o f s u c h t y p e o f d i s ti n g u i s h e d v e c t o r s u p e r b u n d l e s p a c e s.

A v e c t o r s u p e r s p a c e b ( z ) o f d i m e n s i o n ( m , l ) i s a d i s t in g u i s h e d v e c t o r s u p e r -s p a c e ( d v s - s p a c e ) i f i t i s d e c o m p o s e d i n t o an i n v a r i a n t o ri e n t e d d i re c t s u m b ( z ) =~"(1) @ ~ ' ( 2 ) @ ' ' ' @ ~ ( Z ) of v s - spa ces . Y ' (p ) ,d i m . F '( / , ) = ( m ( p ) , l ( p ) ) , w h e r e ( p ) =( 1 ) , ( 2 ) . . . . . ( z ) , ~-]~Pp~ m (p ) = m , ~-~Pp=_~ (p) = l.

C o o r d i n a t e s o n . T" ( p ) w i l l b e p a r a m e t r i z e d a sy(P) = ( y ( l ) , y ( 2) . . . . . y(p) ) = (~(1) , ( (1) , 33(2) , ( (2) . . . . . ~9(p), ( ( p) )

= { y ( A ) = ( ~ ( a ) , ~ - ( a ) ) = ( ~ ( a ) , y ( a ) ) } ,

w h e r e b r a c k e t e d i n d i c e s a r e c o r r e s p o n d i n g l y s p l it o n U f p ) - c o m p o n e n t s :( A ) = ( A ( I ) , A ( 2 ) . . . . . A ( p ) ) , ( a ) = ( a ( 1 ) , a ( 2 ) . . . . . a ( p ) ) ,( ~ ) = ( ~ ( 1 ) ~ ( 2 ) . . . . . ~ ( p ) ) , ( 2 )

F o r s i m p l i c i t y , w e s h a l l a l s o w r i t e ( 2 ) a s ( A ) = ( A 1 , A 2 . . . . . A /, ) , ( a ) = ( a . , a 2 . . . . . a p )a n d ( fi) = ( a l a 2 . . . . . ~ , ) i f t h is w i ll g i v e n o t r is e to a m b i g u i t ie s .


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S.I. Vacaru/Nuclear Physics B 494 [PM] (1997) 590-656 59 7A d i s t i n g u i s h e d v e c t o r s u p e r b u n d l e ( d v s - b u n d l e ) g l z ) = (/ ~ (z ) , r (d ) ,. y -( d ), ~ ) , w i t h

s u r j e c t i v e p r o j e c t i o n r (z) : /~ ( z ) ~ ~ , w h e r e M a n d / ~ ( z ) a r e r e s p e c t i v e l y b a s e a n d to t a ls - s p a c e s a n d t h e d v s - s p a c e f - ( z ) i s t h e s t a n d a r d f ib r e.

A dv s -b un d l e g ( z ) i s con s t ruc t ed a s an o r i en t ed s e t o f v s -bu nd l e s or (p ) : ~ ( i, ) __~ ~ ' (p - J )( w i t h t y p i c a l f i b e r . U ( P ) , p = 1 , 2 . . . . . z ) ; ~ ( 0 ) = / f/ . W e s h a l l u s e i n d e x z ( p ) a s t od e n o t e th e to t al ( i n t e r m e d i a t e ) n u m b e r s o f c o n s e q u e n t v s - b u n d l e c o v e r i n g s o f M .

L o c a l c o o r d i n a t e s o n g ( P ) a r e d e n o t e d a sU ( p ) = ( X , y ( p ) ) = ( X , y ( 1 ) , Y ( 2) . . . . . Y ( p ) )

= ( X , O , y ( p ) , ( ( p ) ) = ( - ~ , 0 , y ( I ) , ~ ' ( I ) , Y ( 2 > , ~ ' ( 2 ) . . . . . y ( p ) , ( ( p ) ): { U ( a ) = ( X I , y ( a ) ) = ( ~ i , 0 ~ , ~ : ( a ) , ( ( a ) ) = ( 2 i , x ~ i, ~ ( a ) , y(a) )

( X I A o A l A p= = y , y . . . . . y . . . . . y a : ) }( i n o u r f u r t h e r c o n s i d e r a t i o n s w e s h a l l c o n s i d e r d i f f e r e n t v a r i a n t s o f s p l i t t i n g o f i n d i c e so f g e o m e t r i c o b j e c t s )

I n s t e a d o f ( 1 ) t h e c o o r d i n a t e t r a n s f o r m s f o r d v s - b u n d l e s { u (" ) = ( x l , y ( A ) ) }{U(a ' ) = ( X 1 ', y ( a ' ) ) } a r e g i v e n b y r e c u r r e n t m a p s :

: 0 x ,' t ' 1 = ( n , k ) , ( 3 )x ' = x ( x ) , s r a n k ~ , 0 x I /a ' _ K A ' I tx ~ . A 1 K A ' l ( x ) C G ( m ( l ) , l ( l ) , A )Y ( l ) - A1 " : Y ( 1 ) ' A I

A/ ' , = Kaa! (u (p -1 A p A ;y(p ) ) ) y (p ) , K A, ( U (p_l ) ) G ( m( p ) , l (p ) , A ) ,

a'. 'y ( ;) = K 2 : ( U ( z _ l ) ) y : z ) , KaA: ( U ( z _ , ) ) E G ( m ( z ) , l (: ) , A ) .In b r i e f w e w r i t e t r a n s f o r m s ( 3 ) as

y (A ' ) = i ( ( A ' ) a (A}X f : _ x f ( x l ) , . ~ { A > Y M or e g ene ra l l y , w e sha l l con s i de r m a t r i ce s K ( '~ ') 1 ' r,,) = (K I ''~(A)/C(AZ)~:'w h e r e K t - O x l ' / O x / .

A s a c o n s e q u e n c e , t h e lo c a l c o o r d i n a t e b a s e s o f t h e m o d u l e o f d s - v e c t o r fi el d s= ( g < z > ) ,O ~(a) = ( ~ I , O g ( A ) ) = ( 0 1 , 0 ( A I ) , 0 ( A 2 ) . . . . . O ( A :)

0 o9 a a 0- O u ( '~ ) - ( ~ x t ' O y ~ , ') O y ( A ~ ) O y ; )

( t h e d u a l c o o r d i n a t e b a s e s a r e d e n o t e d a sd (~) = ( d l , d ( a ) ) = ( d l , d ( a , ) , d ( & ) . . . . . d (A '-) )

( 4 )


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598 S.I. Vacaru/Nuclear Physics B 494 [ P M ] ( 1 9 97 ) 5 9 0 - 6 5 6= du (a) = ( dx I, dy (a l ), dy(a2) . . . . . dy (A=)) ) (5 )

a re t r a n s fo rm e d a s

O ( a ) - ~ ( C g l , C ~ ( A } ) - ~ ( C ~ I C g (A , ) , O ( A 2 ) . . . . . O ( a z ) ) " "+ O ( a )= ( O 3 / , t ~ ( A } ) = ( O 3 1 ,O ~ ( A ,) ,O ~ ( A 2 ) . . . . . O ~ ( a : ) )

O3 1' Oq ' O ' O . A '. Oe X ] = K ] O x ]" ---7 7 1- Y i4 'lo )l~l{) Y (a2 ')/y (A - ~ 2 2 )~ ' ' ' " J I - Y ( z : O ) I a y ( - - A '0 _ KA' I O........~ y A~ ~ + . . . _ . ~ ~ , A h ) A , oaye,') A, a y ~ , l ) (2,I)AI ayeS) ay(za ' 'a KA~ 9 , ~ a ~.y, A3 _m2 A 2 7 "~- ( 3, 2 )A 2_ a ~ + ' ' " " ~- Y ( z ~ 2 ) a 2 ~ - - A t , 'aY (2) aY(2) 0Y(3) Y ( z )

O ' 0 1 A I C ~__ K A z - 1 7 ~ - Y ( z : z - 1 ) A , _ l ~ A tA " ~ ' -I a . - I A ' ~ '- I3y( z_ l ) " Oy (z_l ) ay(~)a =/qA: ,9A'. " (6 )Oyez ) " By(z )

Y -m a t r ic e s f ro m (6 ) a r e p a r ti a l d e ri v a t io n s o f c o r r e s p o n d i n g c o m b i n a t i o n s o f K -c o e f f ic i e n t s f r o m c o o rd i n a t e t ra n s fo rm s (3 ) ,tyA,', 0 ( K AAVp yAp )

Ay -- o~yAf ' f < P"I n b r i e f w e d e n o t e r e sp e c t iv e l y d s - c o o rd in a t e t ra n s fo rm s o f c o o rd i n a t e b a s es ( 4 ) a n d

( 5 ) as, < o / +

w he re m at rix K (d)(,~) its s-in ve rse K I: ! ) , a s we l l Y( (~} and Y ~ a re pa ramet r ized accord ingt o (6 ) . I n o rd e r t o i l lu s t r a te g e o m e t r ic p ro p e r ti e s o f s o m e o f o u r t r a n s fo rm s i t is u s e fu lt o i n t ro d u c e m a t r i x o p e ra t o r s a n d t o c o n s i d e r i n e x p l i c i t f o rm t h e p a r a m e t r i z a t i o n s o fm a t r i c e s u n d e r c o n s i d e r a ti o n . F o r i n s t a n c e , in o p e ra t o r fo rm t h e t r a n s fo rm s (6 )

a =~O' ,a re c h a ra c te r i z e d b y m a t r i c e s o f t h e t y p e


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S.L Vacaru/Nuc lear Physics B 494 [PM] (1997) 590-656 59 9

0 = c9(,~) = 2A2 =OA0

Ox0

OY~l )0A 2OY}2)0

\ OY'~z)

0 1 = 0 ( , ~ , ) =C g 'O A ~ =

O A ' .

C ' ~ X t0

Oy'/l'O

OyeZ)0

A ~.\ Oy(~) /a n d

/ ' y A't A~ A~.K t ( 1 , 0 ) 1 Y ( 2 , 0 ) / ' ' ' Y ( z ~ o ) l0 K A'~ a'2 A'.AI Y(2,1 )a l Y( z~l)a l

= 0 0 K a~ At"A2 V(z~2)A2t0 0 0 . . . KAA~

W e n o t e t h a t w e o b t a i n a s u p e r s y m m e t r i c g e n e r a l i z a t io n o f t h e M i r o n - A t a n a s i u [ 7 7 ]o s c u l a t o r b u n d l e (OscZ if4, Tr, if4) i f t h e f i b e r s p a c e i s t a k e n t o b e a d i r e c t s u m o f zv e c t o r s - s p a c e s o f t h e s a m e d i m e n s i o n d i m 5 " = d i m / ~ , i .e . 5 < d ) = f " 5 - . . . 5 ' . I nt h is c a s e th e K a n d Y m a t r i c e s f r o m ( 3 ) a n d ( 6 ) s a t is f y t h e id e n t i t i e s

KArl _ KA ; A:At - - A 2 = " ' " = K A y 'A ' A t A 'Y(1,0) a = Y(2,I )A . . . . . Yi z , z - l )a '

A' A' AY ( l , , O ) A = Y i p + l , l ) a . . . . . Y ( z , z - 1 ) A , ( p = 2 . . . . . Z - - 1 ) .

F o r s = 1 t h e O s c l ~ l i s t h e t s - b u n d l e T / 1 4 .I n t r o d u c i n g p r o j e c t i o n ~-~ - ~ . (z ) : { ( z ) ~ ~ w e c a n a l s o c o n s i d e r p r o j e c t i o n sr l " " ~ ( m ) ___, ~ ( m ) ( P 2 < P l ) d e f i n e d a sP2

77.szSl x , y( l ) ' " .. , Y(P' ) ) = ( x , y ( 1 ) , . . . , y ( P 2 ) ) .T h e s - d i f f e r e n t i a l s d ~ P~ : T ( g (m)) ~ T (g (p :)) o f m a p s zrP~ i n t u r n d e f i n e v e r t i -c a l d v s - s u b b u n d l e s Vh+l = K erd crt~ h ' ( h = 0 , 1 . . . . . p l - 1 ) o f t h e t a n g e n t d v s - b u n d l eT ( g ( z )) ( t h e d v s - s p a c e Vl = V i s t h e v e r t ic a l d v s - s u b b u n d l e o n { ( z ) . T h e l o c a l f i b r e s o fd v s - s u b b u n d l e s Vh d e t e r m i n e s t h i s r e g u l a r s - d i s t r i b u t i o n Vh+t : u E ~(z) ~ Vh+l (u ) CT ( g ( z )) f o r w h i c h o n e h o l d s i n c l u s i o n s Vz C Vz-1 C . . . C V t, T h e e n u m e r a t e d p r o p e r -t i es o f v e rt i c a l d v s - s u b b u n d l e s a r e e x p l i c i t l y i l l u s t ra t e d b y t r a n s f o r m a t i o n l a w s ( 6 ) f o rd i s t i n g u i s h e d l o c a l b a s e s


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60 0 S.L Vacaru/Nuclear Physics B 494 [PM] (1997) 590-6563 . N o n - l i n e a r c o n n e c t i o n s i n d v s - b u n d l e s

T h e p u r p o s e o f t h is s e c ti o n i s to p r e s e n t a n i n t ro d u c t i o n in t o g e o m e t r y o f th e n o n -l i n e a r c o n n e c t i o n s t r u c t u r e s i n d v s - b u n d l e s . T h e c o n c e p t o f n o n - l i n e a r c o n n e c t i o n ( N -c o n n e c t i o n ) w a s i n t r o d u c e d in t h e f r a m e w o r k o f F in s l e r g e o m e t r y [ 3 0 , 2 9 ,6 0 ] ( t h eg l o b a l d e f i n i t i o n o f N - c o n n e c t i o n is g i v e n i n [ 1 2 ] ) . I t s h o u l d b e n o t e d h e r e th a t t h eN - c o n n e c t i o n ( s p l i t t i n g ) f i e l d c o u l d p l a y a n i m p o r t a n t r u l e i n m o d e l i n g v a r i o u s v a r i a n t so f d y n a m i c a l r e d u c t io n f r o m h i g h e r d i m e n s i o n a l t o l o w e r d i m e n s i o n a l s -s p a ce s w i t h( o r n o t ) d i f f e r e n t t y p e s o f l o c a l a n i s o t r o p y . I n m o n o g r a p h s [ 7 5 , 7 6 ] t h e r e a r e c o n t a i n e dd e t a i l e d i n v e s t ig a t i o n s o f g e o m e t r i c a l p r o p e r t i e s o f N - c o n n e c t i o n s t ru c t u r e s in v - b u n d l e sa n d d i f f e r e n t g e n e r a l i z a t i o n s o f F i n s l e r g e o m e t r y a n d s o m e p r o p o s a l s ( s e e C h . X I Ii n R e f . [ 7 5 ] , w r i t te n b y S . I k e d a ) o n p h y s i c a l in t e r p r e t a ti o n o f N - c o n n e c t i o n in t h ef r a m e w o r k o f " u n i f ie d " f ie l d t h e o r y w i th i n t er a c ti o n s n o n - l o c a li z e d b y y - d e p e n d e n c i e sa r e d is c u s s e d . W e e m p h a s i z e t h a t a n N - c o n n e c t i o n i s a d i f f e r e n t g e o m e t r i c a l o b j e c t f r o mt h a t i n t r o d u c e d b y u s i n g n o n - l i n e a r r e a l i z a t i o n s o f g a u g e g r o u p s a n d s u p e r g r o u p s ( s e e ,f o r i n s t a n c e , th e c o l l e c t i o n o f w o r k s o n s u p e r g r a v i ty [ 9 2 ] a n d a p p r o a c h e s t o g a u g eg r a v i t y [ 9 8 , 8 6 ] ) . T o m a k e t h e p r e s e n t a t i o n t o a i d r a p id a s s im i l a ti o n w e s h a ll h a v er e a l i z e d o u r g e o m e t r i c c o n s t r u c t i o n s f i r s t f o r v s - b u n d l e s a n d t h e n w e s h a l l e x t e n d t h e mf o r h i g h e r o r d e r e x t e n s i o n s , i . e . f o r g e n e r a l d v s - b u n d l e s .3 .1 . N - c o n n e c t i o n s i n v s - b u n d l e s

L e t u s c o n s i d e r t h e d e f i n i t i o n s o f N - c o n n e c t i o n s t r u c tu r e [ 1 0 6 ] i n a v s - b u n d l e ~ =( /~ , r rE , aqr ) w ho se ty pe f ibre i s b . an d ~r : T~--+T191 i s t he superd i f f e r en t i a l o f t hem a p r rE ( I t 7 i s a f i b r e -p r e s e r v i n g m o r p h i s m o f t h e t s- b u n d l e ( T ~ , re ,/1 7/) t o / ~ a n d o ft s -b u n d l e ( T M , r, 37 /) t o M ) . T h e k e r n e l o f th is v s - b u n d l e m o r p h i s m b e i n g a s u b b u n d l eo f ( T E , r E , / ~ ) i s c a ll e d t h e v e r ti c a l s u b b u n d l e o v e r ~ a n d d e n o t e d b y V ~ = ( V /~ , r v , / ~ ) .I t s to ta l spa ce i s V ~ = [. Ju~g vu, wh ere V~ = ke r~ 7 , uC ~. A v ecto r

y = y a c9 t ~ a 0 = y i O . . ~ _ y ' i O y a 0 y a 0+ V A ox , + eo , + +

t a n g e n t t o ~ i n t h e p o i n t u E ~ i s lo c a l l y r e p r e s e n t e d a s( u , Y ) = ( u '~ , Y ~') = ( x l , y A , y * , Y A ) = (sci, o ~ , p " , ( a , ? i , Y ~ , ? " , Y a ) .

A n o n - l in e a r c o n n e c t i o n , N - c o n n e c t i o n , in v s - b u n d l e g i s a s p l it ti n g o n t h e l e ft o f t h ee x a c t s e q u e n c e

O , , V { , i T g , , , T g / V g , , 0, (7)i .e . a m o r p h i s m o f v s - b u n d l e s N : T ~ E V ~ s u c h t h a t N o i i s th e i d e n t i t y o n V ~ .T h e k e r n e l o f t h e m o r p h i s m N i s ca l le d t h e h o r iz o n t a l s u b b u n d l e a n d d e n o t e d b y

( H E , r E , E ) .


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S.I. Vacaru/NuclearPhysics B 494 [PM] (1997) 590-656 601F r o m t h e e x a c t s e q u e n c e ( 7 ) i t f o l l o w s t h a t t h e N - c o n n e c t i o n s tr u c t u r e c a n b e e q u iv a -l e n t l y d e f i n e d a s a d i s t r i b u t i o n { /~ u ~ H u E , T u/ ~ = H uE '@ V u~:} on ~ ; de f i n i ng a g l oba ld e c o m p o s i t i o n , a s a W h i t n e y s u m ,

T g = n g + V g . ( 8 )T o a g i v e n N - c o n n e c t i o n w e c a n a s s o c i a t e a c o v a r i a n t s - d e r i v a t i o n o n M :

Xt { OYAV x Y = Ox + N A ( x , Y ) } s a , ( 9 )w h e r e SA are l o c a l i n d e p e n d e n t s e c t i o n s o f g , Y = YASA a n d X = X t s t .

S - d i f f e r e n t i a b l e f u n c t i o n s N a f r o m ( 3 ) w r i tt e n a s f u n c t i o n s o n x t a n d y A , N a ( x , y ) ,a r e c a ll e d t h e c o e f f i c ie n t s o f t h e N - c o n n e c t i o n a n d s a t is f y th e f o l l o w i n g t r a n s f o r m a t i o nl aw s u n d e r t h e c o o r d i n a t e t ra n s f o r m s ( 1 ) i n :

A' tgXt' = A/IA'MA O M a ' (X) yA.Nt ' ~X l ' " A ' ' l OXI f th e c o e f f i c ie n t s o f a g i v e n N - c o n n e c t i o n a r e s - d i f f e re n t i a b le w i t h r e sp e c t t o t h e

c o o r d i n a t e s y a w e c a n i n t r o d u c e ( a d d i t i o n a l l y t o c o v a r i a n t n o n - l i n e a r s- d e r i v a ti o n ( 9 ) )a l in e a r c o v a r i a n t s - d e r i v a t i o n / 9 ( w h i c h i s a g e n e r a l i z a t io n f o r v s - b u n d l e s o f th e B e r w a l dc o n n e c t i o n [ 2 1 ] ) g i v e n a s fo l lo w s :

w h e r e

a n d

I ~I A/ (x , y ) -- o N A t ( x ' Y ) ( 1 0 )Oy B

~ c ( X , y ) = o.I a A a = BA(y)O@ x b e i n g a s e c t i o n i nor a vec tor f i e ld on g , Z = Z 0-Trx+ Y 0y--X a n d B

t h e v e r t i c a l s - b u n d l e ( V E , r v , F . ) t h e l i n e a r c o n n e c t i o n ( 1 0 ) d e f i n e s t h e s - d e r i v a t i o n( c o m p a r e w i t h ( 9 ) )

L) z n ~ [Z l ( aR t -~- ~[AIBB) ~- yB anA 1 \ ~ x ~ c)y B J a y A "A n o t h e r i m p o r t a n t c h a r a ct e ri s ti c o f a n N - c o n n e c t i o n is i ts c u r v a tu r e ( N - c o n n e c t i o n

c u r v a t u r e ) :1 j ,2Ajdx! A d x ] Q a1-2 = ~ Oy---

w i t h l o c a l c o e f f i c i e n t s


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6 02 S.L Vacaru/Nuclear Physics B 494 [PM] (1997) 590 -65 6B A B ^ A_~ltJI oN:] + Nt I~Bj ( _ ) l t J I ( 1 1 )a : j ~ g g : g A= - N j N m ,OnX ", ., OX1

w h e r e f o r s i m p l i c i t y w e h a v e w r i tt e n ( _ ) I K I I J I = ( _ ) I K J IW e n o t e t h a t l i n e a r c o n n e c t i o n s a r e p a r t i c u l a r c a s e s o f N - c o n n e c t i o n s l o c a l l y p a r a -

m e t r i z e d a s N ' ~ ( x , y ) = N A t ( x ) x t y B, w h e r e t h e f u n c t i o n s N A t ( x ) , d e f i n e d o n M , a r ec a l l e d t h e C h r i s t o f f e l c o e f f i c i e n t s .3.2 . N -conne ct ions in dvs-bun dles

I n o r d e r to d e f in e a n N - c o n n e c t i o n i n t o d v s - b u n d l e ~ (z ) w e c o n s i d e r a n s - s u b b u n d l eN ( ~ ( z ) ) o f t h e t s -b u n d l e T ( g ( z ) ) f o r w h i c h o n e h o l d s ( s e e R e fs . [ 9 4 , 7 7 ] r e s p ec t iv e l yf o r j e t a n d o s c u l a to r b u n d l es ) t h e W h i tn e y s u m ( c o m p a r e w i t h ( 8 ) )\ /

N ( f f ( z )) c a n b e a l s o i n t e r p r e t e d a s a r e g u l a r s - d i s t r i b u t i o n ( h o r i z o n t a l d i s t r i b u t i o n b e i n gs u p p l e m e n t a r y t o t h e v e r t i c a l s - d i s t r i b u t i o n V ( g ( z )) ) d e t e r m i n e d b y t h e m a p s N : u Eg ( z > _ ~ N ( u ) c T ~ ( g ( z> ) .

T h e c o n d i t i o n o f e x i s te n c e o f a n N - c o n n e c t i o n i n a d v s - b u n d l e ~ ( z ) c a n b e p r o v e d a sin R e f s . [ 7 5 - 7 7 ] : I t i s r e q u i r e d th a t g ( z ) i s a p a r a c o m p a c t s - d if f e re n t i a b l e ( in o u r c a s e )m a n i f o l d .

L o c a l l y a n N - c o n n e c t i o n i n g ( z ) i s g i v e n b y i ts c o e f fi c ie n t sr .tA l (U ) , (N:~2 ) t (u ) , A2 At, AI,l v (01 ) l , N(12)A, ( U ) , . . . ( N(ov )t( u ), N( [p)A 1( U ) . . . .

ApN ( /, _l p) A p _ ' ( u ) ) . . . . .A= A=( N ( o z ) / ( u ) , N ( l z ) A I ( U ) . . . . . N :p z )A t, ( U ) . . . . . N : z _ I z ) A : _ , ( U ) ) ,

ap a , Ap ( U ) are c o m p o n e n t s o fh e r e , f o r i n s t a n c e , ( N ( o p ) l ( U ) , N ( l p )A I (U ) . . . . . N ( p_ lp )A ~ ,_ lt h e N - c o n n e c t i o n i n t h e v s - b u n d l e 7r (p ) : E (P ) ~ ~ ( p - l ) . H e r e w e n o t e t h a t i f a n N - c o n -n e c t i o n s t r u c t u r e i s d e f i n e d w e m u s t c o r r e l a t e t o i t t h e l o c a l p a r t i a l d e r i v a t i v e s o n ~ ( z )b y c o n s i d e r i n g i n s t ea d o f t h e l o c a l c o o r d i n a t e b a s e s ( 4 ) a n d ( 5 ) t h e so - c a ll e d l o c a l lya d a p t e d b a s e s ( l a - b a s e s )

~(o~) = ( 8 1 , 8 ( A ) ) ~ - ( 8 1 , 8 ( A I ) , 8 ( A 2 ) . . . . . 8 ( A ~ ) )8 8 ~ 8 8

' ' A2 ' ' ' ' '- ~ ( ~ O y : l ' ) OY(2) OY:z) )( t h e d u a l l a - b a s e s a r e d e n o t e d a s

~ (o t) = ( ~ I , 8 ( A ) ) : ( S I , ~ ( A I ) , 8 ( A 2 ) . . . . . ~ ( A : ) )= 8 u(,~ ) = ( S x t , ~ y ( A O B y (A 2 ) . . . , ~Sy(A~ )

( 1 2 )

) (13 )w i t h c o m p o n e n t s p a r a m e t r i z e d a s


14/67

S . I . V ac ar u / Nuc l e a r P hy s i c s B 494 [ P M] ( 1997 ) 590 - 656 60 3A[ A2 A: _ , NA:O A ,S I = 3 1 - - N l t ~ a I - - N t OA2 -- - - N I O a : 1 - - :

A 2 ~ A - A 3 ~ . - - A z - I A :~A I = O A I - - IVA I 2 - - NA t ~3 - - N At aA:_t -- N & a A : ,A3 A4 Az - t A.~A2 = C~A2 -- N A 2 0A ~ - - g a 2 0 A 4 - - - - N A 2 C~A:- I - - N A 2 0 A : ,

( 1 4 )

A.~ A : _ I = O A : _ I - - N A ~ _ I O A : '

o r , i n m a t r i x f o r m , a sa . = N ( u ) x O , ,

w h e r e

O x /6("1A t O Y ' ~ t[ t~A2 = 8 O y ( ~\aA=

\ O Y e z /1 - N A t - N A2 . . . - N A :

N A :0 1 - - N A ~ " ' " - A,N A :0 0 1 " ' ' - - A 2

0 0 0 . . . 1

O . = 0 ( 4 ) =:A ' l )A2 = (0__ ,# X 1O y ~0

A2O Y ( 2c)

~(~ ) w h e re , fo r i n s t an ce , ]V~ =n g e n e r a l i ze d i n d ex f o r m w e w r it e th e m a t r ix ( 6 ) a s " ( 8 ) '~ ]j , A A I A I N I _ N A I ~ a z N A : ~ a : _ _ N A :N B j = ~ B t . . . . , = ' " " ' A t = - - A t ' A 2 = A 2 ' " "

S o i n e v e r y p o i n t u E ~ ( z ) w e h a v e t h e f o l l o w i n g i n v a r ia n t d e c o m p o s i t i o n :T u / \~(d)) = N o ( u ) ( ~ N , ( u ) . . . N z - , ( u ) G V z ( u ) , (15)

w h e r e 8 t E N O , S A ~ C N l . . . . . 8 a : _ t E g z _ l , a A = E V z .W e n o t e t h a t f o r t h e o s c u l a t o r s - b u n d l e ( O s c Z l f l , r r, 1 (4 ) t h e r e is a n a d d i t i o n a l ( w e

c o n s i d e r th e N - a d a p t e d v a r i a n t ) s - t a n g e n t s t ru c t u r eJ: x(OscZ ) x(o sczr4 )

d e f i n e d a s


15/67

6 0 4 S .I . V a c ar u / N uc l e a r P hy s i c s B 494 [ P M ] ( 1997 ) 590 - 656

3 y / l ) - J ~x l . . . . . 3y[z_l-----~) - J 3y/~-_2 3y[z ) J ( 1 6 )( i n t h i s c a s e I - a n d A - i n d i c e s t a k e t h e s a m e v a l u e s a n d w e c a n n o t d i s t i n g u i s h b e t w e e nt h e m ) , b y c o n s i d e r i n g t h e v e r t i c a l J - d i s t r i b u t i o n s

N o = N , N I = J ( N o ) . . . . . N z - l = J ( N z - 2 ) .I n c o n s e q u e n c e , f o r t h e l a - a d a p t e d b a s e s o n (Osc Z ifl, Tr, if'l) w e h a v e t h e f o l l o w i n gN - c o n n e c t i o n m a t r i x :

N = ~ ( J )" " ( 1 ) =

I _NJ(1)I J -NJ (z)1N ( 2 ) I . .J - - ( z - l ) 1 [1 -N(1) I N s

0 . . .1 -NJ~ -2) 1 ] .0 0 0

( 1 7 )

T h e r e i s a u n i q u e d i s t i n g u i s h e d l o c a l d e c o m p o s i t i o n o f e v e r y s - v e c t o r X E X (~ (z ))o n t h e l a - b a s e ( 1 2 ) :

X = X ( H ) + X ( ~ ) + . . . + X ( ~ ) , ( 1 8 )b y u s i n g t h e h o r i z o n t a l , h , a n d v e r ti c a ls , v l , v 2 . . . . vz p r o j e c t i o n s :

X (H) =h X= X1 t3 l, X (V~) _ ~ u 1 X = X ( A I ) 6 A I . . . . . X ( V :) _ ~ U z X = X ( A z ) ~ A z .W i th r e s p e c t to c o o r d i n a t e t r a n s f o r m s ( 4 ) t h e l a -b a s e s ( 1 2 ) a n d d s - v e c t o r c o m p o n e n t s

( 1 8 ) a r e c o r r e s p o n d i n g l y t r a n s fo r m e d a s

and

6 3x 1' 6 6 = KA'p B- ' Ap av a , ' , ' ( 1 9 )c g x l c g x l O x l l O y ( p ) O y ( p )

X 1 ' = O x 1 ' , A ~ A IX t x(Av ) = Ka~ X( , , ) , V p = 1 , 2 , . z .cgX1 ' , . ,U n d e r c h a n g i n g o f c o o r d i n a te s ( 3 ) t h e lo c a l co e f fi c ie n t s o f a n o n - l in e a r c o n n e c t io n

t r a n s f o r m a s f o l l o w s :v


16/67

S . I . V a c a r u / N u c l e a r P h y s i c s B 4 9 4 [ P M ] ( 1 9 9 7 ) 5 9 0 - 6 5 6 6 0 5(~X I = d x I ,

}l.,~A~ A . . I~y a l = dy~l, ) + " " ( 1 ) t ~'~ ,A2 i t4A2 t l ,~Al I IAA2 t i l lt ~ Y a z = dY(2 ) + " ' " ( 2 ) A I e . r ( I ) " q- " ( 2 ) 1 " a " '

6 y A ~ = d y ~ j ~ ) - . . A s . A ! _ , , A s . A ~ A . I~Vl(s)AtaY(l) ~ ~Vt(s)A2aY(~) + . . . + M ( z ) td x , ( 2 0 )w h e r e M ~ ) a r e th e d u a l c o e f f i c ie n t s o f t h e N - c o n n e c t i o n w h i c h c a n b e e x p r e s s e de x p l i c i t ly b y r e c u r r e n t f o r m u l a s t h r o u g h th e c o m p o n e n t s o f th e N - c o n n e c t i o n ~ ,(t) T o' ' ( A ) "d o t h i s w e s h a l l re w r i t e f o r m u l a s ( 2 0 ) i n m a t r ix f o r m :

6 * = d * x M ( u ) ,w h e r e

a = ( ~ X 1 t ~ y a ' t ~ y A 2 . . . t $ y a ` ) ,

an d

M =

A i A 2 A .1 M ( 1 ) I M ( 2 ) l . . . M ( ~ ) tA2 A.0 1 M ( 2 ) A I . . . M ( ~ ) A t

A .0 0 1 . . . M ( ~ A 2

0 0 0 . . . 1

A2 .. . (~Y~s ) = ( d x I d y ~ ' ) t~y(2 )

t ~X I = d x I ,

6y[2 ~ dY[2 ) 1 J M ~ 2 ) j d x J ,q - M ( 1 ) j d y ( l ) q -1 J 1 J6 y[ ~) = d y [z ) + M o ) j d y ( s _ l ) + M ( 2 ) j d y ( z _ 2 ) + . . . + M ~ z ) j d x J ,

w i t h t h e M - c o e f f i c i e n t s c o m p u t e d b y t h e r e c u r r e n t f o r m u l a s :IM~I )~ = N (1)s ,

I 1 1 KM ( 2 ) j = N ( 2 ) j - ~- N ( 1 ) K M ( 1 ) j ,( 2 1 )

a n d t h e n , t a k in g i n t o c o n s i d e r a t i o n t h a t b as e s O o ( 8 . ) a n d d * ( 8 ) a r e m u t u a l l y d u a l , toc o m p u t e t h e c o m p o n e n t s o f m a t r i x M b e i n g s -i n v e rs e t o m a t ri x N ( s e e ( 1 7 ) ) . W e o m i tt h e s e s i m p l e b u t t e d i o u s c a l c u l u s f o r g e n e r a l d v s - b u n d l e s a n d , f o r s i m p l ic i ty , w e p r e s e n tt h e b a s i c f o r m u l a s f o r o s c u l a t o r s -b u n d l e (O sc z , (4 , 7r , , (4) w h e n J - d i s t r i b u t i o n p r o p e r t i e s( 1 6 ) a n d ( 1 7 ) a l le v ia te s t h e p r o b l em . F o r c o m m o n t y p e o f in d i c e s o n / ( / a n d h i g h e ro r d e r e x t e n s i o n s o n O s c z ~1 t he dua l l a -base i s expres sed as


17/67

1 l , ~ A t ,f 2 ( p ) = ~ ( p ) O t p _ l f l p _ l ~ U O t t , - I / ~ ( ~ U l p - I @ - -

w i t h l o c a l c o e f f i c i e n t s

606 S.l. Vacaru/Nuclear Physics B 494 [P M] (1997) 590-656

1 1 1 K 1 KM ( s ) J = N (s ) J + N ( s _ l ) r M ( i ) j + . . . + N ( 2 ) K M ( z _ 2 ) j + N ~ l rK M ( z -1 ) J "T h e s e t r a n s f o r m a t i o n l a w s f o r d u a l c o e f f i c i e n t s ( 2 1 ) h o l d w i t h r e s p e c t t o c o o r d i n a t et r a n s f o r m s ( 3 ) :

K " f f K ' I 'M(I)J Y(o ,o)K - - M(1)K , Y io ,o )J + Y( 1,0)J,K 1 ' _ 1 / K ' I j K t I tM(2)J Yio ,o )K - M(2)K 'Y io ,o )J + M(1)K'Y(I ,O)J + r(2,0)J,

+ +. . . + +( T h e p r o o f i s a s t ra i g h t fo r w a r d re g r o u p i n g o f te r m s a f te r w e h a v e p u t ( 3 ) i n to ( 2 1 ) ) .

F i n a l l y , w e n o t e t h a t c u r v a t u r e s o f a n N - c o n n e c t i o n i n a d v s - b u n d l e ~ ( z ) c a n b ei n t r o d u c e d i n a m a n n e r s i m i l a r to t h at f o r u s u a l v s - b u n d l e s ( s e e ( 1 1 ) ) b y a c o n s e q u e n ts t e p b y s t e p i n c l u s io n o f h i g h e r d i m e n s i o n a n i s o tr o p ie s :

8At, ' p = l , 2 . . . . . Z,Oy(p)

A t ' t ~ N ~ P - I ) [ f l l , - l y t , - i ] ~ N ~ A ; - |J ' ~ ( p ) f l t , - t ' g p - i - ,Q y p - i ( - ~ u f l p _ Iv U ( p _ l ) t p - 1 )

^ 7w h e r e ga',yt,_~ At' De yAo X I6N: j t ,_ , /Oy


18/67

S .I . V a c a r u / N u c l e a r P h y s i c s B 4 9 4 [ P M ] ( 1 9 9 7 ) 5 9 0 - 6 5 6 6 0 7

O AI ~mp l 3 B I3 B I @ " @ 3 80 1 OE1 @ @ 3 E t , @ ~C1 @ . . . 3 ca2 OD~ . 3Dp: 6 F~ 3 e ': . ( 2 2 )

I n a d d i t i o n t o d s - t e n s o r s w e c a n i n t r o d u c e d s - o b j e c t s w i t h v a r i o u s s - g r o u p a n d c o o r -d i n a t e t r a n s f o r m s a d a p t e d t o t h e g l o b a l s p l i t t i n g ( 1 5 ) .

A l i n e a r d i s t i n g u i s h e d c o n n e c t i o n , d - c o n n e c t i o n , in a d v s - b u n d l e ~ ( z ) i s a li n e a r c o n -n e c t i o n D o n ~ ( z ) w h i c h p r e s e r v e s b y p a r a l l e l i s m t h e h o r i z o n t a l a n d v e r t i c a l d i s t r ib u t i o n si n g ( z ) .

B y a l in e a r c o n n e c t i o n o f a n s - m a n i f o l d w e u n d e r s t a n d a l i n e a r c o n n e c t i o n in i tst a n g e n t b u n d l e .

L e t us d e n o t e b y ~ ( M ) a n d ~ ( g ( P ) ) , r es p ec ti ve ly , t h e m o d u l e s o f v e c to r f ie ld s o n s -m a n i f o l d / Q a n d v s - b u n d l e g ( P ) an d b y . T ' ( M ) a n d . T ' ( g ( t ' )) , r e s p e c t i v e l y , t h e s - m o d u l e so f fu n c t io n s o n M a n d o n g ( P ).

I t i s c l e a r t h a t f o r a g i v e n g l o b a l s p l i t t i n g i n t o h o r i z o n t a l a n d v e r t i c a l s s - s u b b u n d l e s( 1 5 ) w e c a n a s s o c i a te o p e r a t o r s o f h o ri z o n t a l a n d v e r ti c al c o v a r ia n t d e ri v a t io n s ( h - a n dv - d e r i v a ti o n s , d e n o t e d r e s p e c t iv e l y a s D (h) a n d D ( " : 2 " : ) ) w i t h t h e p r o p e r t i e s

D x Y = ( X D ) Y = D h x Y + D . ~ x Y + D , , z x Y + + D . :x Y ,w h e r e

D ( x h ) Y = D h x Y ,, D ( x h) f = ( h X ) fa n d

D ~ " ) Y = D v p x Y , D ~ [ P ) f = ( v p X ) f , ( p = l . . . . . z )f o r e v e r y f E 9 r(2 17 /) w i t h d e c o m p o s i t i o n o f v e c to r s X , Y C E ( g ( z ) ) i n t o h o r iz o n t a l a n dve r t i c a l pa r t s , X = h X + v i X + . . . + v z X a n d Y = h Y + v ~ Y + . . . + v z Y .

T h e l o c a l c o e f f ic i e n t s o f a d - c o n n e c t i o n D i n g ( z ) w i t h r e s p e c t to t h e l o c a l a d a p t e df r a m e ( 5 ) s e p a r a t e i n t o c o r r e s p o n d i n g d i s t in g u i s h e d g r o u p s . W e in t r o d u c e h o r iz o n t a l

: I t r (A ) ~ = ( L I j K ( u ) , L A , x ( u ) , L B 2 K ( ) . . . . . L A : K ( U ) ) o f D h)o c a l c o e f f i c i e n t s ~ . ~ J K ' ~ ( B ) K : A2 bls u c h t h a t

' - ~ - ( p = 1 . . . . . z ) ,~(~.~, ) 3 x g = L t j K ( U ) 3 X 1 , ~ ( 8 . ~ ) 3 y ~ t ~ ) . , , , )D ( h ) 3 q

a n d p - v e r t i c a l l o c a l c o e f f i c i e n t s", I t ' - , ( A ) t"~ a I ( U ) A2 a. ~ j i c > , , ~ ( s > ~ c > ) = ( C J c p ( U ) , , . . B , c , ' , C B 2 c p ( U ) . . . . . C B : c , , ( u ) ) ( p = 1 . . . . z )

s u c h t h a t


19/67

608 S.L Vacaru/Nuclear Phys ics B 494 [PM] (1997) 590-656D (Vr ) ~_ 8~ _~ 7 ) 8 x J = C ~ c , ( u ) ~ x ~ ,

D( V . ) ~ 8 q( ~ ) q - OyC,, 'w h e r e q c . T ( g ( z ) ) , f = 1 . . . . z .

D(Vp) ~ Af( ~ ) ~ . ,B: = CB:c, , A : 'oy I' ~ y f f ) 6 y ( f )

T h e c o v a r i a n t d s - d e r i v a t i o n a l o n g v e c t o r X = X 1 ~6 + yAl 6 + + yA~ ~ o f a, ,d s - t e n s o r f i e l d Q , f o r i n s t a n c e , o f t y p e ( p P r ~ , l ~ < r ~< z , s e e ( 2 2 ) , c a n b e w r i t t e n\ q q r /a s

( h ) 0 O~ x " , ) a . D~ ' = ) Q ,x Q = D x ~ + + . . +w h e r e t h e h - c o v a r i a n t d e r i v a t i v e is d e f in e d a s

D ~xh Q = X K Q tj r K ~ @ ~ar Q d l @ ~By,w i th c o m p o n e n t s

~g")lAr r l z ' ~ H A r Z A, ['llCr -- r H I-liAr IC y D I A rQ JB~ IKl A y c g XK~JBr _~_ ~ HK~a~JBx -~- CIK ~J BI L, J K ~H B r --~ B rK ~ J C ~a n d t h e v t , - c o v a r i a n t d e r i v a t i v e s a r e d e f i n e d a s

O~ 'P) a Cp tAr l n r= X Q j s ~ c p 6 1 O a ~ d 8 ,w i th c o m p o n e n t s~l ' ) lAy= ,, C 1 ,, '.hHA~ CA r t ')lFr (' ,H g'liAr t ,",F~ [')IA~OJBrCplAr ~JBRoycp "~- HC t' ~ J B R + FrCp~ J B R - - ~ J C v ~ H F R - - ~ B r C e ~'~JFR"

T h e f o r m u l a s p re s e n t ed a b o v e s h o w t h a t D F = ( L . . . . . L ( p ) . . . . . C . . . . . C ( p ) . . . . ) a r et he l oca l coe f f i c i en t s o f t he d - connec t i on D wi th r e spec t t o t he l oca l f r ame (~ -~ r~ , a ) .I f a c h a n g e ( 3 ) o f l o ca l c o o r d i n a t e s o n g Iz > i s p e r f o r m e d , b y u s i n g t h e la w o f t r a n s fo r -m a t i o n o f l o c a l fr a m e s ( 1 9 ) , w e o b t a i n t h e f o l lo w i n g t r a n s f o r m a t i o n l aw s o f th e l o c a lc o e f f ic i e n t s o f a d - c o n n e c t i o n :

1 ' O x l t O x J O XM 1 O x l / 0 2 x ML j tM ' - - ~ X I OXj- '7 cgxM, L JM '~ c~xM c~XJ, O x M , , ( 2 3 )

p ~, . , a _ M A f a K ~ j,A f = l ( . , q f ( B f u A [ K A 'Y-L A . . . . = " W. .,- - + C s ~ ' - sL(f )B ~f M' S t~ f OXM ( f ) B I M

1' Ox 1' Ox s c~ t ' '- - - - C t t( p )J ' C /, Ox I ~ x j ' K c~ C ( P ) - t c , , ' A S = K A I K B I, K C , ; cA I ,. . . . B f C ~ A f B / C~ B I C v " ' "

A s i n t h e u s u a l c a s e o f t e n s o r c a l c u l u s o n l o c a l l y i s o t r o p i c s p a c e s , t h e t r a n s f o r m a t i o nl a w s ( 2 3 ) f o r d - c o n n e c t i o n s d i f f e r f r o m t h o s e f o r d s - te n s o r s , w h i c h a r e w r it te n ( f o ri n s t a n c e , w e c o n s i d e r t r a n s f o r m a t i o n l aw s f o r t h e d s - t e n s o r ( 2 2 ) ) a s


20/67

S .l . V a c a r u / N u c l e a r P h y s i c s B 4 9 4 [ P M ] ( 1 9 9 7 ) 5 9 0 - 6 5 6 6 0 91 i t , / , t ~ t = t ~ t ~ t1... p.~ t i ...t~plL~l...~p2...L~,l....IJpsQ y, j t B t o t ~ t f ' t ~ t w tI ... q I ""Dql t~ I '''"~q2 "" rl ""rqs

Ox t; Ox J~ O'- - k, A t, ld.BI Ko t t l ; IdFp , D II . . . lp Ai . . .A, , , E l . ..Ep2 . . .DI . . .D .. ..C )X II C )X J ( . . . . . A I " = B~ . . . . . F p ~ $ J l .. .J q B I . .. B q l C I. .. Cq 2 .. .F I. .. Fq ~ "T o o b t a i n l o c a l f o r m u l a s o n t h e u s u a l h i g h e r o r d e r a n i s o t r o p i c s p a c e s w e h a v e t or e s tr i ct o u r s e l v e s t o e v e n c o m p o n e n t s o f g e o m e t r i c o b j e c t s b y c h a n g i n g , f o r m a l ly , ca p i ta l

i n d i c e s ( I , J , K , . . . ) i n t o ( i , j , k , a . . . . ) a n d s - d e ri v a ti o n a n d s - c o m m u t a t i o n r u l e s i n t ot h o s e f o r r e a l n u m b e r f i e l d s o n u s u a l m a n i f o l d s .4 . 2. T o r s io n s a n d c u r v a t u r e s o f d - c o n n e c t i o n s

L e t g ( z ) b e a d v s - b u n d l e e n d o w e d w i th N - c o n n e c t i o n a n d d - c o n n e c t i o n s t ru c t u re s .T h e t o r s i o n o f a d - c o n n e c t i o n i s i n t r o d u c e d a s

T ( X , Y ) = [X , D Y } - [ X , Y } , X , Y C ~ ( h 7 " / ).T h e f o l l o w i n g i n v a r ia n t d e c o m p o s i t i o n ( b y u s i n g h - a n d v - p r o j e c t i o n s a s s o c ia t e d to N )h o l d t r u e :

T ( X , Y ) = T ( h X , h Y ) + T ( h X , v lY ) + T ( v t X , hX ) + T ( v lX , V l Y) + , . .+ T ( V p - l X , V p - l Y ) + T ( V p - IX , v p Y ) + T ( v p X , V p - l X )+ T ( u p X , v p Y ) + . . . + T ( V z _ I X , v z - 1 Y ) + T ( v z - I X , v z Y )+ T ( v z X , V z - l X ) "~- T ( v z X , v z Y ) .

T a k i n g i n t o a c c o u n t t h e s k e w s u p e r s y m m e t r y o f T a n d t h e e q u a t i o n sh [ v p X , v p Y } = 0 . . . . . v f[vpX, vpY} = 0 , f ~ p ,

w e c a n v e r i f y th a t t h e to r s i o n o f a d - c o n n e c t i o n i s c o m p l e t e l y d e t er m i n e d b y t h e f o l -l o w i n g d s - t e n s o r f i e l d s :

h T ( h X , h Y )v t, T ( hX , hY )h T ( h X , VpY)

v p T ( h X , v p Y )v f T ( v f X , v f Y )v p T ( v f X , v f Y )v f T ( u f X , U p Y )v p T ( v fX , v p Y)v z - l T ( v z - l X , V x - l Y )

v z T ( V z - l X , V z - l Y )

= [ X ( D ( h ) h ) Y } - h [ h X , h Y } . . . . .= - v p [ h X , h Y } . . . . .= -D y~ (V P)h X - h [ h X , v p Y } . . . .= D( xh ) vpY - vp [hX , v t, Y } . . . . .= [ X ( D ( ' J ) v f ) Y } - v f [ v f X , v f Y } . . . . .= - - V l , [ v f X , v f Y } . . . . .= - D ( y ' P ) v f x - v f [ v f X , v l, Y } . . . . .= D(x 'I) vp Y - vp [ v fX , Vp Y } . . . . . f < p ,= [ X ( D ( V : - ' ) V z - t )V} - v : --1 [Vz --I X , UZ I Y } . . . . .= - - V z [ V z -- 1 X , V z - I Y } ,


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610 S.I. Vacaru/Nuclear Physics B 494 [PM] (1997) 590-656V z - l T ( V z - l X , v z Y ) = - D CyV:)V z - I X - V z - l [ V z - l X , v z Y } . . . . .

v z T ( v z - l X , v z Y ) = D (xV= -OV z Y - V z [ V z - l X , v z Y } .w h e r e X , Y C ~ ( ~ ( z ) ) . I n o r d e r t o g e t t h e l o c a l f o r m o f th e d s - t e n s o r f i e ld s w h i c hd e t e r m i n e t h e to r s i o n o f t h e d - c o n n e c t i o n D F ( t h e t o r s i o n s o f D F ) w e u s e

8x .1 ax g I = R~A) d y (a ) ' ~X J ' ay(B) } - ay(B ) ay(A) 'w h e r e

R S> = ( - > ' " i a N :a x x a x J ,

a n d i n t r o d u c e t h e n o t a t i o n sh T ( ~ K , t~ ) = T t j K ~t~XJ I '

8 8 ) -a , 8ax .

8 8 ) p l 6h T ( a y (a ) , a X J = J (a ) ~X I . . . . .t~ t~ p ( a } t~v p T ( ) = J Bp . . . . .ayB, , ' 8X J ay( A}

v , T ( 8 8 ~ , ( A } & ( 2 4 )ayC~, ' OyB: ) = oBiCe cgy(a) "N OW w e c a n c o m p u t e t h e l o ca l c o m p o n e n t s o f t h e t o rs io n s , in t r o d u c e d i n ( 2 4 ) , w i t h

r e s p e c t t o a l a - f r a m e o f a d - c o n n e c t i o n D F = ( L . . . . . L ( p ) . . . . . C . . . . . C ( p ) . . . . ) :TI j K = L l j K - - ( - - ) IJ K I L t K . t,~,(jA) R ( A ) j K ' l ( B > = C I j ( B ) ,

8 N ) A)P ( A ) j ( B ) - cgy(B) L ( A ) ( B ) J 's (A) (B)(C) = C(A) (B)(C) - - ( - - ) I (B) (C)Ic (A) (C)(B) . ( 2 5 )

T h e e v e n a n d o d d c o m p o n e n t s o f th e t o rs i o n s ( 2 5 ) c a n b e s p e c if ie d in a n e x p l ic i t f o r mb y u s i n g d e c o m p o s i t i o n s o f i n d i c e s i n t o e ve n a n d o d d p a r t s ( 1 = ( i , 7 ), J = ( y , j ) . . . . ) ,f o r i n s t a n c e ,

Ti j k = L i j k - - L 'k j , T i j k = L i j k + L^ A ^T i j k = L ij k - - L i k j . . . . .

a n d s o o n ( s e e R e fs . [ 7 5 , 7 6 ] ) .A n o t h e r i m p o r t a n t c h a r a c te r is t ic o f a d - c o n n e c t i o n D F i s i t s cu rva t u re :R ( X , Y ) Z = D [ x D r ) - Dtx,r}Z,


22/67

S .I . V a c a r u / N u c l e a r P h y s i c s B 4 9 4 [ P M ] ( 1 9 9 7) 5 9 0 - 6 5 6 6 1 1

w h e r e X , y Z E ~ ( / ~ ( z ) ) . F r o m t h e p r o p e r t i e s o f h - a n d v - p r o j e c t i o n s i t f o l l o w s t h atv p R ( X , Y ) h Z = O . . . . . h R ( X , Y ) v p Z = O , v f R ( X , Y ) v r Z = O , f 4= p , ( 2 6 )

a n dR ( X , Y ) Z = h R ( X , Y ) h Z + v I R ( X , Y ) v l Z + . . . + v z R ( X ,Y ) v z Z ,

w h e r e X , y Z E ~ ( / ~ ( z ) ) . T a k i n g i n t o a c c o u n t p r o p e r t i e s ( 2 6 ) a n d t h e e q u a t i o n sR ( X , Y ) = - ( - ) lX rI R ( y X ) ,

w e p r o v e t h a t t h e c u r v a t u r e o f a d - c o n n e c t i o n D i n t h e t o t a l s p a c e o f a v s - b u n d l e / ~ ( : )i s c o m p l e t e l y d e t e r m i n e d b y t h e f o l l o w i n g d s - t e n s o r fi e ld s :

{ l ~ ( h ) l ) ( h ) _ F I ( h )R ( h X , h Y ) h Z = ~ [ x ~ r } ~ [hX , hY}_ D ( V O D ( V : - t )[ h X , h } ' } - " " [ h X , h } ' } - Dlh~hV.} ) h Z ,

R ( h X, h Y ) v p Z = ( 1 ) ( h ) l ) ( h ) - - D (h)" ~ [ X ~ Y } [ h X , h Y }- D ( v l) ) v pZ - . . . - D (vp -~) ~" Z (vp) Z[ hX , h Y } [ hX , h Y } ' V l ' - - D [ h X , h Y } ) U P '

R ( v pX , h Y ) h Z [ r ~ ( V p ) r~ ( h ) - - l ) ( h )= I,z.,I[X ~ ' y } ~ [ v v X , h Y }_ r ~ ( vl) a h T _ . . . _ /3 ( v "- l) ( " ) h ) h Z~ [ v t , X , h Y } : . . . . [vpX,hY} - D[..x,_Y } '

R ( v f X , h Y ) v p Z : n ( v /) r ~ (h ) -- D (h)= t ~ ' [ X ~ Y ) [ v fX , h Y }[ v f X , h Y } I U p L ,- U [ v / X , h Y } ) V 1 L . . .

- - [ F } ( U f ) T~ (O P ) - - F ) ( U l ) )hZ -R(vfm, v p Y ) h Z - ~ [ x - - r } ~ [ . : x , . ~ r } _ D ( v : - | ) _ D ( v : )[ , , :x . , , , , r } ~ [ , , :x , , , . r } ) Z ,

- - ( r ~ ( V / ) l - ~ ( v q ) r -~ (v l ) x , ' 7R ( v f X , v q Y ) v p Z - ~ ~ [ x ~ y } - I J [ v / X , v q y ) ) U l ~ - " " "rdVe-l) ~ . 7 - ( v l ,) ) v ~ Z ,

- - I " I [ v f X , v q Y } ) u p - I L " - - D [ v f X t _ q Y } Iw h e r e

D ( h ) r ( h )Fl(h) Fl(h) ( ~ l X Y [ F l ( h ) l ) ( h )I X ~ Y ) ~ ' X ~ Y - - ~ , - - : ~ Y ~ X 'D U O n(v e) ~.l)(h)Fl(Vp) _)Lxv , ,r lo(rVe)o(xh),IX ~ r } ~ x ~ r - (D ( . , , ) n ( h ) = n ( v , , ) n ( h ) ", I . p x } ' l n ( h ) n ( v e )I x ~ r } u x ~ r - - ( - - : "-- '~ " ~ X .D ( V f ) 1 "~ (v v) D ( v f ) F I ( vP ) ~lvyX.erl n( vp ) r~(V/)[ x ~ r } ~ x ~ r - ( - : " - ' r " - ' x

T h e l o c a l c o m p o n e n t s o f t h e d s - t e n s o r f ie l d s ( 2 7 ) a r e i n t r o d u c e d a s f o l l o w s :R ( 6K, 6J )6H = RH IjK SI , R((3K, 6J )~(B) = Ri(BA.jK6(A ,

( 2 7 )

( 2 8 )


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6 1 2 S.L Vacaru/Nuclear Physics B 494 [PM] (1997) 590-6561 . ( a )R ( 6 ( c ) , t ~X )t ~J -~ P J K ( C ) 8 1 , R ( t S ( c ) , t S g ) 6 ( B ) = P(B) .K(C) tS(A) '

. I -~R ( 6 ( c ) 8 ( B ) ) S j = Sj . (B)(C)~I , R ( t S ( o ) , t ~ ( c ) ) t ~ ( B ) t " ( A ), O(B).(C)(D) O(A)"P u t ti n g t h e c o m p o n e n t s o f a d - c o n n e c ti o n D F = ( L . . . . . L(p) . . . . . C . . . . C (v ) . . . . ) in( 2 8 ) , b y a d i r e c t c o m p u t a t i o n , w e o b t a i n t h e f o l l o w i n g l o c a l l y a d a p t e d c o m p o n e n t s o ft h e c u r v a t u r e ( c u r v a t u r e s ) :

R H I j K = tSKLIHj -- ( - - ) IKJ I t~ jL IHK+ L M n j L tM K _ ( _ ) Iral L M n r L t M j + CIH (A)R (A) JK,

R ' ( A) _ t ~ . L ( a)B ) . J K - - ^ ( B ) J - - ~ , - - , IKJ I6 jL (A) (B)K+ L (C ) (B)J L(A ) (C)K -- ( -- ) IKJ I L (C) (B)K "~ C (a) (B) (C) R (c) J r ,

P J ( K (A ) = ~ ( a ) L I j K - - C I j ( A ) I K 4 - C I j ( B ) P ( B) K ( A ),P(B) (A)K(C) = t~(C) L (A) (B)g - - c (A ) (B) (C) IK 4 - c (a ) (B) (D )p (D ) K(C) ,

.IS j .(B )(C) = ~( c )C I j ( n ) - - ( - - ) I (B ) (C)I tS (B)CI j (c )+ c ( H ) IJ ( B) C ( H )( C) -- ( - - ) I ( B ) ( C ) I c ( H ) j ( c ) C I ( H ) ( B ) ,

S(B) (A) (C)(D) ---- S(D) C( A) (O)(C) -- (-- )I( C) (O )I t~(c )C( A) (B)(O)"~ -c (E ) (B) ( c )C(A) (E ) (D) - - ( - - ) I (C ) (D ) J c ( E ) ( B ) (D ) C ( A ) E ) (C ) . (29)

T h e e v e n a n d o d d c o m p o n e n t s o f t h e c u r v a t u re s ( 2 9 ) c a n b e w r i tt e n o u t b y s p l it ti n g t h ei n d i c e s i n t o e v e n a n d o d d p a r t s , f o r i n s t a n c e ,Rh j k - - t~kL ihj t~ jL ihk + L mh jL im k - - L mhkL im j + C ih (a )R (a ) j k ,

i _ " i L m L i f m ^L i . "R h j k - t ~ k L t h j - J - t ~ j L h k ' 4 - h j mk - '} - ~ h k m j + C ' h ( a ) R ( a ) j k . . . .( w e o m i t f o r m u l a s f o r t h e e v e n - o d d c o m p o n e n t s o f th e c u rv a tu r e s b ec a u s e w e s h al l n o tu s e t h e m i n t h i s w o r k ) .4 .3 . B ianch i and R icc i iden t i t i e s

T h e t o r s i o n s a n d c u r v a t u r e s o f e v e ry l in e a r c o n n e c t i o n D o n a v s - b u n d l e ~ (z ) s a t is f yt h e f o l l o w i n g g e n e r a l i z e d B i a n c h i i d e n t i t i e s :

[ ( D x T ) ( Y , Z ) - R ( X , Y ) Z + T ( T ( X , Y ) , Z ) ] = 0 ,s c

[ ( D x R ) ( U , Y , Z ) + R ( T ( X , Y ) Z ) U ] = 0 , ( 3 0 )s c

w h e r e )- -]s c m e a n s t h e r e s p e c t i v e s u p e r s y m m e t r i c c y c l i c s u m o v e r X , Y , Z a n d U . I f Di s a d - c o n n e c t i o n , t h e n b y u s in g ( 2 6 ) a n dv p ( D x R ) (U , Y , h Z ) = 0 , h ( D x R ( U , Y , v p Z ) = 0 , v f ( D x R ) ( U , Y , v p Z ) = 0 ,


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S.I. Vacaru/Nuclear Physics B 494 [P M ] (1997) 590-656 613t h e i d e n ti ti e s ( 3 0 ) b e c o m e

Z [ h ( D x T ) (Y ,Z ) - h R (X , Y ) Z + hT (h T (X , Y ), Z )s c + h T ( v 1 T ( X , Y ) ,Z ) + . . . + h T ( v z T ( X , Y ) , Z ) ] = 0 ,Z [ v f (D x T ) ( y Z ) - v f R ( X , Y ) ZSC

+v fT ( hT( X , Y ) , Z )+ Z v fT ( vpT( X , Y ) , Z ) ] = 0 ,p ~ f

Z [h (DxR) (U , Y ,Z ) + hR(hT(X , Y ) , Z )Us c+ h R ( v I T ( X , Y ) , Z ) U + . . + h R ( V z T (X , Y ), Z ) U ] = O,

Z [ v f ( D x R ) (U , Y ,Z ) + v f R ( h T ( X , Y ) , Z )USC Z v f R ( v p T ( X , Y ), Z ) U ] = 0 . ( 3 1 )

p>~fI n o r d e r t o g e t t h e c o m p o n e n t f o r m o f t h e s e i d e n t i t i e s w e i n s e r t c o r r e s p o n d i n g l y i n

( 3 1 ) t h e s e v a l u e s o f tr i p le s ( X , y Z ) , ( = ( t 3 j, 6 K , 6 L ) , o r ( t ~ ( o ) , 6 ( c ) , 6 (B ) ) ) , a n d p u ts u c c e s s i v e l y U = tSH a n d U = 6 ( a ) . T a kin g i n to a c co u n t ( 2 4 ) , ( 2 5 ) a nd ( 2 7 ) , ( 2 8 ) w eo b t a i n

Z [ T I jK I H T M j K T I H M R ( A ) jK C I H ( A ) - - R j I K H ] = O ,SC [L,K,JZ [ R(A) JKIH TM jK R(A) HM R (B) K P(A) H (B) ] = O,

SC I L,K,J}ICj (B ) IK - - ( - - ) I JK ICIK (B) I j - - T I jK I (B ) cM j (B )T IKM

- - ( - - ) I JK ICMK(B)T1 jM TMjKC IM(B) P (D) j (B )CIK(D)-(--)IKJIp(D)K(8)Ctj(D) + P j I K ( B ) - - ( - - ) I K J I P K I j ( B ) = 0 ,

p (A) j (B ) IK - - ( - - ) IKJ I p (A ) K (B)I J - - R (A) jK I (B ) cM j (B )R( A) KM-- ( -- ) IgJI CM K(B ) R(A) JM TM jK P (m) M(B ) P (O) J(B) P (a) X(O)

~ . ( a ) _-- ( - - ) IKJ Ip(D ) K(B) P(A ) J(D) -- R (> K S(A) B(D ) "'(B).JK -- O,

C I j ( B) I (C) - - ( - - )I (B)(C)Ic Ij (c )_I_(O) - I- cM j( c ) C IM (B ). I = 0 ,- ( - - ) I ( B ) ( C ) I c M j ( B ) C I M ( c ) s (D ) ( B ) ( c ) C I j (D ) - - S j .( B )( C )

p (A) J(B)A_(C) -- ( -- ) I ( B ) ( c ) I p (a) J(C)J_(B) S (a) (B) (C) IJ c M j( c ) P (A) M(B)_ ( _ ) I ( B ) (C )Ic M j(B ) p (A) M(C) P (D) J(8) S(a) (C) (D)- - ( - - ) I (C ) (B ) Ip (D) j ( c ) S (A ) (B )(D) s (D) (B ) ( c )p (A ) j (D )


25/67

61 4 S . I. V ac aru /N uc lear P hy s i c s B 494 [ P M ] ( 1997) 590 - 656+ P ( n ) ( A ) J ( C ) - - ( - - ) [ ( C ) ( B ) I P ( c ) ( A ) J ( B ) = O ,

Z [ s(A) (B)(C).L(D)SC[ (B),(C),(D ) }

"q-s(F)(B)(C) S(A)(D)(F) - - S(B)(A )(c)(D) ] ~' 0'E [ RKIHJ I L - - TMH JRKI LM - - R ( A) Hj P~ 'L ( A) ] = O ,

SC [H,J,LZ I-p'(A) "r'M D'(A) -- R(c) (A) ] = O,t " '(D).HJ IL -- I Hj,X(D).LM HJ P(D) L(C)SC l H,J,L

1 ILJI . ;PK.J(D)[L - - ( - - ) PK.L(D)IJ + R K I L J - L ( D ) "}- CML (D) RK IJ MILJI M I M .I- ( - ) C j ( o ) R r LM - - T JLP~.M(D)

IZJ In(a) n . l+P (A ) L(D)PKI. j (A) - - ( - - ) r j (D)I-K.L(A) - - R(A) JLS4.(A)(D) = O,_ _ l ? ' ( A )P(c) (A) j (D)IL ( - - ) IL J Ip( c) (A)L(D)I j + " ' (C) .LJI(D)

+CML(D)n ( c ) (m ) J M - - ( - ) I tJ I c g j < o > R < C ) ( a) LM_ T M j L P ( c ) (a ) M ( O ) + p ( e ) L (O ) P ( c ) ( a) J < r)- - ( - - ) I L J I p ( F ) j ( D ) P ( c ) ( A )L ( F ) - - R ( F ) j L S ( c ) < A ) r (O ) = O ,

_ ( _ ) I ( C ) ( D ) I D . I IPK.J< D)_I_(C) r K.J(C)-L(D) "~- S x (D)(C)IJM . I+ C j (D)P~.M(C) - - ( - - ) I (C) (D) IcMj (c)PK I .M(D)

+p {A)j ( c )SK. (D) (A. 1 __ (_ )I(C )(D )I p( A) j(D)S.IK.(C)(A).1-I-S (a) (C)(D )P~ .j(A) = 0 ,

P(B) (A)J(D)-L(C) - - ( _ ) I ( c ) ( O ) l P(B) A)J(C)_l_(D) q- S(B) (a) (C)(O)lJ- [ -cM j(D)P (B) (A) M(C) -- ( - - ) [(C)D )I c M j ( c ) p


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a n d

S.L Vacaru/NuclearPhysics B 494 [PM] (1997) 590-656D(h) r~(h) t'7 = R (hX , h Y)h Z + D l~ ) hy hZ + ~ D O'DI X ' -" Y } " ' ~ , } ~ l h X . h r } h Z ,

f= lz

( ' > ) / ) ( h ) h 7 = R(vp X, hY )h Z + D (hI D O's) - hZ,D Ix ~ Y } " ~ [v vX ,hr} h Z -Jr- Z IveX, Y)f= lz D (~'j) ,,~ h ZI~ )r~O'')~v} = R (v p X , v p Y )h Z + Z I, , , ,x,, ', ,- ,

f= l

615

( 3 2 )

zD(h) ( t , f )n h ) n ( h ) , , 7 -- R ( h X , h Y ) v p Z + + Z D[hX,hY}OPZ'~IX t .Zy} ~p ~ - - [hX,h y } V p Zf= lz zD( V t ) , -.(h ) . 7 R ( v f X , h Y ) v p Z + v " ,-.(,, ,,) ~ X T M r~o,,,) . .7IX 1)y} U p/ -, = 2 . .. aLIl v fX , hy} VpL + Z_ ~ l v rX , hy}Vpl~ ,

q=l q=lD 0'~) n(~'D ~ r~( t ' )IX ~ Y } v p Z = R ( o q X , v f Y ) v p Z + ~ { v rx , v r y} v p Z .

s = l( 3 3 )

a n d

x ( A ) I K IL - - ( - - ) IK LIx ( A ) IL IK = R( B) ( A) KL X ( B} _ _ T H K L X( A) IH- - R ( B ) K L X ( A ) ( B ) ,

X I I K ( D ) - - ( - - )] K ( D ) I xI . L (D ) [ x = P g . K ( D ) X H - - C H K ( D ) X I I H- - P ( A ) K ( D ) X I ( A ) ,

X I ( B ) ( C ) - - ( - - ) I ( B ) ( C ) I x I i ( c ) & ( B ) = S 'IH .(B )(C )X H - - s ( A ) (B ) (C ) X I i ( A )

x ( A ) I K IL - - ( - - ) IK LIX( A ) IL IK = R{ B) ( A)KL X( B ) - - T H K L X( A ) IH- R ( n ) KL X ( A ) ( B ) ,

x ( A ) I g ( n ) -- ( - - ) I ( n ) g l x ( a ) ( B ) I K = P ( B ) ( a ) K c X C _ c H K ( B ) X ( A ) I H_ p { D ) K ( B )X(A) I ( D ) ,

x ( A ) ( B )A _ (C - - ( - -) I ( C ) ( B ) I x ( A ) { C ) ( B ) = S ( D ) ( A )( B ) (c ) X ( D )

- - s ( D ) ( B )( C ) X ( A ) ( D ) .W e n o t e th a t t h e a b o v e - p r e s e n te d f o r m u l a s [ 1 1 0 ] g e n e r a l i z e f o r h ig h e r o rd e r a n i s o t r o p yt h e s im i l a r o n e s f o r l o c a l l y a n i s o t r o p i c s u p e r sp a c e s [ 1 0 6 ] .

P u t t i n g X = X t ( u ) ~ X (A)Tx' + ( u ) ~ a n d t a k i n g i n t o a c c o u n t t h e l o c a l f o r m o f t h e h -a n d v - c o v a r i a n t s - d er i v a ti v e s a n d ( 2 4 ) , ( 2 5 ) , ( 2 7 ) , ( 2 8 ) w e c a n , r e s p e ct i v el y , e x p r e s si d e n t it i es ( 3 2 ) a n d ( 3 3 ) i n t h e f o l l o w i n g f o rm :


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616 S.I. Vacaru/Nuclear Physics B 494 [PM] (1997) 590-6564 .4 . C a r t a n s t r u c tu r e e q u a t i o n s i n d v s - b u n d l e s

L e t u s c o n s i d e r a d s - t e n s o r f i e l d o n ~ ( z ) :t = t ~ a ) ~ l ~ ( a ).

T h e d - c o n n e c t i o n 1 - f o r m s w / a n d - ( a )o (B ) a r e i n t ro duc ed a sD t = ( D t l a ) )~It~ (A)

w i t hI J ~ . (B), I . = t la ) l jd XJ + t~A)J_(e)~y(O)t ~ a ) = d t ~A ) + w j t ( A ) - - t ( a ) ' ( B )

F o r t h e d - c o n n e c t i o n 1 - f o r m s o f a d - c o n n e c t i o n D o n { ( z ) d e f i n e d b y w ~ a n d U ,(n~'cA) thef o l l o w i n g s t r u c t u r e e q u a t i o n s h o l d :

d ( d 1) - d n A tolH = --$'2, d( t~ (a )) - t~ B) i ~(B)~(A) --- _ j ~ ( A ) ,1 I r/%(A) ~(C ) f ~,(A) __t~(A)d w ~ - t o y A w H = - O J , ~ ( B ) - ~ ( B) ~ ( c ) = "~(~) '

i n w h i c h t h e t o r s i o n 2 - f o r m s s 2 a n d ~ ( A ) a r e g i v e n r e s p e c t i v e l y b y t h e f o r m u l a sf2 t 1 w! aJ d x 1 , -.I , J 8 (c )-~ ~, t J K" i + ~t . . j (C)U i ,

~ ( A ) ~ 2 R ( A ) j K d J / \ dK - -[ - ~ e ( A ) j ( c ) d J / \ ~ ( c ) --[- I s ( A )( B ) ( C )~ ( B ) / \ ~ (C ) 'a n d

$21 = 1 o 1 .4X d H 1 o.1 .~x A 6 (c) 1 c . t ~(a) 8(C)- ~ x j K HU A + -~I j .K(C) t~ "~- ~ , j .K (C ) t , A , R i ! K . A d " l p A ( c l 1 o( o ) = + -2 ( B ) i a ) K ( c ) d K + 2 ( n ) (C)(O) "

W e h a v e d e f i n e d t h e e x t e r i o r p r o d u c t o n s - s p a c e t o s a t i s f y t h e p r o p e r t y8 (~) A 8 (t~) = - ( - ) I -) (~)1 t3(~) A 6 ('~) .

4 .5 . M e t r i c s i n d v s - b u n d l e sT h e b a s e / 9/ o f d v s - b u n d l e g ( z ) i s c o n s i d e r e d t o b e a c o n n e c t e d a n d p a r a c o m p a c t

s - m a n i f o l d .A m e t r i c s t r u c t u r e o n t h e t o ta l s p a c e / ~ ( z ) o f a d v s - b u n d l e g ( z ) i s a s u p e r s y m m e t r i c ,

s e c o n d - o r d e r , c o v a r i a n t s - t e n s o r f i e l dG = G(,~)(#)c~ ('~) 0 (8)

w h i c h i n e v e r y p o i n t u E g ( z ) i s g i v e n b y t h e n o n - d e g e n e r a t e s u p e r s y m m e t r i c m a t r i xG(,~)( /3) = G (O (~ ) , 0 ( ~ )) ( w i t h n o n - v a n i s h i n g s u p e r d e t e r m i n a n t , s d e t G 4= 0 ) .


28/67

S.L Vacaru/Nuclear Physics B 494 [PM] (1997) 590-656 61 7T h e m e t r i c a n d N - c o n n e c t i o n s t r u c tu r e s o n ~ ( z ) a r e c o m p a t i b l e i f t h e r e th e f o l l o w i n g

c o n d i t i o n s a r e m e t :G ( ~ I , a ( A ) ) = 0 , G ( ~ a f , C g m j , )= 0 , Z >~ p > f >~ 1,

o r , i n c o n s e q u e n c e

superstrings in higher order extensions of finsler superspaces

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