Affine Algebraic Geometry by Gutierrez J., Shpilrain V., Yu J.-T. (eds.)

By Gutierrez J., Shpilrain V., Yu J.-T. (eds.)

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A family Todorov and of a E u c l i d e a n projective ~ that of is s a t i s f i e d the E i n s t e i n Then any fixed (it is of on o n the p a r a m e t e r s of the E i n s t e i n of f o r m s be d e f i n e d . 13) ~ (~] (2,2). = O to a t r a n s f o r m a t i o n to b e f i x e d , addition the a f f i n e d ~ not depend ~ involution If in if we have with an solution for , with kernels ~ Q and ~ ~ 0 , so t h a t Euclidean satisfying complex ~ is i n d u c e d . e straightforward. let us equation (i,i) ~,~c ~ to the p a r a m e t e r where form.

A holomorphic Yang-Mills f i e l d , on a domain U c ~ M is by d e f i n i t i o n a pair consis- t i n g of a holomorphic vector bundle E over U and a holomorphic connection on i t , defined by the covariant d e r i v a t i v e ~ : E ~ E ®1")_1 . ) to consider also i t s The curvature from F of the connection V , the components of which are j u s t the strengths of the Yang-Mills f i e l d , is a section of the bundle is: ~ End E ®~r'z2. The f i e l d equation, f o l lo w in g from the Yang-Mills Lagrangian F = j , where j ~ End E ® ~ 3 is the a x ial current, which can be f o r example constructed using sections of the matter bundle.

9] Witten, Phys. Lett. 77B (1978), 394-397. , Phys. Lett. 78B (1978), 462-464. , in: Complex Manifold Techniques in Theor. , Pitman, London (1979), 180-206. , in: Proc. Zvenigorod Symposium on Group Theory in Physics, Nauka, 1980. , in: Sovremennye Problemy Matematiki 17 (1980), VlNITI. , Doklady AN USSR, 255 (1980), 256. B. 49 Hungary i. BASIC PHILOSOPHY T h e r e h a v e been i n s t a n c e s in the past of p a r t i c l e p h y s i c s w h e n the v a l i d - ity of some f u n d a m e n t a l l a w of n a t u r e was v i c t o r i u s l y r e a f f i r m e d and s p e c u l a t i o n s d o u b t i n g that law w e r e put aside.

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