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**Extra resources for Algebraic Geometry - Bowdoin 1985, Part 2**

**Example text**

Following : if the manifold vector fields Da = e m a m ~ M In this example we point out the is endowed with a Riemannian metric g and if the e a = eamdx TM) (which are the duals of the vielbein forms are chosen to be orthogonal with respect to g, the indices a and m transform with different groups, namely the orthogonal and the general linear groups: 51 /' where ( -~ 0 c 0(n,R) 0 and the prepotentials ~ L )" m. L, L e GL(n,~). ~,~" ~ L" ,~. 52)). 3 (i) Geometric framework of SYM-theories 154) Generalities As internal syrm~etry group we consider a c~mpact matrix Lie group algebra % and suppose that its (hermitean) The corresponding structure constants f qrs generators Tr are represented by G with Lie are normalized by are given by and are assumed to be totally antisymmetric.

61), chap. -4 (Tr in the adj. 63)). to indicate explicitly the connection with respect to which the covariant derivative is defined. 22b) graded c y c l i c permutations of (A,B,C) Matter f i e l d s are described by m u l t i p l e t s of s u p e r f i e l d s definition depends on the representation chosen for tive of ~ A ; ¢. 1 whose p r e c i s e the YM-covariant deriva- is defined by (Here the group generators of the Lie algebra ~). 26b) for the ~-valued fields Q and the matter fields speaking the covariant derivatives ~DA ~ (Strictly respectively.

39). e. on a manifold (pseudo-) Riemannian metric g. M which is equipped with some Thus it seems natural to attempt a generalization of this framework to supersymmetric theories and to introduce some supermetric on rigid superspace body manifold SP/L ~4. in order to "extend" the Minkowski metric of the underlying Such a metric has actually been proposed shortly after the invention of superspace 159)'129)', thereafter it was considered in particular by R. Arnowitt and P. Nath as a vacuum solution in their Riemannian approach to supergravity ~).