By Bloch S. (ed.)

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**Extra info for Algebraic Geometry - Bowdoin 1985, Part 1**

**Sample text**

Wess and his collaborators 153'65)", with a different emphasis from ours these topics are also discussed in refs 57), 66), 128), 155). I The original approach and the WZ-supergauge This chapter ~) is intended for motivation and orientation; one of the aims of the systematic study presented in the following chapters will be to recover the formulation given here from a more geometric approach. Originally 152'51) SYM-theories have been developped in the gauge chiral representation where the matter fields are described by a multiplet fields q~l''''' ~N valued superfield ~ of chiral super- whereas the gauge field is represented by a real Lie algebraV = v(r)T r (T r = generator of the internal symmetry group); the latter contains a large number of component fields usually denoted by X~, M, N, Vm, % , D.

10) where These transformations do not really define a representation of the susy algebra, but only a "representation modulo an ordinary gauge transformation with field-dependent parameter ~,, 152,40,128). 11) ~)A more detailed discussion can be found in refs 130), 57). Note that this situation is analogous to the one encountered in electrodynamics when the non Lorentzcovariant Coulomb gauge is chosen and Lorentz transformations are considered. 10) from a different point of view and analyse the structure of the associated BRS differential algebra.

80) it is often more convenient to define these fields by projecting the superfield # to space-time with the left-invariant differential operators D , D& : 42 Here ~1 ~ ~(x,0,0). Since {D , D~} ~ ~ ~m the so-defined component fields are not all independent of each other and one has to determine the independent ones while taking into account eventual constraints on ~. The first few of these space-time fields coincide with those of the @-expansion, but the higher components differ by space-time derivatives of lower ones ; this corresponds to a fieldredefinition without physical significance.