By Wallwork A.
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GRISARU L = 02[(D2~}pJ = [(0 2 D2~)~ + 2(OaD2~)(Oa~) + 02~D2~JI8=0 • But, on an antichiral superfield, using the anticommutation relations so that L -1/2 1/J a aaä 1/Jä + (0 a)a + Tf which, rewritten in terms of A,B,1/J,F,G gives the usual kinetic Lagrangian of the scalar multiplet. This procedure is faster, and less apt to lead to algebraic mistakes than the one which explicitly substitutes the forms of ~ and ~ and evaluates the 8-integrals. The r~st requires very simple algebra. ,... 2 , ... 02 ...
Then one insists that further transformations should keep us in this gauge. This is a restriction on A which eliminates almost all of the unwanted part of the group. What is left are first of all some transformations which could be used (if so desired) to take the remaining Um to a Wess-Zumino type gauge in which the first few components are zero. In general, care must be taken to define the remaining components so that they transform covariantly under the residual symmetries. These consist of local space-time and supersymmetry transformations (in the sense of component supergravity) and superscale (conformal) transformations.
In general the above integral is the only invariant one can construct. An exception is when the integrand is purely chiral or antichiral. In that case the above integral is zero: We have used the important fact that the difference between d and D is a total derivative which we can drop upon x-integration. Then anihilates the chiral integrand it acts upon. (Another way of seeing the same result is to realize that any chiral expression can be written in the form of e-iUO times an expression depending only on 8.
Business Options: Workbook by Wallwork A.