We analyze the structure of multiloop supergraphs contributing to the effective Lagrangians in 4d supersymmetric gauge theories and in the models obtained from them by dimensional reduction. When d = 4 , this gives the renormalization of the effective charge. For d < 4 , the low-energy effective Lagrangian describes the metric on the moduli space of classical vacua. These two problems turn out to be closely related. In particular, we establish the relationship between the 4d nonrenormalization theorems (in minimal and extended supersymmetric theories) and their low-dimensional counterparts.
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We are indebted to Misha Shifman and Misha Voloshin for illuminating discussions and to Marc Grisaru for useful correspondence. A.S. acknowledges kind hospitality extended to him during his frequent visits to TPI, where this work was mostly done. The work of A.V. was supported in part by DOE grant DE-FG02-94ER408.