We investigate 2d sigma-models with a 2+N dimensional Minkowski signature
target space metric and Killing symmetry, specifically supersymmetrized, and
see under which conditions they might lead to corresponding exact string vacua.
It appears that the issue relies heavily on the properties of the vector
Mμ​, a reparametrization term, which needs to possess a definite form for
the Weyl invariance to be satisfied. We give, in the n=1 supersymmetric
case, two non-renormalization theorems from which we can relate the u
component of Mμ​ to the βuuG​ function. We work out this (u,u)
component of the βG function and find a non-vanishing contribution at
four loops. Therefore, it turns out that at order α′4, there
are in general non-vanishing contributions to Mu​ that prevent us from
deducing superstring vacua in closed form.Comment: 9 pages, latex, CERN-TH.6946/9