  Many classes of non-linear sigma models (NLSMs) are known to contain
composite operators with an arbitrary number 2s of derivatives ("high-gradient
operators") which appear to become strongly relevant within RG calculations at
one (or fixed higher) loop order, when the number 2s of derivatives becomes
large. This occurs at many conventional fixed points of NLSMs which are
perturbatively accessible within the usual epsilon-expansion in d=2+\epsilon
dimensions. Since such operators are not prohibited from occurring in the
action, they appear to threaten the very existence of such fixed points. At the
same time, for NLSMs describing metal-insulator transitions of Anderson
localization in electronic conductors, the strong RG-relevance of these
operators has been previously related to statistical properties of the
conductance of samples of large finite size ("conductance fluctuations"). In
this paper, we analyze this question, not for perturbative RG treatments of
NLSMs, but for 2d Wess-Zumino-Witten (WZW) models at level k, perturbatively in
the current-current interaction of the Noether current. WZW models are special
("Principal Chiral") NLSMs on a Lie Group G with a WZW term at level k. In
these models the role of high-gradient operators is played by homogeneous
polynomials of order 2s in the Noether currents, whose scaling dimensions we
analyze. For the Lie Supergroup G=GL(2N|2N) and k=1, this corresponds to
time-reversal invariant problems of Anderson localization in the so-called
chiral symmetry classes, and the strength of the current-current interaction, a
measure of the strength of disorder, is known to be completely marginal (for
any k). We find that all high-gradient (polynomial) operators are, to one loop
order, irrelevant or relevant depending on the sign of that interaction.
