  We construct a non-chiral current algebra in two dimensions consistent with
conformal invariance. We show that the conformal current algebra is realized in
non-linear sigma-models on supergroup manifolds with vanishing dual Coxeter
number, with or without a Wess-Zumino term. The current algebra is computed
using two distinct methods. First we exploit special algebraic properties of
supergroups to compute the exact two- and three-point functions of the currents
and from them we infer the current algebra. The algebra is also calculated by
using conformal perturbation theory about the Wess-Zumino-Witten point and
resumming the perturbation series. We also prove that these models realize a
non-chiral Kac-Moody algebra and construct an infinite set of commuting
operators that is closed under the action of the Kac-Moody generators. The
supergroup models that we consider include models with applications to
statistical mechanics, condensed matter and string theory. In particular, our
results may help to systematically solve and clarify the quantum integrability
of PSU(n|n) models and their cosets, which appear prominently in string
worldsheet models on anti-deSitter spaces.
