  Werner's conformally invariant family of measures on self-avoiding loops on
Riemann surfaces is determined by a single measure $\mu_0$ on self-avoiding
loops in ${\mathbb C} \setminus\{0\}$ which surround $0$. Our first major
objective is to show that the measure $\mu_0$ is infinitesimally invariant with
respect to conformal vector fields (essentially the Virasoro algebra of
conformal field theory). This makes essential use of classical variational
formulas of Duren and Schiffer, which we recast in representation theoretic
terms for efficient computation. We secondly show how these formulas can be
used to calculate (in principle, and sometimes explicitly) quantities (such as
moments for coefficients of univalent functions) associated to the conformal
welding for a self-avoiding loop. This gives an alternate proof of the
uniqueness of Werner's measure. We also attempt to use these variational
formulas to derive a differential equation for the (Laplace transform of) the
"diagonal distribution" for the conformal welding associated to a loop; this
generalizes in a suggestive way to a deformation of Werner's measure
conjectured to exist by Kontsevich and Suhov (a basic inspiration for this
paper).
