  We present a higher index theorem for a certain class of etale
one-dimensional complex-analytic groupoids. The novelty is the use of the local
anomaly formula established in a previous paper, which represents the bivariant
Chern character of a quasihomomorphism as the chiral anomaly associated to a
renormalized non-commutative chiral field theory. In the present situation the
geometry is non-metric and the corresponding field theory can be renormalized
in a purely conformal way, by exploiting the complex-analytic structure of the
groupoid only. The index formula is automatically localized at the automorphism
subset of the groupoid and involves a cap-product with the sum of two different
cyclic cocycles over the groupoid algebra. The first cocycle is a trace
involving a generalization of the Lefschetz numbers to higher-order fixed
points. The second cocycle is a non-commutative Todd class, constructed from
the modular automorphism group of the algebra.
