  John Lott defined an integer-valued signature $\sigma_{S^1}(M)$ for the orbit
space of a compact orientable manifold with a semi-free $S^1$-action but he did
not construct a Dirac-type operator which has this signature as its index. We
construct such operator on the orbit space and we show that it is essentially
unique and that its index coincides with Lott's signature, at least when the
stratified space satisfies the so-called Witt condition. For the non-Witt case,
this operator remains essentially self-adjoint (in contrast to the Hodge
de-Rham operator) and it has a well defined index which we conjecture will also
compute $\sigma_{S^1}(M)$. This article is a condensed version of the original
author's PhD Thesis where the theory of induced Dirac-Schr\"odinger-type
operators is developed in detail.
