Abstract: The path integral formula for the index of the Dirac operator can be interpreted as a localization formula for U(1)-equivariant cohomology of the free loop space of the manifold. In this lecture I shall first recall the Cartan model of equivariant differential forms of a finite dimensional manifold and the localization formula of Berline-Vergne. We shall then see that the loop space analogue of this result will give the A hat genus. This can be regarded as the bosonic component of the index formula. The corresponding localization formula in the supersymmetric case gives the full index formula.