2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61372Let $M$ be a $Spin$-manifold with $S^1$-action and let $σ\in S^1$ be of finite order. We show that the indices of certain twisted Dirac operators vanish if the action of $σ$ has sufficiently large fixed point codimension. These indices occur in the Fourier expansion of the elliptic genus of $M$ in one of its cusps. As a by-product we obtain a new proof of a theorem of Hirzebruch and Slodowy on involutions.9 pages, revised versionGeometric TopologyAlgebraic Topology57S15, 57R20 (Primary) 19L47, 57S25 (Secondary)text