Witten gave a nonrigorous proof of a remarkable index theorem concerning the Dirac operator on an infinite-dimensional loop space. This discovery is one of the big motivations for introducing the String structures and for search for geometric interpretation of elliptic cohomology.
String Coffee Table blog
E. Witten, The index of the Dirac operator in loop space, Proc. of Conf. on Elliptic Curves and Modular Forms in Algebraic Topology, Princeton, N.J., Sep 1986.
E. Witten, Elliptic genera and quantum field theory, Commun.Math.Phys. 109:525, 1987