The Dirac-Ramond operator in string theory and loop space index theorems

The index of the Direc-Ramond operator is computed and analyzed. It is shown to be the extension of the Atiyah-Singer index theorem for loop space. It can also be seen as a generating function for the Atiyah-Singer index for the states of the string. Its existence depends on the Green-Schwarz anomaly cancellation condition, p₁ (M) = 0, which defines an analog of a spin structure for the loop space. One also finds topological invariants for the loop space which correspond to different twistings of the Dirac-Ramond operator. All of them can be expressed in terms of Jacobi elliptic functions.