Appears in collection : Mathematics Inspired by Physics

Differential operators of infinite order (DOI) are infinite series in derivatives with holomorphic coefficients decaying so fast that the action on holomorphic functions converges and preserves the domain of definition. Thus $\exp(d/dx)$ (shift operator) is not a DOI but $\cos(\sqrt{d/dx})$ is. In 1973 M. Sato gave a characterization of theta-zerovalues by a manifestly modular invariant system of DOI in the modular variables alone, thus deducing modularity from local conditions. This has been developed by several authors (Kashiwara, Kawai, Takei, Yoshida and others) since. I will present a "supersymmetric" approach to Sato's theory based on two observations: The exponential of any odd supersymmetry generator is a DOI. In some cases such odd generators, acting "on-shell" (in the space of solutions of equations of motion), satisfy even-style commutation relations.