Speaker
Mikhail Kapranov
(Kavli IPMU)
Description
Differential operators of infinite order (DOI) are series in derivatives
which converge and define local operators of analytic functions. For
example, $\exp(d/dx)$ is not allowed but $\cos\sqrt{d/dx}$ is. Starting
from 1972, Sato, Kashiwara, Kawai, Takei, Yoshida and others developed a
characterization of $\theta(T)$, the Riemann theta-zerovalue, by a system
of DOI in modular variables alone, thus deducing modularity from local
properties.
In the talk I will explain how supersymmetry gives rise to DOI and how a
supersymmetric thickening of the Siegel upper half plane gives a natural
construction of the modular invariant system of DOI for $\theta(T)$.