Local algebra and string theory

The $\beta \gamma$ system on the cone of pure spinors is an integral part of the string theory developed by N. Berkovits. This $\beta \gamma$ system offer a number of questions for pure mathematicians: what is a precise definition of the space of states of the theory? Is there a mathematical explanations for various dualities (or pairings) predicted by physicists? Can a formula for partition function be written? To help me to answer these questions I use local algebra in an essential way.