We study various aspects of supergeometry, including obstruction,

Atiyah, and super-Atiyah classes. This is applied to the geometry of the

moduli space of super Riemann surfaces and to its Deligne-Mumford compactification. For genus greater than or equal to 5, this moduli space is not projected (and in particular it is not split): it cannot be holomorphically projected to its underlying reduced manifold. Physically, this means that certain approaches to superstring perturbation theory that are very powerful in low orders have no close analog in higher orders. Mathematically, it means that the moduli space of super Riemann surfaces cannot be constructed in an elementary way starting with the moduli space of ordinary Riemann surfaces. It has a life of its own.

When we examine the Deligne-Mumford compactification of moduli space, and especially the Ramond boundary divisors, we find that the interesting new phenomena start already in genus 1. This is interpreted as the mechanism that allows supersymmetry to remain unbroken at tree level in superstring perturbation theory, but to be spontaneously broken at one loop.

(Joint work with E. Witten)

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