Supermoduli Workshop: May 18 – 22, 2015

Organized by Ron Donagi, Samuel Grushevsky, Sheldon Katz, and Edward Witten

Dates: May 18 – 22, 2015

Attendee List Download Talk ScheduleView Videos
The goal of the workshop is to catalyze rapid development of the extension to super Riemann surfaces of the theory of Riemann surfaces and their moduli. We plan to bring together mathematicians and physicists working on Riemann surfaces and their moduli, string amplitudes, and supergeometry.

The workshop will feature lecture courses by:
Pierre Deligne: Institute for Advanced Study
Eric D’Hoker: University of California, Los Angeles
Ron Donagi: University of Pennsylvania
Edward Witten: Institute for Advanced Study

Main topics will include: supermanifolds, supersymmetry, super Riemann surfaces, moduli spaces, Deligne-Mumford compactifications, novel issues related to Ramond punctures, super Mumford isomorphisms, super period matrices, the D’Hoker-Phong work in genus 2, recent explicit results in genus 3, and how these all fit into perturbative superstring theory: the measure on super moduli space, the Neveu-Schwarz and Ramond sectors, the propagator, BRST anomalies, spacetime supersymmetry, and the spontaneous breaking of supersymmetry at 1 loop.

Superstring perturbation theory expresses scattering amplitudes as integrals over the moduli of punctured super Riemann surfaces. The moduli space of super Riemann surfaces has many analogies with the moduli space of ordinary Riemann surfaces. For example, there is a super analog of the Deligne-Mumford compactification, and it is important in understanding the qualitative properties of superstring scattering amplitudes. There is also a superanalog of the Mumford isomorphism between certain line bundles over moduli space, and it plays a role analogous to the role that the ordinary Mumford isomorphism plays in bosonic string theory.

Integration over supermoduli space gives a powerful framework for understanding important properties of string theory such as spacetime supersymmetry. But in practice the understanding of superstring perturbation theory via super Riemann surfaces has been relatively little-developed, beyond elementary examples in genus 0 and 1. Only in the early 2000s did D’Hoker and Phong succeed in determining some genus 2 amplitudes explicitly, by exploiting the super period matrix of a super Riemann surface. Likewise there has been only limited mathematical work on super Riemann surfaces in recent years, one recent result being that despite the analogies between them, super moduli space cannot be reduced to the ordinary moduli space in the sense that there is no holomorphic projection from super moduli space to its underlying reduced space.

More generally, a goal of the workshop is to further develop the field of super Riemann surfaces, requiring the development of appropriate algebro-geometric foundations, exploration of the most important open problems, and connecting with the known physics results. This will be the first workshop in this important emerging field.

[box, type=”download”]A video introduction by Ron Donagi: Watch the Video.
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