18.158: Analysis on loop spaces

Richard Melrose

This is the home page for the course 18.158 for Spring 2013.

Lecture notes: All.pdf 5 February, 2013.

Basic facts:

I plan to develop what I believe are the basic tools needed for analysis on the loop space of a manifold -- the space of smooth maps from the circle into the manifold. Loop spaces are very special examples of Fréchet manifolds but unlike the general case do seem to be amenable to analysis, although not much has actually been done. By the end of the semester I hope to have defined the Dirac-Ramond operator corresponding to a string structure on a manifold and with luck will have started to analyse it. Here is a rough list of topics, each requiring a lecture or two. Things may evolve a little when I get going. I plan to write up lecture notes as I go along but could do with some help if anyone wants to volunteer.
  1. Overview: Orientation/Spin/String and the Whitehead Tower; Dirac operators, index theory and cohomology theories
  2. Loop spaces as Fréchet manifolds
  3. Fusion structures
  4. Central extensions and bundle gerbes
  5. Fusion orientation and spin
  6. Fusion line bundles and gerbes
  7. Loop groups and central extensions
  8. String structures and the first spin-Pontryagin class
  9. Fusion loop-spin structures
  10. Loop-spin representation
  11. Dirac-Ramond operators
  12. Witten genus