# 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.

- Introduction
- Lectures1-2 5th and 7th February. v9 8 February, 2013 -- corrections from Chris on nb incorporated. Log of changes
- Lectures3-4 12th and 14th February. v4 Extended Toeplitz started
- On Tuesday 5th Feb I got as far as subsect 1.11 -- I will start again with this on Thursday, join of paths and fusion condition on orientations on the loop space for an oriented, but not necessarily spin, manifold.

- Appendix 13 February, 2013. v1 Moved the general description of compact manifolds to an Appendix
- References

Basic facts:

- Lectures: TH 9:30-11 in 2-151.
- No, there are no exams, tests or anything. I welcome comments before, during or after lectures on the notes, the lectures or whatever. I will have at least one exercise in each lecture -- I would be very happy if anyone submits solutions to me!

*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.

- Overview: Orientation/Spin/String and the Whitehead Tower; Dirac operators, index theory and cohomology theories
- Loop spaces as Fréchet manifolds
- Fusion structures
- Central extensions and bundle gerbes
- Fusion orientation and spin
- Fusion line bundles and gerbes
- Loop groups and central extensions
- String structures and the first spin-Pontryagin class
- Fusion loop-spin structures
- Loop-spin representation
- Dirac-Ramond operators
- Witten genus

- A. Pressley and G. Segal, Loop groups, The Clarendon Press, Oxford University Press, 1986
- Brylinski90
- Murray1
- Witten-loop
- Segal-Bourbaki
- Stolz-Teichner2005
- Redden2011
- Waldorf2012
- McLaughlin92
- Stevenson2001