### Mathematical loop space literature

#### Posted by Urs Schreiber

Readers of my writings will have noticed that I mention the tem *loop space* from time to time.

Now, I have to admit that my understanding of loop space is very much that of a physicists, not that of a mathematician. My main tool to deal with the subtleties of the very concept of loop space is that I know how the exterior geometry on loop space is related to 2d superconformal field theory (by way of switching from Heisenberg to Schrödinger QFT picture), which is well understood.

Even though I have the suspicion that this, from the mathematical point of view rather unsophisticated, attitude is actually quite useful, my recent occupation with nonabelian connections on loop space indicates that I should try to get a more high-brow understanding of this space.

In particular, thinking about flat connections on loop space - and my claim is that a nonabelian 2-form on target space does correspond to a flat connection on loop space (which also follows from work by Girelli and Pfeiffer) - made me wonder about contractibility of paths in loop space.

A torus in target space is a closed path in loop space. This closed path can certainly be contracted to a point in loop space (this point corresponding to a constant function from the circle into target space) by shrinking the torus to a point.

But there are several closed paths in loop space that correspond to the same torus in target space, one corresponding to every foliation of that torus by circles.

All of these paths can be deformed into each other, but not all can be deformed into each other without using the constant path (in loop space). Namely the ‘horizontal’ and the ‘vertical’ slicings of the torus, corresponding to its 2 1-homotopy classes, can apparently never be continuously deformed into each other without using the ‘vanishing torus’.

Is that right?

See, that’s the level of understanding of global properties of loop space that I am currently sitting at…

For my own good I’ll list a couple of math books on loop space that certainly contain the answer to this and many other questions:

Jean-Luc Brylinski: Loop spaces, characteristic classes, and geometric quantization (1993)

John Frank Adams Infinite Loop Spaces (1978)

Hans Baues: Geometry of Loop Spaces and the Cobar Construction (1980)

L. Feher, A. Stipsicz: Topological Quantum Field Theories & Geometry of Loop Spaces

(1992)

- Sergeev, A. G. [Ed.] Loop spaces and groups of diffeomorphisms (1997)

Posted at July 15, 2004 3:00 PM UTC
## Re: Mathematical loop space literature

Hi Urs,

The basic mathematical fact is that

\pi_1(\Omega M)=\pi_2(M)

so if the second homotopy group of your space is trivial, then you can contract a loop in your loop space, otherwise not.

Here \Omega M is based loops.

For the basics of homotopy and cohomology, Bott and Tu “Differential Forms in Algebraic Topology” is hard to beat.