Question

My question is very naive maybe, I don't have a deep knowledge about string topology. I wanted to ask (explanation or a reference) for the geometric interpretation of free loop space (continues maps) $\mathrm{Map}(S^{1}, G)$, when G is a topological group (Lie group). Does it classify something geometric?

Thank you.

Answer 1

The free loop space sits in a fibration

$$ \Omega M \to M^{S^1} \to M $$

and in the case where $M = G$ is a Lie-group, I understand the main point of Richard Hepworths paper http://arxiv.org/abs/0905.1199 as saying that the fibration trivializes to a product $G^{S^1} \cong G \times \Omega G$. This triviality then makes it more evident how the string topology operations is a mixture of the two geometric entities: the intersection product of $\mathbb{H}_*(G)$ and the $E_1$-structure on the based loops $\Omega G$.

Answer 2

I want to give you a different answer about the group $LG=map(S^1,G)$. Let us take a $G$-bundle over $M$ (we suppose that $M$ is a smooth manifold): $G\rightarrow E\rightarrow M$, we can apply the loop functor and get a new bundle $$LG\rightarrow LE\rightarrow LM$$ this bundle is a $LG$-bundle.
Many geometric properties of $M$ can be traduced as geometric properties of $LM$ (the important concept here is "transgression"). Let me give one example. If you consider a Riemmanian structure on $M$: $$SO(n)\rightarrow Fr(TM)\rightarrow M$$ it is a $SO(n)$-bundle. Then the free loop bundle $$LFr(TM)\rightarrow LM$$ is a $LSO(n)$-bundle. We can define an orientation bundle for $LM$: $$LFr(TM)\times_{LSO(n)}\mathbb{Z}/2\mathbb{Z}\rightarrow LM.$$ An orientation of $LM$ is a section of this bundle. P. Teichner and S. Stolz have introduced a new structure on this bundle called the fusion product and they have proved that fusion preservind orientations of $LM$ are in bijection with spin structures on $M$ (preprint available on P. Teichner's homepage). You can have a look at K. Waldorf's papers: "Spin structures on loop spaces that characterize string manifolds" and "A Loop Space Formulation for Geometric Lifting Problems".

There is a huge litterature on that subject that involves among other things loop groups, non-abelian cohomology, higher geometric objects like gerbes. A good starting point in this subject could be J.-L. Brylinski's book "Loop spaces, characteristic classes and geometric quantization".

Answer 3

Let us suppose that $G$ is a Lie group. The space of maps $map(S^1,G)$ is a topological group called a loop group, maybe it is better to consider smooth loops, in that case we have an infinite dimensional Lie group. A very nice and classical reference is Pressley, Segal "Loop groups", Oxford Mathematical Monographs.

From the point of view of string topology, I can say a few things:

• R. Hepworth, L. Menichi, and S. Kupers have done some computations (I am very sorry if I have forgotten someone). The computation of the Chas-Sullivan BV-structure of the loop homology is very fun.

• In the case of the Lie group $U(n)$, you can consider spaces of polynomial loops and they have a nice geometric filtration given by the polynomial degree. This filtration is related to Morse theory of loop spaces associated to the energy functional. The Chas-Sullivan BV-structure is compatible with this polynomial filtration.

Computations of these homology groups can be very useful if you want to compute the BV structure of the loop homology of $\mathbb CP^n$ and $S^n$. Chas-Sullivan operations give some informations on closed geodesics you can have a look at Goresky-Hingston's paper "Loop products and closed geodesics" Mark Goresky and Nancy Hingston Duke Math. J. Volume 150, Number 1 (2009), 117-209.