String topology for a Lie group   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.
 A: 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.    
A: 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 $\Omega G$.
A: 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".  
