String topology for a Lie group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T00:32:04Z http://mathoverflow.net/feeds/question/113853 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/113853/string-topology-for-a-lie-group String topology for a Lie group Fedotov 2012-11-19T18:14:37Z 2012-11-27T00:48:57Z <p>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? </p> <p>Thank you.</p> http://mathoverflow.net/questions/113853/string-topology-for-a-lie-group/113855#113855 Answer by David C for String topology for a Lie group David C 2012-11-19T18:41:01Z 2012-11-19T18:41:01Z <p>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.</p> <p>From the point of view of string topology, I can say a few things:</p> <ul> <li><p>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.</p></li> <li><p>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.</p></li> </ul> <p>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. </p> http://mathoverflow.net/questions/113853/string-topology-for-a-lie-group/113914#113914 Answer by David C for String topology for a Lie group David C 2012-11-20T06:52:54Z 2012-11-20T06:52:54Z <p>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.<br> 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 <em>fusion product</em> 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". </p> <p>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". </p> http://mathoverflow.net/questions/113853/string-topology-for-a-lie-group/114615#114615 Answer by Tarje Bargheer for String topology for a Lie group Tarje Bargheer 2012-11-27T00:48:57Z 2012-11-27T00:48:57Z <p>The free loop space sits in a fibration</p> <p>$$\Omega M \to M^{S^1} \to M$$</p> <p>and in the case where $M = G$ is a Lie-group, I understand the main point of Richard Hepworths paper <a href="http://arxiv.org/abs/0905.1199" rel="nofollow">http://arxiv.org/abs/0905.1199</a> 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 $\Omega G$.</p>