Loop Groups and Diffeomorphism Groups of the Circle as Colimits
Communications in Mathematical Physics
March 2019, Volume 366, Issue 2, pp 537–565 | Cite as
Loop Groups and Diffeomorphism Groups of the Circle as Colimits
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André Henriques
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First Online: 06 March 2019
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Abstract
In this paper, we show that loop groups and the universal cover of \({{\rm Diff}_+(S^1)}\) can be expressed as colimits of groups of loops/diffeomorphisms supported in subintervals of S1. Analogous results hold for based loop groups and for the based diffeomorphism group of S1. These results continue to hold for the corresponding centrally extended groups. We use the above results to construct a comparison functor from the representations of a loop group conformal net to the representations of the corresponding affine Lie algebra. We also establish an equivalence of categories between solitonic representations of the loop group conformal net, and locally normal representations of the based loop group.
Communicated by Y. Kawahigashi
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Acknowledgements
We thank Sebastiano Carpi for many useful discussions and references. This research was supported by the ERC Grant No. 674978 under the European Union’s Horizon 2020 research innovation programme.
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