Loop Groups and Diffeomorphism Groups of the Circle as Colimits

Communications in Mathematical Physics, Mar 2019

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.

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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 AuthorsAuthors and affiliations André Henriques Open Access Article First Online: 06 March 2019 77 Downloads 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 Download to read the full article text Notes 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. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. References BDH15. Bartels, A., Douglas, C.L., Henriques, A.: Conformal nets I: Coordinate-free nets. Int. Math. Res. Not. 2015(13), 4975–5052 (2015)Google Scholar BE98. Böckenhauer, J., Evans, D.E.: Modular invariants, graphs and \({\alpha}\)-induction for nets of subfactors. Commun. Math. Phys. 197(2), 361–386 (1998).Google Scholar Bot77. Bott R.: On the characteristic classes of groups of diffeomorphisms. Enseign. Math. 23(3–4), 209–220 (1977)MathSciNetzbMATHGoogle Scholar BSM90. Buchholz D., Schulz-Mirbach H.: Haag duality in conformal quantum field theory. Rev. Math. Phys. 2(1), 105–125 (1990)MathSciNetCrossRefzbMATHGoogle Scholar Car04. Carpi S.: On the representation theory of Virasoro nets. Commun. Math. Phys. 244(2), 261–284 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar CDVIT18. Carpi, S., Del Vecchio, S., Iovieno, S., Tanimoto, Y.: Positive energy representations of sobolev diffeomorphism groups of the circle. arXiv:1808.02384 (2018) CW05. Carpi S., Weiner M.: On the uniqueness of diffeomorphism symmetry in conformal field theory, Comm. Math. Phys. 258(1), 203–221 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar CW16. Carpi, S., Weiner, M.: Local energy bounds and representations of conformal nets, Unpublished(2016)Google Scholar DFK04. D’Antoni C., Fredenhagen K., Köster S.: Implementation of conformal covariance by diffeomorphism symmetry. Lett. Math. Phys. 67(3), 239–247 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar DVIT18. Del Vecchio, S., Iovieno, S., Tanimoto, Y.: Solitons and nonsmooth diffeomorphisms in conformal nets. arXiv:1811.04501 (2018) GF68. Gel’fand I.M., Fuks D.B.: Cohomologies of the Lie algebra of vector fields on the circle. Funkcional. Anal. i Priložen. 2(4), 92–93 (1968)MathSciNetzbMATHGoogle Scholar GF93. Gabbiani F., Fröhlich J.: Operator algebras and conformal field theory. Commun. Math. Phys. 155(3), 569–640 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar GW84. Goodman R., Wallach N.R.: Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle. J. Reine Angew. Math. 347, 69–133 (1984)MathSciNetzbMATHGoogle Scholar Hen15. Henriques, A.: What Chern–Simons theory assigns to a point.arXiv:1503.06254 (2015) Hen16. Henriques, A.: The classification of chiral WZW models by \({{H}^4_+({B}{G},\mathbb{Z})}\), Contemporary Mathematics (2016)Google Scholar Hen17. Henriques, A.: Bicommutant categories from conformal nets, (2017) arXiv:1701.02052 Kac90. Victor G.K.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)Google Scholar Kaw02. Kawahigashi Y.: Generalized Longo-Rehren subfactors and \({\alpha}\)-induction. Comm. Math. Phys. 226(2), 269–287 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar KLM01. Kawahigashi Y., Longo R., Müger M.: Multi-interval subfactors and modularity of representations in conformal field theory. Comm. Math. Phys. 219(3), 631–669 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar KW09. Khesin, B., Wendt, R.: The geometry of infinite-dimensional groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge.In: A Series of Modern Surveys in Mathematics. Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics, vol. 51, Springer, Berlin (2009)Google Scholar Lok94. Loke, T.: Operator algebras and conformal field theory of the discrete series representations of Diff(s 1), Ph.D. thesis, Trinity College, Cambridge (1994)Google Scholar LR95. Longo, R., Rehren, K.-H.: Nets of s (...truncated)


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André Henriques. Loop Groups and Diffeomorphism Groups of the Circle as Colimits, Communications in Mathematical Physics, 2019, pp. 537-565, Volume 366, Issue 2, DOI: 10.1007/s00220-019-03394-8