# American Institute of Mathematical Sciences

January  2012, 19: 58-76. doi: 10.3934/era.2012.19.58

## Integration of exact Courant algebroids

 1 Department of Mathematics, University of Toronto, 40 St George Street, Toronto, Ontario M4S2E4, Canada 2 Department of Mathematics, Université de Genève, Geneva, Switzerland

Received  February 2011 Revised  January 2012 Published  June 2012

In this paper, we describe an integration of exact Courant algebroids to symplectic 2-groupoids, and we show that the differentiation procedure from [32] inverts our integration.
Citation: David Li-Bland, Pavol Ševera. Integration of exact Courant algebroids. Electronic Research Announcements, 2012, 19: 58-76. doi: 10.3934/era.2012.19.58
##### References:
 [1] C. Arias Abad and M. Crainic, Representations up to homotopy of Lie algebroids,, 2009. Available from: \url{http://arxiv.org/pdf/0901.0319v2}., (). Google Scholar [2] C. Arias Abad and F. Schaetz, The $A_\infty$ de Rham theorem and integration of representations up to homotopy,, 2010. Available from: \url{http://arxiv.org/pdf/1011.4693}., (). Google Scholar [3] M. Artin and B. Mazur, On the van Kampen theorem,, Topology, 5 (1966), 179. Google Scholar [4] C. Blohmann, M. C. B. Fernandes and A. Weinstein, Groupoid symmetry and constraints in general relativity,, March, (2010). Google Scholar [5] H. Bursztyn and A. Cabrera, Multiplicative forms at the infinitesimal level,, (2010), (2010), 1. Google Scholar [6] H. Bursztyn, A. Cabrera and C. Ortiz, Linear and multiplicative 2-forms,, Letters in Mathematical Physics, 90 (2009), 59. Google Scholar [7] H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets,, Duke Mathematical Journal, 123 (2004), 549. Google Scholar [8] A. S. Cattaneo, Integration of twisted Poisson structures,, Journal of Geometry and Physics, 49 (2004), 187. Google Scholar [9] A. S. Cattaneo, B. Dherin and A. Weinstein, Symplectic microgeometry I: Micromorphisms,, The Journal of Symplectic Geometry, 8 (2010), 205. Google Scholar [10] A. M. Cegarra and J. Remedios, The relationship between the diagonal and the bar constructions on a bisimplicial set,, Topology and its Applications, 153 (2005), 21. doi: 10.1016/j.topol.2004.12.003. Google Scholar [11] M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes,, Commentarii Mathematici Helvetici, 78 (2003), 681. Google Scholar [12] M. Crainic and R. L. Fernandes, Integrability of Lie brackets,, Annals of Mathematics (2), 157 (2003), 575. Google Scholar [13] M. Crainic and R. L. Fernandes, Integrability of Poisson brackets,, Journal of Differential Geometry, 66 (2004), 71. Google Scholar [14] P. G. Goerss and J. F. Jardine, "Simplicial Homotopy Theory," Reprint of the 1999 edition,, Modern Birkhäuser Classics, (2009). Google Scholar [15] A. Gracia-Saz and R. A. Mehta, Lie algebroid structures on double vector bundles and representation theory of Lie algebroids,, Advances in Mathematics, 223 (2010), 1236. Google Scholar [16] A. Gracia-Saz and R. A. Mehta, VB-groupoids and representation theory of Lie groupoids,, (2011), (2011), 1. Google Scholar [17] A. Henriques, Integrating $L_\infty$-algebras,, Compositio Mathematica, 144 (2008), 1017. Google Scholar [18] D. Iglesias Ponte, C. Laurent-Gengoux and P. Xu, Universal lifting theorem and quasi-Poisson groupoids,, (2005), (2005), 1. Google Scholar [19] D. Kochan, Differential gorms and worms,, in, (2005), 128. Google Scholar [20] M. Kontsevich, Deformation quantization of Poisson manifolds,, Letters in Mathematical Physics, 66 (2003), 157. Google Scholar [21] Y. Kosmann-Schwarzbach, Quasi, twisted, and all that$\ldots$in Poisson geometry and Lie algebroid theory,, in, (2005), 363. Google Scholar [22] K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids,, Duke Mathematical Journal, 73 (1994), 415. doi: 10.1215/S0012-7094-94-07318-3. Google Scholar [23] K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids,, Topology, 39 (2000), 445. doi: 10.1016/S0040-9383(98)00069-X. Google Scholar [24] R. A. Mehta and X. Tang, From double Lie groupoids to local Lie 2-groupoids,, Bulletin of the Brazilian Mathematical Society (New Series), 42 (2011), 651. Google Scholar [25] J. W. Milnor, Microbundles. I,, Topology, 3 (1964), 53. Google Scholar [26] D. Roytenberg, "Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds,", Ph.D. thesis, (1999). Google Scholar [27] D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, in, 315 (2002), 169. Google Scholar [28] D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds,, Letters in Mathematical Physics, 61 (2002), 123. Google Scholar [29] D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie algebras,, Letters in Mathematical Physics, 46 (1998), 81. Google Scholar [30] P. Ševera, "Letters to A. Weinstein.", Available from: \url{http://sophia.dtp.fmph.uniba.sk/~severa/letters/}., (). Google Scholar [31] P. Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one,, in, (2005), 121. Google Scholar [32] P. Ševera, $L_\infty$-algebras as first approximations,, in, 956 (2007), 199. Google Scholar [33] P. Ševera and A. Weinstein, Poisson geometry with a 3-form background,, Progress of Theoretical Physics Suppl., 144 (2001), 145. Google Scholar [34] Y. Sheng and C. Zhu, Higher extensions of lie algebroids and application to courant algebroids,, 2011. Available from: \url{http://arxiv.org/pdf/1103.5920}., (). Google Scholar [35] Y. Sheng and C. Zhu, Semidirect products of representations up to homotopy,, Pacific Journal of Mathematics, 249 (2011), 211. Google Scholar [36] P. Xu, On Poisson groupoids,, International Journal of Mathematics, 6 (1995), 101. Google Scholar [37] C. Zhu, Kan replacement of simplicial manifolds,, Letters in Mathematical Physics, 90 (2009), 383. Google Scholar

show all references

##### References:
 [1] C. Arias Abad and M. Crainic, Representations up to homotopy of Lie algebroids,, 2009. Available from: \url{http://arxiv.org/pdf/0901.0319v2}., (). Google Scholar [2] C. Arias Abad and F. Schaetz, The $A_\infty$ de Rham theorem and integration of representations up to homotopy,, 2010. Available from: \url{http://arxiv.org/pdf/1011.4693}., (). Google Scholar [3] M. Artin and B. Mazur, On the van Kampen theorem,, Topology, 5 (1966), 179. Google Scholar [4] C. Blohmann, M. C. B. Fernandes and A. Weinstein, Groupoid symmetry and constraints in general relativity,, March, (2010). Google Scholar [5] H. Bursztyn and A. Cabrera, Multiplicative forms at the infinitesimal level,, (2010), (2010), 1. Google Scholar [6] H. Bursztyn, A. Cabrera and C. Ortiz, Linear and multiplicative 2-forms,, Letters in Mathematical Physics, 90 (2009), 59. Google Scholar [7] H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets,, Duke Mathematical Journal, 123 (2004), 549. Google Scholar [8] A. S. Cattaneo, Integration of twisted Poisson structures,, Journal of Geometry and Physics, 49 (2004), 187. Google Scholar [9] A. S. Cattaneo, B. Dherin and A. Weinstein, Symplectic microgeometry I: Micromorphisms,, The Journal of Symplectic Geometry, 8 (2010), 205. Google Scholar [10] A. M. Cegarra and J. Remedios, The relationship between the diagonal and the bar constructions on a bisimplicial set,, Topology and its Applications, 153 (2005), 21. doi: 10.1016/j.topol.2004.12.003. Google Scholar [11] M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes,, Commentarii Mathematici Helvetici, 78 (2003), 681. Google Scholar [12] M. Crainic and R. L. Fernandes, Integrability of Lie brackets,, Annals of Mathematics (2), 157 (2003), 575. Google Scholar [13] M. Crainic and R. L. Fernandes, Integrability of Poisson brackets,, Journal of Differential Geometry, 66 (2004), 71. Google Scholar [14] P. G. Goerss and J. F. Jardine, "Simplicial Homotopy Theory," Reprint of the 1999 edition,, Modern Birkhäuser Classics, (2009). Google Scholar [15] A. Gracia-Saz and R. A. Mehta, Lie algebroid structures on double vector bundles and representation theory of Lie algebroids,, Advances in Mathematics, 223 (2010), 1236. Google Scholar [16] A. Gracia-Saz and R. A. Mehta, VB-groupoids and representation theory of Lie groupoids,, (2011), (2011), 1. Google Scholar [17] A. Henriques, Integrating $L_\infty$-algebras,, Compositio Mathematica, 144 (2008), 1017. Google Scholar [18] D. Iglesias Ponte, C. Laurent-Gengoux and P. Xu, Universal lifting theorem and quasi-Poisson groupoids,, (2005), (2005), 1. Google Scholar [19] D. Kochan, Differential gorms and worms,, in, (2005), 128. Google Scholar [20] M. Kontsevich, Deformation quantization of Poisson manifolds,, Letters in Mathematical Physics, 66 (2003), 157. Google Scholar [21] Y. Kosmann-Schwarzbach, Quasi, twisted, and all that$\ldots$in Poisson geometry and Lie algebroid theory,, in, (2005), 363. Google Scholar [22] K. C. H. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids,, Duke Mathematical Journal, 73 (1994), 415. doi: 10.1215/S0012-7094-94-07318-3. Google Scholar [23] K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids,, Topology, 39 (2000), 445. doi: 10.1016/S0040-9383(98)00069-X. Google Scholar [24] R. A. Mehta and X. Tang, From double Lie groupoids to local Lie 2-groupoids,, Bulletin of the Brazilian Mathematical Society (New Series), 42 (2011), 651. Google Scholar [25] J. W. Milnor, Microbundles. I,, Topology, 3 (1964), 53. Google Scholar [26] D. Roytenberg, "Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds,", Ph.D. thesis, (1999). Google Scholar [27] D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, in, 315 (2002), 169. Google Scholar [28] D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds,, Letters in Mathematical Physics, 61 (2002), 123. Google Scholar [29] D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie algebras,, Letters in Mathematical Physics, 46 (1998), 81. Google Scholar [30] P. Ševera, "Letters to A. Weinstein.", Available from: \url{http://sophia.dtp.fmph.uniba.sk/~severa/letters/}., (). Google Scholar [31] P. Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one,, in, (2005), 121. Google Scholar [32] P. Ševera, $L_\infty$-algebras as first approximations,, in, 956 (2007), 199. Google Scholar [33] P. Ševera and A. Weinstein, Poisson geometry with a 3-form background,, Progress of Theoretical Physics Suppl., 144 (2001), 145. Google Scholar [34] Y. Sheng and C. Zhu, Higher extensions of lie algebroids and application to courant algebroids,, 2011. Available from: \url{http://arxiv.org/pdf/1103.5920}., (). Google Scholar [35] Y. Sheng and C. Zhu, Semidirect products of representations up to homotopy,, Pacific Journal of Mathematics, 249 (2011), 211. Google Scholar [36] P. Xu, On Poisson groupoids,, International Journal of Mathematics, 6 (1995), 101. Google Scholar [37] C. Zhu, Kan replacement of simplicial manifolds,, Letters in Mathematical Physics, 90 (2009), 383. Google Scholar
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