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, Université de Genève, 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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