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}., ().

[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}., ().

[3]

M. Artin and B. Mazur, On the van Kampen theorem,, Topology, 5 (1966), 179.

[4]

C. Blohmann, M. C. B. Fernandes and A. Weinstein, Groupoid symmetry and constraints in general relativity,, March, (2010).

[5]

H. Bursztyn and A. Cabrera, Multiplicative forms at the infinitesimal level,, (2010), (2010), 1.

[6]

H. Bursztyn, A. Cabrera and C. Ortiz, Linear and multiplicative 2-forms,, Letters in Mathematical Physics, 90 (2009), 59.

[7]

H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets,, Duke Mathematical Journal, 123 (2004), 549.

[8]

A. S. Cattaneo, Integration of twisted Poisson structures,, Journal of Geometry and Physics, 49 (2004), 187.

[9]

A. S. Cattaneo, B. Dherin and A. Weinstein, Symplectic microgeometry I: Micromorphisms,, The Journal of Symplectic Geometry, 8 (2010), 205.

[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.

[11]

M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes,, Commentarii Mathematici Helvetici, 78 (2003), 681.

[12]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets,, Annals of Mathematics (2), 157 (2003), 575.

[13]

M. Crainic and R. L. Fernandes, Integrability of Poisson brackets,, Journal of Differential Geometry, 66 (2004), 71.

[14]

P. G. Goerss and J. F. Jardine, "Simplicial Homotopy Theory," Reprint of the 1999 edition,, Modern Birkhäuser Classics, (2009).

[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.

[16]

A. Gracia-Saz and R. A. Mehta, VB-groupoids and representation theory of Lie groupoids,, (2011), (2011), 1.

[17]

A. Henriques, Integrating $L_\infty$-algebras,, Compositio Mathematica, 144 (2008), 1017.

[18]

D. Iglesias Ponte, C. Laurent-Gengoux and P. Xu, Universal lifting theorem and quasi-Poisson groupoids,, (2005), (2005), 1.

[19]

D. Kochan, Differential gorms and worms,, in, (2005), 128.

[20]

M. Kontsevich, Deformation quantization of Poisson manifolds,, Letters in Mathematical Physics, 66 (2003), 157.

[21]

Y. Kosmann-Schwarzbach, Quasi, twisted, and all that$\ldots$in Poisson geometry and Lie algebroid theory,, in, (2005), 363.

[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.

[23]

K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids,, Topology, 39 (2000), 445. doi: 10.1016/S0040-9383(98)00069-X.

[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.

[25]

J. W. Milnor, Microbundles. I,, Topology, 3 (1964), 53.

[26]

D. Roytenberg, "Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds,", Ph.D. thesis, (1999).

[27]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, in, 315 (2002), 169.

[28]

D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds,, Letters in Mathematical Physics, 61 (2002), 123.

[29]

D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie algebras,, Letters in Mathematical Physics, 46 (1998), 81.

[30]

P. Ševera, "Letters to A. Weinstein.", Available from: \url{http://sophia.dtp.fmph.uniba.sk/~severa/letters/}., ().

[31]

P. Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one,, in, (2005), 121.

[32]

P. Ševera, $L_\infty$-algebras as first approximations,, in, 956 (2007), 199.

[33]

P. Ševera and A. Weinstein, Poisson geometry with a 3-form background,, Progress of Theoretical Physics Suppl., 144 (2001), 145.

[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}., ().

[35]

Y. Sheng and C. Zhu, Semidirect products of representations up to homotopy,, Pacific Journal of Mathematics, 249 (2011), 211.

[36]

P. Xu, On Poisson groupoids,, International Journal of Mathematics, 6 (1995), 101.

[37]

C. Zhu, Kan replacement of simplicial manifolds,, Letters in Mathematical Physics, 90 (2009), 383.

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}., ().

[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}., ().

[3]

M. Artin and B. Mazur, On the van Kampen theorem,, Topology, 5 (1966), 179.

[4]

C. Blohmann, M. C. B. Fernandes and A. Weinstein, Groupoid symmetry and constraints in general relativity,, March, (2010).

[5]

H. Bursztyn and A. Cabrera, Multiplicative forms at the infinitesimal level,, (2010), (2010), 1.

[6]

H. Bursztyn, A. Cabrera and C. Ortiz, Linear and multiplicative 2-forms,, Letters in Mathematical Physics, 90 (2009), 59.

[7]

H. Bursztyn, M. Crainic, A. Weinstein and C. Zhu, Integration of twisted Dirac brackets,, Duke Mathematical Journal, 123 (2004), 549.

[8]

A. S. Cattaneo, Integration of twisted Poisson structures,, Journal of Geometry and Physics, 49 (2004), 187.

[9]

A. S. Cattaneo, B. Dherin and A. Weinstein, Symplectic microgeometry I: Micromorphisms,, The Journal of Symplectic Geometry, 8 (2010), 205.

[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.

[11]

M. Crainic, Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes,, Commentarii Mathematici Helvetici, 78 (2003), 681.

[12]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets,, Annals of Mathematics (2), 157 (2003), 575.

[13]

M. Crainic and R. L. Fernandes, Integrability of Poisson brackets,, Journal of Differential Geometry, 66 (2004), 71.

[14]

P. G. Goerss and J. F. Jardine, "Simplicial Homotopy Theory," Reprint of the 1999 edition,, Modern Birkhäuser Classics, (2009).

[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.

[16]

A. Gracia-Saz and R. A. Mehta, VB-groupoids and representation theory of Lie groupoids,, (2011), (2011), 1.

[17]

A. Henriques, Integrating $L_\infty$-algebras,, Compositio Mathematica, 144 (2008), 1017.

[18]

D. Iglesias Ponte, C. Laurent-Gengoux and P. Xu, Universal lifting theorem and quasi-Poisson groupoids,, (2005), (2005), 1.

[19]

D. Kochan, Differential gorms and worms,, in, (2005), 128.

[20]

M. Kontsevich, Deformation quantization of Poisson manifolds,, Letters in Mathematical Physics, 66 (2003), 157.

[21]

Y. Kosmann-Schwarzbach, Quasi, twisted, and all that$\ldots$in Poisson geometry and Lie algebroid theory,, in, (2005), 363.

[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.

[23]

K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids,, Topology, 39 (2000), 445. doi: 10.1016/S0040-9383(98)00069-X.

[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.

[25]

J. W. Milnor, Microbundles. I,, Topology, 3 (1964), 53.

[26]

D. Roytenberg, "Courant Algebroids, Derived Brackets and Even Symplectic Supermanifolds,", Ph.D. thesis, (1999).

[27]

D. Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids,, in, 315 (2002), 169.

[28]

D. Roytenberg, Quasi-Lie bialgebroids and twisted Poisson manifolds,, Letters in Mathematical Physics, 61 (2002), 123.

[29]

D. Roytenberg and A. Weinstein, Courant algebroids and strongly homotopy Lie algebras,, Letters in Mathematical Physics, 46 (1998), 81.

[30]

P. Ševera, "Letters to A. Weinstein.", Available from: \url{http://sophia.dtp.fmph.uniba.sk/~severa/letters/}., ().

[31]

P. Ševera, Some title containing the words "homotopy" and "symplectic", e.g. this one,, in, (2005), 121.

[32]

P. Ševera, $L_\infty$-algebras as first approximations,, in, 956 (2007), 199.

[33]

P. Ševera and A. Weinstein, Poisson geometry with a 3-form background,, Progress of Theoretical Physics Suppl., 144 (2001), 145.

[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}., ().

[35]

Y. Sheng and C. Zhu, Semidirect products of representations up to homotopy,, Pacific Journal of Mathematics, 249 (2011), 211.

[36]

P. Xu, On Poisson groupoids,, International Journal of Mathematics, 6 (1995), 101.

[37]

C. Zhu, Kan replacement of simplicial manifolds,, Letters in Mathematical Physics, 90 (2009), 383.

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