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December  2013, 5(4): 473-491. doi: 10.3934/jgm.2013.5.473

A Poincaré lemma in geometric quantisation

Eva Miranda 1, and  Romero Solha 2,

1. 

Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, EPSEB, Avinguda del Doctor Marañón, 44-50, 08028, Barcelona, Spain
2. 

Departament de Matemàtica Aplicada I, Universitat Politècnica de Catalunya, ETSEIB, Avinguda Diagonal 647, 08028, Barcelona, Spain

Received  June 2013 Revised  December 2013 Published  December 2013

This article presents a Poincaré lemma for the Kostant complex, used to compute geometric quantisation, when the polarisation is given by a Lagrangian foliation defined by an integrable system with nondegenerate singularities.
Keywords: singularities, foliated cohomology., Geometric quantisation, moment maps, integrable system.
Mathematics Subject Classification: Primary: 53D50; Secondary: 53D2.
Citation: Eva Miranda, Romero Solha. A Poincaré lemma in geometric quantisation. Journal of Geometric Mechanics, 2013, 5 (4) : 473-491. doi: 10.3934/jgm.2013.5.473
References:
[1]

M. Bertelson, Remarks on a Künneth formula for foliated de Rham cohomology, Pacific J. Math., 252 (2011), 257-274. doi: 10.2140/pjm.2011.252.257.  Google Scholar

[2]

R. Bott and L. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, 82, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[3]

G. Bredon, Sheaf Theory, Second edition, Graduate Texts in Mathematics, 170, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0647-7.  Google Scholar

[4]

L. H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, Ph.D. Thesis, Stockholm University, 1984. Google Scholar

[5]

L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals-elliptic case, Comment. Math. Helv., 65 (1990), 4-35. doi: 10.1007/BF02566590.  Google Scholar

[6]

A. Grothendieck, Séminaire Schwartz de la Faculté des Sciences de Paris, 1953/1954. Produits tensoriels topologiques d'espaces vectoriels topologiques, in Espaces Vectoriels Topologiques Nucléaires. Applications, Exposé 24, Secrétariat Mathématique, 11 rue Pierre Curie, Paris, 1954. Google Scholar

[7]

V. Guillemin and S. Sternberg, The Gel'fand-Cetlin system and quantization of the complex flag manifolds, J. Funct. Anal., 52 (1983), 106-128. doi: 10.1016/0022-1236(83)90092-7.  Google Scholar

[8]

V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Inventiones Mathematicae, 67 (1982), 515-538. doi: 10.1007/BF01398934.  Google Scholar

[9]

M. Hamilton, Locally toric manifolds and singular Bohr-Sommerfeld leaves, Mem. Amer. Math. Soc., 207 (2010). doi: 10.1090/S0065-9266-10-00583-1.  Google Scholar

[10]

M. Hamilton and E. Miranda, Geometric quantization of integrable systems with hyperbolic singularities, Ann. Inst. Fourier (Grenoble), 60 (2010), 51-85. doi: 10.5802/aif.2517.  Google Scholar

[11]

L. Kaup, Eine Künnethformel für Fréchetgarben, Math. Z., 97 (1967), 158-168.  Google Scholar

[12]

B. Kostant, Quantization and unitary representations. Part I. Prequantization, in Lectures in Modern Analysis and Applications, III, Lecture Notes in Mathematics, Vol. 170, Springer, Berlin, 1970.  Google Scholar

[13]

K. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147. doi: 10.1112/blms/27.2.97.  Google Scholar

[14]

E. Miranda, On Symplectic Linearization of Singular Lagrangian Foliations, Ph.D. Thesis, Universitat de Barcelona, 2003. Google Scholar

[15]

E. Miranda and R. Solha, On a Poincaré lemma for foliations, in Foliations 2012, World Scientific, 2013, 115-137. Google Scholar

[16]

E. Miranda and S. Vũ Ngoc, A singular Poincaré lemma, International Mathematics Research Notices, 2005 (2005). Google Scholar

[17]

E. Miranda and N. T. Zung, Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. École Norm. Sup. (4), 37 (2004), 819-839. doi: 10.1016/j.ansens.2004.10.001.  Google Scholar

[18]

E. Miranda and F. Presas, Geometric quantization of real polarizations via sheaves,, to appear in The Journal of Symplectic Geometry, ().   Google Scholar

[19]

H.-J. Petzsche, On E. Borel's theorem, Math. Ann., 282 (1988), 299-313. doi: 10.1007/BF01456977.  Google Scholar

[20]

J. Rawnsley, On the cohomology groups of a polarisation and diagonal quantisation, Trans. Amer. Math. Soc., 230 (1977), 235-255. doi: 10.1090/S0002-9947-1977-0648775-2.  Google Scholar

[21]

J. Śniatycki, On cohomology groups appearing in geometric quantization, in Differential Geometric Methods in Mathematical Physics (Proc. Sympos., Univ. Bonn, Bonn, 1975), Lecture Notes in Mathematics, Vol. 570, Springer, Berlin, 1977, 46-66.  Google Scholar

[22]

R. Solha, On Geometric Quantisation of Integrable Systems with Singularities, Ph.D. thesis, Universitat Politècnica de Catalunya, 2013. Google Scholar

[23]

H. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc., 180 (1973), 171-188. doi: 10.1090/S0002-9947-1973-0321133-2.  Google Scholar

[24]

J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163. doi: 10.2307/2371062.  Google Scholar

[25]

N. Woodhouse, Geometric Quantization, Second edition, Oxford Mathematical Monographs, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992.  Google Scholar

show all references

References:
[1]

M. Bertelson, Remarks on a Künneth formula for foliated de Rham cohomology, Pacific J. Math., 252 (2011), 257-274. doi: 10.2140/pjm.2011.252.257.  Google Scholar

[2]

R. Bott and L. Tu, Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, 82, Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[3]

G. Bredon, Sheaf Theory, Second edition, Graduate Texts in Mathematics, 170, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0647-7.  Google Scholar

[4]

L. H. Eliasson, Hamiltonian Systems with Poisson Commuting Integrals, Ph.D. Thesis, Stockholm University, 1984. Google Scholar

[5]

L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals-elliptic case, Comment. Math. Helv., 65 (1990), 4-35. doi: 10.1007/BF02566590.  Google Scholar

[6]

A. Grothendieck, Séminaire Schwartz de la Faculté des Sciences de Paris, 1953/1954. Produits tensoriels topologiques d'espaces vectoriels topologiques, in Espaces Vectoriels Topologiques Nucléaires. Applications, Exposé 24, Secrétariat Mathématique, 11 rue Pierre Curie, Paris, 1954. Google Scholar

[7]

V. Guillemin and S. Sternberg, The Gel'fand-Cetlin system and quantization of the complex flag manifolds, J. Funct. Anal., 52 (1983), 106-128. doi: 10.1016/0022-1236(83)90092-7.  Google Scholar

[8]

V. Guillemin and S. Sternberg, Geometric quantization and multiplicities of group representations, Inventiones Mathematicae, 67 (1982), 515-538. doi: 10.1007/BF01398934.  Google Scholar

[9]

M. Hamilton, Locally toric manifolds and singular Bohr-Sommerfeld leaves, Mem. Amer. Math. Soc., 207 (2010). doi: 10.1090/S0065-9266-10-00583-1.  Google Scholar

[10]

M. Hamilton and E. Miranda, Geometric quantization of integrable systems with hyperbolic singularities, Ann. Inst. Fourier (Grenoble), 60 (2010), 51-85. doi: 10.5802/aif.2517.  Google Scholar

[11]

L. Kaup, Eine Künnethformel für Fréchetgarben, Math. Z., 97 (1967), 158-168.  Google Scholar

[12]

B. Kostant, Quantization and unitary representations. Part I. Prequantization, in Lectures in Modern Analysis and Applications, III, Lecture Notes in Mathematics, Vol. 170, Springer, Berlin, 1970.  Google Scholar

[13]

K. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc., 27 (1995), 97-147. doi: 10.1112/blms/27.2.97.  Google Scholar

[14]

E. Miranda, On Symplectic Linearization of Singular Lagrangian Foliations, Ph.D. Thesis, Universitat de Barcelona, 2003. Google Scholar

[15]

E. Miranda and R. Solha, On a Poincaré lemma for foliations, in Foliations 2012, World Scientific, 2013, 115-137. Google Scholar

[16]

E. Miranda and S. Vũ Ngoc, A singular Poincaré lemma, International Mathematics Research Notices, 2005 (2005). Google Scholar

[17]

E. Miranda and N. T. Zung, Equivariant normal form for nondegenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. École Norm. Sup. (4), 37 (2004), 819-839. doi: 10.1016/j.ansens.2004.10.001.  Google Scholar

[18]

E. Miranda and F. Presas, Geometric quantization of real polarizations via sheaves,, to appear in The Journal of Symplectic Geometry, ().   Google Scholar

[19]

H.-J. Petzsche, On E. Borel's theorem, Math. Ann., 282 (1988), 299-313. doi: 10.1007/BF01456977.  Google Scholar

[20]

J. Rawnsley, On the cohomology groups of a polarisation and diagonal quantisation, Trans. Amer. Math. Soc., 230 (1977), 235-255. doi: 10.1090/S0002-9947-1977-0648775-2.  Google Scholar

[21]

J. Śniatycki, On cohomology groups appearing in geometric quantization, in Differential Geometric Methods in Mathematical Physics (Proc. Sympos., Univ. Bonn, Bonn, 1975), Lecture Notes in Mathematics, Vol. 570, Springer, Berlin, 1977, 46-66.  Google Scholar

[22]

R. Solha, On Geometric Quantisation of Integrable Systems with Singularities, Ph.D. thesis, Universitat Politècnica de Catalunya, 2013. Google Scholar

[23]

H. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc., 180 (1973), 171-188. doi: 10.1090/S0002-9947-1973-0321133-2.  Google Scholar

[24]

J. Williamson, On the algebraic problem concerning the normal forms of linear dynamical systems, Amer. J. Math., 58 (1936), 141-163. doi: 10.2307/2371062.  Google Scholar

[25]

N. Woodhouse, Geometric Quantization, Second edition, Oxford Mathematical Monographs, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992.  Google Scholar

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