Abstract
We define gauge transformations of Jacobi structures on a manifold. This is related to gauge transformations of Poisson structures via the Poissonization. Finally, we discuss how the contact structure of a contact groupoid is effected by a gauge transformation of the Jacobi structure on its base.
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Bruce, A.J., Grabowska, K., Grabowski, J.: Remarks on contact and Jacobi geometry. SIGMA Symmetry Integrability Geom. Methods Appl. 13, Paper No. 059 22 pp (2017)
Bursztyn, H.: On gauge transformations of Poisson structures. In: Quantum Field Theory and Noncommutative Geometry, vol. 89–112, Lecture Notes in Phys., p 662. Springer, Berlin (2005)
Bursztyn, H., Radko, O.: Gauge equivalence of Dirac structures and symplectic groupoids. Ann. Inst. Fourier (Grenoble) 53(1), 309–337 (2003)
Crainic, M., Salazar, M.A.: Jacobi structures and Spencer operators. J. Math. Pures Appl. 103, 504–521 (2015)
Crainic, M., Zhu, C.: Integrability of Jacobi and Poisson structures. Ann. Inst. Fourier (Grenoble) 57(4), 1181–1216 (2007)
Das, A.: On generalized Lie bialgebroids and Jacobi groupoids. Extracta Math. 31(2), 199–225 (2016)
Das, A.: Reduction of Nambu-Poisson manifolds by regular distributions. Math. Phys. Anal. Geom. 21(1), 21:5 (2018)
Grabowski, J., Marmo, G.: Jacobi structures revisited. J. Phys. A 34(49), 10975–10990 (2001)
Iglesias, D., Marrero, J.C.: Generalized Lie bialgebroids and Jacobi structures. J. Geom. Phys. 40(2), 176–199 (2001)
Iglesias, D., Marrero, J.C.: Lie algebroid foliations and \(\mathcal {E}^{1}(M)\)-Dirac structures. J. Phys. A 35(18), 4085–4104 (2002)
Iglesias-Ponte, D., Marrero, J.C.: Jacobi groupoids and generalized Lie bialgebroids. J. Geom. Phys. 48(2-3), 385–425 (2003)
Iglesias-Ponte, D., Wade, A.: Contact manifolds and generalized complex structures. J. Geom. Phys. 53(3), 249–258 (2005)
Iglesias-Ponte, D., Wade, A.: Integration of Dirac-Jacobi structures. J. Phys. A 39(16), 4181–4190 (2006)
Jurco, B., Schupp, P., Wess, J.: Noncommutative line bundle and Morita equivalence. Lett. Math. Phys. 61(3), 171–186 (2002)
Kerbrat, Y., Souici-Benhammadi, Z.: Variétés de Jacobi et groupoides de contact. C. R. Acad. Sci. Paris Ser. I Math. 317(1), 81–86 (1993)
Kirillov, A.: Local Lie algebras. Russ. Math. Surv. 31(4), 55–75 (1976)
León, M., de López, B., Marrero, J. C., Padrón, E.: Lichnerowicz-Jacobi cohomology and homology of Jacobi manifolds: modular class and duality, arXiv:http://arXiv.org/abs/math/9910079v1
Lichnerowicz, A.: Les variétés de Jacobi et leurs algébres de Lie associées. (French). J. Math. Pures Appl. (9) 57(4), 453–488 (1978)
Mackenzie, K.C.H., Xu, P.: Lie bialgebroids and Poisson groupoids. Duke Math. J. 73(2), 415–452 (1994)
Marle, C.-M.: On Jacobi Manifolds and Jacobi Bundles, Symplectic Geometry, Groupoids, and Integrable Systems (Berkeley, CA, 1989), Math. Sci. Res. Inst Publ., vol. 20, pp 227–246. Springer, New York (1991)
Nunes da Costa, J.M., Clemente-Gallardo, J.: Dirac structures for generalized Lie bialgebroids. J. Phys. A 37(7), 2671–2692 (2004)
Ortiz, C.: Multiplicative Dirac structures, PhD thesis IMPA (2009)
Severa, P., Weinstein, A.: Poisson geometry with a 3-form background. Progr. Theoret. Phys. Suppl. 144, 145–154 (2001)
Vaisman, I.: Locally conformal symplectic manifolds. Internat. J. Math. Math. Sci. 8(3), 521–536 (1985)
Vitagliano, L.: Dirac-Jacobi bundles. J. Symplectic Geom. 16(2), 485–561 (2018)
Vitagliano, L., Wade, A.: Generalized contact bundles. C. R. Acad. Sci. Paris, Ser. I(354), 313–317 (2016)
Wade, A.: Conformal Dirac structures. Lett. Math. Phys. 53(4), 331–348 (2000)
Acknowledgements
The research has been carried out when the author was a research fellow at Indian Statistical Institute, Kolkata (India). He wishes to thank the Institute for their support.
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Das, A. Gauge Transformations of Jacobi Structures and Contact Groupoids. Math Phys Anal Geom 22, 11 (2019). https://doi.org/10.1007/s11040-019-9309-5
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DOI: https://doi.org/10.1007/s11040-019-9309-5