Abstract
We introduce symplectic structures on “Lie pairs” of (real or complex) Lie algebroids as studied by Chen et al. (From Atiyah classes to homotopy Leibniz algebras. arXiv:1204.1075, 2012), encompassing homogeneous symplectic spaces, symplectic manifolds with a \({\mathfrak{g}}\)-action, and holomorphic symplectic manifolds. We show that to each such symplectic Lie pair are associated Rozansky–Witten-type invariants of three-manifolds and knots, given respectively by weight systems on trivalent and chord diagrams.
Similar content being viewed by others
References
Atiyah M.F.: Complex analytic connections in fibre bundles. Trans. Am. Math. Soc. 85(1), 181–207 (1957)
Bar-Natan D.: On the Vassiliev knot invariants. Topology 34(2), 423–472 (1995)
Bordemann M.: Atiyah classes and equivariant connections on homogeneous spaces. Travaux mathématiques 20, 29–82 (2012)
Chen, Z., Stiénon, M., Xu, P.: From Atiyah classes to homotopy Leibniz algebras (2012). arXiv:1204.1075
Collingwood, D.H., McGovern, W.M.: Nilpotent Orbits in Semisimple Lie Algebras. Van Nostrand Reinhold Mathematics Series. Van Nostrand Reinhold Co., New York (1993)
Garoufalidis S., Ohtsuki T.: On finite type 3-manifold invariants III: manifold weight systems. Topology 37(2), 227–243 (1998)
Hai N.V.: Conditions nécessaires et suffisantes pour qu’un espace homogène admette une connexion linéaire invariante. C. R. Acad. Sci. Paris 259, 49–52 (1964)
Jantzen, J.C.: Nilpotent orbits in representation theory. In: Lie Theory, vol. 228. Progress in Mathematics, pp. 1–211. Birkhäuser Boston, Boston (2004)
Kamber F.W., Tondeur P.: Characteristic invariants of foliated bundles. Manuscripta Mathematica 11(1), 51–89 (1974)
Kapranov M.: Rozansky–Witten invariants via Atiyah classes. Compositio Mathematica 115(1), 71–113 (1999)
Kapustin, A.: Topological field theory, higher categories, and their applications. In: Proceedings of the International Congress of Mathematicians, vol. III, pp. 2021–2043. Hindustan Book Agency, New Delhi (2010)
Kapustin A., Rozansky L.: Three-dimensional topological field theory and symplectic algebraic geometry II. Commun. Number Theory Phys. 4(3), 463–549 (2010)
Kontsevich M.: Rozansky–Witten invariants via formal geometry. Compositio Mathematica 115(1), 115–127 (1999)
Laurent-Gengoux, C., Stiénon, M., Xu, P.: Kapranov dg-manifolds and Poincaré–Birkhoff–Witt isomorphisms (2014). arXiv:1408.2903
Laurent-Gengoux, C., Voglaire, Y.: Invariant connections and PBW theorem for Lie algebroid pairs (2014, in preparation)
Le T.T.Q., Murakami J., Ohtsuki T.: On a universal perturbative invariant of 3-manifolds. Topology 37(3), 539–574 (1998)
Liu Z.-J., Weinstein A., Xu P.: Dirac structures and Poisson homogeneous spaces. Commun. Math. Phys. 192(1), 121–144 (1998)
Liu Z.-J., Weinstein A., Xu P.: Manin triples for Lie bialgebroids. J. Differ. Geom. 45(3), 547–574 (1997)
Mackenzie, K.C.H.: General theory of Lie groupoids and Lie algebroids, vol. 213. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2005)
Mackenzie K.C.H., Xu P.: Lie bialgebroids and Poisson groupoids. Duke Math. J. 73(2), 415–452 (1994)
Markarian N.: The Atiyah class, Hochschild cohomology and the Riemann–Roch theorem. J. Lond. Math. Soc. Second Ser. 79(1), 129–143 (2009)
Moerdijk I., Mrčun J.: On the integrability of Lie subalgebroids. Adv. Math. 204(1), 101–115 (2006)
Mokri T.: Matched pairs of Lie algebroids. Glasg. Math. J. 39(2), 167–181 (1997)
Molino P.: Classe d’Atiyah d’un feuilletage et connexions transverses projetables. C. R. Acad. Sci. Paris Sér. A-B 272, A779–A781 (1971)
Pikulin S.V., Tevelev E.A.: Invariant linear connections on homogeneous symplectic varieties. Transform. Groups 6(2), 193–198 (2001)
Qiu J., Zabzine M.: Knot invariants and new weight systems from general 3D TFTs. J. Geom. Phys. 62(2), 242–271 (2012)
Ramadoss A.C.: The big Chern classes and the Chern character. Int. J. Math. 19(6), 699–746 (2008)
Roberts J., Willerton S.: On the Rozansky–Witten weight systems. Algebr. Geom. Topol. 10(3), 1455–1519 (2010)
Rozansky L., Witten E.: Hyper-Kähler geometry and invariants of three-manifolds. Selecta Mathematica. New Ser. 3(3), 401–458 (1997)
Sawon, J.: Rozansky–Witten invariants of hyperKähler manifolds. Thesis. University of Cambridge (1999)
Vaisman I.: Sur l’existence des opérateurs différentiels feuilletés à symbole donné. C. R. Acad. Sci. Paris Sér. A-B 276, A1165–A1168 (1973)
Vinberg É.B.: Invariant linear connections in a homogeneous space. Trudy Moskovskogo Matematičeskogo Obščestva 9, 191–210 (1960)
Wang H.C.: On invariant connections over a principal fibre bundle. Nagoya Math. J. 13, 1–19 (1958)
Weinstein A., Xu P.: Extensions of symplectic groupoids and quantization. Journal für Die Reine Und Angewandte Mathematik 417, 159–189 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Ooguri
Y. Voglaire’s research was partially supported by the Fonds National de la Recherche, Luxembourg, through the AFR Grant PDR 2012-1 (Project Reference 3966341).
P. Xu’s research was partially supported by the National Science Foundation Grant DMS-1101827.
Rights and permissions
About this article
Cite this article
Voglaire, Y., Xu, P. Rozansky–Witten-Type Invariants from Symplectic Lie Pairs. Commun. Math. Phys. 336, 217–241 (2015). https://doi.org/10.1007/s00220-014-2221-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-014-2221-8