Abstract
For a smooth manifold M, it was shown in Broojerdian et al. (Iran J Math Sci Inform 6(1):79–96, 2011) that every affine connection on the tangent bundle TM naturally gives rise to covariant differentiation of multivector fields (MVFs) and differential forms along MVFs. In this paper, we generalize the covariant derivative of Broojerdian et al. (Iran J Math Sci Inform 6(1):79–96, 2011) and construct covariant derivatives along MVFs which are not induced by affine connections on TM. We call this more general class of covariant derivatives higher affine connections. In addition, we also propose a framework which gives rise to non-induced higher connections; this framework is obtained by equipping the full exterior bundle \({\wedge^\bullet TM}\) with an associative bilinear form \({\eta}\). Since the latter can be shown to be equivalent to a set of differential forms of various degrees, this framework also provides a link between higher connections and multisymplectic geometry.
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References
Baez J., Rogers C.: Categorified symplectic geometry and the string Lie 2-algebra. Homol. Homotopy Appl. 12(1), 221–236 (2010)
Baez J., Hoffnung A., Rogers C.: Categorified symplectic geometry and the classical string. Commun. Math. Phys. 293, 701–725 (2010)
Broojerdian N., Peyghan E., Heydari A.: Differentiation along multivector fields. Iran. J. Math. Sci. Inform. 6(1), 79–96 (2011)
Cantrign F., Ibort A., de León M.: On the geometry of multisymplectic manifolds. J. Aust. Math. Soc. Ser. A 66(3), 303–330 (1999)
Forger M., Paufler C., Römer H.: The Poisson bracket for Poisson forms in multisymplectic field theory. Rev. Math. Phys. 15, 705–744 (2003)
Green, M., Schwarz, J., Witten, E.: Superstring Theory. Cambridge University Press, Cambridge, (1987)
Hitchin, N.: Generalized geometry-an introduction. In: Handbook of Pseudo–Riemannian Geometry and Supersymmetry, pp. 185–208. EMS, Zürich (2010)
Hitchin, N.: Lectures on generalized geometry. arXiv:1008.0973 [math.DG]
Lee, J.: Introduction to Smooth Manifolds. Springer, Berlin (2003)
Marle C.-M.: Schouten–Nijenhuis bracket and interior products. J. Geom. Phys. 23, 350–359 (1997)
Nijenhuis A.: Jacobi-type identities for bilinear differentiation concomitatants of certain tensor fields. Indag. Math. 17, 390–403 (1955)
Ritter P., Sämann C.: Lie 2-algebra models. JHEP 04, 066 (2014)
Rogers, C.L.: Higher symplectic geometry. Ph.D. thesis, University of California, Riverside (2011)
Sämann, C., Szabo, R.: Groupoids, loop spaces, and quantization of 2-plectic manifolds. Rev. Math. Phys. (2013). doi:10.1142/S0129055X13300057
Zwiebach, B.: A First Course in String Theory. Cambridge University Press, Cambridge (2004)
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Pham, D.N. Higher Affine Connections. Mediterr. J. Math. 13, 1227–1262 (2016). https://doi.org/10.1007/s00009-015-0559-6
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DOI: https://doi.org/10.1007/s00009-015-0559-6