Korean J. Math. Vol. 32 No. 1 (2024) pp.83-95
DOI: https://doi.org/10.11568/kjm.2024.32.1.83

A new quarternionic dirac operator on symplectic submanifold of a product symplectic manifold

Main Article Content

Rashmirekha Patra
Nihar Ranjan Satapathy


The Quaternionic Dirac operator proves instrumental in tackling various challenges within spectral geometry processing and shape analysis. This work involves the introduction of the quaternionic Dirac operator on a symplectic submanifold of an exact symplectic product manifold. The self adjointness of the symplectic quaternionic Dirac operator is observed. This operator is verified for spin $\frac{1}{2}$ particles. It factorizes the Hodge Laplace operator on the symplectic submanifold of an exact symplectic product manifold. For achieving this a new complex structure and an almost quaternionic structure are formulated on this exact symplectic product manifold.

Article Details


[1] Alekseevsky, D., Marchiafava, S. and Pontecorvo, M., Compatible complex structures on almost quaternionic manifolds, Transactions of the American Mathematical Society, 351 (3),(1999), 997–1014. https://www.jstor.org/stable/117913 Google Scholar

[2] Chern, A., Kn ̈oppel, F., Pedit, F., Pinkall, U. and Schr ̈oder, P.,, March, Finding Conformal and Isometric Immersions of Surfaces, Minimal surfaces: Integrable Systems and visualisation. Springer Proceedings in Mathematics and statistics, 349, (2021), 13–33. https://doi.org/10.1007/978-3-030-68541-6_2 Google Scholar

[3] Crane, K., Pinkall, U. and Schr ̈oder, P., Spin Transformations of Discrete Surfaces, ACM Transactions on Graphics. 30 (4),(2011), 104. https://doi.org/10.1145/2010324.1964999 Google Scholar

[4] Datta, M., Immersions in a symplectic manifold, Proc. Indian Acad. Sci. (Math. Sci.), 108 (2),(1998), 137–149. https://doi.org/10.1007/BF02841547 Google Scholar

[5] Datta, M., Immersions in a manifold with a pair of symplectic forms, Journal of Symplectic geometry, 9 (1) (2011), 11–32. Google Scholar

[6] Haydys, A., Nonlinear Dirac operator and quaternionic analysis, Communications of Mathematical Physics. 281 (2008), 251–261. https://doi.org/10.1007/s00220-008-0445-1 Google Scholar

[7] Haydys, A., Dirac operators in gauge theory, In New ideas in low-dimensional topology edited by L. Kauffman and V.Manturov, volume 56 of Ser. Knots Everything, (2015), 161-188. World Sci. Publ., Hackensack, NJ. https://doi.org/10.1142/9789814630627_0005 Google Scholar

[8] Hoffmann, T. and Ye, Z., A discrete extrinsic and intrinsic Dirac operator, Experimental Mathematics. 31 (3),(2022), 920–935. https://doi.org/10.1080/10586458.2020.1727798 Google Scholar

[9] Joyce, D., Hypercomplex Algebraic Geometry, Quart. J. Math. Oxford, 49,(1998) , 129–162. https://doi.org/10.1093/qmathj/49.2.129 Google Scholar

[10] Joyce, D., A theory of quaternionic algebra with applications to hypercomplex geometry, Quaternionic Structures in Mathematics and Physics, (2001), 143–194. https://doi.org/10.1142/9789812810038_0009 Google Scholar

[11] Liu,H. D., Jacobson, A. and Crane, K., A Dirac Operator for Extrinsic Shape Analysis, Eurographics Symposium on Geometry Processing. 36 (5), (2017), 139–149. https://doi.org/10.1111/cgf.13252 Google Scholar

[12] L. Karp, On the Stoke’s theorem for non compact manifolds, Proceedings of the AMS 82 (3), 487–490, 1981. Google Scholar

[13] Massey, W.S., Obstructions to the existance of almost complex structures, Bull. Amer. Math. Soc..67 (1961), 559–564. Google Scholar

[14] Perez, H. J., A Quaternionic Structure as a Landmark for Symplectic Maps, arXiv preprint, arXiv:1910.13031v2, 2019. https://doi.org/10.48550/arXiv.1910.13031 Google Scholar

[15] Tanisli, M., Kansu, M. E. and Demir, S., Supersymmetric quantum mechanics and euclidean Dirac operator with complexified quaternions, Modern Physics Letters A. 28 (8), (2013), (15 pages). https://doi.org/10.1142/S0217732313500260 Google Scholar

[16] Wang, Y. and Solomon, J., Intrinsic and extrinsic operators for shape analysis, In Handbook of Numerical Analysis 20 (2019), 41–115, Elsevier. https://doi.org/10.1016/bs.hna.2019.08.003 Google Scholar

[17] Widdows, D., Quaternion Algebraic Geometry, D. Phil Thesis. (2000), Oxford University. https://people.maths.ox.ac.uk/joyce/theses/WiddowsDPhil.pdf Google Scholar

[18] Ye, Z., Diamanti, O., Tang, C., Guibas, L. and Hoffmann, T., A unified discrete framework for intrinsic and extrinsic Dirac operators for geometry processing, Eurographics Symposium on Geometry Processing. 37 (5) (2018), 93–106. https://doi.org/10.1111/cgf.13494 Google Scholar